Mathematical Sciences: Real and Complex Analysis on Domains

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In this project the principal investigator will continue working on three problems involving real and complex analysis on domains. The first problem concerns an examination of function spaces designed for boundary value problems for partial differential operators defined on domains in real Euclidean space. The second problem concerns the study of automorphism groups of domains in complex n-space and how their group- theoretic properties are determined by the geometry of the boundary of the domain. The third problem concerns various subproblems in the function theory of several complex variables; to wit, boundary uniqueness theorems for holomorphic mappings, biholomorphic exhaustions of domains by other domains, and the boundary behavior of holomorphic functions. Complex analysis is one of the oldest branches of modern mathematics. The principal objects of study in complex analysis are functions with special properties called holomorphic or analytic functions. Holomorphic functions are capable of expressing relationships and equivalences between complex quantities in two or more space dimensions. In this project the principal investigator will study the behavior of holomorphic functions at the boundaries of their domains of definition. The boundary behavior of holomorphic functions is very often the most interesting (and difficult|) aspect of the theory.