Rigidity and superrigidity in von Neumann algebras

冯诺依曼代数中的刚性和超刚性

• 批准号: 1161047
• 负责人: Adrian Ioana
• 金额: $ 34.78万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161047&HistoricalAwards=false
• 关键词: Rigidity; superrigidity; von; Neumann; algebras

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.

这个项目的目的是发展新的技术来研究和分类冯诺依曼代数产生的群体和行动的群体的概率空间。 提出的研究是由以下基本问题的动机：多少冯诺依曼代数记得有关集团或集团行动，它是从？这个问题是密切相关的主题遍历理论，群论和描述集理论，并在最近产生了一系列的互动与这些领域。在过去的十年中，Popa的变形/刚度理论导致了丰富的刚度结果。这些表明，许多关于群或群作用的信息可以通过观察它们的冯诺依曼代数来阅读。最近，第一类群作用和群是冯·诺依曼超刚性的（即可以完全从它们的冯·诺依曼代数重构）已经被发现。研究人员打算继续这条研究路线，并开发新的方法来扩大范诺依曼代数的刚性和超刚性的范围。拟议的研究将结合联合收割机冯诺依曼代数技术从变形/刚性理论与工具遍历理论和表示理论的群体。 研究人员预计，拟议中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，理解物体的部分结构就足以解开物体的整个结构。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依曼代数作为一个肥沃的框架研究刚性的作用。它不仅解决了冯诺依曼代数中的许多长期存在的问题，而且也解决了轨道等价遍历理论和描述集合论中的许多问题。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的联系。 该项目非常适合研究生的培养。 它包括许多方向的研究和开放的问题，从变形/刚度理论。 该项目的成果将通过出版物、系列讲座和会谈广泛传播。研究人员打算继续通过积极参与组织会议和研讨会，促进冯诺依曼代数和刚性广泛主题内的其他学科之间的互动。