Abstracts of all talks

Jon Aaronson, Tel Aviv University, Israel

ON MIXING PROPERTIES OF INFINITE MEASURE PRESERVING TRANSFORMATIONS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

I'll review the Hopf-Krickeberg mixing property with examples and discuss related ergodic properties such as rational weak mixing.

Christian Aarset, Alpen-Adria Universität Klagenfurt, Austria

BIFURCATIONS IN PERIODIC IDEs

Joint work with Christian Pötzsche

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

In theoretical ecology, one often models population growth with the help of discrete-time difference equations. One method to account for the effects of dispersal throughout the habitat is to employ integrodifference equations [1], or IDEs, as opposed to employing scalar difference equations. Given a compact habitat \(\Omega\subset\mathbb{R}^d\), usually with \(d=1,2,3\), together with some appropriate parameter space \(\Lambda\), we consider IDEs on the form \[u_{t+1} := \int_\Omega f(\cdot,y,u_t(y),\alpha)dy \tag{1} \] with \(u_t\in C(\Omega)\) for all \(t\in\mathbb{N}\), where \(f: \Omega\times\Omega\times\mathbb{R}\times\Lambda\rightarrow\mathbb{R}\) is some appropriate function; a commonly employed form for such \(f\) is e.g. \(f(x,y,z,\alpha):=k(x,y)g(y,z,\alpha)\), where \(k\) is some dispersal kernel (e.g. Laplace, Gaussian) and \(g\) is some parameter-dependent growth function (e.g. Beverton-Holt, Ricker).

One is frequently interested in the stability behaviour of fixed points, solutions \(u^*\) of (1). However, certain IDEs, in particular - but not limited to - those using the Ricker growth function, may feature transfer of stability from a branch of fixed points to a branch of two- or higher- periodic solutions, solutions of the iterated equation. We explore such flip bifurcations in details, and generalize this theory to cover bifurcations of periodic solutions of any integer period, with the particular goal of formulating our assumptions so that they can easily be verified numerically.

References

1. M. Kot, W.M. Schaffer, Discrete-Time Growth-Dispersal Models, Mathematical Biosciences 80 (1986), 109-136.

Thomas Alazard, École Normale Supérieure Paris-Saclay, France

OBSERVABILITY, CONTROL AND STABILIZATION OF WATER WAVES

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

The main questions studied in this talk are the generation and the absorption of water waves in a wave tank. We want to understand which water waves can be generated by blowing above a localized portion of the free surface of a liquid. We also want to understand how to damp water waves when they reach the boundary of a numerical wave tank.

For \(2D\) water waves in a rectangular tank, the fluid domain \(\Omega(t)\) is of the form \[ \Omega(t)=\left\{\, (x,y)\,:\, x\in [0,L], h\le y\le \eta(t,x)\,\right\}, \] where \(x\) (resp. \(y\)) is the horizontal (resp. vertical) space variable, \(L\) is the width of the tank, \(h\) its depth and \(\eta\) is the free surface elevation. The equations which dictate the motion are the incompressible Euler equations with free surface. This is a system of two nonlinear equations: the incompressible Euler equation for the velocity potential \(\phi\colon \Omega\rightarrow \mathbb{R}\) (so that the velocity is \(v=\nabla_{x,y}\phi\)) and a kinematic equation for \(\eta\) which states that the free surface moves with the fluid. Zakharov discovered that \(\eta\) is conjugated to the trace \(\psi(t,x)=\phi(t,x,\eta(t,x))\) of the velocity potential on the free surface: the equations have the hamiltonian form \[ \frac{\partial \eta}{\partial t}=\frac{\delta\mathcal{H}}{\delta \psi},\quad \frac{\partial \psi}{\partial t} =-\frac{\delta\mathcal{H}}{\delta \eta}-P_{ext},\tag{1} \] where \(P_{ext}\) is an external pressure and \(\mathcal{H}\) is the energy \[ \mathcal{H}=\frac{g}{2}\int_0^L\eta^2(t,x)\, dx+\int_0^L \big(\sqrt{1+(\partial_x \eta(t,x))^2}-1\big)\, dx+ \frac{1}{2}\iint_{\Omega(t)}\left\vert \nabla_{x,y}\phi(t,x,y)\right\vert^2\, dx dy. \]

\(\bullet\) The first problem is the following : given a time \(T>0\), a final state \((\eta_{final},\psi_{final})\) in some space of regular functions, a non empty interval \(\omega=(a,b)\subset [0,L]\), is-it possible to find a function \(P_{ext}(t,x)\) supported in \([0,T]\times \omega\) such that the solution to (1) with initial data \((\eta_{in},\psi_{in})=(0,0)\) satisfies \((\eta,\psi)\arrowvert_{t=T}=(\eta_{final},\psi_{final})\)? I will present a local controllability result obtained with Pietro Baldi and Daniel Han-Kwan ([1]).

\(\bullet\) We then consider the stabilization problem. The goal here is to find a pressure law, relating \(P_{ext}\) to the unknown \((\eta,\psi)\) and supported inside a small subset of \([0,L]\), such that \(\mathcal{H}\) is decreasing and converges to zero. I will explain how to use the multiplier method of C. Morawetz and J.L. Lions to study this problem ([2, 3, 4]).

References

1. T. Alazard, P. Baldi, D. Han-Kwan, Control for water waves, J. Eur. Math. Soc. 20 (2018), 657–745.
2. T. Alazard, Boundary observability of gravity water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 751–779.
3. T. Alazard, Stabilization of gravity water waves, J. Math. Pures Appl. 114 (2018), 51–84.
4. T. Alazard, Stabilization of the water-wave equations with surface tension, Annals of PDE 2 (2017), 41 pp.

Giovanni S. Alberti, University of Genoa, Italy

INFINITE-DIMENSIONAL INVERSE PROBLEMS WITH FINITE MEASUREMENTS

Joint work with Matteo Santacesaria

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing, may be applied to inverse problems for partial differential equations. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, but the approach is very general and allows to handle many other classes of inverse problems. I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinite-dimensional compressed sensing for deterministic and non-isometric operators, which will be briefly surveyed.

References

1. G.S. Alberti, M. Santacesaria, Calderón’s Inverse Problem with a Finite Number of Measurements, arXiv:1803.04224, 2018.
2. G.S. Alberti, M. Santacesaria, Infinite Dimensional Compressed Sensing from Anisotropic Measurements and Applications to Inverse Problems in PDE, arXiv:1710.11093, 2017.
3. G.S. Alberti, M. Santacesaria, Infinite-Dimensional Inverse Problems with Finite Measurements, in preparation.

Fabio Ancona, University of Padua, Italy

ONE-SIDE BOUNDARY CONTROLLABILITY OF THE \(P\)-SYSTEM

Joint work with Olivier Glass and Khai T. Nguyen

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building B-8, room 0.18.

Abstract

We consider the equations for one-dimensional isentropic compressible gases on an interval, in Eulerian or in Lagrangian coordinates (known as the the \(p\)-system). On one side of the interval it is imposed a fixed boundary condition (for instance the null velocity), while on the other side of the interval the boundary condition is treated as a control that one can choose to influence the system. We prove a result of controllability toward constants states in the context of (discontinuous) weak entropy solutions. Namely, we prove that it is possible, starting from an initial state small in \(BV\), to reach any constant state compatible with the boundary conditions. This type of result was previously obtained in [2] in the context of boundary controls acting on both boundaries. These results are in sharp contrast with what happens for some other \(2 \times 2\) strictly hyperbolic systems with genuinely nonlinear characteristics fields for which Bressan and Coclite [1] showed that, in general, exact controllability to constant states is not possible, even when controlling on both sides of the interval.

References

1. A. Bressan, G.M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim. 41 (2002), 607-622.
2. O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9(3) (2007), 427-486.

Matthieu Astorg, Université d'Orléans, France

WANDERING DOMAINS ARISING WITH FROM LAVAURS MAPS WITH SIEGEL DISKS

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

A famous theorem of Sullivan asserts that polynomials in one complex variables have no wandering Fatou components. On the other hand, in a joint work with Buff, Dujardin, Peters and Raissy, we constructed the first examples of polynomial maps in two complex variables having such components. The construction relies on parabolic implosion, and involves the dynamics of non-autonomous perturbations of a Lavaurs map with an attracting fixed point. In this talk, we will present a more recent work with Boc-Thaler and Peters, in which we classify the local dynamics in the case where the Lavaurs map has a Siegel fixed point. In particular, we prove that wandering domains may also arise in that setting. Time permitting, we will introduce the notion of parabolic curves and how their existence simplifies the proof

Tim Austin, University of California, Los Angeles, USA

RECENT PROGRESS ON STRUCTURE AND CLASSIFICATION IN ERGODIC THEORY

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

A basic formulation of the 'classification problem' asks for criteria to determine when two ergodic measure-preserving systems are isomorphic. It goes back to the foundational work of von Neuman and Halmos.

In more recent decades, this problem has evolved from 'pure' ergodic theory into a point of overlap with descriptive set theory. Culminating in work of Foreman, Rudolph and Weiss, this connection has shown the impossibility of any reasonable such classification. More recently, Foreman and Weiss have also shown that the important restriction of the classification problem to 'classical systems' - that is, smooth, volume-preserving maps of compact manifolds - is equally intractable.

However, some rich positive results are available in the direction of partial structure. These have the flavour that, for all ergodic measure-preserving systems, some 'soft' feature can be turned into a factor map or isomorphism to another system of a special kind. For instance, rotations on compact Abelian groups account for any failure of weak mixing (Halmos-von Neumann), and positive entropy can be fully realized by a Bernoulli-shift factor (Sinai). The second of these results was recently strengthened to show that all ergodic systems have Thouvenot's weak Pinsker property: they can always be split as a direct product of (i) a system with arbitrarily little entropy and (ii) a Bernoulli shift.

I will give a rough overview of some recent developments in this area and of some related settings in which many questions remains open. For the latter, I will especially emphasize the world of measure-preserving actions of sofic, non-amenable groups such as free groups.

Artur Avila, Universität Zürich, Switzerland & IMPA, Brazil

GENERIC CONSERVATIVE DYNAMICS

Date: 2019-09-20 (Friday); Time: 16:20-17:20; Location: building U-2, auditorium.

Abstract

Ellen Baake, Universität Bielefeld, Germany

A PROBABILISTIC VIEW ON THE DETERMINISTIC MUTATION-SELECTION EQUATION

Joint work with Fernando Cordero and Sebastian Hummel

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

We reconsider the prototype version of the mutation-selection differential equation of population genetics. It describes the dynamics of the type composition of an infinite-size population of individuals with two possible types that undergo selection and recurrent mutation. We develop the genealogical point of view to this equation by tracing back the lines of descent of individuals from the population at some (forward) time \(t\). We revisit the ancestral selection graph, which is the standard tool to trace back ancestral lines in populations under selection. Based on this, we introduce the killed ancestral selection graph, which yields the type of a random individual, based on the individual’s potential ancestry and the mutations that define the individual’s type. The result is a stochastic representation of the solution of the (deterministic) differential equation, which is formulated in terms of a duality theorem. This way, the well-known stationary behaviour of the differential equation (and its transcritial bifurcation) translates into the asymptotic behaviour of a (killed) birth-death process.

References

1. E. Baake, F. Cordero, S. Hummel, A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent, J. Math. Biol. 77 (2018), 795-820.
2. E. Baake, A. Wakolbinger, Lines of descent under selection, J. Stat. Phys. 172 (2018), 156-174.

Artur Babiarz, Silesian University of Technology, Poland

LYAPUNOV SPECTRUM ASSIGNABILITY PROBLEM OF DYNAMICAL SYSTEMS

Joint work with Irina Banshchikova, Adam Czornik, Evgenii Makarov, Michał Niezabitowski, and Svetlana Popova

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

For discrete linear time-varying systems with bounded coefficients, the pole assignment problem utilizing linear state feedback is discussed. It is shown that uniform complete controllability is sufficient for the Lyapunov exponents being arbitrarily assignable by choosing a suitable feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have assignable Lyapunov spectrum if and only if the original system is uniformly completely controllable. Using an appropriate time-varying linear feedback we obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system. Moreover, we prove that diagonalizability, Lyapunov regularity and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.

References

1. A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, Pole placement theorem for discrete time-varying linear systems, SIAM Journal on Control and Optimization 55 (2017), 671– 692.
2. A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems, IEEE Transactions on Automatic Control 63 (2018), 3825–3837.
3. A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems, SIAM Journal on Control and Optimization 57 (2019), 1355–1377.

Stephen Baigent, University College London, UK

CONCAVE AND CONCAVE CARRYING SIMPLICES

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

The carrying simplex is a codimension-one invariant hypersurface that is the common boundary of the basins of repulsion of the origin and infinity of both continuous- and discrete-time competitive Kolmogorov systems [1, 2].

Geometrically carrying simplices are ‘nice’ invariant manifolds. They project radially one-to-one and onto the unit probability simplex and are graphs of locally Lipschitz functions. Moreover, in some cases they may be graphs of convex, concave or saddle-like functions [4, 3, 5, 6].

I will introduce the carrying simplex and discuss how the bending of hyperplanes under the map can be used to determine when the carrying simplex is convex or concave.

References

1. M.W. Hirsch, Systems of differential equations which are competitive or cooperative: III Competing species, Nonlinearity 1 (1988), 51–71.
2. M.W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, Journal of Biological Dynamics, 2(2) (2008), 169–179.
3. S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity 26 (2013), 1001–1029.
4. S. Baigent, Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex, Proceedings of the Edinburgh Mathematical Society, 55 (2012), 53–63.
5. S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, Journal of Difference Equations and Applications, 22(5) (2016), 1–14.
6. S. Baigent,. Convex geometry of the carrying simplex for the May-Leonard map, Discrete and Continuous Dynamical Systems Series B, 24(4) (2018) 1697–1723.

Viviane Baladi, CNRS & Sorbonne Université, France

THE FRACTIONAL SUSCEPTIBILITY FUNCTION FOR THE QUADRATIC FAMILY

Joint work with Daniel Smania

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building A-3/A-4, room 103.

Abstract

For \(t\) in a set \(\Omega\) of positive measure, maps in the quadratic family \( f_t(x)=t-x^2 \) admit an SRB measure \(\mu_t\). On the one hand, the dependence of \(\mu_t\) on \(t\) has been shown [1] to be no better than \(1/2\) Hölder, on a subset of \(\Omega\), for \(t_0\) a suitable Misiurewicz-Thurston parameter. On the other hand, the susceptibility function \(\Psi_t(z)\), whose value at \(z=1\) is a candidate for the derivative of \(\mu_t\) with respect to \(t\), has been shown [2] to admit a holomorphic extension at \(z=1\) for \(t=t_0\). Our goal is to resolve this paradox. For this, we introduce and study a fractional susceptibility function.

References

1. V. Baladi, M. Benedicks, and D. Schnellmann, Whitney Hölder continuity of the SRB measure for transversal families of smooth unimodal maps, Invent. Math. 201 (2015), 773-844.
2. Y. Jiang, D. Ruelle, Analyticity of the susceptibility function for unimodal Markovian maps of the interval, Nonlinearity 18 (2005), 2447-2453.

Oscar Bandtlow, Queen Mary University of London, UK

EXPLICIT RESONANCES FOR ANALYTIC HYPERBOLIC MAPS

Joint work with Wolfram Just and Julia Slipantschuk

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

In a seminal paper Ruelle showed that the long-time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the eigenvalues, also known as Pollicott-Ruelle resonances, of the so-called Ruelle transfer operator, a compact operator acting on a suitable Banach space of holomorphic functions.

In this talk I will report on recent work with Wolfram Just and Julia Slipantschuk on how to construct analytic expanding circle maps or analytic Anosov diffeomorphisms on the torus with explicitly computable non-trivial Pollicott-Ruelle resonances. I will also discuss applications of these results.

Weizhu Bao, National University of Singapore, Singapore

MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC EQUATION IN THE NONRELATIVISTIC LIMIT REGIME

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building B-8, room 0.10b.

Abstract

In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation [4]. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter [3]. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation [3]. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime [3]. We find that the time-splitting spectral method performs super-resolution in temporal discretization when the Dirac equation has no magnetic potential [5]. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter [1]. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic limit regime [2]. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.

References

1. W. Bao, Y. Cai, X. Jia and Q. Tang, A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime, SIAM J. Numer. Anal. 54 (2016), 1785-1812.
2. W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, Sci. China Math. 59 (2016), 1461-1494.
3. W. Bao, Y. Cai, X. Jia and Q. Tang, Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime, J. Sci. Comput. 71 (2017), 1094-1134.
4. W. Bao and J. Yin, A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation, Res. Math. Sci. 6 (2019), article 11.
5. W. Bao, Y. Cai and J. Yin, Improved stability of optimal traffic paths, Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic limit regime, arXiv: 1811.02174.

Krzysztof Barański, University of Warsaw, Poland

SLOW ESCAPING POINTS FOR TRANSCENDENTAL MAPS

Joint work with Bogusława Karpińska

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

Let \(f \colon \mathbb{C} \to \mathbb{C}\) be a transcendental entire map. The set \[ I(f) = \{z \in \mathbb{C} : f^n(z) \to \infty\} \] is called the escaping set of \(f\). In relation with the papers [1, 2], we study the dimension of the sets of points in \(I(f)\) which escape to infinity in a given rate.

References

1. W. Bergweiler, J. Peter, Escape rate and Hausdorff measure for entire functions, Math. Z. 274 (2013), 551-572.
2. D.J. Sixsmith, Dimensions of slowly escaping sets and annular itineraries for exponential functions, Ergodic Theory Dynam. Systems 36 (2016), 2273-2292.

Giovanni Bellettini, University of Siena & ICTP, Italy

ON THE RELAXED AREA OF THE GRAPH OF NONSMOOTH MAPS FROM THE PLANE TO THE PLANE

Joint work with Alaa Elshorbagy, Maurizio Paolini, and Riccardo Scala

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

In various problems concerning area-minimizing surfaces, such as the non parametric Plateau problem, it is natural to have at hand a concept of area for graphs of nonsmooth scalar functions. In this scalar context, the correct notion to consider turns out to be the \(L^1(\Omega)\)-relaxation of the classical area functional \(f \in \mathcal C^1(\Omega) \to \int_\Omega \sqrt{1 + \vert \nabla f\vert^2}~dx\); such a notion has been characterized and admits an integral representation in the space of functions of bounded variation in \(\Omega\) [5, 2]. We are interested in a similar problem for vector-valued maps, more precisely for maps \(u = (u_1,u_2) \to \mathbb{R}^2\) from a plane domain \(\Omega\) to the plane. In this situation the classical area functional is polyconvex and, provided the map \(u\) is sufficiently smooth, say \( u \in \mathcal C^1(\Omega; \mathbb{R}^2)\), reads as \[ A(u) := \int_\Omega \sqrt{1 + \vert \nabla u_1\vert^2 + \vert \nabla u_2\vert^2 + \Big( \frac{\partial u_1}{\partial x} \frac{\partial u_2}{\partial y} - \frac{\partial u_1}{\partial y} \frac{\partial u_2}{\partial x} \Big)^2 } ~dx dy. \] Its \(L^1(\Omega;\mathbb{R}^2)\)-relaxation \(\overline A\) is the object of our interest (see also [4], [1]), in particular evaluated at discontinuous maps, the graphs of whice are therefore nonsmooth two-dimensional surfaces of codimension two. Assuming for simplicity \(\Omega\) to be a disk centered at the origin in the source plane, we shall present some recent results [3] concerning piecewise constant maps \(u_T : \Omega \to \{\alpha, \beta, \gamma\}\) taking three values and having a triple junction as a jump set; here \(\alpha, \beta, \gamma\) are three noncollinear vectors in the target plane. The appearence of solutions of certain Plateau-type problems, inducing a nonlocality phenomenon on \(\overline A(u_T)\), will be pointed out.

References

1. E. Acerbi, G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations 2 (1994), 329-371.
2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.
3. G. Bellettini, A. Elshorbagy, M. Paolini, R. Scala, On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions, submitted.
4. E. De Giorgi, On the relaxation of functionals defined on cartesian manifolds, in Developments in Partial Differential Equations and Applications in Mathematical Physics (Ferrara 1992), Plenum Press, New York 1992.
5. E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984.

Niels Benedikter, Institute of Science and Technology, Austria

CORRELATION ENERGY OF THE MEAN-FIELD FERMI GAS BY THE METHOD OF COLLECTIVE BOSONIZATION

Joint work with Phan Thành Nam, Marcello Porta, Benjamin Schlein, and Robert Seiringer

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 0.10b.

Abstract

Quantum correlations play an important role in interacting systems; however, their mathematical description is a highly non-trivial task. I explain how correlations in fermionic systems can be described by bosonizing collective pair excitations. This leads us to an effective quadratic bosonic Hamiltonian. We establish a theory of approximate bosonized Bogoliubov transformations by which we derive a Gell-Mann-Brueckner-type formula for the fermionic ground state energy.

References

1. N. Benedikter, P.T. Nam, M. Porta, B. Schlein, R. Seiringer, Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime, to appear in Communications in Mathematical Physics, 2019.

Joackim Bernier, Université Paul Sabatier, France

EXACT SPLITTINGS OF LINEAR QUADRATIC PDES

Joint work with Paul Alphonse, Fernando Casas, Nicolas Crouseilles, and Yingzhe Li

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

Usually, the higher the order of a splitting method is, the larger its number of steps is. However, we will see that for many linear ODEs, the resolution of an inverse problem provides an exact splitting, involving some modified vector fields, with the same number of steps as the usual low order methods (i.e. Lie or Strang splittings). Applying the Fourier integral operators theory (developed by Hörmander in [4]), we will see how these decompositions can be transposed at the level of the quadratic linear PDEs. I will show how this construction provides some new efficient splittings for many PDEs like the Schrödinger equations in rotating frames or the Vlasov equations with a rotation motion.

References

1. P. Alphonse, J. Bernier, Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects, in preparation.
2. J. Bernier, F. Casas, N. Crouseilles, Splitting methods for rotations: application to Vlasov equations, preprint available on HAL.
3. J. Bernier, Y. Li, N. Crouseilles, Splitting methods for Schrödinger equations in rotating frames, in preparation.
4. L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.

Martin Berz, Michigan State University, USA

RIGOROUS INTEGRATION OF FLOWS OF ODES USING TAYLOR MODELS

Joint work with Kyoko Makino

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

Taylor models combine the advantages of numerical methods and their efficiency under tightly controlled computational resources even for complex problems, with the advantages of symbolic approaches and their ability to be rigorous and to allow the treatment of functional dependencies instead of merely points. The result is a local representation of a function by its Taylor expansion and a mathematically rigorous bound for the approximation error.

The resulting differential algebraic calculus comprising an algebra with differentiation and integration is particularly amenable to the study of ODEs and PDEs based on fixed point problems from functional analysis. We describe the development of rigorous tools to determine enclosures of flows of general nonlinear differential equations based on Picard iterations.

The methods can be used for the computation of enclosures of flows over large domains to prescribed accuracy via domain decomposition methods. We study the behavior of the methods for several dynamical systems, and in particular analyze suitable parameter settings to  efficiently balance local domain size versus local order. Comparisons to other recently proposed computational approaches are given, showing the advantages of the Taylor model methods for large domains.

Fabrizio Bianchi, CNRS & Université de Lille, France

LATTES MAPS AND THE HAUSDORFF DIMENSION OF THE BIFURCATION LOCUS

Joint work with François Berteloot

Date: 2019-09-20 (Friday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

Given a holomorphic family of endomorphisms of \(P^k (C)\), the bifurcation locus is the set of parameters of instability for the Julia set, the support of the measure of maximal entropy. This locus coincides with the non-harmonicity locus of the Lyapunov function, the sum of the Lyapunov exponents. Lattes maps can be characterised as the minima of the Lyapunov function, and thus lie in the bifurcation locus. We will prove that near isolated Lattes maps the Hausdorff dimension of the bifurcation locus is maximal in any direction.

Stefano Bianchini, SISSA, Italy

DIFFERENTIABILITY OF THE FLOW FOR BV VECTOR FIELDS

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

We show that the Regular Lagrangian Flow \(X(t,y)\) generated by nearly incompressible BV vector fields admits a derivative \(\nabla X(t,y)\) in the sense of measure. This matrix satisfies the ODE \[ \frac{d}{dt} \nabla X(t,y) = (D\mathbf{b}(t))_y \nabla X(t-,y) \] where \((D\mathbf{b})_y\) is the disintegration of the measure \(\int D\mathbf{b}(t) dt\) w.r.t. the trajectories \(X(t,y)\).

Piotr Biler, University of Wrocław, Poland

GLOBAL EXISTENCE VS FINITE TIME BLOWUP IN KELLER--SEGEL MODEL OF CHEMOTAXIS

Joint work with Grzegorz Karch and Jacek Zienkiewicz

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

We consider the parabolic-elliptic model for the chemotaxis with fractional (anomalous) diffusion (\(\alpha\in(0,2)\)) in several space dimensions (\(d\ge 3\)) \begin{align} u_t+(-\Delta)^{\alpha/2} u+\nabla\cdot(u\nabla v)&= 0,\ \ &x\in {\mathbb R}^d,\ t>0,\nonumber\\ \Delta v+u &= 0,\ \ & x\in {\mathbb R}^d,\ t>0,\label{eqv}\\ u(x,0)&= u_0(x)\ge0,\ \ &x\in {\mathbb R}^d.\nonumber \end{align} Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

References

1. P. Biler, G. Karch, J. Zienkiewicz, Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834-875.
2. P. Biler, J. Zienkiewicz, Blowing up radial solutions in the minimal Keller-Segel model of chemotaxis, J. Evol. Equ. 19 (2019), 71-90.
3. P. Biler, Singularities of solutions in chemotaxis systems, De Gruyter, Berlin, 2019, ISBN 978-3-11-059789-9.

Alexander Blokh, University of Alabama at Birmingham, USA

LOCATION OF SIEGEL CAPTURE POLYNOMIALS IN PARAMETER SPACES OF CUBIC POLYNOMIALS

Joint work with Arnaud Chéritat, Lex Oversteegen, and Vladlen Timorin

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

Consider cubic polynomials with a Siegel disk containing an eventual image of a critical point and call them IS-capture polynomials ("IS" stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture polynomial belongs to the boundary of a unique bounded hyperbolic component determined by the rational lamination of the map.

Dirk Blömker, Universität Augsburg, Germany

STOCHASTIC INTERFACE MOTION IN THE CAHN-HILLIARD EQUATION

Joint work with Dimitra Antonopoulou and Georgia Karali

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

We study the two and three dimensional stochastic Cahn–Hilliard equation in the sharp interface limit. It is given by\begin{equation*}\partial_t u = -\Delta v+ \varepsilon^\sigma\partial_t {W},\\v = -\displaystyle\frac{F^{\prime}(u)}{\varepsilon}+\varepsilon \Delta u,\end{equation*}subject to Neumann boundary conditions on a bounded domain \(\mathcal{D}\).

Here \(u: \mathcal{D}\times[0,T]\to [-1,1]\) is the scalar concentration field of one of the components in a separation process,for example of binary alloys. The function \(F\) is a double well potential, for example with \(F'(u)=u-u^3\). The noise is given by a spatially smooth Wiener process \(W\).The small parameter \(\varepsilon>0\) measures the width of transition layers generated during phase separation.

Using formal asymptotic expansions, in \([1]\) we identify the limit. In the case \(\sigma=1\) our results indicate that the stochastic Cahn–Hilliard equation converges to a two-phase Hele-Shaw (or Mullins-Sekerka) problem with stochastic forcing on the transition layers. For the interface \(\Gamma(t)=\{ x\in\mathcal{D}\ :\ u(t,x)=0\}\) in the limit \(\varepsilon\to0\) we obtain \begin{equation*}\left\{\begin{aligned}&\Delta v=0 \,\, \mbox{ in } \mathcal{D} \backslash\Gamma,\\&\partial_n v=0 \,\, \mbox{ on }\partial\mathcal{D},\\&v=\lambda H+ W\,\, \mbox{ on } \Gamma,\\&V= [\partial_n v] \,\, \mbox{ on } \Gamma,\end{aligned}\right.\end{equation*}where \(H\) is the mean curvature and \(V\) the normal velocity of \(\Gamma\).The jump term \([\partial_n v]\) denotes the average of the normal derivative of \(v\) from both sides of \(\Gamma\).

In a joint work with S. Yokoyama (Tokyo) we show that the stochastic Hele-Shaw problem has a local smooth solution,given that the initial surface \(\Gamma(0)\) is a smooth closed hypersurface that does not touch the boundary of \(\mathcal{D}\).

In the case when the noise is sufficiently small (i.e. \(\sigma>1\) sufficiently large), in \([1]\) we can prove rigorously that the limit is a deterministic Hele-Shaw problem without \(W\). The main reason for \(\sigma\) being very large for this result is due to the lack of sufficiently good spectral estimates for the linearized Cahn-Hilliard operator. These are currently only verified in \(H^{-1}\).

References

1. D.C. Antonopoulou, D. Blömker, G.D. Karali, The sharp interface limit for the stochastic Cahn–Hilliard equation, Ann. Inst. H. Poincaré Probab. Statist. 54(1) (2018), 280-298.

Daniele Boffi, University of Pavia, Italy

A FICTITIOUS DOMAIN APPROACH TO FLUID-STRUCTURE INTERACTION PROBLEMS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

We review a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method for the numerical approximation of the interaction between fluids and solids (see [1, 2]). The discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. Optimal convergence estimates are proved for its finite element space discretization. The model, originally introduced for the coupling of incompressible fluids and solids, can be extended to include the simulation of compressible structures [3].

Recent research [4] investigates several time marching strategies for the proposed method.

References

1. D. Boffi, N. Cavallini, L. Gastaldi, The Finite Element Immersed Boundary Method with Distributed Lagrange multiplier, SIAM Journal on Numerical Analysis 53(6) (2015), 2584–2604.
2. D. Boffi, L. Gastaldi, A fictitious domain approach with distributed Lagrange multiplier for fluid-structure interactions, Numerische Mathematik 135(3) (2017), 711–732.
3. D. Boffi, L. Gastaldi, L. Heltai, A distributed Lagrange formulation of the Finite Element Immersed Boundary Method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, D. Boffi, L. Pavarino, G. Rozza, S. Sacchi, C. Vergara eds., SEMA SIMAI Springer Series 16 (2018), 1–21.
4. D. Boffi, L Gastaldi, S. Wolf, Higher order time stepping schemes for fluid-structure interaction problems, in preparation.

Barbara Boldin, University of Primorska, Slovenia

ECO-EVOLUTIONARY CYCLES OF VIRULENCE UNDER SELECTIVE PREDATION

Joint work with Eva Kisdi and Stefan A.H. Geritz

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

In nature, diseased animals do not die peacefully on their deathbeds. Rather, they are often more likely to fall victim to predators, who selectively prey upon the weak. Such disease-induced selective predation brings about changes in prey populations, which may in turn alter the evolutionary dynamics of pathogens. The resulting eco-evolutionary system may exhibit rich dynamics with multiple attractors and limit cycles. We introduce a novel technique to provide a constructive proof that the system can exhibit stable limit cycles under realistic assumptions about model ingredients and perform a detailed bifurcation analysis for a concrete example of a predator-prey-pathogen system.

References

1. E. Kisdi, S.A.H. Geritz, B. Boldin, Evolution of pathogen virulence under selective predation: A construction method to find eco-evolutionary cycles, Journal of Theoretical Biology 339 (2013), 140-150.

Everaldo Bonotto, University of São Paulo, Brazil

NEW TRENDS ON GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

Joint work with Marcia Federson and Rodolfo Collegari

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

The theory of generalized ordinary differential equations lies on the fact that these equations encompass various types of other equations such as ordinary differential equations (ODEs), impulsive differential equations (IDEs), measure differential equations (MDEs), functional differential equations and dynamic equations on time scales. Moreover, the theory of generalized ordinary differential equations deals with problems, especially, when the functions involved have many discontinuities and/or are of unbounded variation.

In this talk, we present the general theory of generalized ordinary differential equations and also the most recent results on this topic. In special, we show that this theory also encompass the stochastic differential equations.

References

1. M. Federson, Š. Schwabik, Generalized ODE approach to impulsive retarded differential equations, Differential Integral Equations 11 (2006), 1201-1234.
2. Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, Series in real Anal., vol. 5, 1992.

Jan Boroński, AGH University of Science and Technology, Poland

ALL MINIMAL CANTOR SYSTEMS ARE SLOW

Joint work with Jiří Kupka and Piotr Oprocha

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

A Cantor set is a 0-dimensional compact metric space without isolated points, and Cantor system is a dynamical system on the Cantor set. A minimal system is the one that has all orbits dense. We are interested in the following question: Can every minimal Cantor system be embedded into \(\mathbb{R}\) with vanishing derivative everywhere? A particular instance of that question was raised by Samuel Petite at the Workshop on Aperiodic Patterns in Crystals, Numbers and Symbols that took place in Lorentz Center in June of 2017, who asked if expansive minimal Cantor systems have this property. It was conjectured that the expansive systems lack such a property, because some kind of expanding must take place in these systems. In contrast, I shall discuss a postive answer to the above question on all Cantor minimal systems, obtained in [2]. There are more reasons for which this result seems surprising. By the Margulis-Ruelle inequality the topological entropy of a piecewise Lipschitz differentiable map \(f\), with an invariant measure \(\mu\), is bounded from above by the integral over the support of \(\mu\) of the Lyapunov characteristic of \(f\). In the case of derivative zero, all Lyapunov exponents, and as a result Lyapunov characteristic of \(f\) are all equal to \(0\). Therefore it is natural to expect that vanishing derivative on an invariant set will imply zero entropy on that set. Such an intuition was supported by the zero entropy examples in [3] and [1]. However our result shows that no such connection exists. Note that if \(f\) is \(C^1\) on \(\mathbb{R}\) and \(f(P)\subset P\) for some perfect compact subset \(P\subset \mathbb{R}\), then there is \(x\in P\) with \(|f'(x)|\geq 1\) (see [4] for a nice survey on this and related topics). For systems with positive entropy it is also a consequence of Margulis-Ruelle inequality mentioned above, so in this case the map is not even Lipschitz continuous. This gives rise to the following question: Can the differentiable extensions of minimal Cantor systems to \(\mathbb{R}\), guaranteed by our result, be additionally required to be \(\alpha\)-Hölder continuous for some \(0<\alpha<1\)?

References

1. J. Boroński, J. Kupka, P. Oprocha, Edrei's Conjecture Revisited, Ann. Henri Poincaré 19 (2018), 267-281.
2. J. Boroński, J. Kupka, P. Oprocha, All Minimal Cantor Systems Are Slow, arXiv:1902.10641.
3. K.C. Ciesielski, J. Jasiński, An auto-homeomorphism of a Cantor set with derivative zero everywhere, J. Math. Anal. Appl. 434 (2016), 1267-1280.
4. K.C. Ciesielski, J. Seoane-Sepúlveda, Differentiability versus continuity: restriction and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56(2) (2019), 211-260.

Alberto Boscaggin, University of Turin, Italy

GENERALIZED PERIODIC SOLUTIONS TO PERTURBED KEPLER PROBLEMS

Joint work with Walter Dambrosio, Duccio Papini, Rafael Ortega, and Lei Zhao

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

For the perturbed Kepler problem \[ \ddot x = - \frac{x}{\vert x \vert^3} + \nabla_x W(t,x), \qquad x \in \mathbb{R}^d, \] with \(d =2\) or \(d = 3\), we discuss the existence of periodic solutions, possibly interacting with the singular set (\(x = 0\)). First, a suitable notion of generalized solution is introduced, based on the theory of regularization of collisions in Celestial Mechanics; second, existence and multiplicity results are provided, with suitable assumptions on the perturbation term \(W\), by the use of sympletic and variational methods.

References

1. A. Boscaggin, W. Dambrosio, D. Papini, Periodic solutions to a forced Kepler problem in the plane, Proc. Amer. Math. Soc., to appear.
2. A. Boscaggin, R. Ortega, L. Zhao, Periodic solutions and regularization of a Kepler problem with time-dependent perturbation, Trans. Amer. Math. Soc., to appear.

Bruno Bouchard, CEREMADE, Paris-Dauphine, PSL Research University, France

PERFECT REPLICATION WITH MARKET IMPACT: TOWARDS A DUAL FORMULATION FOR A CLASS OF SECOND ORDER COUPLED FBSDEs

Joint work with Grégoire Loeper, Halil Mete Soner, and Chao Zhou

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 2.19.

Abstract

We first extend the study of [2, 3] to stochastic target problems with general market impacts. The perfect hedging problem amounts to solving a second order coupled FBSDEs. Unlike [2, 3], the related fully non-linear PDE is not concave and the regularization/verification approach of [2] can not be applied. In place, we need to generalize the a priori estimates of [3] and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the super-hedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation suggests a dual formulation for the Brownian diffusion Markovian setting. We shall explain how this dual formulation can indeed be exploited to solve a general class of non-Markovian second order coupled FBSDEs driven by general continuous martingales.

References

1. B. Bouchard, G. Loeper, M. Soner and C. Zhou, Second order stochastic target problems with generalized market impact, arXiv:1806.08533, (2018).
2. B. Bouchard, G. Loeper and Y. Zou, Hedging of covered options with linear market impact and gamma constraint, SIAM Journal on Control and Optimization 55(5) (2017), 3319–3348.
3. G. Loeper, Option pricing with linear market impact and nonlinear Black-Scholes equations, The Annals of Applied Probability 28(5) (2018), 2664–2726.

Abed Bounemoura, CNRS & CEREMADE, Université Paris-Dauphine, France

ANALYTIC INVARIANT CURVES FOR ANALYTIC TWIST MAPS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

Yoccoz proved that the Bruno arithmetical condition is optimal for the analytic linearization of a circle diffeomorphism close to a rotation. We will explain how to use this result to show that the same condition is optimal for the analytic preservation of quasi-periodic invariant curves for twist maps of the annulus, as well as new questions in this context.

Tom Britton, Stockholm University, Sweden

EPIDEMICS IN STRUCTURED COMMUNITIES WITH SOCIAL DISTANCING

Joint work with Frank Ball, KaYin Leung, and Dave Sirl

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

Consider a large community, structured as a network, in which an epidemic spreads. Infectious individuals spread the disease to each of their susceptible neighbors, independently, at rate \(\lambda\), and each infectious individual recovers and becomes immune at rate \(\gamma\). The social distancing is modeled by each susceptible who has an infectious neighbor rewires away from this individual to a randomly chosen individual at rate \(\omega\). Our main result is surprising and says: the rewiring is rational from an individual perspective since it reduces the risk of being infected, but at the same time it may be harmful for the community at large in that the outbreak may get bigger compared to no rewiring (\(\omega=0\)).

References

1. F. Ball, T. Britton, KY. Leung, and D. Sirl, Individual preventive measures during an epidemic may have negative population-level outcomes, Journal of Royal Society: Interface. 15 (2018), 20180296.
2. F. Ball, T. Britton, KY. Leung, and D. Sirl, A stochastic SIR epidemic model with preventive dropping of edges, J. Math. Biol. 75 (2019), 1875-1951.

Martin Brokate, Technische Universität München, Germany

RATE INDEPENDENT EVOLUTIONS: DIFFERENTIAL SENSITIVITY

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

Rate independent evolutions are inherently nonsmooth. Mappings involving rate independent evolutions do not possess classical derivatives. However, basic rate independent processes like the scalar play or stop hysteresis operator have weak derivatives in the sense of Bouligand. Moreover, they are semismooth. As a consequence, the control-to-state mapping of certain evolutions which include rate independent elements also enjoy these properties. This work is published in [1, 2], based on the previous work [3].

References

1. M. Brokate, Newton and Bouligand derivatives of the scalar play and stop operator, arXiv:1607.07344v2 [math.FA] (2019), 41 pp., submitted to Math. Model. Nat. Phenom.
2. M. Brokate, K. Fellner, M. Lang-Batsching, Weak differentiability of the control-to-state mapping in a parabolic equation with hysteresis, arXiv:1905.01863 [math.AP] (2019), 18 pp., submitted to Nonlinear Differ. Equ. Appl.
3. M. Brokate, P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Contin. Dyn. Syst. 35 (2015), 2405-2421.

Henk Bruin, Universität Wien, Austria

ON UNIMODAL INVERSE LIMIT SPACES

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

Inverse limit spaces of unimodal maps are used to model attractors of maps (on the plane). As such, the classification of unimodal inverse limit spaces has an impact on out understanding of e.g. Henon-like attractors. In this talk I want to give an update on how topological properties of unimodal inverse limit spaces relate to dynamical properties of the underlying map.

Zoltán Buczolich, Eötvös Loránd University, Hungary

GENERIC BIRKHOFF SPECTRA

Joint work with Balázs Maga and Ryo Moore

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

Suppose that \(\Omega = \{0, 1\}^ {\mathbb {N}}\) and \(\sigma\) is the one-sided shift. The Birkhoff spectrum \( \displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},\) where \(\dim_{H}\) is the Hausdorff dimension. It is well-known that the support of \(S_{f}( {\alpha})\) is a bounded and closed interval \(L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]\) and \(S_{f}( {\alpha})\) on \(L_{f}\) is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical \(f\in C( \Omega)\) in the sense of Baire category. For a dense set in \(C( \Omega)\) the spectrum is not continuous on \(\mathbb {R}\), though for the generic \(f\in C(\Omega)\) the spectrum is continuous on \( \mathbb {R}\), but has infinite one-sided derivatives at the endpoints of \(L_{f}\). We give an example of a function which has continuous \(S_{f}\) on \(\mathbb {R}\), but with finite one-sided derivatives at the endpoints of \(L_{f}\). The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions \(f\) and \(g\) are close in \(C(\Omega)\) then \(S_{f}\) and \(S_{g}\) are close on \(L_{f}\) apart from neighborhoods of the endpoints.

Inese Bula and Diāna Mežecka, University of Latvia, Latvia

ON THE SOLUTIONS OF RICCATI DIFFERENCE EQUATION VIA FIBONACCI NUMBERS

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 2.2.

Abstract

A difference equation of the form \[x_{n+1}= \frac{\alpha + \beta x_n}{A+ Bx_n}, \quad n=0,1,...,\] where the parameters \(\alpha\), \(\beta\), \(A\), \(B\) and the initial condition \(x_0\) are real numbers is called a Riccati difference equation. This equation has been studied in many articles (for example, see general review in [1]). In [2, 3] authors studied special cases of Riccati difference equation whose solutions can be expressed via Fibonacci numbers.

In our talk we consider a Riccati difference equation in the form \[ x_{n+1}= \frac{F_m + F_{m-1} x_n}{F_{m+1} + F_m x_n}, \quad n=0,1,..., \tag{1} \] where \(F_0=0\), \(F_1=1\), ..., \(F_{m+1}=F_{m}+F_{m-1}\), \(m=1,2,...\), are Fibonacci numbers. We show some properties of equations (1), including following result.

Theorem. For every \(m=1,2,...\), and every initial condition \(x_0 \neq - \frac{F_{k+1}}{F_{k}}, \, k=1,2,...\), the solution of equation (1) is in the form \[x_n=\frac{F_{mn}+F_{mn-1}x_0}{F_{mn+1}+F_{mn} x_0},\quad n=1, 2,... \, .\]

References

1. M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRCBoca, Raton, Florida, 2002.
2. D.T. Tollu, Y. Yazlik, N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations 174 (2013), 1-7.
3. D.T. Tollu, Y. Yazlik, N. Taskara, The Solutions of Four Riccati Difference Equations Associated with Fibonacci Numbers, Balkan Journal of Mathematics 02 (2014), 163-172.

Pál Burai, University of Debrecen, Hungary

MEAN LIKE MAPS GENERATED BY PATHOLOGICAL FUNCTIONS

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

The goal of this talk is the investigation of quasi-arithmetic expressions (close relatives of quasi-arithmetic means) generated by invertible (not necessary continuous) functions.

The resulted class can contain maps, which are not means, and which are not regular. However, it contains the whole class of quasi-arithmetic means.

References

1. S. Balcerzyk, Wstęp do algebry homologicznej, Państwowe Wydawnictwo Naukowe, Warsaw, 1970, Biblioteka Matematyczna, Tom 34.
2. K. Baron, On additive involutions and Hamel bases, Aequationes Math., 87(1-2) (2014), 159-163.
3. P. Burai, An extension theorem for conditionally additive functions and its application for the equality problem of Quasi-arithmetic expressions, submitted.
4. J.G. Dhombres and R. Ger, Conditional Cauchy equations, Glas. Mat. Ser. III 13(33)(1) (1978), 39-62.
5. Z. Daróczy and L. Losonczi, Über die Erweiterung der auf einer Punktmenge additiven Funktionen, Publ. Math. Debrecen 14 (1967), 239-245.
6. W. Jabłoński, Additive involutions and Hamel bases, Aequationes Math. 89(3) (2015), 575-582.
7. M. Kuczma, An introduction to the theory of functional equations and inequalities, Birkhäuser Verlag, Basel, second edition, 2009. Cauchy’s equation and Jensen’s inequality, Edited and with a preface by Attila Gilányi.
8. I. Makai, Über invertierbare Lösungen der additiven Cauchy-Funktionalgleichung, Publ. Math. Debrecen 16 (1969), 239-243.

David Burguet, CNRS & Sorbonne Université, France

SYMBOLIC EXTENSIONS AND UNIFORM GENERATORS FOR TOPOLOGICAL REGULAR FLOWS

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

Building on the theory of symbolic extensions and uniform generators for discrete transformations we develop a similar theory for topological regular flows. In this context a symbolic extension is given by a suspension flow over a subshift.

References

1. D. Burguet, Symbolic extensions and uniform generators for topological regular flows, Journal of Differential Equations, (to appear), https://arxiv.org/abs/1812.04285.

Leo Butler, University of Manitoba, Canada

INVARIANT TORI FOR A CLASS OF THERMOSTATED SYSTEMS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-4, room 120.

Abstract

A thermostated hamiltonian system is a model of a mechanical system immersed in a heat bath at constant temperature. A fundamental question is whether the system reaches thermal equilibrium. Even for 1-degree-of-freedom hamiltonians this question is non-trivial, and an answer has only been known in the simplest case by work of Legoll, Luskin and Moeckel [4, 5].

Most of the literature on thermostats focuses on specific examples. I propose a mathematical definition of a thermostat that captures the content of these examples. Under some conditions, the properties of a thermostat lead to the existence of a thermostatic equilibrium when it is only weakly coupled with the hamiltonian. Under a modest additional hypothesis on the non-degeneracy of this equilibrium, one obtains the existence of invariant tori in a neighbourhood of the thermostatic equilibrium. The existence of such KAM tori frustrates "thermalization".

These conditions are verified for four well-known examples in the literature [1, 2, 3, 7, 8], when the hamiltonian is real-analytic and "well-behaved".

If time permits, I will discuss related results about variants including non-equilibrium thermostats, and multiple/recursive thermostats.

References

1. W. Hoover, Canonical dynamics: equilibrium phase space distributions, Phys. Rev. A. 31 (1985), 1695–1697.
2. W. Hoover, Nosé-Hoover nonequilibrium dynamics and statistical mechanics, Mol. Simul. 33 (2007), 13–19.
3. W.G. Hoover, J.C. Sprott, and C.G. Hoover, Ergodicity of a singly-thermostated harmonic oscillator, Communications in Nonlinear Science and Numerical Simulation, 32(Supplement C):234 – 240, 2016.
4. F. Legoll, M. Luskin, and R. Moeckel, Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator, Arch. Ration. Mech. Anal. 184(3) (2007), 449–463.
5. F. Legoll, M. Luskin, and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity 22(7) (2009), 1673–1694.
6. S. Nosé, A unified formulation of the constant temperature molecular dynamics method, J. Chem. Phys. 81 (1984), 511–519.
7. D. Tapias, A. Bravetti, and D.P. Sanders, Ergodicity of one-dimensional systems coupled to the logistic thermostat, CMST 23, 11 2016.
8. H. Watanabe and H. Kobayashi, Ergodicity of a thermostat family of the Nosé-Hoover type, Phys. Rev. E 75:040102, Apr 2007.

Sun-Sig Byun, Seoul National University, South Korea

GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC PROBLEMS WITH NONSTANDARD GROWTH

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

We are concerned with a quasilinear elliptic equation with non-standard growth condition over a non-smooth domain. The nonlinearity involves a non-uniformly ellipticity property and the boundary of the domain is sufficiently flat. We prove a global regularity estimate for the gradient of solutions in the frame of a generalized Sobolev space under substantially more general assumptions.

Liviu Cădariu-Brăiloiu, Politehnica University of Timisoara, Romania

APPLICATIONS OF FIXED POINT RESULTS TO THE GENERALIZED HYERS-ULAM STABILITY OF A FUNCTIONAL EQUATION

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

In the last time there are emphasised several methods which allows to obtain Hyers-Ulam stability results for large classes of functional, differential and integral equations, in various spaces. For example, some fixed points theorems for operators (not necessarily linear) satisfying suitable very general properties have been proved recently. These results were used to obtain properties of generalized Hyers-Ulam stability, hyperstability, superstability, best constant, for different classes of functional equations.

The aim of this talk is to present an application of such a fixed point theorem for proving generalized Hyers-Ulam stability properties of a functional equation.

References

1. J. Brzdęk, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Analysis - TMA 74 (2011), 6728–6732.
2. J. Brzdęk, L. Cădariu, K. Ciepliński, Fixed point theory and the Ulam stability, J. Function Spaces 2014 (2014), Article ID 829419, 16 pp.
3. J. Brzdęk, L. Cădariu, Stability for a family of equations generalizing the equation of p-Wright affine functions, Appl. Math. Comput. 276 (2016), 158–171.
4. K. Ciepliński, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations – a survey, Ann. Funct. Anal. 3(1) (2012), 151–164.

Russel Caflisch, New York University, USA

ACCELERATED SIMULATION FOR PLASMA KINETICS

Joint work with Denis Silantyev and Bokai Yann

Date: 2019-09-17 (Tuesday); Time: 15:00-15:40; Location: building B-8, room 0.10b.

Abstract

This presentation will discuss the kinetics of Coulomb collisions in plasmas, as described by the Landau-Fokker-Planck equation, and its numerical solution using a Direct Simulation Monte Carlo (DSMC) method. Acceleration of this method is achieved by coupling the particle method to a continuum fluid description. Efficiency of the resulting hybrid method is greatly increased by inclusion of particles with negative weights. This complicates the simulation, and introduces difficulties have plagued earlier efforts to use negatively weighted particles. This talk will describe significant progress that has been made in overcoming those difficulties.

Renato Calleja, National Autonomous University of Mexico, Mexico

TORUS KNOT CHOREOGRAPHIES IN THE \(N\)-BODY PROBLEM

Joint work with Eusebius Doedel, Carlos García Azpeitia, Jason Mireles-James, and Jean-Philippe Lessard

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

\(N\)-body choreographies are periodic solutions to the \(N\)-body equations in which \(N\) equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for proving the existence of spatial choreographies in the gravitational \(N\) body problem with the help of the digital computer. These arise from the polygonal system of \(N\) bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies.

References

1. R. Calleja, E. Doedel, and C. García-Azpeitia, Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the \(n\)-body problem, Celestial Mech. Dynam. Astronom. 130 (2018), 130:48.
2. R. Calleja, C. García-Azpeitia, J.P. Lessard, and J.D. Mireles-James, Torus knot choreographies in the \(n\)-body problem,Preprint, available at http://cosweb1.fau.edu/~jmirelesjames/torusKnotChoreographies.html

Juan Calvo, University of Granada, Spain

A BRIEF PERSPECTIVE ON TEMPERED DIFFUSION EQUATIONS

Joint work with Antonio Marigonda and Giandomenico Orlandi

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

Tempered diffusion equations (also termed ''flux-saturated'' or ''flux-limited'' diffusion equations) are a class of non-linear versions of the standard diffusion equation displaying a mixture of parabolic and hyperbolic features. After reviewing the entropy solution framework and the qualitative properties of such equations, we will discuss the way in which those models fit in the variational framework developed by Jordan, Kinderlehrer and Otto to study evolution problems in terms of optimal transport theory. Then we will present some anisotropic versions of the foregoing theory and conclude with a tentative application to Developmental Biology: cytoneme networks.

Giacomo Canevari, University of Verona, Italy

THE SET OF TOPOLOGICAL SINGULARITIES OF VECTOR-VALUED MAPS

Joint work with Giandomenico Orlandi

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

We introduce [1] an operator \(\mathbf{S}\) on vector-valued maps \(u\colon\Omega\subseteq\mathbb{R}^d\to\mathbb{R}^m\), which has the ability to capture the relevant topological information carried by \(u\). In particular, this operator is defined on maps that take values in a closed submanifold \(\mathcal{N}\) of the Euclidean space \(\mathbb{R}^m\), and coincides with the distributional Jacobian in case \(\mathcal{N}\) is a sphere. More precisely, the range of \(\mathbf{S}\) is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. We use \(\mathbf{S}\) to characterise strong limits of smooth, \(\mathcal{N}\)-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière [2]. We present applications to the study of manifold-valued maps of bounded variation, and to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with \(\mathcal{N}\)-well potentials.

References

1. G. Canevari, G. Orland, opological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces, Calc. Var. Partial Dif. 58(2) (2019), 58-72.
2. M.R. Pakzad, T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal. 13(1) (2003), 223-257.

Yves Capdeboscq, Université de Paris, France

SOLVING HYBRID INVERSE PROBLEMS MOST OF THE TIME

Joint work with Giovanni S. Alberti

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We show that combining a projection method attributed to H. Whitney in geometry, and a unique continuation principle for solution of 2nd order PDEs, a small number of boundary condition can be selected so that the vector space generated by the gradient fields of electrical potentials be of maximal rank everywhere in the domain. This result applies in particular when the conductivity of the medium varies in space. This provides a generalization of the so-called thermal coordinates beyond the two dimensional case.

Maciej Capiński, AGH University of Science and Technology, Poland

ARNOLD DIFFUSION IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM

Joint work with Marian Gidea

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

We present a topological mechanism for diffusion in Hamiltonian systems and apply it to the Planar Elliptic Restricted Three-Body Problem. We treat the elliptic problem as a perturbation of the circular problem, where the perturbation parameter \(\varepsilon\) is the eccentricity of the orbits of the primaries. We measure the energy as the Hamiltonian \(H\) of the circular problem. Our objective is to prove that for any \(\varepsilon>0\) there exist orbits which start with some value \(H=h\) and finish with \(H=h+c\), where \(c>0\) is independent from \(\varepsilon\). Our method is based on topological shadowing of trajectories along homoclinic intersections of invariant manifolds. We perform a geometric construction, which allows us to obtain orbits for which we can control the increase in energy. The method is suitable for computer assisted proofs and can be used to obtain explicit bounds on the energy changes, for explicit ranges of the perturbation parameter. We apply it to the setting of the Neptune-Tryton system.

The construction also leads to symbolic dynamics in energy and to stochastic properties of the diffusing orbits. This will be the subject of the talk by Marian Gidea in the same session.

Marco Caroccia, University of Florence, Italy

INTEGRAL REPRESENTATION OF LOCAL ENERGIES ON \(BD\)

Joint work with Matteo Focardi and Nicolas Van Goethem

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

In collaboration with Matteo Focardi and Nicolas Van Goethem we make use of the result on the structure of the \(\mathcal{A}\)-free measure provided in [1] to give a full integral representation result for local energies defined on the space of bounded deformation maps \(BD(\Omega)\) which accounts also for the Cantor part of the symmetric gradient measure. We employ the global method introduced in [2] and exploited also in [3] to give an integral representation Theorem for energies defined on \(SBD(\Omega)\). A full integral representation result for energy on \(BD(\Omega)\) has been missing until now due to a lack of information on the structure of the Cantor part of the symmetric gradient measure. This piece of information has been finally provided by Rindler, De Philippis in the celebrated result of 2016 [1] and it represents the crucial ingredient of our proof, together with a double blow-up procedure and a fine analysis of the blow-up limits around Cantor points. I will briefly describe the methodology of the proof and the consequences of our result in fracture mechanics and in damage models.

References

1. G. De Philippis and F. Rindler, On the structure of A-free measures and applications, Annals of Mathematics, (2016) pp. 1017–1039.
2. G. Bouchitté, I. Fonseca, and L. Mascarenhas, A global method for relaxation, Archive for rational mechanics and analysis 145(1) (1998) pp. 51-98.
3. F. Ebobisse and R. Toader, A note on the integral representation of functionals in the space SBD (O), Rend. Mat. Appl. 7 (2001).

Bernardo Carvalho, Federal University of Minas Gerais, Brazil & Friedrich-Schiller-Universität Jena, Germany

CONTINUUM-WISE HYPERBOLICITY

Joint work with Alfonso Artigue, Welington Cordeiro, and José Vieitez

Date: 2019-09-16 (Monday); Time: 11:05-11:25; Location: building A-4, room 120.

Abstract

Hyperbolicity is one of the most important concepts in the theory of chaotic dynamical systems. Since the seminal works of Anosov [1] and Smale [5] it has been a main topic of research among many mathematicians. In hyperbolic systems, each tangent space splits into two invariant subspaces, the first being uniformly contracted, and the second uniformly expanded, by the action of the derivative map. The dynamics of such systems can be well described in both topological and statistical viewpoints, so many effort is being made to understand the dynamics beyond uniform hyperbolicity and many generalizations have been considered.

In this talk I will discuss one specific notion of hyperbolicity introduced in a joint work with A. Artigue, W. Cordeiro and J. Vieitez [3] called continuum-wise hyperbolicity. Examples of these systems are the Anosov diffeomorphisms [1], the topologically hyperbolic homeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere. We discuss the dynamics of cw-hyperbolic homeomorphisms, proving that some dynamical properties that are present in the hyperbolic theory, such as the shadowing property and a spectral decomposition, are still present in cw-hyperbolic ones, while enlightening the differences and peculiarities of these systems, such as wilder local stable sets containing cantor sets and the existence of arbitrarily small horseshoes.

