Rigidity and superrigidity in von Neumann algebras

这个项目的目的是发展新的方法来研究和分类由概率空间上的群和群的作用产生的von Neumann代数。这项拟议的研究是由以下基本问题推动的：冯·诺依曼代数对构造它的群或群行动记得多少？这个问题与遍历理论、群论和描述集合论中的主题密切相关，最近产生了一系列与这些领域的互动。在过去的十年里，波帕的变形/刚性理论已经导致了丰富的刚性结果。这表明，许多关于群或群行动的信息可以通过查看它们的von Neumann代数来阅读。最近，人们发现了第一类von Neumann超刚性(即可以从von Neumann代数完全重构)的群作用和群。作者打算继续这方面的研究，并发展新的方法来扩大von Neumann代数的刚性和超刚性的范围。这项研究将结合形变/刚性理论中的von Neumann代数技巧，以及遍历理论和群表示理论中的工具。研究人员预计，拟议的研究项目还将导致这些领域与冯·诺依曼代数理论之间的新的互动。在数学中，刚性指的是一种类似理想的情况，在这种情况下，了解对象的部分结构足以解开对象的整个结构。多年来，刚性结果已经出现在数学的各个领域。最近，形变/刚性理论巩固了von Neumann代数作为研究刚性的肥沃框架的作用。它不仅解决了von Neumann代数中的许多长期存在的问题，而且解决了轨道等价遍历理论和描述集合论中的许多长期存在的问题。在接下来的几年里，该理论将有巨大的潜力在这些领域找到新的应用，并与其他领域建立令人惊讶的联系。建议的项目非常适合研究生的培养。它包括许多研究方向和变形/刚度理论产生的公开问题。该项目的成果将通过出版物、系列讲座和讲座广泛传播。调查员打算通过积极参与组织会议和研讨会，继续促进冯·诺依曼代数与其他学科之间在僵化这一广泛主题下的互动。

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.