Title: Optimal results in Diophantine approximation on homogeneous spaces
Abstract: Let G be a Lie group, L a lattice in G, and H a closed subgroup of G.
Suppose that L acts on the homogeneous space G/H with dense orbits. We would like to measure how dense these orbits actually are, or equivalently, gauge the efficiency of approximation of a general point on G/H by a lattice orbit. Departing from classical Diophantine approximation, our discussion will include such examples as the group of isometries of hyperbolic space, or the general linear group. We will begin by describing the extensive scope of this set-up and demonstrate its connection to Diophantine approximation on algebraic varieties. We will then present an approach to the optimal solution of the problem of estimating the speed of approximation for lattice actions for a large class of homogeneous spaces. Finally, we will describe some more refined problems related to equidistribution and discrepancy of lattice orbits, as time permits.