References

1. D.V. Anosov, Geodesic flows on compact manifolds of negative curvature, Trudy mat. Inst. V.A Steklova, 90 (1967).
2. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Beyond topological hyperbolicity: the L-shadowing property, arXiv:1902.07578.
3. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Continuum-wise hyperbolicity, preprint.
4. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Countably and entropy expansive homeomorphisms with the shadowing property, preprint.
5. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.

Álvaro Castañeda, University of Chile, Chile

DICHOTOMY SPECTRUM AND ALMOST TOPOLOGICAL CONJUGACY ON NONAUTONOMOUS UNBOUNDED DIFFERENCE SYSTEM

Joint work with Gonzalo Robledo

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We will consider the nonautonomous linear system \[ x(n + 1) = A(n)x(n)\tag{1} \] where \(x(n)\) is a column vector of \(\mathbb{R}^d\) and the matrix function \(n \mapsto A(n) \in \mathbb{R}^{d\times d}\) is non singular. We also assume that (1) has an exponential dichotomy on \(\mathbb{Z}\) with projector \(P = I\) (see [1] for a formal definition). We also consider the perturbed system \[ w(n + 1) = A(n)w(n) + f (n, w(n))\tag{2} \] where \(f : \mathbb{Z} \times \mathbb{R}^d \to \mathbb{R}^d\) is continuous in \(\mathbb{R}^d\) is a Lipschitz function such that \(n \mapsto f(n,0)\) is bounded for any \(n \in \mathbb{Z}.\) We will present a result with sufficient conditions ensuring that (1) and (2) are almost topologically equivalent, namely the existence of a map \(H : \mathbb{Z} \times \mathbb{R}^d \to \mathbb{R}^d\) with the following properties: i) For each fixed \(n \in \mathbb{Z},\) the map \(u \mapsto H(n,u)\) is a bijection. ii) For any fixed \(n \in \mathbb{Z},\) the maps \(u \mapsto H(n,u)\) and \(u \mapsto H^{-1}(n,u) = G(n,u)\) are continuous with the possible exception of a set with Lebesgue measure zero. iii) If \(x(n)\) is a solution of (1), then \(H[n,x(n)]\) is a solution of (2). Similarly, if \(w(n)\) is a solution of (2), then \(G[n, w(n)]\) is a solution of (1). This result can also be seen as a generalization of a continuous result obtained by F. Lin in [2].

References

1. B. Aulbach, S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Diff. Eqs. Appl. 7 (2001), 895–913.
2. F. Lin, Hartman’s linearization on nonautonomous unbounded system, Nonlinear Analysis 66 (2007), 38–50.

Dongho Chae, Chung-Ang University, South Korea

ON THE TYPE I BLOW-UP FOR THE INCOMPRESSIBLE EULER EQUATIONS

Joint work with Jörg Wolf

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 0.10b.

Abstract

In this talk we discuss the Type I blow up and the related problems in the 3D Euler equations. We say a solution \(v\) to the Euler equations satisfies Type I condition at possible blow up time \(T_*\) if \(\lim\sup_{t\nearrow T_*} (T_*-t) \|\nabla v(t)\|_{L^\infty} <+\infty\). The scenario of Type I blow up is a natural generalization of the self-similar(or discretely self-similar) blow up. We present some recent progresses of our study regarding this. We first localize previous result that "small Type I blow up" is absent. After that we show that the atomic concentration of energy is excluded under the Type I condition. This result, in particular, solves the problem of removing discretely self-similar blow up in the energy conserving scale, since one point energy concentration is necessarily accompanied with such blow up. We also localize the Beale-Kato-Majda type blow up criterion. Using similar local blow up criterion for the 2D Boussinesq equations, we can show that Type I and some of Type II blow up in a region off the axis can be excluded in the axisymmetric Euler equations.

Nishant Chandgotia, Hebrew University of Jerusalem, Israel

SOME RESULTS ON PREDICTIVE SEQUENCES

Joint work with Benjamin Weiss

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

A sequence of natural numbers \(P\) is called predictive if for any zero-entropy stationary process \(X_i, X_0\) is measurable with respect to \(X_{-i}\); \(i\in P\). In this talk, we will discuss several necessary conditions and sufficient conditions for sequences to be predictive.

Ajay Chandra, Imperial College London, UK

STOCHASTIC NON-ABELIAN YANG-MILLS HEAT FLOW

Joint work with Martin Hairer and Hao Shen

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

I will start by describing how a mathematician can think of the problem of constructing a Yang-Mills quantum field theory and how one approach to this problem involves working with singular SPDE. I will then describe results obtained in ongoing joint work with Martin Hairer and Hao Shen regarding local existence and gauge covariance for a singular SPDE that should correspond to a non-Abelian Yang-Mills quantum field theory.

Jean-François Chassagneux, Université Paris Diderot, France

WEAK ERROR EXPANSION FOR MEAN-FIELD SDE

Joint work with Lukasz Szpruch and Alvin Tse

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

In this work, we study the weak approximation error by particle system of Mean Field SDE. We prove an expansion of this error in terms of the number of particle. Our strategy of proof follows the approach of Talay-Tubaro for weak approximation of SDE by an Euler Scheme. We thus consider a PDE on the Wasserstein space (called the Master Equation in mean-field games literature) and, relying on smoothness properties of the solution, obtain our expansion. We also prove the required smoothness properties under sufficient conditions on the coefficient function.

Hayato Chiba, Tohoku University, Japan

A BIFURCATION OF THE KURAMOTO MODEL ON NETWORKS

Joint work with Georgi Medvedev

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

For the mean-field limit of a system of globally coupled phase oscillators defined on networks, a bifurcation from the incoherent state to the partially locked state at the critical coupling strength is investigated based on the generalized spectral theory. This reveals that a network topology affects the dynamics through the eigenvalue problem of a certain Fredholm integral operator which defines the structure of a network.

References

1. H. Chiba, G.S. Medvedev, The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas, Discret. Contin. Dyn. S.-A 39 (2019), 131-155.
2. H. Chiba, G.S. Medvedev, The mean field analysis ofthe Kuramoto model on graphs II. Asymptotic stability of the incoherentstate, center manifold reduction, and bifurcations, Series, Discret. Contin. Dyn. S.-A, (2019).
3. H. Chiba, G.S. Medvedev, M.S. Muzuhara, Bifurcations in the Kuramoto model on graphs, Chaos 28 (2019), 073109.

Luigi Chierchia, Roma Tre University, Italy

KAM THEORY FOR SECONDARY TORI

Joint work with Luca Biasco

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

As well known, classical KAM (Kolmogorov, Arnold, Moser) theory deals with the persistence, under small perturbations, of real-analytic (or smooth) Lagrangian tori for nearly-integrable non-degenerate Hamiltonian systems. In this talk I will present a new uniform KAM theory apt to deal also with secondary tori, i.e., maximal invariant tori (with different homotopy) "generated" by the perturbation (and that do not exist in the integrable limit). The word "uniform" means that primary and secondary tori are constructed simultaneously; in particular, in the case of Newtonian mechanical systems on \(\textbf{T}^d\), it is proven that, for generic perturbations, the union of primary and secondary tori leave out a region of order \(\varepsilon |\log \varepsilon|^a\), if \(\varepsilon\) is the norm of the perturbation, in agreement (up to the logarithmic correction) with a conjecture by Arnold, Kozlov and Neishtadt.

Some of these results have been announced in the note [1].

References

1. L. Biasco, L. Chierchia, On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems, Rend. Lincei Mat. Appl. 26 (2015), 423-432.

Jacek Chudziak, University of Rzeszów, Poland

ON SOME APPLICATIONS OF QUASIDEVIATION MEANS

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We show that willingness to accept (WTA) and willingness to pay (WTP) are particular cases of quasideviation means, introduced in [2]. Using this fact and applying some results in [3], we investigate the properties of WTA and WTP related to the experimentally observed disparity between them [1, 4].

References

1. J.L. Knetsch, J.A. Sinden, Willingness to pay and compensation demanded: experimental evidence of an unexpected disparity in measures of value, The Quarterly Journal of Economics 99 (1984), 507-521.
2. Zs. Páles, Characterization of quasideviation means, Acta. Math. Sci. Hungar. 40 (1982), 243-260.
3. Zs. Páles, General inequalities for quasideviation means, Aequationes Math. 36 (1988), 32-56.
4. R. Thaler, Toward a positive theory of consumer choice, Journal of Economic Behavior and Organization 1 (1980), 39-60.

Jernej Činč, AGH University of Science and Technology, Poland & University of Ostrava, Czech Republic

PRIME ENDS DYNAMICS IN PARAMETRISED FAMILIES OF ROTATIONAL ATTRACTORS

Joint work with Jan P. Boroński and Xiao-Chuan Liu

Date: 2019-09-19 (Thursday); Time: 17:45-18:05; Location: building A-3/A-4, room 103.

Abstract

The prime ends rotation number induced by surface homeomorphisms restricted to open domains is one of the important tools in the study of boundary dynamics. Parametrised families of dynamical systems can provide a clearer view of both the surface dynamics and the boundary dynamics in many situations. Our study [1] serves as a contribution in this direction, by providing new examples in various contexts, by investigating the prime ends rotation numbers arising from parametrized BBM embeddings of inverse limits of topological graphs [2].

First, motivated by a topological version of the Poincaré-Bendixson Theorem obtained recently by Koropecki and Passeggi [4], we show the existence of homeomorphisms of \(S^2\) with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux.

Second, with the help of a reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where except for countably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This contrasts with the negative resolution of Walker's Conjecture from [5] by Koropecki, Le Calvez and Nassiri [3], and implies that our examples induce Denjoy homeomorphisms on the circles of prime ends.

Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

References

1. J. Boroński, J. Činč, X.-C. Liu, Prime ends dynamics in parametrised families of rotational attractors, arXiv:1906.04640 [math.DS] (2019).
2. P. Boyland, A. de Carvalho, T. Hall, Prime ends dynamics in parametrised families of rotational attractors, Bull. Lond. Math. Soc. 45(5) (2013), 1075–1085.
3. A. Koropecki, P. Le Calvez, M. Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J. 164 (2015), 403-472.
4. A. Koropecki, A. Passeggi, A Poincaré–Bendixson theorem for translation lines and applications to prime ends, Comment. Math. Helv. 94 (2019), 141–183.
5. R.B. Walker, Periodicity and decomposability of basin boundaries with irrational maps on prime ends, Trans. Amer. Math. Soc. 324 (1991), 303–317.

Trevor Clark, Imperial College London, UK

CONJUGACY CLASSES OF REAL ANALYTIC MAPPINGS

Joint work with Sebastian van Strien

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

I will discuss recent results on the manifold structure of the topological conjugacy classes of real-analytic mappings. These results are based on the construction of a “pruned polynomial-like mapping" associated to a real mapping. This gives us an ''external structure'' for a real-analytic mapping.

Albert Cohen, Sorbonne Université, France

OPTIMAL SAMPLING AND RECONSTRUCTION IN HIGH DIMENSION

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building B-8, room 0.10b.

Abstract

Motivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider the general problem of approximating an unknown arbitrary function in any dimension from the data of point samples. The approximants are picked from given or adaptively chosen finite dimensional spaces. One principal objective is to obtain an approximation which performs as good as the best possible using a sampling budget that is linear in the dimension of the approximating space. We will show that this object if can is met by taking a random sample distributed according to a well chosen probability measure, and reconstructing by appropriate least-squares or pseudo-spectral methods.

Welington Cordeiro, Polish Academy of Sciences, Poland

BEYOND TOPOLOGICAL HYPERBOLICITY

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. First of all, we will talk about some of these generalizations of expansivity and show some interesting examples. In particular, we will define entropy expansivity, \(N\)-expansivity and measure expansivity and we will give a overview about the recent results for these systems.

References

1. A. Artigue, Dendritations of surfaces, Ergodic Theory Dynam. Systems 38 (2018), 2860-2912.
2. A. Artigue, M.J. Pacífico, J. Vieitez, \(N\)-expansive homeomorphisms on surfaces, Communications in Contemporary Mathematics 19 (2017).
3. R. Bowen, Entropy-Expansive Maps, Transactions of the American Mathematical Society 164 (1972), 323-331.
4. B. Carvalho, W. Cordeiro, \(N\)-expansive homeomorphisms with the shadowing property, Journal of Differential Equations 261 (2016), 3734-3755.
5. B. Carvalho, W. Cordeiro, Positively \(N\)-expansive homeomorphisms and the \(L\)-shadowing property, J. Dyn. Diff. Equat. 261 (2018), https://doi.org/10.1007/s10884-018-9698-3.
6. W. Cordeiro, M. Denker, X. Zhang, On specification and measure expansiveness, Discrete and Continuous Dynamical Systems 37 (2017), 1941-1957.
7. H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598.
8. J. Li, R. Zhang, Levels of Generalized Expansiveness, Journal of Dynamics and Differential Equations, (2015), 1-18.
9. R. Mañé, Expansive homeomorphisms and topological dimension, Trans, AMS 252 (1979), 313–319.
10. C.A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32(1) (2012), 293-301.

Michele Coti Zelati, Imperial College London, UK

SUFFICIENT CONDITIONS FOR TURBULENCE SCALING LAWS IN 2D AND 3D

Joint work with Jacob Bedrossian, Sam Punshon-Smith, and Franziska Weber

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 0.10b.

Abstract

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws for both 2d [1] and 3d [2] stochastically forced Navier-Stokes equations. These conditions, which we name weak anomalous dissipation, replace the classical anomalous dissipation condition. For statistically stationary solutions, weak anomalous dissipation appear to be very effective and not too far from being necessary as well.

References

1. J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, F. Weber, Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations, arXiv 1905.03299 (2019).
2. J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, F. Weber, A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier-Stokes Equations, Comm. Math. Phys. 367 (2019), 1045-1075.

Petr Čoupek, Charles University, Czech Republic

AN ERGODIC CONTROL PROBLEM WITH ROSENBLATT NOISE

Joint work with Tyrone E. Duncan, Bohdan Maslowski, and Bożenna Pasik-Duncan

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

The talk will be devoted to an infinite time horizon linear-quadratic control problem for a stochastic differential equation with additive Rosenblatt noise.

Rosenblatt processes arise naturally as limits of suitably normalized sums of long-range dependent random variables in a non-central limit theorem. These continuous processes are self-similar, have stationary increments and exhibit long memory; however, unlike the family of regular fractional Brownian motions, they are not Gaussian. This last property makes their analysis somewhat intriguing and it is also the reason why they received considerable attention in recent years.

Initially, some recent results on stochastic calculus for Rosenblatt and related fractional processes will be presented in the talk. Subsequently, the ergodic control problem will be formulated and solved; and the optimal control as well as the optimal cost will be given explicitly.

Matteo Cozzi, University of Bath, UK

REGULARITY AND RIGIDITY RESULTS FOR NONLOCAL MINIMAL GRAPHS

Joint work with Xavier Cabré, Alberto Farina, and Luca Lombardini

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre & Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification.

Graziano Crasta, Sapienza University of Rome, Italy

BERNOULLI FREE BOUNDARY PROBLEM FOR THE INFINITY LAPLACIAN

Joint work with Ilaria Fragalà

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

We study the following interior Bernoulli free boundary problem for the infinity Laplacian:\[\begin{cases}\Delta_\infty u = 0 & \text{ in } \Omega^+(u) := \{x\in\Omega:\ u(x) > 0\},\\u = 1 & \text{ on } \partial \Omega,\\ |\nabla u| = \lambda & \text{ on } F(u) := \partial\Omega^+(u) \cap \Omega\, , \end{cases}\]where \(\Omega\) is an open bounded connected domain in \(\mathbb{R}^n\) (\(n\geq 2\)), and \(\Delta_\infty\) is the infinity Laplace operator.

Our results cover existence, uniqueness, and characterization of viscosity solutions (for \(\lambda\) above a threshold representing the ''infinity Bernoulli constant''), their regularity, and their relationship with the solutions to the interior Bernoulli problem for the \(p\)-Laplacian.

References

1. G. Crasta, I. Fragalà, Bernoulli free boundary problem for the infinity Laplacian, arXiv:1804.08573.
2. G. Crasta, I. Fragalà, On the supremal version of the Alt-Caffarelli minimization problem, arXiv:1811.12810.

Daan Crommelin, CWI & University of Amsterdam, Netherlands

UNRESOLVED SCALES AND STOCHASTIC PARAMETERIZATION IN ATMOSPHERE-OCEAN MODELING

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

The question how to represent or parameterize processes at unresolved scales remains an important issue in atmosphere-ocean modeling, posing both practical and theoretical challenges. For multiscale dynamical systems such as atmosphere, ocean and climate, the theoretical framework of the Mori-Zwanzig (MZ) formalism can help to guide the development of reduced (or coarse-grained) models in which processes at small/fast scales are no longer explicitly resolved but instead parametrized by stochastic models terms [1]. In practice it is often not feasible to derive these reduced models including their stochastic closure analytically following MZ. However, the formalism gives insight in suitable functional forms of the reduced models and in particular in the role of memory terms. This insight can be useful in data-based approaches, where model closures or parameterizations are extracted from available data. I will discuss work where such a data-based approach is developed and used in atmosphere-ocean modeling. As part of this approach, both resampling methods [2, 4] and discrete models [3] have been explored.

References

1. G. Gottwald, D. Crommelin, C. Franzke, Stochastic Climate Theory [in:] Nonlinear and Stochastic Climate Dynamics (ed. C. Franzke and T. O'Kane), 209-240, Cambridge University Press, 2017.
2. N. Verheul, D. Crommelin, Data-driven stochastic representations of unresolved features in multiscale models, Comm. Math. Sci. 14 (2016), 1213-1236.
3. J. Dorrestijn, D. Crommelin, P. Siebesma, H. Jonker, F. Selten, Stochastic convection parameterization with Markov chains in an intermediate complexity GCM, J. Atmos. Sci. 73 (2016), 1367-1382.
4. W. Edeling, D. Crommelin, Towards data-driven dynamics surrogate models for ocean flow, to appear in Proceedings of PASC 2019 conference (Zurich, 2019).

Jacek Cyranka, University of California, San Diego, USA & University of Warsaw, Poland

CONTRACTIBILITY OF A PERSISTENCE MAP PREIMAGE

Joint work with Konstantin Mischaikow

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the original dynamical system? In this paper we provide a definition of a persistence diagram for a point in \(\mathbb{R}^N\) modeled on piecewise monotone functions. We then provide conditions under which time series of persistence diagrams can be used to guarantee the existence of a fixed point of the flow on \(\mathbb{R}^N\) that generates the time series. To obtain this result requires an understanding of the preimage of the persistence map. The main theorem of this paper gives conditions under which these preimages are contractible simplicial complexes.

References

1. J. Cyranka and K. Mischaikow, Contractibility of a persistence map preimage, arXiv e-prints 2018, arXiv:1810.12447.

Adam Czornik, Silesian University of Technology, Poland

DISCRETE TIME-VARYING FRACTIONAL LINEAR EQUATIONS AS VOLTERRA CONVOLUTION EQUATIONS

Joint work with Pham The Anh, Artur Babiarz, Michał Niezabitowski, and Stefan Siegmund

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

We study the discrete-time fractional linear systems. We show how the different type (Caputo, Riemann-Liouville, forward and backward) of fractional linear difference equation may be converted to Volterra convolution equation. Using this representation we obtain some results about rate of convergency and divergency of solutions and variation of constant formulae. Moreover we show that the norm of difference between two different solution can not tends to infinity faster than a polynomial which degree depends of the fractional order of difference.

Acknowledgements

The research was funded by the National Science Centre in Poland granted according to decisions DEC-2015/19/D/ST7/03679 (A.B.) and DEC-2017/25/B/ST7/02888 (A.C.), respectively. The research was supported by the Polish National Agency for Academic Exchange according to the decision PPN/BEK/2018/1/00312/DEC/1 (M.N.). The research was partially supported by an Alexander von Humboldt Polish Honorary Research Fellowship (S.S.).

References

1. P. Anh, A. Babiarz, A. Czornik, M. Niezabitowski, S. Siegmund, Variation of constant formulas for fractional difference equations, Archives of Control Sciences 28 (2018), 617-633.
2. P. Anh, A. Babiarz, A. Czornik, M. Niezabitowski, S. Siegmund, Asymptotic properties of discrete linear fractional equations, Bulletin of the Polish Academy of Sciences: Technical Sciences, accepted for publication.

Danijela Damjanović, Royal Institute of Technology (KTH), Sweden

CENTRALIZER CLASSIFICATION FOR SOME PARTIALLY HYPERBOLIC MAPS

Joint work with Amie Wilkinson and Disheng Xu

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

In this talk I will survey recent advances on classification of centralizer for some conservative partially hyperbolic diffeomorphisms with one dimensional center foliation. I will describe how disintegration of volume along the center foliation [4] together with classification of higher rank partially hyperbolic abelian actions [1], [2] lead to classification results for centralizers [3]. I will also mention several conjectures concerning classification of centralizers.

References

1. D. Damjanović and D. Xu, On conservative partially hyperbolic abelian actions with compact center foliation, arXiv 1706.03626.
2. D. Damjanović, A. Wilkinson, D. Xu, Global rigidity of conservative partially hyperbolic actions with compact center foliation, in preparation.
3. D. Damjanović, A. Wilkinson, D. Xu, Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms, arXiv:1902.05201.
4. A. Avila, M. Viana, A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity II, in preparation.

Alexandre Danilenko, Institute for Low Temperature Physics and Engineering, NAS, Ukraine

GENERIC NONSINGULAR POISSON SUSPENSION IS OF TYPE \(III_1\)

Joint work with Emmanuel Roy and Zemer Kosloff

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

Let \((X,\mu)\) be a standard measure space equipped with a non-atomic \(\sigma\)-finite infinite measure and let Aut\((X,\mu)\) denote the group of all \(\mu\)-nonsingular transformations of \(X\). The Poisson suspension \((X^*,\mu^*)\) of \((X,\mu)\) is a well defined Lebesgue space. Then \[ \text{Aut}_2(X,\mu):= \left\{T\in \text{Aut}(X,\mu)\mid \sqrt{\frac{d\mu\circ T}{d\mu}}-1\in L^2(X,\mu)\right\} \] is the largest subgroup of Aut\((X,\mu)\) consisting of those \(T\) for which the Poisson suspension \(T_*\) is \(\mu^*\)-nonsingular [1]. It contains strictly the group \[ \text{Aut}_1(X,\mu):= \left\{T\in \text{Aut}(X,\mu)\mid {\frac{d\mu\circ T}{d\mu}}-1\in L^1(X,\mu)\right\} \] introduced in [2]. Aut\(_j(X,\mu)\) admits a natural Polish topology \(d_j\) stronger than the weak topology, \(j=1,2\), and \(d_1\) is stronger then \(d_2\) [1]. There is a continuous homomorphism \(\chi:\operatorname{Aut}_1(X,\mu)\to \Bbb R\), \(\chi(T):=\int_X(\frac{d\mu\circ T}{d\mu}-1)d\mu\) [1, 2].

Theorem 1. \(\{T\in \operatorname{Aut}_2(X,\mu)\mid T\text{ is ergodic of type \(III_1\) and \(T_*\) is ergodic of type \(III_1\)}\}\) is a dense \(G_\delta\) in \(d_2\).

Theorem 2. \(\{T\in \text{Ker} \chi\mid T\text{ is ergodic of type \(III_1\) and \(T_*\) is ergodic of type $III_1$}\}\) is a dense \(G_\delta\) in \((\text{Ker }\chi,d_1)\).

Theorem 3. If \(T\in\operatorname{Aut}_1(X,\mu)\) and \(\chi(T)\ne 0\) then \(T_*\) is totally dissipative.

Example. There is a totally dissipative \(T\in \operatorname{Aut}_1(X,\mu)\) such that for each real \(t\in (0,\frac 1{4})\), the Poisson suspension \((X^*,(t\mu)^*,T_*)\) is conservative but for each \(t>2\), the Poisson suspension \((X^*,(t\mu)^*,T_*)\) is totally dissipative.

References

1. A.I. Danilenko, E. Roy, Z. Kosloff, Nonsingular Poisson suspensions, in preparation.
2. Yu A. Neretin, Categories of symmetries and infinite-dimensional groups, Oxford University Press, 1996.

Charles Dapogny, Université Grenoble-Alpes, France

SHAPE AND TOPOLOGY OPTIMIZATION VIA A LEVEL-SET BASED MESH EVOLUTION METHOD

Joint work with Grégoire Allaire, Florian Feppon, and Pascal Frey

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

The purpose of this presentation is to introduce a robust front-tracking method for dealing with arbitrary motions of shapes, even dramatic ones (e.g. featuring topological changes); although this method is illustrated in the particular context of shape optimization, it naturally applies to a wide range of inverse problems and reconstruction algorithms.

The presented method combines two different means of representing shapes: on the one hand, they are meshed explicitly, which allows for efficient mechanical calculations by means of any standard Finite Element solver; on the other hand, they are represented by means of the level set method, a format under which it is easy to track their evolution. The cornerstone of our method is a pair of efficient algorithms for switching from either of these representations to the other.

Several numerical examples are discussed in two and three space dimensions, in the 'classical' physical setting of linear elastic structures, but also in more involved situations involving e.g. fluid-structure interactions.

(Left) Optimal design of a mast withstanding an incoming flow; (right) optimal design of a bridge.

References

1. G. Allaire, F. Jouve and A.M. Toader, Structural optimization using shape sensitivity analysis and a level-set method, J. Comput. Phys. 194 (2004), 363-393.
2. G. Allaire, C. Dapogny and P. Frey, Shape optimization with a level set based mesh evolution method, Comput. Meths. Appl. Mech. Engrg. 282 (2014), 22-53.
3. F. Feppon, G. Allaire, F. Bordeu, J. Cortial and C. Dapogny, Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework, SeMA journal (2019), HAL preprint: hal-01686770.

Udayan B. Darji, University of Louisville, USA

SOME APPLICATIONS OF LOCAL ENTROPY THEORY

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

In this talk we discuss some applications of local entropy theory, in particular results of Kerr-Li, to problems in topological dynamics and induced dynamics on the space of probability measures.

In the setting of topological dynamics, we discuss how local entropy theory can be used to show that in certain settings, the complexity of a dynamical system implies indecomposability in the inverse limit space of the dynamical system [2, 3], settling some old problems stated in [1].

A topological dynamical system \((X,f)\) induces natural dynamics on \(P(X)\), the space of probability measure on \(X\) defined by \(\tilde{f} :P(X) \rightarrow P(X)\) by \(\tilde{f} (\mu) = \mu f^{-1}\). A nontrivial and a remarkable result is that \(f\) has topological entropy zero if and only if \(\tilde{f}\) has measure zero [4]. Using techniques of [4], recently it was shown [5] that one can sharpen this result to null systems. We discuss how local entropy theory can be used to prove theorems of these types with ease.

References

1. M. Barge, J. Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289(1) (1985), 355-365.
2. C. Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, Proc. Amer. Math. Soc. 139(8) (2011), 2783-2791.
3. U.B. Darji, H. Kato, Chaos and Indecomposability, Adv. Math. 304 (2017), 793–808.
4. E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc. 8(3) (1995), 665–686.
5. Y. Qiao and X. Zhou, Zero sequence entropy and entropy dimension, Discrete and Continuous Dynam. Systems, vol 37, Number 1 (2017), no. 2, 435–448.

Rafael de la Llave, Georgia Institute of Technology, USA

SOME GEOMETRIC MECHANISMS FOR ARNOLD DIFFUSION

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects. Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate. This can be understood by noting that the small perturbations generate some invariant structures that, with their stable and unstable manifolds can cover a large region in phase space. We will present recent developments in identifying these invariant objects, both in finite and in infinite dimensions.

Thierry de la Rue, CNRS & Université de Rouen Normandie, France

SARNAK CONJECTURE IN DENSITY

Joint work with Alexander Gomilko and Mariusz Lemańczyk

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

The talk will be based on a recent joint work with Alexander Gomilko and Mariusz Lemańczyk, in which we establish the following result related to Sarnak conjecture. (Here \(\mu\) denotes the classical Möbius arithmetic function.)

If \((X,T)\) is a zero entropy topological dynamical system with at most countably many invariant measures, then there exists a subset \(A=A(X,T)\) of full logarithmic density in the set of natural integers, such that for any \(f\) continuous on \(X\), \[ \sup_x \frac{1}{N} \sum_{1\le n\le N} \mu(n) f(T^n x) \longrightarrow 0, \quad\text{as }N\to\infty,\ N\in A. \]

The main tools are the results about logarithmic Furstenberg systems of the Möbius function proved by Frantzikinakis and Host [1], the logarithmic version of the so-called strong MOMO property [2], and an argument inspired by Tao to pass from logarithmic averages to classical averages along a subsequence of full logarithmic density.

References

1. N. Frantzikinakis, B. Host, The logarithmic Sarnak conjecture for ergodic weights, Annals Math. 187 (2018), 869–931.
2. E. H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math. 228 (2018), 707–751.

Antonio De Rosa, New York University, USA

SOLUTIONS TO TWO CONJECTURES IN BRANCHED TRANSPORT: STABILITY AND REGULARITY OF OPTIMAL PATHS

Joint work with Maria Colombo and Andrea Marchese

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 0.10a.

Abstract

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power \(\alpha \in (0,1)\) of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when \(\alpha\) is bigger than the critical threshold \(1-\frac 1n\), where \(n\) is the dimension of the ambient space \(\mathbb R^n\). In [2, 3] we prove it holds for every exponent \(\alpha \in (0,1)\) and we provide a counterexample for \(\alpha=0\). Thus we completely solve a conjecture of Bernot, Caselles and Morel, see [1, Problem 15.1]. Moreover, the robustness of our argument allows us to prove stability for more general lower semicontinuous cost functionals, called \(H\)-masses, introduced in [6] and also studied in [5]. Furthermore, in [4] we prove the stability for the mailing problem, which was completely open in the literature, solving a second conjecture in [1, Remark 6.13]. We use the latter result to show the regularity of the optimal networks, partially answering [1, Problem 15.5].

References

1. M. Bernot, V. Caselles, and J.M. Morel, Optimal transportation networks. Models and theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.
2. M. Colombo, A. De Rosa, and A. Marchese, Improved stability of optimal traffic paths, Calc. Var. Partial Differential Equations 57 (2018), 28 pp.
3. M. Colombo, A. De Rosa, and A. Marchese, On the well-posedness of branched transportation, Available on arXiv: https://arxiv.org/abs/1904.03683, (2019).
4. M. Colombo, A. De Rosa, and A. Marchese, Stability for the mailing problem, J. Math. Pures Appl. (2018).
5. M. Colombo, A. De Rosa, A. Marchese, and S. Stuvard On the lower semicontinuous envelope of functionals defined on polyhedral chains, Nonlinear Analysis 163 (2017), 201-215.
6. T. De Pauw and R. Hardt, Size minimization and approximating problems, Calc. Var. Partial Differential Equations 17 (2003), 405-442.

Luz de Teresa, National Autonomous University of Mexico, Mexico

A STACKELBERG STRATEGY TO CONTROL A HEAT EQUATION

Joint work with Enrique Fernández-Cara and José Antonio Villa

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

In this conference we present the control problem that arises with the application of multiple strategies to the control of parabolic equations. We assume that we can act on the system through a hierarchy of controls. The leader control has an optimization objective and the follower a null control objective. We discuss the differences that arise when the leader control has a null controllability objective while the follower an optimization one. Literature, as far as we know, treats the second problem, see [1, 2, 3, 4].

References

1. F.D. Araruna, E. Fernández-Cara and M.C. Santos, Stackelberg-Nash Exact Controllability For Linear And Semilinear Parabolic Equations, ESAIM:COCV 21 (2015), 835-856.
2. F.D. Araruna, E. Fernández-Cara, S. Guerrero and M.C. Santos, New results on the Stackelberg–Nash exact control of linear parabolic equations, System and Control Letters 104 (2017), 78-85.
3. V. Hernández-Santamaría, L. de Teresa, Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs, Capítulo en “Recent Advances in PDEs: Analysis, Numerics and Control". Editado por: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. Springer 2018.
4. C. Montoya, L. de Teresa, Robust Stackelberg controllability for the Navier-Stokes equations, Nonlinear Differ. Equ. Appl. 25(46) (2018), https://doi.org/10.1007/s00030-018-0537-3.

Jana de Wiljes, Universität Potsdam, Germany

ACCURACY OF NONLINEAR FILTERS

Joint work with Xin Tong

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In the context of nonlinear high dimensional filtering problems ensemble based techniques such as the Ensemble Kalman Filter are still consider state of the art despite the lack of mathematical foundation in this setting. In a recent study long time stability and accuracy results for the deterministic Ensemble Kalman Bucy Filter were derived for a setting with fully observed state subject to small measurement noise [1]. Here these results are extended to the case where the ensemble size is larger than the state space for a localized deterministic Ensemble Kalman Bucy Filter. In contrast to the previously derived bounds the bounds for the localized filter are independent of the ensemble size. Further a dimension independent bound is obtained for the individual components of the error and a Laplace type condition holds.

References

1. J. de Wiljes, S. Reich, W. Stannat, Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise, SIAM J. APPLIED DYNAMICAL SYSTEMS 17 (2018), 1152-1181.

Félix del Teso, BCAM, Spain

A UNIFIED PDE/NUMERICAL FRAMEWORK FOR NONLOCAL AND LOCAL EQUATIONS OF POROUS MEDIUM TYPE

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

We consider the following problem of porous medium type: \[ \left\{ \begin{split} \partial_t u(x,t)-(L^\sigma +\mathcal{L}^{\mu})\left[\varphi(u)\right](x,t)=f(x,t), \qquad & (x,t)\in \mathbb{R}^N\times (0,\infty),\\ u(x,t)=u_0(x),\hspace{5cm}& x\in \mathbb{R}^N, \end{split} \right. \] where \(\varphi: \mathbb{R}^N\to\mathbb{R}\) is continuous and nondecreasing, and \[ \begin{split} L^\sigma[v](x)&=\text{Tr}\left(\sigma\sigma^T D^2v(x)\right), \hspace{2.8cm} \textbf{(local diffusion)}\\ \mathcal{L}^\mu[v](x)&=\text{P.V.} \int_{|z|>0}\left(v(x+z)-v(x)\right)d\mu(z), \textbf{ (nonlocal diffusion)} \end{split} \] with \(\sigma\in \mathbb{R}^{N\times p}\) and \(\mu\) symmetric measure s.t. \( \int \min\{|z|^2,1\}d\mu(z)<+\infty\).

We will present a general overview of some of the results obtained in collaboration with J. Endal and E.R. Jakobsen:

\(\bullet\) Uniqueness of distributional solutions.

\(\bullet\) Continuous dependence on \(L^\sigma +\mathcal{L}^{\mu}\), \(\varphi\) and \(u_0\).

\(\bullet\) Unified theory of monotone numerical schemes of finite difference type. Here we use the fact that operators in the class of \(\mathcal{L}^\mu\) includes discretizations of \(L^\sigma+\mathcal{L}^\mu\). This fact allows us to use a pure PDE approach.

\(\bullet\) We also propose a branch of discretizations and schemes and analyze their accuracy.

\(\bullet\) As a consequence of numerics, we obtain existence of distributional solutions together with interesting properties like \(L^1\)-contraction, \(C([0,T],L^1_{\text{loc}}(\mathbb{R}^N))\) regularity, energy estimates, ...

François Delarue, Université Nice-Sophia-Antipolis, France

MEASURE-VALUED DIFFUSION PROCESSES AND MEAN FIELD GAMES

Joint work with Alekos Cecchin

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 2.18.

Abstract

The purpose of the talk is to address smoothing effect of diffusion processes with values in the space of probability measures, especially when the latter is constructed above a finite set. The motivation comes from the theory of mean field games and of mean field control, which is dedicated to the analysis of equilibria within large population of rational agents and which has been growing fast since the earlier works of Lasry and Lions [3, 4] (see also the recent monographs [1,2]). One key fact is that such equilibria may be described by a stochastic measure-valued process when the whole population is subjected to a common source of noise. In this framework, equilibria turn out to be unique if the common noise induces sufficiently strong regularizing properties onto the space of probability measures. While the latter mostly regards the influence of a common noise onto the equilibria, it also raises interesting questions on the case without common noise: We here show that, by letting the influence of the common noise tend to zero, we may select, in some cases (known as potential cases), some specific equilibria among all the possible ones.

References

1. R. Carmona and F. Delarue, robabilistic Theory of Mean Field Games: vol. I, Mean Field FBSDEs, Control, and Games, Stochastic Analysis and Applications. Springer Verlag, 2018.
2. R. Carmona and F. Delarue, robabilistic Theory of Mean Field Games: vol. II, Mean Field Games with Common Noise and Master Equations, Stochastic Analysis and Applications. Springer Verlag, 2018.
3. J.M. Lasry and P.L. Lions, Jeux à champ moyen I. Le cas stationnaire, Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(9) (2006).
4. J.M. Lasry and P.L. Lions, Jeux à champ moyen II. Horizon fini et contrôle optimal, Comptes Rendus de l’Académie des Sciences de Paris, ser. A 343(10) (2006).

Jan Dereziński, University of Warsaw, Poland

PROPAGATORS ON CURVED SPACETIMES

Joint work with Daniel Siemssen and Adam Latosiński

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 0.10b.

Abstract

Quantum Field Theory on curved spacetimes has many interesting links to various branches of mathematics, such as differential geometry, symplectic dynamics, partial differential equations, pseudodifferential calculus, symmetric spaces and operator theory. I will discuss some of these links.

Gregory Derfel, Ben-Gurion University of the Negev, Israel

ON THE ASYMPTOTIC BEHAVIOUR OF THE ZEROS OF THE SOLUTIONS OF THE PANTOGRAPH EQUATION

Joint work with Peter Grabner and Robert Tichy

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

We study asymptotic behaviour of the solutions of the pantograph equation. From this we derive asymptotic formula for the zeros of these solutions.

References

1. C. Zhang, An asymptotic formula for the zeros of the deformed exponential function, J. Math. Anal. Appl. 441(2) (2016), 565-573.
2. L. Wang, C. Zhang, Zeros of the deformed exponential function, Advances in Mathematics 332 (2018), 311-348.
3. G. Derfel, P. Grabner and R. Tichy, On the asymptotic behaviour of the zeros of the solutions of a functional-differential equation with rescaling, Operator Theory: Advances and Applications 263 (2018), 281-295.

Andreas Deutsch, Technische Universität Dresden, Germany

BIOLOGICAL LATTICE-GAS CELLULAR AUTOMATON MODELS FOR THE ANALYSIS OF COLLECTIVE EFFECTS IN CANCER INVASION

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 1.11.

Abstract

Cancer invasion may be viewed as collective phenomenon emerging from the interplay of individual biological cells with their environment. Cell-based mathematical models can be used to decipher the rules of interaction. In these models cells are regarded as separate movable units. Here, we introduce an integrative modelling approach based on mesoscopic biological lattice-gas cellular automata (BIO-LGCA) to analyse collective effects in cancer invasion. This approach is rule- and cell-based, computationally efficient, and integrates statistical and biophysical models for different levels of biological knowledge. In particular, we provide BIO-LGCA models to analyse mechanisms of invasion in glioma and breast cancer cell lines.

References

1. A. Deutsch and S. Dormann, Cellular automaton modeling of biological pattern formation: characterization, applications, and analysis, Birkhäuser, Boston, 2018.

Giulia Di Nunno, University of Oslo, Norway

OPTIMAL STRATEGIES IN A MARKET WITH MEMORY

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 2.19.

Abstract

We consider a market model driven by Volterra type dynamics driven by a time-changed Levy noise. These dynamics allow both for memory features and clustering effects in the trading times. In this framework, we study an optimal portfolio problem, which is then tackled via maximum principle. To produce such results we use different kind of information flows that take care of the time-change in adequate way and we rely on the non-anticipating stochastic derivative for random fields. Moreover, we study the solutions of Volterra type SDEs and Volterra type BSDEs driven by time-changed Levy noises.

Tien-Cuong Dinh, National University of Singapore, Singapore

UNIQUE ERGODICITY FOR FOLIATIONS ON COMPACT KAEHLER SURFACES

Joint work with Viet-Anh Nguyen and Nessim Sibony

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

Let \(F\) be a holomorphic foliation by Riemann surfaces on a compact Kaehler surface. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed \((1,1)\)-current, or equivalently, no invariant measure. Then there exists a unique (up to a multiplicative constant) positive ddc-closed \((1,1)\)-current directed by \(F\), or equivalently, a unique harmonic measure. This is a very strong ergodic property showing that all leaves of \(F\) have the same asymptotic behavior. Our proof uses an extension of the theory of densities to a new class of currents. A complete description of the cone of directed positive ddc-closed \((1,1)\)-currents (i.e. harmonic measures) is also given when \(F\) admits directed positive closed currents (i.e. invariant measures).

Matúš Dirbák, Matej Bel University, Slovakia

PRODUCT-MINIMAL SPACES

Joint work with Ľubomír Snoha and Vladimír Špitalský

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

We call a compact metric space \(Y\) product-minimal (respectively, homeo-product-minimal) if for every minimal system \((X,T)\) there is a continuous map (respectively, a homeomorphism) \(S\colon Y\to Y\) such that the product system \((X\times Y,T\times S)\) is minimal. Every homeo-product-minimal space is product-minimal and every product-minimal space is minimal, while the converse implications do not hold. In the talk we shall present examples of (homeo-)product-minimal spaces and list some operations, under which the class of all (homeo-)product-minimal spaces is closed.

References

1. M. Dirbák, Ľ. Snoha, V. Špitalský, Minimal direct products, preprint (2019), 40 pp.

Sebastián Donoso, University of Chile, Chile

TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF FINITE RANK MINIMAL SUBSHIFTS

Joint work with Fabien Durand, Alejandro Maass, and Samuel Petite

Date: 2019-09-16 (Monday); Time: 17:45-18:05; Location: building A-3/A-4, room 103.

Abstract

I will discuss topological and combinatorial properties of finite rank minimal systems, establishing a clear connection with the \(S\)-adic subshifts, under recognizability assumptions. I will also mention results concerning the asymptotic components of a finite rank subshift and show that there is a rank two minimal subshift with superlinear complexity. I will mention results concerning the automorphism group of a finite rank subshift and state some open questions.

References

1. V. Berthé, W. Steiner, J. Thuswaldner and R. Yassawi, Recognizability for sequences of morphisms, Ergodic Theory Dynam. Systems, to appear.
2. S. Donoso, F. Durand, A. Maass and S. Petite, From bounded rank to \(S\)-adic subshifts: complexity and automorphism groups, preprint 2019.
3. T. Downarowicz and A. Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), 739-747.

Zuzana Došlá, Masaryk University, Czech Republic

DISCRETE BOUNDARY VALUE PROBLEMS ON UNBOUNDED DOMAINS

Joint work with Mauro Marini and Serena Matucci

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We study the boundary value problem \[ \begin{cases} \Delta(a_{n}\Phi(\Delta x_{n}))=b_nF(x_{n+1}), \ \ n\in\mathbb{N}& \\ x_{1}=c,\quad x_{n}>0,\quad \lim_{n\to\infty} x_n=d,\ & \end{cases} \tag{1}\] where \(\Delta\) is the forward difference operator \(\Delta x_{n}=x_{n+1}-x_{n}\), \(\Phi\) is an increasing odd homeomorphisms, \(\Phi :(-\rho,\rho)\rightarrow(-\sigma,\sigma)\) such that \(\Phi(u)u>0\) for \(u\neq0,\) and \(\rho,\sigma\leq\infty.\) We assume that the sequences \((a_n)\), \((b_n)\) are positive, and boundary conditions satisfy \(c>0\) and \(d\geq 0.\) Solutions of (1) with the terminal condition \(\lim_{n\to\infty} x_n=0\) are usually called decaying solution.

Problem (1) appears in the discretization process for searching spherically symmetric solutions of certain nonlinear elliptic differential equations with generalized phi-Laplacian. The case of noncompact domains seems to be of particular interest in view of applications to radially symmetric solutions to PDEs on the exterior of a ball.

Prototypes of \(\Phi\) are the classical \(\Phi\)-Laplacian, \[ \Phi_{p}(u)=|u|^{p-2}u, \quad p\geq1\, ; \] and when \(\sigma <\infty\) and \(\rho<\infty\) operators \[ \Phi_{C}(u)=\frac{u}{\sqrt{1+|u|^{2}}} \quad \text {and}\quad \Phi_{R}(u)=\frac{u}{\sqrt{1-|u|^{2}}}\, \] arising in studying radial symmetric solutions of partial differential equations with the mean curvature and the relativity operator, respectively.

If \(\Phi\) is the classical \(\Phi\)-Laplacian, the solvability of (1) has been investigated in [2], using properties of the recessive solution to suitable half-linear difference equations, a half-linearization technique and a fixed point theorem in Frechét spaces (see also [3]). Problem (1) is also motivated by [1] where general \(\Phi\) has been considered and extremal solutions have been investigated in case that \((b_n)\) is negative.

References

1. M. Cecchi, Z. Došlá, M. Marini, Regular and extremal solutions for difference equations with generalized phi-Laplacian, J. Difference Equ. Appl. 18 (2012), 815-831.
2. Z. Došlá, M. Marini, S. Matucci, Decaying solutions for discrete boundary value problems on the half line, J. Difference Equ. Appl. 22 (2016), 1244-1260.
3. M. Marini, S. Matucci, P. Řehák, Boundary value problems for functional difference equations on infinite intervals, Adv. Difference Equ. 2006 Article 31283 (2006), 14 pp.

Marie Doumic, CNRS & Sorbonne Université, France

ESTIMATING THE FRAGMENTATION CHARACTERISTICS IN GROWING AND DIVIDING POPULATIONS

Joint work with Miguel Escobedo, Magali Tournus and Wei-Feng Xue

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

Growing and dividing populations may be described either by stochastic branching processes or by integro-differential equations, both related by the fact that the integro-differential equation may be seen as the Kolmogorov equation of the branching process. Denoting \(u(t,x)\) the concentration of individuals of size \(x\) at time \(t,\) a typical growth-fragmentation equation may be written as \[\frac{\partial}{\partial t} u(t,x) + \frac{\partial}{\partial t} (g(x) u(t,x) ) + B(x) u(t,x) =\int\limits_0^1 B(\frac{x}{z}) u(t,\frac{x}{z}) \frac{dk_0 (z)}{z},\] where \(g(x)\) is the growth rate, \(B(x)\) the division rate, and \(k_0\) is called the (self-similar) fragmentation kernel, which characterizes the probability for a dividing particle of size \(\frac{x}{z}\) to give rise to an offspring of size \(x\). During the last decade, using the asymptotic behaviour of this equation or of the related stochastic process to estimate the division rate (\(B(x)\) in the equation) of a population has led to many interesting questions and results, in mathematics as well as in biology. In this talk, I will review some of them, and focus on the question of estimating the fragmentation kernel \(k_0\), which revealed a much more ill-posed problem than estimating the division rate \(B(x)\). We then applied our methods to fragmenting protein fibrils, following the experiments done by W.F. Xue's laboratory in the university of Kent.

References

1. D. Béal, M. Tournus, R. Marchante, T. Purton, D. Smith, M.F. Tuite, M. Doumic, and W.-F. Xue, The Division of Amyloid Fibrils, bioRxiv, 2018.
2. T. Bourgeron, M. Doumic, and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems 30(2) (2014).
3. M. Doumic, M. Escobedo, and M. Tournus, Estimating the division rate and kernel in the fragmentation equation, Annales de l’Institut Henri Poincaré (C) Non Linear Analysis 35(7) (2018).

Tomasz Downarowicz, Wrocław University of Science and Technology, Poland

A CRUSH COURSE ON SYMBOLIC EXTENSIONS OF \(\mathbb{Z}\)-ACTIONS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

Given a dynamical system \((X,T)\), where \(T\) is a homeomorphism of a compact metric space \(X\), we seek for its lossless digitalization in form of a subshift \((Y,\sigma)\), where \(Y\subset\Lambda^\mathbb Z\) (\(\Lambda\) is a finite alphabet) and \(\sigma\) denotes the standard shift, such that \((X,T)\) is a topological factor of \((Y,\sigma)\). It is obvious that a symbolic extension exists only for systems with finite topological entropy. But this condition is not sufficient. It turns out that in order to decide which systems admit symbolic extensions and how small can be their entropy one needs to study subtle entropy properties captured by the so-called entropy structure. In my talk I will try to present the most crucial definitions and facts around this topic.

References

1. M. Boyle, T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), 119-161.
2. T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, Cambridge, 2011.

Davor Dragičević, University of Rijeka, Croatia

NEW CHARACTERIZATIONS OF HYPERBOLICITY FOR LINEAR COCYCLES

Joint work with Adina Luminita Sasu and Bogdan Sasu

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

We will describe some new characterizations of stability, expansivity and hyperbolicity of linear cocycles developed in [1] which are based on the ideas from subadditive ergodic theory.

References

1. D. Dragičević, A.L. Sasu, B. Sasu, On the asymptotic behavior of discrete dynamical systems - an ergodic theory approach, submitted.

Romain Dujardin, Sorbonne Université, France

DYNAMICS OF UNIFORMLY HYPERBOLIC HÉNON MAPS

Joint work with Eric Bedford and Misha Lyubich

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

In this talk I will review some new recent results on the dynamics of uniformly hyperbolic Hénon maps. Topics covered will include the \(J=J^*\) problem, some new geometric and topological criteria for hyperbolicity, and the topological structure of the Julia set for hyperbolic maps.

Hugo Duminil-Copin, IHÉS, France & University of Geneva, Switzerland

MARGINAL TRIVIALITY OF THE SCALING LIMITS OF CRITICAL ISING AND \(\varphi^4\) MODELS IN 4D

Joint work with Michael Aizenman

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building B-8, room 0.10b.

Abstract

The question of constructing a non-Gaussian field theory, i.e. a field with non-zero Ursell functions, is at the heart of Euclidean (quantum) field theory. While non-triviality results in \(d<4\) and triviality results in \(d>4\) were obtained in famous papers by Glimm, Jaffe, Aizenman, Frohlich and others, the crucial case of dimension 4 remained open. In this talk, we show that any continuum \(\varphi^4\) theory constructed from Reflection Positive lattice \(\varphi^4\) or Ising models is inevitably free in dimension \(4\). The proof is based on a delicate study of intersection properties of a non-Markovian random walk appearing in the random current representation of the model.

Roxana Dumitrescu, King's College London, UK

MEAN-FIELD GAMES OF OPTIMAL STOPPING: A RELAXED SOLUTION APPROACH

Joint work with Géraldine Bouveret and Peter Tankov

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 2.19.

Abstract

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.

Petr Dunin-Barkowski, National Research University Higher School of Economics, Russia

LOOP EQUATIONS AND A PROOF OF ZVONKINE’S \(qr\)-ELSV FORMULA

Joint work with Reinier Kramer, Alexandr Popolitov, and Sergey Shadrin

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 0.10b.

Abstract

The talk is devoted to the outlining of the proof of the 2006 Zvonkine's conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called \(r\)-ELSV formula. In fact, this proof works in even a bit more general setting, namely it works for the \(qr\)-ELSV formula (which is the orbifold generalization of the \(r\)-ELSV formula), conjectured recently in [1]. The proof relies on expressing both the aforementioned Hurwitz and intersection numbers in terms of expansions of multi-point functions resulting from the application of the spectral curve topological recursion procedure on a particular spectral curve.

The talk is based on [2].

References

1. R. Kramer, D. Lewanski, A. Popolitov, S. Shadrin, Towards an orbifold generalization of Zvonkine’s \(r\)-ELSV formula, arXiv:1703.06725, 1–20.
2. P. Dunin-Barkowski, R. Kramer, A. Popolitov, S. Shadrin, Loop equations and a proof of Zvonkine’s \(qr\)-ELSV formula, arXiv:1905.04524, 1–17.

Christophe Dupont, Université de Rennes 1, France

DYNAMICS OF FIBERED ENDOMORPHISMS OF \(\mathbb C \mathbb P(2)\)

Joint work with Johan Taflin

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building A-3/A-4, room 105.

Abstract

The talk concerns the endomorphisms of \(\mathbb C \mathbb P(2)\) preserving a pencil of lines, those maps generalize the polynomial skew products of \(\mathbb C^2\) studied by Jonsson. We show that the equilibrium measure of those endomorphisms decomposes (Fubini's formula relative to the invariant pencil) and we study its Lyapunov exponents. One of them is equal to the exponent of the rational map acting on the pencil. We provide for the other one a formula involving a relative Green function and the critical set. In particular, that exponent is larger than the logarithm of the degree of the endomorphism. This is a joint work with Johan Taflin.

Semyon Dyatlov, University of California, Berkeley & Massachusetts Institute of Technology, USA

CONTROL OF EIGENFUNCTIONS ON NEGATIVELY CURVED SURFACES

Joint work with Jean Bourgain, Long Jin, and Stéphane Nonnenmacher

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 0.10b.

Abstract

Given an \(L^2\)-normalized eigenfunction with eigenvalue \(\lambda^2\) on a compact Riemannian manifold \((M,g)\) and a nonempty open set \(\Omega\subset M\), what lower bound can we prove on the \(L^2\)-mass of the eigenfunction on \(\Omega\)? The unique continuation principle gives a bound for any \(\Omega\) which is exponentially small as \(\lambda\to\infty\). On the other hand, microlocal analysis gives a \(\lambda\)-independent lower bound if \(\Omega\) is large enough, i.e. it satisfies the geometric control condition.

This talk presents a \(\lambda\)-independent lower bound for any set \(\Omega\) in the case when \(M\) is a negatively curved surface, or more generally a surface with Anosov geodesic flow. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves.

Aurelia Dymek, Nicolaus Copernicus University in Toruń, Poland

\(\mathcal{B}\)-FREE NUMBERS FROM DYNAMICAL POINT OF VIEW

Joint work with Stanisław Kasjan, Joanna Kułaga-Przymus and Mariusz Lemańczyk

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

Let \(\mathcal{B}\subset\{2,3,\ldots\}\). We call an integer \(n\) a \(\mathcal{B}\)-free number if \(n\) has no factor in \(\mathcal{B}\). We denote the set of all \(\mathcal{B}\)-free integers by \(\mathcal{F}_{\mathcal{B}}\). We consider the characteristic function of \(\mathcal{F}_\mathcal{B}\) in the space of binary sequences and denote it by \(\eta\). The subshift given by the orbit closure of \(\eta\) is called \(\mathcal{B}\)-free system and denoted by \(X_\eta\). A prominent example of a such system is the square-free system which is studied since 2010 [2]. In this case the frequencies of blocks yields a natural shift-invariant ergodic measure on \(\{0,1\}^\mathbb{Z}\). It is called the Mirsky measure.

During the talk I will concentrate on same ergodic properties of \(\mathcal{B}\)-free systems (genericity, entropy, invariant measures) and give some combinatorial applications [1].

References

1. A. Dymek, S. Kasjan, J. Kułaga-Przymus, M. Lemańczyk, \(\mathcal{B}\)-free sets and dynamics, Trans. Amer. Math. Soc. 370(8) (2018), 5425–5489.
2. P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

Martin Eigel, WIAS, Germany

A STATISTICAL LEARNING APPROACH FOR PARAMETRIC PDES

Joint work with Reinhold Schneider and Philipp Trunschke

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

Parametric PDEs (as encountered in the popular field of Uncertainty Quantification) are computationally complex due to the high dimensionality of the models describing random data. Common numerical approaches are Monte Carlo methods for statistical quantities of interest and functional approximations, representing the entire solution manifold in some function space. Assuming sufficient regularity (or sparsity), the latter attain high theoretical convergence rates. In practice, this can be realised e.g. by employing some kind of (a posteriori) error control in the computations. However, the implementation usually is non-trivial and does not generalise easily.

We examine a non-intrusive "Variational Monte Carlo'' (VMC) method based on statistical learning theory. This provides a combination of deterministic and statistical convergence results. The Galerkin solution can be computed with high probability using a tensor recovery algorithm on a training set of generated solution realisations. The representation in efficient hierarchical tensor formats tames the "curse of dimensionality''. Similarly, a residual a posteriori error estimator can be reconstructed easily, steering all discretisation parameters.

References

1. M. Eigel, R. Schneider, P. Trunschke, S. Wolf, Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional PDEs, Advances in Comp. Math., (to be published 2019).

Manfred Einsiedler, ETH Zürich, Switzerland

MEASURE RIGIDITY FOR HIGHER RANK DIAGONALIZABLE ACTIONS

Joint work with Elon Lindenstrauss

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

We review old and recent measure rigidity results for higher rank diagonalizable actions on homogeneous spaces and contrast these results with the rank one and unipotent case. After this we consider higher rank actions on irreducible arithmetic quotients of \(\operatorname{SL}_2(\mathbb{R})^k\) for \(k\geq 2\). If the quotient is compact, positive entropy of an ergodic invariant measure \(\mu\) implies algebraicity of \(\mu\) with semisimple stabiliser. For non-compact quotients there are more possibilities. The main novelty here is that the acting group does not have to be maximal or in a special position. The main new idea is to use a quantitative recurrence phenomenon to transport positivity of entropy for one acting element to another.

László Erdős, Institute of Science and Technology, Austria

FROM WIGNER-DYSON TO PEARCEY: UNIVERSAL EIGENVALUE STATISTICS OF RANDOM MATRICES

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building B-8, room 0.10b.

Abstract

E. Wigner's revolutionary vision postulated that the local eigenvalue statistics of large random matrices are independent of the details of the matrix ensemble apart from its basic symmetry class. There have recently been a substantial development to prove Wigner's conjecture for larger and larger classes of matrix ensembles motivated by applications. They include matrices with entries with a general correlation structure and addition of deterministic matrices in a random relative basis. We also report on three types of universality, commonly known as the bulk, edge and cusp universality, referring to the behaviour of the density of states in the corresponding energy regime. While bulk and edge universalities have been subject to intensive research, the cusp universality has been studied only in very special cases before. Our recent work settles the question of this third and last type of universality in full generality.

