Title: The Pisot Conjecture

Abstract: In its narrowest form, the Pisot conjecture asserts that the shift on a substitutive system of irreducible, unimodular, Pisot type has pure discrete spectrum. Versions of this conjecture arise in the study of diffraction properties of quasicrystals, geometric models of tiling spaces, beta expansions, and arithmetical coding of hyperbolic toral automorphisms. The conjecture has been verified in the quadratic (two letter) case, and, if irreducibility is dropped, there is a family of counterexamples based on pseudo-Anosov diffeos that are branched covers of toral automorphisms. The appropriate setting and scope for the conjecture remain elusive; I'll survey what is known.

Title: Generalized polynomials, translations on nilmanifolds and Ergodic Ramsey Theory

Abstract: Generalized polynomials are functions which are obtained from regular polynomials via iterated use of the floor function. We will discuss some recent results on the connections between the generalized polynomials and translations on nilmanifolds (joint work with A. Leibman) and new applications to combinatorics (joint work with Randall McCutcheon).

Title: Z^d Sofic and Finite Type Shifts: subsystems and quotients

Abstract: We consider Z^d sofic and finite type shifts. For d=1, there is a familiar wealth of subsystems, and much of the structure of factor maps is well understood. For d > 1, I'll describe how the situation is very different. (For example, there is a positive entropy sofic shift whose unique minimal subsystem is a single point.) These results are joint work with Michael Schraudner and Ronnie Pavlov, and rely heavily on results and constructions of Mike Hochman and Tom Meyerovitch together and Mike Hochman individually.

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Title: Orbit equivalence in a hybrid toplogical and measure-theoretic category.

Abstract: We will briefly review various notions of orbit equivalence in both measure-theoretic and topological settings. The prototype is Dye's theorem asserting that the orbit relations of any two ergodic measure-preserving maps are equivalent via an a.e. defined measure-preserving map. Then we will discuss some recent results in which the equivalence is actually an a.e. defined homeomorphism.

Title: Rates of mixing for flows and skew extensions.

Abstract: We describe some of the results and outstanding problems on rates of mixing for hyperbolic flows, skew extensions and the distribution of prime periods. Parts of the work presented is joint with Ian Melbourne and ANdrew Torok.

Title: IS CHAOS COMPATIBLE WITH UNIFORM DISTRIBUTION? (preliminary report)

Abstract: A more technical reformulation of the question in the title is: Can a dynamical system with positive topological entropy be uniquely ergodic? The answer may depend on the class of systems under consideration. For example, it is positive for both homeomorphisms of compact manifolds ( M. Rees) and for symbolic systems (Schwatzbauer). A really interesting question is one for classical systems, i.e. diffeomorphisms and smooth flows on compact manifolds. This question appeared in my discussions with Michel Herman in the late seventies and it led to two significant woks at the time: Herman's example showing that the answer is positive if ``uniquely ergodic'' is weakened to ``minimal'', and mine that implies in particular that the answer is negative in dimension two. The latter was based an application of smooth ergodic theory and uniform hyperbolicity. While I always believed that the answer is negative for smooth systems in any dimension and that the proof should use nonuniform partial hyperbolicity, several attempts to implement this approach failed and the problem looked quite resistant. About a year ago my Ph. D. student Sun Peng started to work on a model problem dealing with non-uniformly hyperbolic skew products over uniquely ergodic systems. Recently he succeeded in proving that such a system always has an invariant measure with atomic conditionals in the fibers. This implies negative answer for skew products. It looks like combining a modification of Peng's method with my old idea gives a key to the old problem in the general setting.

Title: Recurrence along generalized polynomials

Abstract: A generalized polynomial is a real-valued function which is obtained from the conventional polynomials by use of the operations of addition, multiplication, and taking the greatest integer part. Generalized polynomials show up naturally in ergodic theory. In particular, there are strong relations between bounded generalized polynomials and translations on nilmanifolds (recently established by Bergelson and Leibman). We will survey some results concerning uniform distribution of generalized polynomials, and discuss questions related to sets of recurrence and how weak mixing implies weak mixing of all orders along generalized polynomials.

Title: Actions of the Discrete Heisenberg Group and Noncommutative Mahler Measure

Abstract: The discrete Heisenberg group is the first interesting nonabelian group to study when trying to extend the commutative theory of actions by automorphisms of a compact abelian group. In joint work with Klaus Schmidt, I will describe some of the fascinating dynamical and algebraic problems, and also some partial progress on them. One outcome is a natural notion of Mahler measure for polynomials in noncommuting variables, whose value can be computed in special cases.

Title: Generic Riemann structures

Abstract: Various generic properties of Riemann structures are discussed, including the analytic case.

Title: Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis

Abstract: I shall consider the system of n (n>1) elastically colliding hard balls of masses m_1, m_2, ... , m_n and radius r on the flat unit torus T^d, d>1. My goal is to sketch the main dynamical and geometric ideas behind the proof of the so called Boltzmann-Sinai Ergodic Hypothesis for such hard ball systems, i. e. the full hyperbolicity and ergodicity for every selection (m_1,..., m_n;r) of the external geometric parameters, without the assumption that almost every singular trajectory is geometrically hyperbolic (sufficient), i. e. that the so called Chernov-Sinai Ansatz holds true for the model. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.

