Abstracts. Furstenberg The Dynamics of Some Arithmetically Generated Sequences

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1 CHAOS AND DISORDER IN MATHEMATICS AND PHYSICS Monday 10:00-11:00 Okounkov Algebraic geometry of random surfaces 11:30-12:30 Furstenberg Dynamics of Arithmetically Generated Sequences 12:30-14:30 lunch 14:30-15:30 Liverani Coupled Markov Chains and Coupled Map Lattices 16:00-17:00 Schlag On stable manifolds for orbitally unstable evolution equations Tuesday 9:00-10:00 Shlosman How the random 3D Ising crystal grows? 10:10-11:10 Presutti On phase diagram of continuous particle systems 11:30-12:30 Zeitouni Random walk in random environments 13:00-19:00 Walk / Lunch Wednesday 9:30-10:30 Viana Lyapunov exponents 11:00-12:00 Ornstein TBA 12:00-14:00 lunch 14:00-15:00 Forni Weakly mixing interval exchange transformations 15:00-16:00 Young Detecting strange attractors 18:00-19:30 Discussion (Galgani et el) 19:30-24:00 Conference dinner Thursday 9:30-10:30 Kupianen Fourier Law and Boltzman equation 11:00-12:00 Mahalov Global Regularity of 3D Navier-Stokes Equations with Large Initial Vorticity and Rotating Turbulence 12:00-14:00 lunch 14:00-15:00 Shnirelman Strange properties of 2-dimensional fluid 15:30-16:30 Zakharov On the Dynamics of Vortex Lines in Ideal Fluid 18:00-19:30 Discussion TBA Friday 9:00-10:00 Kaloshin Diffusion for Hamiltonian ODE s and PDE s 10:10-11:10 Kleinbock Chaos and disorder in homogeneous dynamics and number theory 11:40-12:40 Lindenstrauss Classification of (some) invariant measures under Cartan flows and applications. 1

2 2 Abstracts. Furstenberg The Dynamics of Some Arithmetically Generated Sequences We will survey the application and the applicability of dynamics to questions on the asymptotic behavior of certain arithmetically defined sequences. The notorious 3n+1 problem is a candidate for such treatment. Also the issue of well-approximability of algebraic numbers as well as the distribution of fractional parts of powers of a rational > 1 will be examined in this context. Galgani Sinai and Italian Mathematics Kaloshin Diffusion for certain Hamiltonian ODE s and PDE s Let H(θ, I) = H 0 (I) + εh 1 (θ, I), θ T n, I U R n be a smooth/analytic Hamiltonian written in action-angle variables (θ, I) respectively. Equations of motion are θ = I H = I H 0 + ε I H 1 I = θ H = ε I H 1. To establish Arnold diffusion for a nearly integrable Hamiltonian system means show existence of trajectories (θ, I)(t) whose action I(t) substantially changes in time, i.e. sup I(t) I(0) O(1) t>0 independently how small ε is. During the talk we shall discuss three examples, where Arnold diffusion has been established. An abstract convex Hamiltonian H 0 and a perturbation H 1 from an open set of perturbations; Restricted Planar Circular 3 Body Problem with small mass ratio; time-periodic 1-dimensional Nonlinear Schrödinger Equation (NLS). Kleinbock Chaos and Disorder in homogenuous dynamics and number theory I will survey several recent results and open problems at the interface between dynamics of subgroup actions on homogeneous spaces and simultaneous Diophantine approximation. Kupiainen Fourier Law and Boltzmann equation We review the problem of heat conduction in Hamiltonian systems and discuss the derivation of Fourier s law from a truncated set of

3 equations for the stationary state of a mechanical system coupled to boundary noise. Lindenstrauss Classification of (some) invariant measures under Cartan flows and applications. In my talk I will discuss measures on a locally homogeneous spaces such as SL n (Z)\SL n (R) invariant under diagonalizable Abelian groups (a.k.a. R-split tori), particularly the case of maximal R-split tori. Under the assumption that the Kolmogorov-Sinai entropy of the measure with respect to some element of the acting group is positive one can essentially classify such measures in many cases. This classification has many applications, for example in studying closed orbits of maximal R-split tori, in the study of simultaneous Diophantine approximations (in particular, giving a substantial partial result towards Littlewood conjecture), and Arithmetic Quantum Chaos. Liverani Coupled Markov Chains and Coupled Map Lattices In many situations one needs to study extended systems obtained by weakly coupling local dynamics with good mixing properties. I will discuss an approach to such situations, recently developed together with G.Keller, by applying it to some relevant examples. Mahalov Global Regularity of the 3D Navier-Stokes Equations with Uniformly Large Initial Vorticity and Rotating Turbulence We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold. The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data. With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large nitial vorticity. For the same class of initial data we study energy transfer in the 3D 3

