Mathematical Sciences: Real and Complex Analysis on Domains

在这个项目中，首席研究员将继续 研究涉及真实的和复分析的三个问题， 域. 第一个问题涉及对功能的考察 偏微分方程边值问题空间 定义在真实的欧氏区域上的微分算子 空间 第二个问题是关于自同构的研究 复n空间中的域群，以及它们的群- 理论性质是由几何形状的 域的边界。 第三个问题涉及各种 多复变函数论中的子问题; 即全纯映射的边界唯一性定理， 域的双全纯穷竭，以及 全纯函数的边界行为 复分析是现代数学中最古老的分支之一， 数学 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 表示复合体之间的关系和等价关系 在两个或多个空间维度的数量。 本工程 首席研究员将研究全纯 在其定义域的边界上发挥作用。 的 全纯函数的边界行为往往是最 有趣（和困难）|）理论方面。

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.