Dynamical and descriptive aspects of groups and their actions

该项目位于逻辑和分析的接口，更具体地说，描述性集合论，测量群论，图论，遍历论，概率论，和算子代数。近年来，这些领域的数学家已经认识到，通过描述组合和图论手段，可以对可数群和等价关系的代数、动态和描述结构复杂性问题进行富有成效的研究。首席研究员和合作者采用这种组合的观点来创造新的工具，用于回答这些领域的几个开放问题。该研究项目旨在产生更多新的通用工具，并促进这些领域之间富有成效的互动。该项目涉及研究生和本科生的培训。PI提出运用描述集合论和逻辑学的观点和见解，研究群作用的轨道等价关系、测度等价和拟等距、顺从和内顺从、循环超刚性、可树性、弱等价、刚性和反刚性等问题。提案中的问题涉及许多领域，包括描述集合论、可测群论、几何群论、可测组合学、遍历论、概率论和算子代数。PI和合作者已经抓住了解决许多提出的问题的第一个立足点，这个项目可能会推进这些领域的知识。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。

The project lies at the interface of logic and analysis, more specifically, 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 groups 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. This project involves training of graduate and undergraduate students.The PI proposes to study problems around orbit equivalence relations, measure equivalence and quasi-isometry, amenability and inner amenability, cocycle superrigidity, treeability, weak equivalence, and rigidity and anti-rigidity for group actions by applying the perspective and insights of descriptive set theory and logic. The problems in the proposal are of interest to many fields, including descriptive set theory, measured group theory, geometric group theory, measurable combinatorics, ergodic theory, probability theory, and operator algebras. The PI and collaboratorshave already seized the first foothold toward a resolution to many of the proposed problems, and this project is likely to advance knowledge in 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.