Rigidity, Measured Group Theory, and Dynamics

获奖：DMS 1611765，首席研究员：Alexander furman对称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，并进一步分析在各种变换下定律和对象是如何变化的。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在大群体存在时发生的动力学和几何现象（即，具有大量显式或隐式对称性的情况）。这项研究延续了过去几十年的重要工作，将几何、概率论、动力系统和数论等主题联系起来。更具体地说，该建议侧重于三个方面：(1)测量群论-几何群论的遍历理论模拟；(2)高阶超刚度现象的进一步研究；(3)在存在大群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度类比先前建立的高秩格的刚度结果。另一个有希望的方向是发展格罗莫夫-双曲群的测量模拟。第二个主题包括研究与高阶格相似但不同的群的刚性现象，但承认富Weyl群。第三个主题发展了一种利用某些群作用建立李雅普诺夫谱的简单性的方法——这项研究将现在关于矩阵随机积的经典结果、负弯曲流形上的测地流和Teichmuller动力学的最新发展联系起来。

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.