Function Theory on Symmetric Spaces

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AbstractKoranyiThe proposal is concerned with several lines of investigation, The first one of these is the mapping problem of domains in several complex variables with the aid of symplectic quasiconformal maps (with respect to the Bergman metric). It has already been proved that bounded simply connected smooth strongly pseudoconvex domains can always be mapped ontoeach other by such maps. The present project is concerned with developing geometric methods for computing or estimating the minimal qusiconformal distortion. The second line of investigation has to do with Poisson transforms of sections of vector bundles. Such Poisson transforms havebeen used to extend infinitesimally quasiconformal maps from the complex unit ball to the interior. They are now to be further investigated for their injectivity properties and for pairs of differential operators intertwined by them. Here the machinery of semisimple Lie groups will be used to its full capacity. A further subject to be studied is generalized conformal maps of generalized flag manifolds; it is expected that Liouville's classic theorem will extend to this situation. (This question is related to the preceding one through the notion of parabolic geometry.) In addition, three further, more or less related problems will be investigated; these concern analysis on two-step nilpotent groups, harmonic functions on spaces with negative curvature, and an applicationof the theory of reflection groups to statistics.The main goal of the proposal is to discover new mathematical facts in the field of analysis of functions. The theory of functions of one complex variable has been the most central field of mathematics in the last 150 years. It has innumerable applications to physics and othersciences. The theory of functions of several complex variables has had important successes but has not up to now matched in perfection and applicability the one-variable theory. The first goal of the proposal is to try to improve on this situation by studying the so-called mapping properties of functions of several complex variables. It will also be investigated to what extent the fundamental notions and facts of this theory can be extended to more general situations. The project also includes the investigation of some further related problems, one of these concerns an application to statistics. A secondary goal of the proposal is to understand more clearly the connections between certain already known facts in the main subject. This should make them easier to learn; in this way a contribution to mathematical education is expected to be made.