Speaker: Andreas Wieser (Institute for Advanced Study)

Title: Effective equidistribution of periodic orbits of semisimple groups

Abstract: Qualitatively, actions of unipotent groups on homogeneous spaces are very well understood after breakthrough works of Ratner from the 90's. The past two decades have seen dramatic progress in providing rates in these results. In this talk, we will be interested in the distributional properties of periodic orbits of semisimple groups. Here, the desire is to show that such a periodic orbit 'approximates' a periodic orbit of a larger group with a rate that is polynomial in the volume.

In joint work with Einsiedler, Lindenstrauss, and Mohammadi, we prove such a result for compact (congruence) quotients with a rate this is explicit in all aspects (including the volume of the ambient space). This relies heavily on a variety of tools including uniformity of the spectral gap for periodic orbits in congruence quotients and polynomial properties of unipotent flows. We will discuss these results with an emphasis on the dynamical methods and mention applications in number theory briefly.