Emre Esenturk, University of Warwick, UK

MATHEMATICAL ANALYSIS OF EXCHANGE DRIVEN GROWTH: FUNDAMENTALS AND LARGE TIME BEHAVIOUR

Joint work with Juan Velazquez

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

Exchange-driven growth is a process (represented by an infinite set of ODEs) in which pairs of clusters interact and exchange single unit of mass at a time. In the recent years EDG has been used to model several natural and social phenomena The rate of exchange is given by an interaction kernel \(K(j,k)\) which depends on the masses of the two interacting clusters (of sizes \(j\) and \(k\)). Despite its wide use first mathematical analyses of this system were provided only recently [1].

In this talk we present results on the fundamentals properties (existence, uniqueness, nonexistence and etc) and the large time behaviour. For the existence, we show two different sets of results depending on whether \(K(j,k)\) is symmetric or not. In the case of non-symmetric kernels we present global existence and uniqueness of solutions for kernels satisfying \(K(j,k)\leq Cjk\). This result is optimal in the sense that for faster growing kernels the solutions cannot exist (up to some technical assumptions). On the other hand, in the case of symmetric kernels we show that global unique solutions exist for kernels satisfying \(K(j,k)\leq j^{\mu}k^{\nu}+j^{\nu}k^{\mu}\) (\(\mu +\nu\leq3\) and \(\mu,\nu\leq2)\) and that the nonexistence is also "delayed". For the large time behavior, we again show two sets of result for separable type kernels. Under some technical assumptions, we show that the system admits equilibrium solutions up to a critical mass above which there is no equilibrium. We show that if the system has an initial mass above the critical mass then the solutions converge to critical equilibrium distribution in a weak sense while strong convergence can be shown when initial mass is below.

References

1. E. Esenturk, Mathematical theory of exchange-driven growth, Nonlinearity 31 (2018), 3460-3483.

Peyman Eslami, University of Rome Tor Vergata, Italy

INDUCING SCHEMES FOR PIECEWISE EXPANDING MAPS OF \(\mathbb{R}^{n}\)

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

For piecewise expanding maps of \(\mathbb{R}^{n}\) I will show how to construct an inducing scheme where the base map is Gibbs-Markov and the return times have exponential tails. The existence of such a structure has many consequences in regards to the statistical properties of systems with discontinuities and non-uniform expansion.

Heiðar Eyjólfsson, Reykjavík University, Iceland

HILBERT SPACE-VALUED STOCHASTIC VOLATILITY MODELS AND AMBIT FIELDS

Joint work with Fred Espen Benth

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

We study Hilbert space-valued stochastic volatility models, and discuss representation and approximation of such processes. A typical application of such a model is the modelling of forward curves as an element in a given Hilbert space. Specifically, in a separable Hilbert space, a Lévy process driven variance process is introduced. We discuss ways of approximating the variance process in this setting. A problem of specific interest is how one obtains the square-root of the variance process. We discuss ways of obtaining and approximating the square root in an infinite dimensional Hilbert space. We moreover relate these models to the class Hilbert space- valued volatility modulated Volterra processes we call Hambit fiels [2]. Hambit fields are Hilbert space-valued analogues of ambit fields as introduced by Barndorff-Nielsen and Schmiegel [1]. Hambit fields can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes, for which Ornstein-Uhlenbeck process constitute a particular case. Hambit fields can moreover be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation.

References

1. O.E. Barndorff-Nielsen and J. Schmiegel, Lévy-based tempo-spatial modelling; with applications to turbulence, Uspekhi Mat. NAUK 59 (2004), 65-91.
2. F.E. Benth and H. Eyjolfsson, Representation and approximation of ambit fields in Hilbert space, Stochastics 89 (2017), 311-347

Giorgio Fabbri, CNRS, France

HJB EQUATIONS IN INFINITE DIMENSION AND OPTIMAL CONTROL OF STOCHASTIC EVOLUTION EQUATIONS VIA GENERALIZED FUKUSHIMA DECOMPOSITION

Joint work with Francesco Russo

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 0.18.

Abstract

In this talk we present the result of the paper [1].

We consider a stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space. Thanks to the identification of the mild solution of the state equation as \(\nu\)-weak Dirichlet process, the value processes is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem.

Through that technique some of the required assumptions are milder than those employed in previous contributions about non-regular solutions of Hamilton-Jacobi-Bellman equations. We present some explicit example.

References

1. G. Fabbri, F. Russo, HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition, SIAM Journal on Control and Optimization 55(6) (2016), 4072-4091.

William Fagan, University of Maryland, College Park, USA

IMPROVED FORAGING BY SWITCHING BETWEEN DIFFUSION AND ADVECTION: BENEFITS FROM MOVEMENT THAT DEPENDS ON SPATIAL CONTEXT

Joint work with Tyler Hoffman, Daisy Dahiya, Eliezer Gurarie, Robert Stephen Cantrell, and Chris Cosner

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

Animals use different modes of movement at different times, in different locations, and on different scales. Incorporating such context-dependence in mathematical models represents a substantial increase in complexity, but creates an opportunity to more fully integrate key biological features. Here we consider the spatial dynamics of a population of foragers with two subunits. In one subunit, foragers move via diffusion (random search) whereas in the other, foragers move via advection (gradient-following search). Foragers switch back and forth between the subunits as functions of their spatial context (i.e., depending on whether they are inside or outside of a patch, or depending on whether or not they can detect a gradient in resource density). We consider a one dimensional binary landscape of resource patches and non-habitat and gauge success in terms of how well the mobile foragers overlap with the distribution of resources. Actively switching between dispersal modes can sometimes greatly enhance this spatial overlap relative to the spatial overlap possible when foragers merely blend advection and diffusion modes at all times. Switching between movement modes is most beneficial when an organism’s gradient-following abilities are weak compared to its overall capacity for movement, but switching can actually be quite detrimental for organisms that can rapidly follow resource gradients. An organism’s perceptual range plays a critical role in determining the conditions under which switching movement modes benefits versus disadvantages foragers as they seek out resources.

Núria Fagella, University of Barcelona, Spain

WANDERING DOMAINS IN TRANSCENDENTAL FUNCTIONS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

Wandering domains are Fatou components that only exist in the transcendental setting. Although important progress has taken place in the past few years, there are still many open questions. In this talk I will review the state of the art on the existence and classification of wandering domains, and their relation with the singularities of the inverse map. I shall present some recent results on these topics.

Teresa Faria, University of Lisbon, Portugal

GLOBAL DYNAMICS FOR NICHOLSON’S BLOWFLIES SYSTEMS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

We study the global asymptotic behaviour of solutions for a Nicholson's blowflies system with patch structure and multiple discrete delays: \[ \begin{split} x_i'(t)=-d_i(t)x_i(t)+\sum_{j=1,j\ne i}^n a_{ij}(t)x_j(t)+\sum_{k=1}^m \beta_{ik}(t) x_i(t-\tau_{ik}(t))e^{-c_{i}(t)x_i(t-\tau_{ik}(t))},\\ \hskip 30mm i=1,\dots,n,\ t\ge 0, \end{split}\tag{1} \] where all the coefficient and delay functions are continuous, nonnegative and bounded, \(d_i(t)>0,c_i(t)\ge c_i>0\) and \(\beta_i(t):=\sum_{k=1}^m \beta_{ik}(t) >0\) for \(t\ge 0,\) \(i,j=1,\dots, n, k=1,\dots, m\). For the autonomous version of (1), an overview of results concerning the total or partial extinction of the populations, uniform persistence, existence and absolute global asymptotic stability of a positive equilibrium is presented, see [3, 4]. A criterion for the global attractivity of the positive equilibrium depending on the size of delays is also given [2], extending results in [1]. Most of these results rely on some properties of the so-called community matrix and on the specific shape of the nonlinearity.

For non-autonomous systems (1), sufficient conditions for both the extinction of the populations in all the patches and the permanence of the system were established in [3]. In this case, (1) is treated as a perturbation of the linear homogeneous cooperative ODE system \(x_i'(t)=-d_i(t)x_i(t)+\sum_{j=1,j\ne i}^n a_{ij}(t)x_j(t)\ (1\le i\le n)\), for which conditions for its asymptotic stability are imposed; although the nonlinearities in (1) are non-monotone, techniques of cooperative DDEs are used.

References

1. H.A. El-Morshedy, A. Ruiz-Herrera, Geometric methods of global attraction in systems of delay differential equations, J. Differential Equations 263 (2017), 5968-5986.
2. D. Caetano, T. Faria, Stability and attractivity for Nicholson systems with time-dependent delays, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), Paper no. 63, 19 pp.
3. T. Faria, R. Obaya, A.M. Sanz, Asymptotic behaviour for a class of non-monotone delay differential systems with applications, J. Dynam. Differential Equations 30 (2018), 911-935.
4. T. Faria, G. Röst, Persistence, permanence and global stability for an \(n\)-dimensional Nicholson system, J. Dynam. Differential Equations 26 (2014), 723-744.

Bassam Fayad, CNRS & Université Paris Diderot, France

INFINITE LEBESGUE SPECTRUM FOR SURFACE FLOWS

Joint work with Giovanni Forni and Adam Kanigowski

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

We study the spectral measures of conservative mixing flows on the two torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients : 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows, such as horocyclic flows on the unit tangent bundle \(M\) of a compact hyperbolic surface.

References

1. B. Fayad, G. Forni , A. Kanigowski, Lebesgue spectrum of countable multiplicity for conservative flows on the torus, arXiv 2019.

Włodzimierz Fechner, Lodz University of Technology, Poland

NEW INEQUALITIES FOR PROBABILITY FUNCTIONS IN THE TWO-PERSON RED-AND-BLACK GAME

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 1.8.

Abstract

We discuss a model of a two-person, non-cooperative stochastic game, inspired by the discrete version of the red-and-black gambling problem presented by Dubins and Savage [3]. Assume that two players hold certain amounts of money. At each stage of the game they simultaneously bid some part of their current fortune and the probability of winning or loosing depends on their bids. In many models of the red-and-black game it is assumed that the win probability is a function of the quotient of the bid of the first player and the sum of both bids. In the literature some additional properties, like concavity or super-multiplicativity, are assumed in order to ensure that bold and timid strategy is the Nash equilibrium (e.g. in works of Chen and Hsiau [1, 2]). In the talk we propose a generalization in which the probability of winning is a two-variable function which depends on both bids. We introduce two new functional inequalities whose solutions lead to win probability functions for which a Nash equilibrium is realized by the bold-timid strategy.

References

1. M.R. Chen, S.R. Hsiau, Two-person red-and-black games with bet-dependent win probability functions, J. Appl. Probab. 43(4) (2006), 905-915.
2. M.R. Chen, S.R. Hsiau, wo new models for the two-person red-and-black game, J. Appl. Probab. 47(1) (2010), 97-108.
3. L.E. Dubins, L.J. Savage, How to gamble if you must. Inequalities for stochastic processes, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965.
4. W. Fechner, New inequalities for probability functions in the two-person red-and-black game, arXiv:1811.00359 [math.PR].

Żywilla Fechner, Lodz University of Technology, Poland

BASIC FUNCTIONS ON HYPERGROUP-TYPE STRUCTURES

Joint work with László Székelyhidi

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

The aim of the talk is to present a characterization of functions like exponential monomials, polynomials and moment functions. We are interested in functions defined on some special type of hypergroups like affine groups, double coset hypergroups and hypergroup joins. We also discuss a connection of these functions with spectral synthesis problems.

References

1. L. Székelyhidi, Functional equations on hypergroups, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
2. Ż. Fechner, L. Székelyhidi, Spherical and moment functions on the affine group of \(SU(n)\), Acta Mathematica Hungarica 157(1) (2019), 10–26.
3. Ż. Fechner, L. Székelyhidi, Functional equations on double coset hypergroups, Annals of Functional Analysis, 8(3) (2017), 411–423,
4. Ż. Fechner, L. Székelyhidi, Sine functions on hypergroups, Archiv der Mathematik, 106(4) (2016), 371–382.

Eduard Feireisl, Czech Academy of Sciences, Czech Republic

DISSIPATIVE SOLUTIONS TO THE COMPRESSIBLE EULER SYSTEM

Joint work with Dominic Breit and Martina Hofmanová

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building B-8, room 0.10b.

Abstract

We introduce the concept of (generalized) dissipative solutions to the compressible Euler system and review their basic properties:

\(\bullet\) Existence. Dissipative solutions exist globally in time for any finite energy initial data.

\(\bullet\) Maximal dissipation, semigroup selection. One can select a solution semigroup among dissipative solutions. The selected solution maximizes the energy dissipation (entropy production), see [1].

\(\bullet\) Weak-strong uniqueness. A dissipative and a weak solution emanating from the same initial data coincide as soon as the weak solution belongs to certain Besov class and its velocity gradient satisfies a one sided Lipschitz condition, see [2].

\(\bullet\) Convergence of numerical schemes. Cesaro avarages produced by suitable numerical schemes converge strongly to a dissipative solution, see [3].

References

1. D. Breit, E. Feireisl, M. Hofmanová, Solution semiflow to the isentropic Euler system, Arxive Preprint Series, arXiv 1901.04798, 2019.
2. E. Feireisl, S.S. Ghoshal, A. Jana, On uniqueness of dissipative solutions to the isentropic Euler system, Arxive Preprint Series, arXiv 1903.11687, 2019.
3. E. Feireisl, M. Lukáčová–Medviďová, H. Mizerová, \(\mathcal{K}\)-convergence as a new tool in numerical analysis, Arxive Preprint Series, arXiv 1904.00297, 2019.

Guglielmo Feltrin, University of Udine, Italy

PARABOLIC ARCS FOR TIME-DEPENDENT PERTURBATIONS OF THE KEPLER PROBLEM

Joint work with Alberto Boscaggin, Walter Dambrosio, and Susanna Terracini

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

We prove the existence of parabolic arcs with prescribed asymptotic direction for the equation \[ \ddot{x} = - \dfrac{x}{\lvert x \rvert^{3}} + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \] where \(d \geq 2\) and \(W\) is a (possibly time-dependent) lower order term, for \(\vert x \vert \to +\infty\), with respect to the Kepler potential \(1/\vert x \vert\). The result applies to the elliptic restricted three-body problem and, more generally, to the restricted \((N+1)\)-body problem. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space.

Alessio Figalli, ETH Zürich, Switzerland

ON THE REGULARITY OF STABLE SOLUTIONS TO SEMILINEAR ELLIPTIC PDES

Date: 2019-09-18 (Wednesday); Time: 09:00-10:00; Location: building U-2, auditorium.

Abstract

Stable solutions to semilinear elliptic PDEs appear in several problems. It is known since the 1970’s that, in dimension \(n \gt 9\), there exist singular stable solutions. In this talk I will describe a recent work with Cabré, Ros-Oton, and Serra, where we prove that stable solutions in dimension \(n \le 9\) are smooth. This answers also a famous open problem, posed by Brezis, concerning the regularity of extremal solutions to the Gelfand problem.

Galina Filipuk, University of Warsaw, Poland

ASPECTS OF SPECIAL FUNCTIONS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

Special functions often solve ordinary differential equations. The well-known hypergeometric and Heun functions solve linear second order differential equations, whereas the Painlevé transcendents solve nonlinear second order differential equations. In this talk I shall overview some aspects of linear and nonlinear special functions and their differential equations. I shall also describe connections of linear equations to Okubo type systems.

References

1. G. Filipuk, A. Lastra, On the solutions of Okubo-type systems, Preprint 2019.
2. G. Filipuk, A. Ishkhanyan, J. Dereziński, On the Heun equation (tentative title), Preprint 2019.

Todd Fisher, Brigham Young University, USA

EQUILIBRIUM STATES FOR CERTAIN PARTIALLY HYPERBOLIC ATTRACTORS

Joint work with Krerley Oliveira

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-4, room 120.

Abstract

I will discuss partially hyperbolic attractors introduced by Castro and Nascimento and show they have unique equilibrium states for natural classes of potentials. If the system is \(C^2\), then there is a unique equilibrium states for the geometric potential and its 1-parameter family. The proofs follow by applying general techniques developed by Climenhaga and Thompson. This is joint work with Krerley Oliveira.

Matthew D. Foreman, University of California, Irvine, USA

PROGRESS ON THE SMOOTH REALIZATION PROBLEM

Joint work with Benjamin Weiss

Date: 2019-09-19 (Thursday); Time: 11:50-12:10; Location: building A-4, room 106.

Abstract

We discuss a Global Structure Theorem for measure preserving transformations that has two corollaries:

1. For all Choquet simplices \(\mathcal K\) there is an ergodic Lebesgue-measure preserving diffeomorphism of the 2-torus that has \(\mathcal K\) as its simplex of invariant measures.

2. For all countable ordinals \(\alpha\) there is a measure distal, measure preserving diffeomorphism of the 2-torus that has distal height \(\alpha\).

The first result changes Toeplitz systems built by Downarowicz into transformations that can be realized as diffeomorphisms. The second result stands in contrast to work of Rees, who showed that in the category of topological distality, the distal height is bounded by the dimension of the manifold.

References

1. J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. 33(2) (1932), 587-642.
2. D.V. Anosov, A.B. Katok, New examples in smooth ergodic theory, Trudy Moskov. Mat. Obšč. 23 (1970), 3-36.
3. T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), 241-256.
4. M. Rees, On the structure of minimal distal transformation groups with topological manifolds as phase spaces, Ph.D. Thesis, University of Warwick (1977).
5. M. Foreman and B. Weiss, A Symbolic Representation of Anosov-Katok systems, Journal d’Analyse Mathématique 137 (2019), 603-661.
6. M. Foreman and B. Weiss, From Odometers to Circular Systems: A Global Structure Theorem, ArXiv 1703.07093, March (2017).

Ana Cristina Moreira Freitas, University of Porto, Portugal

DYNAMICAL COUNTEREXAMPLES FOR THE USUAL INTERPRETATION OF THE EXTREMAL INDEX

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building A-4, room 120.

Abstract

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The existence of an Extremal Index less than 1 is associated to the occurrence of periodic phenomena, which is responsible for the appearance of clusters of exceedances. The Extremal Index usually coincides with the reciprocal of the mean of the limiting cluster size distribution. We build dynamically generated stochastic processes with an Extremal Index for which that relation does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point.

Jorge Milhazes Freitas, University of Porto, Portugal

RARE EVENTS FOR FRACTAL LANDSCAPES

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building A-4, room 120.

Abstract

We consider the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to fractal sets of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal geometrical structure.

Bartosz Frej, Wrocław University of Science and Technology, Poland

FACTORING GROUP SHIFTS ONTO THE FULL SHIFT

Joint work with Dawid Huczek

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

It is known that any subshift of finite type with the action of \(\mathbb{Z}\) and entropy greater or equal than \(\log n\) factors onto the full shift over \(n\) symbols (see [7] and [1] for the cases of equal and unequal entropy, respectively). Extending these results for actions of other groups has been difficult, and it is known that a factor map onto a full shift of equal entropy may not exist in this case (see [2]). Johnson and Madden showed in [6] that any SFT with the action of \(\mathbb{Z}^d\), which has entropy greater than \(\log n\) and satisfies an additional mixing condition (known as corner gluing), has an extension which is finite-to-one (hence of equal entropy) and maps onto the full shift over \(n\) symbols. This result was later improved by Desai in [4] and finally by Boyle, Pavlov and Schraudner in [3].

I will prove that in the case of actions of a countable amenable group, any strongly irreducible symbolic dynamical system with entropy greater than \(\log n\) has an equal-entropy symbolic extension which factors onto the full shift over \(n\) symbols. The construction uses tilings of amenable groups as presented in [5].

References

1. M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Systems 3(4) (1983), 541–557.
2. M. Boyle, M. Schraudner, \(\mathbb{Z}^d\) shifts of finite type without equal entropy full shift factors, J. Differ. Equations Appl. 15 (2009), 47–52.
3. M. Boyle, R. Pavlov, M. Schraudner, Multidimensional sofic shifts without separation and their factors, Trans. Amer. Math. Soc. 367, 5 (2015), 3371–3421.
4. A. Desai, A class of \(\mathbb{Z}^d\) shifts of finite type which factors onto lowe entropy full shifts, Proc. Amer. Math. Soc. 27(8) (2009), 2613-2621.
5. T. Downarowicz, D. Huczek, G. Zhang, Tilings of amenable groups, J. Reine Angew. Math 747 (2019), 277-298.
6. A. Johnson, K. Madden, Factoring higher-dimensional shifts of finite type onto the full shift, Ergodic Theory Dynam. Systems 25 (2005), 811-822.
7. B. Marcus, Factors and extensions of full shifts, Monatsh. Math. 88(3) (1979), 239-247.

Marlène Frigon, Université de Montréal, Canada

EXISTENCE AND MULTIPLICITY RESULTS FOR SYSTEMS OF FIRST ORDER DIFFERENTIAL EQUATIONS VIA THE METHOD OF SOLUTION-REGIONS

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

We present existence and multiplicity results for systems of first order differential equations of the form: \[ \begin{aligned} &u'(t) = f(t,u(t)) \quad \text{for a.e. } t\in [0,T], \\ &u \in \mathcal{B}; \end{aligned} \tag{1} \] where \(f :[0,T]\times \mathbb{R}^N\to \mathbb{R}^N\) is a Carathéodory function and \(\mathcal{B}\) denotes a boundary value condition. No growth conditions will be imposed on \(f\). Even though this problem was widely treated, few multiplicity results can be found in the literature.

In the case where there is only one equation (\(N=1\)), the method of upper and lower solutions is well known and very useful to obtain existence and multiplicity results. In particular, this was done in the pioneering work of Mawhin [4].

Very few multiplicity results were obtained in the case where the system (1) has more than one equation (\(N > 1\)). In [3], Frigon and Lotfipour introduced the notion of strict solution-tubes on which rely their multiplicity results. This method was used in [1] to obtain multiplicity results for systems of differential equations with a nonlinear differential operator.

We will present the method of solution-regions to establish existence and multiplicity results for the system (1). A solution-region will be a suitable set \(R\) in \([0,T] \times \mathbb{R}^N\) for which we will deduce that it contains the graph of viable solutions. We will show that this method generalizes the methods of upper and lower solutions and of solution-tubes. We will introduce also the notion of strict solution-regions and we will give conditions insuring the existence of at least three viable solutions of (1). Many non trivial examples will be presented throughout this presentation to show that the method of solution-regions is a powerful tool to establish the existence of solutions of systems of differential equations.

References

1. N. El Khattabi, M. Frigon, N. Ayyadi, Multiple solutions of problems with nonlinear first order differential operators, J. Fixed Point Theory Appl. 17 (2015), 23-42.
2. M. Frigon, Existence and multiplicity results for systems of first order differential equations via the method of solution-regions, Adv. Nonlinear Stud. 18 (2018), 469-485.
3. M. Frigon, M. Lotfipour, Multiplicity results for systems of first order differential inclusions, J. Nonlinear Convex Anal. 16 (2015), 1025-1040.
4. J. Mawhin, First order ordinary differential equations with several periodic solutions, Z. Angew. Math. Phys. 38 (1987), 257-265.
5. F. Adrián, F. Tojo, A constructive approach towards the method of solution-regions, J. Math. Anal. Appl. 472 (2019), 1803-1819.

Peter K. Friz, Technische Universität Berlin & WIAS, Germany

ON ROUGH SEMIMARTINGALES

Joint work with K. Le, A. Hocquet, and P. Zorin-Kranich

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

We introduce the new class of rough semimartingales (RSM) with motivation from filtering and SPDE theory. Under natural assumptions, RSM have a unique decomposition. Moreover, RSM are stable under composition with regular functions and stochastic / rough integration. RSM provide further a natural framework to study classes of Markov processes (which are not semimartingales) and we introduce the rough martingale problem.

Marco Fuhrman, University of Milan, Italy

OPTIMAL CONTROL OF STOCHASTIC EVOLUTION EQUATIONS VIA RANDOMIZATION AND BACKWARD SDES

Joint work with Emanuela Gussetti

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-8, room 0.18.

Abstract

Backward Stochastic Differential Equations (BSDEs) have been successfully applied to represent the value of optimal control problems for controlled stochastic differential equations. Since in the classical framework several restrictions on the scope of applicability of this method remained, in recent times several approaches have been devised to obtain the desired probabilistic representation in more general situations.

We will review the so called randomization method, originally introduced by B. Bouchard in the framework of optimal switching problems, which consists in introducing an auxiliary, "randomized" problem with the same value as the original one, where the control process is replaced by an exogenous random point process, and optimization is performed over a family of equivalent probability measures. The value of the randomized problem is then represented by means of a special class of BSDEs with a constraint on one of the unknown processes.

This methodology will be applied in the framework of controlled evolution equations (with immediate applications to controlled SPDEs), a case for which very few results are known so far.

Gabriel Fuhrmann, Imperial College London, UK

TAME IMPLIES REGULAR

Joint work with Eli Glasner, Tobias Jäger, and Christian Oertel

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

The last decade saw an increased interest in tame systems revealing their connections to different areas of mathematics like Banach spaces, substitutions and tilings, quasicrystals, cut and project schemes and even model theory and logic. A major breakthrough in the general understanding of tameness was achieved by Glasner's recent structural result for tame minimal systems [1]. One of its consequences is that a tame minimal dynamical system which has an invariant measure is almost automorphic, uniquely ergodic and measure-theoretically isomorphic to its maximal equicontinuous factor.

In this talk, we prove that tame minimal dynamical systems \((X,G)\) with an invariant measure are actually regularly almost automorphic, that is, they allow for a factor map \(\pi\) from \((X,G)\) to an equicontinuous system \((\mathbb T,G)\) such that almost every point in \(\mathbb T\) (with respect to the unique invariant measure on \(\mathbb T\)) has a unique preimage under \(\pi\), see [2].

References

1. E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math. 211 (2018), 213-244.
2. G. Fuhrmann, E. Glasner, T. Jäger, C. Oertel, Irregular model sets and tame dynamics, arXiv:1811.06283, (2018), 1-22.

Masaaki Fukasawa, Osaka University, Japan

3R HYBRID SCHEME FOR BROWNIAN SEMISTATIONARY PROCESSES

Joint work with Asuto Hirano

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

The Brownian semistationary process has attracted much attention recently in quantitative finance in the context of rough volatility modeling. We propose a new numerical approximation scheme, 3R hybrid scheme, which refines the hybrid scheme proposed by Bennedsen et al. [1] for Brownian semistationary processes. The mean squared error is shown to be significantly reduced while computational costs remain almost the same. The key idea is to reuse random variables through orthogonal projections.

References

1. M. Bennedsen, A. Lunde and M.S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21 (2017), 931-965.

José Pedro Gaivão, University of Lisbon, Portugal

BILLIARDS INSIDE STRICTLY CONVEX BODIES WITH POSITIVE TOPOLOGICAL ENTROPY

Joint work with M. Bessa, G. Del Magno, J.L. Dias, and M.J. Torres

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building A-4, room 120.

Abstract

In this talk we discuss the topological entropy of billiards inside strictly smooth convex bodies. We show that in a \(C^2\)-open and dense set of strictly convex bodies, the associated multidimensional billiard maps have positive topological entropy.

Zbigniew Galias, AGH University of Science and Technology, Poland

ENCLOSURE OF THE DOUBLE SCROLL ATTRACTOR FOR THE CHUA’S CIRCUIT WITH A CUBIC NONLINEARITY

Joint work with Warwick Tucker

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-7, room 2.4.

Abstract

We consider the Chua's circuit with a cubic nonlinearity. The dynamics of the circuit is defined by \[ C_1\dot{x}_1 = (x_2-x_1)/R-g(x_1), \quad C_2\dot{x}_2 = (x_1-x_2)/R+x_3, \quad L \dot{x}_3 = -x_2-R_0 x_3, \] where \(g(x_1)=g_1x_1+g_2x_1^3\). The system is considered with the following parameter values \(C_1=0.7\), \(C_2=7.8\), \(L=1.891\), \(R_0=0.01499\), \(g_1=-0.59\), \(g_2=0.02\), and \(R=2\), for which in simulations one observes the double scroll attractor.

Let us define \(\Sigma=\Sigma_1\cup\Sigma_2\), where \(\Sigma_1=\{x\colon x_1=2.1647\}\) and \(\Sigma_2=\{x\colon x_1=-2.1647\}\). The return map \(P:\Sigma\mapsto\Sigma\) is defined as \(P(x)=\varphi(\tau(x),x)\), where \(\varphi(t,x)\) is the trajectory based at \(x\), and \(\tau(x)\) is the return time after which the trajectory \(\varphi(t,x)\) returns to \(\Sigma\).

A candidate \(T\subset \Sigma\) for a trapping region enclosing the numerically observed attractor of the return map \(P\) is constructed. The return map \(P\) is not defined on the whole set \(T\). This is a consequence of the fact that the double scroll attractor contains the origin-an unstable equilibrium. For some initial points in \(x\in T\) the corresponding trajectories converge to the origin, i.e. \(\varphi(t,x)\to(0,0,0)\) for \(t\to\infty\). It follows that standard rigorous integration procedures cannot be used to study the dynamics of the system over the whole set \(T\). A method to handle trajectories passing arbitrarily close to an equilibrium is needed. Such trajectories may have arbitrarily large return times. The Jacobian matrix \(J\) at the origin has one real positive eigenvalue \(\lambda\approx0.2066\) and a pair of complex eigenvalues with negative real parts \(\alpha\pm\beta i\approx-0.075\pm 0.1966\). Normal form theory is used to develop a method to find enclosures of trajectories in a neighborhood of an unstable fixed point of a spiral type.

We prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity. More precisely, we prove the following theorem: for each \(x\in T\) either \(P(x)\in T\) or the trajectory \(\varphi(t,x)\) converges to the origin without intersecting \(\Sigma\), i.e., \(\varphi(t,x)\to(0,0,0)\) for \(t\to\infty\) and \(\{\varphi(t,x)\colon t>0\}\cap \Sigma=\emptyset\).

In the computer assisted proof, to handle trajectories passing close to the origin, we define the cylinder \(C\) centered at the origin. We also define the entry set, which is a part of the cylinder side and the exit set consisting of two parts each enclosed in one of the cylinder bases.

The proof of the main results is composed of three parts. In the first part the set \(T\) is covered by boxes. For each box we prove that either the image of this box under \(P\) is enclosed in \(T\) or all trajectories based in this box enter the cylinder \(C\) through the entry set. In the second part, we show that all trajectories based at the exit set reach \(T\). In the third part of the proof, we show that trajectories based at the entry set either converge to the origin or exit the cylinder through the exit set.

The first two parts of the proof are carried out using the CAPD library for the computation of trajectories and the evaluation of the return map \(P\). The third part of the proof is carried out using the normal form theory.

Juan Galvis, National University of Colombia, Colombia

ROBUST SOLVERS FOR HIGH-CONTRAST MULTISCALE PROBLEMS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

We present some recent developments in the numerical approximation of elliptic partial differential equations with high-contrast multiscale coefficients. In particular we review recently introduced robust upscaling technique known as the generalized multiscale finite element method (GMsFEM). We also present the design of robust two-levels domain decomposition methods that use the GMsFEM method as a second level. In order to show the benefits of using the proposed methodology several appliations are cosidered: two-phase flow in high-contrast multiscale porous media, the free boundary dam problem in heterogeneous media and an elasticity problem in topology optimization.

References

1. J. Galvis, C. Vásquez, L.F. Contreras, Numerical upscaling of the free boundary dam problem in multiscale high-contrast media, Submitted. 2019.
2. E. Abreu, C. Diaz, J. Galvis, A convergence analysis of Generalized Multiscale Finite Element Methods, 1990.
3. B. Lazarov, S. Serrano, M. Zambrano, J. Galvis, Fast multiscale contrast independent preconditioner for linear elastic topology optimization problems, in preparation, 2019.
4. Y. Efendiev, J. Galvis, T.Y. Hou, Generalized multiscale finite element methods (GMsFEM), Journal of Computational Physics 251 (2013), 116-135.

Ábel Garab, Alpen-Adria Universität Klagenfurt, Austria

DELAY DIFFERENCE EQUATIONS: PERMANENCE AND THE STRUCTURE OF THE GLOBAL ATTRACTOR

Joint work with Christian Pötzsche

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

In the first part of the talk we give sufficient conditions on the uniform boundedness and permanence of non-autonomous multiple delay difference equations of the form \[x_{k+1}=x_k f_k(x_{k-d},\dots,x_{k-1},x_k),\] where \(f_k\colon D \subseteq (0,\infty)^{d+1}\to (0,\infty)\). This also implies the existence of the global (pullback) attractor, provided the right-hand side is continuous.

In the second part, under some feedback conditions the right-hand side, we give a so-called Morse decomposition of the global attractor for equations of the form \(x_{k+1}=g(x_{k-d}, x_{k}).\) The decomposition is based on an integer valued Lyapunov functional introduced by J. Mallet-Paret and G. Sell.

Both results are applicable for a wide range of single species discrete time population dynamical models, such as models by Ricker, Pielou, Mackey-Glass, Wazewska-Lasota, and Clark.

Felipe García-Ramos, Autonomous University of San Luis Potosí, Mexico

TOPOLOGICAL MODELS OF KRONECKER SYSTEMS

Joint work with Tobias Jäger, Xiangdong Ye, and Dominik Kwietniak

Date: 2019-09-17 (Tuesday); Time: 10:40-11:00; Location: building A-3/A-4, room 103.

Abstract

I will talk about the range of behaviours of topological models of Kronecker systems (a.k.a. discrete spectrum systems) and loosely Kronecker systems (a.k.a. zero entropy loosely Bernoulli).

References

1. F. García-Ramos, T. Jäger and X. Ye, Mean equicontinuity, almost automorphy and regularity, Preprint, 2019.
2. F. García-Ramos, D. Kwietniak, Topological models of loosely Kronecker systems, Preprint, 2019.

Maurizio Garrione, Polytechnic University of Milan, Italy

PERIODIC SOLUTIONS OF THE BRILLOUIN ELECTRON BEAM FOCUSING EQUATION: SOME RECENT RESULTS

Joint work with Roberto Castelli and Manuel Zamora

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We present some recent results dealing with the existence of (positive) \(2\pi\)-periodic solutions of the Brillouin electron beam focusing equation \[ \ddot{x} + b(1+\cos t) x = \frac{1}{x}, \tag{1} \] in dependence on the real parameter \(b\). In literature, such a study was particularly stimulated by a conjecture that arose some decades ago, saying that for every \(b \in (0, 1/4)\) there exists a \(2\pi\)-periodic solution of (1). So far, the conjecture was neither shown to be true nor disproved, but the existence results were improved and refined step by step, reaching existence for \(b \in (0, b_0]\), with \(b_0 \approx 0.1645\), having a strong relation with the first stability interval of the associated Mathieu equation \(\ddot{x} + b(1+\cos t) x=0\). In this talk, we will try to make a little step further in understanding the picture for the \(2\pi\)-periodic solvability of (1). On the one hand, we will see that existence may hold true also for values of \(b\) belonging to stability intervals of the Mathieu equation other than the first, explicitly exhibiting one of such intervals. On the other hand, we will show that there exists \(b^* \approx 0.248\) such that existence holds for \(b \in (0, b^*]\) and for \(b=b^*\) the branch of solutions obtained through symmetry extension of Neumann ones undergoes a fold bifurcation, so that, as a by-product, multiplicity of \(2\pi\)-periodic solutions for \(b\) close to \(b^*\) is proved. This raises further questions about (1) and the validity of the related conjecture. The techniques used rely, respectively, on careful winding number estimates and computer-assisted proofs.

References

1. V. Bevc, J.L. Palmer, C. Süsskind, On the design of the transition region of axisymmetric, magnetically focused beam valves, J. British Inst. Radio Engineer. 18 (1958), 696-708.
2. R. Castelli, M. Garrione, Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions, J. Differential Equations 265 (2018), 2502-2543.
3. M. Garrione, M. Zamora, Periodic solutions of the Brillouin electron beam focusing equation, Commun. Pure Appl. Anal. 13 (2014), 961-975.
4. P.J. Torres, Mathematical models with singularities. A zoo of singular creatures, Atlantis Press, Paris, 2015.

Paul Gassiat, CEREMADE, Université Paris-Dauphine, France

ASYMPTOTIC FORMULAE IN ROUGH VOLATILITY MODELS

Joint work with Peter Friz and Paolo Pigato

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

Stochastic volatility models where the volatility behaves similarly to a fractional Brownian motion of Hurst index \(H < 1/2\) ("rough volatility") have recently been the subject of considerable interest from the mathematical finance community, due to their ability to reproduce important features observed in market prices. In this talk I will present a result on asymptotics of short-dated call option prices in such models. The proof is based on combining the Laplace method on Wiener space with rough path type techniques.

Pierre Germain, New York University, USA

ON THE DERIVATION OF KINETIC WAVE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 0.10b.

Abstract

Consider a nonlinear dispersive equation with random initial data. In the appropriate regime, it is conjectured that its dynamics are described, after averaging over the random data, by a kinetic wave equation. I will present recent progress towards the proof of this conjecture.

Marian Gidea, Yeshiva University, USA

SYMBOLIC DYNAMICS AND STOCHASTIC BEHAVIOR IN THE ELLIPTIC RESTRICTED THREE-BODY PROBLEM

Joint work with Maciej Capiński

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

We study Hamiltonian instability in the elliptic restricted three-body problem, in the context of a concrete model, on the motion of a small body (e.g., asteroid or spaceship) relative the Neptune-Triton system.

The elliptic restricted three-body problem can be regarded as a perturbation of the circular restricted three-body problem, with the perturbation parameter \(\varepsilon\) being the eccentricity of the orbits of the primaries. When the perturbation parameter is set to zero, the total energy \(H_\varepsilon\) of the system is preserved. When the perturbation parameter is non-zero, the total energy may vary. We provide two global instability results concerning the variation of energy along trajectories.

First, we show that for every suitably small, non-zero perturbation parameter, there exist trajectories along which the energy makes chaotic jumps. That is, given a sequence of energy level sets \((h^\sigma)_{\sigma\geq 0}\), with \(\|h^{\sigma+1}-h^\sigma\|>2\eta\), for some suitable \(\eta>0\), there exists a trajectory with \(\|H_\varepsilon (t^\sigma)-h^\sigma\|<\eta\), for some times \(t^\sigma>0\) and all \(\sigma\geq 0\).

Second, we show that the distributions of energies along orbits starting from some sets of initial conditions converge to a Brownian motion with drift as the perturbation parameter tends to zero. Moreover, we can obtain any desired values of the drift and of the variance for the limiting Brownian motion, for appropriate sets of initial conditions. That is, if we consider the stochastic process \(X_{t}^{\varepsilon }(z)\) representing the evolution of the energy along a trajectory starting from some point \(z\), with appropriately rescaled time \(t\), then, for every \(\mu,\sigma\in \mathbb{R}\) there exists a set \(\Omega_\varepsilon\) of initial points \(z\) for which \(X_{t}^{\varepsilon}-X_{0}^{\varepsilon }\) converges in distribution to \(\mu t+\sigma W_{t}\) as \(\varepsilon\to 0\), where \(W_{t}\) is the standard Wiener process.

In both cases we obtain an explicit range of the perturbation parameter \(\varepsilon\) for which the above phenomena occur. The proof of the results is based on topological methods and validated numerics.

Our results address conjectures made by Arnold [1] and Chirikov [2].

References

1. V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady 5 (1964), 581-565.
2. B. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), 264-379.

Yoshikazu Giga, University of Tokyo, Japan

ON THE LARGE TIME BEHAVIOR OF SOLUTIONS TO BIRTH AND SPREAD TYPE EQUATIONS

Joint work with Hiroyoshi Mitake, Takeshi Ohtsuka, and Hung V. Tran

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

We consider a level-set eikonal-curvature flow equation with an external force. Such a problem is considered as a model to describe an evolution of height of crystal surface by two-dimensional nucleation or possibly some class of growths by screw dislocations. For applications, it is important to estimate growth rate. Without an external source term the solution only spreads horizontally and does not grow vertically so the source term plays a key role for the growth.

Although the large time behavior of parabolic equations are well studied, the equations we study are degenerate parabolic equations where no diffusion effect exists in the normal to each level-set of a solution. Thus, very little is known even for growth rate. Our goal is to describe our recent progress on such type of problems. Ealier results are presented in the paper by H. Mitake, H.V. Tran and the lecturer published in SIAM Math. Anal. in 2016. A review paper is published in Proc. Int. Cong. of Math. in 2018.

In this talk, we first show the existence of asymptotic speed called growth rate. We also discuss asymptotic profile as well as estimates on growth rate.

Ewa Girejko, Bialystok University of Technology, Poland

ON CONSENSUS UNDER DoS ATTACK IN THE MULTIAGENT SYSTEMS

Joint work with Agnieszka B. Malinowska

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

In the paper multiagent systems under Denial-of-Service (DoS) attack are considered. We provide convergence results to ensure the consensus in the system under the attack. Since DoS attack is usually unpredictable with respect to duration of time and lasts one second or more, we examine the problem on various time domains.

Dorota Głazowska, University of Zielona Góra, Poland

EMBEDDABILITY OF PAIRS OF WEIGHTED QUASI-ARITHMETIC MEANS INTO A SEMIFLOW

Joint work with Justyna Jarczyk and Witold Jarczyk

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

Let \(I \subset \mathbb{R}\) be an interval. Given any continuous strictly monotonic function \( f: I \rightarrow \mathbb{R}\) and \(p \in \left( 0,1\right)\) the formula \[ A^f_p(x,y)=f^{-1}\left(pf(x)+(1-p)f(y)\right), \] defines a mean on \(I\) called the quasi-arithmetic mean generated by \(f\) and weighted by \(p\).

We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as those continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.

Franz Gmeineder, Universität Bonn, Germany

ON TRACES AND SOME FINE PROPERTIES OF FUNCTIONS OF BOUNDED \(A\)-VARIATION

Joint work with Lars Diening

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

In this talk I address an interplay between trace theorems and selected fine properties of functions of bounded \(A\)-variation. In the case of \(BV\)-functions, the singular part of the gradients splits into the jump- and the Cantor part. We establish that, within the framework of functions of bounded \(A\)-variation, such a splitting requires refinement, and explain the connections to the corresponding trace theory.

Diogo Gomes, KAUST, Saudi Arabia

A MEAN-FIELD GAME PRICE MODEL

Joint work with João Saúde

Date: 2019-09-16 (Monday); Time: 10:40-11:00; Location: building B-8, room 2.19.

Abstract

Here, we introduce a price-formation model where a large number of small players can store and trade electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly and compare our model with real data.

Igors Gorbovickis, Jacobs University, Germany

ON RENORMALIZATION OF CRITICAL CIRCLE MAPS WITH NONINTEGER EXPONENTS

Date: 2019-09-19 (Thursday); Time: 10:40-11:00; Location: building A-3/A-4, room 105.

Abstract

We discuss some results and ongoing developments in the study of renormalization of critical circle maps with non-integer critical exponents sufficiently close to odd integers.

References

1. I. Gorbovickis, M. Yampolsky, Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents, to appear in Ergodic Theory Dynam. Systems.
2. I. Gorbovickis, M. Yampolsky, Renormalization for unimodal maps with non-integer exponents, Arnold Math. Journal 4(2) (2018), 179-191.

Matthew Griffiths, King’s College London, UK

CYLINDRICAL LÉVY PROCESSES AND LÉVY SPACE-TIME WHITE NOISES

Joint work with Markus Riedle

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

It is well known (e.g. [1]) that the canonical cylindrical Brownian motion and the Gaussian space-time white noise correspond to each other. In this talk we consider the analogue relation between cylindrical Lévy processes and Lévy space-time white noises. Since there does not exist a “canonical” cylindrical Lévy process the situation is quite different from the Gaussian case. We then apply the established relations by embedding cylindrical Lévy processes in certain Besov spaces, which may be seen as a first result analysing the regular (or irregular) behaviour of the jumps of a cylindrical Lévy process.

References

1. G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monographs 26 Institute of Mathematical Statistics, Hayward, California, 1995.

Marcel Guardia, Polytechnic University of Catalonia, Spain

DIFFUSIVE BEHAVIOR ALONG MEAN MOTION RESONANCES IN THE RESTRICTED 3 BODY PROBLEM

Joint work with Vadim Kaloshin, Pau Martín, and Pau Roldán

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

Consider the Restricted Planar Elliptic Three Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. For eccentricity of Jupiter \(e_0\) small enough we show that there exists a family of probability measures \(\nu_{e_0}\) supported at the \(3 : 1\) mean motion resonance such that the pushforward under the associated Hamiltonian flow has the following property. At the time scale \(te_0^{-2}\), the distribution of the eccentricity of the Asteroid weakly converges to an (Ito stochastic) diffusion process on the line as \(e_0\to 0\). This resonance corresponds to the biggest of the Kirkwood gap on the Asteroid belt in the Solar System.

Thirupathi Gudi, Indian Institute of Science, India

APPROXIMATION OF DIRICHLET BOUNDARY CONTROL PROBLEM USING ENERGY SPACES

Joint work with Sudipto Chowdhury and Akambadath K. Nandakumaran

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

In this talk, we present an alternative energy space based approach for formulating the Dirichlet boundary control problem and then propose a finite element based numerical method for approximating its solution numerically. A priori error estimates of optimal order in the energy norm as well as in the \(L_2\) norm will be discussed. Furthermore, we discuss on deriving a reliable and efficient a posteriori error estimator using an auxiliary problem for adaptive mesh refinement. The theoretical results will be illustrated by some numerical experiments.

References

1. S. Chowdhury, T. Gudi, A.K. Nandakumaran, Error bounds for a Dirichlet boundary control problem based on energy spaces, Journal Math. Comp. 86 (2017), 1103-1126.

Olivier Guéant, Université Paris 1 Panthéon-Sorbonne, France

IT’S ALL RELATIVE: MEAN FIELD GAME EXTENSIONS OF MERTON’S PROBLEM

Joint work with Alexis Bismuth

Date: 2019-09-16 (Monday); Time: 11:05-11:25; Location: building B-8, room 2.19.

Abstract

Merton's problem deals with the optimal investment and consumption choices of economic agents. The classical results of Merton have been extended to add many features, but never, as far as we know, to take account of jealousy. In this talk, we show how the introduction of jealousy modifies Merton's problem and results in a problem of the mean field game (MFG) type (a mean field game of controls in fact). Interestingly, many analytical results can be obtained and will be presented, along with applications.

Yonatan Gutman, Polish Academy of Sciences, Poland

A PROBABILISTIC TAKENS THEOREM

Joint work with Krzysztof Barański and Adam Śpiewak

Date: 2019-09-19 (Thursday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

Let \(X \subset \mathbb{R}^N\) be a Borel set, \(\mu\) a Borel probability measure on \(X\) and \(T:X \to X\) a Lipschitz and injective map. Fix \(k \in \mathbb{N}\) greater than the (Hausdorff) dimension of \(X\) and assume that the set of \(p\)-periodic points has dimension smaller than \(p\) for \(p=1, \ldots, k-1\). We prove that for a typical polynomial perturbation \(\tilde{h}\) of a given Lipschitz map \(h : X \to \mathbb{R}\), the \(k\)-delay coordinate map \(x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))\) is injective on a set of full measure \(\mu\). This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required measurements from \(2\dim X\) to \(\dim X\) and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results and settle conjectures in the physics literature.

Johnny Guzmán, Brown University, USA

SMOOTH FINITE ELEMENT EXACT SEQUENCE ON POWELL-SABIN SPLITS

Joint work with Anna Lischke and Michael Neilan

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Starting with  \(C^1\) finite elements on a Powell-Sabin split we show how to construct a de Rham exact sequence. The lowest order FEM in our family of FEM is piecewise quadratics. An interesting feature in the exact sequence is that Powell-Sabin splits introduce  singular vertices and naturally we have to constrain the space of 2-forms in two dimensions. We will indicate possible generalization to higher dimensions.

Mats Gyllenberg, University of Helsinki, Finland

FINITE DIMENSIONAL STATE REPRESENTATION OF STRUCTURED POPULATION MODELS

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building B-8, room 0.10b.

Abstract

Structured population models can be formulated as delay systems. We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. The ideas and results are illustrated by models for infectious diseases and physiologically structured populations.

Leszek Hadasz, Jagiellonian University in Kraków, Poland

FROM CFT TO QUANTUM CURVES AND SUPER-AIRY STRUCTURES

Joint work with Vincent Bouchard, Paweł Ciosmak, Zbigniew Jaskólski, Masahide Manabe, Kento Osuga, Błażej Ruba, and Piotr Sułkowski

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 0.10b.

Abstract

During the talk I will discuss how the methods and notions developed in the area of two-dimensional, quantum conformal field theory allow to solve efficiently some problems related to matrix models: finding families of quantum curves related to a given classical algebraic curve and constructing interesting examples of super-Airy structures (algebras of differential operators engineered to solve topological recursion relations).

References

1. P. Ciosmak, L. Hadasz, Z. Jakólski, M. Manabe and P. Sułkowski, From CFT to Ramond super-quantum curves, JHEP 1805 133 (2018).
2. V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba and P. Sułkowski, Super Airy Structures, work in progress.

Christian Hainzl, Universität Tübingen, Germany

LOWER BOUND ON THE HARTREE-FOCK ENERGY OF THE ELECTRON GAS

Joint work with David Gontier and Mathieu Lewin

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 0.10b.

Abstract

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the free Fermi Gas is always unstable under the formation of spin or charge density waves. I present the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

Martin Hairer, Imperial College London, UK

RANDOM LOOPS

Date: 2019-09-19 (Thursday); Time: 09:00-10:00; Location: building U-2, auditorium.

Abstract

Helmut Harbrecht, University of Basel, Switzerland

ANALYSIS OF TENSOR APPROXIMATION SCHEMES FOR CONTINUOUS FUNCTIONS

Joint work with Michael Griebel

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

This talk is concerned with the analysis of tensor approximation schemes for continuous functions in high dimensions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate weights.

References

1. M. Griebel, H. Harbrecht, Analysis of tensor approximation schemes for continuous functions, arXiv:1509.09058, (2019).

Àlex Haro, University of Barcelona, Spain

SINGULARITY THEORY FOR KAM TORI: FROM SYMPLECTIC GEOMETRY TO APPLICATIONS THROUGH ANALYSIS

Joint work with Rafael de la Llave and Alejandra González

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

We present a method to find nontwist KAM tori. These are tori for which the twist condition fails. Our method also leads to a natural classification of KAM tori which is based on Singularity Theory. This talk aims to illustrate the main ideas of our approach, going from rigorous results to numerical computations up to the verge of breakdown.

Jan Haskovec, KAUST, Saudi Arabia

RIGOROUS CONTINUUM LIMIT FOR THE DISCRETE NETWORK FORMATION PROBLEM

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 1.26.

Abstract

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their \(\Gamma\)-convergence towards a continuum energy functional.

Erika Hausenblas, Montanuniversität Leoben, Austria

A SCHAUDER-TYCHANOFF TYPE STOCHASTIC FIXPOINT THEOREM AND A COUPLED STOCHASTIC SYSTEM FOR PATTERN FORMATION

Joint work with Mechthild Thalhammer, Tsiry Avisoa Randrianasolo, and Jonas Toelle

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in a specific situation, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray–Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear.

In the derivation of a macroscopic model such as the deterministic Gray–Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations.

In the talk, we first present a stochastic Schauder-Tychanoff type Theorem, then we present as an application the existence of solution of the stochastic Gray-Scott system. Finally, we present some numerical results of the stochastic Gray–Scott equations driven by independent spatially time-homogeneous Wiener processes. The numerical simulations based on the application of a time-adaptive first-order operator splitting method and the fast Fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic noise.

Patrick Henning, Royal Institute of Technology (KTH), Sweden

THE NUMERICAL SOLUTION OF NONLINEAR SCHRÖDINGER EQUATIONS WITH APPLICATIONS TO SUPERFLUIDITY

Joint work with Daniel Peterseim and Johan Wärnegård

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

In this talk we consider the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass- and energy-conserving variant of the Crank-Nicolson method in time. The usage of finite elements becomes necessary if the equation contains terms that dramatically reduce the overall regularity of the exact solution. Examples of such terms are rough potentials or disorder potentials as appearing in many physical applications. We present some analytical results that show how the reduced regularity of the exact solution could affect the expected convergence rates and how it results in possible coupling conditions between the spatial mesh size and the time step size. We will also demonstrate the importance of numerical energy-conservation in applications with low-regularity by simulating the phase transition of a Mott insulator into a superfluid.

References

1. P. Henning, D. Peterseim, Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials, M3AS Math. Models Methods Appl. Sci. 27(11) (2017), 2147-2184.
2. P. Henning, J. Wärnegård, Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation, ArXiv e-print 1804.10547, 2018.
3. P. Henning, D. Peterseim, Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency, ArXiv e-print 1812.00835, 2018.

Michael Herrmann, Technische Universität Braunschweig, Germany

LONGTIME BEHAVIOR AND ASYMPTOTIC REGIMES FOR SMOLUCHOWSKI EQUATIONS

Joint work with Barbara Niethammer and Juan J.L. Velázquez

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

Smoluchowski's coagulation equation is the most fundamental dynamical model for mass aggregation and appears in many different branches of physics, chemistry, biology, and materials science. The mathematical properties of this nonlinear integral equation, however, are only partially understood and depend intimately on the chosen coagulation kernel.

In this talk we discuss several asymptotic regimes for kernel functions and investigate the longtime behavior of solutions by combining asymptotic analysis, heuristic arguments, and numerical simulations. In particular, we study traveling waves and self-similar profiles for near-diagonal kernels with homogeneity one and provide analytical or numerical evidence for the onset of instabilities and the formation of oscillations. We further sketch the challenges in the numerical computation of self-similar solutions and initial value problems.

Cartoon of the self-similar solutions in two different asymptotic regimes.

References

1. M. Herrmann, B. Niethammer, J.J.L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math. 75(1) (2016), 105–130.
2. M. Bonacini, B. Niethammer, J.J.L. Velázquez, Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one, Arch. Rational. Mech. Anal. 233(1) (2019) 1–43.
3. Ph. Laurençot, B. Niethammer, J.J.L. Velázquez, Oscillatory dynamics in Smoluchowski’s coagulation equation with diagonal kernel, Kinet. Relat. Models 11(4) (2018) 933–952.

Desmond Higham, University of Edinburgh, UK

DIFFERENTIAL EQUATIONS FOR NETWORK CENTRALITY

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

I will derive and discuss two circumstances where ODEs arise in the study of large, complex networks. In both cases, the overall aim is to identify the most important nodes in a network-this task is useful, for example, in digital marketing, security and epidemiology. In one case, we define our node centrality measure using the concept of nonbacktracking walks. This requires us to derive an expression for an exponential-type generating function associated with the walk counts of different length. Solving the ODE leads to a computationally useful characterisation of the centrality measure. In another case, we are presented with a time-ordered sequence of networks; for example, recording who emailed who over each one-minute time-window. Here, by considering the asymptotic limit as the window size tends to zero, we arrive at a limiting ODE that may be treated with a numerical method.  Results for both algorithms will be illustrated on real network examples.

Jonas Hirsch, Universität Leipzig, Germany

DIMENSIONAL ESTIMATES AND RECTIFIABILITY FOR MEASURES SATISFYING LINEAR PDE CONSTRAINTS

Joint work with Adolfo Arroyo-Rabasa, Guido De Philippis, and Filip Rindler

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

We are going to present an rectifiability result for measures satisfying a linear PDE constraint. The presented rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. For instance it includes the the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD).

More precisely Let \(\mathcal{A}\) be a \(k^{\text{th}}\)-order linear constant-coefficient PDE operator acting on \(\mathbb R^m\)-valued functions on \(\mathbb{R}^d\) via \[ \mathcal{A} \varphi := \sum_{|\alpha|\le k}A_{\alpha} \partial^\alpha\varphi \text{ for all }\varphi\in C^\infty(\mathbb{R}^d;\mathbb{R}^m), \] where \(A_{\alpha}\in \mathbb{R}^{n \times m}\) are (constant) matrices, \(\alpha=(\alpha_1,\dots,\alpha_d)\in (\mathbb N \cup \{0\})^d\) is a multi-index and \(\partial^\alpha:=\partial_1^{\alpha_1}\ldots\partial_d^{\alpha_d}\). We also assume that at least one \(A_\alpha\) with \(\lvert \alpha\rvert = k\) is non-zero. An \(\mathbb{R}^m\)-valued Radon measure \(\mu \in \mathcal M(U;\mathbb{R}^m)\) defined on an open set \(U\subset \mathbb{R}^d \) is said to be \(\mathcal{A}\)-free if \[ \mathcal{A} \mu=0 \qquad\text{in the sense of distributions on \(U\).}\tag{1} \]

Using the Lebesgue-Radon-Nikodým theorem we may define the polar of \(\mu\) by \[ \frac{{\rm d}\mu}{{\rm d}|\mu|}(x):=\lim_{r\to 0} \frac{\mu(B_r(x))}{|\mu|(B_r(x))}\,. \]

In the pioneering work [1], G. De Philippis and F. Rindler established a strong constraint on the direction on the polar on the singular part of an \(\mathcal{A}\)-free measure.

In this talk we are going to present a refinement of this pioniering result, the direction of the polar is further constrained on "lower dimensional parts" of the measure, [2].

As a consequence in the particular case of divergence-free tensors we are able to obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

References

1. G. De Philippis, F. Rindler, On the structure of \(\mathcal{A}\)-free measures and applications, Ann. of Math. (2) 184(3) (2016), 1017–1039.
2. A. Arroyo-Rabasa, G. De Philippis and F. Rindler, Dimensional estimates and rectifiability for measures satisfying linear PDE constraints, Geometric and Functional Analysis 184 (2019), 1420-8970.

Michael Hochman, Hebrew University of Jerusalem, Israel

EQUIDISTRIBUTION FOR COMMUTING MAPS

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

In two classical papers circa 1960, J. Cassels and W. Schmidt proved that a.e. numbers in the ternary Cantor set (with respect to Cantor-Lebesgue measure) eqidistributes for Lebesgue measure under the map \(Tx=bx \bmod 1\), whenever \(b\) is an integer that is not a power of \(3\). This phenomenon has since been established in much greater generality on the interval, e.g. Host's theorem, according to which one can replace Cantor-Lebesgue measure by any \(\times 3\)-ergodic measure of positive entropy, provided \(\gcd(3,b)=1\). In this talk I will describe a new and heuristically simple proof of such results, and then discuss how it can be extended to give new results in multi-dimensional settings.

