Algebra, Number Theory and Algebraic Geometry

Harvard University, Cambridge MA

Investigators

Abstract

The investigator (Gross) and his colleagues (Kazhdan and Sommers) are working on an assortment of problems in number theory, representation theory, and algebraic geometry. Gross will study Fourier coefficients for modular forms on quaternionic real groups (with N. Wallach), the exceptional theta correspondences in the style of Siegel (with W.T. Gan), and various theories of modular forms modulo a prime. He will also investigate subvarieties of Shimura varieties in the middle dimension and motives with a fixed Galois group (with G. Savin). Kazhdan will work on Langlands' lifting and the theory of unipotent crystals, perverse sheaves on loop spaces, and the theory of algebraic integration. Sommers will study representations arising from covers of nilpotent orbits and attempt to resolve the normality question for the closure of nilpotent orbits in a complex Lie algebra. He will also study connections with unitary representations of complex Lie groups. The proposal deals with several questions in the subfields of mathematics known as number theory, representation theory, and algebraic geometry. Many of these questions are motivated by the philosophy that algebraic information can be obtained by geometric methods. At the center of the work is the use of a symmetry group, or algebraic group, which is an object that is both algebraic and geometric in nature. These symmetry groups arise naturally in physics and chemistry. It is not too ambitious to say that the solution to the problems in this proposal will one day affect research in cryptography, theoretical physics, and quantum computing.