Derivations, quantum Dirichlet forms, and deformation/rigidity theory in von Neumann algebras

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The theory of von Neumann algebras is a non-commutative integration theory which was first developed by F. Murray and J. von Neumann as a tool for understanding representation theory for groups as well as providing a mathematical framework for quantum mechanics. Recently the appearance of new rigidity phenomena in von Neumann algebras has led to deeper understanding and to the solutions to many longstanding problems both in von Neumann algebras and in other areas such as orbit equivalence ergodic theory and measurable equivalence relations. There has also emerged recently a connection between deformation/rigidity theory, free probability, group cohomology, and derivations/quantum Dirichlet forms on von Neumann algebras. It is the intent of this author to investigate these new connections in order to gain more understanding of the relationship which is developing, and to use this understanding in order to approach some deep problems from new perspectives.