Helge Holden, Norwegian University of Science and Technology, Norway

ON TRAFFIC MODELING AND THE BRAESS PARADOX

Joint work with Nils Henrik Risebro and Rinaldo Colombo

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

We will discuss models for vehicular traffic flow on networks. The models include both the Lighthill-Whitham-Richards (LWR) model and Follow-the-Leader (FtL) models. Emphasis will be on the Braess paradox [1] in which adding a road to a traffic network can make travel times worse for all drivers, and we will show one way of studying the Braess paradox with an LWR model [2].

Furthermore, we will show how we can consider the FtL model as a discretization of the LWR model [3, 4]. Finally, we will also discuss a novel model for multi-lane traffic [5].

References

1. D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung 12 (1968), 258-268.
2. R. Colombo, H. Holden, On the Braess paradox with nonlinear dynamics and control theory, J. Optimization Theory and Appl. 168 (2016), 216-230.
3. H. Holden, N.H. Risebro, Continuum limit of Follow-the-Leader models - a short proof, Discrete Contin. Dyn. Syst. 38 (2018), 715-722.
4. H. Holden, N.H. Risebro, Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Networks & Heterogeneous Media 13 (2018), 409-421.
5. H. Holden, N.H. Risebro, Models for dense multilane vehicular traffic, arXiv:1812.01361.

Gustav Holzegel, Imperial College London, UK

NON-LINEAR STABILITY OF THE SCHWARZSCHILD FAMILY OF BLACK HOLES

Joint work with Mihalis Dafermos, Igor Rodnianski, and Martin Taylor

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 0.10b.

Abstract

I will discuss the statement and the proof of the finite-codimension non-linear stability of the Schwarzschild family as solutions to the vacuum Einstein equations. The proof relies crucially on our previous work [1] on the linear stability of the Schwarzschild family and makes use of many analytical techniques developed over the years in the analysis of hyperbolic equations on black hole spacetimes, including control of the non-linearities of the Einstein equations in the radiation zone.

References

1. M. Dafermos, G. Holzegel, I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Mathematica 222 (2019), 1-214.

Dietmar Hömberg, WIAS & Technische Universität Berlin, Germany

MATHEMATICS FOR STEEL PRODUCTION AND MANUFACTURING

Joint work with Manuel Arenas, Prerana Das, and Robert Lasarzik

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In my presentation I will discuss some results from the European Industrial Doctorate project "MIMESIS - Mathematics and Materials Science for steel production and manufacturing". Applications cover tube welding, induction hardening and flame cutting.

The mathematical models considered combine a vector potential formulation of Maxwell's equations with a nonlinear heat equation and an evolution equation for the change of microstructure. In the presentation we analyse the well-posedness of these multi-field problems, discuss related optimal control problems and show some simulation results related to real industrial use cases.

References

1. D. Hömberg, T. Petzold, E. Rocca, Analysis and simulations of multifrequency induction hardening, Nonlinear Analysis: Real World Applications 22 (2015), 84-97.
2. D. Hömberg, Q. Liu, J. Montalvo-Urquizo, D. Nadolski, T. Petzold, A. Schmidt, A. Schulz, Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects, Finite Elements in Analysis and Design 121 (2016), 86-100.
3. J.I. Asperheim, P. Das, B. Grande, D. Hömberg, T. Petzold, Numerical simulation of high-frequency induction welding in longitudinal welded tubes, WIAS Preprint No. 2600, (2019).

Chun-Hsiung Hsia, National Taiwan University, Taiwan

ON THE MATHEMATICAL ANALYSIS OF SYNCHRONIZATION FOR THE TIME-DELAYED KURAMOTO OSCILLATORS

Joint work with Chang-Yeol Jung, Bongsuk Kwon, and Yoshihiro Ueda

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 0.10b.

Abstract

We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.

References

1. C.-H. Hsia, C.-Y. Jung, B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, Journal of Differential Equations 267 (2019), 742-775.
2. C.-H. Hsia, C.-Y. Jung, B. Kwon, Y. Ueda, Synchronization of Kuramoto oscillators with time-delayed interactions and phase lag effect, preprint.

Yanghong Huang, University of Manchester, UK

SPECTRAL RELATIONS OF THE GENERATOR TO ALPHA-STABLE PROCESSES AND RELATED SPECTRAL METHODS

Joint work with Xiao Wang

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

The majority of special functions and orthogonal polynomials are well-known to be associated with second order differential equations arising from mathematical physics, and are usually have to be extended to Merjie G function or Fox H function. In this talk, classical Jacobi polynomials are shown to establish spectral relations of the generator of alpha-stable processes, generalising the fractional Laplacian in one dimension to the non-symmetric case. The resulting spectral relations will be used to characterised the singularity near the boundary and the regularity of the solution to the Dirichlet problem, together with the development of a higher order spectral methods.

Hermen Jan Hupkes, University of Leiden, Netherlands

DYNAMICS ON LATTICES

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

We study dynamical systems posed on lattices, with a special focus on the behaviour of basic objects such as travelling corners, expanding spheres and travelling waves. Such systems arise naturally in many applications where the underlying spatial domain has a discrete structure. Think for example of the propagation of electrical signals through nerve fibres, where the the myeline coating has gaps at regular intervals. Or the study of magnetic spins arranged on crystal lattices.

Throughout the talk we will explore the impact that the spatial topology of the lattice has on the dynamical behaviour of solutions. We will discuss lattice impurities, the consequences of anistropy and make connections with the field of crystallography.

Renato Huzak, Hasselt University, Belgium

FINITE CYCLICITY OF THE CONTACT POINT IN SLOW-FAST INTEGRABLE SYSTEMS OF DARBOUX TYPE

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-7, room 1.9.

Abstract

Using singular perturbation theory and family blow-up we prove that nilpotent contact points in deformations of slow-fast Darboux integrable systems have finite cyclicity. The deformations are smooth or analytic depending on the region in the parameter space. This paper is a natural continuation of [M. Bobieński, P. Mardesic and D. Novikov, 2013] and [M. Bobieński and L. Gavrilov, 2016] where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the ''integrable'' direction in the parameter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the ''slow-fast'' direction in the parameter space.

Radu Ignat, Université Paul Sabatier, France

PROGRESS IN THE ANALYSIS OF DOMAIN WALLS IN THIN FERROMAGNETIC FILMS

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 3.22.

Abstract

Ferromagnetic materials are nowadays widely used as technological tools, especially for magnetic data storage. The modeling of small ferromagnets is based on the micromagnetic theory, an intriguing example of multiscale, nonconvex and nonlocal variational problems. One of the main challenges consists in understanding the pattern formation of the magnetization, in particular the domain walls. The aim of my talk is to present recent progress in the analysis of domain walls such as Bloch and Néel walls. I will present several results concerning their structure, their properties (stability, symmetry etc.) as well as the interaction energy of domain walls. The proof of these results is based on methods coming from geometric analysis and harmonic maps, elliptic regularity theory, variational methods and hyperbolic conservation laws.

References

1. L. Döring, R. Ignat, Asymmetric domain walls of small angle in soft ferromagnetic films, Arch. Ration. Mech. Anal. 220 (2016), 889-936.
2. L. Döring, R. Ignat, F. Otto, A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types, J. Eur. Math. Soc. (JEMS) 16 (2014), 1377-1422.
3. R. Ignat, Singularities of divergence-free vector fields with values into \(\mathbb{S}^1\) or \(\mathbb{S}^2\). Applications to micromagnetics, Confluentes Mathematici 4 (2012), 1-80.
4. R. Ignat, B. Merlet, Lower bound for the energy of Bloch walls in micromagnetics, Arch. Ration. Mech. Anal. 199 (2011), 369-406.
5. R. Ignat, A. Monteil, A DeGiorgi type conjecture for minimal solutions to a nonlinear Stokes equation, Comm. Pure Appl. Math., accepted (2019).
6. R. Ignat, R. Moser, A zigzag pattern in micromagnetics, J. Math. Pures Appl. 98 (2012), 139-159.
7. R. Ignat, R. Moser, Interaction energy of domain walls in a nonlocal Ginzburg-Landau type model from micromagnetics, Arch. Ration. Mech. Anal. 221 (2016), 419-485.
8. R. Ignat, R. Moser, Energy minimisers of prescribed winding number in an \(\mathbb{S}^1\)-valued nonlocal Allen-Cahn type model, preprint arXiv:1810.11427 (2018).
9. R. Ignat, F. Otto, The magnetization ripple: a nonlocal stochastic PDE perspective, J. Math. Pures Appl., online (2019).

Emanuel Indrei, Purdue University, USA

THE GEOMETRY OF THE FREE BOUNDARY NEAR THE FIXED BOUNDARY GENERATED BY A FULLY NONLINEAR UNIFORMLY ELLIPTIC OPERATOR

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

The dynamics of how the free boundary intersects the fixed boundary has been the object of study in the classical dam problem which is a mathematical model describing the filtration of water through a porous medium split into a wet and dry part. By localizing around a point at the intersection of free and fixed boundary, one is led to the following problem \[ \begin{cases} F(D^2 u)=\chi_\Omega & \text{in }B_{1}^{+}\\ u=0 & \text{on }B'_{1} \end{cases} \] where \(\Omega = \big(\{u \ne 0\} \cup \{\nabla u \neq 0\} \big) \cap \{x_n>0\}\subset \mathbb{R}_+^n\), \(B'_{1}=\{x_n=0\}\cap \overline{B_1^+}\), and \(F\) is a convex \(C^1\) fully nonlinear uniformly elliptic operator. This talk focuses on the regularity problem for the free boundary \(\Gamma=\partial \Omega \cap B_{1}^{+}\).

References

1. E. Indrei, Boundary regularity and non-transversal intersection for the fully nonlinear obstacle problem, Comm. Pure Appl. Math. 72 (2019), 1459-1473.

Flaviana Iurlano, Sorbonne Université, France

CONCENTRATION VERSUS OSCILLATION EFFECTS IN BRITTLE DAMAGE

Joint work with Jean-François Babadjian and Filip Rindler

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 3.22.

Abstract

This work is concerned with an asymptotic analysis, in the sense of \(\Gamma\)-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero Lebesgue measure and, at the same time and to the same order \(\varepsilon\), the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at \(\varepsilon\) fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as \(\varepsilon\to 0\), concentration of damage and worsening of the elastic properties are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at \(\varepsilon\) fixed) to being linear (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. We explicitly identify the \(\Gamma\)-limit in two and three dimensions for isotropic Hooke tensors. In the limit, a surprising expression for the bulk density appears involving now a continuum damage variable. We further consider the regime where the divergence is square-integrable, which in the limit leads to a Tresca-type plasticity model.

Jean-François Jabir, National Research University Higher School of Economics, Russia

LAGRANGIAN STOCHASTIC MODELS FOR TURBULENT FLOWS AND RELATED PROBLEMS

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Lagrangian stochastic models for turbulence define a particular family of Langevin stochastic differential equations, endowing some specific nonlinearities of McKean type, that were originally introduced in the framework of computational fluid dynamics to describe and simulate the motions of a generic particle of a fluid flow. Although these stochastic models are currently applied in various engineering problems, Lagrangian stochastic models for turbulent flows display a certain number of original mathematical problems broadly linked to existence and uniqueness problems for singular McKean-Vlasov dynamics and the validation of related particle approximations; the modeling of boundary conditions for Langevin models; the introduction of distributions constraints ... The first part of the talk will be dedicated to a short presentation of practical interest and the characteristic theoretical problems related to these Lagrangian stochastic models while some resolutions to these problems, in simplified situations, will be discussed in the rest of the talk.

References

1. M. Bossy, J.-F. Jabir, D. Talay, n conditional McKean Lagrangian stochastic models, Probab. Theory Related Fields 151(1-2) (2011), 319-351.
2. M. Bossy, J.-F. Jabir, Lagrangian stochastic models with specular boundary condition, Journal of Functional Analysis 268(6) (2015), 1309-1381.
3. M. Bossy, J. Fontbona, J.-F. Jabir, P.-E. Jabin, Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain, Communications in Partial Differential Equations 38(7) (2013), 1141-1182.
4. M. Bossy, J.-F. Jabir, Particle approximation for Lagrangian Stochastic Models with specular boundary condition, Electron. Commun. Probab. 23 (2018), 1-14.

Joanna Janczewska, Gdańsk University of Technology, Poland

BIFURCATION OF EQUILIBRIUM FORMS OF AN ELASTIC ROD ON A TWO-PARAMETER WINKLER FOUNDATION

Joint work with Marek Izydorek, Nils Waterstraat, and Anita Zgorzelska

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 1.9.

Abstract

Bifurcation theory is one of the most powerful tools in studying deformations of elastic rods, plates and shells. Numerous works have been devoted to the study of bifurcation in elasticity theory. We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation of Winkler's type. The rod is being compressed by forces at the ends. The left end is free, but we require the shear force to vanish. At the right end, we assume the rod is simply supported. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions.

References

1. M. Izydorek, J. Janczewska, N. Waterstraat, A. Zgorzelska, Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation, Nonlinear Anal. Real World Appl. 39 (2018), 451-463.
2. J. Janczewska, Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point, Cent. Eur. J. Math. 2 (2004), 561-572.
3. A. Borisovich, J. Dymkowska, Elements of Functional Analysis with Applications in Elastic Mechanics, Gdańsk University of Technology, Gdańsk, 2003 (in Polish).

Justyna Jarczyk, University of Zielona Góra, Poland

GAUSSIAN ALGORITHM FOR MAPPINGS BUILT OF PARAMETRIZED MEANS

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

After recalling the notion of iterate of a function depending on a parameter, introduced by K. Baron and M. Kuczma in [1], we present a counterpart of Gaussian algorithm for mappings built of parametrized means. We consider also a special case of the so-called random means and describe some specific properties of the limit of their Gauss iterates.

References

1. K. Baron, M. Kuczma, Iteration of random-valued functions on the unit interval, Colloq. Math. 37 (1977), 263-269.

Witold Jarczyk, John Paul II Catholic University of Lublin, Poland

GENERALIZED GAUSSIAN ALGORITHM

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

The classical Gaussian algorithm runs as follows: taking any \(x,y \in (0, +\infty)\) put \(x_1:=x\), \(y_1:=y\) and \[ x_{n+1}:=A\left(x_n,y_n\right), \quad y_{n+1}:=G\left(x_n,y_n\right), \qquad n \in {\mathbb N}, \] where \(A\) and \(G\) state for the arithmetic and geometric mean, respectively. Gauss proved that both the sequences converge to a common limit, say \(A\otimes \hspace{-0,1cm}G(x,y)\). The function \(A\otimes G\) is a mean on \((0, +\infty)\), i.e. it satisfies \[ \min \{x,y\} \leq A\otimes G(x,y) \leq \max \{x,y\}, \qquad x,y \in (0, +\infty), \] and has nice properties. Iterating the map \((A,G): (0, +\infty)^2\rightarrow (0, +\infty)^2\) one can write down the convergence of Gaussian iterates to \(A\otimes G\) as \[ (A,G)^i\rightarrow \left(A\otimes G,A\otimes G \right). \] The Gauss procedure has been fairly extended to a pretty large class of pairs \((M,N)\) of means on an arbitrary interval \(I\). The talk is a survey of results concerning the convergence of iterates \((M,N)^i\) and properties of the mean \[ M\otimes N:= \lim_{i\rightarrow \infty} (M,N)^i. \] Starting with ideas and results aggregated by J.M. Borwein and P.B. Borwein from different papers more than 30 years ago we come to those proved by J. Matkowski.

Xavier Jarque, Universitat de Barcelona & IMUB, Catalonia

UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS

Joint work with Núria Fagella and Kirill Lazebnik

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

We use the Folding Theorem of [1] to construct an entire function \(f\) in class \(\mathcal{B}\) and a wandering domain \(U\) of \(f\) such that \(f\) restricted to \(f^n(U)\) is univalent, for all \(n\geq 0\). The components of the wandering orbit are bounded and surrounded by the postcritical set.

References

1. C. Bishop, Constructing entire functions by quasiconformal folding, Acta Mathematica 214(1) (2015), 1-60.

Piotr Jaworski, University of Warsaw, Poland

FROM APPLICATIONS TO EQUATIONS

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

In mathematical finance, the standard approach is to model the logarithms of prices of financial assets as Wiener processes, which are correlated in a deterministic way. In my talk I will deal with a pair of Wiener processes \(W^1\) and \(W^2\) which are randomly correlated. Under the assumption that the quadratic covariation of \(W^1\) and \(W^2\) can be described by a deterministic function of \(W^1\) and \(W^2\), \[ d\langle W^1,W^2\rangle_t = A(t,W_t^1,W_t^2) dt,\] I will show that the joint distribution function \(F(t,x_1,x_2)\) and the copula \(C(t,u,v)\) of the pair \((W^1,W^2)\) are generalized weak solutions of parabolic partial differential equations.

Jacek Jendrej, CNRS & Université Paris 13, France

MULTI-BUBBLES FOR A CRITICAL WAVE EQUATION

Joint work with Yvan Martel

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 0.10b.

Abstract

If a solution of an evolution partial differential equation resembles, near some point in space, a rescaled copy of a fixed profile, with the scale tending to zero in finite or infinite time, we say that a bubble is created at this point. Such behavior can be possible only if the rescaling preserves the energy of the profile, which is called the energy-critical setting.

We consider the focusing nonlinear wave equation in the energy-critical case, in space dimension \(5\). Given any finite set of \(K\) points in space, we construct a solution for which a bubble is created at each of these points in infinite time. The energy of the solution is equal to the energy of the profile multiplied by \(K\), which means that no energy is radiated in the process. To our knowledge, we provide the first construction of bubbling at multiple points for a wave equation.

David Jerison, Massachusetts Institute of Technology, USA

THE TWO HYPERPLANE CONJECTURE

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building B-8, room 0.10a.

Abstract

I will introduce a conjecture that I call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. Emanuel Milman has shown that in its strongest, dimension-independent form, my conjecture implies the Hyperplane Conjecture of Kannan, Lovász and Simonovits in theoretical computer science, which says that the area of such an isoperimetric surface is comparable, by an absolute constant independent the convex body and its dimension, to the area of some hyperplane dividing the convex body in half. Their conjecture is closely related to several famous unsolved problems in high dimensional convex geometry. But unlike the hyperplane conjecture, the two-hyperplane conjecture has significance even in low dimensions.

I will relate the conjecture to qualitative and quantitative connectivity properties and regularity of area-minimizing surfaces, free boundaries and level sets of eigenfunctions, and report on work in progress with Guy David. The main theme of the talk is that the level sets of least energy solutions to scalar variational problems should be as simple as possible, but no simpler.

Zhuchao Ji, Sorbonne Université, France

NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

The dynamics of Topological Collet-Eckmann rational maps on Riemann sphere are well understood, due to the work of Przytycki, Rivera-Letelier and Smirnov [1, 2]. In this talk we study the dynamics of polynomial skew products of \(\mathbb{C}^2\). Let \(f\) be a polynomial skew products with an attracting invariant line \(L\), such that \(f\) restricted on \(L\) satisfies Topological Collet-Eckmann condition and a Weak Regularity condition. We show that the the Fatou set of \(f\) in the basin of \(L\) equals to the union of the basins of attracting cycles, and the Julia set of \(f\) in the basin of \(L\) has Lebesgue measure zero. As a consequence there are no wandering Fatou components in the basin of \(L\) (we remark that for some polynomial skew products with a parabolic invariant line \(L\), there can exist a wandering Fatou component in the basin of \(L\) [3, 4]).

References

1. F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Inventiones mathematicae 151 (2003), 29–63.
2. F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Annales Scientifiques de l’École Normale Superieure 40 (2007), 135–178.
3. M. Astorg, X. Buff, R. Dujardin, H. Peters and J. Raissy, A two-dimensional polynomial mapping with a wandering Fatou component, Annals of mathematics 184 (2016), 263– 313.
4. M. Astorg, L. Boc-Thaler and H. Peters, Wandering domains arising from Lavaurs maps with Siegel disks, arXiv preprint arXiv:1907.04140 (2019).
5. Z. Ji, Non-uniform hyperbolicity in polynomial skew products, In preparation.

Víctor Jiménez López, University of Murcia, Spain

ON THE MARKUS-NEUMANN THEOREM

Joint work with José Ginés Espín Buendía

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

According to a well-known result by L. Markus [2], extended by D.A. Neumann in [3], two continuous surface flows are equivalent if and only if there is a homeomorphism preserving orbits and time directions of their separatrix configurations. In this talk, based on the paper [1], some examples are shown to illustrate that the Markus-Neumann theorem, as stated in the original papers, needs not work. Also, we show how the (nontrivial) gap of the proof can be amended to get a correct (and somewhat more general) version of the theorem.

References

1. J. G. Espín Buendía, V. Jiménez López, On the Markus-Neumann theorem, J. Differential Equations 265 (2018), 6036-6047.
2. L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc. 76 (1954), 127-148.
3. D.A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73-81.

Bangti Jin, University College London, UK

TIME-STEPPING SCHEMES FOR FRACTIONAL DIFFUSION

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

Overall the last decade, a large number of time stepping schemes have been developed for time-fractional diffusion problems. These schemes can be generally divided into: finite difference type, convolution quadrature type and discontinuous Galerkin methods. Many of these methods are developed by assuming that the solution is sufficiently smooth, which however is generally not true. In this talk, I will describe recent works in analyzing and developing robust numerical schemes that do not assume solution regularity directly, but only data regularity.

Thomas M. Jordan, University of Bristol, UK

MULTIFRACTAL ANALYSIS FOR PLANAR SELF-AFFINE SETS

Joint work with Balázs Bárány, Antti Käenmäki, and Michał Rams

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

There is a standard problem in multifractal analysis of looking at level sets determined by the Birkhoff average of a suitable function. We look at the problem for self-affine sets on the plane. We show how recent work by Bárány, Hochman and Rapaport combined with results on approximation of pressure functions on suitable subsystems can give fairly complete solutions to this problem under certain generic algebraic assumptions and suitable separation assumptions.

Vesa Julin, University of Jyväskylä, Finland

CLASSICAL SOLUTION TO THE FRACTIONAL MEAN CURVATURE FLOW

Joint work with Domenico La Manna

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 0.10a.

Abstract

I will discuss about recent developments in the study of fractional mean curvature flow and introduce our recent work which is the first short time existence result of the smooth solution to the fractional mean curvature flow.

Sigrid Källblad, Technische Universität Wien, Austria

STOCHASTIC CONTROL OF MEASURE-VALUED MARTINGALES WITH APPLICATIONS TO ROBUST FINANCE

Joint work with A. Cox, M. Larsson, and S. Svaluto.

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

Motivated by robust pricing problems in mathematical finance, we consider in this talk a specific constrained optimisation problem. Our approach is based on reformulating this problem as an optimisation problem over so-called measure-valued martingales (MVMs) enabling the problem to be addressed by use of dynamic programming methods. In the emerging stochastic control problem MVMs appear as weak solutions to a specific SDE for which we prove existence of solutions; we then show that our control problem satisfies the Dynamic Programming Principle and relate the value function to a certain HJB-type equation. A key motivation for the study of control problems featuring MVMs is that a number of interesting probabilistic problems can be formulated as such optimisation problems; we illustrate this by applying our results to optimal Skorokhod embedding problems as well as robust pricing problems.

Vadim Kaloshin, University of Maryland, College Park, USA

ON DYNAMICAL SPECTRAL RIGIDITY OF PLANAR DOMAINS

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building A-3/A-4, room 103.

Abstract

Consider a convex domain on the plane and  the associated billiard inside. The length spectrum is the closure  of the union of perimeters of all period orbits. The length spectrum is  closely related to the Laplace spectrum, through so-called the wave trace. The well-known question popularized by M. Kac: "Can you hear  the shape of a drum?" asks if the Laplace spectrum determines a domain  up to isometry. We call a domain dynamically spectrally rigid (DSR) if any  smooth deformation preserving the length spectrum is an isometry. During  the talk I will discuss recent results on DSR of convex planar domains.

Laura Kanzler, Universität Wien, Austria

KINETIC MODELING OF COLONIES OF MYXOBACTERIA

Joint work with Sabine Hittmeir, Gerhard Kitzler, Angelika Manhart, Christian Schmeiser, and Joachim Schöberl

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

Myxobacteria are rod-shaped, social bacteria that are able to move on flat surfaces by ’gliding’ and form a fascinating example of how simple cell-cell interaction rules can lead to emergent, collective behavior. Observed movement patterns of individual bacteria in such a colony include straight runs with approximately constant velocity, alignment interactions and velocity reversals [1], [2], [3]. Experimental evidence shows that above mentioned behavior is a consequence of direct cell-contact interaction rather than diffusion of chemical signals, which indicates the suitability of kinetic modeling.

In this talk a new kinetic model of Boltzmann-type for such colonies of myxobacteria will be introduced and investigated. For the spatially homogeneous case an existence and uniqueness result will be shown, as well as exponential decay to an equilibrium for the Maxwellian collision operator. The methods used for the analysis combine several tools from kinetic theory, entropy methods as well as optimal transport. The talk will be concluded with numerical simulations confirming the analytical results.

References

1. A. Baskaran, M.C. Marchetti,, Nonequilibrium statistical mechanics of self propelled hard rods, J. Stat. Mech. 2010 (2010), P04019.
2. E. Bertin, M. Droz, G. Gregoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis, J. Phys. A: Math. Theor. 42 (2009), 445001.
3. O.A. Igoshin, G. Oster, Improved stability of optimal traffic paths, Rippling of Myxobacteria 188 (2004), 221-233.

Petr Kaplický, Charles University, Czech Republic

ON UNIQUENESS OF GENERALIZED NEWTONIAN FLOWS

Joint work with Miroslav Bulíček, Frank Ettwein, and Dalibor Pražák

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 0.10b.

Abstract

In [3] a new model for fluid dynamics was suggested by O.A. Ladyzhenskaya. The inner properties of the fluid were described by the constitutive relation \[ \mathbb S = ( 1 + |\mathbb D|^{p-2}) \mathbb D \] where \(p\gt 1\) was a given parameter - power-law index, \(\mathbb S\) was the stress tensor and \(\mathbb D\) the symmetric part of the velocity gradient. If \(p=2\) the model reduces to the Navier-Stokes model. If \(p \gt 2\) one expects that the model exhibits better properties. It is indeed so. In particular, if \( p\ge 11/5\), one is allowed to test weak formulation of the equations with the weak solution itself. Consequently, any weak solution satisfies energy equality. A further motivation for this model was uniqueness of weak solutions. Already in [3], it is established provided \(p\ge 5/2\) or in case of smooth initial condition for \(p\ge 12/5\). The range \(p\in [11/5, 12/5)\) however remained untouched except the case of spatial periodic condition, for which one can improve spatial regularity.

I will present results on uniqueness of the weak solutions to a class of systems, including the one mentioned above, in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index \(p\ge 11/5\), some regularity of a solution with respect to time variable was established. Consequently, this information can be used for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices \(p\ge 11/5\), see [1, 2]. Such a result was available for \(p\ge 12/5\) and therefore the result extends the uniqueness to the whole range of \(p\)’s for which the energy equality holds.

References

1. M. Bulíček, F. Ettwein, P. Kaplický, and D. Pražák, The dimension of the attractor for the 3D flow of a non-Newtonian fluid, Commun. Pure Appl. Anal. 8(5) (2009), 1503–1520.
2. M. Bulíček, P. Kaplický, D. Pražák, Uniqueness and regularity of flows of non-Newtonian fluids with critical power-law growth, Mathematical Models and Methods in Applied Sciences 29 (2019), 1207–1225.
3. O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

Olena Karpel, AGH University of Science and Technology, Poland & Institute for Low Temperature Physics and Engineering, NAS, Ukraine

THE NUMBER OF ERGODIC INVARIANT MEASURES FOR BRATTELI DIAGRAMS

Joint work with Sergey Bezuglyi and Jan Kwiatkowski

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

We study the simplex \(\mathcal{M}_1(B)\) of probability measures on a Bratteli diagram \(B\) which are invariant with respect to the tail equivalence relation. Equivalently, \(\mathcal{M}_1(B)\) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We prove a criterion of unique ergodicity of a Bratteli diagram. In the case of a finite rank \(k\) Bratteli diagram \(B\), we give a criterion for \(B\) to have exactly \(1 \leq l \leq k\) ergodic invariant measures and describe the structures of the diagram and the subdiagrams which support these measures. We also find sufficient conditions under which a Bratteli diagram of arbitrary rank has a prescribed number (finite or infinite) of probability ergodic invariant measures.

Theodoros Katsaounis, KAUST, Saudi Arabia & University of Crete, Greece

A POSTERIORI ERROR CONTROL AND ADAPTIVITY FOR SCHRÖDINGER EQUATIONS

Joint work with Irene Kyza

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

In this talk I will present some recent results on aposteriori error estimation for linear and nonlinear Schrodinger equations. We use finite element discretizations and the Crank Nicolson time stepping scheme. For the derivation of the estimates we use the reconstruction technique and linear and nonlinear stability arguments as in the continuous problem. Based on these aposteriori estimators we further design and analyse a time-space adaptive algorithm. Various numerical experiments verify and complement our theoretical results.

Alexey O. Kazakov, National Research University Higher School of Economics, Russia

WILD PSEUDOHYPERBOLIC ATTRACTOR IN A FOUR-DIMENSIONAL LORENZ MODEL

Joint work with Sergey Gonchenko and Dmitry Turaev

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

We present an example of a new strange attractor. We show that it belongs to a class of wild pseudohyperbolic spiral attractors. A theory of pseudohyperbolic spiral attractors was proposed in [1], however examples of concrete systems of differential equations with such attractors were not known.

We consider the following system of differential equation \[ \left\{ \begin{array}{l} \dot x = \sigma (y - x), \\ \dot y = x (r-z) - y, \\ \dot z = xy - bz + \mu w, \\ \dot w = -b w - \mu z, \end{array} \right. \tag{1} \] where \(\sigma,r,b\) and \(\mu\) are parameters. This system can be viewed as a four-dimensional extension of the classical Lorenz model. We perform a series of numerical experiments with the strange attractor which exists in the system at \(\mu = 7, \sigma = 10, b = 8/3, r = 25\). We demonstrate that this attractor is indeed pseudohyperbolic and wild.

The pseudohyperbolicity [1, 2] is a key word here. It means that certain conditions hold which guarantee that every orbit in the attractor is unstable (i.e. it has a positive maximal Lyapunov exponent). Moreover, this instability property persists for all small perturbations of the system.

The wildness of the observed attractor means that it contains a "wild hyperbolic set" [3 ,4] - a uniformly hyperbolic invariant set which has a pair of orbits such that the unstable manifold of one orbit has a nontransversal intersection with the stable manifold of the other orbit in the pair and this property is preserved for all \(C^2\)-small perturbations.

Acknowledgments

This work was supported by RSF grant No. 17-11-01041.

References

1. D. Turaev, L.P. Shilnikov, An example of a wild strange attractor, Sbornik: Mathematics 189(2) (1998), 291–314.
2. D. Turaev, L.P. Shilnikov, Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors, Doklady Mathematics 77(1) (2008), 17–21.
3. S.E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci. 50 (1979), 101–151.
4. S.V. Gonchenko, D. Turaev, L.P. Shilnikov, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Russian Acad. Sci. Dokl. Math. 47(2) (1993), 268–283.

Gerhard Keller, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

PERIODS AND FACTORS OF WEAK MODEL SETS

Joint work with Christoph Richard

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

Weak model sets are defined by a cut-and-project scheme (CPS) \((G,H,\mathcal{L},W)\), where \(G\) and \(H\) are locally compact second countable abelian groups, \(\mathcal{L}\subset G\times H\) is a cocompact lattice, and \(W\subset H\) is a compact set called the window. Denote by \(\pi_G\) and \(\pi_H\) the canonical projections from \(G\times H\) to \(G\) and \(H\), respectively. It is assumed that \(\pi_G|_{\mathcal{L}}\) is \(1-1\) and that \(\pi_H(\mathcal{L})\) is dense in \(H\). Typical cases to think of are \(G=\mathbb{Z}^d\) or \(\mathbb{R}^d\), while \(H\) could be \(\mathbb{R}^k\) or an odometer group.

Denote \(\mathbb{T}:=(G\times H)/\mathcal{L}\). For \(t\in\mathbb{T}\) the set \(\Lambda_t:=\pi_G\left((G\times W)\cap(t+\mathcal{L})\right)\subset G\) is a weak model set. The structure of \(\Lambda_t\) can be studied (besides many other possibilities) using dynamical systems methods: To that end define \(X:=\overline{\{\Lambda_t:t\in\mathbb{T}\}}\) and its subset \(X_0:=\overline{\{\Lambda_0+g:g\in G\}}\), where the topology stems from the vague topology on the space of locally finite measures on \(G\) when \(\Lambda\subset G\) is identified with the Dirac comb \(\sum_{x\in\Lambda}\delta_x\). In this way both spaces are compact metrizable, and \(G\) acts on them by translation.

Model sets, i.e. the case when \(\overline{\operatorname{int}(W)}=W\), were originally studied by Y. Meyer [4], motivated by problems in harmonic analysis. There dynamical aspects are much studied and well understood. If \(W\) is aperiodic (\(W+h=W\) \(\Rightarrow\) \(h=0\), always true if \(H=\mathbb{R}^d\)), then \((X,G)\) is an almost 1-1 extension of its maximal equicontinuous factor (MEF) \((\mathbb{T},G)\), and if also \(|\partial W|=0\), then Haar-a.e. fibre of this factor map is a singleton. Examples are Sturmian sequences, Toeplitz sequences, the set of vertices of a Penrose tiling, and many others. See [5, 1] for reference.

But also the situation when \(\overline{\operatorname{int}(W)}\) is a strict subset of \(W\) is of considerable interest; it suffices to mention the set of square free integers or the visible lattice points, which are weak model sets with compact groups \(H\) and \(\operatorname{int}(W)=\emptyset\). Sets of \(\mathcal{B}\)-free numbers provide many other, intermediate examples. In [2] we prove among others:

Theorem A. \((\mathbb{T}/_{\mathbb{H}_{\operatorname{int}(W)}},G)\) is the MEF of \((X,G)\), where \(\mathbb{H}_{\operatorname{int}(W)}=\{(0,h)\in G\times H: \operatorname{int}(W)+h=\operatorname{int}(W)\}\). If \(\mathbb{H}_W=\mathbb{H}_{\operatorname{int}(W)}\), this is an almost 1-1 extension.

Remark. If \(\operatorname{int}(W)=\emptyset\), the MEF is thus trivial. But if \(W\) is aperiodic and Haar regular, the maximal equicontinuous generic factor is still \((\mathbb{T},G)\), see [3].

Theorem B. \((X,G,Q)\) is measure theoretically isomorphic to \((\mathbb{T}/_{{\mathbb{H}}_W^{Haar}},G,|\,.\,|)\), where \(|\,.\,|\) denotes Haar measure, \({{\mathbb{H}}_W^{Haar}}=\{(0,h)\in G\times H: |(W+h)\triangle W|=0\}\), and \(Q\) is the image of the Haar measure on \(\mathbb{T}\) under the map \(t\mapsto \Lambda_t\) (called Mirsky measure in arithmetic contexts).

References

1. M. Baake, D. Lenz, R.V. Moody, Characterization of model sets by dynamical systems, Ergod. Th. & Dynam. Sys. 27 (2007), 341-382.
2. G. Keller, C. Richard, Periods and factors of weak model sets, Israel J. Math. 229 (2019), 85-132.
3. G. Keller, Maximal equicontinuous generic factors and weak model sets, ArXiv:1610.03998 (2016).
4. Y. Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland, Amsterdam, 1972.
5. M. Schlottmann, Generalized model sets and dynamical systems in: Directions in Mathematical Quasicrystals, Eds. M.~Baake, R.V.~Moody, CMR Monograph Series 13 (2000), 143-159.

Marek Kimmel, Rice University, USA

EXPECTED SITE FREQUENCY SPECTRA OF CELLS: COALESCENT VERSUS BIRTH-AND-DEATH-PROCESS APPROACH

Joint work with Khanh Dinh, Roman Jaksik, Amaury Lambert, and Simon Tavaré

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 1.11.

Abstract

Recent years have produced a large amount of work on inference about cancer evolution from mutations identified in cancer samples. Much of the modeling work has been based on classical models of population genetics, generalized to accommodate time-varying cell population size [1]. Reverse-time, genealogical, views of such models, commonly known as coalescent theory, have been used to infer aspects of the past of growing populations [2]. Another approach is to use branching processes, the simplest scenario being the linear birth-death process (lbdp), a binary fission Markov age-dependent branching process. A genealogical view of such models is also available [3]. As will be seen in the sequel, the two approaches lead to similar but not identical results.

Inference from evolutionary models of DNA often exploits summary statistics of the sequence data, a common one being the so-called Site Frequency Spectrum. In a sequencing experiment with a known number of sequences, we can estimate for each site at which a novel somatic mutation has arisen, the number of cells that carry that mutation. These numbers are then grouped into sites which have the same number of copies of the mutant. Consider genealogy of a sample of \(n = 20\) cells that . includes 13 mutational events out of which 7 mutations are present in a single cell, 3 are present in 3 cells, 2 are present in 6, and 1 mutation is present in 17 cells. If we denote the numberof mutations present in \(k\) cells by \(\eta_k\), \(\eta_1=7\), \(\eta_3 = 3\), \(\eta_6 = 2\), and \(\eta_{17} = 1\), with all other \(\eta_k\) equal to 0. The vector \(\eta\) is called the Site Frequency Spectrum (SFS). It can be computed from the statistics of mutations in a sample of cells, in which DNA has been sequenced.

We examine how the SFS based on birth-death processes differ from those based on the coalescent model. This may stem from the different sampling mechanisms in the two approaches. However, we also show that despite this, they can be made quantitatively comparable at least for the range of parameters typical for tumor cell populations. We also touch upon the "singleton estimation problem" and the "self-renewal fraction versus proliferation rate" controversy.

References

1. M.J. Williams, B. Werner, T. Heide, C. Curtis, C.P. Barnes, A. Sottoriva and T.A. Graham, Quantification of subclonal selection in cancer from bulk sequencing data, Nature Genetics 50 (2018), 895-903.
2. R. Griffiths and S. Tavaré, The age of a mutation in a general coalescent tree, Stochastic Models 14 (1998), 273-295.
3. A. Lambert and T. Stadler, Birth-death models and coalescent point processes: The shape and probability of reconstructed phylogenies, Theoretical Population Biology 90 (2013), 113-128.

Kristin Kirchner, ETH Zürich, Switzerland

CONVERGENCE OF GALERKIN APPROXIMATIONS FOR FRACTIONAL STOCHASTIC PDES

Joint work with David Bolin, Mihály Kovács, and Sonja G. Cox

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

Many models in spatial statistics are based on Gaussian Matérn fields. Motivated by the relation between this class of Gaussian random fields and stochastic partial differential equations (PDEs), we consider the numerical solution of stochastic PDEs with additive spatial white noise on a bounded Euclidean domain \(\mathcal{D}\subset\mathbb{R}^d\). The non-local differential operator is given by the fractional power \(\mathcal{A}^\beta\), \(\beta \gt 0\), of a second-order elliptic differential operator \(\mathcal{A}\).

We propose an approximation which combines recent Galerkin techniques for deterministic fractional-order PDEs with an efficient way to simulate white noise. Under minimal regularity assumptions on the differential operator \(\mathcal{A}\), in [1, 2, 3] we perform an error analysis for this approximation showing (i) strong mean-square convergence in \(L_2(\mathcal{D})\), (ii) weak convergence, and (iii) convergence in Sobolev spaces: for the approximation of the random field in \(L_q(\Omega;H^s(\mathcal{D}))\), where \(q\in(0,\infty)\) and \(s\in[0,1]\), as well as for the covariance function of the approximation in the mixed Sobolev space \(H^{s,s}(\mathcal{D}\times\mathcal{D})\) at explicit and sharp rates.

For the motivating example of Gaussian Matérn fields, where \(\mathcal{A} =-\Delta + \kappa^2\) and \(\kappa \equiv \operatorname{const.}\), we perform several numerical experiments for various values of the fractional exponent \(\beta\gt 0\) in dimensions \(d\in\{1,2\}\), which attest the theoretical results.

References

1. D. Bolin, K. Kirchner, M. Kovács, Numerical solution of fractional elliptic stochastic PDEs with spatial white noise, IMA J. Numer. Anal., electronic (2018), n/a–n/a.
2. D. Bolin, K. Kirchner, M. Kovács, Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise, BIT Numer. Math. 58 (2018), 881–906.
3. S.G. Cox, K. Kirchner, Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields, Preprint, arXiv:1904.06569, 2019.

Eva Kisdi, University of Helsinki, Finland

ADAPTIVE DYNAMICS OF PATHOGENS AND THEIR HOSTS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

This talk will review how pathogens evolve to exploit their hosts, and how hosts in turn evolve under selective pressure from their pathogens. Special attention is paid to nonlinear feedbacks, which are responsible for a wide range of dynamical phenomena including the evolution of diversity and the emergence of evolutionary cycles via Hopf bifurcations. I discuss whether pathogens can evolve so virulent that they drive their hosts extinct, and whether hosts can escape their pathogens by evolving such that the pathogen is no longer viable.

Gergely Kiss, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Hungary

FUNCTIONAL EQUATIONS, FIELD HOMOMORPHISMS AND DERIVATIONS IN THE LIGHT OF SPECTRAL THEORY

Joint work with Eszter Gselmann and Csaba Vincze

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

First, in my talk I discuss the solutions of linear functional equations on fields of the form \[ \sum_{i=1}^n f_i(b_ix+c_iy)=0 ~~~~~\forall x,y\in K, \] where \(b_i,c_i\) are given constants, \(K\) is the field and \(f_i\) are unknown functions. I present that typically the set of solutions is a linear space containing field homomorphisms and higher order derivations. This result is based on spectral synthesis. Here I recall the theoretic background and discuss the main tools that we of use. In the second part of my presentation I study functional equations \[ \sum_{i=1}^n f_i^{p_i}(x^{q_i})=0 ~~~~~\forall x\in K, \] and \[ \sum_{i=1}^n x^{p_i}f_i(x^{q_i})=0 ~~~~~\forall x\in K, \] where \(p_i, q_i\) are positive integers and \(f_i\) are additive functions, that characterize field homomorphisms and higher order derivations, respectively. Among other techniques these results deliberately use spectral theory. Finally, I mention some further directions of research in this area. My talk is based on the [1, 2, 3].

References

1. G. Kiss, M. Laczkovich, Linear functional equations, differential operators and spectral synthesis, Aequationes Mathematicae 89(2) (2015), 301-328.
2. E. Gselmann, G. Kiss, Cs. Vincze, On functional equations characterizing derivations: methods and examples, Results in Mathematics 74 (2018), 27 pp.
3. E. Gselmann, G. Kiss, Cs. Vincze, Characterization of field homomorphisms through Pexiderized functional equations, Journal of difference equations and applications 25 (2019), 26 pp.

Sergiu Klainerman, Princeton University, USA

ON THE NONLINEAR STABILITY OF BLACK HOLES

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Black holes are precise mathematical solutions of the Einstein field equations mainly represented by the famous two parameter Kerr family including, as a particular case, the Schwarzschild solution. To correspond to physical reality, i.e. to be more than mathematical artifacts, these solutions have to be stable under small perturbations. While there is today no doubt concerning the physical reality of black holes, based both on observational data and numerical simulations, an actual proof of stability remains a fundamental challenge of Mathematical and Geometric Analysis.

In my talk I will formulate the precise mathematical problem of the nonlinear stability of the Kerr family and describe the main results known so far. In the second part of the talk I will describe my recent result with J. Szeftel "Global Nonlinear Stability of Schwarzschild Spacetime under Polarized perturbations" - arXiv:1711.07597. The result establishes the full nonlinear stability of Schwarzschild spacetime under axially symmetric, polarized perturbations, i.e. stability of solutions of the Einstein vacuum equations for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admit a hyper-surface orthogonal space-like Killing vector-field with closed orbits. While building on the remarkable advances made in last 15 years on establishing quantitative linear stability, the paper introduces a series of new ideas among which we emphasize the \(\textit{general covariant modulation}\) (GCM) procedure which allows us to construct, dynamically, the center of mass frame of the final state. The mass of the final state itself is tracked using the well known Hawking mass relative to a well adapted foliation itself connected to the center of mass frame. Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space.

Tomasz Klimsiak, Nicolaus Copernicus University in Toruń, Poland

ASYMPTOTICS FOR LOGISTIC-TYPE EQUATIONS WITH DIRICHLET FRACTIONAL LAPLACIAN

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

We will present results on the asymptotics, as \(t\rightarrow \infty\) and \(p\rightarrow \infty\), of solutions of the following problem: \[ \frac{\partial u}{\partial t}-(\Delta^{\alpha/2})_{|D}u=a u-bu^p,\quad u(0,\cdot)=\varphi.\quad(*) \] Here \(D\) is a bounded Lipschitz domain in \(\mathbb R^d\), \(a>0, p>1, \alpha\in (0,2)\), and \(b,\varphi\) are bounded positive nontrivial Borel functions on \(D\). We show that for suitable \(a\) the limit function does not depend on the order of limits and is a unique solution of an obstacle problem.

Equations and systems of type \((*)\) with nonlocal operators appear in many models of population biology. Dirichlet fractional Laplacian in \((*)\) is designed to describe nonlocal dispersal strategy of animals (see [3]). Stationary equations of type \((*)\) have been studied recently in [1] with strictly positive \(b\).

Asymptotics as \(p\rightarrow \infty\) for solutions to stationary counterpart to \((*)\) with classical Laplacian was studied for the first time in [2]. In this paper it is observed that for large \(p\) solutions of this stationary problem behave like solutions of certain steady-state predator-pray models. The methods used in [2] extensively exploit the local character of Dirichlet Laplacian and can not be applied to the case of fractional Laplacian. We present a new method (see [4], [5]) based on the notion of ultracontractivity and probabilistic potential theory. The method we introduce may be applied to a wide class of nonlocal operators.

References

1. L. Caffarelli, S. Dipierro, E, Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models 10 (2017), 141–170.
2. E. Dancer, Y. Du, On a free boundary problem arising from population biology, Indiana Univ. Math. J. 52 (2003), 51–67.
3. N.E. Humphries, N. Queiroz, J.R.M. Dyer et al., Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature 465 (2010), 1066–1069.
4. T. Klimsiak, Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator, arXiv:1905.01667v1 (2019).
5. T. Klimsiak, Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian, arXiv:1905.01666v1 (2019).

Hans Knüpfer, Universität Heidelberg, Germany

DOMAIN PATTERN FORMATION IN FERROMAGNETIC SAMPLES

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We investigate the optimal shape and patterns for magnetic domains from the perspective of Calculus of Variations. These patterns are driven mainly by the competition of local energies such as interfacial energies and anisotropy energy and the nonlocal magnetostatic interaction. In particular, we are interested in pattern formation in thin ferromagnetic films with perpendicular anistropy and in the phase transformation for non-trivial states. For the analysis, we do not use any assumptions on the shape of the domain, rather the arguments are based on the derivation of suitable interpolation inequalities.

Jakub Konieczny, Hebrew University of Jerusalem, Israel

AUTOMATIC SEQUENCES, NILSYSTEMS, AND HIGHER ORDER FOURIER ANALYSIS

Joint work with Jakub Byszewski and Clemens Müllner

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

Automatic sequences give rise to one of the basic models of computation and have remarkable links to many areas of mathematics, including dynamics, algebra and logic. Distribution of these sequences has long been studied. During the talk we will explore this topic from the point of view of higher order Fourier analysis. As it turns out, many of the classical automatic sequences are highly Gowers uniform, while others can be expressed as the sum of a structured component and a uniform component much more efficiently than guaranteed by the arithmetic regularity lemma. We investigate the extent to which this phenomenon extends to general automatic sequences and consider some closely related problems that make sense for sparse sequences. 

Jozef Kováč, Comenius University in Bratislava, Slovakia

DISTRIBUTIONAL CHAOS IN RANDOM DYNAMICAL SYSTEMS

Date: 2019-09-19 (Thursday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

Consider two continuous interval maps \(f,g:[0,1]\to [0,1]\) and the random dynamical system given by \[ x_{n+1}= \begin{cases} f(x_n) & \text{ with probability } p,\\ g(x_n) & \text{ with probability } 1-p, \end{cases} \] where \(p\in (0,1)\). Distributional chaos for such systems was defined in [1]. We will discuss some of its properties (for example stability, types of distributional chaos, etc.).

References

1. J. Kováč, K. Janková, Distributional chaos in random dynamical systems, J. Difference Equ. Appl. 25(4) (2019), 455–480.

Michał Kowalczyk, University of Chile, Chile

MAXIMAL SOLUTION OF THE LIOUVILLE EQUATION IN DOUBLY CONNECTED DOMAINS

Joint work with Angela Pistoia and Giusi Vaira

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

In this talk I will discuss a new existence result for the widely studied Liouville problem \(\Delta u+\lambda^2 e^u=0\) in a bounded, two dimensional, doubly connected domain with Dirichlet boundary conditions. I will show that for a sequence of \(\lambda_n \to 0\) this equation has solutions that blow-up in in the whole domain. Profiles of the blowing-up solutions are related to a free boundary problem which gives a solution to an optimal partition problem for the given domain. I will also describe the role of the free boundary problem in other classical equations such as the mean field model or the prescribed Gaussian curvature equation.

Bernd Krauskopf, University of Auckland, New Zealand

A HETERODIMENSIONAL CYCLE IN A 4D FLOW

Joint work with Andy Hammerlindl, Gemma Mason and Hinke M. Osinga

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. Recent theoretical work for diffeomorphisms of dimension at least three has shown that the existence of heterodimensional cycles may be a \(C^1\)-robust property. We study a concrete example of a heterodimensional cycle in a flow, specifically in a four-dimensional Atri model of intracellular calcium dynamics. For suitable parameter values, this model has two saddle periodic orbits of different index. We employ a boundary-value problem setup to compute their global invariant manifolds to show that and how they intersect in a connecting orbit of codimension one and an entire cylinder of connecting orbits. We present the different invariant objects in different projections of the four-dimensional phase space, as well as in intersection with a three-dimensional Poincaré section. In this way, we examine how this heterodimensional cycle arises and organises the nearby dynamics.

Raphaël Krikorian, CNRS & Université de Cergy-Pontoise, France

ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

A real analytic hamiltonian or a real analytic exact symplectic diffeomorphism admitting a non resonant elliptic fixed point is always formally conjugated to a formal integrable system, its Birkhoff Normal Form (BNF). Siegel proved in 1954 that the formal conjugation reducing a hamiltonian to its BNF is in general divergent and Hakan Eliasson has asked whether the BNF itself could be divergent. Perez-Marco proved in 2001 that for any fixed non resonant frequency vector the following dichotomy holds: either any real analytic hamiltonian system admitting this frequency vector at the origin has a convergent BNF or for a prevalent set of hamiltonians admitting this frequency vector the BNF generically diverges. It is possible to exhibit examples of hamiltonian systems with diverging BNF (X. Gong 2012 or the recent examples of B. Fayad in 4 degrees of freedom). The aim of this talk is to give a complete answer to the question of the divergence of the BNF (in the setting of exact symplectic diffeomorphisms): for any non resonant frequency vector, the BNF of a real analytic exact symplectic diffeomorphism admitting this frequency vector at the origin, is in general divergent. This theorem is the consequence of the remarkable fact that the convergence of the formal object that is the BNF has dynamical consequences, in particular an abnormal abundance of invariant tori.

Wojciech Kryński, Polish Academy of Sciences, Poland

ODES AND GEOMETRIC STRUCTURES ON SOLUTION SPACES

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 1.9.

Abstract

We shall consider ordinary differential equations (ODEs) from the geometric vewpoint. Our aim is study geometric structures appearing on the solution spces to ODEs. In particular, for the third order ODEs one can get canonical conformal structures on the solution spaces. Higher order generalizations lead to \(GL(2)\)-geometry and, in general, to the so-called causal or cone geometry. In the talk, we shall also present applications to the twistor theory and to the integrable systems of partial differential equaitons.

Piotr Krzyżanowski, University of Warsaw, Poland

ROBUST PRECONDITIONING FOR DISCONTINUOUS GALERKIN DICRETIZATIONS OF DIFUSSION PROBLEM WITH HIGH CONTRAST COEFFICIENTS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

We consider a diffusion problem in a heterogeneous medium, with prescribed transmission properties. We discuss preconditioners for iterative solutions of algebraic systems arising from problem discretizations of discontinuous Galerkin type.

In particular, a diffusion problem through a thin membrane is discussed. A nonoverlapping domain decomposition based preconditioner is introduced, and its convergence properties are discussed and verified in numerical experiments. In particular, the convergence rate is shown independent of the contrast in the diffusion coefficient, the number of inclusions and of the transmission parameter as well.

Hieronim Kubica, AGH University of Science and Technology, Poland

PERSISTENCE OF NORMALLY HYPERBOLIC INVARIANT MANIFOLDS IN THE ABSENCE OF RATE CONDITIONS

Joint work with Maciej Capiński

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

Normally hyperbolic invariant manifolds (NHIMs) [8] are persistent [2, 3, 4, 5]. The converse is also true - persistent manifolds are precisely the normally hyperbolic ones [9]. These results take advantage of analytical assumptions. Features like rate conditions associated with the map driving the dynamics, smooth structure on the invariant set, or \(C^1\) topology on the space of admissible perturbations play a key role. In a recent result, we prove that a weaker form of persistence is viable also in a setting which relies only on topological, qualitative assumptions about the dynamics [1].

We consider perturbations of normally hyperbolic invariant manifolds, under which they can loose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system preserves the properties of topological expansion and contraction, described in terms of covering relations [6, 7], then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. The method used to show this is not itself perturbative. It can be applied to establish the existence of invariant sets within a prescribed neighborhood also in the absence of a normally hyperbolic invariant manifold prior to perturbation. The dynamics is assumed to be given by a continuous map, without the assumption of invertibility.

References

1. M. Capiński, H. Kubica, Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions, presented for publication, 2018
2. N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193–226.
3. N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/74), 1109– 1137.
4. N. Fenichel, Asymptotic stability with rate conditions for dynamical systems, Bull. Amer. Math. Soc. 80 (1974), 346–349.
5. N. Fenichel, Asymptotic stability with rate conditions II, Indiana Univ. Math. J. 26 (1977), 81–93.
6. M. Gidea, P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 32–58.
7. M. Gidea, P. Zgliczyński, Covering relations for multidimensional dynamical systems II, J. Differential Equations 202 (2004), 59–80.
8. M. Hirsch, C. Pugh, M. Shub Invariant Manifolds, Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2006.
9. R. Mañé, Persistent manifolds are normally hyperbolic, Trans. Amer. Math. Soc. 246 (1978), 261– 283.

Toshikazu Kuniya, Kobe University, Japan

GLOBAL BEHAVIOR OF A MULTI-GROUP SIR EPIDEMIC MODEL WITH AGE STRUCTURE AND ESTIMATION OF \(\cal R_0\) FOR THE CHLAMYDIA EPIDEMIC IN JAPAN

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 1.26.

Abstract

In this talk, we study the global behavior of a multi-group SIR epidemic model with age structure. Under the assumptions that the disease transmission coefficient is independent of the age of infective individuals and the sum of the mortality and the recovery rate is constant, the model can be rewritten into a multi-group SIR model with age-dependent susceptiblity. We then define the basic reproduction number \(\cal{R}_0\) by the spectral radius of the next generation matrix and show that \(\cal{R}_0\) completely determines the global behavior of the model: if \(\cal{R}_0 \lt 1\), then the disease-free equilibrium is globally attractive, whereas if \(\cal{R}_0 \gt 1\), then the endemic equilibrium is globally attractive. In the application, we estimate \(\cal{R}_0\) for the chlamydia epidemic in Japan in 2015 by comparing four special cases of our model: a homogeneous model, an age-independent two-sex model, an age-dependent one-sex model and an age-dependent two-sex model. In conclusion, we see that \(\cal{R}_0\) is in the range \(1.0148\)-\(1.0535\), the age structure has more influence on the estimation result than the two-sex structure and disregarding the age structure could lead to the underestimation of \(\cal{R}_0\).

References

1. T. Kuniya, Global behavior of a multi-group SIR epidemic model with age structure and an application to the chlamydia epidemic in Japan, SIAM J. Appl. Math. 79 (2019), 321-340.
2. T. Kuniya, J. Wang, H. Inaba, A multi-group SIR epidemic model with age structure, Disc, Cont. Dyn. Syst. Series B 21 (2016), 3515-3550.

Krystyna Kuperberg, Auburn University, USA

A MEASURE PRESERVING PL MODIFICATION OF THE JONES-YORKE BOUNDED ORBIT FLOW ON \(\mathbb{R}^3\)

Joint work with Jeffrey Ford

Date: 2019-09-19 (Thursday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

We present a modification of the example by G. Stephen Jones and James A. Yorke of a smooth, bounded orbit, dynamical system on \(\mathbb{R}^3\). Our example is piecewise linear and measured. We use methods of piecewise linear dynamical system developed by Greg~Kuperberg, in particular his notion of a slanted suspension.

References

1. G.S. Jones, J.A. Yorke, The existence and nonexistence of critical points in bounded flows, J. Differential Equations 6 (1969), 238-246.
2. G. Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), 70-97.

Dominik Kwietniak, Jagiellonian University in Kraków, Poland

ON THE ABUNDANCE OF (NON)TAME GROUP ACTIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

Tame dynamical systems were introduced by Köhler [3] in 1995. Tameness is a topological notion roughly corresponding to compactness appearing in the compactness vs weak mixing dichotomy that underlines the structure theory for measure preserving actions.

