Mathematical Sciences: Analytic and Geometric Questions for Smooth Group Actions

该项目的主要目的是探索与李群和叶的作用有关的微分几何和动力系统之间的一些联系。特别研究了实数阶至少为2的半单李群及其格的光滑作用。考虑的其他问题涉及服从群胚的表示和转换方法，以及具有Kazhdan性质T的叶层。动力系统的一般理论提供了用于理解过程的长期发展的工具，无论它们是物理的、生物的还是纯粹的数学的。通常，过程演化被认为是单个参数的函数，通常被解释为时间。然而，在多参数系统中出现了一些定性的新现象。这个项目特别关注的是一种“刚性”性质。(参数空间由‘’群‘’或‘’叶‘’的数学概念表示。这里讨论的一些“群”，即具有“性质T”的格群，与诸如信息网络的最佳连通性这样的主题有着惊人的联系)。这个正在进行的项目的基本目的是通过几何和概率技术来探索这个主题的几个更基本和最基本的问题。

Abstract Feres The main purpose of the project is to explore a number of connections between Differential Geometry and Dynamical Systems associated to actions of Lie groups and foliations. Particular attention is given to smooth actions of semisimple Lie groups of real rank at least 2 and their lattices. Other problems considered are concerned with representations of amenable groupoids and the method of Transference, and foliations with Kazhdan's property T. The general theory of Dynamical Systems provides tools used to understand the long term development of processes, whether they are physical, biological, or purely mathematical. Ordinarily, the process evolution is considered as a function of a single parameter, usually interpreted as time. However, some qualitatively new phenomena arise in multiparameter systems. Of particular concern in this project is a kind of ''rigidity'' property. (The parameter space is represented by the mathematical notions of a ''group'' or a ''foliation.'' Some of the ''groups'' considered here, namely lattice groups with ''property T,'' have surprising connections with such a subject as optimal connectivity properties of information networks). The basic aim of this ongoing project is to explore a few of the more basic and fundamental questions of the subject by means of geometric and probabilistic techniques.