Wallach will investigate invariant differential operators acting on invariant distributions on a reductive Lie algebra, diagonal invariants and harmonics for Coxeter groups, Automorphic forms related to the error term in asymptotic behavior of lattice point problem and the Kloosterman-Selberg zeta function. Wallach will also work to develop parallel computer algorithms for the calculation of Groebner bases and a general software package to be used in the study of diagonal invariants and harmonics for Coxeter groups. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.