In recent years several authors developed the theory of tame systems revealing connections to other areas of mathematics like Banach spaces, circularly ordered systems, substitutions and tilings, quasicrystals, cut and project schemes and even model theory and logic.

During my talk, I will discuss results of [1], where we study tameness and nullness of regular almost automorphic \(G\)-actions utilising a generalised notion of semi-cocycle extensions. In particular, we show that every ergodic equicontinuous \(G\)-action on a compact metric space admits a regular almost automorphic extension which is non-tame as well as tame but non-null extension. In some sense, this complements a recent result of Glasner [2]. We prove that such examples appear in well-studied families of group actions including Delone dynamical systems and symbolic systems (including Toeplitz flows over arbitrary \(G\)-odometers).

References

1. G. Fuhrmann, D. Kwietniak, On tameness of almost automorphic dynamical systems for general groups, preprint, arXiv:1902.10780 [math.DS] (2019).
2. E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math.211(1) (2018), 213-244.
3. A. Köhler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. Sect. A 95(2) (1995), 179-191.

Kei Fong Lam, Chinese University of Hong Kong, Hong Kong

TOTAL VARIATION AND PHASE FIELD REGULARISATIONS OF AN INVERSE PROBLEM WITH QUASILINEAR MAGNETOSTATIC EQUATIONS

Joint work with Irwin Yousept

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 3.22.

Abstract

We tackle the inverse problem of reconstructing a discontinuous coefficient in magnetostatic equations from measurements in a subdomain. This problem is motivated from identifying the location of magnetic materials (e.g. iron) in a bounded domain containing also non-magnetic materials (e.g. copper), and can be viewed as an idealised problem for non-invasive/non-destructive testing based on electromagnetic phenomena. The magnetic material produces a stronger response compared to the non-magnetic material in the presence of an applied current field, and the situation can be well-described by quasilinear H(curl) magnetostatic equations. As the inverse problem is likely to be ill-posed, we reformulate it into a constraint minimisation problem with perimeter penalisation. Existence of minimisers, stability with respect to data perturbation, and consistency as the penalisation parameter tends to zero are discussed. We then introduce a further phase field approximation of the minimisation problem and derive the first order necessary optimality conditions. Then, we investigate the sharp interface limit to demonstrate the phase field approximation is a meaningful method to solve the inverse problem.

Martin Larsson, Carnegie Mellon University, USA

THE EXPRESSIVENESS OF RANDOM DYNAMICAL SYSTEMS

Joint work with Christa Cuchiero and Josef Teichmann

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 2.19.

Abstract

Deep neural networks perform exceedingly well on a variety of learning tasks, in particular in finance where they are quickly gaining importance. Training a deep neural network amounts to optimizing a nonlinear objective over a very large space of parameters. This would seem a hopeless task if a globally optimal solution were required. The fact that this can succeed suggests that the result is largely insensitive to the detailed structure of the selected locally near-optimal solution, a perspective that is supported by empirical evidence. In this work we attempt a step toward a theoretical understanding of this phenomenon. In a model of deep neural networks as discretizations of controlled dynamical systems, we rigorously prove that any learning task can be accomplished even if a majority of the parameters are chosen at random

Stig Larsson, Chalmers University of Technology & University of Gothenburg, Sweden

BACKWARD EULER-MARUYAMA METHOD FOR SDES WITH MULTIVALUED DRIFT COEFFICIENT

Joint work with Monika Eisenmann, Raphael Kruse, and Mihály Kovács

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 2.18.

Abstract

We consider the numerical approximation of a multivalued SDE \[ \begin{cases} \mathrm{d} X(t) + f(X(t)) \, \mathrm{d} t \ni g(t) \, \mathrm{d} W(t), \quad t \in (0,T],\\ X(0) = X_0, \end{cases} \] where the mapping \(f \colon \mathbf{R}^d \to 2^{\mathbf{R}^d}\) is maximal monotone, of at most polynomial growth, coercive, and fulfills the condition \[ \langle f_v - f_z , z-w \rangle \leq \langle f_v - f_w , v-w \rangle, \] for every \(v,w,z \in \mathbf{R}^d\), \(f_v \in f(v)\), \(f_w \in f(w)\), and \(f_z \in f(z)\) as proposed in [1]. Under these low regularity assumptions on the drift coefficient, we can prove well definedness of the backward Euler method, as well as the strong convergence with a rate of \(\frac{1}{4}\), if \(g\) lies in a suitable Hölder space.

References

1. R.H. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), 525-589.

Irena Lasiecka, University of Memphis, USA

GLOBAL SOLUTIONS AND STABILITY FOR A 3-D FLUID-STRUCTURE INTERACTION

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building B-8, room 0.18.

Abstract

We consider an interface problem consisting of a 3-D fluid equation interacting with a 3-D dynamic elasticity. The interface is moving according to the speed of the fluid. The PDE system is modeled by system of partial differential equations describing motion of an elastic body inside an incompressible fluid. The fluid is governed by Navier-Stokes equation while the structure is represented by the system of dynamic elasticity with weak dissipation. The interface between the two environments undergoes oscillations which lead to moving frame configuration, the latter giving rise to a quasilinear system. Short time local existence of solutions has been shown in [1]. The aim of the talk is to discuss global [in time] solutions under small disturbance hypothesis. Stability [in time] of such solutions is also considered along with some control problems related to minimization of the vorticity. The problem is motivated by applications arising in bio-mechanics, aeroelasticity and industrial processes. In the presence of weak damping affecting the solid the control-to-observation map is proved global-so that the size of the data can be chosen uniformly in time. This allows consideration of an infinite time horizon control problem. The latter depends on the global existence results obtained in [2].

References

1. D. Coutand, S. Shkoller, Motion of an elastic solid inside an incompressible fluid, Archives of Rational Mechanics and Analysis 176 (2008), 1173-1207.
2. M. Ignatova, I. Kukavica, I. Lasiecka, A. Tuffaha, Small data global existence for a fluid structure model, Nonlinearity; 30 (2017), 848-898.

Antoine Laurain, University of São Paulo, Brazil

SHAPE DESIGN FOR SUPERCONDUCTORS GOVERNED BY H(CURL)-ELLIPTIC VARIATIONAL INEQUALITIES

Joint work with Malte Winckler and Irwin Yousept

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 0.18.

Abstract

We study a shape optimization problem governed by an elliptic \(\operatorname{curl}\)-\(\operatorname {curl}\) variational inequality (VI) of the second kind. We present a Moreau-Yosida type regularization for the dual formulation of the VI that guarantees the Gâteaux-differentiability of the regularized dual variable. Then, for a fixed regularization parameter, the existence of an optimal shape for the corresponding problem is proved by means of a compactness theorem. Then we analyze the sensitivity of the regularized objective functional by rigorously computing the corresponding shape derivative using the averaged adjoint method, a lagrangian-type formulation. We also give a stability estimate for the shape derivative with respect to the regularization parameter, and show the strong convergence of the optimal shapes and the corresponding states for the regularized problem towards a solution to the problem without regularization. Finally, we present the numerical algorithm based on the distributed shape derivative coupled with the level set method, and we apply it to problems stemming from the type-II (HTS) superconductivity.

References

1. I. Yousept, Hyperbolic Maxwell Variational Inequalities for Bean’s Critical-State Model in Type-II Superconductivity, SIAM J. Numer. Anal. 55 (2017), 2444-2464.
2. A. Laurain, K. Sturm, Distributed shape derivative via averaged adjoint method and applications, ESAIM Math. Model. Numer. Anal. 50 (2016), 1241-1267.

Kirill Lazebnik, California Institute of Technology, USA

UNIVALENT POLYNOMIALS AND HUBBARD TREES

Joint work with Nikolai G. Makarov and Sabyasachi Mukherjee

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

We study the space of ''external polynomials'' \[ \Sigma_d^* := \left\{ f(z)= z+\frac{a_1}{z} + \cdots +\frac{a_d}{z^d} : a_d=-\frac{1}{d}\textrm{ and } f|_{\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}} \textrm{ is conformal}\right\}. \] It is proven that a simple class of combinatorial objects (bi-angled trees) classify those \(f\in\Sigma_d^*\) with the property that \(f(\mathbb{T})\) has the maximal number \(d-2\) of double points. We discuss a surprising connection with the class of anti-holomorphic polynomials of degree \(d\) with \(d-1\) distinct, fixed critical points and their associated Hubbard trees.

References

1. Lazebnik, Kirill, Makarov, Nikolai, Mukherjee, Sabyasachi, Univalent Polynomials and Hubbard Trees, arXiv, 2019.

Yong-Hoon Lee, Pusan National University, South Korea

BIFURCATION OF A MEAN CURVATURE PROBLEM IN MINKOWSKI SPACE ON AN EXTERIOR DOMAIN

Joint work with Rui Yang

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

We study the existence of positive radial solutions for a mean curvature problem in Minkowski space on an exterior domain. Based on \(C^1\)-regularity of solutions, which is closely related to the property of nonlinearity \(f\) near 0, we make use of the global bifurcation theory to establish some existence results of positive radial solutions when \(f\) is sublinear at \(\infty\).

Martin Leguil, University of Toronto, Canada

SPECTRAL DETERMINATION OF OPEN DISPERSING BILLIARDS

Joint work with Péter Bálint, Jacopo De Simoi, and Vadim Kaloshin

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-4, room 120.

Abstract

In an ongoing project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift of finite type that provides a natural labeling of all periodic orbits. One direction we have investigated in [1] is whether it is possible to recover from the Marked Length Spectrum (i.e., the set of lengths of all periodic orbits together with their labeling) the local geometry near periodic points. In particular, we show in [1] that the Marked Length Spectrum determines the curvatures of the scatterers at the base points of \(2\)-periodic orbits, and the Lyapunov exponents of each periodic orbit. In a second work [2], we show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum.

References

1. P. Bálint, J. De Simoi, V. Kaloshin, and M. Leguil, Marked Length Spectrum, homoclinic orbits and the geometry of open dispersing billiards, Communications in Mathematical Physics (2019), 1-45.
2. J. De Simoi, V. Kaloshin, and M. Leguil, Marked Length Spectral determination of analytic chaotic billiards with axial symmetries, arXiv preprint arXiv:1905.00890 (2019).

Mariusz Lemańczyk, Nicolaus Copernicus University in Toruń, Poland

MULTIPLICATIVE FUNCTIONS AND DISJOINTNESS IN ERGODIC THEORY

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

In 2010, P. Sarnak formulated the Möbius orthogonality conjecture stating that the classical Möbius function does not correlate with any continuous observable in a (topological) zero entropy dynamical system. This conjecture has deep connections with analytic number theory and joinings in ergodic theory. My talk will be devoted to present some of these connections and an overview of the latest achievements.

Zbigniew Leśniak, Pedagogical University of Cracow, Poland

ON FRACTIONAL ITERATES OF A BROUWER HOMEOMORPHISM

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We present a method for finding continuous (and consequently homeomorphic) orientation preserving iterative roots of a Brouwer homeomorphism for which there exists a family of pairwise disjoint invariant lines covering the plane.

To obtain the roots we use properties of the equivalence classes of the codivergency relation. In particular, the key role plays the fact that each of the invariant lines of the considered family is contained either in the set of regular points or in the set of irregular points of the given Brouwer homeomorphism.

References

1. Z. Leśniak, On fractional iterates of a free mapping embeddable in a flow, J. Math. Anal. Appl. 366 (2010), 310-318.
2. Z. Leśniak, On properties of the set of invariant lines of a Brouwer homeomorphism, J. Difference Equ. Appl. 24 (2018), 746-752.

Shingyu Leung, Hong Kong University of Science and Technology, Hong Kong

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS ON MANIFOLDS AND POINT CLOUDS

Joint work with Hongkai Zhao, Meng Wang, and Ningchen Ying

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 3.21.

Abstract

We present recent numerical methods for solving partial differential equations on manifolds and point clouds. In the first part of the talk, we introduce a new and simple discretization, named the Modified Virtual Grid Difference (MVGD), for numerical approximation of the Laplace-Beltrami operator on manifolds sampled by point clouds. We first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. The new discretization provides more diagonal dominance to the resulting linear system and improves its conditioning. In the second part, we present a local regularized least squares radial basis function (RLS-RBF) method for solving partial differential equations on irregular domains or on manifolds. The idea extends the standard RBF method by replacing the interpolation in the reconstruction with the least squares fitting approximation.

References

1. M. Wang, S. Leung and H. Zhao, Modified Virtual Grid Difference for Discretizing the Laplace-Beltrami Operator on Point Clouds, SIAM J. Sci. Comput. 40 (2018).
2. N. Ying and S. Leung, A Local Regularized Least Squares Radial Basis Function (RLS-RBF) Method for Differential Operators in Meshless Domain, in preparation.

Dongchen Li, Imperial College London, UK

PERSISTENT HETERODIMENSIONAL CYCLES IN PERIODIC PERTURBATIONS OF LORENZ-LIKE ATTRACTORS

Joint work with Dmitry Turaev

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of \(C^{\infty}\) diffeomorphisms. This implies the existence of a \(C^2\)-open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in \(C^{\infty}\). In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small perturbation of any such flow can belong to this domain - in this case the corresponding heterodimensional cycles belong to a chain-transitive attractor of the perturbed flow.

Jian Li, Shantou University, China

RECENT DEVELOPMENTS ON MEAN EQUICONTINUITY AND MEAN SENSITIVITY

Date: 2019-09-17 (Tuesday); Time: 11:05-11:25; Location: building A-3/A-4, room 103.

Abstract

In this talk we will discuss recent developments on mean equicontinuity and mean sensitivity both on topological and measure-theoretic settings, based on the following three papers and related works.

References

1. J. Li, S. Tu, X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems 35 (2015), 2587-2612.
2. J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity 29 (2016), 2133-2144.
3. W. Huang, J. Li, J. Thouvenot, L. Xu, X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory and Dynamical Systems, preprint, arXiv:1806.02980.

Xue-Mei Li, Imperial College London, UK

AVERAGING DYNAMICS DRIVEN BY FRACTIONAL NOISE

Joint work with Martin Hairer

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

Two scale stochastic equations model evolutions of random motions under the influence of fast motions. An overwhelming amount of efforts have been devoted to where the fast motion is assumed to have a Markov property or have independent increments, time series data begs to differ. I hope to explain the effective dynamics for a slow/fast systems where the slow system is driven by a fractional Brownian motion, and perhaps also to touch on further developments.

Pierre Lissy, CEREMADE, Université Paris-Dauphine, France

LOCAL CONTROLLABILITY TO THE TRAJECTORIES OF THE FOKKER-PLANCK EQUATION WITH A LOCALIZED CONTROL

Joint work with Michel Duprez

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 0.18.

Abstract

We will present a new result on the control of the Fokker-Planck equation, posed on a smooth bounded domain of \(\mathbb{R}^d\) \((d \ge 1)\). More precisely, the control is the drift force, localized on a small open subset. We prove that this system is locally null controllable to regular nonzero trajectories, with potentially strictly less that \(d\) controls. The results are obtained thanks to a linearization method based on a standard inverse mapping procedure and the fictitious control method. The main novelties of the present article are twofold. Firstly, we propose an alternative strategy to the standard fictitious control method: the algebraic solvability is performed and used directly on the adjoint problem. Secondly, we prove a new Carleman inequality for the heat equation with one order space-varying coefficients: the right-hand side is the gradient of the solution localized on a subset (rather than the solution itself), and the left-hand side contains arbitrary high derivatives of the solution.

Aleksandr Logunov, Princeton University, USA

ZERO SETS OF LAPLACE EIGENFUCNTIONS

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Laplace eigenfunctions on two dimensional sphere are restrictions of homogeneous harmonic polynomials of three variables onto the sphere. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree \(n\) is comparable to \(n\).

References

1. A. Logunov, E. Malinnikova, Review of Yau’s conjecture on zero sets of Laplace eigenfunctions, to appear in Current Developments in Mathematics.
2. A. Logunov, E. Malinnikova, Quantitative propagation of smallness for solutions of elliptic equations, Proceedings of the International Congress of Mathematicians–Rio de Janeiro, 2357–2378, 2018.
3. A. Logunov, E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, 50 Years with Hardy Spaces, A Tribute to Victor Havin, 333–344, 2018.
4. A. Logunov, E. Malinnikova, Lecture notes on quantitative unique continuation for solutions of second order elliptic equations, to appear in IAS/Park City Mathematics series, AMS.
5. A. Logunov, 2018, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Annals of Mathematics, 187(1), 221–239.
6. A. Logunov, 2018, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Annals of Mathematics, 187(1), 241–262.

Ilaria Lucardesi, Université de Lorraine, France

ENERGY RELEASE RATE IN PLANAR ELASTICITY IN PRESENCE OF REGULAR CRACKS

Joint work with Stefano Almi and Giuliano Lazzaroni

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 3.22.

Abstract

In this talk, we first analyze the singular behavior of the displacement of a linearly elastic body in dimension 2 close to the tip of a \(C^\infty\) crack, extending the well-known results for straight fractures [4]. As conjectured by Griffith [3], the displacement behaves as the sum of an \(H^2\)-function and a linear combination of two singular functions, whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimal fracture elongation and we compute the energy release rate, enlightening its dependence on the stress intensity factors [2]. In the last part of the talk we present the generalization to \(C^{1,1}\) fractures and an application to crack evolution [1].

References

1. S. Almi, G. Lazzaroni, I. Lucardesi, Paper in preparation.
2. S. Almi, I. Lucardesi, Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks, Nonlinear Differ. Equ. Appl. 25 (2018).
3. A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Royal Soc. A 221 (1921), 163-198.
4. P. Grisvard, Singularities in Boundary Value Problems, Research in Applied Mathematics 22 Springer, Berlin, 1992.

Junling Ma, University of Victoria, Canada

HOST CONTACT STRUCTURE IS IMPORTANT FOR THE RECURRENCE OF INFLUENZA A

Joint work with Juan M. Jaramillo, Pauline van den Driessche, Sanling Yuan

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

An important characteristic of influenza A is its ability to escape host immunity through antigenic drift. Individuals infected by a strain of influenza A during an epidemic have decreased immunity (estimated to be 12-25% in literature) to the drifted strains. In this talk, we compute the required decrease in immunity so that a drifted strain can invade after a pandemic. By comparing the model predictions of a homogeneously mixing mode, a heterogeneously mixing model, and a network model, we show that, for realistic loss of immunity, the establishment of a drift strain is only possible on contact networks. This suggests that stable contacts like classmates, coworkers and family members are a crucial path for the spread of influenza in human populations.

Dominika Machowska, University of Łódź, Poland

COMPETITION IN DEFENSIVE AND OFFENSIVE ADVERTISING STARTEGIES IN A SEGMENTED MARKET

Joint work with Andrzej Nowakowski

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building B-8, room 0.18.

Abstract

We propose the new goodwill model à la Nerlove-Arrow defined on a competitive segmented market. Based on the dual dynamic approach, we give the sufficient condition under which the open-loop equilibrium exists for the new game. We also introduce \(\varepsilon\)-open loop equilibrium as a basis for the numerical algorithm using a construction of the optimal solution in the finite steps. The numerical algorithm enables an analysis of how the level of the homogeneity of given competitive products and customer recommendations modify optimal goodwill and the total profit of each player.

Ezequiel Maderna, University of the Republic, Uruguay

HYPERBOLIC EXPANSIONS WITH ARBITRARY LIMIT SHAPE

Joint work with Andrea Venturelli

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

A well-known fact in the classical \(N\)-body problem is that if we normalize by the size of the configuration a completely parabolic motion, then the normalized configuration converge to the set of central configurations.

We will show that there is no such restriction for motions with positive energy. Moreover, we will show the existence of hyperbolic motions with arbitrarily chosen limit shape, and this for any given initial configuration of the bodies. The energy level \(h>0\) of the motion can also be chosen arbitrarily. The proof uses variational methods and represents a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

Kyoko Makino, Michigan State University, USA

RIGOROUS GLOBAL SEARCH, DETERMINATION OF MANIFOLDS AND THEIR HOMOCLINIC POINTS, AND ENTROPY ESTIMATES

Joint work with Martin Berz

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

Taylor models provide enclosures of functions over a domain within a relaxation band of their Taylor expansion around a point inside the domain. They are obtained automatically by evaluating the code list of the underlying function in Taylor model arithmetic, and under minimal requirements on the underlying floating  point arithmetic, the enclosures are mathematically rigorous. The widths of the resulting band scales with a high order of the width of the domain, and so in practice enclosures are obtained that are usually much sharper than those from conventional rigorous methods like intervals, centered forms, and related linearizations for all but the simplest cases. In general, the complexity and nonlinearity of the underlying function dictates optimum order and domain widths to achieve a desired accuracy. The resulting rigorous relaxations can be used for the local description of objective functions and constraints in rigorous global search. Furthermore, the resulting representations can be used efficiently for higher order domain reduction based on conditions on the objective function and the constraints.

  The methods can be applied to various problems in dynamical systems. First, Taylor models allow for the computation of tight enclosures of manifolds of dynamical systems. Once these enclosures are given, they can be used with the Taylor model-based rigorous global optimizer to find and isolate all homoclinic points. From these it is possible to determine so-called homoclinic tangles, which contain information on lower bounds of topological entropy of the underlying systems. Various examples of the practical use of the methods are given.

Agnieszka Malinowska, Bialystok University of Technology, Poland

OPTIMAL CONTROL OF FRACTIONAL MULTI-AGENT SYSTEMS

Joint work with Tatiana Odzijewicz

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We deal with control strategies for discrete-time fractional multi-agent systems. By using the discrete fractional order operator we introduce memory effects to the considered problem. Necessary optimality conditions for discrete-time fractional optimal control problems with single- and double-summator dynamics are proved. We demonstrate the validity of the proposed control strategy by numerical examples.

A.B. Malinowska is supported by the Bialystok University of Technology grant S/WI/1/2016 and funded by the resources for research by Ministry of Science and Higher Education.

Jens Marklof, University of Bristol, UK

KINETIC THEORY FOR THE LOW-DENSITY LORENTZ GAS

Joint work with Andreas Strombergsson

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strombergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

David Martí-Pete, Polish Academy of Sciences, Poland

FINGERS IN THE PARAMETER SPACE OF THE COMPLEX STANDARD FAMILY

Joint work with Mitsuhiro Shishikura

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

We investigate the parameter space of the complex standard family \[F_{\alpha,\beta}(z)=z+\alpha+\beta \sin z,\] where the parameter \(0<\beta\ll 1\) is considered to be fixed and the bifurcation is studied with respect to the parameter \(\alpha\in\mathbb{C}\). This two parameter family of entire functions are lifts of holomorphic self-maps of \(\mathbb{C}^*\) that arise as the complexification of the Arnol'd standard family of circle maps. In the real axis of the \(\alpha\)-parameter plane one can observe the so-called Arnold tongues, given by the real parameters \((\alpha, \beta)\) such that \(F_{\alpha,\beta}\) has a constant rotation number. Their complex extension contain some finger-like structures which were observed for the first time by Fagella in her PhD thesis [1] that increase in number as \(\beta\to 0\). We study the qualitative and quantitative aspects of such fingers via parabolic bifurcation. In particular, we show that for every \(0<\beta\ll 1\) the number of fingers is finite and give an estimate of this quantity as \(\beta \to 0\). This is a very general capture phenomenon that appears in the parameter spaces of many families of holomorphic functions with more than one critical point.

References

1. N. Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl. 229(1) (1999), 1-31.

Bohdan Maslowski, Charles University, Czech Republic

FILTERING AND OPTIMAL CONTROL AND FOR GAUSS-VOLTERRA PROCESSES IN HILBERT SPACES

Joint work with Tyrone E. Duncan, Bożenna Pasik-Duncan and Vít Kubelka

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

Kalman-Bucy type filter and some methods of parameter estimation are studied in the case when signals are Hilbert space-valued Gaussian processes.  The corresponding integral equations are derived for the optimal estimate and covariance of the error. Some basic properties of the filter are discussed. These general results are illustrated by examples  of  linear SPDEs where the noise terms are Gauss-Volterra processes (in particular, fractional Brownian motions). Also, some optimal control results for such systems are recalled for the case of quadratic cost functionals. In both cases, the results are compared with the standard ones for Gauss-Markov systems.

Serena Matucci, University of Florence, Italy

ASYMPTOTIC PROBLEMS FOR SECOND ORDER NONLINEAR DIFFERENCE EQUATIONS WITH DEVIATING ARGUMENT

Joint work with Zuzana Došlá

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-7, room 2.2.

Abstract

A fixed point approach, based on Schauder linearization device in the Frechét space of all the sequences, is presented and compared with corresponding results in the space of continuous functions, extending results in [3]. As an applications, the problem of the existence of the so- called intermediate solutions is analyzed for the half-linear and sublinear Emden-Fowler type equations with deviating argument \[ \Delta(a_n |\Delta x_n|^\alpha \, \mathrm{ sgn } \, \Delta x_n) + b_n |x_{n+q}|^\beta \, \mathrm{ sgn } \, x_{n+q}=0,\tag{1} \] where \(\Delta\) is the forward difference operator, \(a=\{a_n\}, \, b=\{b_n\}\) are positive real sequences, \(0< \beta \le \alpha\) and \(q \in \mathbb Z\). In particular, we analyze the effect of the deviating argument on the existence of unbounded nonoscillatory solutions for (1), by means of a comparison with the equation \[ \Delta(a_n |\Delta y_n|^\alpha \, \mathrm{ sgn } \, \Delta y_n) + b_n |y_{n+1}|^\beta \, \mathrm{ sgn } \, y_{n+1}=0.\tag{2} \] As a consequence, necessary and sufficient conditions for the existence of intermediate solutions for (1) (that is, eventually positive solutions \(x\) s.t. \(\lim_n x_n=+\infty\), \(\lim_n a_n |\Delta x_n|^\alpha=0\)) are given. The results presented generalize some in [1] in case \(\alpha=\beta\), and in [2] when \(\alpha>\beta\).

References

1. M. Cecchi, Z. Došlá, M. Marini, On the growth of nonoscillatory solutions for difference equations with deviating argument, Adv. Difference Equ. (2008), Art. ID 505324, 15 pp.
2. M. Cecchi, Z. Došlá, M. Marini, Intermediate solutions for nonlinear difference equations with p- Laplacian, Adv. Stud. Pure Math., 53 (2009), 33—40.
3. M. Marini, S. Matucci, P. Řehàk, Boundary value problems for functional difference equations on infinite intervals, Adv. Difference Equ. (2006), Art. 31283, 14 pp.

John Mayer, University of Alabama at Birmingham, USA

CRITICAL PORTRAITS, SIBLING PORTRAITS, THE CENTRAL STRIP, AND NEVER CLOSE SIDES OF POLYGONS IN LAMINATIONS

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

Laminations of the unit disk were introduced by William Thurston as a topological/combinatorial vehicle for understanding the (connected) Julia sets of polynomials, and, in particular, the parameter space of quadratic polynomials. Though the problem that Thurston was interested in has not been solved, the local connectedness of the Mandelbrot set (the analytic parameter space of quadratic polynomials), his excursion into laminations eventually gave birth to laminations as a way of understanding higher degree polynomials and their corresponding laminations. Much work has been done for cubic polynomials and their parameter spaces (analytic and laminational). In this talk we will decribe some work in progress on understanding phenomena that can occur with laminations, and consequently with Julia sets (maybe), of higher degree, \(d\ge 3.\) In particular, we are interested in laminational phenomena that cannot occur for \(d=2\), but can occur for \(d=3,\) cannot occur for \(d\le 3,\) but can occur for \(d=4,\) and so on. The topics mentioned in the title are on the route of discovery.

References

1. A. Blokh, D. Mimbs, L. Oversteegen, K. Valkenburg, Laminations in the language of leaves, Trans. Amer. Math. Soc. 365 (2013), 5367-5391.
2. A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen, D. Parris, Rotational subsets of the circle under \(z^n\), Topology and Its Appl. 153 (2006), 1540-1570.
3. D. Childers, Wandering polygons and recurrent critical leaves, Ergodic Theory Dynam. Systems 27(1) (2007), 87-107.
4. D. Cosper, J. Houghton, J. Mayer, L. Mernik, J. Olson, Central Strips of sibling leaves in laminations of the unit disk, Topology Proc. 48 (2016), 69-100.
5. J. Mayer, L. Mernik, Central Strip Portraits, preprint (2017).

Dario Mazzoleni, Catholic University of Brescia, Italy

ASYMPTOTIC SPHERICAL SHAPES IN SPECTRAL OPTIMIZATION PROBLEMS

Joint work with Benedetta Pellacci and Gianmaria Verzini

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 0.18.

Abstract

We study the positive principal eigenvalue of a weighted problem associated with the Neumann-Laplacian settled in a box \(\Omega\subset \mathbb{R}^N\), which arises from the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the sign-changing weight, one is lead to consider a shape optimization problem, which is known to admit spherical optimal shapes only in trivial cases. We investigate if spherical shapes can be recovered in the limit when the negative part of the weight diverges. First of all, we show that the shape optimization problem appearing in the limit is the so called \(\textit{spectral drop}\) problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Thanks to \(\alpha\)-symmetrization techniques on cones, it can be proved that optimal shapes for the spectral drop problem are spherical for suitable choices of the box, the most interesting case being when \(\Omega\) is a convex polytope, and in this case a quantitative analysis of the convergence can be performed. Finally, for a smooth \(\Omega\), we show that small volume spectral drops are asymptotically spherical, with center at points with high mean curvature.

References

1. D. Mazzoleni, B. Pellacci, G. Verzini, Asymptotic spherical shapes in some spectral optimization problems, Preprint arXiv:1811.01623.
2. D. Mazzoleni, B. Pellacci, G. Verzini, Quantitative analysis of a singularly perturbed shape optimization problem in a polygon, Preprint arXiv:1902.05844.

Alberto Mercado, Federico Santa María Technical University, Chile & Institut de Mathématiques de Toulouse, France

BOUNDARY CONTROLLABILITY OF A COUPLED FOURTH-SECOND ORDER PARABOLIC SYSTEM

Joint work with Nicolás Carreño and Eduardo Cerpa

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

We study a control system coupling fourth and second-order parabolic equations, when we only control the second-order partial differential equation through a boundary condition \(h\): \[ \left\lbrace \begin{array}{ll} u_t(t,x) + u_{xxxx}(t,x) = v(t,x),&t\in(0,T),\,x\in(0,\pi), \\ v_t(t,x) - d v_{xx}(t,x) = 0, &t\in(0,T),\,x\in(0,\pi), \\ u(t,0)=u_{xx}(t,0) = 0,&t\in(0,T),\\ u(t,\pi)=u_{xx}(t,\pi) = 0,&t\in(0,T), \\ v(t,0) = h(t),\, v(t,\pi)=0,&t\in(0,T). \end{array}\right.\tag{1}\]

Following the methods introduced in [1], we obtain positive and negative results for approximate- and null-controllability, depending on the diophantine approximation properties of the diffusion coefficient \(d\). In particular, we prove that, if \(\sqrt d\) has finite irrationality measure (also called Liouville-Roth constant), then system (1) is null controllable in any time \(T>0\).

References

1. F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, and L. de Teresa, Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal. 267 (2014), 2077-2151.
2. N. Carreño, E. Cerpa, and A. Mercado, Boundary controllability of a cascade system coupling fourth- and second-order parabolic equations, Preprint.

Alpár R. Mészáros, University of California, Los Angeles, USA

SOBOLEV ESTIMATES FOR FIRST ORDER MEAN FIELD GAMES AND PLANNING PROBLEMS

Joint work with Jameson Graber, Francisco Silva, and Daniela Tonon

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

In this talk we discuss Sobolev estimates for weak solutions of first order variational Mean Field Game systems (in the sense of Lasry-Lions) with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity conditions on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates in the space variable are fully global in time, while the ones involving the time variable are local in time. In the same time we show how to obtain the same estimates for the mean field planning problem (introduced by P.-L. Lions). Our methods have their roots in Brenier's work on the regularity of the pressure field arising in weak solutions of the incompressible Euler equations (see [2]), which was improved later by Ambrosio-Figalli in [1]. The talk is based on the works [3, 4].

References

1. L. Ambrosio, A. Figalli, On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations 31(4) (2008), 497–509.
2. Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Commun. Pure Appl. Math. 52(4) (1999), 411–452.
3. P.J. Graber, A.R Mészáros, Sobolev regularity for first order Mean Field Games, Ann. Inst. H. Poincaré Anal. Non Linéaire 35(6) (2018), 1557–1576.
4. P.J. Graber, A.R Mészáros, F.J. Silva, D. Tonon, The planning problem in Mean Field Games as regularized mass transport, Calc. Var. Partial Differential Equations (2019), to appear.

Małgorzata Migda, Poznań University of Technology, Poland

ASYMPTOTIC PROPERTIES OF SOLUTIONS TO SUM-DIFFERENCE EQUATIONS OF VOLTERRA TYPE

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

Volterra difference equations appeared as a discretization of Volterra integral and integro-differential equations. They also often arise during the mathematical modeling of some real life situations where the current state is determined by the whole previous history. In this talk we consider some difference equations of Volterra type. In particular we discuss the equations of the form \[ \Delta(r_n\Delta x_n)=b_n+\sum_{k=1}^{n}K(n,k)f(x_k). \] We give sufficient conditions for the existence of a solution \(x\) of the above equation with the property \[ x_n=y_n+{\mathrm{o}} (n^s), \] where \(y\) is a given solution of the equation \(\Delta(r_n\Delta y_n)=b_n\) and \(s\in(-\infty,0]\). We show also applications of the obtained results to a linear Volterra equation. Sufficient conditions for the existence of asymptotically periodic solutions will be discussed as well.

References

1. J. Migda, M. Migda, Asymptotic behavior of solutions of discrete Volterra equations, Opuscula Math. 36 (2016), 265-278.
2. J. Migda, M. Migda, M. Nockowska-Rosiak, Asymptotic properties of solutions to second-order difference equations of Volterra type, Discrete Dynamics in Nature and Society (2018), Article ID 2368694, 10 pp.
3. J. Migda, M. Migda, Z. Zbąszyniak, Asymptotically periodic solutions of second order difference equations, Appl. Math. Comput. 350 (2019), 181-189.

Jason D. Mireles James, Florida Atlantic University, USA

VALIDATED NUMERICS FOR STABLE/UNSTABLE MANIFOLDS OF DELAY DIFFERENTIAL EQUATIONS

Joint work with Jean-Philippe Lessard and Olivier Hénot

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

The parameterization method is a general functional analytic framework for studying invariant manifolds. The idea is to formulate a chart or covering map for the manifold as the solution of an appropriate invariance equation. Studying the invariance equation leads to both numerical schemes for approximating the invariant manifold and to a posteriori methods for quantifying discretization and truncation errors. This talk considers the parameterization method for unstable manifolds of delay differential equations (DDEs), focusing on the numerical implementation as well as the derivation of mathematically rigorous computer assisted error bounds. One challenge is the fact that the solution of a DDE depends on both present and past states, so that a DDE generates an infinite dimensional dynamical system. The invariant manifolds studied here play an important role in describing the global dynamics of this system.

Michał Misiurewicz, Indiana University - Purdue University Indianapolis, USA

RENORMALIZATION TOWERS AND THEIR FORCING

Joint work with Alexander Blokh

Date: 2019-09-16 (Monday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

Over half a century ago, Sharkovsky proved his theorem about periodsof periodic orbits of continuous interval maps. Existence of someperiods force existence of some other periods, and the orderingobtained in such a way is linear. Later, people noticed that insteadof looking at periods, one can take into account permutations.Unfortunately, this gives only a partial order, which is verycomplicated and impossible to describe in simple terms. We propose themiddle ground: to look at the block structures of permutations (theycan be also understood in terms of renormalizations). This is a finerclassification of periodic orbits than just by periods, but stillresults in a linear ordering.

Reza Mohammadpour, Polish Academy of Sciences, Poland

ON HAUSDORFF DIMENSION OF THIN NONLINEAR SOLENOIDS

Joint work with Michał Rams and Feliks Przytycki

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-4, room 120.

Abstract

Let \(M=S^1\times \mathbb{D}\) be the solid torus, where \(\mathbb{D}=\{ v\in \mathbb{R}^{2} | |v|<1 \}\) carries the product distance \(d=d_{1}\times d_{2}\), and suppose \(f:M\rightarrow M\) such that \[ (x,y,z)\mapsto (\eta(x,y,z)\! \!\!\!\mod 2\pi , \lambda(x,y,z)+u(x) , \nu(x,y,z)+v(x)) \tag{1}\] is a smooth embedding map.

Bothe [1] was the first who obtained results on the dimension of the attractor of a thin linear solenoid where contraction rates are strong enough. Barriera, Pesin and Schemeling [2] established a dimension product structure of invariant measures in the course of proving the Eckmann Ruelle conjecture.

Conjecture. The fractal dimension of a hyperbolic set is (at least generically or under mild hypotheses) the sum of those of its stable and unstable slices, where fractal can mean either Hausdorff or upper box dimension.

In spite of the difficulties due to possible low regularity of the holonomies, indeed, Schmeling [4] found that solenoids often lack regular holonomies but the set of non-liptchitz points seemed to be rather small in the measure scene. Hasselblat and Schmeling [3] proved the conjecture for a class of thin linear solenoids. We prove the conjecture for a class of thin nonlinear solenoids of map (1).

References

1. H. Bothe, The dimension of some solenoids, Ergodic Theory and Dynamical Systems, 15 (1995), 449-474.
2. L. Barreira, Y. Pesin, J. Schmeling, Dimension and product structure of hyperbolic measures, Annals of Mathematics, 3 (1999), 755-783.
3. B. Hasselblatt, J. Schmeling, Dimension product structure of hyperbolic sets, In Modern dynamical systems and applications, 331-345. Cambridge Univ. Press, Cambridge, 2004.
4. J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets, II. Math. Nachr. 170 (1994), 211-225.

Janusz Morawiec, University of Silesia in Katowice, Poland

ON BETWEENNESS-PRESERVING MAPPINGS

Joint work with Wiesław Kubiś and Thomas Zürcher

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 1.8.

Abstract

We are interested in a Euclidean version of betweenness. We say that a point \(z\) is between two points \(x\) and \(y\) if and only if \(z\) is in the convex hull of \(x\) and \(y\). In this setting, we call a betweenness-preserving map monotone. The aim of this talk is to present regularity results for monotone mappings in the plane.

Clément Mouhot, University of Cambridge, UK

QUANTITATIVE LINEAR STABILITY (HYPOCOERCIVITY) FOR CHARGED PARTICLES IN A CONFINING FIELD

Joint work with K. Carrapatoso, J. Dolbeault, F. Hérau, S. Mischler, and C. Schmeiser

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building B-8, room 0.10b.

Abstract

We report on recent joint results in which we develop quantitative methods for proving the existence of a spectral gap and estimating the gap, for hypocoercive kinetic equations that combine the local conservation laws of fluid mechanics and a confining potential force. The proofs involve a cascade of correctors and global commutator estimates, as well as new quantitative inequalities of Korn type. The latter extend to the case of the whole space with a potential force the classical Korn inequality in a bounded domain of elasticity theory. These results are a step towards constructing global solutions near equilibrium to the full nonlinear Boltzmann equation for charged particles subject to a confining potential.

Iván Moyano, University of Cambridge, UK

SPECTRAL INEQUALITIES FOR THE SCHRÖDINGER OPERATOR \(-\Delta_x + V(x)\) in \(\mathbb{R}^d\)

Joint work with Gilles Lebeau

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building B-8, room 0.18.

Abstract

In this talk, we will first review some classical results on the so-called ’spectral inequalities’, which yield a sharp quantification of the unique continuation of the spectral family associated with the Laplace-Beltrami operator in a compact manifold. In a second part, we will discuss how to obtain the spectral inequalities associated to the Schrodinger operator \(-\Delta_x + V(x)\), in \(\mathbb{R}^d\), in any dimension \(d\geq 1\), where \(V=V(x)\) is a real analytic potential. In particular, we can handle some long- range potentials.

Boris Muha, University of Zagreb, Croatia

EXISTENCE AND REGULARITY FOR THE NON-LINEAR KOITER SHELL INTERACTING WITH THE \(3D\) INCOMPRESSIBLE FLUID

Joint work with Sebastian Schwarzacher

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 3.22.

Abstract

We study the unsteady Navier Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter Type. The latter one constitutes a moving part of the boundary of the physical domain of the fluid. This leads to a coupled system of non-linear PDEs with the moving boundary. We study weak solution to the corresponding fluid-structure interaction (FSI) problem. We introduce new methods that allow to prove higher regularity estimates for the shell. Due to the improved regularity estimates it is then possible to extend the known existence theory of weak solutions to the FSI problem with non-linear Koiter shell. The regularity result holds for arbitrary weak solution under certain geometric condition on the deformation of the boundary.

Michael Multerer, USI Lugano, Switzerland

MULTILEVEL QUADRATURE FOR ELLIPTIC PARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Michael Griebel and Helmut Harbrecht

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor product approximation between the spatial variable and the parameter. In this presentation, we employ this fact and reverse the multilevel quadrature method via the sparse grid construction by applying differences of quadrature rules to finite element discretizations of increasing resolution. Besides being algorithmically more efficient if the underlying quadrature rules are nested, this way of performing the sparse tensor product approximation enables the easy use of non-nested and even adaptively refined finite element meshes. Especially, we present an error and regularity analysis of the fully discrete solution, taking into account the effect of polygonal approximations to a curved physical domain and the numerical approximation of the bilinear form. Numerical results in three spatial dimensions are provided to illustrate the approach.

References

1. M. Griebel, H. Harbrecht and M. Multerer, Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains, arXiv:1509.09058, 2018.

Cosimo Munari, Universität Zürich, Switzerland

DUALITY FOR RISK FUNCTIONALS ON ORLICZ SPACES

Joint work with Niushan Gao, Denny Leung, Foivos Xanthos

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

A well-known result by Delbaen states that every convex risk measure defined on the space of bounded positions is automatically weak-star lower semicontinuous whenever it satisfies the Fatou property. This allows to derive a nice dual representation where the constraint set of dual elements consists of countably-additive measures. This result is no longer true if one abandons the bounded setting. The objective of the talk is to show that a dual representation with countably-additive measures holds in a general Orlicz space if the risk measure is additionally assumed to be either law invariant (the risk measure depends only on the probability law of a risky position) or surplus invariant (the risk measure depends, in a suitable way, only on the downside of a risky position).

References

1. N. Gao, D. Leung, C. Munari, F. Xanthos, Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces, Finance and Stochastics 22 (2018), 395–415.
2. N. Gao, C. Munari, Surplus-invariant risk measures, Mathematics of Operations Research, to appear.

Gulcin M. Muslu, Istanbul Technical University, Turkey

THE GENERALIZED FRACTIONAL BENJAMIN-BONA-MAHONY EQUATION: ANALYTICAL AND NUMERICAL RESULTS

Joint work with Goksu Oruc and Handan Borluk

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 3.21.

Abstract

The generalized fractional Benjamin-Bona-Mahony (gfBBM) equation models the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The equation involves two fractional terms unlike the well-known fBBM equation. In this talk, we present the local existence and uniqueness of the solutions for the Cauchy problem. The sufficient conditions for the existence of solitary wave solutions are discussed. The Petviashvili method is proposed for the generation of the solitary wave solutions and their evolution in time is investigated by Fourier spectral method, numerically. The efficiency of the numerical methods is tested and the relation between nonlinearity and fractional dispersion will be presented by various numerical experiments.

This work was supported by Research Fund of the Istanbul Technical University. Project Number:42257.

Akambadath Keerthiyil Nandakumaran, Indian Institute of Science, India

ASYMPTOTIC ANALYSIS OF OPTIMAL CONTROL PROBLEMS IN OSCILLATORY DOMAINS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 0.18.

Abstract

Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the present application is much far and wide. It has applications in composite media, porous domains, laminar structures, domains with rapidly oscillating boundaries, to name a few. The PDE problems posed on such complicated domains lead to the analysis of homogenization. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic analysis. There are various methods developed in the last 50 years to understand the mathematical homogenization theory.

In this talk, we discuss the asymptotic analysis of various optimal control problems defined in domains who boundary is rapidly (highly) oscillating. Such complex domains appears in many real life applications like heat radiators, flows in channels with rough boundaries, propagation of electro-magnetic waves in regions having rough interface, absorption diffusion in biological structures, acoustic vibrations in medium with narrow channels etc. In the first part, we briefly present the work which we are carrying out in my group (see and later we present some specific results. We introduce the so called unfolding operators which we have developed for the problems under study through which we characterize the optimal controls. Finally, we do a homogenization process and obtain the limit control problem.

References

1. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Locally periodic unfolding operator for highly oscillating rough domains, Annali di Matematica Pura ed Applicata (1923 -), https://doi.org/10.1007/s10231-019-00848-7.
2. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Semi-linear optimal control problem on a smooth oscillating domain, Communications in Contemporary Mathematics, DOI: 10.1142/S0219199719500299 (26 pages).
3. R. Mahadevan, A.K. Nandakumaran, R. Prakash, Homogenization of an elliptic equation in a domain with oscillating boundary with non-homogeneous non-linear boundary conditions, Appl. Math. Optim., https://doi.org/10.1007/s00245-018-9499-4 (34 pages).
4. S. Aiyappan, A.K. Nandakumaran, R. Prakash, Generalization of Unfolding Operator for Highly Oscillating Smooth Boundary Domains and Homogenization, Calculus of Variations and PDE, (2018), 57-86.
5. S. Aiyappan, A.K. Nandakumaran, Optimal Control Problem in a Domain with Branched Structure and Homogenization, Mathematical Methods in Applied Sciences 40(8) (2017), 3173-3189.
6. A.K. Nandakumaran, R. Prakash and B.C. Sarda, Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization, SIAM Journal on Control and Optimization 53(5) (2015), 3245-3269.

Péter Nándori, Yeshiva University, USA

THE LOCAL LIMIT THEOREM FOR HYPERBOLIC DYNAMICAL SYSTEMS AND APPLICATIONS

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-4, room 120.

Abstract

We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that the MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Examples include reward renewal processes, Axiom A flows, as well as the systems admitting Young's tower, such as Sinai's billiard with finite horizon, suspensions over Pomeau-Manneville maps and geometric Lorenz attractors. Then we discuss two applications in infinite ergodic theory. First, we prove the mixing of global observables by some infinite measure preserving hyperbolic systems that are well approximated by periodic systems (examples include billiards with small potential field and various ping pong models). Here, global observables are functions that are not integrable with respect to the infinite invariant measure, but have convergent average values over large boxes. Second, we discuss the Birkhoff theorem for such global observables in the simplest case: iid random walks. The talk is based on joint work with Dmitry Dolgopyat and in parts with Marco Lenci.

References

1. D. Dolgopyat, M. Lenci, P. Nándori, Global observables for random walks: law of large numbers, http://arxiv.org/abs/1902.11071 (2019).
2. D. Dolgopyat, P. Nándori, Infinite measure mixing for some mechanical systems, http://arxiv.org/abs/1812.01174 (2018).
3. D. Dolgopyat, P. Nándori, On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory and Dynamical Systems, to appear, https://doi.org/10.1017/etds.2018.29.

Marcin Napiórkowski, University of Warsaw, Poland

NORM APPROXIMATION FOR MANY-BOSON QUANTUM DYNAMICS

Joint work with Christian Brennecke, Phan Thành Nam, and Benjamin Schlein

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 0.10b.

Abstract

Because of the complexity of the many-body Schrödinger equation, to gain insight into the properties of many-body quantum systems it is necessary to use effective theories.

In my talk, I will review recent advances [1, 2] in the derivation of effective equations that govern the dynamics of Bose-Einstein condensates.

References

1. C. Brennecke, P.T. Nam, M. Napiórkowski, B. Schlein, Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, in press.
2. P.T. Nam, M. Napiórkowski, Norm approximation for many-body quantum dynamics: focusing case in low dimensions, Adv. Math. 350, 547-587.

Markus Neher, KIT, Germany

INTERVAL AND TAYLOR MODEL METHODS FOR ODES

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 2.4.

Abstract

Verified integration methods for ODEs are methods that compute rigorous bounds for some specific solution or for the flow of some initial set of a given ODE. Interval arithmetic has been used for calculating such bounds for solutions of initial value problems. The origin of these methods dates back to Ramon Moore [2].

Unfortunately, interval methods sometimes suffer from overestimation. This can be caused by the \(\textit{dependency problem}\), that is the lack of interval arithmetic to identify different occurrences of the same variable, and by the \(\textit{wrapping effect}\), which occurs when intermediate results of a calculation are enclosed into intervals. In verified integration this happens when enclosures of the flow at intermediate time steps of the interval of integration are computed. Overestimation may then degrade the computed enclosure of the flow, enforce miniscule step sizes, or provoke premature abortion of the integration.

Taylor models, developed by Martin Berz in the 1990s, combine interval arithmetic with symbolic computations [1]. A Taylor model consists of a multivariate polynomial and a remainder interval. In all computations, the polynomial part is handled by symbolic calculations, which are essentially unaffected by the dependency problem or the wrapping effect. Only the interval remainder term and polynomial terms of high order, which are usually small, are bounded using interval arithmetic. Taylor models also benefit from their capability of representing non-convex sets. For nonlinear ODEs, this increased flexibility in admissible boundary curves for the flow is an intrinsic advantage over traditional interval methods.

In our talk, we analyze Taylor model mehods for the verified integration of ODEs and compare these methods with interval methods.

References

1. M. Berz, From Taylor series to Taylor models, AIP Conference Proceedings 405 (1997), 1-23.
2. R.E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, N.J., 1966.

Michael Neilan, University of Pittsburgh, USA

EXACT SEQUENCES OF PIECEWISE POLYNOMIALS ON ALFELD SPLITS

Joint work with Guosheng Fu and Johnny Guzmán

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 3.21.

Abstract

We develop exact polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an \(n\)-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.

Robin Neumayer, Northwestern University, USA

ANISOTROPIC LIQUID DROP MODELS

Joint work with Rustum Choksi and Ihsan Topaloglu

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 0.10a.

Abstract

Anisotropic surface energies are a natural generalization of the perimeter functional that arise, for instance, in scaling limits for certain probabilistic models on lattices. Smoothness and ellipticity assumptions are sometimes imposed on the energy to improve analytic aspects of associated isoperimetric problems, but these assumptions are not always desirable for some applications nor checkable when the problem comes from a scaling limit. We consider an anisotropic variant of a model for atomic nuclei and show that minimizers behave in a fundamentally different way depending on whether or not the energy is smooth and elliptic. This is joint work with Choksi and Topaloglu.

Daniel Nicks, University of Nottingham, UK

THE ITERATED MINIMUM MODULUS AND EREMENKO’S CONJECTURE

Joint work with Phil Rippon and Gwyneth Stallard

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

Eremenko has conjectured that for any transcendental entire function \(f\), the escaping set \(I(f):= \{ z : f^n(z)\to\infty \mbox{ as } n\to\infty\}\) is connected. This talk will focus on real entire functions of finite order with only real zeroes. We show that Eremenko's conjecture holds for such a function \(f\) (and in fact \(I(f)\) has a "spider's web" structure) if there exists \(r>0\) such that the iterated minimum modulus \(m^n(r)\to\infty\) as \(n\to\infty\). Here \(m(r):=\min_{|z|=r}|f(z)|\). We will briefly discuss examples of families of functions for which this minimum modulus condition does, and does not, hold.

Barbara Niethammer, Universität Bonn, Germany

DYNAMIC SCALING IN COAGULATION EQUATIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

In 1916 Smoluchowski derived a mean-field model for mass aggregation in order to develop a mathematical theory for coagulation processes. Since Smoluchowski's groundbreaking work his model and variants of it including fragmentation terms have been used in a diverse range of applications such as aerosol physics, polymerization, population dynamics, or astrophysics.

In this talk I will give an overview on recent work, studying the long-time behaviour of such equations. A key question is that of dynamic scaling, that is whether solutions develop a universal self-similar form for large times. This issue is only understood for some exactly solvable cases, while in the general case most questions are still completely open. I will give an overview of the main results in the past decades and explain why we believe that in general the scaling hypothesis is not true.

Fabio Nobile, Ecole Polytechnique Fédérale de Lausanne, Switzerland

DYNAMICAL LOW RANK APPROXIMATION OF RANDOM TIME DEPENDENT PDES

Joint work with Yoshihito Kazashi and Eva Vidlicková

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Partial differential equations with stochastic coefficients and input data arise in many applications in which the data of the PDE need to be described in terms of random variables/fields due either to a lack of knowledge of the system or to its inherent variability. The numerical approximation of statistics of the solution poses several challenges when the number of random parameters is large and/or the parameter-to-solution map is complex, and effective surrogate or reduced models are of great need in this context.

In this talk we consider time dependent PDEs with few random parameters and seek for an approximate solution in separable form that can be written at each time instant as a linear combination of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation) in the spirit of a truncated Karhunen-Loève expansion. Since the optimal deterministic and stochastic modes can significantly change over time, we consider here a dynamical approach where those modes are computed on the fly as solutions of suitable evolution equations. From a geometrical point of view, this corresponds to constraining the original dynamics to the manifold of fixed rank functions, i.e. functions that can be written in separable form with a fixed number of terms. Equivalently, the original equations are projected onto the tangent space to the manifold of fixed rank functions along the approximate trajectory, similarly to the Dirac-Frenkel variational principle in quantum mechanics.

We discuss the construction of the method, present an existence result for a random semi-linear evolutionary equation, and discuss practical numerical aspects for several time dependent PDEs with random parameters, including the heat equation with a random diffusion coefficient; the incompressible Navier-Stokes equations with random Dirichlet boundary conditions; the wave equation with random wave speed.

References

1. E. Musharbash, F. Nobile, T. Zhou, Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comp. 37(2) (2015), A776-A810.
2. E. Musharbash, F. Nobile, Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, J. Comput. Physics 354 (2018), 135-162.
3. E. Musharbash, F. Nobile, ymplectic dynamical low rank approximation of wave equations with random parameters, MATHICSE Technical Report 18.2017 École Polytechnique Fédérale de Lausanne.
4. Y. Kazashi, F. Nobile, Improved stability of optimal traffic paths, Existence of dynamical low rank approximation for random semi-linear evolutionary equations on the maximal interval, in preparation.

Andrzej Nowakowski, University of Łódź, Poland

CONTROL OF BLOWUP AND APPROXIMATE OPTIMALITY CONDITIONS FOR THE WAVE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 0.18.

Abstract

The model problem we study is a semilinear wave equation to which we add a control \(u\) in a source function and on the boundary: \[x_{tt} -\Delta x =f (t ,z ,x ,u)\;\;\text{in}\;\mathbb{R}^{ +} \times D , \] \[x (t ,z) =u(t,z)\;\;\text{\ }\;\text{on}\;\mathbb{R}^{ +} \times \; \partial D , \] \[x (0 ,z) =v_{0} (z), \ x_{t}(0,z)=v_{1}(z). \] This type of problems arise in a great variety of situations. In the paper we propose the problem of controlling a system which may blow up in finite time. We want to minimize the blowup time. To this effect sufficient optimality conditions for controlled blowup time are derived in terms of new dynamic programming methodology. We define \(\varepsilon \)-optimal value function and we construct sufficient \(\varepsilon \)-optimal conditions for that function again in terms of new dynamic programming inequality.

Tatiana Odzijewicz, SGH Warsaw School of Economics, Poland

OPTIMAL LEADER-FOLLOWER CONTROL FOR THE FRACTIONAL OPINION FORMATION MODEL

Joint work with Ricardo Almeida and Agnieszka B. Malinowska

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

This work deals with an opinion formation model, that obeys a nonlinear system of fractional-order differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader-follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented.

References

1. R. Almeida, A. B. Malinowska, T. Odzijewicz, Optimal leader–follower control for the fractional opinion formation model, J. Optimiz. Theory App. 182 (2019), 1171–1185.

Kazuki Okamura, Shinshu University, Japan

SOME RESULTS FOR CONJUGATE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

I will talk about conjugate maps between two iterated function systems driven by several weak contractions, depending on [3] and [4]. Specifically, a special case of our framework is as follows: Let \(X\) and \(Y\) be compact metric spaces. Let \(I\) be a finite set. Assume that for each \(i \in I\), weak contractions \(f_i : X \to X\) and \(g_i : Y \to Y\) are given. Consider the solution \(\varphi : X \to Y\) satisfying that \[\varphi (f_i (x)) = g_i (\varphi(x)), \ i \in I, x \in X.\tag{1}\]

Conjugate equations of this kind are a certain generalization of de Rham’s functional equations [5]. They are considered by Zdun [8], Girgensohn-Kairies-Zhang [2], Shi-Yilei [7], Serpa- Buescu [6] and Bárány-Kiss-Kolossváry [1].

I will mainly talk about regularity of a unique solution \(\varphi\) of (1). Then, I will give examples to which our results are applicable. If time is permitted, I will also discuss existence and uniqueness of a more general class of conjugate equations than above.

References

1. B. Bárány, G. Kiss, and I. Kolossváry, Pointwise regularity of parameterized affine zipper fractal curves, Nonlinearity, 31 (2018) 1705-1733.
2. R. Girgensohn, H.-H. Kairies, W. Zhang, Regular and irregular solutions of a system of functional equations, Aequationes Math. 72 (2006), 27-40.
3. K. Okamura, Some results for conjugate equations, to appear in Aequationes Math.
4. K. Okamura, Hausdorff dimensions for graph-directed measures driven by infinite rooted trees, preprint.
5. G. de Rham, Sur quelques courbes définies par des équations fonctionalles, Univ. E Politec. Horino. Rend. Sem. Mat. 16 (1957), 101-113.
6. C. Serpa, J. Buescu, Constructive solutions for systems of iterative functional equations, Constr. Approx. 45 (2017), 273-299.
7. Y.-G. Shi, T. Yilei, On conjugacies between asymmetric Bernoulli shifts, J. Math. Anal. Appl. 434 (2016) 209-221.
8. M. C. Zdun, On conjugacy of some systems of functions, Aequationes Math. 61 (2001) 239-254.

Guillaume Olive, Jagiellonian University in Kraków, Poland

MINIMAL CONTROL TIME FOR ONE-DIMENSIONAL FIRST-ORDER HYPERBOLIC SYSTEMS

Joint work with Long Hu

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building B-8, room 0.18.

