The Pluri-Potential Theory and Its Applications

建议编号：DMS-0200743PI：Evgeny Poletsky吸引本项目主要通过多势理论对几何复分析进行进一步的研究。将在以下主题进行研究：包含所有必要背景的“一般”势理论的存在性；与多次调和函数的经典问题有关的无限维空间的复几何；通过这类集合上的多势理论的紧集的复几何；区域和多极格林函数的几何(经典的模拟是Martin-Caratheodory紧致)和广义Bernstein-Walsh和马尔可夫多项式的几何。还将考虑其他几何问题。在本项目中，将对几何复数分析进行研究。几何作为数学的一部分，旨在描述不同对象之间的定性联系。例如，平行线不相交，三角形中的高度在同一点相交。当欧几里得对象：点、线和面被更复杂的结构：函数、面和集取代时，研究就会有更微妙的味道。这是因为提供对象之间联系的机制不透明。在我们的提议中，我们将以类似于电荷能级的势的形式来寻找这样一种机制。

Proposal Number: DMS-0200743PI: Evgeny PoletskyABSTRACTThis project proposes the further study of the geometric complex analysis, mostly, through pluri-potential theory.Research will be conducted in the following topics:the existence of a "general" potential theory containingall the necessary background; the complex geometry of infinite-dimensional spaces related to classical problemsfor plurisubharmonic functions; the complex geometry ofcompact sets via the pluri-potential theory on suchsets; the geometry of domains and multipole Green functions (the classical analog is the Martin-Caratheodorycompactification) and Bernstein-Walsh and Markov inequalities for generalized polynomials. Other geometricquestions will also be considered.In this project research will be conducted on geometriccomplex analysis. Geometry, as a part of mathematics,aims to describe qualitative links between different objects. For example, parallel lines do not meet and theheights in a triangle meet at the same point. When Euclidean objects: points, lines and planes are replacedby more complicated structures: functions, surfaces andsets, the research is of a more delicate flavor.It happens because the mechanism providing links betweenobjects is not transparent. In our proposal we will lookfor such a mechanism in the form of potentials similar tothe energy levels of electrical charges.