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

群是抽象的数学实体，它捕捉了一个对象的对称结构。群体是通过它们作为各种其他数学对象的对称性来研究的。这些行为在数学和科学中无处不在，它们有不同的分辨率。该项目考虑了群体行为在保持可定义集合的非常广泛的背景下，以及在保持接近概念和保持集合大小的更狭窄的背景下。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训练的机会。该项目将解决三种类型的问题，所有这些问题都与拓扑群的动力学有关。第一部分是对Borel等价关系描述集合论中一个重要二分法的研究进展；第二部分将研究子测度，测度在mm空间相关序列中的集中，以及相关拓扑群的动力学；第三部分，利用可测量的动态和幺正表示，讨论由一般测度保持变换生成的闭子群的性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.