Deformation/rigidity theory in von Neumann algebras and ergodic theory

Vanderbilt University, Nashville TN

Investigators

Abstract

Von Neumann algebras were introduced in the 1930s and 1940s in part as a tool for developing a mathematical foundation for quantum physics. Von Neumann algebras have since become a field of independent interest with further applications to areas of mathematics such as ergodic theory, Voiculescu's free probability theory, Jones's theory of subfactors and planar algebras, knot theory, and many others. The development of von Neumann algebras has also historically been closely connected to the study of measurable dynamics, and these connections have recently begun to reemerge in the presence of a newly developed rigidity phenomenon. The investigation of this rigidity phenomenon has since led to new connections between von Neumann algebras and other areas of mathematics. Furthering the development of rigidity theory will in turn lead to new insights and connections among these various fields, and it will provide opportunities to exploit this phenomenon in other areas. This project will investigate the newly emerging deformation/rigidity theory in von Neumann algebras, as well as continue to explore the deep connection between this theory and ergodic theory. Deformation/rigidity theory, initiated by Popa in the early 2000s, has been extremely successful over the last decade in answering a number of longstanding problems in von Neumann algebras and ergodic theory. The juxtaposition between deformability properties such as Haagerup's property, free products, or unbounded cocycles, with rigidity properties such as so-called property T or the spectral gap allows one to discover hidden structure in a von Neumann algebra where both types of phenomena occur. This has led to new insight in the structural properties of these von Neumann algebras, and in turn has found applications to other areas such as measured group theory, or the theory of invariants. Developing alongside deformation/rigidity theory is the corresponding techniques applied to ergodic theory of group actions. This has already led to a number of striking results, and yet here the surface has only been scratched. This project will also investigate more fully these interactions.