Rigidity, Measured Group Theory, and Dynamics

AbstractAward：DMS 1611765，首席研究员：亚历山大弗曼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来研究数学定律和各种相关对象（即，不改变它们的变换），并进一步分析在各种变换下规律和对象如何变化。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在大群体存在下发生的动态和几何现象（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中连接几何、概率论、动力系统和数论的重要工作，更具体地说，这项研究计划集中在三个领域：（1）测量群论-几何群论的遍历理论模拟;（2）进一步研究高阶超刚性现象;（3）进一步研究高阶超刚性现象;（4）进一步研究高阶超刚性现象。（3）在大群存在的乘法遍历定理的经典背景下研究李雅普诺夫指数。第一个主题包括研究刚性的秩一格在类比先前建立的刚性结果为更高的秩格。另一个有前途的方向是Gromov双曲群的测量模拟的发展。第二个主题包括研究类似但不同于高阶格的群的刚性现象，但承认丰富的Weyl群。第三个主题开发了一种方法，建立简单的李雅普诺夫谱使用某些群体行动-这项调查连接现在的经典结果随机产品的矩阵，测地线流负弯曲的流形，和最近的事态发展Teichmuller动态。

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.