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 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 unipotents flows. However they had to assume that $H$ is maximal and reductive (in particular 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 $SL(n,R)$. 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 measure supported on $U SL(n,Z)/SL(n,Z)$, where $U$ is the set of upper triangular unipotent matrices.
Joint work with A. Mohammad.