Descriptive Set Theory and Measured Group Theory

本项目旨在解决描述组合学和轨道等价领域的几个问题。这些主题是在描述集合论，测量群论，图论，遍历论，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，通过描述组合和图论的方法，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。除了组合的观点，这些领域的研究也被全局的观点所促进，从全局的观点来看，遍历理论中的问题，例如，可以被看作是关于标准概率空间中自同构群的行动的拓扑动力学和描述性问题。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。可测群论试图通过其在标准概率空间上的可测作用的遍历理论性质，特别是通过由这些作用产生的轨道等价关系的结构性质来理解可数无限群。激励现象是作用群的代数性质通常通过关联等价关系的可测量性质来表示。这种现象的一种极端形式可以在轨道等价、循环刚性和超刚性定理中看到，这些定理表明，在某些情况下，等价关系完全记住了产生它的群。另一个极端是所谓的轨道等价反刚性定理，它指出某些群和作用不能通过观察它们产生的等价关系来区分。这个研究项目涉及这两个极端的现象。这些问题是由主要研究者和合作者建立的刚性/反刚性结果引起的，这导致了本项目中解决的新问题。该项目将解决有关可数群和等价关系的代数、动力学和描述性结构复杂性的问题，特别是在轨道等价、可树性、循环超刚性、弱等价刚性以及图和群作用的可测量组合性质和参数等领域。该项目采用描述性组合和图论方法来解决这些问题。

This project aims to address several problems in the areas of descriptive combinatorics and orbit equivalence. These topics lie at the interface 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. In addition to a combinatorial perspective, research in these fields is also facilitated by a global perspective, from which problems in ergodic theory, for example, may be seen as topological-dynamical and descriptive problems concerning actions of the group of automorphisms of a standard probability space. This research project aims to generate more new general tools and to promote fruitful interactions among these fields. Measured group theory seeks to understand countably infinite groups through ergodic theoretic properties of their measurable actions on standard probability spaces, and particularly through structural properties of the orbit equivalence relations generated by these actions. The motivating phenomenon is that algebraic properties of an acting group are often expressed through measurable properties of the associated equivalence relation. An extreme form of this phenomenon is seen in orbit equivalence and cocycle rigidity and superrigidity theorems, which state that in certain settings an equivalence relation completely remembers the group from which it was generated. At the other extreme are what might be called orbit equivalence anti-rigidity theorems, stating that certain groups and actions cannot be distinguished by looking at the equivalence relations they generate. This research project touches upon phenomena at both of these extremes. The questions are motivated by rigidity/anti-rigidity results established by the principal investigator and collaborators, which have led to new questions addressed in this project. The project will address problems concerning algebraic, dynamical, and descriptive structural complexity of countable groups and equivalence relations, specifically in the areas of orbit equivalence, treeability, cocycle superrigidity, weak equivalence rigidity, and measurable combinatorial properties and parameters of graphs and group actions. The project pursues a descriptive combinatorial and graph theoretic approach to these problems.