Geometric Methods in Representation Theory

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This is a project in Lie theory and geometry. Named after the mathematician Sophus Lie, Lie theory may be described as the study of symmetries of spaces of solutions of systems of differential equations. Especially important are systems of equations that model classical and quantum mechanics. Geometric properties of solution spaces encode properties of the physical systems being modeled by the differential equations. This project will develop the connections between the geometric properties and symmetries of these solution spaces, using mainly algebraic techniques. The research will focus on four specific projects. The first project will find a twisted analog of conformal blocks and the Verlinde formula, the second project is to determine an irredundant set of Inequalities for the saturated tensor cone of a Kac-Moody Lie algebra, the third project will generalize the classical Jacobson-Morozov Theorem to symmetrizable Kac-Moody Lie Algebras, and the fourth project is to develop a general topological analog of Fulton's Conjecture for group restrictions. 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.