Professor Barbasch will investigate properties of unipotent representations of reductive groups such as their unitarity and their relation to automorphic forms and the orbit method. He will continue his joint work with A. Moy and S. Evens on related topics. Professor Speh will continue her joint work with J. Rohlfs on Eisenstein cohomology aiming to compute Tamagawa numbers and obtain rationality results for periods integrals. Jointly with Dan Barbasch she will continue work on lifting automorphic forms from twisted endoscopic groups. Birgit Speh will also continue her work on Seiberg-Witten equations and twisted torsion. This proposal falls in the general area of representation theory of Lie groups and its applications to automorphic forms. Lie groups (named in honor of the Norwegian mathematician Sophus Lie) and their representation theory, have 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 in analysis and theoretical physics, especially quantum mechanics and elementary particle physics. The theory of automorphic forms in its modern form was initiated by Gelfand, Piatetski-Shapiro, Langlands and Borel and relies heavily on the representation theory of Lie groups. It has had a significant impact on problems arising in classical number theory and is expected to continue to do so. These theories are very abstract, but have surprising important applications in areas like theoretical computer science and coding theory.
