Mathematical Sciences: The Pluri-Potential Theory and Its Application

在这个项目中，首席研究员将研究多势理论中应用于复杂分析的各种问题。本文的最终目的是建立多元次调和函数和全纯函数的不变边界理论。特别是，主要研究者将研究关于Green-Martin拓扑的多次谐波函数的边界性质，多次谐波函数分解为势函数和极大函数，多项式壳的解析结构和Wermer紧凑。次调和函数理论由于其固有的美以及对偏微分方程理论等其他领域的重要意义，在复变函数理论中占有特殊的地位。在这个项目中，首席研究员将继续研究所谓的“多次谐波”函数的类似理论。多次谐波函数出现在许多不同的环境中，了解它们的性质将有助于我们理解微分几何和复函数理论中的许多问题。

In this project the principal investigator will study various problems in pluri-potential theory that have applications to complex analysis. The ultimate goal of this work is to construct an invariant boundary theory for plurisubharmonic functions and holomorphic functions. In particular, the principal investigator will study boundary properties of pluri- subharmonic functions with respect to the Green-Martin topology, the decomposition of plurisubharmonic functions into potentials and maximal functions, the analytic structure of polynomial hulls and Wermer compacta. The theory of subharmonic functions occupies a special place in complex function theory because of its inherent beauty and because it is important to various other areas such as the theory of partial differential equations. In this project the principal investigator will continue working on an analogous theory for what are called "plurisubharmonic" functions. Plurisubharmonic functions arise in many different contexts, and a knowledge of their properties will help us understand a number of problems in differential geometry and complex function theory.