Abstract

The goal of this talk is to present some recent results in [2] concerning the exact controllability of one-dimensional first-order linear hyperbolic systems when all the controls are acting on the same side of the boundary. We show that the minimal time needed to control the system is given by an explicit and easy-to-compute formula with respect to all the coupling parameters of the system. The proof relies on the introduction of a canonical UL-decomposition and the compactness-uniqueness method.

References

1. M. Duprez, G. Olive, Compact perturbations of controlled systems, Math. Control Relat. Fields 8 (2018), 397-410.
2. L. Hu, G. Olive, Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls, preprint (2019), 47 pages.
3. A.F. Neves, H.S. Ribeiro, and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal. 67 (1986), 320-344.
4. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), 639-739.
5. N. Weck, A remark on controllability for symmetric hyperbolic systems in one space dimension, SIAM J. Control Optim. 20 (1982), 1-8.

Dietmar Ölz, University of Queensland, Australia

MODELLING AND SIMULATION OF COLLECTIVE MIGRATION IN EPITHELIAL LAYERS

Joint work with Z. Neufeld, A. Czirok, and H. Khatae

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 1.26.

Abstract

The mechanochemical coupling between the cells aligned in epithelial layers impacts characteristic features of their collective migration. I will introduce a 1D model for collective cell migration in epithelial sheets in which the cytoskeletons of adjacent cells are coupled both, mechanically and through mechanochemical feedback. The travelling wave analysis of the mathematical model can be made explicit predicting a polarization wave and associated wave speed which we can be observed in experiments. Finally I will also talk about recent results based on particle simulations of collective cell migration.

Piotr Oprocha, AGH University of Science and Technology, Poland

ON THE ENTROPY CONJECTURE OF MARCY BARGE

Joint work with Jan Boroński and Jernej Činč

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

I shall discuss a positive solution to the following problem, obtained in a joint work with J. Boroński and J. Činč.

Question (M. Barge, 1989 [8]) Does there exist, for every \(r\in [0,\infty]\), a pseudo-arc homeomorphism whose topological entropy is \(r\)?

Until now all known pseudo-arc homeomorphisms have had entropy \(0\) or \(\infty\). Recall that the pseudo-arc is a compact and connected space (continuum) first constructed by Knaster in 1922 [6]. It can be seen as a pathological fractal. According to the most recent characterization [5] it is topologically the only, other than the arc, continuum in the plane homeomorphic to each of its proper subcontinua. The pseudo-arc is homogeneous [2] and played a crucial role in the classification of homogeneous planar compacta [4]. Lewis showed that for any \(n\) the pseudo-arc admits a period \(n\) homeomorphism that extends to a rotation of the plane, and that any \(P\)-adic Cantor group action acts effectively on the pseudo-arc [7] (see also [10]). We adapt Lewis' inverse limit constructions, by combining them with a Denjoy-Rees scheme [1] (see also [9], [3]). The positive entropy homeomorphisms that we obtain are periodic point free, except for a unique fixed point.

I am going to present various results related to the problem, to conclude with a discussion of its solution.

References

1. F. Béguin, S. Crovisier, F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. École Norm. Sup. 40 (2007), 251-308.
2. R.H. Bing, homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742.
3. J.P. Boroński, J. Kennedy, X. Liu, P. Oprocha, Minimal noninvertible maps on the pseudocircle, arXiv:1810.07688.
4. L.C. Hoehn, L.G. Oversteegen, A complete classification of homogeneous plane continua, Acta Math. 216 (2016), 177-216.
5. L.C. Hoehn, L.G. Oversteegen, A complete classification of hereditarily equivalent plane continua, arXiv:1812.08846.
6. B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
7. W. Lewis, Periodic homeomorphisms of chainable continua, Fund. Math. 117 (1983), 81-84.
8. W. Lewis, Continuum theory and dynamics problems, Continuum theory and dynamical systems (Arcata, CA, 1989), 99–101, Contemp. Math., 117, Amer. Math. Soc., Providence, RI, 1991.
9. M. Rees, A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 2 (1981), 537-550.
10. J. Toledo, Inducible periodic homeomorphisms of tree-like continua, Trans. Amer. Math. Soc. 282 (1984), 77–108.

Giandomenico Orlandi, University of Verona, Italy

ENERGY MINIMIZING MAPS INTO FINSLER MANIFOLDS AND OPTIMAL ONE-DIMENSIONAL NETWORKS

Joint work with Sisto Baldo and Annalisa Massaccesi

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

We consider certain natural variational problems for maps valued into manifolds equipped with a Finsler norm, study their relaxation and show the emergence of optimal one dimensional networks where energy concentrates, providing a link with the classical Steiner Tree problem or, more generally, with Gilbert-Steiner irrigation-type problems.

Hinke Osinga, University of Auckland, New Zealand

ROBUST CHAOS: A TALE OF BLENDERS, THEIR COMPUTATION, AND THEIR DESTRUCTION

Joint work with Stephanie Hittmeyer, Bernd Krauskopf, and Katsutoshi Shinohara

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building B-7, room 1.8.

Abstract

A blender is an intricate geometric structure of a three- or higher-dimensional diffeomorphism [1]. Its characterising feature is that its invariant manifolds behave as geometric objects of a dimension that is larger than expected from the dimensions of the manifolds themselves. We introduce a family of three-dimensional Hénon-like maps and study how it gives rise to an explicit example of a blender [2, 3]. We employ our advanced numerical techniques to present images of blenders and their associated one-dimensional stable manifolds. Moreover, we develop an effective and accurate numerical test to verify what we call the \(\textit{carpet property}\) of a blender. This approach provides strong numerical evidence for the existence of the blender over a large parameter range, as well as its disappearance and geometric properties beyond this range. We conclude with a discussion of the relevance of the carpet property for chaotic attractors.

References

1. C. Bonatti, S. Crovisier, L.J. Díaz, A. Wilkinson, What is... a blender?, Not. Am. Math. Soc. 63 (2016), 1175-1178.
2. L.J. Díaz, S. Kiriki, K. Shinohara, Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations, Nonlinearity 27 (2014), 353-378.
3. S. Hittmeyer, B. Krauskopf, H.M. Osinga, K. Shinohara, Existence of blenders in a Hénon-like family: geometric insights from invariant manifold computations, Nonlinearity 31 (2018), R239-R267.

Alexander Ostermann, University of Innsbruck, Austria

LOW-REGULARITY INTEGRATORS FOR NONLINEAR SCHRÖDINGER EQUATIONS

Joint work with Frédéric Rousset, Marvin Knöller and Katharina Schratz

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

Nonlinear Schrödinger equations are usually solved by pseudo-spectral methods, where the time integration is performed by splitting schemes or exponential integrators. Notwithstanding the benefits of this approach, its successful application requires additional regularity of the solution. For instance, second-order Strang splitting requires four additional derivatives for the solution of the cubic nonlinear Schrödinger equation. Similar statements can be made about other dispersive equations like the Korteweg–de Vries or the Boussinesq equation.

In this talk, we introduce as an alternative low-regularity Fourier integrators. They are obtained from Duhamel’s formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For cubic nonlinear Schrödinger equations, first-order convergence of such methods only requires the boundedness of one additional derivative of the solution, and second-order convergence the boundedness of two derivatives. For details, see [1, 2].

Moreover, a filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation is presented. This scheme has better convergence rates at low regularity than any known scheme in the literature so far. To prove this superior error behavior, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of \(L^2\). We are able to establish a global error estimate in \(L^2\) for \(H^1\) solutions, which is roughly of order \(\tau^{ \frac12 + \frac{5-d}{12} }\) in dimension \(d \leq 3\) with \(\tau\) denoting the time step size. For details, see [3].

References

1. M. Knöller, A. Ostermann, K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data, Preprint, arXiv:1807.01254, to appear in SIAM J. Numer. Anal. (2019).
2. A. Ostermann, K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math. 18 (2018), 731-755.
3. A. Ostermann, F. Rousset, K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Preprint, arXiv:1902.06779 (2019).

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Germany

EFFECTIVE BEHAVIOR OF RANDOM MEDIA

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

In engineering applications, heterogeneous media are often described in statistical terms. This partial knowledge is sufficient to determine the effective, i.e. large-scale behavior. This effective behavior may be inferred from the Representative Volume Element (RVE) method. I report on last years' progress on the quantitative understanding of what is called stochastic homogenization of elliptic partial differential equations: optimal error estimates of the RVE method, leading-order characterization of fluctuations, effective multipole expansions. Methods connect to elliptic regularity theory and to concentration of measure arguments.

Lex Oversteegen, University of Alabama at Birmingham, USA

SLICES OF THE PARAMETER SPACE OF CUBIC POLYNOMIALS

Joint work with Alexander Blokh and Vladlen Timorin

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

In this talk we consider slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the main cubioid in this parameter space. The main cubioid is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials \(z^2+c\) for \(c\) in the filled main cardioid.

Ivan Ovsyannikov, Universität Hamburg, Germany

BIRTH OF DISCRETE LORENZ ATTRACTORS IN GLOBAL BIFURCATIONS

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building A-4, room 120.

Abstract

Discrete Lorenz attractors are chaotic attractors, which are the discrete-time analogues of the well-known continuous-time Lorenz attractors. They are genuine strange attractors, i.e. they do not contain simpler regular attractors such as stable equilibria, periodic orbits etc. In addition, this property is preserved under small perturbations. Thus, Lorenz attractors, discrete and continuous, represent the so-called robust chaos.

In the talk a list of global (homoclinic and heteroclinic) bifurcations [1, 2, 3, 4] is presented, in which it was possible to prove the appearance of discrete Lorenz attractors. The proof is based on the study of first return (Poincare) maps, which are defined in a small neighbourhood of the homoclinic or heteroclinic cycle. The first return map can be transformed to the form asymptotically close to the three-dimensional Hénon map via smooth transformations of coordinates and parameters. According to [1, 5, 6, 7], Henon-like maps possess the discrete Lorenz attractor in an open subset of the parameter space.

References

1. S. Gonchenko, J. Meiss, I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn. 11 (2006), 191–212.
2. S. Gonchenko, I. Ovsyannikov, On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors, Math. Model. Nat. Phenom. 8 (2013), 71–83.
3. S. Gonchenko, I. Ovsyannikov, J. Tatjer, Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points, Regul. Chaotic Dyn. 19 (2014), 495–505.
4. S. Gonchenko, I. Ovsyannikov, Homoclinic tangencies to resonant saddles and discrete Lorenz attractors, Discrete Contin. Dyn. Syst. S 10 (2017), 273–288.
5. S. Gonchenko, I. Ovsyannikov, C. Simo, D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), 3493–3508.
6. S. Gonchenko, A. Gonchenko, I. Ovsyannikov, D. Turaev, Examples of Lorenz-like attractors in Hénon-like maps, Math. Model. Nat. Phenom. 8 (2013), 32–54.
7. I. Ovsyannikov, D. Turaev, Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model, Nonlinearity 30 (2017), 115–137.

Fedor Pakovich, Ben-Gurion University of the Negev, Israel

COMMUTING RATIONAL FUNCTIONS REVISITED

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-7, room 1.8.

Abstract

Let \(A\) and \(B\) be rational functions on the Riemann sphere. The classical Ritt theorem states that if \(A\) and \(B\) commute and do not have an iterate in common, then up to a conjugacy they are either powers, or Chebyshev polynomials, or Lattès maps. This result however provides no information about commuting rational functions which do have a common iterate. On the other hand, non-trivial examples of such functions exist and were constructed already by Ritt. In the talk we present new results concerning this class of commuting rational functions. In particular, we describe a method which permits to describe all rational functions commuting with a given rational function.

Zsolt Páles, University of Debrecen, Hungary

ON DERIVATIONS WITH ADDITIONAL PROPERTIES

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-7, room 1.8.

Abstract

A function \(d:\mathbb{R}\to \mathbb{R}\) is called a derivation if, for all \(x,y\in \mathbb{R} \), \[ d(x+y)=d(x)+d(y) \qquad\mbox{and}\qquad d(xy)=yd(x)+xd(y). \] It is a nontrivial fact that, for any non-algebraic number \(t\in \mathbb{R}\), there exists a derivation which does not vanish at \(t\). Nonzero derivations have many striking applications in the theory of functional equations and functional inequalities. Derivations derivate many of the elementary functions. For instance, if \(f:I\to \mathbb{R}\) is the ratio of two polynomials with algebraic coefficients, then, for every \(x\in I\), \[ d(f(x))=f'(x)d(x). \] It has been an old problem of the theory of functional equations whether there exists a nonzero derivation which derivates the exponential function or any of the trigonometric functions in the above sense. Our main result shows that the answer to this problem is affirmative.

Kenneth James Palmer, National Taiwan University, Taiwan

EXPONENTIAL DICHOTOMY AND SEPARATION IN LINEAR DIFFERENCE EQUATIONS

Joint work with Flaviano Battelli

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We consider linear difference equations \(x(n+1)=A(n)x(n)\), in which \(A(n)\) may not be invertible or bounded. The main issues considered here are robustness (or roughness) and the relation between a triangular system and its corresponding diagonal system. In general, exponential separation is weaker than exponential dichotomy but, for certain systems, it turns out that in some sense exponential separation implies exponential dichotomy. Differences between the differential equations case and the difference equations case are highlighted.

Olivier Menoukeu Pamen, University of Liverpool, UK & African Institute for Mathematical Sciences, Ghana

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS OF SINGULAR SDES

Joint work with Salah Mohammed and Ludovic Tangpi

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 2.18.

Abstract

In this talk, we are interested in existence and uniqueness of strong solutions for stochastic differential equations with irregular drift coefficients. The driving noise is a \(d\)-dimensional Brownian motion. The method relies on Malliavin calculus and as a byproduct, we obtain Malliavin differentiability of the solutions. Existence of Sobolev differentiable flows in small time will also be discussed.

References

1. V.E. Benes, Existence of optimal stochastic control laws, SIAM J. Control Optim., 9 (1971), 446–475.
2. G. Da Prato, P. Malliavin, D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris, Sér. I 315 (1992), 1287–1291.
3. A.M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. 24, Article ID rnm 124, (2007), 26 pp.
4. H.-J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one- dimensional stochastic differential equations, I, II, III. Math. Nachr., 143, 144, 151 (1989, 1991), 167–184, 241–281, 149–197.
5. E. Fedrizzi and F. Flandoli, Hölder Flow and Differentiability for SDEs with Nonregular Drift, Stochastic Analysis and Applications, 31 (2013), 708–736.
6. I. Gyöngy, N. V. Krylov, Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Relat. Fields 105 (1996), 143-158.
7. N.V. Krylov, M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Prob. Theory Rel. Fields 131(2) (2005), 154-196.
8. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.
9. A. Lanconelli, F. Proske, On explicit strong solutions of Itô-SDE’s and the Donsker delta function of a diffusion, Infin. Dimen. Anal. Quant. Prob. related Topics, 7 (2004).
10. O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske, T. Zhang, A variational approach to the construction and malliavin differentiability of strong solutions of SDE’s, Math. Ann. 357 ( 2013), 761–799.
11. T. Meyer-Brandis, F. Proske, Construction of strong solutions of SDE’s via Malliavin calculus, Journal of Funct. Anal. 258 (2010), 3922–3953.
12. S.E.A. Mohammed, M.K.R. Scheutzow, Spatial estimates for stochastic flows in Euclidean space, Annals of Probability, 26(1) (1998), 56–77.
13. S.E.A. Mohammed, T. Nilssen, F. Proske, Sobolev differentiable stochastic flows for sde’s with singular coefficients: Applications to the stochastic transport equation, Annals of Probability, 43(3) (2015), 1535–1576.
14. T. Nilssen, One-dimensional SDE’s with discontinuous, unbounded drift and continuously differentiable solutions to the stochastic transport equation, Technical Report 6, University of Oslo, 2012.
15. D. Nualart, The Malliavin Calculus and Related Topics, Springer, 1995.
16. F. Proske, Stochastic differential equations-some new ideas, Stochastics 79 (2007), 563-600.
17. A.Y. Veretennikov, On the strong solutions of stochastic differential equations, Theory Probab. Appl. 24 (1979), 354–366.
18. A.K. Zvonkin, A transformation of the state space of a diffusion process that removes the drift, Math.USSR (Sbornik) 22 (1974), 129–149.

David Pardo, University of the Basque Country & BCAM, Spain

EFFECTIVE COMPRESSIONAL WAVE VELOCITY ESTIMATION FOR POROUS ROCKS

Joint work with Ángel Javier Omella, Julen Alvarez-Aramberri, Magdalena Strugaru, Vincent Darrigrand, Carlos Santos, and Héctor González

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building B-8, room 3.21.

Abstract

In geophysics, it is of paramount importance to characterize the effective compressional wave velocity of the Earth's crust layers. In this work, we propose a set of numerical methods and techniques to estimate the effective compressional wave velocities of highly heterogeneous porous rocks along the entire frequency spectrum [1, 2]. To do so, we incorporate the internal structure of the rock at the pore scale and the properties of each of its constituents (density and primary wave velocity). For the low/medium frequency spectrum, we solve the acoustic equation in the frequency domain by the Finite Element Method (FEM), and we postprocess the solution along straight lines to estimate the homogenized compressional wave velocity. To obtain accurate results, we show the necessity to extend the domain by repeating the rock sufficient times with respect to the excitation frequency. Due to this requirement on the computational domain size, we consider non-fitting meshes [3], in which each finite element includes highly-discontinuous material properties. The use of non-fitting meshes allows us to reduce the number of degrees of freedom with respect to the use of traditional conforming fitting meshes. We take advantage of having to repeat the rock when precomputing blocks of the stiffness matrix to reduce the computational cost. At high frequencies, we solve the Eikonal equation by the Fast Marching Method (FMM) [4] to estimate the effective compressional wave velocity. The performance of the proposed methods are illustrated with different numerical experiments over synthetic and real porous rocks where the formations are provided by X-ray micro computed tomography.

References

1. Á.J. Omella, J. Alvarez-Aramberri, M.Strugaru, V. Darrigrand, D. Pardo, H. González, C. Santos, A Simulation Method for the Computation of the Effective P-Wave Velocity in Porous Rocks. Part 1: 1D Analysis, Submitted to Computational Geosciences (2019).
2. Á.J. Omella, M.Strugaru, J. Alvarez-Aramberri, V. Darrigrand, D. Pardo, H. González, C. Santos, A Simulation Method for the Computation of the Effective P-Wave Velocity in Porous Rocks. Part 2: 2D and 3D Analysis, Submitted to Computational Geosciences (2019).
3. T. Chaumont-Frelet, D. Pardo, Á. Rodríguez-Rozas, Finite element simulations of logging-while-drilling and extra-deep azimuthal resistivity measurements using non-fitting grids, Computational Geosciences 22(5) (2018), 1161-1174.
4. J.A. Sethian, A.M. Popovici, 3-D traveltime computation using the fast marching method, Geophysics 64 (1999), 516-523.

Andrea Pascucci, University of Bologna, Italy

ON STOCHASTIC LANGEVIN AND FOKKER-PLANCK EQUATIONS

Joint work with Antonello Pesce

Date: 2019-09-16 (Monday); Time: 11:30-11:50; Location: building B-8, room 2.19.

Abstract

We study existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of a degenerate SPDE satisfying the weak Hörmander condition. Our method is based on a Wentzell's reduction of the SPDE to a PDE with random coeffcients to which we apply the parametrix technique to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Applications to stochastic filtering are also discussed.

References

1. N.V. Krylov, Hörmander's theorem for stochastic partial differential equations, Algebra i Analiz 27(3) (2015), 157-182.
2. N.V. Krylov and A. Zatezalo, A direct approach to deriving filtering equations for diffusion processes, Appl. Math. Optim. 42(3) (2000), 315-332.
3. H. Kunita, Stochastic flows and stochastic differential equations, Vol. 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
4. E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52(1) (1994), 29-63.
5. P.L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 1679 (1994), 191-204.
6. A. Pascucci and A. Pesce, The parametrix method for parabolic SPDEs, (2019), arXiv:1803.06543v3.

Paweł Pasteczka, Pedagogical University of Cracow, Poland

QUASI-ARITHMETIC GAUSS-TYPE ITERATION

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-7, room 1.8.

Abstract

For a sequence of continuous, monotone functions \(f_1,\dots,f_n \colon I \to \mathbb{R}\) (\(I\) is an interval) we define the mapping \(M \colon I^n \to I^n\) as a Cartesian product of quasi-arithmetic means generated by \(f_j\)-s, that is functions \(A^{[f_j]}(v_1,\dots,v_n):=f_j^{-1}\big(\tfrac1n(f_j(v_1)+\cdots+f_j(v_n))\big)\). It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of \(I^n\).

We prove that whenever all \(f_j\)-s are \(\mathcal{C}^2\) with nowhere vanishing first derivative, then this convergence is quadratic. We present both qualitative- and quantitative-type results concerning this iteration. In particular, we deliver an effective upper estimation of the value \(\text{Var}\, M^{k}(v)\) and calculate the limit \(\frac{\text{Var}\, M^{k+1}(v)}{(\text{Var}\, M^{k}(v))^2}\) in a nondegenerated case.

References

1. P. Pasteczka, Iterated quasi-arithmetic mean-type mappings, Colloq. Math. 144 (2016), 215–228.
2. P. Pasteczka, On the quasi-arithmetic Gauss-type iteration, Aequationes Math. 92 (2018), 1119– 1128.

Grigorios Pavliotis, Imperial College London, UK

LONG TIME BEHAVIOUR, PHASE TRANSITIONS AND FLUCTUATIONS FOR THE McKEAN-VLASOV EQUATION

Joint work with José A. Carrillo, Matias Gonzalo Delgadino, Susana N. Gomes, Rishabh S. Gvalani, and André Schlichting

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

We study the long time behaviour, the number and structure of stationary solutions and fluctuations for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction. Our main objectives are the study of convergence to a stationary state, the construction of the bifurcation diagram for the stationary problem and the study of fluctuations around the McKean-Vlasov limit, in particular past the phase transition. The application of our work to the study of models for opinion formation and of synchronization for Kuramoto-type models is also discussed. This talk is based on the recent works [1, 2].

References

1. J.A. Carrillo, R.S. Gvalani, G. A. Pavliotis, and A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus, (2018).
2. S.N. Gomes and G.A. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Science (2018).

Ronnie Pavlov, University of Denver, USA

NON-UNIFORM SPECIFICATION PROPERTIES ON SUBSHIFTS

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

A celebrated result of Bowen implies uniqueness of equilibrium state for certain potentials on expansive systems with the specification property. In the setting of symbolic dynamics, this property is equivalent to the existence of a constant \(N\) such that any two \(n\)-letter words \(v,w\) in the language can be combined into a new word in the language given a gap between them of length at least \(N\). Several weakenings of specification have seen recent activity, among them non-uniform specification, in which one allows the gap to have size controlled by an increasing function \(f(n)\). I will summarize some known results about non-uniform specification, including a provable threshold on \(f(n)\) below which one can prove generalizations of Bowen's result on uniqueness of equilibrium state.

References

1. R. Pavlov, On non-uniform specification and uniqueness of the equilibrium state in expansive systems, Nonlinearity 32 (2019), 2441-2466.
2. R. Pavlov, On intrinsic ergodicity and weakenings of the specification property, Adv. Math. 295 (2016), 250-270.

Nicolas Perkowski, Max Planck Institute for Mathematics in the Sciences, Germany

A ROUGH SUPERBROWNIAN MOTION

Joint work with Tommaso Cornelis Rosati

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

We consider a 1- or 2-dimensional branching random walk in a small random potential and show that its large scale behavior is described by a new stochastic process, which we call rough superBrownian motion. This process is a mixture of the classical superBrownian motion and the continuous parabolic Anderson model (PAM), where the superBrownian part captures fluctuations caused by the branching mechanism and the PAM part describes the large scale behavior of the random potential. We use pathwise arguments to deal with the PAM-singularity, and martingale tools to deal with the singularity from the superBrownian part. We also study the survival properties of the rough superBrownian motion and show that it behaves very differently from its classical counterpart.

Daniel Peterseim, Universität Augsburg, Germany

SPARSE COMPRESSION OF EXPECTED SOLUTION OPERATORS

Joint work with Michael Feischl

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 3.21.

Abstract

We show that the expected solution operator of a prototypical linear elliptic partial differential operator with random diffusion coefficient is well approximated by a computable sparse matrix. This result holds true without structural assumptions on the random coefficient such as stationarity, ergodicity or any characteristic length of correlation. The constructive proof is based localized orthogonal multiresolution decompositions of the solution space for each realization of the random coefficient. The decompositions lead to a block-diagonal representation of the random operator with well-conditioned sparse blocks. Hence, an approximate inversion is achieved by a few steps of some standard iterative solver. The resulting approximate solution operator can be reinterpreted in terms of classical Haar wavelets without loss of sparsity. The expectation of the Haar representation can be computed without difficulty using appropriate sampling techniques. The overall construction leads to a computationally efficient method for the direct approximation of the expected solution operator which is relevant for stochastic homogenization and uncertainty quantification.

References

1. M. Feischl, D. Peterseim, Sparse Compression of Expected Solution Operators, ArXiv e-prints 1807.01741 (2018), 1-18.

Gabriella Pinzari, University of Padua, Italy

RENORMALIZABLE INTEGRABILITY OF THE PARTIALLY AVERAGED NEWTONIAN POTENTIAL

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 2.4.

Abstract

Definition Let \(h\), \(g\) be two (commuting) functions of the form \[ h(p, q, y, x)=\widehat h({\rm I}(p,q), y, x)\ ,\qquad g(p, q, y, x)=\widehat g({\rm I}(p,q), y, x)\] where \[(p, q, y, x)\in {\cal D}:={\cal B}\times U\] with \(U\subset {\mathbb R}^2\), \({\cal B}\subset{\mathbb R}^{2n}\) open and connected, \((p,q)=(p_1, \cdots, p_n, q_1, \cdots, q_n)\) conjugate coordinates with respect to the two-form \(=dy\wedge dx+\sum_{i=1}^{n}dp_i\wedge dq_i\) and \({\rm I}(p,q)=({\rm I}_1(p,q), \cdots, {\rm I}_n(p,q))\), with \[{\rm I}_i:\ {\cal B}\to {\mathbb R}\ ,\qquad i=1,\cdots, n\] pairwise Poisson commuting: \[\{{\rm I}_i, {\rm I}_j\}=0 \qquad \forall 1\le i \lt j\le n \qquad i+1, \cdots, n.\] We say that \(h\) is renormalizably integrable via \(g\) if there exists a function \[\widetilde h:\qquad {\rm I}({\cal B})\times g(U)\to {\mathbb R}\ , \] such that \[h(p,q,y,x)=\widetilde h({\rm I}(p,q), \widehat g({\rm I}(p,q),y,x))\] for all \((p, q, y, x)\in {\cal D}\).

It is proved that the partial average i.e., the Lagrange average with respect to just one of the two mean anomalies, of the Newtonian part of the perturbing function in the three-body problem Hamiltonian is renormalizably integrable. Consequences on the dynamics of the three-body problem are briefly discussed. The talk is based on [1], ArXiv: arXiv:1808.07633 and work in progress.

References

1. G. Pinzari, A first integral to the partially averaged Newtonian potential of the three-body problem, Celest Mech. Dyn. Astr. 131(22) (2019).

Marcin Pitera, Jagiellonian University in Kraków, Poland

RISK SENSITIVE DYADIC IMPULSE CONTROL FOR UNBOUNDED PROCESSES

Joint work with Łukasz Stettner

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

Dyadic impulse control of continuous time Feller-Markov processes with risk-sensitive long-run average cost is considered. The uncontrolled process is assumed to be bounded in the weighted norm and to be ergodic; the process could be unbounded in the supremum norm and do not necessarily satisfy uniform ergodicity property. The existence of solution to suitable Bellman equation using local span contraction method is shown, and link to optimal problem solution is established with the help of Hölder’s (entropic) inequalities.

Mihály Pituk, University of Pannonia, Hungary

ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS OF LINEAR DIFFERENCE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

In this talk, we will summarize some results on the asymptotic behavior of the positive solutions of linear difference equations. Under appropriate assumptions, we will study the growth rates and the existence of wieghted limits of the positive solutions.

References

1. R. Chieocan, M. Pituk, Weighted limits for Poincaré difference equations, Applied Mathematics Letters 49 (2015), 51-57.
2. R. Obaya, M. Pituk, A variant of the Krein-Rutman theorem for Poincaré difference equations, Journal of Difference Equations and Applications 18 (2012), 1751-1762.
3. M. Pituk, C. Pötzsche, Ergodicity beyond asymptotically autonomous linear difference equations, Applied Mathematics Letters 86 (2018), 149-156.

Mark Pollicott, University of Warwick, UK

INFLECTION POINTS FOR LYAPUNOV SPECTRA

Joint work with Oliver Jenkinson and Polina Vytnova

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

The Lyapunov spectra for a dynamical system describes the size (Hausdorff dimension) of the set of points which have a given Lyapunov exponent. H. Weiss conjectured that the associated graph is convex, but Iommi and Kiwi constructed a simple counter example. We explore this problem further, constructing examples with any given number of points of inflection.

Marcello Porta, Universität Tübingen, Germany

ON THE CORRELATION ENERGY OF INTERACTING FERMI GASES IN THE MEAN-FIELD REGIME

Joint work with Christian Hainzl, Felix Rexze, Niels Benedikter, Phan Thành Nam, Benjamin Schlein, and Robert Seiringer

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 0.10b.

Abstract

In this talk I will discuss the ground state properties of a homogeneous, interacting Fermi gas, in the mean-field regime. I will focus on the correlation energy, defined as the difference between many-body and Hartree-Fock ground state energies. It is a long-standing open problem in mathematical physics to rigorously compute this quantity, for large quantum systems. I will present upper and lower bounds for the correlation energy, that are optimal in their dependence on the number of particles, and that agree for small interactions. The lower bound captures the corrections to the energy predicted by second-order perturbation theory; it is based on the combination of Bogoliubov theory and on correlation inequalities for the many-body interaction. The upper bound establishes the validity of the random-phase approximation as a rigorous upper bound to the ground state energy; it is based on a suitable choice of the trial state, and on a rigorous bosonization scheme.

Dylan Possamaï, Columbia University, USA

A GENERAL APPROACH TO NON–MARKOVIAN TIME-INCONSISTENT STOCHASTIC CONTROL FOR SOPHISTICATED PLAYERS

Joint work with Camilo Hernández

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 2.19.

Abstract

This paper is the first attempt at a general non-Markovian theory of time--inconsistent stochastic control problems in continuous-time. We consider sophisticated agents who are aware of their time-inconsistency and take into account in future decisions. We prove here that equilibria in such a problem can be characterised through a new type of multi-dimensional system of backward SDEs, for which we obtain wellposedness. Unlike the existing literature, we can treat the case of non-Markovian dynamics, and our results go beyond verification type theorems, in the sense that we prove that any (strict) equilibrium must necessarily arise from our system of BSDEs. This is a joint work with Camilo Hernández, Columbia University.

Christian Pötzsche, Alpen-Adria Universität Klagenfurt, Austria

GLOBAL ATTRACTIVITY AND DISCRETIZATION IN INTEGRODIFFERENCE EQUATIONS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

Integrodifference equations are popular models in theoretical ecology to describe the temporal evolution and spatial dispersal of populations having nonoverlapping generations. As a contribution to the numerical dynamics of such infinite-dimensional dynamical systems, we establish that global attractivity of periodic solutions is robust under a wide class of spatial discretizations. Beyond robustness also the convergence order of the numerical schemes is preserved.

References

1. M. Kot, W. Schaffer, Discrete-time growth-dispersal models, Math. Biosci. 80 (1986), 109-136.
2. C. Pötzsche, Numerical dynamics of integrodifference equations: Basics and discretization errors in a \(C^0\)-setting, Appl. Math. Comput. 354 (2019), 422-443.
3. C. Pötzsche, Numerical dynamics of integrodifference equations: Global attractivity in a \(C^0\)-setting, submitted (2019).

Vojtěch Pravec, Silesian University in Opava, Czech Republic

REMARKS ON DEFINITIONS OF PERIODIC POINTS FOR NONAUTONOMOUS DYNAMICAL SYSTEM

Date: 2019-09-16 (Monday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

Let \((X,f_{1,\infty})\) be a nonautonomous dynamical system. In this talk we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new definition of asymptotic periodicity. This definition is not only very natural but also resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for uniformly convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We also show that even for uniformly convergent systems the nonautonomous analog of Sharkovsky’s Theorem is not valid for most definitions of periodic points.

References

1. J.S. Cánovas, Li-Yorke chaos in a class of nonautonomous discrete systems, Journal of Difference Equations and Applications 17 (2011), 479–486.
2. A. Miralles, M. Murillo-Arcila, M. Sanchis, Sensitive dependence for nonautonomous disrcete dynamical systems, Journal of Mathematical Analysis and Applications 463 (2018), 268–275.
3. Y. Shi, G. Chen, Chaos of time-varying discrete dynamical systems, Journal of Difference Equations and Applications 15 (2009), 429–449.

Bruno Premoselli, Université libre de Bruxelles, Belgium

COMPACTNESS OF SIGN-CHANGING SOLUTIONS TO SCALAR CURVATURE-TYPE EQUATIONS WITH BOUNDED NEGATIVE PART

Joint work with Jérôme Vétois

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 0.10b.

Abstract

We consider the equation \(\Delta_g u+hu=|u|^{2^*-2}u\) in a closed Riemannian manifold \((M,g)\), where \(h\in C^{0,\theta}(M)\), \(\theta \in (0,1)\) and \(2^* = \frac{2n}{n-2}\), \(n:=\dim(M)\ge3\). We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is \(\textit{a priori}\) bounded. We obtain this result under the conditions that \(n\ge7\) and \(h<\frac{n-2}{4 (n-1) }\textrm{Scal}_g\) in \(M\), where \(\textrm{Scal}_g\) is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function \(h\).

Yannick Privat, Université de Strasbourg, France

OPTIMAL RESOURCES CONFIGURATIONS IN POPULATION DYNAMICS

Joint work with Jimmy Lamboley, Antoine Laurain, and Grégoire Nadin

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 0.18.

Abstract

In this work, we are interested in the analysis of optimal resources configurations (typically foodstuff) necessary for a species to survive. For that purpose, we use a logistic equation to model the evolution of population density involving a term standing for the heterogeneous spreading (in space) of resources. The principal issue investigated in this talk writes: How to spread in an optimal way resources in a closed habitat? This problem can be recast as the one of minimizing the principal eigenvalue of an operator with respect to the domain occupied by resources, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. In particular, we investigate the optimality of balls.

David Prömel, University of Oxford, UK

MARTINGALE OPTIMAL TRANSPORT DUALITY AND ROBUST FINANCE

Joint work with Patrick Cheridito, Matti Kiiski, and H. Mete Soner

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 2.19.

Abstract

Without assuming any probabilistic price dynamics, we consider a frictionless financial market given by the Skorokhod space, on which some financial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the fundamental theorem of asset pricing. For this purpose we study the corresponding martingale optimal transport (MOT) problem: We obtain a dual representation of the Kantorovich functional (super-replication functional) defined for functions (financial derivatives) on the Skorokhod space using quotient sets (hedging sets). Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. As an immediate consequence of the duality result, we deduce a general robust fundamental theorem of asset pricing.

References

1. P. Cheridito, M. Kiiski, D. J. Prömel, H. M. Soner, Martingale Optimal Transport Duality, ArXiv Preprint arXiv:1904.04644.

Vladimir Protasov, University of L'Aquila, Italy & Lomonosov Moscow State University, Russia

THE JOINT SPECTRAL RADIUS AND FUNCTIONAL EQUATIONS: A RECENT PROGRESS

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

Joint spectral radius of matrices have been used since late eighties as a measure of stability of linear switching dynamical systems. Nearly in the same time it has found important applications in the theory of refinement equations (linear difference equations with a contraction of the argument), which is a key tool in the construction of compactly supported wavelets and of subdivision schemes in approximation theory and design of curves and surfaces. However, the computation or even estimation of the joint spectral radius is a hard problem. It was shown by Blondel and Tsitsiklis that this problem is in general algorithmically undecidable. Nevertheless recent geometrical methods [1,2,3,4] make it possible to efficiently estimate this value or even find it precisely for the vast majority of matrices. We discuss this issue and formulate some open problems.

References

1. N. Guglielmi, V.Yu. Protasov, Exact computation of joint spectral characteristics of matrices, Found. Comput. Math 13 (2013), 37-97.
2. C. Möller, U. Reif, A tree-based approach to joint spectral radius determination, Linear Alg. Appl. 563 (2014), 154-170.
3. N. Guglielmi, V.Yu. Protasov, Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions, SIAM J. Matrix Anal. 37 (2016), 18-52.
4. T. Mejstrik, Improved invariant polytope algorithm and applications, arXiv:1812.03080.

Sergio Pulido, ENSIIE & Université Paris-Saclay, France

STOCHASTIC VOLTERRA EQUATIONS

Joint work with Eduardo Abi Jaber, Christa Cuchiero, and Martin Larsson

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 2.19.

Abstract

We obtain general weak existence and stability results for Stochastic Convolution Equations (SVEs) with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. The motivation to study SVEs comes from the literature on rough volatility models. Our approach relies on weak convergence in \(L^p\) spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

Jasmin Raissy, Université Paul Sabatier, France

A DYNAMICAL RUNGE EMBEDDING OF \(\mathbb{C}\times\mathbb{C}^*\) IN \(\mathbb{C}^2\)

Joint work with Filippo Bracci and Berit Stensønes

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

In this talk, I will present the construction of a family of automorphisms of \(\mathbb{C}^2\) having an invariant, non-recurrent Fatou component biholomorphic to \(\mathbb{C}\times \mathbb{C}^*\) and which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such component is obtained by globalizing, thanks to a result of Forstneric, a local construction, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. Such Fatou coordinate is a fiber bundle map on \(\mathbb{C}\), whose fiber is \(\mathbb{C}^*\), forcing the global basin to be biholomorphic to \(\mathbb{C}\times\mathbb{C}^*\). The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Pöschel’s results about existence of local Siegel discs and suitable estimates for the Kobayashi distance. This construction gives an example of a Runge embedding of \(\mathbb{C}\times \mathbb{C}^*\) in \(\mathbb{C}^2\), since attracting Fatou components are Runge domains.

References

1. F. Bracci, J. Raissy, B. Stensønes, Automorphisms of \(\mathbb{C}^k\) with an invariant non-recurrent attracting Fatou component biholomorphic to \(\mathbb{C}\times(\mathbb{C}^*)^{k-1}\), to appear in Journal of the European Mathematical Society, www.ems-ph.org/journals/forthcoming.php?jrn=jems, arXiv:1703.08423.

Pavel Řehák, Brno University of Technology, Czech Republic

REFINED DISCRETE REGULAR VARIATION AND ITS APPLICATIONS IN DIFFERENCE EQUATIONS

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

We introduce a new class of the so-called regularly varying sequences with respect to an auxiliary sequence \(\tau\), and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to \(\tau\). We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, as a by-product, a knew view on the Kummer type test for convergence of series, which generalizes, among others, Raabe's test and Bertrand's test.

References

1. P. Řehák, Refined discrete regular variation and its applications, to appear in Math. Meth. Appl. Sci., DOI: 10.1002/mma.5670.

Filip Rindler, University of Warwick, UK

THEME & VARIATIONS ON \({\rm div}\, \mu = \sigma\)

Joint work with A. Arroyo-Rabasa, G. De Philippis, J. Hirsch, and A. Marchese

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 0.10a.

Abstract

The PDE \({\rm div}\, \mu = \sigma\) for (vector) measures \(\mu\) and \(\sigma\) appears - sometimes in a slightly hidden way - in many different problems of geometric measure theory and the calculus of variations, for instance in the structure theory of normal currents, Lipschitz functions and varifolds. In this talk I will survey a number of recent results about this equation and other related PDEs. As applications, I will discuss the structure of singularities of solutions, dimensional estimates, and several versions of Rademacher's theorem (both in Euclidean and non-Euclidean settings).

Philip Rippon, Open University, UK

CONSTRUCTING BOUNDED SIMPLY CONNECTED WANDERING DOMAINS WITH PRESCRIBED DYNAMICS

Joint work with Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, and Gwyneth Stallard

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

We give a new general technique for constructing transcendental entire functions with bounded simply connected wandering domains, which allows us to prescribe the type of long term dynamics within the wandering domains. In some cases we can show that the wandering domains have Jordan curve boundaries.

Matteo Rizzi, University of Chile, Chile

THE CAHN-HILLIARD EQUATION: EXISTENCE RESULTS AND QUALITATIVE PROPERTIES

Joint work with Michal Kowalczyk

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-8, room 0.10a.

Abstract

In the talk I will present the construction of a family \(\{u_\varepsilon\}\) of solutions to the Cahn-Hilliard equation \[-\varepsilon\Delta u_\varepsilon=\varepsilon^{-1}(u_\varepsilon-u_\varepsilon^3)-\ell_\varepsilon, \qquad\ell_\varepsilon\in\mathbb{R},\] whose zero level set is prescribed and approaches, as \(\varepsilon\to 0\), a given complete, embedded, \(k\)-ended constant mean curvature surface. Moreover, I will present some classification results, dealing with properties such as boundedness, monotonicity and radial symmetry.

References

1. M. Kowalczyk, M. Rizzi, Multiple Delaunay ends solutions of the Cahn-Hilliard equation, accepted by Communications in Partial differential equations.
2. M. Rizzi, Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation, submitted to Calculus of variations and PDEs.

Elisabetta Rocca, University of Pavia, Italy

RECENT RESULTS ON ADDITIVE MANUFACTURING GRADED-MATERIAL DESIGN BASED ON PHASE-FIELD AND TOPOLOGY OPTIMIZATION

Joint work with Ferdinando Auricchio, Elena Bonetti, Massimo Carraturo, Dietmar Hoemberg, and Alessandro Reali

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

A novel graded-material design for additive manufacturing based on phase-field and topology optimization is introduced by means of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Different numerical examples are discussed and first order optimality conditions are obtained including possible stress constraints in the objective functional. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.

References

1. M. Carraturo, E. Rocca, E. Bonetti, D. Hoemberg, A. Reali, A. Auricchio, Additive Manufacturing Graded-material Design based on Phase-field and Topology Optimization, arXiv:1811.07205v2 (2018).

Pablo Roldán, Yeshiva University, USA

CONTINUATION OF PERIODIC SOLUTIONS FROM THE CLASSICAL TO THE CURVED THREE-BODY PROBLEM

Joint work with Abimael Bengochea and Ernesto Perez-Chavela

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 2.4.

Abstract

In the classical 3-body problem it is known that any three masses lying on the vertices of an equilateral triangle generate a relative equilibria, which is a periodic solution. We will discuss the possible continuation of this periodic solution to the curved 3-body problem. (The curved problem is the extension of the classical one to a manifold of constant curvature.)

Emanuela Rosazza Gianin, University of Milano-Bicocca, Italy

RISK MEASURES AND PROGRESSIVE ENLARGEMENT OF FILTRATIONS: A BSDE APPROACH

Joint work with Alessandro Calvia

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-8, room 2.19.

Abstract

The aim of the talk is to investigate dynamic risk measures in the case of enlargement of filtration and its impact on the corresponding risk measure and on its properties. More precisely, we show how to induce a dynamic risk measure from a BSDE whose noise is given by a Brownian motion and a marked point process. In terms of the underlying information flow, this corresponds to a progressive enlargement of a Brownian filtration with information brought by the occurrence of random events at random times. This may describe the presence of defaults. The class of BSDEs with jumps considered was introduced in [1]. In the single jump case, we show that dynamic risk measures induced by these BSDEs admit a decomposition into two risk measures, one before and the other after the default. Furthermore, we prove that standard properties of dynamic risk measures are guaranteed by similar properties of the driver of these BSDEs and that time-consistency holds. From a financial point of view, the decomposition of the "global” risk measure into different "local” ones is reasonable. Before and after a default time, indeed, the risk measure should be updated in order to take into account the new information.

References

1. I. Kharroubi, T. Lim, Progressive enlargement of filtrations and backward stochastic differential equations with jumps, J. Theoret. Probab. 27 (2014), 683-724.

Lionel Rosier, MINES ParisTech, PSL Research University, France

EXACT CONTROLLABILITY OF NONLINEAR HEAT EQUATIONS IN SPACES OF ANALYTIC FUNCTIONS

Joint work with Camille Laurent

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 0.18.

Abstract

It is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. That issue has obtained very recently almost sharp results in the linear case (see [4, 1, 2]). In this talk, we investigate the set of reachable states for a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable \(x\), the unknown \(y\), and its derivative \(y_x\). By investigating carefully a nonlinear Cauchy problem in \(x\) in some space of Gevrey functions, and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. It time allows, works in progress about the reachable states for KdV and for ZK will be outlined.

References

1. J. Dardé, S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim. 56(3) (2018), 1692-1715.
2. A. Hartman, K. Kellay, M. Tucsnak, From the reachable space of the heat equation to Hilbert spaces of holomorphic functions, to appear in JEMS.
3. C. Laurent, L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, arXiv:1812.06637, submitted.
4. P. Martin, L. Rosier, P. Rouchon, On the reachable states for the boundary control of the heat equation, Appl. Math. Res. Express. AMRX 2 (2016), 181-216.
5. P. Martin, I. Rivas, L. Rosier, P. Rouchon, Exact controllability of a linear Korteweg-de Vries equation by the flatness approach, to appear in SIAM J. Control Optim.

Gergely Röst, University of Szeged, Hungary & University of Oxford, UK

BLOWFLY EQUATIONS: HISTORY, CURRENT RESEARCH AND OPEN PROBLEMS

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

The nonlinear delay differential equation today known as Nicholson's blowfly equation was introduced in 1980 to offer an explanation for a curious dataset that had been found in experiments with a laboratory insect population. Complex dynamics arises due to the interplay of the time delay and a non-monotone feedback.

In addition to being an elegant biological application, this equation has inspired the development of a large number of analytical and topological tools for infinite dimensional dynamical systems, including local and global Hopf-bifurcation analysis for delay differential equations, asymptotic analysis, stability criteria, invariant manifolds, singular perturbation techniques, invariance principles, order preserving semiflows by non-standard cones in Banach spaces, and the study of slowly and rapidly oscillatory solutions.

In this talk we give an overview of these developments, and discuss three current research directions, namely

(i) a more refined model of age-dependent intraspecific competition in pre-adult life stages and its effects on adult population dynamics;

(ii) the effect of environmental heterogeneity on nonlinear oscillations;

(iii) the evolution of maturation periods.

Ionel Roventa, University of Craiova, Romania

OPTIMAL APPROXIMATION OF INTERNAL CONTROLS FOR A WAVE-TYPE PROBLEM WITH FRACTIONAL LAPLACIAN USING FINITE-DIFFERENCE METHOD

Joint work with Pierre Lissy

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 0.18.

Abstract

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. Even if the initial condition is filtered, the control will excite all frequencies. This creates a lot of technical difficulties, because the spectral is not uniform with respect to the discretization step \(h\). The proof is mainly based on a (non-classic) moment method. For more general uniform controllability results by using filtered spaces and resolvent estimates, the interested reader is referred to [2, 3, 7, 8].

Mathematically speaking, our model can be seen as an intermediate case between the cases of the wave equation and the beam equation. Our strategy consists of an appropriate filtering technique, introduced in [5] and notably used in [1, 4, 6] in the context of wave or beam equation, which consists in relaxing the control requirement by controlling only the low-frequency part of the solution. This approach will be considered here.

References

1. N. Cîndea, S. Micu, I. and Rovenţa, Boundary controllability for finite-differences semidiscretizations of a clamped beam equation, SIAM J. Control Optim. 55(2) (2017), 785–817.
2. S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes, Numer. Math. 113 (2009), 377–415.
3. S. Ervedoza, Observability in arbitrary small time for discrete conservative linear systems, Some Problems in Nonlinear Hyperbolic Equations, ed. Tatsien Li, Yuejun Peng and Bopeng Rao, Series in Contemporary Mathematics CAM15, 283–309.
4. P. Lissy and I. Rovenţa, Optimal filtration for the approximation of boundary controls for the one- dimensional wave equation using finite-difference method, Math. Comp. 88(315) (2019), 273–291.
5. S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math. 91 (2002), 723–768.
6. S. Micu, I. Rovenţa, and L.E. Temereanca, Approximation of the controls for the linear beam equation, Math. Control Signals Systems 28(2) (2016), Art. 12, 53 pp.
7. L. Miller, Resolvent conditions for the control of unitary groups and their approximations, Journal of Spectral Theory 2 (2012), 1-55.
8. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47 (2005), 197-243 (electronic).

Maciej Sablik, University of Silesia in Katowice, Poland

ON FUNCTIONAL EQUATIONS CHARACTERIZING GENERALIZED POLYNOMIALS

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

We present some results on solving functional equations that characterize generalized polynomials. The results are essentially coming from our earlier works but we are going to investigate some new problems. We will give a description of solutions defined on Abelian groups and ask about the possible solutions in the case where semigroups (Abelian) are considered.

References

1. M. Sablik, T. Riede, Characterizing polynomial functions by a mean value property, Publ. Math. Debrecen 52 (1998), 597-609.
2. M. Sablik, Taylor's theorem and functional equations, Aequationes Math. 60 (2000), 258-267.
3. M. Sablik, An elementary method of solving functional equations, Annales Univ. Sci. Budapest., Sect. Comp. 48 (2018), 181-188.
4. L. Székelyhidi, Convolution type functional equations on topological commutative groups, World Scientific Publishing Co. Inc., Teaneck, NJ, 1991.
5. T. Szostok, Functional equations stemming from numerical analysis, Dissertationes Math. (Rozprawy Mat.) 508 (2015), 57 pp.

Joan Saldaña, University of Girona, Spain

EPIDEMIC OSCILLATIONS AND THE SPREAD OF AWARENESS

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-8, room 1.26.

Abstract

In this talk we consider an epidemic model with a preventive behavioural response triggered by the risk perception of infection among individuals. The analysis of models combining disease and behaviour dynamics has mostly focused on the impact of the latter on the initial growth of an outbreak (computation of \(R_0\)) and the existence of endemic equilibria (see, for instance, [2, 3, 5] for mean-field models). Here we ask whether the interplay between behaviour and disease spreading is always able to prevent periodic re-emergence of a communicable disease when awareness decays over time.

In a recent work [4], it was shown that oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible deterministic models with a single compartment of alerted hosts and no demographics, but they can occur when one considers two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts. Progressive levels of awereness in epidemic modelling were considered in [1], whereas the introduction of an additionnal compartment of active alerted hosts in an SIS model (without awareness decay) was made in [6].

In a deterministic context, such epidemic flare-ups translate into solutions of ODE models showing sustained oscillations which approach zero when there is a high enough fraction of alerted individuals in the population (see [4]). The question we will address here is how robust these oscillations are when we move away from such a deterministic framework and consider stochastic simulations of the epidemic dynamics.

References

1. S. Funk, E. Gilad, C. Watkins, V.A.A. Jansen, The spread of awareness and its impact on epidemic outbreaks, PNAS 21 (2009), 6872–6877.
2. S. Funk, E. Gilad, V.A.A. Jansen, Endemic disease, awareness, and local behavioral response, Journal Theoretical Biology 264(5) (2010), 01–509.
3. D. Juher, I.Z. Kiss, J. Saldaña, Analysis of an epidemic model with awareness decay on regular random networks, Journal of Theoretical Biology 365 (2015), 457–468.
4. W. Just, J. Saldaña, Y. Xin, Oscillations in epidemic models with spread of awareness, Journal of Mathematical Biology 62 (2018), 1027–10574.
5. I.Z. Kiss, J. Cassell, M. Recker, P.L. Simon, The impact of information transmission on epidemic outbreaks, Mathematical Biosciences 225 (2010), 1–10.
6. F.D. Sahneh, F.N. Chowdhury, C.M. Scoglio, On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading, Scientific Reports 2 (2012), 632.

Bernardo San Martín, Catholic University of the North, Chile

THE ROVELLA ATTRACTOR IS ASYMPTOTICALLY SECTIONAL-HYPERBOLIC

Joint work with Kendry Vivas

Date: 2019-09-16 (Monday); Time: 10:40-11:00; Location: building A-4, room 120.

Abstract

The Rovella attractor is a compact invariant set for a vector field \(X_0\) constructed in a similar way as the geometric Lorenz attractor, but replacing the central expansive condition at the singularity by a central contracting condition plus two additional geometric hypothesis: the unstable manifold of the singularity is contained in the stable manifold of hyperbolic periodic orbits and the one dimensional reduction for the first return Poincaré map has negative Schwarzian derivative. Rovella showed that although this attractor is non robust, it is almost 2-persistent in the \(C^3\) topology. In this paper we will prove that for a generic two-parameter family of vector fields that contains \(X_0\), asymptotically sectional-hyperbolicity is an almost 2-persistent property. In particular, we will prove that the Rovella attractor is asymptotically sectional-hyperbolic.

References

1. D. Carrasco-Olivera, B. San Martín, On the \(\mathcal{K}^*-\)expansiveness of the Rovella attractor, Bull. Braz. Math. Soc., 48 (2017), 649—662.
2. R.J. Metzger and C.A. Morales, The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37(206) (2006), 89—101.
3. C.A. Morales and B. San Martín, Contracting Singular Horseshoe, Nonlinearity, 30 (2017), 4208–4219.
4. E.M. Muñoz Morales, B. San Martín and J.A. Vera Valenzuela, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Lineaire, 45–67, Series, 20 (2003), 867–888.
5. A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Brasil. Mat, 24 (1993), 233-–259.
6. B. San Martín and K. Vivas, Asymtoticaly sectional-hiperbolic attractors, Discrete Contin. Dyn. Syst., 39 (2019), 4057-4071.

Marco Sansottera, University of Milan, Italy

ANALYTIC STUDY OF THE SECULAR DYNAMICS OF EXOPLANETARY SYSTEMS

Joint work with Anne-Sophie Libert, Ugo Locatelli, and Antonio Giorgilli

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 2.4.

Abstract

The search for exoplanets around nearby stars has produced a massive amount of observational data, pointing out the peculiar character of the Solar system. To date, more than 600 multiple planet systems have been found and the number of discovered exoplanets with unexpected orbital properties (such as highly eccentric orbits, mutually inclined planetary orbits, hot Jupiters, compact multiple systems) constantly increases.

The Laplace-Lagrange secular theory uses the circular approximation as a reference, thus its applicability to extrasolar systems can be doubtful. In this talk we aim to show that perturbation theory reveals very efficient for describing the long-term evolution of extrasolar systems.

First we study the long-term evolution of coplanar extrasolar systems with two planets by extending the Laplace-Lagrange theory (see [1, 2]). We identify three categories of systems: (i) secular systems, whose long-term evolution is accurately described using high order expansions in the eccentricities; (ii) near a mean-motion resonance systems, for which an approximation at order two in the masses is required; (iii) really close to or in a mean-motion resonance systems, for which a resonant model has to be used.

Then, being the inclinations of exoplanets detected via radial velocity method essentially unknown, we show how perturbation theory can be used in order to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. We propose a novel procedure (see [3]): a reverse KAM approach by using normal forms depending on a free parameter related to the unknown mutual inclinations of the exoplanets. Our approach can interestingly complement the concept of AMD-stability (see [4, 5]) to analyze the dynamics of the multiple-planet extrasolar systems.

References

1. A.-S. Libert, M. Sansottera, On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems, Celest. Mech. Dyn. Astr. 117 (2013), 149–168.
2. M. Sansottera, A.-S. Libert, Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance, Celest. Mech. Dyn. Astr., to appear (2019).
3. M. Volpi, U. Locatelli, M. Sansottera, A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems, Celest. Mech. Dyn. Astr. 130(36) (2018).
4. J. Laskar, A.C. Petit, AMD-stability and the classification of planetary systems, Astron. & Astroph. 605 (2017), A72.
5. A.C. Petit, J. Laskar, G. Boué, AMD-stability in the presence of first-order mean motion resonances, Astron. & Astroph., 607 (2017), A35.

Lucjan Sapa, AGH University of Science and Technology, Poland

MATHEMATICAL MODELS OF INTERDIFFUSION WITH VEGARD RULE

Joint work with Bogusław Bożek and Marek Danielewski

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 3.22.

Abstract

We study the diffusional transport in an \(s\)-component solid solution. Let \(\Omega\subset\mathbb{R}^n\) be an open and bounded set with a piecewise smooth boundary \(\partial\Omega\) and let \(T>0\) be fixed. Moreover, let \(\Omega_i\), \(D_i\), \(j_i\) and \(c_{0i}\) mean the partial molar volumes, the diffusion coefficients, the evolution of a mass through \(\partial\Omega\) and the initial concentrations. The unknowns are the concentrations of the components of a mixture \(c_i\) and the potential \(F\) of a drift velocity.

The local mass conservation law for fluxes with the Darken drift term and the Vegard rule lead to the parabolic-elliptic system of strongly coupled nonlinear differential equations \[ \left\{ \begin{array} {lcc} \partial_tc_i+{\rm{div}}\bigl(-D_i(c_1,...,c_s)\nabla c_i+c_i\nabla F\bigl)=0 & \text{on} & [0,T]\times\Omega,\\ \triangle F={\rm{div}}\bigl(\sum_{k=1}^s\Omega_k D_k(c_1,...,c_s)\nabla c_k\bigl) & \text{on} & [0,T]\times\Omega,\\ \int_\Omega Fdx=0 & \text{on} & [0,T],\\ \end{array} \right. \tag{1}\] with the nonlinear coupled initial-boundary conditions \[ c_i(0,x)=c_{0i}(x) \quad\text{on}\quad \Omega, \tag{2}\] \[ \left\{ \begin{array} {lcc} -D_i(c_1,...,c_s)\frac{\partial c_i}{\partial{\bf{n}}}+c_i\frac{\partial F}{\partial{\bf{n}}}=j_i(t,x) & \text{on} & [0,T]\times\partial \Omega,\\ \frac{\partial F}{\partial{\bf{n}}}=\sum_{k=1}^s\Omega_k\bigl(D_k(c_1,...,c_s)\frac{\partial c_k}{\partial{\bf{n}}}+j_k(t,x)\bigl) & \text{on} & [0,T]\times\partial\Omega, \end{array} \right.\tag{3} \] for \(i=1,...,s\). This model was introduced in [3] and a some special case in [4]. In the one-dimensional case it can be transformed to the well-known model studied for example in [2, 5].

