Ergodic Theory, Groups, and Geometry

本研究将探讨李群的作用以及相关的动力学、拓扑学和几何学问题。 重点将放在非紧半单李群在流形上的作用，以及群，在作用下不变的几何结构和流形的拓扑结构，特别是基本群之间的关系。 使用过神理论，即，群作用的测度理论结构和行为，将是广泛的。 特别是，这些作用的定常测度和不变测度的结构，以及这些一般测度与标准算术或射影模型的相似程度，将发挥重要作用。对对称性的研究在科学和数学中发挥了重要作用，是理解许多现象的基本的、往往是不可缺少的工具。 与科学和技术的联系涵盖了从基本粒子物理学到材料的晶体结构、到网络设计、到动力系统的各个领域。 研究对称性的基础数学是群论。 这项研究将集中在有关大对称群的基本数学问题上，即所考虑的结构具有许多对称性的情况。 预计这项工作将有助于对对称性的基本理解，从而增加其在科学中潜在应用的力量。

AbstractZimmerThis research will investigate actions of Lie groups and related questionsof dynamics, topology, and geometry. A focus will be on the actions ofnon-compact semisimple Lie groups on manifolds, and the relationshipbetween the group, geometric structures invariant under the action, and thetopology of the manifold, in particular the fundamental group. Use ofergodic theory, i.e., the measure theoretic structure and behavior of groupactions, will be extensive. In particular, the structure of stationarymeasures and invariant measures for these actions, and the extent to whichsuch general measures are similar to standard arithmetic or projectivemodels, will play a major role.The study of symmetry has played a major role in science and mathematics,and is a fundamental, often indispensable, tool for understanding manyphenomena. The connections to science and technology span the gamut fromelementary particle physics, to crystal structure of materials, to networkdesign, to dynamical systems. The underlying mathematics for studyingsymmetry is group theory. This research will focus on fundamentalmathematical questions concerning large symmetry groups, i.e. situations inwhich the structure under consideration has many symmetries. It isanticipated that this work will contribute to the fabric of fundamentalunderstanding of symmetry, thereby increasing the power of its potentialapplication across science.