Rigidity, Measured Group Theory, and 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：人们通过分析它们的对称性(即不改变它们的变换)来研究数学规律和各种相关对象，并进一步分析规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。该项目的主要主题是研究动力学和几何学中出现的现象，这些现象发生在大群体存在的情况下(即具有大量显性或隐含对称性的情况)。在过去的几十年里，这项研究继续着几何、概率论、动力系统和数论等学科的重要工作。更具体地说，该建议集中在三个领域：(1)被测群论-几何群论的遍历理论的类比；(2)进一步研究高阶超刚性现象；以及(3)在大群存在的乘法遍历定理的经典背景下研究Lyapunov指数。第一个主题包括研究秩一格的刚性，类似于以前建立的高阶格的刚性结果。另一个有希望的方向是发展一种可测量的格罗莫夫-双曲群的类似物。第二个主题包括研究类似但不同于高阶格但允许丰富的Weyl群的群的刚性现象。第三个主题发展了一种利用某些群作用建立Lyapunov谱的简单性的方法--这项研究将现在关于矩阵的随机积的经典结果、负曲线流形上的测地线流以及TeichMuller动力学的最新发展联系在一起。

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.