Speaker: Amos Nevo (University of Chicago) 

Title: Optimal Diophantine exponents and spectral estimates in the automorphic representation

Abstract:

Our main topic will be the action of an irreducible lattice subgroup in a semisimple group G on a homogeneous variety G/H, and specifically a natural notion of a Diophantine approximation exponent for lattice orbits on the variety. We will begin by describing a sufficient condition, in terms of spectral estimates in the automorphic representation, for the optimality of the exponent. The condition holds in a surprising abundance of cases we will describe, but in others its validity depends on the Selberg conjecture. This inevitably raises the basic problem of establishing spectral estimates in the automorphic representation which can serve as an alternative to the temperedness condition the Selberg conjecture guarantees.