Title: On the dynamics of periodically perturbed homoclinic solutions

Abstract: This talk on the dynamics of periodically perturbed homoclinic solutions, a classical scenario that has substantially influenced the modern chaos theory. People have looked at the time-T map and they have seen two cases: (1) homoclinic tangles, and (2) stable and unstable manifold pulled apart. (1) is a classical case for chaos and (2) is thought as a simple case. A standard procedure in proving the existence of chaos in a given system of differential equation is to evaluate Melnikov's integral to confirm (1). Then we conclude that there exists a horseshoe. Anything beyond the existence of a horseshoe has been somewhat a myth. For instance, one could ask the following questions: Let $\mu$ be the small parameter representing the magnitude of the perturbation (i) As $\mu \to 0$, how does the dynamics structure of Melnikov's tangles evolve? (ii) For small $\mu$, does a homoclinic tangle contain more than a horseshoe in general? and how much more? These two questions are among the many I will answer in this talk.

Title: Anosov Diffeomorphisms are transitive

Abstract: We solve a long standing problem and show that all Anosov diffeomorphisms are topologically transitive

Title: Random sources and sinks for stochastic dynamical systems

Abstract: Phase portraits have proved to be an effective tool for the description of the long term behavior of deterministic dynamical systems. Here we describe some results from an ongoing program to develop similar techniques for stochastic dynamical systems. We illustrate some of the essential differences between the deterministic and stochastic theories by presenting examples of stochastic dynamical systems (generated by stochastic differential equations) on the circle and the two-dimensional torus.

Title: Long time behavior and related topics to stochastic Shell models

Abstract: Some stochastic shell models are studied. Their long time behavior is studied and the existence os a stochastic attractor is proved as for its finite dimension. Some features about their statistical properties will be discussed.

Title: Phase Noise and Fokker Planck Equation

Abstract: A local moving orthonormal transformation has been introduced to rigorously study phase noise in stochastic differential equations arising from nonlinear oscillators. A general theory of phase and amplitude noise equations and its corresponding Fokker–Planck equations are derived to characterize the dynamics of phase and amplitude error. As an example, a van der Pol oscillator is considered by using the general theory.

Title: Large deviations in fluid systems under random fluctuations

Abstract: The speaker presents recent work on variational approaches for estimating the likelihood of small probability events (such as large deviations) for fluid systems under uncertainty.

Title: Thermodynamic formalism for random transformations revisited

Abstract: We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron--Frobenius type theorem for certain random positive linear operators treating these models in a unifying way relying on the Hilbert projective norms formalism.

Title: The Principal Spectrum and Principal Lyapunov Exponent for Random Parabolic PDEs of Second Order

Abstract: Random linear parabolic partial differential equations (PDEs) of second order are considered. The concepts of principal spectrum and principal Lyapunov exponent are introduced. Their properties are investigated, and applications to reaction-diffusion equations of Kolmogorov type are considered. This is a joint work with Wenxian Shen.

Title: Traveling Waves in Diffusive Random Media

Abstract: The current talk is concerned with traveling waves in diffusive random media. It first introduces a rigorous definition of traveling wave solutions in diffusive random media. Various fundamental properties of such traveling wave solutions are then presented. Finally, applications of the general theory developed to some specific model problems are discussed.

Title: A propagation-of-chaos type result in stochastic averaging

Abstract: Stochastic averaging goes back to Khasminskii in the 1960's. The standard result is that, given a separation of scales, one can find effective dynamics for slow components. We investigate the motion of two particles in such a system, in particular in a randomly-perturbed twist map. The nub of the issue is how two points escape from a 1-1 resonance zone. Results of Pinsky and Wihstutz indicate that there is a third scale at work, which we can use to study the escape from resonance.

Title: Topology-Guided Sampling of Complicated Random Patterns

Abstract: Many stochastic partial differential equation models arising in applications generate complex time-evolving patterns which are hard to quantify due to the lack of any underlying regular structure. The influence of stochasticity leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. In many of these instances, such as for example in phase-field type models in materials science, one is interested in the geometry of sublevel sets of a function in terms of their topology, in particular, their homology. Recent computational advances make it possible to compute the homology of discrete structures efficiently and fast. Such methods can be applied to the above situation if the sublevel sets of interest are approximated using an underlying discretization of the considered evolution equation. Yet, this method immediately raises the question of the accuracy of the computed homology. In this talk, I will present a probabilistic approach which gives insight into the suitability of the above method in the context of random fields. We will obtain explicit probability estimates for the correctness of the homology computations, which in turn yield a-priori bounds for the suitability of certain grid sizes as well as information on the optimal location of sampling points.

Title: Basic theory of stochastic functional differential equations

Abstract: Our main aim is to develop the basic theory of existence, uniqueness and continuation for stochastic functional differential equations.

Title: Stationary Solutions and Random Periodic Solutions of Random Dynamical Systems

Abstract: In this talk, I will present some basic concepts in random dynamical systems, especially the two types of invariant solutions: stationary solutions and random periodic solutions. The talk will be ended by giving a result on the existence of the random periodic solutions.