DMS-9501238 PI: Gross Gross will continue his study of the natural Dirichlet form operator over loop spaces associated to Brownian bridge measure. The techniques used involve harmonic analysis of the square integrable functions on a Lie group in terms of a completion of its universal enteloping algebra. He expects to extend this harmonic analysis to noncompact groups and symmetric spaces. This is part of a long range goal of proving a Hodge-deRham theorem for loop spaces. 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.
