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.
