Spectral theory and dynamics on hyperbolic manifolds

数论中的许多问题（例如，理解多项式方程的整数解的数量和几何）可以通过研究对称空间上某些流动的动力学来解决。另一方面，解析数论中的经典工具可以被用来更好地理解这些空间的几何和动力学。本项目研究了利用谱理论和解析数论的工具来研究双曲流形上某些群作用的几何和动力学的一些问题。双曲流形的动力学出现在数学的许多领域：在数论中，它们是自同构形式理论的核心；在低维拓扑中，它们有助于证明几何化定理；在动力学中，双曲流形上的测地线流是双曲（Anosov）动力系统的一个经典例子；在数学物理中，双曲流形的谱可以看作是量子化混沌系统的一个例子。本项目研究了利用谱理论和解析数论的工具来研究双曲流形上具有和不具有算术结构的群作用的几何和动力学问题。具体来说，该研究研究了关于封闭测地线长度谱的问题，以了解从其封闭测地线长度的信息中可以获得多少双曲流形的几何形状；双曲空间上单参数流的逃逸至无穷远速率问题（源自丢番图近似问题）以及关于局部对称空间上强谱隙性质的问题，这对双曲动力学和数论的许多应用都是至关重要的。

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.