Since Furstenberg gave a proof of Szemeredi's Theorem in additive combinatorics by showing its equivalence to a multiple recurrence result in ergodic theory, the nonconventional ergodic averages that are the subjects of this result and its later generalizations have attracted considerable interest. In this talk we will discuss some recent progress in their analysis, showing how an extension of an initially-given system of commuting probability-preserving transformations can be used in a proof of the norm convergence of some such averages.

In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation. Entropy has since been central to the classification theory of measurable dynamics. In the 70s and 80s researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss). My recent work generalizes the entropy concept to actions of sofic groups; a class of groups that contains for example, all countable subgroups of GL(n,C). Applications include the classification of Bernoulli shifts over a free group. This answers a question of Ornstein and Weiss.

In this talk we will explain some ideas behind our answer to a question of J. Bourgain, which was motivated by earlier questions of A. Bellow and H. Furstenberg. We show that the sequence of square integers is universally bad for the pointwise ergodic theorem in the class of integrable functions.

We discuss cohomological equations related to a multiplicative functional equation and to an equation (Helson's problem) considered by C.C. Moore and K. Schmidt in 1980.

Some applications will be given. In particular in a second part (joint work with Albert Raugi), we consider the following equation: H_{τ x} = φ(x) H_x (φ(x))^-1, for μ-a.e. x ∈ X,   (*) where τ is an ergodic measure preserving transformation on a probability space (X, μ), (H_x)_{x ∈ X} a measurable family of subgroups of a locally compact group G and φ a measurable map from X to G. This equation appears in the ergodic decomposition of a G-valued cocycle.
A question is: Does (*) imply the existence of a fixed subgroup H and a measurable map a such that for μ-a.e. x∈X : H_x = a(x) H a(x)^-1 ? The answer is positive when G is a nilpotent connected Lie group. In the solvable case, a counter-example related to the first part of the talk will be constructed.

By the problem of spectral multiplicities we mean the following: which subsets of positive integers are realizable as the set of essential values of the spectral multiplicity function for the Koopman unitary operator associated with an ergodic measure preserving transformation? A similar question can be asked in the subclass of mixing transformations. In general, these two basic problems of the spectral theory of dynamical systems remain open. I will talk about a recent (2009) progress in the field.

On prouve qu'un cocycle de Schrödinger quasi-périodique en une dimension avec un petit potentiel analytique est toujours conjugué à un cocycle de rotations. Ceci étend à toutes les fréquences un résultat prouvé par Dinaburg et Sinai dans le cas de fréquence diophantienne.

For a given non-periodic Bohr subset of integers B, we give a sharp lower bound for the upper Banach density of A+B for any subset A of integers in terms of densities of A and B. If the bound is attained then A contains a Bohr substructure and B is essentially a one-dimensional Bohr set. To obtain the combinatorial results we study recurrence properties of ergodic measure preserving systems along Bohr sets. Joint work with Michael Björklund (KTH).

Rather than presenting recent results, this will document how the relatively recent applications of ergodic theory to combinatorial problems arose naturally in an attempt to formulate a theory of prediction meaningful for a single sample sequence of the past of a stationary process.

Given a compact set X in the plane, the image of X under orthogonal projection to almost every line has the maximal possible Hausdorff dimension, i.e. min{1,dim(X)}. An old conjecture of Furstenberg's predicts that when X=A×B, and A,B are ×2 and ×3 invariant sets, respectively, then this should hold for every line except the trivial exceptions (those parallel to the axes). I will describe a proof of this and its measure equivalent. This is joint work with Pablo Shmerkin.

The Erdos-Renyi random graph G(n,p) is the probability space of all graphs on the vertex set {1,2,...n}, with each edge inserted independently with probability p. Usually p is defined to be a function of n, and one asks whether a graph in G(n,p) is likely to have some (monotone) property as n goes to infinity. In their seminal 1959 paper, Erdos and Renyi showed that if p << log(n)/n then G(n,p) is almost always disconnected, but if p >> log(n)/n, then it is almost always connected.

We consider this as a topological statement and in this talk we will study the topology of random two dimensional simplicial complexes. Linial, Meshulam and Wallach showed that homology of random two dimensional simplicial complexes undergoes a phase transition at p=2 log(n)/n, which is analogous to the phase transition for random graphs. We show that homotopy has a different phase transition. When p>>n^-1/2 then the complex is almost always simply connected and when p << n^-1/2 then the complex almost always an infinite hyperbolic fundamental group.

This is joint work with Eric Babson and Matt Kahle.

Non-uniform measure rigidity was initiated in a joint 2007 paper with Boris Kalinin where we showed that any action of Z^k on T^k+1, k≥ 2, whose elements are homotopic to an elements of linear action diagonalizable over R, has an absolutely continuous invariant measure; later we showed jointly with Federico Rodriguez Hertz that this is the only measure that projects into Lebesgue under the semi-conjugacy with the linear action.

In this talk I will describe an extension of these results to a considerably more general situation: we only require that for the linear action the Lyapunov exponents are simple and there are no proportional Lyapunov exponents. (2009, joint with F. Rodriguez Hertz.) The key difference is that this condition requires no connection between rank and dimension.

We use main technical devises from the earlier papers but add significant new ingredients: the graph argument, the new entropy inequality, and index argument in the proof of uniqueness.

I will also discuss applications of nonuniform measure rigidity to the Zimmer program: we prove that any action of SL(n,Z) on Tⁿ whose elements are homotopic to those of the standard linear action, is differentiably conjugate to the latter on an open set that projects under the semi-conjugacy to a complement to a finitely many periodic orbits.

From the operator theory point of view joinings of measure-preserving systems are - intertwining the corresponding Koopman representations - Markov operators between the underlying L²-spaces. Using some ideas from the operator theory, A.M. Vershik proposed in 2004 to study equivalence of dynamical systems based on quasi-equivalence of operators in the Markov category. I will present some results concerning this new equivalence notion contained in a joint work with K. Frączek.

I will present some recent work related to Furstenberg's theorem and conjecture regarding the action nonlacunary semigroups of integers on the one torus, including qualitative form of Furstenberg's theorem obtained in joint write with Bourgain, Michel, Venkatesh.

We introduce the notion of topological and metric entropy dimensions to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential growth rate of orbits. We define the dimension set of a dynamical system to study the complexity of its factors.

Let G be a simple Lie group, Γ be a Zariski dense subgroup of G and Λ be a lattice of G. In this joint work with Yves Benoist we prove that the Γ-invariant subsets of G/Λ are finite or dense. This topological description relies on a metric result : we prove that if μ is a compactly supported probability measure on G which support spans a Zariski dense subsemigroup of G, the extremal μ-stationary measures of G/Λ are the Haar measure and finitely supported measures.

The abelian sandpile models are lattice models which were introduced in 1988 by Bak, Tang and Wiesenfeld, and subsequently explored by Deepak Dhar. We discuss this interesting class of models and their connections with certain other Z^d-actions, like dimer and harmonic models. This is joint work with Evgeny Verbitskiy.