Deformation/rigidity theory in von Neumann algebras and ergodic theory

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Deformation/rigidity theory, initiated by Sorin Popa in the early 2000's, 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 property (T) or 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 into 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 L2-invariants. The Principal Investigator will continue to investigate and develop these ideas, focusing on their close connection to aspects of ergodic theory. Von Neumann algebras were introduced in the 1930's and 40's 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' 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 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 will in turn lead to new insights and connections among these various fields.