Rigidity in von Neumann Algebras; Connections and Applications to Orbit Equivalence and Geometric Group Theory

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Within the general context of Popa's deformation/rigidity theory and building on principal investigator's prior research results, the goal of this project is to find new rigidity results for von Neumann algebras associated with group actions on probability spaces. The project revolves around the following main problems: (1) find new examples of actions of more "exotic" groups on probability spaces that can be completely reconstructed from their von Neumann algebras; (2) find additional examples of group actions that lead to von Neumann algebras with unique Cartan subalgebras; (3) obtain new applications to ergodic theory (particularly orbit equivalence and structural properties for equivalence relations) and probability theory (percolation on graphs); and (4) deepen the connections with geometric group theory and representation theory. The principal investigator plans to achieve these goals by further refining his previous techniques and continuing to expand the cohomological and geometric group theory perspective that he and his coauthors have brought to the study of rigidity in von Neumann algebras. He expects these techniques to reveal new aspects of the theory that will enable an even more fruitful interplay between these fields.The study of von Neumann algebras was initiated in the 1930s by Murray and von Neumann as a tool to study quantum mechanics, and it has progressively morphed into a stand-alone discipline. It also created a basis for the development of powerful mathematical theories that ultimately brought valuable insight to areas of physics (statistical mechanics), biology (DNA structure), and engineering (cell-phone design). This project, which continues the study of rigidity phenomena in von Neumann algebras, is expected to generate novel applications and to establish new bridges to other active research areas in mathematics (probability, ergodic theory, and geometric aspects in group theory). On a broader scale, the principal investigator anticipates that applications of rigidity phenomena outside the mathematical spectrum will continue to unveil themselves, particularly in engineering and computer science (e.g., error-correcting codes). The project will offer numerous development opportunities for graduate students or young researchers just entering this very active research field. While the topic of the project is focused heavily on advanced trends, it also contains substantial research directions and open problems that can be excellent subjects for Ph.D. theses. The principal investigator is currently teaching a two-semester introductory course on this subject with the purpose of attracting new young talent to the area. He will continue to disseminate his findings from the project through publications, lecture series, and seminar and colloquium talks, and to promote the connections with related areas of mathematics and beyond.