Probability, Analysis and Dynamics 18

Transcription

1 Probability, Analysis and Dynamics 18 Bristol, 4-6 April, 2018 Abstracts of talks

2 Viviane Baladi (CNRS / IMJ-PRG / UPMC, Paris, FR): Analytical tools for dynamics with singularities, including Sinai billiards Thursday 16:30 In the past 15 years, tools from analysis, in particular new Banach spaces of anisotropic distributions on manifolds, have allowed substantial progress in dynamical systems. After briefly explaining how a spectral gap for a transfer operator furnishes ergodic information, I shall focus on Sinai billiards maps and flows. These natural but technically challenging systems are uniformly hyperbolic and volume preserving - however grazing orbits give rise to singularities. New analytic tools allowed us (with M. Demers and C. Liverani) to obtain exponential mixing for the natural volume and finite horizon Sinai billiard flows. With M. Demers (WIP), we have constructed a measure of maximal entropy for the billiard map. I will compare the two results. Krzysztof Burdzy (University of Washington, US): Collisions of billiard balls and foldings Wednesday 9:30 In lieu of an abstract I offer a problem. Consider three billiard balls of the same radius and mass, undergoing totally elastic reflections on a billiard table with no walls (the whole plane). All three balls can be given non-zero initial velocities. What is the maximum (supremum) possible number of collisions among the three balls? The supremum is taken over all initial positions and initial velocities. I will discuss this problem and its generalization to any finite family of balls in one, two and higher dimensions. Joint work with Jayadev Athreya and Mauricio Duarte. 2

3 Laura DeMarco (Northwestern University, US): Complex dynamics and arithmetic equidistribution Thursday 14:00 I will explain a notion of arithmetic equidistribution that has found application in the study of complex dynamical systems. It was first introduced about 20 years ago, by Szpiro-Ullmo-Zhang, to analyze the geometry and arithmetic of abelian varieties. In this talk, I will relate these ideas to the Mandelbrot set, and I will present recent work (joint with N.M. Mavraki) using dynamics and equidistribution to re-examine their unsolved problems about abelian varieties. Dmitry Dolgopyat (University of Maryland, US): Local Limit Theorem for Nonstationary Markov chains Friday 11:30 Dobrushin and Sethuraman-Varadhan have proved a sharp Central Limit Theorem for additive functionals of sufficiently fast mixing non-stationary Markov chains. We discuss the Local Limit Theorem in the same setting and give some extensions and applications. Joint work with Omri Sarig. Tatjana Eisner (University of Leipzig, DE): Weighted ergodic theorems Wednesday 14:10 We present an overview on good weights for the pointwise ergodic theorem. 3

4 Alison Etheridge (University of Oxford, UK): Modelling evolution in a spatial continuum Thursday 9:00 Since the pioneering work of Fisher, Haldane and Wright at the beginning of the 20th Century, mathematics has played a central role in theoretical population genetics. In turn, population genetics has provided the motivation both for important classes of probabilistic models, such as coalescent processes, and for deterministic models, such as the celebrated Fisher-KPP equation. Whereas coalescent models capture relatedness between genes, the Fisher KPP equation captures something of the interaction between natural selection and spatial structure. What has proved to be remarkably difficult is to combine the two, at least in the biologically relevant setting of a two-dimensional spatial continuum. In this talk we describe some of the challenges of modelling evolution in a spatial continuum and then, as time permits, turn to some results concerning the interplay between natural selection and spatial structure. Geoffrey Grimmett (University of Cambridge, UK): The 1-2 model: dimers, polygons, and the complex Ising model Thursday 10:00 The 1-2 model is a disordered interacting system in two dimensions which can be mapped to a number of well known processes including the dimer, Ising, and polygon systems of probability theory and mathematical physics. These connections will be explained in this lecture, and the exact solution will be summarised (joint work with Zhongyang Li). Martin Hairer (Imperial College London, UK): A new universality class in 1+1 dimensions Friday 9:00 The title says it all. 4

