The counting formula of Eskin and McMullen

Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). The key insight is to relate the counting problems to the equidistribution of circles under geodesic flow. In this talk I will discuss how to deduce circle equidistribution and counting problem asymptotics from mixing. The talk will involve many pictures and focus on the case of hyperbolic surfaces, however, the arguments presented will be general and their application to counting on general affine symmetric spaces will be explained at the end of the talk.