We will present theorems on existence, uniqueness and properties of weak solutions in the suitable Sobolev spaces. Moreover, finite implicit difference methods (FDM) and theorems concerned convergence and stability will be given. The agreement between the theoretical results, numerical simulations and experimental data will be shown.

References

1. B. Bożek, L. Sapa, M. Danielewski, Difference methods to one and multidimensional interdiffusion models with Vegard rule, Math. Model. Anal. 24 (2019), 276–296.
2. B. Bożek, L. Sapa, M. Danielewski Existence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rule, Topol. Methods Nonlinear Anal. 52 (2018), 423–448.
3. L. Sapa, B. Bożek, M. Danielewski, Weak solutions to interdiffusion models with Vegard rule, 6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017) (Budapest, Hungary, 2017), 020039-1–020039-9, AIP Conference Proceedings 1926, American Institute of Physics, https://doi.org/10.1063/1.5020447, 2018.
4. B. Wierzba, M. Danielewski, The lattice shift generated by two dimensional diffusion process, Comp. Mater. Sci. 95 (2014), 192–197.
5. K. Holly, M. Danielewski, Interdiffusion and free-boundary problem for r-component \((r \ge 2)\) one- dimensional mixtures showing constant concentration, Phys. Rev. B 50 (1994), 13336–13346.

Maria Saprykina, Royal Institute of Technology (KTH), Sweden

INSTABILITY AND DIFFUSION IN HAMILTONIAN SYSTEMS VIA THE APPROXIMATION BY CONJUGATION METHOD

Joint work with Bassam Fayad

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

We present examples of nearly integrable Hamiltonian systems with several strong diffusion properties. In particular, we construct a real-analytic near integrable Hamiltonian system whose flow is topologically weakly mixing on the energy surface.

Our constructions are obtained by a version of the successive conjugation scheme à la Anosov-Katok. The talk is based on a joint work with Bassam Fayad.

Marcus Sarkis, Worcester Polytechnic Institute, USA

ROBUST MODEL REDUCTIONS FOR DARCY EQUATIONS WITH HIGH-CONTRAST COEFFICIENTS

Joint work with Alexandre Madureira

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 3.21.

Abstract

Major progress has been made recently to make preconditioners robust with respect to variation of coefficients. A reason for this success is the adaptive selection of primal constraints based on localized generalized eigenvalue problems. In this talk we discuss how to transfer this technique to the field of discretizations. Given a target accuracy, we design a robust model reduction by delocalizing multiscale basis functions and establish a priori energy error estimates with such target accuracy with hidden constants independently of the coefficients.

Jan Sbierski, University of Oxford, UK

GENERIC BLOW-UP RESULTS FOR LINEAR WAVES IN THE INTERIOR OF A SCHWARZSCHILD BLACK HOLE

Joint work with Grigorios Fournodavlos

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building B-8, room 0.10b.

Abstract

I will discuss recent work [1], joint with Grigorios Fournodavlos, on the behaviour of generic solutions to the wave equation in the interior of a Schwarzschild black hole. We derive an asymptotic expansion of a general solution near the singularity at \(r=0\) and show that it is characterised by its first two leading order terms in \(r\), a principal logarithmic term and a bounded second order term. Based on results [2], [3], [4] by Angelopoulos, Aretakis, and Gajic on the late time asymptotics of generic solutions to the wave equation in the exterior of a Schwarzschild black hole we then show that the principal logarithmic term is non-vanishing in a neighbourhood of the asymptotic endpoints of the singular hypersurface \(r=0\).

References

1. G. Fournodavlos, J. Sbierski, Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole, J. Arch. Rational Mech. Anal. (2019), https://doi.org/10.1007/s00205-019-01434-0.
2. Y. Angelopoulos, S. Aretakis, D. Gajic, A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes, Ann. PDE, 4(2) (2018), Art. 15, 120 p.
3. Y. Angelopoulos, S. Aretakis, D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes, Adv. Math. 323 (2018), 529–621.
4. Y. Angelopoulos, S. Aretakis, D. Gajic, A proof of Price’s late-time asymptotics for all angular frequencies, in preparation.

Walter Schachermayer, Universität Wien, Austria

ASYMPTOTIC SYNTHESIS OF CONTINGENT CLAIMS IN A SEQUENCE OF DISCRETE-TIME MARKETS

Joint work with M. Kreps

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 2.19.

Abstract

We prove a connection between discrete-time models of financial markets and the celebrated Black-Scholes-Merton continuous-time model in which "markets are complete." Specifically, we prove that if (a) the probability law of a sequence of discrete-time models converges (in the functional sense) to the probability law of the Black-Scholes-Merton model, and (b) the largest possible one-period step in the discrete-time models converges to zero, then every bounded and continuous contingent claim can be asymptotically synthesized with bounded risk: For any \(\epsilon > 0\), a consumer in the discrete-time economy far enough out in the sequence can synthesize a claim that is no more than \(\epsilon\) different from the target contingent claim \(x\) with probability at least \(1 -\epsilon\), and which, with probability 1, has norm less or equal to the norm of the target claim. This shows that, in terms of important economic properties, the Black-Scholes-Merton model, with its complete markets, idealizes many more discrete-time models than models based on binomial random walks.

Dierk Schleicher, Aix Marseille Université, France

THE RATIONAL RIGIDITY PRINCIPLE: TOWARDS RIGIDITY OF RATIONAL MAPS

Joint work with Kostiantyn Drach

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

Rigidity is one of the key goals in holomorphic dynamics; one way to phrase it is to say that any two maps can be distinguished in combinatorial terms. There are deep results and remarkable progress especially about polynomial maps. Much less is known about non-polynomial rational maps. We present recent progress, in joint work with Kostiantyn Drach, in establishing rigidity of a large class of rational maps, in particular Newton maps of polynomials. Similarly, we prove local connectivity of the corresponding Julia sets.

Benjamin Schlein, Universität Zürich, Switzerland

EXCITATION SPECTRUM OF TRAPPED BOSE-EINSTEIN CONDENSATES

Joint work with C. Boccato, C. Brennecke, S. Cenatiempo, and S. Schraven

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 0.10b.

Abstract

In this talk, we will discuss some recent results [1, 2, 3] concerning the ground state energy and the low-energy excitation spectrum of gases of \(N\) bosons trapped in a volume of order one and interacting through a repulsive potential with scattering length of the order \(1/N\) (Gross-Pitaevskii regime). Our results confirm the validity of Bogoliubov theory.

References

1. C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein, Bogoliubov theory in the Gross-Pitaevskii limit, Acta Math. 222 (2019), 219–335.
2. C. Boccato, C. Brennecke, S. Cenatiempo, B. Schlein, Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime, Preprint arXiv:1812.03086.
3. C. Brennecke, B. Schlein, S. Schraven, Excitation spectrum of trapped Bose-Einstein condensates, In preparation.

Volker Schlue, University of Melbourne, Australia

STABILITY OF EXPANDING BLACK HOLE COSMOLOGIES

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 0.10b.

Abstract

In general relativity, an explicit family of solutions to the Einstein equations \(\textrm{Ric}(g)=\Lambda g\) with positive cosmological constant \(\Lambda>0\), the so-called Kerr-de Sitter space-times, describe the equilibrium states of a black hole in an expanding universe. The black hole exterior falls into two components, the stationary (near) zone, and the expanding (far) zone, separated by the cosmological horizon of the black hole. While the near region (bounded by the event and cosmological horizons) was recently proven to be dynamically stable [3], this talk reports on the dynamics of the far region (beyond the cosmological horizon) [5, 6]. Unlike in [3] (or the black hole stability problem for the Kerr solutions with \(\Lambda=0\) [1, 4]), the solution does not globally converge to an explicit family of solutions, but displays genuine asymptotic degrees of freedom; this was first observed for closed de Sitter cosmologies in [2]. Nonetheless we can prove that the conformal Weyl curvature decays [6] due to the expansion of the space-time geometry, which forms an essential part of the analysis of the cosmological region.

References

1. M. Dafermos, G. Holzegel and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math. 222 (2019), 1–214.
2. H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm. Math. Phys. 107 (1986), 587–609.
3. P. Hintz and A. Vasy, The global non-linear stability of the Kerr–de Sitter family of black holes, Acta Math. 220 (2018), 1–206.
4. S. Klainerman and J. Szeftel, Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations, arXiv:1711.07597 [gr-qc], 2019.
5. V. Schlue, Global Results for Linear Waves on Expanding Kerr and Schwarzschild de Sitter Cosmologies, Communications in Mathematical Physics 334 (2015), 977–1023.
6. V. Schlue, Decay of the Weyl curvature in expanding black hole cosmologies, arXiv:1610.04172v1 [math.AP], 2016.

Ewa Schmeidel, University of Bialystok, Poland

CONSENSUS OF MULTI-AGENTS SYSTEMS ON ARBITRARY TIME SCALE

Joint work with Urszula Ostaszewska and Małgorzata Zdanowicz

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 2.2.

Abstract

In my talk an emergence of leader-following model based on graph theory on the arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function of time. We propose and prove conditions ensuring a leader-following consensus for any time scales using Grönwall inequality. The presented results are illustrated by examples.

References

1. U. Ostaszewska, E. Schmeidel, M. Zdanowicz, Exponentially stable solution of mathematical model based on graph theory of agents dynamics on time scales, Adv. Difference Equ., to appear.
2. U. Ostaszewska, E. Schmeidel, M. Zdanowicz, Emergence of consensus of multi-agents systems on time scales, Miskolc Math. Notes, (to appear, article code: MMN-2704).

Christian Schmeiser, Universität Wien, Austria

MATHEMATICAL MODELS OF ACTIN DRIVEN CELL MOTILITY

Joint work with Aaron Brunk, Stefanie Hirsch, Gaspard Jankowiak, Angelika Manhart, Dietmar Oelz, Diane Peurichard, Nikolaos Sfakianakis, and Christoph Winkler

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 1.26.

Abstract

Actin is one of the most important proteins occurring in practically all eukaryotic cells. It has the ability to polymerize and to form microfilaments, an essential part of the cytoskeleton. Among other functions, actin filaments contribute to the motility of crawling cells. In particular, a network of actin filaments supports the lamellipodium, a motility organelle of many cell types. The lamellipodium is a flat cell protrusion, and a first theoretical explanation of its flatness has been provided in [6]. Based on this flatness, a two-dimensional anisotropic two-phase continuum model, the Filament Based Lamellipodium Model, has been formulated and analyzed in [5] and then extended and used for simulations in [3] and [7]. The qualitative behaviour of submodels has been analyzed in [1] and in [4].

Recently, polymerization driven cortical flow has been modelled as an alternative mechanism for cell motility in [2]. The model is able to explain experimental results on adhesion-free motility in artificial micro-channels with structured walls.

References

1. S. Hirsch, A. Manhart, C. Schmeiser, Mathematical modeling of myosin induced bistability of lamellipodial fragments, J. Math. Biol. 74 (2017), 1-22.
2. G. Jankowiak, D. Peurichard, A. Reversat, M. Sixt, C. Schmeiser, Modelling adhesion-independent cell migration, preprint, 2019.
3. A. Manhart, D. Oelz, C. Schmeiser, N. Sfakianakis, An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals, J. Theor. Biol. 382 (2015), 244-258.
4. A. Manhart, C. Schmeiser, Existence of and decay to equilibrium of the filament end density along the leading edge of the lamellipodium, J. Math. Biol. 74 (2017), 169-193.
5. D. Oelz, C. Schmeiser, Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover, Archive Rat. Mech. Anal. 198 (2010), 963-980.
6. C. Schmeiser, C. Winkler, The flatness of lamellipodia explained by the interaction between actin dynamics and membrane deformation, J. Theor. Biol. 380 (2015), 144-155.
7. N. Sfakianakis, D. Peurichard, A. Brunk, C. Schmeiser, Modelling cell-cell collision and adhesion with the Filament Based Lamellipodium Model, Biomath 7 (2018), Article ID: 1811097.

Thomas Schmidt, Universität Hamburg, Germany

OBSTACLE PROBLEMS FOR BV MINIMIZERS, AND WEAK SUPERSOLUTIONS TO THE \(1\)-LAPLACE EQUATION

Joint work with Lisa Beck and Christoph Scheven

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

The talk will be concerned with Dirichlet and obstacle problems for the total variation, the area integral, and possibly more general variational integrals on the space of functions of bounded variation. In particular, it is planned to discuss duality-based connections to (super)solutions of PDEs of \(1\)-Laplace and minimal surface type.

References

1. L. Beck, T. Schmidt, Convex duality and uniqueness for \(\mathrm{BV}\)-minimizers, J. Funct. Anal. 268 (2015), 3061-3107.
2. C. Scheven, T. Schmidt, \(\mathrm{BV}\) supersolutions to equations of \(1\)-Laplace and minimal surface type, J. Differ. Equations 261 (2016), 1904-1932.
3. C. Scheven, T. Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35 (2018), 1175-1207.

Philipp Schoenbauer, Imperial College London, UK

A SUPPORT THEOREM FOR SINGULAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Martin Hairer

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

The purpose of this talk is to present a far-reaching generalization of the support theorem of Stroock and Varadhan. Recall that, given a stochastic differential equation (SDE), this theorem considers all ordinary differential equations (ODEs) which formally look the same as the SDE, but with the noise replaced by an arbitrary smooth function. The support of the SDE is then shown to be the closure of the set of all solutions to these ODEs. We prove a support theorem in the same spirit for stochastic partial differential equations (SPDEs). As part of our analysis we establish the “correct” way to deal with the issue of divergent renormalisation constants in such a description. (This issue makes it difficult to even guess the correct formulation of a support theorem.) Our approach applies to a range of interesting (singular) SPDEs, among them the stochastic quantization equations and the generalised KPZ equations. As an important corollary, we show the uniqueness of the invariant measure for the 3D stochastic quantization equation.

Katharina Schratz, Heriot-Watt University, UK

NONLINEAR FOURIER INTEGRATORS FOR DISPERSIVE EQUATIONS

Joint work with Alexander Ostermann and Frédéric Rousset

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

A large toolbox of numerical schemes for nonlinear dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, non-smooth phenomena play a fundamental role in modern physical modeling (e.g., blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. In this talk I present a new class of nonlinear Fourier integrators which offer strong geometric structure at low regularity and high oscillations. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization – linking the finite dimensional discretization to powerful existence results of nonlinear dispersive PDEs in low regularity spaces.

Tere M. Seara, Polytechnic University of Catalonia, Spain

ON THE BREAKDOWN OF SMALL AMPLITUDE BREATHERS FOR THE REVERSIBLE KLEIN-GORDON EQUATION

Joint work with Marcel Guardia and Otavio Gomide

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-7, room 2.4.

Abstract

Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied.

Nikita Selinger, University of Alabama at Birmingham, USA

PACMAN RENORMALIZATION

Joint work with Dima Dudko and Misha Lyubich

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building A-3/A-4, room 105.

Abstract

In a joint work with Misha Lyubich and Dima Dudko, we develop a theory of Pacman Renormalization inspired by earlier surgery construction by Branner and Douady. We show the hyperbolicity of periodic points of this renormalization with one-dimensional unstable manifold. This yield multiple consequences such as scaling law for centers of hyperbolic components attached to the main cardioid of the Mandelbrot set and local stability of certain Siegel discs.

Peter Šepitka, Masaryk University, Czech Republic

SINGULAR STURMIAN THEORY FOR WEAKLY DISCONJUGATE LINEAR HAMILTONIAN DIFFERENTIAL SYSTEMS

Joint work with Roman Šimon Hilscher

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 2.2.

Abstract

In this talk we introduce several new results in the Sturmian theory of weakly disconjugate (or equivalently, eventually controllable) linear Hamiltonian systems. We present singular comparison theorems on unbounded intervals for two nonoscillatory systems satisfying the Sturmian majorant condition and the Legendre condition. In particular, we show exact formulas and optimal estimates for the numbers of proper focal points of conjoined bases of these systems. This topic was infrequently studied in the literature and the validity of singular comparison theorems on unbounded intervals for general uncontrollable setting is an open problem so far. The presented results complete and generalize the previously obtained (i) singular Sturmian comparison/separation theorems on unbounded intervals by O. Došlý and W. Kratz in [1], and by the author jointly with R. Šimon Hilscher in [3], (ii) as well as the Sturmian comparison theorems on compact intervals by R. Šimon Hilscher in [4] and by J. Elyseeva in [2].

References

1. O. Došlý, W. Kratz, Singular Sturmian theory for linear Hamiltonian differential systems, Appl. Math. Lett. 26 (2013), 1187–1191.
2. J.V. Elyseeva, Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index, J. Math. Anal. Appl. 444 (2016), 1260–1273.
3. P. Šepitka, R. Šimon Hilscher, Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems, J. Differential Equations 266 (2018), 7481–7524.
4. R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, Math. Nachr. 284 (2011), 831–843.

Jacek Serafin, Wrocław University of Science and Technology, Poland

A STRICTLY ERGODIC, POSITIVE ENTROPY SUBSHIFT UNIFORMLY UNCORRELATED TO THE MÖBIUS FUNCTION

Joint work with Tomasz Downarowicz

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

This talk is based on two recent papers [1] and [2], where we show that if \(y=(y_n)_{n\ge 1}\) is a bounded sequence with zero average along every infinite arithmetic progression then for every \(N\ge 2\) there exists a strictly ergodic subshift \(\Sigma\) over \(N\) symbols, with entropy arbitrarily close to \(\log N\), uniformly uncorrelated to \(y\). In particular, for \(y=\mu\) being the Möbius function, there exist subshifts as above which satisfy the assertion of Sarnak’s conjecture ([3]). To the best of our knowledge, no other examples of positive entropy systems uncorrelated to the Möbius sequence are known.

Our result shows, among other things, that (even for strictly ergodic systems) the so-called strong MOMO (Möbius Orthogonality on Moving Orbits) property is essentially stronger than uniform uncorrelation.

References

1. T. Downarowicz, J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, International Mathematics Research Notices (2017), https://doi.org/10.1093/imrn/rnx192.
2. T. Downarowicz, J. Serafin, A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Möbius function, Studia Mathematica (2019), to appear.
3. P. Sarnak, Three lectures on the Möbius function randomness and dynamics, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

Kieran Sharkey, University of Liverpool, UK

CAPTURING THE QUASI-STATIONARY DISTRIBUTION WITHIN A DETERMINISTIC FRAMEWORK FOR STOCHASTIC SIS DYNAMICS

Joint work with Christopher Overton and Robert Wilkinson

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

The stochastic suscetible-infectious-susceptibe (SIS) model represents an important class of epidemic dynamics, and is thought to represent processes such as the spread of sexually transmitted diseases and computer viruses. A feature of this model is the existence of a single absorbing state, corresponding to the disease free state, to which the system will always converge for finite population sizes and finite infection transmission parameters.

There has been a long history of deterministic representations of the SIS model. Relating these models to the stochastic dynamics frequently makes use of mean-field assumptions, which are derived from the infinite population limit [1]. These models provide useful theoretical insight but do not feature the absorbing state, and therefore it is hard to link the insights back to the stochastic model.

In this work we develop novel methods to account for the absorbing state of the stochastic model within a deterministic framework. We do this by obtaining a deterministic approximation to the quasi-stationary distribution (QSD) of the model; i.e. the long-term steady-state behaviour conditional on not having reached the absorbing state [2, 3]. In particular, we build a system of population level equations, which when solved provide an accurate and efficient approximate to the QSD of the Markovian network-based SIS model for a large range of networks and parameter sets.

References

1. I.Z. Kiss, J. Miller, P.L. Simon, Mathematics of Epidemics on Networks, Springer, Pub. Place, 2017.
2. J.N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous time finite Markov chains, Journal of Applied Probability: Series B 2 (1967), 88-100.
3. I. Nåssell, On the time to extinction in recurrent epidemics, Journal of the Royal Statistical Society: Series B 61 (1999), 309-330.

Stefan Siegmund, Technische Universität Dresden, Germany

A HILBERT SPACE APPROACH TO FRACTIONAL DIFFERENCE EQUATIONS

Joint work with Pham The Anh, Artur Babiarz, Adam Czornik, Konrad Kitzing, Michał Niezabitowski, Sascha Trostorff, and Hoang The Tuan

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 2.2.

Abstract

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator \(1 - \tau^{-1}\) with the right shift \(\tau^{-1}\) on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems

References

1. Pham The Anh, A. Babiarz, A. Czornik, K. Kitzing, M. Niezabitowski, S. Siegmund, S. Trostorff, Hoang The Tuan, A Hilbert space approach to fractional difference equations, submitted.

Inbo Sim, University of Ulsan, South Korea

ON THE STUDY OF POSITIVE SOLUTIONS FOR SEMIPOSITONE \(p\)-LAPLACIAN PROBLEMS

Joint work with Lee, Shivaji, and Son

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

In this talk, I introduce what semipositone problems are and some issues on it. I focus on constructions of a subsolution to show the existence of positive solutions for sublinear (infinite) semipositone problems on bounded domains. Moreover, I discuss the existence and uniqueness of positive radial solutions for sublinear (infinite) semipositone \(p\)-Laplacian problems on the exterior of a ball with nonlinear boundary conditions. This talk is mainly based on joint works with Lee, Shivaji and Son.

Roman Šimon Hilscher, Masaryk University, Czech Republic

THE STORY OF FOCAL POINT IN DISCRETE STURMIAN THEORY

Joint work with Peter Šepitka

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-7, room 2.2.

Abstract

We will discuss the development of the concepts of generalized zeros and focal points for second order difference equations and symplectic difference systems in the relation with the validity of the Sturmian separation and comparison theorems. Our aim is to present recent progress in this area by discussing singular Sturmian theory for possibly uncontrollable symplectic difference systems on unbounded intervals. We will also present a simple application of the new concept in disconjugacy criteria for the second order Sturm-Liouville difference equations on unbounded intervals.

References

1. C. D. Ahlbrandt, A. C. Peterson, Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences 16 Kluwer Academic Publishers Group, Dordrecht, 1996.
2. M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199(3) (1996), 804–826.
3. M. Bohner, O. Došlý, W. Kratz, Sturmian and spectral theory for discrete symplectic systems, Trans. Amer. Math. Soc. 361(6) (2009), 3109–3123.
4. O. Došlý, J. V. Elyseeva, R. Šimon Hilscher, Symplectic Difference Systems: Oscillation and Spectral Theory, Birkhäuser, Basel, 2019 (to appear).
5. J. V. Elyseeva, Comparative index for solutions of symplectic difference systems, Differential Equations, 45(3) (2009), 445–459.
6. J. V. Elyseeva, Comparison theorems for symplectic systems of difference equations, Differential Equations, 46(9) (2010), 1339–1352.
7. P. Hartman, Difference equations: disconjugacy, principal solutions, Green’s function, complete monotonicity, Trans. Amer. Math. Soc. 246 (1978), 1–30.
8. W. Kratz, Discrete oscillation, J. Difference Equ. Appl. 9(1) (2003), 135–147.
9. P. Šepitka, R. Šimon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), 243–275.
10. P. Šepitka, R. Šimon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), 657–698.
11. P. Šepitka, R. Šimon Hilscher, Singular Sturmian separation theorems for nonoscillatory symplectic difference systems, J. Difference Equ. Appl. 24(12) (2018), 1894–1934.

Robert Skiba, Nicolaus Copernicus University in Toruń, Poland

A CONTINUATION PRINCIPLE FOR FREDHOLM MAPS AND ITS APPLICATION TO DIFFERENTIAL EQUATIONS

Joint work with Christian Pötzsche

Date: 2019-09-20 (Friday); Time: 12:05-12:25; Location: building B-7, room 2.2.

Abstract

In this talk we are going to present an abstract and flexible continuation theorem for zeros of parametrized Fredholm maps between Banach spaces. It guarantees not only the existence of zeros to corresponding equations but also that they form a continuum for parameters from a connected manifold. Our basic tools will be transfer maps and a specific topological degree. Next, we will explain how using an abstract and flexible continuation theorem to find global branches of homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach will be based on degree-theoretical arguments. In particular, Landesman-Lazer conditions will be proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

References

1. C. Pötzsche, R. Skiba, Global Continuation of Homoclinic Solutions, Zeitschrift fur Analysis und ihre Anwendungen 37(2) (2018), 159-187.
2. C. Pötzsche, R. Skiba, A Continuation Principle for Fredholm maps I: Theory and Basics, submitted.
3. C. Pötzsche, R. Skiba, A Continuation Principle for Fredholm maps II: Application to homoclinic solutions, submitted.

Stanislav Smirnov, University of Geneva, Switzerland & Skoltech, Russia

2D PERCOLATION REVISITED

Joint work with Mikhail Khristoforov

Date: 2019-09-17 (Tuesday); Time: 09:00-10:00; Location: building U-2, auditorium.

Abstract

We will discuss the state of our understanding of 2D percolation, and will present a recent joint work with Mikhail Khristoforov, giving a new proof of its conformal invariance at criticality.

Ľubomír Snoha, Matej Bel University, Slovakia

ON THE PROBLEM OF THE TOPOLOGICAL CLASSIFICATION OF MINIMAL SETS

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

One of the open problems in topological dynamics is the problem of the topological classification of minimal spaces/sets.

A dynamical system \((X,f)\) given by a topological space \(X\) and a continuous map \(f: X\to X\) is called minimal if all forward orbits are dense. A nonempty closed set \(M\subseteq X\) with \(f(M)\subseteq M\) is a minimal set for the system \((X,f)\) or for the map \(f\), if \((M, f|_M)\) is a minimal system. Thus, a system \((X,f)\) is minimal if and only if \(X\) is a minimal set. In every compact system there are minimal sets. A space \(X\) is said to be minimal if it admits a minimal (in general noninvertible) map.

The classification problem has two parts:

(1) Which spaces \(X\) are minimal and which are not?

(2) Given a space \(X\), consider all possible continuous selfmaps of this space and their minimal sets. Describe these sets topologically (find their full topological characterization).

(An illustrating example to (2): The minimal sets in \(I=[0,1]\) are nonempty finite sets and Cantor sets, meaning that if \(f\colon I\to I\) is continuous and \(M\) is a minimal set of \(f\), then \(M\) is either nonempty finite or Cantor and, conversely, if \(M \subseteq I\) is nonempty finite or Cantor, then there is a continuous map \(f\colon I\to I\) such that \(M\) is a minimal set of \(f\).)

In the talk we give a survey of some results on the classification problem.

Gabriel Soler López, Technical University of Cartagena, Spain

MINIMAL INTERVAL EXCHANGE TRANSFORMATIONS AND MINIMAL SURFACES

Joint work with José Ginés Espín Buendía, Antonio Linero Bas ,and Daniel Peralta Salas

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

An interval exchange transformation of \(n\)-interval, abbreviately \(n\)-IET, is an injective map \(T:D\subset[0,1]\to [0,1]\) such that:

\(\bullet\) \(D\) is the union of \(n\) pairwise disjoint open intervals, \(D=\cup_{i=1}^n I_i\), with \(I_i=]a_i,a_{i+1}[\), \(a_1=0\), \(a_{n+1}=1\) and \(n\geq 2\);

\(\bullet\) \(T|_{I_i}\) is a map of constant slope equals to \(1\) or \(-1\);

If \(T\) reverses the orientation in the interval set \(\mathcal{F}=\{I_{f_1},I_{f_2},\dots, I_{f_k}\}\) (the slope is \(-1\) in these intervals) for some \(1\le f_j\le n\) then we stress it by saying that \(T\) is an interval exchange transformation of \(n\)-intervals with \(k\)-flips or an (n,k)-IET; otherwise we say that \(T\) is an interval exchange transformation of \(n\)-intervals without flips or an oriented interval exchange transformation of \(n\)-intervals.

The point \(a_i\) is said to be a false discontinuity if \(\lim_{x\to a_i^+} T(x)=\lim_{x\to a_i^-} T(x)\). We will say that \(T\) is a proper \((n,k)\)-IET when it has not false discontinuities.

Let \(x\in [0,1]\) then the orbit of \(x\) under \(T\) is the set: \[\mathcal{O}_T(x)=\{T^n(x): n \textrm{ is an integer and } T^n(x) \textrm{ makes sense} \}.\] \(T\) is said to be minimal if for any \(x\in[0,1]\) then \(\mathcal{O}_T(x)\) is dense in \([0,1].\)

Let \(\mu\) denote the standard Lebesgue measure on \([0,1]\). It is easy to see that \(\mu\) (and any of its multiples) is an invariant measure for any interval exchange transformation. \(T\) will be said to be uniquely ergodic if it does not admit other invariant measures. We will present some progress in the theory of interval exchange transformations with flips. In particular we will pay attention to the Main Theorem in [2] which assures the existence of minimal, uniquely ergodic, proper \((n,k)\)-IET's for any \(n,k\in\mathbb{N}\) with \(n\ge 4\) and \(1\le k\le n\). Also it will be explained the study non-oriented surfaces admitting minimal flows made in [1] and the work in progress to build minimal non uniquely ergodic flipped IET's.

Acknowledgements

This talk has been partially supported by the grant number MTM 2017-84079-P from Ministerio de Ciencia Innovación y Universidades (Spain)

References

1. J.G. Espín, D. Peralta, and G. Soler, Existence of minimal flows on nonorientable surface, Discrete and Continuous Dynamical Systems. Series A, 37 (2017), 4191–4211.
2. A. Linero, G. Soler López, Minimal interval exchange transformations with flips, Ergodic Theory Dynam. Systems. 38 (2018), 3101–3144.

Francesco Solombrino, University of Naples Federico II, Italy

STABEL CONFIGURATIONS OF PRESTRAINED RODS

Joint work with Marco Cicalese and Matthias Ruf

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-8, room 3.22.

Abstract

We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By \(\Gamma\)-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. We also show, by means of a bifurcation analysis, an exchange of stability between the straight configuration and a branch of local minimizers with so-called hemihelical shape, confirming experimental results.

References

1. M. Cicalese, M. Ruf, and F. Solombrino, On global and local minimizers of prestrained thin elastic rods, Calculus of Variations and Partial Differential Equations 56(4):115 (2017).
2. M. Cicalese, M. Ruf, and F. Solombrino, Hemihelical local minimizers in prestrained elastic bistrips, Zeitschrift für angewandte Mathematik und Physik ZAMP 68(6):122 (2017).

Jan Philip Solovej, University of Copenhagen, Denmark

THE DILUTE LIMIT OF INTERACTING BOSE GASES

Joint work with Birger Brietzke and Søren Fournais

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 0.10b.

Abstract

I will discuss the interacting many-body Bose gase and, in particular, recent progress [2, 3] in understanding the asymptotics of the ground state energy in the dilute limit. The ground state energy of two bosons confined in a large box can be expressed in terms of the zero energy scattering length of the interacting potential. It has been a general belief in the physics literature [1, 4, 5] that the ground state energy in the dilute limit has a two term asymptotic expansion which is universal in the sense that the terms still depend only on the scattering length of the interaction potential. The asymptotics gives the celebrated Lee-Huang-Yang formula [4]. I will discuss recent progress of understanding this formula and the universality.

References

1. N.N. Bogolyubov, On the theory of superfluidity, Proc. Inst. Math. Kiev 9 (1947), 89-103.
2. B. Brietzke, S. Fournais, and J.P. Solovej, A simple 2nd order lower bound to the energy of dilute Bose gases, arXiv:1901.00539.
3. S. Fournais and J.P. Solovej, The energy of dilute Bose gases, arXiv:1904.06164.
4. T.D. Lee, K. Huang, and C.N. Yang, Eigenvalues and eigenfunctions of a bose system of hard spheres and its low-temperature properties, Physical Review 106 (1957), 1135-1145.
5. W. Lenz, Die Wellenfunktion und Geschwindigkeitsverteilung des entarteten Gases, Z. Phys. 56 (1929), 778–789.

Gwyneth Stallard, Open University, UK

CLASSIFYING SIMPLY CONNECTED WANDERING DOMAINS

Joint work with Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, and Phil Rippon

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

For rational functions, a classification of periodic Fatou components, with a detailed description of the dynamical behaviour inside each of the four possible types, was given around 100 years ago. For transcendental entire functions, there is an additional class known as Baker domains that are now well understood, and many examples of wandering domains, which cannot occur for rational functions. Although there is now a detailed description of the dynamical behaviour inside multiply connected wandering domains, there has been no systematic study of simply connected wandering domains. We show that there is in fact a wealth of possibilities for such domains and give a new classification into nine different types in terms of the hyperbolic distance between iterates and by whether orbits approach the boundaries of the domains. We give a new general technique for constructing bounded simply connected wandering domains which can be used to show that all nine types are realisable.

Samuel Stechmann, University of Wisconsin-Madison, USA

CHALLENGES IN DATA ASSIMILATION AND PREDICTION OF TROPICAL WEATHER AND CLIMATE

Joint work with Ying Li

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

Data assimilation is the process of combining (imperfect) observational data and (imperfect) model data in order to estimate the state of a complex system, such as the atmosphere. It is crucial to weather prediction as it provides the initial conditions for a forecast. In this talk, we describe the unique challenges that arise for tropical weather/climate, along with some recent results on estimating the intrinsic limits of predictability in the tropics using observational data.

Łukasz Stettner, Polish Academy of Sciences, Poland

PORTFOLIO OPTIMIZATION WITH TRANSACTION COSTS - DIRECT APPROACH

Joint work with Tomasz Rogala

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building B-8, room 2.19.

Abstract

We consider discrete time market with general bid and ask prices and will be interested in maximization of utility from terminal wealth. Our approach will be direct, we shall not use well developed duality theory (see for example [1] and references therein). The result is based on an analysis of general dynamic programming equation and work with suitable selectors. We show in particular the existence and form of shadow price i.e. the price on the market without proportional transaction costs for which the strategies and the value of the functional (utility from terminal wealth) are the same as in the case of market with transaction costs. Such results are obtained under general conditions on the utility function as well as on the bid and ask prices, both in one as well as in multidimensional case. Results presented are based on the papers [2], [3] and [4].

References

1. C. Czichowski, J. Muhle-Karbe, W. Schachermayer, ransaction costs, shadow prices, and duality in discrete time, SIAM J. Financial Math. 5 (2014), 258-277.
2. T. Rogala, Ł. Stettner, On Construction of Discrete Time Shadow Price, Appl. Math. Optim. 72(3) (2015), 391-433.
3. T. Rogala, Ł. Stettner, Optimal strategies for utility from terminal wealth with general bid and ask prices, Appl. Math. Optim. (2019) published online.
4. T. Rogala, Ł. Stettner, Multidimensional shadow price with general bid and ask prices, in preparation.

Federico Stra, Ecole Polytechnique Fédérale de Lausanne, Switzerland

RECENT ADVANCES IN MULTI-MARGINAL OT APPLIED TO DFT

Joint work with M. Colombo and S. Di Marino

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building B-8, room 0.10a.

Abstract

Multi-marginal optimal transport has been adopted to approximate the electron-electron interaction energy in the context of Density Functional Theory (DFT).

In this talk I will review the OT formulation of the DFT problem and mention what conditions ensure the finiteness and the continuity of multi-marginal optimal transport with repulsive cost (expressed in terms of a suitable concentration property of the measure).

Finally I will present some recent results regarding the \(\Gamma\)-convergence of the functionals for the study of the semiclassical limit of ground states.

References

1. M. Colombo, S. Di Marino, F. Stra, Continuity of multi-marginal optimal transport with repulsive cost, SIAM J. Math. Anal. 51(4) (2019), 2903–2926.
2. M. Colombo, S. Di Marino, F. Stra, Semiclassical limit of ground state in DFT, in preparation.

Ewa Stróżyna, Warsaw University of Technology, Poland

ANALYTIC PROPERTIES OF THE COMPLETE FORMAL NORMAL FORM FOR THE BOGDANOV–TAKENS SINGULARITY

Joint work with Henryk Żołądek

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building B-7, room 1.9.

Abstract

In [19] a complete formal normal forms for germs of 2-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

References

1. A. Algaba, C. García and M. Reyes, Invariant curves and integration of vector fields, J. Diff. Equat. 266 (2019), 1357–1376.
2. V. I. Arnold, Geometrical Methods in the Theory of Differential Equations, Springer–Verlag, Berlin–Heidelberg–New York, 1983, [Russian: Nauka, Moscow, 1978].
3. V.I. Arnold and Yu. S. Ilyashenko, Ordinary differential equations, in: Ordinary Differential Equations and Smooth Dynamical Systems, Springer, Berlin, (1997), 1–148; (in Russian: Fundamental Directions 1, VINITI, Moskva, 1985, 1–146).
4. A. Baider and J. Sanders, Further reduction of the Bogdanov–Takens normal form, J. Differential Equations 99 (1992), 205–244.
5. R.I. Bogdanov, Local orbital normal forms of vector fields on a plane, Trans. Petrovski Sem. 5 (1979), 50–85 [in Russian].
6. N.N. Brushlinskaya, A finiteness theorem for families of vector fields in the neighborhood of a singular point of a Poincaré type, Funct. Anal. Applic. 5 (1971), 10–15 [Russian].
7. A.D. Bryuno, Analytic form of differential equations, I. Trans. Moscow Math. Soc. 25 (1971), 131– 288; II. Trans. Moscow Math. Soc. 26 (1972), 199–239 [Russian: I. Tr. Mosk. Mat. Obs. 25 (1971), 119–262; II. Tr. Mosk. Mat. Obs. 26 (1972), 199–239].
8. P. Bonckaert and F. Verstringe, Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part, Ann. Inst. Fourier 62 (2012), 2211–2225.
9. M. Canalis-Durand and R. Schäfke, Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. 13 (2004), 493–513.
10. X. Gong, Integrable analytic vector field with a nilpotent linear part, Ann. Inst. Fourier 45 (1995), 1449–1470.
11. Yu. Ilyashenko, Divergence of series that reduce an analytic differential equation to linear normal form at a singular point, Funkts. Anal. Prilozh. 13(3) (1979), 87–88 [Russian].
12. Yu. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Math., 86, Amer. Math. Soc., Providence, 2008.
13. E. Lombardi and L. Stolovich, Normal forms of analytic perturbations of quasi-homogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Sup. \(4^{\mathrm{e}}\) série 41 (2010), 659–718.
14. F. Loray, Preparation theorem for codimension one foliations, Ann. Math. 163 (2006), 709–722.
15. R. Pérez-Marco, Total convergence or general divergence in small divisors, Commun. Math. Phys. 223 (2001), 451–464.
16. L. Stolovich and F. Verstringe, Holomorphic normal form of nonlinear perturbations of nilpotent vector fields, Regular Chaotic Dyn. 21 (2016), 410–436.
17. E. Stróżyna and H. Żołądek, The analytic and formal normal form for the nilpotent singularity, J. Diff. Equations 179 (2002), 479–537.
18. E. Stróżyna and H. Żołądek, Divergence of the reduction to the multidimensional Takens normal form, Nonlinearity 24 (2011), 3129–3141.
19. E. Stróżyna and H. Żołądek, The complete normal form for the Bogdanov–Takens singularity, Moscow Math. J. 15 (2015), 141–178.
20. E. Stróżyna, Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case, J. Diff. Equations 259 (2015), 709–722.
21. E. Stróżyna, Normal forms for germs of vector fields with quadratic leading part. The remaining cases, Studia Math. 239 (2017), 6718–6748.
22. F. Takens, Singularities of vector fields, Publ. Math. IHES 43 (1974), 47–100.
23. H. Żołądek, The Monodromy Group, Monografie Matematyczne, 67, Birkhäuser, Basel, 2006.

Endre Süli, University of Oxford, UK

IMPLICITLY CONSTITUTED FLUID FLOW MODELS: ANALYSIS AND APPROXIMATION

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Classical models describing the motion of Newtonian fluids, such as water, rely on the assumption that the Cauchy stress is a linear function of the symmetric part of the velocity gradient of the fluid. This assumption leads to the Navier-Stokes equations. It is known however that the framework of classical continuum mechanics, built upon a simple linear constitutive equation for the Cauchy stress, is too narrow to describe inelastic behavior of solid-like materials or viscoelastic properties of materials. Our starting point in this work is therefore a generalization of the classical framework of continuum mechanics, called the implicit constitutive theory, which was proposed recently in a series of papers by Rajagopal. The underlying principle of the implicit constitutive theory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and, in particular, linear) function of the symmetric part of the velocity gradient, one may allow a nonlinear, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian fluid flow models with implicit and possibly discontinuous power-law-like rheology.

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued graph. Using a variety of weak compactness techniques, we show that when the graph of the stress-strain relationship is maximal monotone a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the discretization parameter, measuring the granularity of the finite element triangulation, tends to zero. When the graph is nonmonotone, a subsequence of the sequence of finite element solutions is shown to converge to a gradient Young-measure solution of the problem. A key new technical tool in the analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions. The talk is based on a series of papers with Miroslav Bulíček and Josef Málek (Prague), Miles Caddick (Oxford), Lars Diening (Bielefeld), Christian Kreuzer (Dortmund), and ongoing research with Alexei Gazca-Orozco (Oxford) and Tabea Tscherpel (Aachen).

References

1. M. Bulíček, J. Málek, E. Süli, Existence of global weak solutions to implicitly constituted kinetic models of incompressible dilute homogeneous polymers, Commun. Part. Diff. Eq. 38(5) (2013), 882-924.
2. M. Caddick, E. Süli, Numerical approximation of Young-measure solutions to parabolic systems of forward-backward type, Appl. Anal. Discr. Math., Accepted for publication on 17/02/2019.
3. L. Diening, C. Kreuzer, E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, SIAM J. Numer. Anal. 51(2) (2013), 984-1015.
4. C. Kreuzer, E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, ESAIM: Math. Model. Numer. Anal. 50(5) (2016), 1333-1369.
5. E. Süli, T. Tscherpel, Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids, IMA J. of Numer. Anal., https://doi.org/10.1093/imanum/dry097.

Piotr Sułkowski, University of Warsaw, Poland & California Institute of Technology, USA

RANDOM MATRIX MODELS AND TOPOLOGICAL RECURSIONS

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 0.10b.

Abstract

Within the past few decades we have witnessed great progress in the theory of random matrices. In particular, in the last one-and-a-half decade a powerful formalism of topological recursions - which can be interpreted as generalization of Ward identities for matrix models - has been developed. Topological recursions have already found a lot of applications in various branches of mathematics (in particular algebraic geometry, enumerative geometry, knot theory), high energy physics, statistical physics, and various other fields. In this talk I will explain what topological recursions are and summarize some of their applications in research areas mentioned above.

Grzegorz Świątek, Warsaw University of Technology, Poland

MANDELBROT SET SEEN BY HARMONIC MEASURE: THE SIMILARITY MAP

Joint work with Jacek Graczyk

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci \({\cal M}_d\) for unicritical polynomials \(f_c(z)=z^d+c\). It is known that these parameters are structurally unstable and have stochastic dynamics. In [3] it was shown that for \(c\) from a set of full harmonic measure in \(\partial{\cal M}_d\) there exists a quasi-conformal similarity map \(\Upsilon_{c}\) between phase and parameter spaces which is conformal at \(c\). In a recent work [2] we prove \(C^{1+\frac{\alpha}{d}-\epsilon}\)-conformality, \(\alpha = \text{HD}({\cal J}_{c})\), of \(\Upsilon_{c}(z):\mathbb{C}\mapsto \mathbb{C}\) at typical \(c\in \partial {\cal M}_d\) and establish that globally quasiconformal similarity maps \(\Upsilon_{c}(z)\), \(c\in \partial {\cal M}_d\), are \(C^1\)-conformal along external rays landing at \(c\) in \(\mathbb{C}\setminus {\cal J}_{c}\) mapping onto the corresponding rays of \({\cal M}_d\). This conformal equivalence leads to a proof that the \(z\)-derivative of the similarity map \(\Upsilon_{c}(z)\) at typical \(c\in \partial {\cal M}_d\) is equal to \(1/{\cal T}'(c)\), where \[ {\cal T}(c)=\sum_{n=0}^{\infty}\left( D_z\left[f_{c}^n(z)\right]_{z=c}\right)^{-1}\] is the transversality function previously studied by Benedicks-Carleson and Levin, see [1, 4]. There are additonal geometric consequences of these results. A typical external radius of the connectedness locus is contained in an asymptotically very nearly linear twisted angle, but nevertheless passes through infinitely many increasingly narrow straits.

References

1. M. Benedicks, L. Carleson, On iterations of \(1-ax^2\) on \((-1,1)\), Ann. of Math. 122 (1985), 1-25.
2. J. Graczyk, G. Świątek, Analytic structures and harmonic measure at bifurcation locus, arXiv 1904.09434 (2019).
3. J. Graczyk, G. Świątek, Fine structure of connectedness loci, Math. Ann. 369 (2017), 49-108.
4. G. Levin, An analytical approach to the Fatou conjecture, Fund. Math. 171 (2002), 177-196.

Tomasz Szarek, University of Gdańsk, Poland

INVARIANT MEASURES FOR RANDOM WALKS ON HOMEO\(^+\)(R)

Joint work with D. Buraczewski and S. Brofferio

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

Let \({g_n}\) be a sequence of i.i.d. Homeo\(^+\)(R)–valued randomvariables whose distribution is a measure \(\mu\). We consider the left randomwalk on Homeo\(^+\)(R) defined by the random variables \(f_n := g_n \circ\cdot\cdot\cdot\circ g_1\). We study the Markov chain \((X_n)\) on the real line corresponding to \({g_n}\), i.e.for any \(x \in \mathbb{R}\) and \(n \in \mathbb{N}\) we consider the random variables defined by \(X^x_n :=f_n(x)\). The main purpose of the talk is to provide suffcient conditions forthe existence of a unique invariant Radon measure (mainly infinite) for\((X_n)\). This research generalizes the results obtained by Deroin, Kleptsyn,Navas and Parvani, who studied similar problems for groups of homeomorphisms.

László Székelyhidi, University of Debrecen, Hungary

FUNCTIONAL EQUATIONS VIA SPECTRAL SYNTHESIS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

Spectral synthesis studies the structure of translation invariant spaces of continuous functions over topological groups. The masterpiece is Laurent Schwartz's theorem stating that on the real line every translation invariant linear space of continuous complex valued functions which is closed under compact convergence is the closure of all exponential polynomials included in the space. As the solution space of a great variety of systems of convolution type functional equations satisfies these conditions spectral synthesis can be applied to describe the solutions. These ideas are worth for generalizations to obtain extensions of classical results of abstract harmonic analysis for functions without growth conditions (boundedness, integrability, etc.) Recently extensions of Schwartz's theorem have been proved over discrete groups using ring-theoretical methods, and spherical versions of the theorem have been obtained on spaces of functions invariant under various subgroups of the general linear group. Also extensions of the basic results to more general situations have been proved by relaxing the group-structure. In this survey talk we present the fundamental methods, ideas and results together with relevant applications to functional equations.

Anders Szepessy, Royal Institute of Technology (KTH), Sweden

LANGEVIN DYNAMICS DERIVED FROM QUANTUM MECHANICS AND ITS RELATION TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Joint work with Håkon Hoel

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In the study of stochastic partial differential equation one may wonder what is the noise? Often the stochasticity modelled in partial differential equations has its origin in thermal fluctuations.

Starting from a quantum formulation of a molecular system coupled to a heat bath, I will show that ab initio Langevin dynamics, with a certain rank one friction matrix determined by the coupling, approximates the quantum system more accurately than any Hamiltonian system, for large mass ratio between the system and heat bath nuclei.

will also give an example  of course-graining a stochastic molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions.

References

1. A. Kammonen, P. Plecháč , M. Sandberg, A. Szepessy, Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics, nn. Henri Poincaré 19 (2018), 2727-2781.
2. P. Plecháč , M. Sandberg, A. Szepessy, The classical limit of quantum observables in conservation laws of fluid dynamics, arXiv:1702.04368.
3. E. von Schwerin, A. Szepessy, A stochastic phase-field model determined from molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis 44 (2010), 627-646.

Lukasz Szpruch, University of Edinburgh, UK

MEAN-FIELD LANGEVIN DYNAMICS AND ENERGY LANDSCAPE OF NEURAL NETWORKS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In this paper we present a probabilistic analysis of the long-time behaviour of the nonlocal, diffusive equations with a gradient flow structure in 2-Wasserstein metric. Our work is motivated by a desire to provide a theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of deep neural networks. The key insight is that the certain class of the finite dimensional non-convex problems becomes convex when lifted to infinite dimensional space of measures. We leverage this observation and show that the flow of marginal laws induced by the Mean Field Langevin Dynamics equation converges to the stationary distribution which is exactly the minimiser of the energy functional. At the heart of our analysis is a pathwise perspective on Otto calculus used in gradient flow literature which is of independent interest. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle.

Zuzanna Szymańska, Polish Academy of Sciences, Poland

MICROSCOPIC DESCRIPTION OF DNA THERMAL DENATURATION

Joint work with Mateusz Dębowski and Mirosław Lachowicz

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

We propose a microscopic model describing the process of DNA thermal denaturation. The process consists of the splitting of DNA base pairs resulting in the separation of two complementary DNA strands. In contrast to the previous modelling attempts we take into account the states of all base pairs of DNA which in fact imposes the microscopic nature of the approach. The model is a linear integro-differential non-autonomous equation describing the dynamics of probability density which characterizes the distances between the bases within individual base pairs. We take into account not only the strength of double and triple hydrogen bonds between the complementary bases but also the stacking interactions between neighborhood base pairs. We show basic mathematical properties of the model and present numerical simulations that reproduce the sigmoid shape of DNA melting curves and reveal the appearance of experimentally observed denaturation bubbles.

References

1. M. Dębowski, M. Lachowicz, Z. Szymańska, Microscopic description of DNA thermal denaturation, Appl Math Comput. 361 (2019), 47-60.

Katarzyna Szymańska-Dębowska, Lodz University of Technology, Poland

CANARD SOLUTIONS IN EQUATIONS WITH BACKWARD BIFURCATIONS

Joint work with Jacek Banasiak and Milaine Seuneu Tchamga

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 1.26.

Abstract

We consider a singularly perturbed initial value problem in the case of intersecting quasi stationary manifolds. The main results are concerned with the asymptotic behavior of solutions as the small parameter tends to zero.

Our results are related to the Tikhonov approach. The main condition for the validity of the Tikhonov theorem is that the quasi steady states be isolated and attractive. In applications, however, we often encounter the situation when two or more quasi steady states intersect. It involves the so called exchange of stabilities: the branches of the quasi steady states change from being attractive to being repelling (or conversely) across the intersection. The assumptions of the Tikhonov theorem fail to hold in the neighbourhood of the intersection but it is natural to expect that any solution that passes close to it follows the attractive branches of the quasi steady states on either side of the intersection. However, in many cases an unexpected behaviour of the solution is observed - it follows the attracting part of a quasi steady state and, having passed the intersection, it continues along the now repelling branch of the former quasi steady state for some prescribed time and only then jumps to the attracting part of the other quasi steady state. Such a behaviour we call the delayed switch of stability. We shall focus on the so called backward bifurcation, in which two of three quasi steady states intersect and exchange stabilities at the intersection.

As an application, we consider two predator-prey models: the Rosenzweig-MacArthur model and the Leslie-Gowers/Holling model. In both these cases the quasi-steady manifolds intersect and a backward bifurcation occurs along their intersection. We give a proof of the existence of canards and provide an exact value of time at which the stability switch occurs.

Johan Taflin, Université de Bourgogne, France

REGULARITY OF ATTRACTING CURRENTS

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

To each attractor of an endomorphism \(f\) of \(\mathbb{CP}^k\) it is possible to associate an analytic object called an attracting current. It can be used to obtain several information on the attractor and in this talk, I will explain how a weak form of regularity of this current is related to the dynamics of \(f\) on the attractor.

Antoine Tambue, Western Norway University of Applied Sciences, Norway

MAGNUS-TYPE INTEGRATOR FOR THE FINITE ELEMENT DISCRETIZATION OF SEMILINEAR PARABOLIC NON-AUTONOMOUS SPDEs DRIVEN BY MULTIPLICATIVE NOISE

Joint work with Jean Daniel Mukam

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 2.18.

Abstract

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square \(L^2\) norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order \(\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+\Delta t^{1/2}\right).\) Numerical simulations to illustrate our theoretical finding are provided.

References

1. A. Tambue, J.D. Mukam, Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise, arXiv:1809.04438v1, 2018.

Satoshi Tanaka, Okayama University of Science, Japan

ON THE UNIQUENESS OF POSITIVE RADIAL SOLUTIONS OF SUPERLINEAR ELLIPTIC EQUATIONS IN ANNULI

Joint work with Naoki Shioji and Kohtaro Watanabe

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

The Dirichlet problem \[ \left\{ \begin{array}{cl} \Delta u + f(u) =0 & \mbox{in} \ x \in A, \\[1ex] u=0 & \mbox{on} \ \partial A, \end{array} \right. \] is considered, where \(A:=\{x\in \mathbb{R}^N : a\lt |x| \lt b \}\), \(N \in \mathbb{N}\), \(N \ge 2\), \(0 \lt a \lt b \lt\infty \), \(f \in C^1[0,\infty)\), \(f(u) \gt 0\) and \(uf'(u) \ge f(u)\) for \(u \gt 0\).

The uniqueness of radial positive solutions is studied. Hence the boundary value problem \[ u'' + \dfrac{N-1}{r} u' + f(u) = 0, \quad r \in (a,b), \qquad %\\[1ex] u(a) = u(b) = 0 \] is considered. The uniqueness results of positive solutions are shown.

Michael Tehranchi, University of Cambridge, UK

DUALITY FOR HOMOGENEOUS OPTIMISATION PROBLEMS

Joint work with David Driver

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building B-8, room 2.19.

Abstract

This talk is concerned with stochastic optimal control problems with a certain homogeneity. For such problems, a novel dual problem is formulated. The results are applied to a stochastic volatility variant of the classical Merton problem. Another application of this duality is to the study the right-most particle of a branching Levy process.

Dalia Terhesiu, Leiden University, Netherlands

LIMIT PROPERTIES FOR WOBBLY INTERMITTENT MAPS

Joint work with Douglas Coates and Mark Holland

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

It is known that finite measure preserving intermittent maps with indifferent fixed points characterised by regular variation satisfy stable laws for sufficiently regular observables that do not vanish at the indifferent fixed points. We consider a finite measure preserving Pomeau Manneville type map, perturb the behaviour at the (only one) indifferent fixed point according to a St. Petersburg type distribution and obtain convergence to a non trivial limit distribution (a semistable law) along subsequences. Also, we obtain lower bounds on the decay of correlation for such modified maps and suitable observables. In this talk I will present these results.

Marita Thomas, WIAS, Germany

ANALYTICAL AND NUMERICAL ASPECTS OF RATE-INDEPENDENT GRADIENT-REGULARIZED DAMAGE MODELS

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-8, room 3.22.

Abstract

This presentation deals with techniques for the spatial and temporal discretization of models for rate-independent damage featuring a gradient regularization and a non-smooth constraint due to the unidirectionality of the damage process. A suitable notion of solution for the non-smooth process is introduced and its corresponding discrete version is studied by combining a time-discrete scheme with finite element discretizations of the domain. Results and challenges on the convergence of the discrete problems in the sense of evolutionary Gamma-convergence in dependence of the choice of the gradient term and the mesh properties are discussed. Directions towards adaptive strategies are pointed out. The presented results are based on collaborations with S. Tornquist (WIAS), Ch. Kuhn and A. Schlüter (Kaiserslautern), S. Bartels and M. Milicevic (U Freiburg) and with M. Walloth and W. Wollner within the Priority Program SPP 1748 of the German Research Foundation.

Vladlen Timorin, National Research University Higher School of Economics, Russia

INVARIANT SPANNING TREES FOR QUADRATIC RATIONAL MAPS

Joint work with Anastasia Shepelevtseva

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

A theorem of W. Thurston (sometimes called the fundamental theorem of complex dynamics) opens a door for algebraic, topological and combinatorial methods into dynamics of rational maps on the Riemann sphere. We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes - the ivy graph representing the pullback relation on (isotopy classes of) spanning trees.

Pedro J. Torres, University of Granada, Spain

PERIODIC SOLUTIONS OF THE LORENTZ FORCE EQUATION

Joint work with David Arcoya and Cristian Bereanu

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building B-7, room 1.9.

Abstract

We consider the existence of \(T\)-periodic solutions of the Lorentz force equation \begin{eqnarray*} \left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'= E(t,q)+q'\times B(t,q) \end{eqnarray*} where the electric and magnetic fields \(E,B\) are written in terms of the electric and magnetic potentials \(V:[0,T]\times\mathbb R^3\to\mathbb R\) and \(W:[0,T]\times\mathbb R^3\to\mathbb R^3\) as \[ E=-\nabla_q V-\frac{\partial W}{\partial t},\qquad B=\mbox{curl}_q\, W. \] Following for instance [3, 4], this is the relativistic equation of motion for a single charge in the fields generated by \(V\) and \(W.\) The above equation is formally the Euler - Lagrange equation of the relativistic Lagrangian \begin{eqnarray*} \mathcal L(t,q,q') = 1 - \sqrt{1 - |q'|^2} + q'\cdot W(t,q) +V(t,q). \end{eqnarray*} and also it is the Hamilton - Jacobi equation of the relativistic Hamiltonian \begin{eqnarray*} \mathcal H(t,p,q)=\sqrt{1+|p-W(t,q)|^2}-1+V(t,q). \end{eqnarray*} The purpose of the talk is to review some methods recently developed for the existence and multiplicity of \(T\)-periodic solutions, by using a topological degree approach [1] or a novel critical point theory [2].

