Mathematical Sciences: Polynomially Convex Hulls and their Applications

9412392斯洛德科夫斯基本方案中的问题涉及多项式凸壳及其应用。提出者打算追求他对单位圆上纤维状多项式壳的长期兴趣。他将特别专注于分析具有非连通纤维的多项式外壳的结构，并找到它们的新应用。他还计划研究二维或更多维Levi方程的弱解及其在描述n维复欧氏空间的子流形的多项式壳和全纯包络方面的应用。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。当方程被定义在多个复变量的区域上时，函数和解的近似因区域的几何形状而变得复杂。这个项目研究复杂几何和近似之间的相互作用。特别是，该项目将应用于复方程解的极值定位。***

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. ***