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

该项目位于逻辑和分析的接口，更具体地说，描述集理论，测量群理论，图论，遍历理论，概率论和算子代数。近年来，数学家们逐渐认识到，可数群的代数复杂性、动力复杂性和描述结构复杂性以及等价关系等问题可以通过描述组合和图论的方法得到有效的研究。主要研究者和合作者采用这种组合的观点来创建新的工具，用于回答这些领域中的几个开放问题。 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。PI建议研究围绕描述集理论及其与轨道等价关系，上循环超刚性，Treasure，内部顺应性，弱等价和弱包含以及刚性/反刚性的特定相互作用的问题。PI的研究在这些领域已经促使了丰富的有趣的和基本的问题，其中交叉的各种领域，包括描述集理论，测量群理论，图论，遍历理论，概率论和算子代数。对这些问题的研究将进一步揭示这些领域之间令人惊讶的联系。该奖项反映了NSF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, 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.