Mathematical Sciences: Real and Complex Analysis on Domains

在这个项目中，首席研究员将继续研究三个问题，涉及域的真实和复杂分析。第一个问题是研究为在真实欧几里德空间上定义的偏微分算子的边值问题设计的函数空间。第二个问题是研究复n空间中域的自同构群及其群的群论性质如何由域边界的几何形状决定。第三个问题涉及复变函数理论中的各种子问题；研究了全纯映射的边界唯一性定理，全纯域对其他域的生物全纯耗尽，以及全纯函数的边界行为。复分析是现代数学最古老的分支之一。复分析的主要研究对象是具有特殊性质的函数，称为全纯函数或解析函数。全纯函数能够表示两个或多个空间维度上复数量之间的关系和等价。在这个项目中，首席研究员将研究全纯函数在其定义域边界处的行为。全纯函数的边界行为通常是理论中最有趣（也是最难）的方面。

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.