Moduli Spaces in Representation Theory and Symplectic Algebraic Geometry

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Mathematical studies of diverse problems, from the dynamics of particles, to spatial symmetries, to quantum physics, rely on the formulation of a particular type of geometry called symplectic geometry. The structures of symplectic geometry that arise in all these different problems possess certain common features that will be further isolated, explored, and catalogued in this project. The project will then develop general techniques to exploit those common features to calculate fundamental quantities in symplectic geometry. The resulting techniques and calculations will be applied to solve a number of outstanding problems in geometry, the mathematical study of symmetry, and mathematical physics. Symplectic algebraic varieties provide the natural geometric setting for basic constructions in geometric representation theory. Such varieties appear naturally in the study of supersymmetric quantum field theories, as moduli spaces of vacua in 3D N=4 theories. The two appearances are deeply inter-related, as quantum field theory provides insight to geometric representation theory, which representation theory repays by answering questions of importance in physics. The project locates diverse symplectic algebraic varieties in a common framework---as moduli spaces in noncommutative Poisson geometry---develops general tools for their exploration, and yields applications to representation theory, topology, geometry, and supersymmetric quantum field theories. In particular, the project will develop and apply new methods to find generators for topological and categorical invariants associated to symplectic algebraic varieties. 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.