Rigidity, Measured Group Theory, and Dynamics

刚性、测量群论和动力学

• 批准号: 1611765
• 负责人: Alexander Furman
• 金额: $ 27.1万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611765&HistoricalAwards=false
• 关键词: Rigidity; Measured; Group; Theory; Dynamics

AbstractAward: DMS 1611765, Principal Investigator: Alexander FurmanStudy of symmetry is one of the central themes in mathematics: one investigates mathematical laws and various associated objects by analysing their symmetries (i.e., transformations that do not change them) and further analyses how the laws and objects change under various transformations. In modern language one studies groups of symmetries and groups of transformations. The overarching theme of the project is to study phenomena in dynamics and geometry that occur in the presence of large groups (i.e., situations with a lot of explicit or implicit symmetries). This research continues important work over the last few decades that connects topics in geometry, probability theory, dynamical systems, and number theory.More specifically, the proposal focuses on three areas: (1) measured group theory - an ergodic-theoretic analogue of geometric group theory; (2) further study of higher-rank super-rigidity phenomena; and (3) study of Lyapunov exponents in the classical context of the multiplicative ergodic theorem in the presence of large groups. The first topic includes study of rigidity for rank-one lattices in analogy with previously established rigidity results for higher-rank lattices. Another promising direction is development of a measured analogue of Gromov-hyperbolic groups. The second theme includes study of rigidity phenomena for groups that resemble but are different from higher-rank lattices, but admit rich Weyl groups. The third topic develops a method of establishing simplicity of the Lyapunov spectrum using certain group actions - this investigation connects now classical results on random products of matrices, geodesic flows on negatively curved manifolds, and recent developments in Teichmuller dynamics.

摘要奖：DMS 1611765，首席研究员：Alexander Furman对称性研究是数学的中心主题之一：通过分析数学定律和各种相关对象的对称性（即不改变它们的变换）来研究它们，并进一步分析定律和对象在各种变换下如何变化。在现代语言中，人们研究对称群和变换群。该项目的首要主题是研究存在大群体（即具有大量显式或隐式对称性的情况）时发生的动力学和几何现象。这项研究继续了过去几十年来连接几何、概率论、动力系统和数论等主题的重要工作。更具体地说，该提案重点关注三个领域：（1）测得群论——几何群论的遍历理论类似物； (2)进一步研究高阶超刚性现象； (3) 在存在大群的乘法遍历定理的经典背景下研究李亚普诺夫指数。第一个主题包括研究一阶晶格的刚性，与先前建立的高阶晶格的刚性结果类似。另一个有希望的方向是开发格罗莫夫双曲群的可测量类似物。第二个主题包括研究与高阶格相似但不同但承认富外尔群的群的刚性现象。第三个主题开发了一种使用某些群作用来建立李雅普诺夫谱的简单性的方法 - 这项研究现在将矩阵随机乘积的经典结果、负弯曲流形上的测地流以及 Teichmuller 动力学的最新发展联系起来。