Equidistribution, Diophantine approximation, and Grassmannians

Dirichlet’s approximation theorem tells us that we can approximate real numbers by rational numbers, with the errors controlled by the size of the denominators. Moreover, for m*n real matrices we can study the simultaneous approximation problem. Due to the work of Dani, this problem is related to the dynamical behavior of flows in the space of unimodular lattices in R^n, and thus to the question asked by Margulis concerning limit distributions of translates of smooth manifolds. In this talk I will present some recent progress on this problem, and I will explain some interesting connections to the geometry of Grassmannian manifolds.