Activities This Week

An IRS on a locally compact group $G$ is defined to be a probability measure on the space of all closed subgroups of $G$, which is invariant under conjugation. This notion generalizes, at the same time: normal subgroups, subgroups of finite index and lattices in $G$. More importantly the extra flexibility offered by the probabilistic becomes useful in proving many theorems even in the classical setting. In the past few years, we have been working in the opposite direction: trying to reprove results about IRSs in a more deterministic setting. This approach requires us to consider a much wider class of subgroups that we refer to as boomerang subgroups. I will show how these new methods give rise to a streamlined and more general version of the Nevo-Stuck-Zimmer theorem for certain lattices.

This is a joint work with Waltraud Lederle.

I will describe various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.

In this talk, for an ergodic probability preserving system (X,B,m,T), we will discuss the existence of a Z^d valued function , whose corresponding cocycle satisfies the d-dimensional local central limit theorem. As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in L^2 of polynomial multiple averages of non-commuting zero entropy transformations. We also provide first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff.