Mathematical Sciences: Analytic and Geometric Questions for Smooth Group Actions

数学科学：平稳群体行动的分析和几何问题

• 批准号: 9623109
• 负责人: Renato Feres
• 金额: $ 4.7万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623109&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analytic; Geometric; Questions

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.

摘要Feres项目的主要目的是探索微分几何与李群和叶理作用相关的动力系统之间的一些联系。重点研究了实秩至少为2的半单李群及其格的光滑作用。考虑的其他问题涉及可服从群的表示和迁移方法，以及具有Kazhdan性质t的叶。动力系统的一般理论提供了用于理解过程长期发展的工具，无论是物理的，生物的还是纯数学的。通常，过程演化被认为是单个参数的函数，通常解释为时间。然而，在多参数系统中出现了一些质的新现象。在这个项目中特别关注的是一种“刚性”属性。(参数空间由“群”或“叶”的数学概念表示。这里考虑的一些“群”，即具有“属性T”的格群，与信息网络的最优连通性等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一些更基本和基本的问题。