Mathematical Sciences: Research in Harmonic Analysis

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Hongming will continue his study of noncommutative harmonic analysis. This will include an investigation of generalized hypergeometric functions on matrix spaces, domains of positivity, and Hermitian symmetric spaces, concentrating on developing the operator-valued theory, extending the context to include Siegel domain of type II, deriving Mellin-Barnes representations, and exploring applications to infinite-dimensional representation theory. The mathematical bedrock of Hongming's research is the theory of Lie groups. Named in honor of the Norwegian mathematician Sophus Lie, these groups have been one 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.DMS-9200774 Hongming. Hongming 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 Hongming'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.