Spectral theory and dynamics on hyperbolic manifolds

双曲流形的谱理论和动力学

基本信息

批准号：
1401747
负责人：
Dubi Kelmer
金额：
$ 15.4万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2017-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401747&HistoricalAwards=false
关键词：
Spectral theory dynamics hyperbolic manifolds

项目摘要

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.

数论中的许多问题（例如，理解多项式方程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，可以采用解析数论的经典工具来更好地理解这些空间的几何和动力学。该项目研究了许多问题，其中使用谱理论和解析数论工具来研究双曲流形上某些群作用的几何和动力学。双曲流形动力学出现在数学的许多领域：在数论中，它们是自守形式理论的核心；在低维拓扑中，它们有助于证明几何定理；在动力学中，双曲流形上的测地流是双曲（阿诺索夫）动力系统的典型例子；在数学物理学中，双曲流形谱可以被视为量化混沌系统的一个例子。该项目研究了许多问题，其中谱理论和解析数论的工具用于研究双曲流形上某些群作用的几何和动力学，有或没有算术结构。具体来说，该研究研究了有关闭合测地线长度谱的问题，以了解双曲流形的几何形状有多少可以从其闭合测地线长度信息中获得；关于双曲空间上一个参数流逃逸到无穷大的速率的问题（源自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特性的问题，这对于双曲动力学和数论的许多应用至关重要。