Title: Self-affine measures with equal Hausdorff and Lyapunov dimensions Abstract: Let μ be the stationary measure on ℝd which corresponds to a self-affine iterated function system Φ and a probability vector p. Denote by A⊂Gl(d,ℝ) the linear parts of Φ. Assuming the members of A contract by more than 12, it follows from a result by Jordan, Pollicott and Simon, that if the translations of Φ are drawn according to the Lebesgue measure, then dimHμ=min{D,d} almost surely. Here D is the Lyapunov dimension, which is an explicit constant defined in terms of A and p.

Abstract:

Title: Topological structures and the pointwise convergence of some averages for commuting transformations Abstract: ``Topological structures'' associated to a topological dynamical system are recently developed tools in topological dynamics. They have several applications, including the characterization of topological dynamical systems, computing automorphisms groups and even the pointwise convergence of some averages.  In this talk I will discuss some developments of this subject, emphasizing applications to the pointwise convergence of some averages.

The Danzer problem and a solution to a related problem of Gowers Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers'

Abstract: Expander graphs have many wonderful properties, and have been an immensely useful and fruitful area of research in both applicative and theoretical fields. There has been a lot of interest recently in the study of higher dimensional generalizations of expanders to d-uniform hypergraphs. Many competing definitions have been proposed, and different definitions may be appropriate depending on the property of expanders that we wish to preserve.

Title: Invariant random subgroups

Title: On the Mixing Property for Hyperbolic Systems [following a paper by Martine Babillot]

Given a Z^d shift of finite type and a finite range shift-invariant interaction, we present sufficient conditions for efficient approximation of pressure and, in particular, topological entropy. Among these conditions, we introduce a combinatorial analog of the measure-theoretic property of Gibbs measures known as strong spatial mixing and we show that it implies many desirable properties in the context of symbolic dynamics. Next, we apply our

Title: What are amenable groups and why are groups non-amenable Abstract: I will give an introduction to amenable groups and explain the result of Kevin Whyte that a countable non-amenable group admits a "translation-like" action by a non-abelian free group. I will also discuss (without proof) a measure theoretic analogue of this result due to D. Gaboriau and R. Lyons.

Title: What are amenable groups and why are groups non-amenable Abstract: I will give an introduction to amenable groups and explain the result of Kevin Whyte that a countable non-amenable group admits a "translation-like" action by a non-abelian free group. I will also discuss (without proof) a measure theoretic analogue of this result due to D. Gaboriau and R. Lyons.

Title: What are amenable groups and why are groups non-amenable Abstract: I will give an introduction to amenable groups and explain the result of Kevin Whyte that a countable non-amenable group admits a "translation-like" action by a non-abelian free group. I will also discuss (without proof) a measure theoretic analogue of this result due to D. Gaboriau and R. Lyons.

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.