4 4 Euler equations and discuss applications to regularity of its solutions and rotating turbulence. Okounkov Algebraic geometry of random surfaces Formation of limit shapes is a fundamental phenomenon (law of large numbers) observed in many random surface models. I will discuss a class of random surface models for which these limit shapes turn out to be algebraic for a dense set of boundary conditions. The tools of algebraic geometry can thus be used to obtain a very explicit information about these limit shape, especially about formation of their singularities. The talk will be based on a joint work with Richard Kenyon. If time permits, I will explain how, conversely, probabilistic ideas yield new results in algebraic geometry. Presutti On the phase diagram of continuous particle systems The Pirogov-Sinai theory of phase transitions is extended to perturbations of mean field hamiltonians with the small parameter of the theory being the inverse interaction range. The result is then used to study a neighborhood of the liquid-vapour branch of the phase diagram of a system of identical point particles in R d interacting via a long but finite range Kac potential. Schlag On stable manifolds for orbitally unstable evolution equations It is well-known that many standard evolution PDEs of the focusing type exhibit both finite-time blow-up as well as special families of global solutions which depend on a finite number of parameters. For example, focusing nonlinear Schroedinger equations have standing wave solutions (solitons) which depend on as many parameters as there are symmetries (either 2n+2 or 2n+4 if n is the dimension). In many cases these special solutions are known to be unstable. We will describe recent work by the author on finite co-dimensional manifolds in the infinite of data (typically codimension one) on which these special solutions are stable. Moreover, their evolution can be described quite explicitly as a bulk term plus a scattering term. In particular, we will describe recent work on the focusing critical wave equation in three dimensions. Some of this work is joint with Joachim Krieger at Harvard. Shnirelman Strange properties of 2-dimensional fluid This talk is devoted to the properties of an ideal incompressible fluid moving in a 2-dimensional domain. This is an infinite-dimensional dynamical system with peculiar properties whose visible manifestation is the inverse energy cascade: the kinetic energy of the flow is transferred from small to large scales, and eventually accumulates at the global scale, so that (almost) any flow, whatever the initial velocity

5 field, ends up as a steady and stable large scale flow. Thus, steady and stable flows form an infinite-dimensional attractor in the space of velocity fields, which is in apparent contradiction with the time reversibility of the Euler equations. The inverse cascade is observed in experiments and computer simulations, but has not been established rigorously. Moreover, the existence of even one solution of Euler equations displaying the inverse cascade is not proved. In the present talk we discuss some results giving a partial justification of the inverse cascade. In particular, we show the existence of weak solutions of the Euler equations displaying the extreme form of the inverse cascade, when the scale of the initial velocity is zero. Viana Lyapunov exponents I ll give an overview of recent results on the theory of Lyapunov exponents of dynamical systems and linear cocycles, especially about existence of non-zero exponents, criteria for simplicity of the Lyapunov spectrum, and continuity of the exponents as a function of the dynamics. As an application of the theory, I ll discuss the recent proof of the Zorich-Kontsevich conjecture on the Lyapunov exponents of the Teichmuller flow. Young Detecting strange attractors First I will attempt to give a mathematical characterization of strange attractors. Then I will specialize to what one might call strange attractors of the simplest kind, namely those with only a single direction of instability (without instability, they wouldn t be strange!) Conditions that imply the presence of this type of attractors will be given, and examples will be discussed to show that they occur naturally. Zeitouni Random walk in random environments Random walks in random environments in dimension 1 exhibit many unusual features: slowdown, aging, stable limits, and most famously, anomalous diffusivity, discovered by Sinai, holding even in the perturbative regime. The higher dimensional case is believed not to exhibit such rich behavior, but the proof of such a statement still presents many challenges, in spite of the considerable progress achieved in recent years. I will describe the background and some recent results, including the breakdown of certain 0-1 laws in ergodic environments on the one hand, and diffusive behavior in the perturbative regime on the other. 5