Rigidity in von Neumann Algebras and Higher Rank Groups

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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 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.Developing alongside the theory of von Neumann algebras has been ergodic theory, and many results in one field has had major applications in the other. A reemergence of this collaboration has occurred in the last ten years with Popa's discovery of deformation/rigidity theory, which juxtaposes deformability properties such as Haagerup's property, free products, or unbounded cocycles, with rigidity properties such as property (T) or spectral gap, allowing one to discover hidden structure in a von Neumann algebra in the case when both types of phenomena occur. This project will investigate more fully these connections, focusing specifically on connections to the deep rigidity theory for ergodic actions of lattices in higher rank groups initiated by Mostow, Margulis, Zimmer, and many others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.