Descriptive Dynamics: Group Actions and Their Measured, Borel, and Topological Structures

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The project lies at the interface of logic and analysis, more specifically, of descriptive set theory, measured group theory, graph theory, ergodic theory, probability theory, and operator algebras. In recent years, mathematicians in these fields have come to realize that problems concerning algebraic, dynamical, and descriptive structural complexity of countable group and equivalence relations can be fruitfully studied via descriptive combinatorial and graph theoretic means. The principal investigator and collaborators have employed this combinatorial perspective to create new tools used to answer several open problems in these fields. This research project aims to generate more new general tools and to promote fruitful interactions among these fields. The PI proposes to study problems around descriptive set theory and its specific interactions with orbit equivalence relations, cocycle superrigidity, treeability, inner amenability, weak equivalence and weak containment, and rigidity/anti-rigidity. The PI's research in these areas has already prompted a wealth of interesting and fundamental questions which intersect a variety of fields including descriptive set theory, measured group theory, graph theory, ergodic theory, probability theory, and operator algebras. Research into these questions will reveal further surprising connections between these fields.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.