Abstract:  A group G is stable in permutations if every almost-action of G on a finite set is close to some actual action. Part of the interest in this notion comes from the observation that a non-residually finite stable group cannot be sofic.  I will show that surface groups are stable in a flexible sense, that is if one is allowed to "add a few extra points" to the action. This is the first non-trivial stability result for a non-amenable group. 

Combinatorial group theory began with Dehn's study of surface groups, where he used arguments from hyperbolic geometry to solve the word/conjugacy problems. In 1984, Cannon generalized those ideas to all "hyperbolic groups", where he was able to give a solution to the word/conjugacy problem, and to show that their growth function satisfies a finite linear recursion. The key observation that led to his discoveries is that the global geometry of a hyperbolic group is determined locally: first, one discovers the local picture of G, then the recursive structure

Abstract: A Markov chain over a finite state space is said to exhibit the total variation cutoff phenomenon if, starting from some Dirac measure, the total variation distance to the stationary distribution drops abruptly from near maximal to near zero. It is conjectured that simple random walks on the family of $k$-regular, transitive graphs with a two sided $\epsilon$ spectral gap exhibit total variation cutoff (for any fixed $k$ and $\epsilon). This is known to be true only in a small number of cases.

For a finitely generated subgroup H of the free group F_r, the Stallings graph of H is a finite combinatorial graph, whose edges are labeled by r letters (and their inverses), so that paths in the graphs correspond precisely to the words in H. Furthermore, there is a map between the graphs of two subgroups H and J, precisely when one is a subgroups of the other. Stallings theory studies the algebraic information which is encoded in the combinatorics of these graphs and maps.

Abstract: In this talk, I shall present a generalization of the lattice point counting problem for Euclidean balls in the context of a certain type of homogeneous groups, the so-called Heisenberg groups.

Abstract: A Markov chain over a finite state space is said to exhibit the total variation cutoff phenomenon if, starting from some Dirac measure, the total variation distance to the stationary distribution drops abruptly from near maximal to near zero. It is conjectured that simple random walks on the family of $k$-regular, transitive graphs with a two sided $\epsilon$ spectral gap exhibit total variation cutoff (for any fixed $k$ and $\epsilon). This is known to be true only in a small number of cases.

The Littlewood and the p-adic Littlewood conjectures are famous open problems on the border between number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that the analogue of the p-adic Littlewood conjecture over \mathbb{F}_3((1/t)) is false. The counterexample is given by the Laurent series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non vanishing of certain Hankel determinants.

I will discuss joint work with S. Starchenko, which combines dynamical systems in the nilmanifold setting with definable objects in o-minimal structures (e.g. semi-algebraic sets): Let G be a real algebraic unipotent group and let L be a lattice in G with p:G->G/L the quotient map. Given a subset X of G which is semi-algerbaic, or more generally definable in an o-minimal structure, we describe the closure of p(X) in terms of finitely many definable families of cosets of positive dimensional algebraic subgroups of G.

Abstract: An observation by Jens Marklof shows that the primitive rational points of a fixed denominator along the periodic unipotent orbit of volume equal to the square of the denominator equidistribute inside a proper submanifold of the modular surface. This concentration as well as the equidistribution are intimately related to classical questions of number theoretic origin. We discuss the distribution of the primitive rational points along periodic orbits of intermediate size. In this case, we can show joint equidistribution with polynomial rate in the modular surface and in the torus.

We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.

Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study.