Definable Equivalence Relations and Dynamics, Topological and Measurable, of Polish Groups

波兰群的可定义等价关系和动力学、拓扑和可测

• 批准号: 1954069
• 负责人: Slawomir Solecki
• 金额: $ 32.5万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954069&HistoricalAwards=false
• 关键词: Definable; Equivalence; Relations; Dynamics; Topological

Groups are abstract mathematical entities that capture the structure of symmetries of an object. Groups are studied through the way they act as symmetries of various other mathematical objects. These actions are ubiquitous in mathematics and in sciences and they come at various levels of resolution. The project considers group actions in the very broad context of preserving definable sets, and more narrow contexts of preserving a notion of being close and of preserving the size of sets. The ideas in the project use and connect several disparate areas of mathematics. In addition, the project provides research training opportunities for graduate students. The project will tackle three types of issues all related to dynamics of topological groups. The first part aims to make progress on an important dichotomy in Descriptive Set Theory of Borel equivalence relations; the second part will investigate submeasures, concentration of measure in associated sequences of mm-spaces, and dynamics of associated topological groups; the third part, using measurable dynamics and unitary representations, will deal with the nature of the closed subgroups generated by generic measure preserving transformations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

群是抽象的数学实体，它们捕捉对象的对称结构。群是通过它们作为各种其他数学对象的对称的方式来研究的。这些行为在数学和科学中无处不在，它们有不同的分辨率。该项目认为，在非常广泛的背景下，保持可定义的集，和更狭窄的背景下，保持一个概念，被关闭，并保持集的大小的群体行动。该项目中的思想使用并连接了几个不同的数学领域。此外，该项目还为研究生提供研究培训机会。 该项目将解决三种类型的问题都与拓扑群的动力学。第一部分主要研究Borel等价关系描述集理论中的一个重要二分法，第二部分主要研究子测度、测度在mm-空间的相伴序列中的集中性以及相伴拓扑群的动力学;第三部分，利用可测动力学和酉表示，该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用的支持基金会的学术价值和更广泛的影响审查标准。