Rigidity in von Neumann Algebras: Connections and Applications to Orbit Equivalence, Geometric Group Theory, and Continuous Model Theory

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The study of von Neumann algebras began in efforts to understand particle physics but subsequently became an independent discipline that has stimulated the development of powerful mathematical theories, bringing valuable insight to other areas of physics (statistical mechanics), biology (DNA structure), and engineering (cell-phone design). This project further advances research in this field and is expected to reveal new applications and establish new bridges to other areas in mathematics (ergodic theory, group theory, logic, and probability). The activities of this project provide ample dissertation subject matter for three graduate students who are involved in the research.This research project explores new boundaries and technical inputs in deformation/rigidity theory towards the classification of von Neumann algebras arising from groups and their actions on probability spaces. The project targets the following specific themes: find new natural examples of actions of groups on probability spaces that are remembered by their von Neumann algebras; construct the first examples of property (T) groups that can be reconstructed from their von Neumann algebras; investigate the structure of von Neumann algebras arising from mapping class groups and outer automorphisms of free groups; find new invariants to classify ultrapowers of factors and explore in depth their applications to continuous model theory. To achieve these goals the principal investigator will develop new methods to implement algebraic, cohomological, geometric, and dynamical information about a group and its actions into the analytic framework of von Neumann algebras. In addition, the project pursues new analytic tools towards the calculation of the symmetry groups of ultrapower factors. The results arising from this project are expected to reveal significant new applications and connections to representation theory, geometric group theory, orbit equivalence, and logic.