Activities This Week

To any group, and more generally a C*-algebra, is associated its simplex of traces. The extreme points are called characters, which are a central notion in harmonic analysis. For a Kazhdan group, the simplex of traces is Bauer - the extreme points are closed. For the free group Fn the simplex of traces is Poulsen - the extreme points are dense. What about the simplex of Out(Fn)-invariant traces on Fn (n>3)? Is it Bauer, Poulsen or something in between? What about free products of finite groups, and free product of matrix algebras? Some answers and proofs will be provided, after an introduction on traces and characters. The talk is based on works with Levit, Orovitz and Slutsky.

The group of diffeomorphisms acts on the space of symplectic structures on a given manifold. Taking a quotient by isotopies, we obtain the mapping class group action on the Teichmuller space of symplectic structures; the latter is a finite-dimensional manifold. The mapping class group action on the Teichmuller space is quite often ergodic, which leads to important consequences for symplectic invariants, such as symplectic packing problems. I would describe some of the problems which were solved using this approach. This is a joint work with Michael Entov.