Mathematical Sciences: Analysis Over Loop Spaces

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Gross will investigate the natural Dirichlet form operator over loop spaces associated to Brownian bridge measure. The particular interest is in proving existence and uniqueness of ground states of the associated Schroedinger operators over homotopy classes. This is a continuation of Gross's recent work on logarithmic Sobolev inequalities on loop groups which has many implications for the representation theory of Lie 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 and theoretical physics, especially quantum mechanics and elementary particle physics.