Discrete Groups in Algebraic Geometry

Harvard University, Cambridge MA

Investigators

Abstract

The investigator will study applications of group theory to algebraic geometry. More specifically, he will study those moduli spaces of algebraic-geometric objects that can be understood in terms of discrete groups acting on Hermitian symmetric spaces. His tools will include the arithmetic properties of the groups and the construction of automorphic forms on the symmetric spaces. Group theory is the study of symmetry, and examples of discrete groups are the symmetries of a chemical or mineral crystal structure. Moduli spaces arise when one considers some geometric objects (say, surfaces in space), and considers two of them to be the same if they differ only in some minor way, such as a change in point of view. It is surprising and deep that discrete groups can help one understand moduli spaces, but many beautiful connections have been discovered by many different people. The investigator will look for more of these connections and try to better understand the ones already known.