Algebra, Number Theory and Algebraic Geometry

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The investigator (Gross) and his colleagues (Kazhdan and Sommers) areworking on an assortment of problems in number theory, representationtheory, and algebraic geometry. Gross will study Fourier coefficients formodular forms on quaternionic real groups (with N. Wallach), theexceptional theta correspondences in the style of Siegel (with W.T. Gan),and various theories of modular forms modulo a prime. He will alsoinvestigate subvarieties of Shimura varieties in the middle dimension andmotives with a fixed Galois group (with G. Savin). Kazhdan will work onLanglands' lifting and the theory of unipotent crystals, perverse sheaveson loop spaces, and the theory of algebraic integration. Sommers willstudy representations arising from covers of nilpotent orbits and attemptto resolve the normality question for the closure of nilpotent orbits in acomplex Lie algebra. He will also study connections with unitaryrepresentations of complex Lie groups. The proposal deals with several questions in the subfields ofmathematics known as number theory, representation theory, and algebraicgeometry. Many of these questions are motivated by the philosophy thatalgebraic information can be obtained by geometric methods. At the centerof the work is the use of a symmetry group, or algebraic group, which isan object that is both algebraic and geometric in nature. These symmetrygroups arise naturally in physics and chemistry. It is not too ambitiousto say that the solution to the problems in this proposal will one dayaffect research in cryptography, theoretical physics, and quantumcomputing.