Theta Functions, Intersection Theory and Representation Theory

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This project will focus on the interconnected areas of non-abelian theta functions, intersection theory and representation theory. The view point is that the theory of theta functions and invariant theory (with relations to the classical Schubert calculus) constitute a common generalization of the classical theory of theta functions. The spaces of theta functions correspond to physical states in the WZW model of conformal field theory, and this correspondence links the subject to modern mathematical physics.The Hitchin connection on the spaces of non-abelian theta functions lies at the heart of the physics view point on theta functions and several outstanding problems concerning its properties remain open (unitarity, whether it is ``motivic''). The aim of the project is to study the Hitchin connection, and to have techniques from conformal field theory bear upon open problems in invariant theory such as the saturation conjecture for the even orthogonal groups. A study of the Hitchin connection will be carried out with potential applications to strange duality questions (that is, relations between non-abelian theta functions for different groups) in their most general formulation. It is hoped that a study of the strange dualities will lead to insights on the Hitchin connection.Algebraic geometry studies the solutions to systems of polynomial equations. Representation theory (the theory of symmetry), combinatorics, and mathematical physics have fundamental links with algebraic geometry. Theta functions, originating in algebraic geometry, are some of the most important functions in mathematics. These functions appear frequently in mathematical physics. The project aims to deepen the links between theta functions and representation theory. It is hoped that the project will lead to better algorithms in various combinatorial questions. Possible applications outside of mathematics are via contributions to the theory of differential equations (the Riemann-Hilbert problem), to theoretical computer science and physics (string theory and conformal field theory).