Mathematical Sciences: Domains of Holomorphy and Their Automorphisms

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This project concerns mathematical research on problems arising in the area of several complex variables. It involves the function-theoretic and geometric properties of bounded pseudoconvex domains in n-dimensional complex space and their automorphism groups. The first goal of the research is to identify domains with noncompact automorphism groups by the local geometry of their boundaries. Particular interest is placed on boundaries which are not entirely smooth. The second component of the work is the extension of studies on the asymptotic behavior of the intrinsic invariant metrics defined on the bounded pseudoconvex domains. The research relates the boundary behavior of the automorphisms and the compactness of the automorphism groups. Finally, work will be done identifying the bounded symmetric domains by their boundary geometry among domains with noncompact automorphism groups. The projects are connected by the theme that the size of the automorphism groups and the boundary geometry of bounded domains are tied together. The study of several complex variables arose at the beginning of the century initially to examine those properties of classical function theory which generalize to several variables. It turned out that most properties do not generalize and research turned instead toward some of the more intriguing geometric and analytic questions which arise only when the domains are of dimension two or greater. This project continues that tradition.