Counting lattice points and asymptotic properties in Z^n

I will discuss questions of counting lattice points in SL(2,R) (a lattice is a discrete subgroup with finite co-volume) and more generally in SL(n,R). The implication that I will focus on in the talk is the equidistribution of certain parameters related to integral vectors: their directions, the integral lattices in their orthogonal space, and the shortest solution to their associated gcd equation. The results are from my PhD thesis, and from ongoing joint work with Y. Karassik.