References

1. C. Bereanu, J. Mawhin, Boundary value problems for some nonlinear systems with singular \(\phi\)-Laplacian, J. Fixed Point Theor. Appl. 4 (2008), 57-75.
2. D. Arcoya, C. Bereanu, P.J. Torres, Critical point theory for the Lorentz force equation, Arch. Rational Mech. Anal. 232 (2019), 1685-1724.
3. R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics. Electrodynamics, vol. 2, Addison-Wesley, Massachusetts, 1964.
4. L.D. Landau, E.M. Lifschitz, he Classical Theory of Fields, Fourth Edition: Volume 2, Butterworth-Heinemann, 1980.

Emmanuel Trélat, Sorbonne Université, France

OPTIMAL CONTROL AND APPLICATIONS TO AEROSPACE

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building B-7, room 1.8.

Abstract

I will report on nonlinear optimal control theory and show how it can be used to address problems in aerospace, such as orbit transfer. The knowledge resulting from the Pontryagin maximum principle is in general insufficient for solving adequately the problem, in particular due to the difficulty of initializing the shooting method. I will show how the shooting method can be successfully combined with numerical homotopies, which consist of deforming continuously a problem towards a simpler one. In view of designing low-cost interplanetary space missions, optimal control can also be combined with dynamical system theory, using the nice dynamical properties around Lagrange points that are of great interest for mission design.

Masato Tsujii, Kyushu University, Japan

COHOMOLOGICAL THEORY OF THE SEMI-CLASSICAL ZETA FUNCTIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

We first review very briefly about recent developments in analysis of transfer operators for hyperbolic dynamical systems. We will then focus on the semi-classical (or Gutzwiller-Voros) zeta functions for geodesic flows on negatively curved manifolds. We show that the semi-classical zeta function is the dynamical Fredholm determinant of a transfer operator acting on the leaf-wise cohomology space along the unstable foliation. This realize the idea presented by Guillemin and Patterson a few decades ago. As an application, we see that the zeros of the semi-classical zeta function concentrate along the imaginary axis, imitating those of Selberg zeta function.

References

1. F. Faure, M. Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, Inventiones Mathematicae 208 (2017), 851-998.
2. M. Tsujii, On cohomological theory of dynamical zeta functions, preprint, arXiv 1805.11992.

Warwick Tucker, Uppsala University, Sweden

SMALL DIVISORS AND NORMAL FORMS

Joint work with Zbigniew Galias

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building B-7, room 1.8.

Abstract

In this talk, we will discuss the computational challenges of computing trajectories of a non-linear ODE in a region close to a saddle-type fixed-point. By introducing a carefully selected close to identity change of variables, we can bring the non-linear ODE into an "almost" linear system. This normal form system has an explicit transfer-map that transports trajectories away from the fixed point in a controlled manner. Determining the domain of existence for such a change of variables poses some interesting computational challenges. The proposed method is quite general, and can be extended to the complex setting with spiral saddles. It is also completely constructive which makes it suitable for practical use. We illustrate the use of the method by a few examples.

Amjad Tuffaha, American University of Sharjah, United Arab Emirates

ON BOUNDARY CONTROL OF THE POISSON EQUATION WITH THE THIRD BOUNDARY CONDITION

Joint work with Alip Mohamed

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 0.18.

Abstract

In this talk, we consider an optimal control problem involving the Poisson equation on the unit disk in \(\mathbb{C}\) subject to the third boundary condition and where the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter \(\lambda\) is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when \(\lambda\) is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

References

1. A. Mohamed, A. Tuffaha, On Boundary Control of the Poisson Equation with the third boundary condition, Journal of Mathematical Analysis and Applications 459(1) (2018), 217–235.

Dmitry Turaev, Imperial College London, UK

LORENZ ATTRACTORS IN FLOWS AND MAPS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

We review a theory of pseudohyperbolic attractors, which serve as a generalization of the classical Lorenz attractor to the case of a higher codimension of the strong-stable foliation. These attractors are genuinely chaotic (every orbit in such attractor has positive maximal Lyapunov exponent) and are robust with respect to small perturbations. The class includes periodically perturbed hyperbolic and Lorenz attractors and attractors in lattice dynamical systems. We show that pseudohyperbolic attractors emerge naturally at local and global bifurcations of codimension 3, hence they are present in a vast set of diverse applications. We also show that robust presence of homoclinic tangencies and heterodimensional cycles is a characteristic feature of pseudohyperbolic attractors.

Walter Van Assche, KU Leuven, Belgium

ORTHOGONAL POLYNOMIALS AND PAINLEVÉ EQUATIONS

Date: 2019-09-17 (Tuesday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

Painlevé equations are nonlinear differential equations for which the branch points do not depend on the initialcondition (no movable branch points). There are also discrete Painlevé equations which are non-linearrecurrence relations with enough structure (symmetry and geometry) that make them integrable.Both the discrete and continuous Painlevé equations appear in a natural way in the theory of orthogonalpolynomials. The recurrence coefficients of certain families of orthogonal polynomials often satisfya discrete Painlevé equation. The Toda equations describing the movement of particles with an exponentialinteraction with their neighbors, is equivalent to an exponential modification \(e^{xt}\, d\mu(x)\) of the orthogonality measure \(d\mu \) for a family of orthogonal polynomials, and the corresponding recurrencecoefficients satisfy the Toda equations, which is a system of differential-difference equations. Combining this with the discrete Painlevé equations then gives a Painlevé differential equation.We will illustrate this by a number of examples. The relevant solutions of these Painlevé equationsare usually in terms of known special functions, such as the Bessel functions, the Airy function, parabolic cylinder functions, or (confluent) hypergeometric functions.

References

1. W. Van Assche, Orthogonal Polynomials and Painlevé Equations, Australian Mathematical Society Lecture Notes 27, Cambridge University Press, (2018).

Nicolas Van Goethem, University of Lisbon, Portugal

INTRINSIC VIEWS IN ELASTO-PLASTICITY

Joint work with Samuel Amstutz

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building B-8, room 3.22.

Abstract

In this talk, I will present a new model of elasto-plasticity based on the elastic strain incompatibility. Existence results for a novel boundary value problem will be reported as well as asymptotic results. Indeed, this new system for incompatible elasticity generalizes the classical linearized elasticity system to which it converges as the incompatibility modulus converges to \(-\infty\). I will also briefly introduce intrinsic views from a historical point of view.

Sebastian van Strien, Imperial College London, UK

CONJUGACY CLASSES OF REAL ANALYTIC MAPS: ON A QUESTION OF AVILA-LYUBICH-DE MELO

Joint work with Trevor Clark

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building A-4, room 120.

Abstract

Avila-Lyubich-de Melo proved that the topological conjugacy classes of unimodal real-analytic maps are complex analytic manifolds, which laminate a neighbourhood of any such mapping without a neutral cycle. Their proof that the manifolds are complex analytic depends on the fact that they have codimension-one in the space of unimodal mappings.

In joint work with Trevor Clark, we show how to construct a “pruned polynomial-like mapping" associated to a real mapping. This gives a new complex extension of a real-analytic mapping.

The additional structure provided by this extension, makes it possible to generalize this result of Avila-Lyubich-de Melo to interval mappings with several critical points. Thus we show that the conjugacy classes are complex analytic manifolds whose codimension is determined by the number of critical points.

Building on these ideas, we will show that in the space of unimodal mappings with negative Schwarzian derivative, the conjugacy classes laminate a neighbourhood of every mapping.

András Vasy, Stanford University, USA

GLOBAL ANALYSIS VIA MICROLOCAL TOOLS: FREDHOLM THEORY IN NON-ELLIPTIC SETTINGS

Joint work with Peter Hintz

Date: 2019-09-17 (Tuesday); Time: 15:00-15:40; Location: building B-8, room 0.10a.

Abstract

In this lecture I will describe a microlocal framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, introduced in [8] and extended in [6] and various other works. Examples in which such a framework (or a similar framework) has recently been useful include wave propagation on black hole spacetimes, which is the key analytic ingredient for showing the stability of black holes [5, 4], analysis of the resolvent of the generator of the flow for dynamical systems [2], which is the key tool for the analysis of the Ruelle zeta function [1], Feynman propagators in quantum field theory [3, 9] and inverse problems, namely boundary rigidity and tensor tomography [7].

References

1. Semyon Dyatlov and Maciej Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér. (4) 49(3) (2016), 543–577.
2. Frédéric Faure and Johannes Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., 308(2) (2011), 325–364.
3. Jesse Gell-Redman, Nick Haber and András Vasy, The Feynman propagator on perturbations of Minkowski space, Comm. Math. Phys. 342(1) (2016), 333–384.
4. Peter Hintz and András Vasy, Linear stability of slowly rotating Kerr black holes, Preprint, arXiv:1906.00860, 2019.
5. Peter Hintz and András Vasy, The global non-linear stability of the Kerr–de Sitter family of black holes, Acta mathematica, 220 (2018), 1–206.
6. Peter Hintz and András Vasy, Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes, Anal. PDE 8(8) (2015), 1807–1890.
7. Plamen Stefanov, Gunther Uhlmann and András Vasy, Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge Preprint, arXiv:1702.03638, 2017.
8. András Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., 194(2):381–513, 2013.
9. András Vasy and Michał Wrochna, Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes, Ann. Henri Poincaré 19(5) (2018), 1529–1586.

Luis Vega, University of the Basque Country & BCAM, Spain

THE VORTEX FILAMENT EQUATION, THE TALBOT EFFECT, AND NON-CIRCULAR JETS

Joint work with Valeria Banica and Francisco De La Hoz

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building B-8, room 0.10a.

Abstract

We will propose the vortex filament equation as a possible toy model for turbulence, in particular because of its striking similarity to the dynamics of non-circular jets. This similarity implies the existence of some type of Talbot effect due to the interaction of non-linear waves that propagate along the filament. Another consequence of this interaction is the existence of a new class of multi-fractal sets that can be seen as a generalization of the graph of Riemann's non-differentiable function. Theoretical and numerical arguments about the transfer of energy will be also given.

Bozhidar Velichkov, Université Grenoble-Alpes, France

ON THE LOGARITHMIC EPIPERIMETRIC INEQUALITY

Joint work with Maria Colombo and Luca Spolaor

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

This talk is dedicted to some recent advances on the regularity of the free boundaries arising in variational minimization problems. In particular, we will present a new variational approach for the analysis of singularities: the logarithmic epiperimetric inequalities, which was already applied to several different free boundary problems: the obstacle problem [2], the thin-obstacle problem [3, 7], the one-phase Alt-Caffarelli problem [4], to area-minimizing currents [5], and to parabolic free boundary problems [8].

The focus of this talk is on the classical obstacle problem:\[\qquad\min\Big\{\int_{B_1} \!\!\big(|\nabla u|^2+u\big)\,dx\,:\, u\in H^1(B_1),\ u\ge 0\ \text{in}\ B_1,\ u\ \text{is prescribed on}\ \partial B_1\Big\}.\qquad{\rm(OB)}\]It is well-known that, given a solution \(u\) and setting \(\Omega_u:=\{u>0\}\), the free boundary \(\partial\Omega_u\cap B_1\) can be decomposed into a regular part, \(Reg\,(\partial\Omega_u)\), and a singular part, \(Sing\,(\partial\Omega_u)\), where

\(\bullet\) \(Reg\,(\partial\Omega_u)\) is a smooth manifold (this result was proved by Caffarelli in [1]);

\(\bullet\) \(Sing\,(\partial\Omega_u)\) are the points, at which the Lebesgue density of the set \(\{u=0\}\) vanishes.

In this talk, we will prove a logarithmic epiperimetric inequality for the Weiss' boundary adjusted energy, from which we will deduce that the singular set \(Sing\,(\partial\Omega_u)\) is \(C^{1,\log}\)-regular, that is, it is contained into the countable union of \(C^{1,\log}\)-regular manifolds. This results was first proved in [2]. Recently, Figalli and Serra [6] showed that this result is also optimal in the sense that the logarithmic modulus of continuity cannot be improved for general singularities.

References

1. L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155–184.
2. M. Colombo, L. Spolaor, B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal. 28 (2018), 1029–1061.
3. M. Colombo, L. Spolaor, B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, Comm. Pure. Appl. Math. (to appear).
4. M. Engelstein, L. Spolaor, B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, Preprint arXiv:1801.09276.
5. M. Engelstein, L. Spolaor, B. Velichkov (Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents, Geom. Topol. 23 (2019), 513–540.
6. A. Figalli, J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math. 215 (2019), 311–366.
7. S. Jeon, A. Petrosyan, Almost minimizers for the thin obstacle problem, Preprint arXiv 1905.11956.
8. W. Shi, An epiperimetric inequality approach to the parabolic Signorini problem, Preprint arXiv 1810.11791.

Sjoerd Verduyn Lunel, Utrecht University, Netherlands

DELAY EQUATIONS AND TWIN SEMIGROUPS

Joint work with Odo Diekmann

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

A delay equation is a rule for extending a function of time towards the future on the basis of the (assumed to be) known past. By translation along the extended function (i.e., by updating the history), one defines a dynamical system. If one chooses as state-space the continuous initial functions, the translation semigroup is continuous, but the initial data corresponding to the fundamental solution is not contained in the state space.

In ongoing joint work with Odo Diekmann, we choose as state space the space of bounded Borel functions and thus sacrifice strong continuity in order to gain a simple description of the variation-of-constants formula.

The aim of the lecture is to introduce the perturbation theory framework of twin semigroups on a norming dual pair of spaces, to show how renewal equations fit in this framework and to sketch how neutral equations can be covered. The growth of an age-structured population serves as a pedagogical example.

Bruno Vergara, ICMAT, Spain

CONVEXITY OF WHITHAM’S HIGHEST WAVE

Joint work with Alberto Enciso and Javier Gómez-Serrano

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

In this talk I will discuss a conjecture of Ehrnström and Wahlén [1] on the profile of solutions of extreme form to Whitham’s model of shallow water waves. This is a non-local dispersive equation featuring travelling waves and singularities. Analogously to Stokes waves for Euler, we will see that there exists a highest, cusped and periodic solution to this model which is convex between consecutive crests [2].

References

1. M. Ehrnström, E. Wahlén, On Whitham's conjecture of a highest cusped wave for a nonlocal shallow water wave equation, Ann. Inst. H. Poincaré Anal. Non. Linéaire, in press (2019), arXiv:1602.05384.
2. A. Enciso, J. Gómez-Serrano, B. Vergara, Convexity of Whitham's highest cusped wave, (2018), arXiv:1810.10935.

Alex Viguerie, University of Pavia, Italy

A FAT BOUNDARY-TYPE METHOD FOR LOCALIZED NONHOMOGENEOUS MATERIAL PROBLEMS

Joint work with Ferdinando Auricchio and Silvia Bertoluzza

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building B-8, room 3.22.

Abstract

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a uniform mesh numerically unfeasible. While nonuniform meshes can be employed, they may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. To address the aforementioned challenges, we employ a technique related to the Fat boundary method [1] as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings with a series of two-dimensional numerical tests.

References

1. A. Viguerie, S. Bertoluzza, F. Auricchio, A Fat Boundary-type Method for Localized Nonhomogeneous Material Problems, submitted.

Jordi Villadelprat, Rovira i Virgili University, Spain

DULAC TIME FOR FAMILIES OF HYPERBOLIC SADDLE SINGULARITIES

Joint work with David Marín

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

We consider a smooth unfolding of a saddle point and we fix two transverse sections, the first one at the stable separatrix and the second one at the unstable separatrix. The Dulac time is the time that spends the flow for the transition from the first to the second transverse section. We present a structure theorem for the asymptotic expansion of the Dulac time, with the principal part expressed in terms of Roussarie's monomial scale, and the remainder having flat properties that are well preserved through the division-derivation algorithm. We also provide explicit formulae for the coefficients of the first monomials in the principal part by means of a new integral operator that generalizes Mellin transform. We explain its applicability in the study of the bifurcation diagram of the period function of the quadratic reversible centers.

Kurt Vinhage, University of Chicago, USA

CLASSIFICATION OF TOTALLY CARTAN ACTIONS

Joint work with Ralf Spatzier

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

We will discuss recent progress on the Katok-Spatzier conjecture, which aims to classify Anosov \(\mathbb{R}^k\) and \(\mathbb{Z}^k\) actions under the assumption that there are no nontrivial smooth rank one factors. Classification is the strongest conclusion in the smooth rigidity program, which assumes nothing about the structure of the underlying manifold or dynamics other than the Anosov hyperbolicity assumptions. We develop new techniques to build homogeneous structures from dynamical ones. The remarkable features of the techniques are their low regularity requirements and their use of metric geometry over differential geometry to build group actions. We apply these techniques to the totally Cartan setting, where bundles associated to the hyperbolic strcuture are one-dimensional. Joint with Ralf Spatzier.

Liz Vivas, Ohio State University, USA

NON-AUTONOMOUS PARABOLIC BIFURCATION

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

Let \(f(z) = z+z^2+O(z^3)\) and \(f_\epsilon(z) = f(z) + \epsilon^2\). A classical result in parabolic bifurcation [1, 2] in one complex variable is the following: if \(N-\frac{\pi}{\epsilon}\to 0\) we obtain \((f_\epsilon)^{N} \to \mathcal{L}_f\), where \(\mathcal{L}_f\) is the Lavaurs map of \(f\). In this paper we study a non-autonomous parabolic bifurcation. We focus on the case of \(f_0(z)=\frac{z}{1-z}\). Given a sequence \(\{\epsilon_i\}_{1\leq i\leq N}\), we denote \(f_n(z) = f_0(z) + \epsilon_n^2\). We give sufficient and necessary conditions on the sequence \(\{\epsilon_i\}\) that imply that \(f_{N}\circ\ldots f_{1} \to \textrm{Id}\) (the Lavaurs map of \(f_0\)). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.

References

1. A.Douady, Does a Julia set depend continuously on the polynomial?, Proc. Sympos. Appl. Math. 49 (1994), 91-138.
2. P. Lavaurs, Systemes dynamiques holomorphes: explosion de points périodiques paraboliques, PhD thesis, Paris 11, 1989.

Anja Voß-Böhme, Hochschule für Technik und Wirtschaft Dresden, Germany

MODEL-BASED INFERENCES ABOUT CELLULAR MECHANISMS OF TUMOR DEVELOPMENT FROM TISSUE-SCALE DATA

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 1.11.

Abstract

Cancer development is widely understood as a multistep process in which cells increase in malignancy through progressive genotypic and phenotypic alterations. Although there is an increasing knowledge about the biology of the involved cellular processes in vitro, the transfer of the results to in-vivo situations remains a challenge. This is due to the fact that the early phase of tumor development, which initially involves only a small number of cells, is hardly observable while the cellular basis of tissue-scale observations is difficult to decipher. Cell-based mathematical models provide a valuable tool to investigate in which way tissue-scale observables depend on cellular mechanisms and intercellular interaction. Here, we analyze the population dynamics of spatial and non-spatial Markov models which describe genetic and phenotypic cell changes and use these findings to calibrate the models by tissue-scale data. This allows to generate quantitative predictions about tumor initiation, progression and regression behavior.

References

1. T. Buder, A. Deutsch, B. Klink, A. Voss-Böhme, Patterns of Tumor Progression Predict Small and Tissue-Specific Tumor-Originating Niches, Frontiers in Oncology 8(668) (2019).
2. T. Buder, A. Deutsch, M. Seifert, A. Voss-Böhme, CellTrans: An R package to quantify stochastic cell state transitions, Bioinformatics and Biology Insights 11 (2017), 1-14.
3. T. Buder, A. Deutsch, B. Klink, A. Voss-Böhme, Model-Based Evaluation of Spontaneous Tumor Regression in Pilocytic Astrocytoma, PLoS Computational Biology 11(12) (2015), e1004662.

Hao Wang, University of Alberta, Canada

ANIMAL MOVEMENT WITH SPATIAL MEMORY

Joint work with Junping Shi, Chuncheng Wang, Qingyan Shi, Yongli Song, and Xiangping Yan

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

Animals often self-organize into territorial structure from movements and interactions of individual animals. Memory is one of the cognitive processes that may affect the movement and navigation of the animals. I will review several mathematical approaches to animal spatial movements, and then introduce our recent work via a modified Fick's law to model and simulate the memory-based movement. Results on bifurcation and pattern formation will be shown for these non-standard reaction-diffusion models.

References

1. J. Shi, C. Wang, H. Wang, X. Yan, Diffusive spatial movement with memory, Journal of Dynamics and Differential Equations, in press, 2019.
2. J. Shi, C. Wang, H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, in press, 2019
3. Q. Shi, J. Shi, H. Wang, Spatial Movement with distributed memory, preprint.

Thomas Ward, University of Leeds, UK

TIME-CHANGES PRESERVING ZETA FUNCTIONS

Joint work with Sawian Jaidee and Patrick Moss

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

A time-change is a function \(h\colon\mathbb{N}\to\mathbb{N} \), and \(h\) is said to 'preserve zeta functions' if, for any dynamical zeta function \(\exp\left(\sum_{n\ge1}a_nz^n/n\right)\), where \(a_n=\vert\{x\in X\mid T^nx=x\}\vert\) for some dynamical system \(T\colon X\to X\), the time-changed function \(\exp\left(\sum_{n\ge1}a_{h(n)}z^n/n\right)\) is the dynamical zeta function of some dynamical system. That is, for any homeomorphism of a compact metric space \(T\colon X\to X\) there is some other homeomorphism of a compact metric space \(S\colon Y\to Y\) with the property that \(\vert\{x\in X\mid T^{h(n)}x=x\}\vert = \vert\{y\in Y\mid S^ny=y\}\vert\) for all \(n\in\mathbb{N}\). The time-changes that preserve zeta functions form a monoid \(\mathcal{P}\), and we show that a polynomial lies in \(\mathcal{P}\) if and only if it is a monomial (meaning that \(\mathcal{P}\) is algebraically small), that \(\mathcal{P}\) is uncountable (meaning that it is set-theoretically large), and that \(\mathcal{P}\) contains no permutations (that is, \(\mathcal{P}\) has no torsion as a monoid).

References

1. S. Jaidee, P. Moss, T. Ward, Time-changes preserving zeta functions, Proc. Amer. Math. Soc. (to appear).
2. A. Pakapongpun, T. Ward, Functorial orbit counting, J. Integer Seq. 12(2) (2009), Article 09.2.4.
3. Y. Puri, T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seq. 4(2) (2001), Article 01.2.1.
4. A. J. Windsor, Smoothness is not an obstruction to realizability, Ergodic Theory Dynam. Systems 28(3) (2008), 1037-1041.

Claude Warnick, University of Cambridge, UK

QUASINORMAL MODES OF BLACK HOLES

Joint work with Dejan Gajic

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 0.10b.

Abstract

In recent years the problem of defining the quasinormal modes of subextremal black holes has been satisfactorily resolved for asymptotically de Sitter [1] and anti-de Sitter black holes [2]. The quasinormal frequencies may be realised as eigenvalues of a Fredholm operator resulting from a natural choice of coordinates on the black hole background. I will report on recent work with Dejan Gajic to extend this approach to treat extremal and asymptotically flat black hole spacetimes.

References

1. A. Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, Invent. Math 194 (2013), 381-513.
2. C. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Comm. Math. Phys. 333 (2015), 959-1035.

Hendrik Weber, University of Bath, UK

A-PRIORI BOUNDS FOR SINGULAR SPDES

Joint work with Ajay Chandra and Augustin Moinat

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 2.18.

Abstract

The theory of regularity structures is a powerful tool to develop a stable solution theory for a whole class of stochastic PDEs arising in statistical mechanics and quantum field theory. Initiated in Hairer's groundbreaking work in 2013, in only a few years an astonishingly general solution theory covering essentially all equations which satisfy a certain scaling condition (subcriticality or super-renormalizability), has been developed. However, up to now, most results only gave control over solutions for small times and on bounded spatial domains.

The aim if this talk is to present a method to prove a priori estimates in the framework regularity structures. These bounds complement the short time existence and uniqueness theory to obtain control of solutions globally in time and on unbounded domains. Our bounds are implemented in the example of the dynamic \(\Phi^4\) equation, which is formally given by \[ (\partial_t - \Delta) u = -u^3 + \infty u +\xi. \] This equation is subcritical if the distribution \(\xi\) is of class \(C^{-3+\frac{\delta}{2}}\) for \(\delta>0\), and we obtain bounds for all such \(\xi\). An analogy to the regularity of white noise suggests to interpret this as a solution theory for \(\Phi^4\) in dimension \(4-\delta\).

References

1. A. Chandra, A. Moinat, H. Weber, A priori bounds for \(\Phi^4\) in the subcritical regime, in preparation.
2. A. Moinat, H. Weber, Space-time localisation for the dynamic \(\Phi^ 4_3\) model, arXiv preprint arXiv:1811.05764 (2018).

Daren Wei, Pennsylvania State University, USA

KAKUTANI EQUIVALENCE OF UNIPOTENT FLOWS

Joint work with Adam Kanigowski and Kurt Vinhage

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-4, room 120.

Abstract

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only standard unipotent flows are given by \(\begin{pmatrix}1&t\\0&1\end{pmatrix}\times \operatorname{id}\) acting on \((\operatorname{SL}(2,\mathbb{R})\times G')/\Gamma'\), where \(\Gamma'\) is an irreducible lattice in \(\operatorname{SL}(2,\mathbb{R})\times G'\) (with the possibility that \(G' = \{e\}\)).

References

1. A. Kanigowski, K. Vinhage, D. Wei, Kakutani equivalence of unipotent flows, preprint arXiv:1805.01501.

Benjamin Weiss, Hebrew University of Jerusalem, Israel

ON THE COMPLEXITY OF SMOOTH SYSTEMS

Joint work with Matthew Foreman

Date: 2019-09-19 (Thursday); Time: 11:15-11:45; Location: building A-4, room 106.

Abstract

About ten years ago, in joint work with the late Dan Rudolph and Matt Foreman, we showed that the isomorphism relation for ergodic measure systems is not Borel, but rather a complete analytic set. In fact we showed that the transformations that are isomorphic to their inverses is already complete analytic. Since the smooth realization problem is still open it was not clear how complex is the class of diffeomorphisms of compact manifolds that preserve a volume element. In more recent work with Matt Foreman we show that already the ergodic diffeomorphisms of the torus that preserve Lebesgue measure is also a complete analytic set.

Adrian Weisskopf, Michigan State University, USA

NORMAL FORM METHODS AND RIGOROUS GLOBAL OPTIMIZATION FOR THE ASTRODYNAMICAL BOUNDED MOTION PROBLEM

Joint work with Martin Berz and Roberto Armellin

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

Due to their common origin and mathematical underpinnings, it is sometimes possible to transfer specific advanced methods to analyze dynamical systems from one field of applications to another. In this work, we illustrate the transfer of differential algebra (DA) based normal form methods and rigorous global optimization for Nekhoroshev-type stability estimates, which were first developed in the field of particle beam physics and accelerator physics, to the field of astrodynamics to design bounded motion in the Earth's zonal problem.

The DA framework [1] and in particular the DA normal form algorithm [4], and their associated techniques are hybrid methods of numerical and analytic calculations and have been established by Berz et al. The methods can be turned mathematically rigorous by fully accounting for expansion errors and floating point inaccuracies, as is done in the Taylor model methods discussed in this session.

Many of the DA tools have been applied in the field of accelerator physics, where they reveal details of those dynamical systems that are otherwise very difficult to obtain by conventional methods. More recently, researchers have begun on the fruitful transfer of those DA methods to the astrodynamics community [3, 6, 5, 2]. An advancement of this transfer to normal form methods provides new possibilities, as it will be demonstrated in this work, including the capability of determining entire sets of bounded motion orbits in the full zonal problem.

Given an origin preserving Poincaré return map of a repetitive Hamiltonian system expanded in its phase space coordinates and system parameters, the DA normal form algorithm provides a nonlinear change of variables by an order-by-order transformation to normal form coordinates in which the motion is rotationally invariant. This circular phase space behavior allows for a straightforward extraction of the phase space rotation frequency and an action-angle parameterization of the normal form motion.

The normal form parameterization is used to find orbit sets satisfying the bounded motion condition, i.e. same average nodal period and drift in the ascending node of the bounded orbits. For the right Poincaré surface of the Poincaré return map, the inverse normal form transformation is used to parameterize the map by the action-angle parameterization of the corresponding normal form motion. Averaging the map over a full phase space revolution by a path integral along the angle-parameterization yields the averaged nodal period and drift in the ascending node for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers and valid for more than ten years.

References

1. M. Berz, Differential algebraic description of beam dynamics to very high orders, Part. Accel. 24 (1988), 109-124.
2. A. Wittig, R. Armellin, High order transfer maps for perturbed Keplerian motion, Celestial Mechanics and Dynamical Astronomy 122 (2015), 333-358.
3. P. Di Lizia, R. Armellin, M. Lavagna, Application of high order expansions of two-point boundary value problems to astrodynamics, Celestial Mechanics and Dynamical Astronomy 102 (2008), 355-375.
4. M. Berz, Modern Map Methods in Particle Beam Physics, Academic Press, 1999.
5. R. Armellin, P. Di Lizia, F. Topputo, M. Lavagna, F. Bernelli-Zazzera, M. Berz, Gravity assist space pruning based on differential algebra, Celestial Mechanics and Dynamical Astronomy 106 (2010), 1-24.
6. R. Armellin, P. Di Lizia, M. Berz, K. Makino, Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization, Celestial Mechanics and Dynamical Astronomy 107 (2010), 377-395.

Zhi-Tao Wen, Shantou University, China

INTEGRABILITY OF DIFFERENCE EQUATIONS WITH BINOMIAL SERIES

Joint work with Katsuya Ishizaki

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building B-7, room 2.2.

Abstract

We consider binomial series \(\sum_{n=0}^\infty a_n z^{\underline{n}}\), where \(z^{\underline{n}}=z(z-1)\cdots(z-n+1)\). Integrability by binomial series is concerned for difference equations. In this talk, we consider a formal solution of a difference equation written by binomial series. Further, we discuss conditions of convergence of these formal solutions to find a sufficient condition for meromorphic solutions, and investigate the order of growth of them. As an application, we construct a difference Riccati equation possessing a transcendental meromorphic solution of order \(1/2\).

Klaus Widmayer, Ecole Polytechnique Fédérale de Lausanne, Switzerland

LONG TIME DYNAMICS IN THE ROTATING EULER EQUATIONS

Joint work with Yan Guo and Benoit Pausader

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building B-8, room 0.10b.

Abstract

We investigate long time dynamics of solutions to the rotating Euler equations in three spatial dimensions. We develop a framework that is adapted to the symmetries and the dispersive properties of this problem and show how it can be used to understand the behavior of small data solutions, uniformly in the parameter of rotation.

The key idea is to use the available symmetries as much as possible, rather than to pursue a more brute force approach. While this streamlines the deduction of some energy type estimates, it also requires a fresh look at the (linear) dispersive estimates, deviating from the classical stationary phase intuition.

Daniel Wilczak, Jagiellonian University in Kraków, Poland

CONTINUATION AND BIFURCATIONS OF HALO ORBITS IN THE CIRCULAR RESTRICTED THREE BODY PROBLEM

Joint work with Irmina Walawska

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building B-7, room 2.4.

Abstract

We propose a general framework for computer-assisted verifcation of isochoronus, period- tupling or touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamiltonian systems.

In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof [1] of the existence of wide branches of halo orbits bifurcating from \(L_{1,2,3}\)-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters (Sun-Jupiter and Earth-Moon systems) we prove, that \(L_{1,2}\) branches of halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.

The computer-assisted proof uses rigorous ODE solvers and algorithms for computation of Poincare maps and their derivatives from the CAPD library [2].

References

1. I. Walawska, D. Wilczak, Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems, Commun. Non. Sci. Num. Simul. 74C (2019), 0-54.
2. CAPD library: C++ package for validated numerics for discrete and continuous dynamical systems, http://capd.ii.uj.edu.pl.

Bettina Wilkens, University of Namibia, Namibia

A RING-THEORETIC APPROACH TO DISCRETE SPECTRAL SYNTHESIS

Joint work with László Székelyhidi

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

Let \(G\) be an Abelian group and let \(\mathcal C(G)\) be the vector space of complex-valued functions on \(G\). With the topology of pointwise convergence, \(\mathcal C(G)\) is a locally convex space. The group \(G\) acts on \({\mathcal C}G\) by translations. Closed submodules of \({\mathcal C}G\) are called varieties. Consider the bilinear product \(\mathcal C(G) \times \mathbb{C} G \rightarrow \mathbb{C}\) given by \[ \langle \sum\limits_{x \in G} a_x x, \, f \rangle =\sum\limits_{x \in G} a_x f(x).\] Assigning the function \(f \mapsto \langle a, \, f \rangle\) to \(a\) in \(\mathbb{C} G\) yields an identification of \(\mathbb{C}G G\) with the space of linear functionals on \({\mathcal C}G \) that are continuous with respect to the topology of pointwise convergence. Assigning the map \(a \mapsto \langle a, \, f \rangle\) to \(f\) provides an identification of \(\mathcal C(G)\) with the algebraic dual \(\mathbb{C}G^{\ast}.\) Defining orthogonal complements in the usual way, the Hahn-Banach theorem yields that the map \(V \mapsto V^{\perp}\) is a one-to-one correspondence between varieties and ideals of \(\mathbb{C}G\).

We exploit this to investigate to characterise the dual \(\mathbb{C}G/V^{\perp}\) when \(V\) is a variety with spectral analysis - each subvariety of \(V\) contains a one-dimensional module - that is syntheziable - topologically generated by its finite-dimensional subvarieties- or possesses spectral synthesis - only has synthesizable subvarieties. We shall see that these properties correspond to well-known and well-researched properties of commutative rings. If \(V\) has spectral synthesis- the strongest of the listed properties, then \(\mathbb{C}G/V^{\perp}\) emerges as "almost" Noetherian. Finally, we discuss how the ring theoretic results may be used to provide a description of the module structure of \(V\).

References

1. R. Gilmer, W. Heinzer, Ideals contracted from a Noetherian extension ring, J. Pure Appl. Algebra 24(2) (1980), 123-144.
2. L. Székelyhidi, Harmonic and Spectral Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.
3. L. Székelyhidi, B. Wilkens, Spectral synthesis and residually finite-dimensional algebras, J. Algebra Appl. 16 (2017), 10 pp.

Carsten Wiuf, University of Copenhagen, Denmark

ON THE STABILITY OF THE STEADY STATES IN THE N-SITE FUTILE CYCLE

Joint work with Elisenda Feliu and Alan Rendall

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building B-8, room 1.26.

Abstract

The multiple or \(n\)-site futile cycle is a biological process that resides in the cell. Specifically, it is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially \(n\) times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. In its standard form it has \(3n+3\) variables (concentrations of species) and \(6n\) parameters. It is known that the system might have at least as many as \(2\lfloor \tfrac{n}{2}\rfloor+1\) steady states (where \(\lfloor x\rfloor\) is the integer part of \(x\)) for particular choices of parameters. Furthermore, for the simple futile cycle (\(n=1\)) there is only one steady state which is globally stable. For the dual futile cycle (\(n=2\)) the stability of the steady states has been determined in the following sense: There exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general \(n\), evidence that the possible number of asymptotically stable steady states increases with \(n\) has been given, which has led to the conjecture that parameter values can be chosen such that \(\lfloor\tfrac{n}{2}\rfloor+1\) out of \(2\lfloor\tfrac{n}{2}\rfloor+1\) steady states are asymptotically stable and the remaining steady states are unstable.

We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity \(2\lfloor\tfrac{n}{2}\rfloor+1\). Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

Christos Xenophontos, University of Cyprus, Cyprus

APPROXIMATION OF FOURTH ORDER TWO-POINT SINGULARLY PERTURBED PROBLEMS OF REACTION-DIFFUSION TYPE

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building B-8, room 3.21.

Abstract

We consider fourth order two-point singularly perturbed problems of reaction-diffusion type and the approximation of their solution by Galerkin's method. We consider both \(hp\) Finite Elements (FEs) and Isogeometric Analysis (IGA). We first present regularity results which show that the solution may be decomposed into a smooth part, two boundary layers at the endpoints and a (negligible) remainder. Estimates for each part in the decomposition are obtained, which are explicit in the order of differentiation and the singular perturbation parameter [1]. Guided by these results, we construct an approximation using the so-called Spectral Boundary Layer mesh in FEs [2] and knot-vector in IGA [3], which converges independently of the singular perturbation parameter. When the error is measured in the energy norm associated with the problem, the convergence rate is exponential, as the degree of the approximating polynomials is increased. Numerical examples illustrating the theory will also be presented.

References

1. P. Constantinou, \(hp\) Finite Element Methods for Fourth Order Singularly Perturbed Problems, Ph.D. Dissertation, Department of Mathematics and Statistics, University of Cyprus, 2019.
2. J.M. Melenk, C. Xenophontos, L. Oberbroeckling, Robust exponential convergence of hp-FEM for singularly perturbed systems of reaction-diffusion equations with multiple scales, IMA J. Num. Anal. 33 (2013), 609-628.
3. K. Liotati, C. Xenophontos, Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study, BAIL2018 Conference (Glascow, Scotland, 2018), 2019 (to appear).

Disheng Xu, University of Chicago, USA

ON THE RIGIDITY OF PARTIALLY HYPERBOLIC \(\mathbb{Z}^k\) ACTION

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

Roughly speaking, a partially hyperbolic diffeomorphism on a manifold \(M\) is a certain type of mapping, from \(M\) to itself, with local directions of “expansion”, “neutral” and “contraction”. The study of the partially hyperbolic system, i.e. the \(\mathbb{Z}\)-action generated by the iterates of a partially hyperbolic diffeomorphism, is one of the central topic in dynamical systems in the last four decades.

On the other hand, in general it is expected that under suitable assumptions, a \(\mathbb{Z}^k\) action by diffeomorphisms on manifold, \(k > 1\) shares some strong rigidity properties (stronger than that of \(\mathbb{Z}\)-action). A \(\mathbb{Z}^k\)-action on a manifold is called partially hyperbolic if the action contains at least one partially hyperbolic diffeomorphism. In this talk we will show some recent rigidity results on the study of partially hyperbolic \(\mathbb{Z}^k\)-action on manifolds, this is a joint work with D. Damjanović and A. Wilkinson.

Jonguk Yang, Stony Brook University, USA

RECURRENCE OF ONE-SIDED SEQUENCES UNDER SHIFT

Date: 2019-09-19 (Thursday); Time: 11:05-11:25; Location: building A-3/A-4, room 105.

Abstract

Consider the shift map acting on the space of one-sided sequences. Under this dynamics, a sequence exhibits one of three types of recurrence: non-recurrence, reluctant recurrence, or persistent recurrence. However, for a given arbitrary sequence, it can be difficult to determine which of these three possibilities will occur. To solve this problem, we introduce an algebraic structure on sequences called filtration that enables us to count recurrences efficiently. This then leads to the characterization of the persistent recurrence property as a kind of infinite renormalizability of the shift map.

Shing-Tung Yau, Harvard University, USA

STABILITY AND NONLINEAR PDES IN MIRROR SYMMETRY

Date: 2019-09-16 (Monday); Time: 09:00-10:00; Location: building U-2, auditorium.

Abstract

I shall give a talk about a joint work that I did with Tristan Collins on an important nonlinear system equation of Monge-Ampère type. It is motivated from the theory of Mirror symmetry in string theory. I shall also talk about its algebraic geometric meaning.

Jiangong You, Nankai University, China

DYNAMICAL SYSTEM APPROACH TO SPECTRAL THEORY OF QUASI-PERIODIC SCHRÖDINGER OPERATORS

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building B-7, room 1.8.

Abstract

The spectral theory of quasiperiodic operators is a fascinating field which continuously attracts a lot of attentions for its rich background in quantum physics as well as its rich connections with many mathematical theories and methods. In this talk, I will briefly introduce the problems in this field and their connections with dynamical system. I will also talk about some recent results joint with Avila, Ge, Leguil, Zhao and Zhou on both spectrum and spectral measure by reducibility theory in dynamical systems.

References

1. A. Avila, J. You and Q. Zhou, Sharp phase transitions for the almost Mathieu operator, Duke Math. J. 166 (2017), 2697-2718.
2. A. Avila, J. You and Q. Zhou, The dry Ten Martini problem in the non-critical case, Preprint.
3. M. Leguil, J. You, Z. Zhao, Q. Zhou, Asymptotics of spectral gaps of quasi-periodic Schrödinger operators, arXiv:1712.04700.
4. L. Ge, J. You and Q. Zhou, Exponential dynamical localization: Criterion and applications, arXiv:1901.04258.
5. L. Ge, J. You, Arithmetic version of Anderson localization via reducibility, Preprint.

Manuel Zamora, University of Oviedo, Spain

SECOND-ORDER INDEFINITE SINGULAR EQUATIONS. THE PERIODIC CASE

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-7, room 1.9.

Abstract

In this talk we will discuss the existence of a \(T-\)periodic solution to the second order differential equation \[ u''=\frac{h(t)}{u^{\lambda}}, \] where \(\lambda>0\) and the weight \(h\in L(\mathbb{R}/T\mathbb{Z})\) is a sign-changing function. When \(\lambda\geq 1\), the obtained results have the form of relation between the multiplicities of the zeroes of the weight function \(h\) and the order of the singularity of the nonlinear term. Nevertheless, when \(\lambda\in (0,1)\), the key ingredient to solve the aforementioned problem is connected more with the oscillation and the symmetry aspects of the weight function \(h\) than with the multiplicity of its zeroes.

References

1. J. Godoy, M. Zamora, A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity, J. Dyn. Diff. Equat. 31 (2019), 451–468.
2. R. Hakl, M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Diff. Equat. 263 (2017), 451–469.

Thaleia Zariphopoulou, University of Texas at Austin, USA

STOCHASTIC MODELING AND OPTIMIZATION IN HUMAN-MACHINE INTERACTION SYSTEMS

Joint work with Agostino Capponi and Svein Olefsson

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building B-8, room 0.10b.

Abstract

I will introduce a family of human-machine interaction (HMI) models in optimal asset allocation, risk management and portfolio choice (robo-advising). Modeling difficulties stem from the limited ability to quantify the human's risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, itself adapts to real-time incoming information that is exogenous to the human. Furthermore, the human's risk preferences and the machine's states may evolve at different scales. This interaction creates an adaptive cooperative game with asymmetric and incomplete information exchange between the two parties.

As a result, challenging questions arise on, among others, how frequently the two parties should communicate, what information can the machine accurately detect, infer and predict, how the human reacts to exogenous events and what are the effects on the machine's actions, how to improve the inter-linked reliability between the human and the machine, and others.

Such HMI models give rise to new, non-standard optimization problems that include well-posed and ill-posed sub-problems, and combine adaptive stochastic control, stochastic differential games, optimal stopping, multi-scales and learning.

References

1. A. Capponi, S. Olefsson and T. Zariphopoulou, Personalized robo-advising, Preprint, 2019.

Anna Zatorska-Goldstein, University of Warsaw, Poland

RENORMALIZED SOLUTIONS FOR PARABOLIC PROBLEMS WITH ANISOTROPIC STRUCTURE AND NONSTANDARD GROWTH

Joint work with Iwona Chlebicka and Piotr Gwiazda

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building B-8, room 0.10a.

Abstract

We investigate nonlinear parabolic problems where the ellipticity and the growth conditions for the leading part of the operator is driven by an inhomogeneous and anisotropic function of the Orlicz type. We establish the existence and the uniqueness of renormalized solutions when merely integrable data are allowed. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Our results cover in particular the case of variable exponent growth, with the exponent depending both on time and space variables.

References

1. P. Gwiazda, I. Skrzypczak, A. Zatorska-Goldstein, Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space, J. Differential Equations 264(1) (2018), 341–377.
2. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Well-posedness of parabolic equations in the non- reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions, J. Differential Equations 265(11) (2018), 5716–5766.
3. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Renormalized solutions to parabolic equations in time and space anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, J. Differential Equations 267(2) (2019), 1129–1166.
4. I. Chlebicka, P. Gwiazda, A. Zatorska-Goldstein, Parabolic equation in time and space anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon, Ann. Inst. H. Poincaré Anal. Non Linéaire (2019), https://doi.org/10.1016/j.anihpc.2019.01.003.

Piotr Zgliczyński, Jagiellonian University in Kraków, Poland

CENTRAL CONFIGURATIONS IN PLANAR \(n\)-BODY PROBLEM FOR \(n = 5,6,7\) WITH EQUAL MASSES

Joint work with Małgorzata Moczurad

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

We give a computer assisted proof of the full listing of central configuration for \(n\)-body problem for Newtonian potential on the plane for \(n=5,6,7\) with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For \(n=8,9,10\) we establish the existence of central configurations without any reflectional symmetry.

References

1. M. Moczurad, P. Zgliczyński, Central configurations in planar \(n\)-body problem for \(n=5,6,7\) with equal masses, arXiv:1812.07279.

Guohua Zhang, Fudan University, China

ASYMPTOTIC \(h\)-EXPANSIVENESS FOR AMENABLE GROUP ACTIONS

Joint work with Tomasz Downarowicz

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

Asymptotic \(h\)-expansiveness for amenable group actions can be introduced respectively using topological conditional entropy in [1] and using entropy structure in [2]. In this talk we will show the equivalence of these two kinds of asymptotic \(h\)-expansiveness.

References

1. N.-P. Chung and G. Zhang, Weak expansiveness for actions of sofic groups, J. Funct. Anal. 268(11) (2015), 3534-3565.
2. T. Downarowicz and G. Zhang, Symbolic extensions of amenable group actions and the comparison property, arXiv:1901.01457, preprint.

Ke Zhang, University of Toronto, Canada

DIFFUSION LIMIT FOR THE SLOW-FAST STANDARD MAP

Joint work with Alex Blumenthal and Jacopo De Simoi

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building B-7, room 2.4.

Abstract

We discuss a simple two-dimensional slow-fast system, which is conjugate to the Chirikov standard map with a large parameter. Consider a random initial condition and view the \(n\)th iterate of the slow variable as a sequence of random variables, we prove a central limit theorem for this sequence, under suitable parameter values and time horizon. Our main motivation for studying this model is a phenomenon called "scattering by resonance" in physical systems.

Tusheng Zhang, University of Manchester, UK

LARGE DEVIATION PRINCIPLES FOR FIRST-ORDER SCALAR CONSERVATION LAWS WITH STOCHASTIC FORCING

Joint work with Zhao Dong, Jianglun Wu, and Rangrang Zhang

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building B-8, room 2.18.

Abstract

In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.

References

1. M. Boué, P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26(4) (1998), 1641-1659.
2. A. Debussche, M. Hofmanová, and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab. 44(3) (2016), 1916-1955.
3. A. Matoussi, W. Sabbagh, T. Zhang, Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs, To appear in Applied Mathematics and Optimization, arXiv:1712.02169.
4. A. Debussche, J. Vovelle, Scalar conservation laws with stochastic forcing (revised version). http://math.univ-lyon1.fr/vovelle/DebusscheVovelleRevised, J. Funct. Anal. 259(4) (2010), 1014-1042.

Weinian Zhang, Sichuan University, China

INVARIANT MANIFOLDS WITH/WITHOUT SPECTRAL GAP

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building B-7, room 1.8.

Abstract

In this talk we discuss invariant manifolds obtained with or without a spectral gap condition, showing approximation to weak hyperbolic manifolds (with gap condition) and giving the existence and smoothness for invariant submanifolds on a center manifold (without gap condition).

Wenmeng Zhang, Chongqing Normal University, China

SMOOTH LINEARIZATION WITH A NONUNIFORM DICHOTOMY

Joint work with Davor Dragičević and Weinian Zhang

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 1.8.

Abstract

In this talk, we give two smooth linearization theorems for \(C^{1,1}\) nonautonomous systems with a nonuniform strong exponential dichotomy. The first theorem concerns \(C^1\) linearization with a gap condition, while the second one concerns simultaneously differentiable and Hölder continuous linearization without any gap conditions. Restricted in the autonomous case, the second result gives the simultaneously differentiable and Hölder linearization of \(C^{1,1}\) hyperbolic systems without any non-resonant conditions.

Huaizhong Zhao, Loughborough University, UK

RANDOM PERIODICITY: THEORY AND MODELLING

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building B-8, room 2.18.

Abstract

Random periodicity is ubiquitous in the real world. In this talk, I will provide the concepts of random periodic paths and periodic measures to mathematically describe random periodicity. It is proved that these two different notions are "equivalent". Existence and uniqueness of random periodic paths and periodic measures for certain stochastic differential equations are proved. An ergodic theory is established. It is proved that for a Markovian random dynamical system, in the random periodic case, the infinitesimal generator of its Markovian semigroup has infinite number of equally placed simple eigenvalues including 0 on the imaginary axis. This is in contrast to the mixing stationary case in which the Koopman-von Neumann Theorem says there is only one eigenvalue 0, which is simple, on the imaginary axis. Geometric ergodicity for some stochastic systems is obtained. Possible applications e.g. in stochastic resonance will be discussed.

This talk is based on a series of work with Chunrong Feng et al.

Linfeng Zhou, Sichuan University, China

NONUNIFORM EXPONENTIAL DICHOTOMY AND ADMISSIBILITY

Joint work with Kening Lu and Weinian Zhang

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 1.8.

Abstract

Nonuniform exponential dichotomy describes nonuniform hyperbolicity for linear nonautonomous dynamical systems. In this talk, we present results on the relationships between nonuniform exponential dichotomies and admissible pairs for classes of weighted bounded functions, and the equivalent relationships between nonuniform exponential dichotomy and admissible pairs of classes of Lyapunov bounded functions.

References

1. L. Zhou, K. Lu, W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility, J. Differential Equations 262 (2017), 682-747.
2. L. Zhou, W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Functional Analysis 271 (2016), 1087-1129.

Falko Ziebert, Universität Heidelberg, Germany

PHASE FIELD APPROACH TO SUBSTRATE-BASED CELL MOTILITY

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-8, room 1.26.

Abstract

I will give an introduction to the substrate-based crawling motility of eukaryotic cells and survey our recent advances in its modeling. A modular approach, based on the phase field method to track the deformable and moving cells [1], allows us to describe, e.g., cell movement on structured substrates with modulated adhesion or stiffness [2], collective cell migration [3], as well as motion in 3D confinement [4]. I will also discuss the example of cellular shape waves [5], where the computational approach allows for additional insight via semi-analytic methods (employing asymptotic reduction and multiple scales).

References

1. F. Ziebert, S. Swaminathan, I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface 9 (2012), 1084.
2. J. Löber, F. Ziebert, I. S. Aranson, Modeling crawling cell movement on soft engineered substrates, Soft Matt. 10 (2014), 1365.
3. J. Löber, F. Ziebert, I. S. Aranson, Collisions of deformable cells lead to collective migration, Sci. Rep. 5 (2015), 9172.
4. B. Winkler, I. S. Aranson, F. Ziebert, Confinement and substrate topography control cell migration in a 3D computational model, submitted (2018).
5. C. Reeves, B. Winkler, F. Ziebert, I. S. Aranson, Rotating lamellipodium waves in polarizing cells, Comms. Phys. 1 (2018), 73.

Henryk Żołądek, University of Warsaw, Poland

INVARIANTS OF GROUP REPRESENTATIONS, DIMENSION/DEGREE DUALITY AND NORMAL FORMS OF VECTOR FIELDS

Joint work with Ewa Stróżyna

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-7, room 1.9.

Abstract

We develop a constructive approach to the problem of polynomial first integrals for linear vector fields. As an application we obtain a new proof of the theorem of Wietzenböck about finiteness of the number of generators of the ring of constants of a linear derivation in the polynomial ring. Moreover, we propose an alternative approach to the analyticity property of the normal form reduction of a germ of vector field with nilpotent linear part in a case considered by Stolovich and Verstringe.

Xingfu Zou, University of Western Ontario, Canada

DYNAMICS OF DDE SYSTEM GENERALIZED FROM NICHOLSON’S EQUATION TO A TWO-PATCH ENVIRONMENT WITH DENSITY-DEPENDENT DISPERSALS

Joint work with Chang-Yuan Cheng and Shyan-Shiou Chen

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We derive a model system that describes the dynamics of a single species over two patches with local dynamics governed by Nicholson's DDE and coupled by density dependent dispersals. Under a biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population, we are able to analyze model to some extent: well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. We also identify a positive invariant set and establish global convergence of solutions under certain conditions. We find that although the levels of the density-dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal-spatial patterns and the final population sizes.

Enrique Zuazua, DeustoTech, Bilbao, Basque Country & Universidad Autónoma de Madrid, Spain

POPULATION DYNAMICS AND CONTROL

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building B-8, room 0.10a.

Abstract

Population dynamics is an old subject. Classical models in this field are written in terms of reaction-diffusion equations.

There is a wide literature concerning the dynamical properties of those systems. But much less is known from a control perspective. And control constitutes the ultimate proof of our understanding of a process.

This lecture will be devoted to present two recent results in this area. We first consider a bistable reaction-diffusion arising in the modelling of bilingual populations and then address models involving age structuring and spatial diffusion (of Lotka-McKendrick type).

As we shall see, both aspects require of an in depth understanding of the dynamics of the systems under consideration.

We shall present sharp results on our ability to steer the dynamics of those systems to a prescribed final configuration. Some open problems and future directions of research will also be presented.

This lecture is based on recent joint work in collaboration with D. Maity, C. Pouchol, E. Trélat, M. Tucsnak and J. Zhu.

References

1. E. Trélat, J. Zhu, E. Zuazua, Allee optimal control of a system in ecology, Mathematical Models and Methods in Applied Sciences 28(9) (2018), 1665-1697.
2. C. Pouchol, E. Trélat, E. Zuazua, Phase portrait control for 1D monostable and bistable reaction-diffusion equations, Nonlinearity, to appear (hal-01800382).
3. D. Maity, M. Tucsnak, E. Zuazua, Sharp Time Null Controllability of a Population Dynamics Model with Age Structuring and Diffusion, J. Math pures et appl., to appear (hal-01764865).

Andy Zucker, Université Paris Diderot, France

BERNOULLI DISJOINTNESS

Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss

Date: 2019-09-16 (Monday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

We consider the concept of disjointness for topological dynamical systems, introduced by Furstenberg. We show that for every discrete group, every minimal flow is disjoint from the Bernoulli shift. We apply this to give a negative answer to the Ellis problem for all such groups. For countable groups, we show in addition that there exists a continuum-sized family of mutually disjoint free minimal systems. Using this, we can identify the underlying space of the universal minimal flow of every countable group, generalizing results of Balcar-Blaszczyk and Turek. In the course of the proof, we also show that every countable ICC group admits a free minimal proximal flow, answering a question of Frisch, Tamuz, and Vahidi Ferdowsi.

Federico Zullo, University of Brescia, Italy

ON THE DYNAMICS OF THE ZEROS OF SOLUTIONS OF AIRY EQUATION (AND BEYOND)

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building B-7, room 1.9.

Abstract

We discuss the dynamics of the zeros of entire functions in the complex plane. In particular we present the dependence of the zeros of solutions of the Airy equation on two parameters introduced in the equation. The parameters characterize the general solution of the equation. A system of infinitely many nonlinear evolution differential equations are obtained, displaying interesting properties. The possibility to extend the approach to other entire functions in the complex plane will be discussed.

Hans Zwart, University of Twente, Netherlands

DIFFERENTIAL ALGEBRAIC PORT-HAMILTONIAN EQUATIONS

Joint work with Volker Mehrmann

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building B-8, room 0.18.

Abstract

Port-Hamiltonian (pH) models can be used to describe physical systems which interact with their environment. Examples include ordinary and partial differential equations. By now the theory is quite complete with results ranging from control, approximation, and well-posedness of partial differential equations, [2]. Recently, the theory has been extended to differential equations with constraints, i.e., differential algebraic equations (DAE). We refer to [3] for the algebraic set-up of these systems and to [1] for (time-varying) DAE's with a finite-dimensional state space. Following on this, we study (time-invariant) port-Hamiltonian DAE on an infinite-dimensional state space. Our main focus is to show existence of (mild) solutions for this class of systems. So we consider the following abstract differential equation \[ E \dot{x}(t) = AQ x(t), \qquad x(0)=x_0 \] where \(E\) and \(Q\) are bounded operators on \(X\) satisfying \(E^*Q = Q^*E\), and \(A\) is the infinitesimal generator of a contraction semigroup on the Hilbert space \(X\). Since \(A\) is the the infinitesimal generator of a contraction semigroup, the above equation possesses a solution when \(E=Q=I\). If \(E\) and \(Q\) are are non-invertible, then this needs not to hold. We present some sufficient conditions under which the above DAE possesses a unique solution. Furthermore, we show that our results are related to boundary triplets and passive systems.

References

1. C. Beattie, Christopher, V. Mehrmann, H. Xu, Hongguo, H. Zwart, Linear port-Hamiltonian descriptor systems,, Mathematics of Control, Signals, and Systems 30 (2018), 17-27.
2. B. Jacob, H. Zwart, Linear port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser/Springer, Basel, 2012.
3. A. van der Schaft, B. Maschke, Generalized port-Hamiltonian DAE systems, Systems & Control Letters 121 (2018), 31-37.

Maciej Zworski, University of California, Berkeley, USA

FROM CLASSICAL TO QUANTUM AND BACK

Date: 2019-09-20 (Friday); Time: 09:00-10:00; Location: building U-2, auditorium.

Abstract

Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a successful tool in spectral theory and partial differential equations. We can say that these two fields lie on the "quantum/wave side".

In the last few years microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic flows. That followed the introduction of specially tailored spaces by Blank-Keller-Liverani, Baladi-Tsujii and other dynamicists and their microlocal interpretation by Faure-Sjoestrand and by Dyatlov and the speaker.

I will explain this microcar/dynamical connection in the context of Ruelle resonances, decay of correlations and meromorphy of dynamical zeta functions. I will also present some recent advances, among them results by Dyatlov-Guillarmou (Smale's conjecture on meromorphy of zeta functions for Axiom A flows), Guillarmou-Lefeuvres (local determination of metrics by the length spectrum) and Dang-Rivière (Ruelle resonances and Witten Laplacian).