Spectral theory and dynamics on hyperbolic manifolds

Boston College, Chestnut Hill MA

Investigators

Abstract

Many problems in number theory (for instance, understanding the number and geometry of integer solutions to polynomial equations) can be approached by studying dynamics of certain flows on symmetric spaces. On the other hand, classical tools from analytic number theory can be adapted and used to better understand the geometry and dynamics of these spaces. This project investigates a number of problems where spectral theory and tools from analytic number theory are used to study geometry and dynamics of certain group actions on hyperbolic manifolds. Dynamics on hyperbolic manifolds come up in many areas of mathematics: In number theory they are central in the theory of automorphic forms; in low dimensional topology they are instrumental in the proof of the geometrization theorem; in dynamics the geodesic flow on hyperbolic manifolds is a classic example of hyperbolic (Anosov) dynamical systems; in mathematical physics the spectrum of hyperbolic manifolds can be viewed as an example of quantized chaotic systems. This project investigates a number of problems where spectral theory and tools from analytic number theory are used to study geometry and dynamics of certain group actions on hyperbolic manifolds, with and without arithmetic structure. Specifically, the research studies questions regarding the length spectrum of closed geodesics to understand how much of the geometry of a hyperbolic manifold can be obtained from information on the lengths of its closed geodesics; questions on the rate of escape to infinity of one parameter flows on hyperbolic spaces (originating from problems in Diophantine approximations); and questions on the strong spectral gap property on locally symmetric spaces, which is crucial to many applications to hyperbolic dynamics and number theory.