Rigidity of group actions in Ergodic Theory, and related topics

遍历理论中群体行为的刚性及相关主题

• 批准号: 0604611
• 负责人: Alexander Furman
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604611&HistoricalAwards=false
• 关键词: Rigidity; group; actions; Ergodic; Theory

The project focuses on three areas of rigidity in the theory of "large" groups and in Ergodic theory: (1) extending Margulis' superrigidity and Zimmer's cocycle superrigidity from Lie groups setting to other situations, including A_2 groups and lattices in products;(2) exploring the connections between recent advances due to Popa in von Neumann Algebras in the context of Ergodic theory;(3) studying measure rigidity questions in the spirit of Ratner's theorem for algebraic actions of non-amenable groups (which have no unipotents).The common phenomenon, termed "rigidity", in the above problems is that certain algebraic structures describing intrinsic symmetries of the studied objects impose very strong constrains on the otherwise flexible characteristics of these objects. One of the most intriguing aspects of what became known as "rigidity theory" is its characteristic synergy of methods and tools from rather different areas: Algebra, Number theory, Geometry and Analysis. This project concerns some problems in Dynamics most related to Lie groups and Analysis.

本项目主要集中在“大”群理论和遍历理论中的三个刚性领域：(1)将Marguis的超刚性和Zimmer的余圈超刚性从Lie群环境推广到其他情形，包括乘积中的A_2群和格；(2)在遍历理论的背景下，探索Popa在von Neumann代数中的最新进展之间的联系；(3)按照Ratner关于非服从群(没有单位元群)的代数作用的Ratner定理的精神，研究度量刚性问题。在上述问题中，常见的现象称为“刚性”，即描述被研究对象的内在对称性的某些代数结构对这些对象的其他柔性特征施加了非常强的约束。“刚性理论”最耐人寻味的一个方面是，它将代数、数论、几何和分析等不同领域的方法和工具结合在一起。这个项目涉及一些与李群和分析最相关的动力学问题。