## Topology & Geometric Group Theory Seminar

## Spring 2008

Tuesday, March 11, 3:00 - 4:00, Rockefeller 104

**Alireza
Golsefidy**, Princeton University

*Translates of horospherical measures and counting problems*

(Joint with A. Mohammadi.) In this talk, I will briefly explain the
relation between some of the counting problems, mixing, and ergodic
theory. The counting problems might be of a geometric or number
theoretic nature.

For instance consider V=G/H a homogeneous variety, and one would like
to study the integer or rational points on V. Eskin, Mozes, and Shah
attacked this problem via unipotent flows. However they had to assume
that H is maximal, reductive and not inside any parabolic subgroup of
G. I will explain an ergodic theoretic approach toward such problem
for a flag variety.

For a geometric example, consider SL(n,Z)-translates of a horosphere
in the symmetric space of SL(n,R). The question is how many of them
intersect a ball of radius R. In fact, Eskin and McMullen answered
this question for n=2, using mixing. I will explain why mixing is not
enough and how one can get such a result for any n.

I will show that the main ingredient for both of the mentioned
questions is understanding the limits of translates of horospherical
measures, i.e. the probability measures supported on U SL(n,Z)/SL(n,Z),
where U is the set of upper triangular unipotent matrices.

Back to seminar home page.