Large centralizers and counting integral points on affine varieties

Abstract: Duke-Rudnick-Sarnak and Eskin-McMullen initiated the use of ergodic methods to count integral points on affine homogeneous varieties. They reduced the problem to one of studying limiting distributions of translates of periods of reductive groups on homogeneous spaces. The breakthrough of Eskin, Mozes and Shah provided a rather complete understanding of this question in the case the reductive group has a “small centralizer” inside the ambient group. In this talk, we describe work in progress giving new results on the equidistribution of generic translates of closed orbits of semisimple groups with “large centralizers”. The key new ingredient is an algebraic description of a partial compactification (for lack of a better word) of the set of intermediate groups which act as obstructions to equidistribution. This allows us to employ tools from geometric invariant theory to study the avoidance problem.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar