Translates of horospherical measures and counting problems.

(Joint with A. Mohammadi) I will talk about the possible limit
measures of the 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. Then I
explain using this
information how one can prove Batyrev-Manin-Tschinkel’s conjecture on
the number of
rational points with height at most T on a flag variety (reproving a
theorem by
Franke-Manin-Tschinkel & Batyrev-Tschinkel), or study the asymptotic
of number of
translates of a horosphere intersecting a ball of radius T in the
symmetric space of
SL(n,R) (generalizing a result of Eskin-McMullen.)