Spectral theory and dynamics on hyperbolic manifolds

数论中的许多问题(例如，了解多项式方程的整数解的数目和几何)可以通过研究对称空间上某些流的动力学来解决。另一方面，解析数论中的经典工具可以被改编和使用，以更好地理解这些空间的几何和动力学。这个项目研究了一些问题，其中谱理论和解析数论的工具被用来研究双曲流形上某些群作用的几何和动力学。双曲流形上的动力学出现在许多数学领域：在数论中，双曲流形是自同构形理论的中心；在低维拓扑学中，双曲流形上的测地线流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)动力系统的经典例子；在数学物理中，双曲流形的谱可以视为量子化混沌系统的一个例子。这个项目研究了一些问题，其中谱理论和解析数论的工具被用来研究双曲流形上某些群作用的几何和动力学，无论有没有算术结构。具体地说，这项研究研究了关于闭测地线的长度谱的问题，以了解一个双曲流形的几何图形的多少可以从它的闭测地线的长度信息中获得；关于双曲空间上的单参数流到无穷远的逃逸率的问题(源于丢番图近似中的问题)；以及关于局部对称空间上的强谱间隙性质的问题，这对于双曲动力学和数论的许多应用是至关重要的。

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.