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

群是抽象的数学实体，捕捉对象的对称性结构。通过群作为各种其他数学对象的对称性来研究群。这些行为在数学和科学中无处不在，而且它们的分辨率各不相同。该项目在保持可定义集合的非常广泛的背景下考虑群体行动，而在保持集合的大小和保持接近的概念的更狭隘的背景下考虑群体行动。该项目中的想法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了研究培训机会。该项目将解决三种类型的问题，这些问题都与拓扑群的动力学有关。第一部分的目的是在Borel等价关系的描述集合论中的一个重要的二分法上取得进展；第二部分将研究子测度、相关mm-空间序列中的测度集中度以及相关拓扑群的动力学；第三部分将利用可测动力学和么正表示来处理由一般保持测量变换生成的闭子群的性质。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.