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Vadim Kaimanovich: Continuity of asymptotic characteristics of random products

In this talk (based on a joint work with A. Erschler) we shall discuss continuity of the basic asymptotic invariants of random products and random walks: entropy, rate of escape (e.g., Lyapunov exponents) and the harmonic measure.

David Kerr: Sofic invariants for group actions

Soficity is a weak kind of finite approximation property for discrete groups that generalizes both amenability and residual finiteness. In a remarkable breakthrough, Lewis Bowen showed how one can define an entropy invariant for measure-preserving actions for countable sofic groups by measuring the exponential growth of the number of models for the dynamics which are compatible with a fixed sequence of sofic models for the group. By applying an operator algebra perspective, Hanfeng Li and I have developed a more general approach to sofic entropy that produces both measure and topological dynamical invariants, and we have established the variational principle in this context. In another direction, one can allow the sofic model for the group to vary at the same time, in the spirit of Voiculescu's free entropy, and for free actions this yields an invariant of orbit equivalence, as Ken Dykema, Mikael Pichot, and I have shown. I will discuss all of these sofic invariants and their behaviour in prototypical examples.

Tullio Ceccherini-Silberstein: On surjunctivity and sofic groups

A category C is said to be SURJUNCTIVE if every injective endomorphism is surjective (and therefore bijective). In symbolic dynamics, we say that a group G is SURJUNCTIVE if the category consisting of shifts A^G where A is a finite alphabet with morphisms the cellular automata T:A^G → A^G (i.e. selfmaps continuous with respect to the prodiscrete topology and commuting with the G-shift) is surjunctive. We outline the proof of the Gromov-Weiss surjunctivity theorem for SOFIC groups and discuss some generalizations to linear cellular automata leading to the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups.

Konstantin Tolmachov: Variance of the Plancherel measure on the set of Young diagrams

In this talk we will discuss asymptotic properties of random partitions with respect to the Plancherel measure and compute the precise asymptotics for the variance of a diagram shape.

Alexey Klimenko: Cesàro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups

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Sevak Mkrtchyan: Asymptotic properties of Schur-Weyl duality

Vershik and Kerov in 1985 gave asymptotic bounds for the maximal and typical dimensions of the irreducible representations of the symmetric group. It was conjectured by Grigori Olshanski that the maximal and typical dimensions of the isotypic components of the representations in the base of Schur-Weyl duality accept similar asymptotic bounds. The isotypic components of this representation are parametrized by certain Young diagrams, and the relative dimensions of these components give rise to a measure on Young diagrams. Philippe Biane in 2001 found the limit shape of a typical Young diagram with respect to this measure. We will discuss a proof of the conjecture which is based on showing that the limit shape found by Biane is the minimizer of a certain functional.

Xavier Bressaud: Self-similar tilings and local rules. The Tribonacci case

The study of infinite words (symbolic dynamics) leads to distinguish two particular classes of dynamical systems dramatically different properties: substitutive systems (symbolic "self-similar" systems) and subshifts of finite type (characterized by local rules).
This distinction is deeply challenged in "size 2", that is to say for the study of tilings of the plane, first by the existence of aperiodic tilings characterized by certain local rules (Robinson) and by results of Moses and Goodman-Strauss showing that large classes of "self-similar" tilings are indeed characterized by local rules.
The self-similar tilings (quasicrystals) appearing in the study of symbolic Pisot substitutions, particularly those representing discrete planes (with cubic slopes), and more specifically the so-called Tribonacci tiling (aperiodic tiling by Rauzy fractals) are not covered by the result of Goodman-Strauss.
I will try to show how, using the same ideas, we can adapt existing results for characterizing Tribonacci tiling with local rules.

Jon Fickensher: Self-Inverses in Rauzy Classes

Rauzy Classes are collections of permutations generated by induction on Interval Exchange Transformations. While functions on an interval, IET's have a connection to flat surfaces. Recently, all Rauzy Classes have been classified by their corresponding flat surfaces. We will use this classification to show that every class contains a permutation that is its own inverse.

Vladlen Timorin: Topological cubic polynomials

We study topological dynamics of polynomials (viewed as dynamical systems on the complex plane). In parameter spaces of cubic polynomials, we describe the subsets corresponding to the simplest dynamical behavior. Based on joint work with A. Blokh, L. Oversteegen, R. Ptacek.