Matrix Approximations, Stability of Groups and Cohomology Invariants

The field of operator algebras emerged from the matrix mechanics formulation of quantum mechanics created by Heisenberg and developed subsequently by von Neumann. Physical properties of particles are interpreted as infinite matrices which evolve in time and can be organized as algebraic structures of linear operators acting on Hilbert spaces. Just like geometric spaces, operator algebras may feature important symmetries corresponding to intrinsic properties that are preserved under groups of transformations. The principal investigator will study discrete finite-dimensional approximations that capture topological properties of these infinite-dimensional structures and their stability properties. In a different direction the principal investigator will study topological invariants arising in the bundle theory of operator algebras. An educational component of the project is devoted to the training of students in an area of operator algebras that has direct connections to group stability and testability problems in computer science and the theory of topological insulators from solid state physics. Three projects concerned with analytical and topological aspects of operator algebras will be investigated. The purpose of the first project is to study the stability of discrete groups with respect to the operator norm and topological obstructions to group stability in various contexts. The second project is devoted to finite-dimensional approximation properties of non-amenable discrete groups, with a focus on quasidiagonality as a tool in the construction of almost flat vector bundles and group quasi-representations that carry topological information. The third project is concerned with invariants of continuous fields of C*-algebras and their applications to C*-dynamical systems and higher twisted K-theory. The aim is to obtain a complete calculation of the cohomology groups that classify the continuous fields of stable strongly self-absorbing C*-algebras as part of the generalized Dixmier-Douady theory that the principal investigator has developed with Pennig. 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.