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

该项目位于逻辑和分析的界面上，更具体地说，是描述集合论、测量群论、图论、遍历理论、概率论和算子代数的接口。近年来，这些领域的数学家已经认识到，通过描述性组合和图论方法，可以很好地研究可数群的代数、动力学和描述性结构复杂性以及等价关系等问题。主要研究人员和合作者利用这种组合观点创建了新的工具，用于回答这些领域的几个公开问题。这一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。PI建议围绕描述集合论及其与轨道等价关系、上循环超刚性、树性、内部适应性、弱等价和弱包容以及刚性/反刚性的具体相互作用来研究问题。PI在这些领域的研究已经引发了许多有趣和基本的问题，这些问题涉及到描述集合论、测度群理论、图论、遍历论、概率论和算子代数等多个领域。对这些问题的研究将揭示这些领域之间令人惊讶的进一步联系。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。

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.