(click titles for abstracts)
Actions of linear groups on sets and on probability spaces
Let A = Z! be the full symmetric group of a countable set
and B = Aut(X,µ) the automorphism group of a standard probability space.
I will discuss the role of the Zariski topology in the study of representation of countable
linear groups into these two Polish groups. I will illustrate the method by discussing the following two theorems.
Theorem 1 (joint with Dennis Gulko). For every sharply
2-transitive linear permutation group Γ, there exists a near
field N (i.e. a skew field which is distributive only from the left) such that Γ is the semidirect product of N* and N.
Theorem 2. If a countable linear group acts by measure preserving transformations on a probability space (X,µ) in such a way that almost every point has an amenable stabilizer. Then the stabilizers are all contained in a common amenable normal subgroup.
- Coffee break
On simple amenable groups
We will discuss amenability of the topological full group of
a minimal Cantor system. Together with the results of H. Matui this
provides examples of finitely generated simple amenable groups. Joint
with N. Monod.
- Lunch break
On percolation characterization of non-amenability
We will discuss Benjamini–Schramm non-amenabitily conjecture in relation with two values measuring non-amenability of a Cayley graph — the spectral radius of the random walk and the isoperimetric constant.
- Coffee break
Quantum correlations and Kirchberg's conjecture
I will talk about Tsirelson's problem on quantum correlations of independent
bipartite systems. The problem is known to be equivalent to Kirchberg's and
Connes's conjectures, and hence has a close connection with Gromov's problem
asking whether every group is sofic.