Rigidity and superrigidity in von Neumann algebras

本项目的目的是开发新的技术来研究和分类冯·诺伊曼代数，这些代数是由群和群在概率空间上的作用产生的。提出的研究是由以下基本问题驱动的：一个冯·诺伊曼代数能记住多少关于它所构建的群体或群体行为的信息？这个问题与遍历论、群论和描述集论的主题密切相关，最近在这些领域产生了一系列的互动。在过去的十年中，Popa的变形/刚度理论导致了大量的刚度结果。这些研究表明，通过观察冯·诺伊曼代数，可以解读大量关于群体或群体行为的信息。最近，发现了第一类冯诺依曼超刚性群和群（即可以完全由其冯诺依曼代数重构的群）。研究者打算继续这条研究路线，并开发新的方法来扩展冯诺依曼代数的刚性和超刚性的范围。提出的研究将结合变形/刚性理论中的冯·诺伊曼代数技术与遍历理论和群的表示理论的工具。研究者期望这项研究计划也将导致这些领域与冯·诺依曼代数理论之间的新的相互作用。在数学中，刚性指的是一种类似理想的情况，在这种情况下，对物体结构的部分理解足以揭示物体的整个结构。多年来，刚性结果出现在数学的各个领域。最近，变形/刚性理论巩固了冯·诺伊曼代数作为研究刚性的肥沃框架的作用。它不仅解决了冯·诺依曼代数中许多长期存在的问题，而且还解决了轨道等价遍历理论和描述集理论中的许多问题。在未来的几年里，该理论有很大的潜力在这些领域找到新的应用，以及与其他领域的惊人联系。建议的项目非常适合研究生的训练。它包含了许多研究方向和从变形/刚性理论中产生的开放问题。这个项目的成果将通过出版物、系列讲座和会谈广泛传播。研究者打算通过积极参与会议和研讨会的组织，继续促进冯·诺伊曼代数与刚性广泛主题内的其他学科之间的相互作用。

The aim of this project is to develop new techniques to study and classify von Neumann algebras arising from groups and actions of groups on probability spaces. The proposed research is motivated by the following fundamental question: how much does a von Neumann algebra remember about the group or group action it was constructed from? This question is intimately related to topics in ergodic theory, group theory and descriptive set theory and has recently generated a flurry of interactions with these fields. During the last decade, Popa's deformation/rigidity theory has led to a wealth of rigidity results. These show that a lot of information about groups or group actions can be read by looking at their von Neumann algebras. Recently, the first classes of group actions and groups which are von Neumann superrigid (i.e. which can be entirely reconstructed from their von Neumann algebras) have been discovered. The investigator intends to continue this line of research and develop new methods to extend the scope of rigidity and superrigidity in von Neumann algebras. The proposed research will combine von Neumann algebraic techniques from deformation/rigidity theory with tools from ergodic theory and representation theory of groups. The investigator expects that the proposed research project will also lead to new interactions between these fields and the theory of von Neumann algebras.In mathematics, rigidity refers to an ideal-like situation in which understanding part of the structure of an object is enough to unravel the object's entire structure. Over the years, rigidity results have appeared in various areas of mathematics. Recently, deformation/rigidity theory has cemented the role of von Neumann algebras as a fertile framework for studying rigidity. It has led to the resolution of many long-standing problems not only in von Neumann algebras, but in orbit equivalence ergodic theory and descriptive set theory as well. In the coming years, the theory has great potential to find new applications to these areas as well as surprising connections with other fields. The proposed project is well suited to the training of graduate students. It includes many directions of study and open problems that sprung from deformation/rigidity theory. The results deriving from this project will be broadly disseminated through publications, lecture series, and talks. The investigator intends to continue promoting interactions between von Neumann algebras and other subjects within the broad theme of rigidity through active involvement in the organization of conferences and seminars.