5 Mike Hochman (The Hebrew University, IL): Dimension of self-affine sets in the plane Wednesday 15:30 I will describe recent work (joint with Rapaport and Bárány) in which we compute the dimension of planar self-affine sets and measures satisfying the strong separation condition. This confirms some old predictions that, under suitable irreducibility assumptions, the dimension is equal to Falconer s classical upper bound. Alexander Holroyd (University of Washington, US): Local constraint solving - how to color without looking (much) Friday 15:30 How can individuals cooperate to satisfy local constraints without a central authority? Examples might include autonomous drones navigating in a swarm, or university departments scheduling their seminars. Individuals can make random choices and communicate with each other, but all must follow the same procedure. To formalize this, we ask for a finitary factor of iid variables satisfying a shift of finite type. How small can we make the coding radius - the distance to which an individual must communicate? In the setting of the integer line Z, there is a surprising universal answer that applies to every non-trivial constraint problem. In d-dimensional Euclidean space, answers are available for the key case of proper coloring; it turns out that there is a big difference between 3 and 4 colors. Finally, I ll mention how changing the question slightly leads to a remarkable mathematical object that seemingly has no right to exist. Konstantin Khanin (University of Toronto, CA): On global solutions to the random heat equation Friday 10:00 We shall discuss the problem of existence and uniqueness of global solutions to the random heat equation in non-compact setting. While we expect positive answers in general, at present rigorous results are available only in a very few cases. We shall mostly discuss results in the 3D case corresponding to the so called weak disorder situation. 5

6 Antti Kupiainen (University of Helsinki, FI): Liouville Field Theory: A Case of Integrable Probability Thursday 11:30 In 1994 Dorn and Otto and independently Zamoldzchikov and Zamoldzchikov conjectured a remarkable formula for certain correlation functions of the Liouville theory, a conjectural scaling limit of planar maps. I will discuss a recent proof of this DOZZ formula which is based on novel integrability properties of the Gaussian Multiplicative Chaos. This is joint work with R. Rhodes and V. Vargas. Jens Marklof (University of Bristol, UK): Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows Thursday 15:30 We show that the time evolution of a quantum particle in a periodic potential converges in a combined high-frequency/boltzmann-grad limit, up to second order in the coupling constant, to terms that are compatible with the linear Boltzmann equation. This complements results of Eng and Erdős for low-density random potentials, where convergence to the linear Boltzmann equation is proved in all orders. Our analysis suggests, however, that the linear Boltzmann equation fails in the periodic setting for terms of order four and higher. The proof uses Floquet-Bloch theory, multi-variable theta series and equidistribution theorems for homogeneous flows. This is joint work with Jory Griffin (Queens University, Canada) Vladimir Markovic (Caltech, US): Randomness in geometry of hyperbolic manifolds and Teichmüller spaces Wednesday 16:30 I will show how Teichmüller dynamics and statistical properties of geodesic flows help us prove theorems about topology of 3-manifolds and shapes of Teichmüller spaces. 6

7 Felix Otto (Max Planck Institut, DE): Effective behavior of random media Wednesday 11:50 We are interested in the large-scale behavior of solutions to elliptic partial differential equations with random coefficients. If the coefficient field is stationary and ergodic, solutions are close to solutions of a deterministic, constant-coefficient elliptic equation, so that this phenomenon is called stochastic homogenization. While this is well-understood on a qualitative level, there has been recent activity on a more quantitative level, which also sheds new light on the original phenomenon. For instance, it is possible to predict the leading-order fluctuations (which turn out to be Gaussian) of macroscopic observables of the solution. Or, it is possible to give a sense to the multipole expansion and its relation to moments for a random medium. Steffen Rohde (University of Washington, US): Conformal laminations Wednesday 10:50 Given an equivalence relation on the circle, is there a conformal map f of the open disc whose extension to the boundary circle realizes the relation (in the sense that f(x) = f(y) if and only if x y)? We discuss analytical, dynamical and probabilistic aspects of this problem and show (joint work with Peter Lin) that the Continuum Random Tree admits such conformal representation. Sasha Sodin (Queen Mary, University of London, UK): Non-Hermitian random Schrödinger operators Friday 14:00 Non-Hermitian random Schrödinger operators were put forth in the mid 1990-s by Hatano and Nelson, and mathematically studied by Goldsheid and Khoruzhenko. We shall describe the model and discuss some results obtained in a recent joint work with I. Goldsheid. 7