Ergodic Theory, Groups, and Geometry

遍历理论、群和几何

基本信息

批准号：
9988774
负责人：
Kevin Corlette
金额：
$ 29.47万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2000
资助国家：
美国
起止时间：
2000-07-01 至 2005-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988774&HistoricalAwards=false
关键词：
Ergodic Theory Groups Geometry

项目摘要

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.

AbstractZimmerthis Research将调查谎言组的行动以及动态，拓扑和几何形状的相关问题。将重点放在非脉冲半径谎言基团对流形的作用上，以及该组之间的关系，在作用下的几何结构和歧管的thetopology，尤其是基本群体。 erergodic理论的使用，即衡量群体的理论结构和行为，将是广泛的。特别是，这些动作的固定测量和不变措施的结构，以及一般措施与标准算术或ProjectiveModels相似的程度，将起主要作用。对对称性的研究在科学和数学中起着重要作用，并且在科学和数学中起了重要作用，并且是一个基本的，通常是必不可少的，可以理解多个菲诺米纳的工具。与科学和技术的连接涵盖了范围从域粒子物理学到材料的晶体结构，网络设计，与动态系统的连接。研究对称的基础数学是群体理论。这项研究将集中于有关大型对称群体的基本数学问题，即所考虑的结构具有许多对称性。概论时，这项工作将有助于对称对称性的基本理解的结构，从而提高其在科学跨科学的能力。