CAREER: Homotopical representation theory and TQFTs

This project will develop novel connections between modern mathematics and quantum field theory. Quantum field theories are an effective tool in modern physics. The project will develop the mathematics needed to rigorously describe and analyze such theories by leveraging recent innovations in abstract algebra and geometry to explicitly describe quantum field theories in mathematical terms. In addition to scientific advances, the project will both develop text-based resources for mathematicians that will advance researchers' abilities to access and deploy cutting edge methods in algebra and geometry and provide support to community-based education programs in the Los Angeles area. More precisely, the PI will use homotopical approaches in representation theory to provide rigorous mathematical realizations of topological quantum field theories whose associated moduli of vacua are non-compact. Topological field theories of this type arise naturally as twists of supersymmetric quantum field theories. In producing the aforementioned realizations, the PI will develop new relationships between representations of quantum groups and the geometry of Springer resolutions. 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.