Hodge Theory and Representation Theory

Texas A&M University, College Station TX

Investigators

Abstract

Abstract Award: DMS 1309238, Principal Investigator: Colleen Robles Hodge theory provides some of the basic invariants of a complex algebraic variety, and has yielded some of the deepest results in algebraic geometry. The study of Hodge structures and their symmetry groups, MumfordñTate groups, lies at the intersection of complex geometry, representation theory and arithmetic. The proposed work addresses the relationship between Hodge theory and representation theory, with a focus on those aspects that may be described using complex geometry. (The Hodge theory of algebraic varieties will not be addressed: the emphasis is on Hodge structures as objects of independent interest.) In the classical case that the Hodge domain D is a Hermitian symmetric space that may be equivariantly embedded in Siegel's upper-half space, the relation between Hodge theory and the geometric and arithmetic properties of a variety is a deep and extensively researched subject. The area is considerably less developed in the non-classical case, and it is generally felt that the principle obstacle to generalizing the theory (arithmetic properties, automorphic forms, theory of Shimura varieties, et cetera) is our limited understanding of the system of differential equations governing variations of Hodge structure. The principle motivation and objective of this project is to better understand this system, the complex geometry of the Hodge domain D, and the associated representation theory in the non-classical setting. Hodge theory lies at the crossroads of several mathematical subjects, including Algebraic Geometry, Complex Geometry, Number Theory and Representation Theory. This rich confluence makes the subject a fertile and influential area of research. The underlying motivation to study variations of Hodge structure (a dominant theme in this proposal) is to understand moduli of algebraic varieties. Algebraic varieties arise as the solution spaces to polynomial equations. The ability to understand and manipulate solutions of systems of polynomial equations is essential in many areas of engineering, science and mathematics.