Large centralizers and counting integral points on affine varieties.

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
report on work in progress giving new results on the equidistribution of
generic translates of certain closed orbits of semisimple groups with
“large centralizers”. The key new ingredient is an algebraic description
of a compactification (in a suitable sense) 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.