Operator Algebra, Dynamics and Geometry

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The principal investigator will investigate problems in dynamics and geometry in the framework of operator algebras. Sample topics include: 1. Study the relation between entropy of principal algebraic actions of countable groups and Fuglede-Kadison determinant; 2. Develop an operator algebraic approach to entropy of actions of sofic groups; 3. Investigate combinatorial independence in measurable and topological dynamics; 4. Develop a convex analysis approach to the study of noncommutative Choquet boundary; 5. Study quantum isometry of compact quantum metric spaces. In order to get a new mathematical tool to study quantum mechanics, von Neumann introduced operator algebras. The theory of operator algebras has grown into a huge exciting area of modern mathematics, and is related to many other areas of physics and mathematics. Ergodic theory and dynamics arose from studying the long-term behavior of complicated processes. This project will deepen and broaden connections between geometry, dynamics, and operator algebras, and has application in physics. The proposed study of entropy of algebraic actions and actions of sofic groups will enhance our understanding of complicated symmetries. The study of the new interrelation between combinatorics and dynamics has application to the local theory of Banach spaces. The development of noncommutative Choquet boundary has application in operator theory. The study of metric geometry will provide a concrete mathematical foundation for certain statements in the theoretical high-energy physics literature.