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可应用于复杂分析的各种问题。本工作的最终目的是建立多次调和函数和全纯函数的不变边界理论。特别是，主要研究者将研究多重次调和函数在Green-Martin拓扑下的边界性质，多重次调和函数的位势和极大函数的分解，多项式壳的解析结构和Wermer紧性。次调和函数理论在复变函数论中占有特殊的地位，因为它具有内在的美，而且对于偏微分方程组等其他领域也很重要。在这个项目中，首席研究人员将继续研究一种类似的理论，即所谓的“多次亚谐”函数。多重亚调和函数产生于许多不同的背景下，了解它们的性质将有助于我们理解微分几何和复变函数论中的许多问题。

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.