Effective Equidistribution and Spectral Gap

Abstract. In these notes we discuss some equidistribution problems with the aim to give reasonable error rates, i.e. we are interested in effective statements. We motivate some arguments by studying a concrete problem on a two-torus, and then describe recent results on the equidistribution of semisimple orbits obtained in joint work with G. Margulis and A. Venkatesh. We end by studying the relationship between equidistribution of closed orbits and mixing properties. This leads to a way of transporting spectral gap from one group — via an effective equidistribution result on a quotient by an irreducible lattice – to another group. The latter topic is ongoing joint work with G. Margulis and A. Venkatesh.