Applications of ergodic theory to geometry: Dynamical Zeta Functions and their applications

遍历理论在几何中的应用：动态 Zeta 函数及其应用

基本信息

批准号：
EP/M001903/1
负责人：
Mark Pollicott
金额：
$ 119.07万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM001903%2F1
关键词：
Applications ergodic theory geometry Dynamical

项目摘要

The proposed research lies at the interface of Ergodic Theory and Dynamical Systems, geometry, number theory, partial differential operators and mathematical physics. Central to this research programme are the the application of ideas from smooth ergodic theory to problems in different areas of mathematics. As such it is a highly intra-disciplinary research program. It also seems very timely, since there has been an explosion of activity in these areas in the last year which has attracted widespread attention. The proposed research is at the cutting edge of this development. In particular, the basis for this project rests on four important inter-related strands in applications of ergodic theory and dynamical systems to other areas: zeta functions and Poincare series (with their connections to number theory and geometry); Decay of correlations and resonances (with applications to the physical sciences); Numerical algorithms (with applications to both Pure and Applied Mathematics); and Teichmuller theory and Weil-Petersson metrics (at the boundary of ergodic theory, analysis and geometry).The study of geometric zeta functions for closed geodesics on negatively curved manifolds was initiated by Fields Medallist A. Selberg in the 1950s (following his earlier work on number theory). Selberg studied the case of constant curvature manifolds, using trace formulae and ideas from representation theory which do not generalise. However, recent work of Giulietti, Liverani and myself used a completely different viewpoint involving ideas in ergodic theory to extend the zeta function for negatively curved manifolds (and even more generally smooth Anosov flows, generalizing the geodesic flow). This provides the starting point for our proposed research on zeta functions, providing both a springboard to a whole host of significant applications and providing the scientific framework via the new ideas and techniques it initiated.

拟议的研究在于厄运理论和动力学系统，几何，数理论，部分差异操作员和数学物理学的界面。该研究计划的核心是从平稳的厄贡理论到数学不同领域的问题的应用。因此，这是一项高度高学科的研究计划。这似乎也很及时，因为去年这些地区的活动爆炸激增，引起了广泛关注。拟议的研究处于这一发展的最前沿。特别是，该项目的基础取决于在千古理论和动态系统到其他领域的应用中的四个重要相关链：zeta函数和庞加雷系列（及其与数字理论和几何学的联系）；相关和共振的衰减（对物理科学的应用）；数值算法（并应用于纯数学和应用数学）；以及Teichmuller理论和Weil-Petersson指标（在Ergodic理论，分析和几何学的边界上）。几何Zeta Zeta函数在1950年代由Fields A. Selberg启动了封闭的大地测量歧管上的封闭地球学的函数（此后，他的早期工作是在数字理论上）。塞尔伯格使用痕量公式和不推广的表示理论的思想研究了恒定曲率歧管的情况。然而，朱利叶蒂，利弗拉尼和我本人的最新工作使用了一个完全不同的观点，涉及厄贡理论中的思想，以扩展zeta函数，以使曲折的流形（甚至更普遍地平滑anosov流动），从而推广了大地测量流）。这为我们提出的关于Zeta功能的研究提供了起点，为整个重要应用程序提供了跳板，并通过其发起的新想法和技术提供了科学框架。