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

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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.