Monday, February 3, 2020
02/03/2020 - 4:00pm
Motivated by Selberg’s 3/16 theorem for congruence covers of the modular surface, there have been many recent developments starting with Bourgain-Gamburd-Sarnak’s work to obtain analogous results for infinite volume hyperbolic manifolds. One such result by Oh-Winter is a generalization of the uniform exponential mixing formulation of Selberg’s theorem for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but also have a wide range of applications including resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.
02/03/2020 - 4:30pm
The Orbit Method is a conjectural correspondence between co-adjoint orbits of a Lie group G and its irreducible unitary representations. When G is nilpotent or simply connected and solvable, this correspondence is perfect and complete. But when G is reductive, serious problems arise. The worst of these problems have to do with the nilpotent orbits of G. As of yet, there is no general method for attaching unitary representations to nilpotent orbits. In this talk, I will attach a finite set of irreducible unitary representations to the principal nilpotent orbits. I will deduce formulas for the K-types and associated varieties of the representations in question.