Mathematical Sciences: Polynomially Convex Hulls and their Applications

本文所讨论的问题是多项式凸包及其应用。提议者打算追求他对单位圆上纤维化的多项式船体的长期兴趣。他将特别专注于分析非连接纤维的多项式壳的结构，并寻找它们的新应用。他还计划研究二维或多维李维方程的弱解及其在n维复欧几里德空间子流形全纯多项式壳和包络的描述中的应用。数学分析的作用是提供有关方程解的定性、定量和几何信息。当方程在多个复杂变量的定义域上定义时，由于定义域的几何性质，函数和解的逼近变得复杂。本项目研究复几何与近似之间的相互作用。特别是，该项目将应用于复杂方程解的极值位置。＊＊＊

9412392 Slodkowski The problems in this proposal are concerned with polynomial convex hulls and their applications. The proposer intends to pursue his long term interest in polynomial hulls fibered over the unit circle. he will concentrate especially on analyzing the structure of polynomial hulls with non-connected fibers and finding their new applications. He also plans to study weak solutions of Levi equations in two or more dimensions and their applications to description of polynomial hulls and envelopes of holomorphy of submanifolds of n-dimensional complex Euclidean space. The role of mathematical analysis is to provide qualitative, quantitative and geometric information concerning the solutions of equations. When the equations are defined over domains in several complex variables the approximation of functions and solutions is complicated by the geometry of the domain. This project studies this interplay between the complex geometry and approximation. In particular, the project will have application to the location of extreme values of solutions to complex equations. ***