Cyclic Cohomology of Toeplitz Operators and Proper Group Actions

This award supports a research project in noncommutative geometry and its applications. Noncommutative geometry is a mathematical field in which spaces are studied through an algebraic lens, in a way analogous to the relationship between a topological space and the algebra of continuous, complex-valued functions on that space. An important feature of the algebraic structures in noncommutative geometry is that – unlike multiplication of functions – their multiplication is not commutative, that is, the order in which elements are multiplied changes the outcome. This property makes for a mathematically rich generalization of classical geometry and enables connections between noncommutative geometry and many other fields, including representation theory, differential geometry, and algebraic topology, as well as the mathematical foundations of quantum mechanics. This project will explore the noncommutative geometry of two systems of non-commuting operators, one arising from the theory of Toeplitz operators, and the other from group symmetries. The PI’s primary aim is to introduce new numerical invariants to distinguish noncommutative spaces, and refine known invariants such as the Connes-Chern character. The project will also provide research opportunities for graduate students and postdoctoral scholars and support the PI’s outreach efforts to middle school and high school students in the community. Cyclic theory, the noncommutative analogue of de Rham theory in differential geometry, will be applied in this project to study an analytic version of the Milnor fibration, a novel concept arising in the investigation of the Arveson-Douglas conjecture. A new method for analyzing variations of traces and cyclic cocycles on the fibration will be developed, building an analytic approach to secondary invariants. Additionally, a geometric model for the cyclic cohomology of a Lie group will be constructed to study invariants for proper cocompact actions. This model will facilitate the exploration of new homotopy invariants associated with diffeomorphism groups. 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.