Mathematical Sciences: Function Theory on Symmetric Spaces

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DMS-9500794 PI: Koranyi Koranyi will continue his investigation of quasiconformal maps in the context of Cauchy-Riemann manifolds. Part of the project is the study of Poisson transforms on vector bundles and their interaction with homogeneous differential operators. A second direction will be the study of boundary behavior for harmonic functions. A third direction will be the use of the harmonic analysis of the symmetric space of positive definite matrices to obtain methods of best estimation of a Gaussain distribution in the presence of some prior information. The analysis involved in this research rests on 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.