Mathematical Sciences: Polynomially Convex Hulls and Evolution of Pseudoconvex Sets by Levi Curvature

摘要 斯洛德科夫斯基 Slodkowski将继续研究多项式外壳， 理解在低正则性假设下的Jordan弧族的壳的结构。建议的第一部分 集中于一系列的问题所产生的关键问题 当这样的船体是拓扑超曲面时。斯洛德科夫斯基计划 在规律性假设下研究它们， 拟共形几何这个程序的实现将产生对全纯运动的应用。 第二部分 的建议涉及子集的演变， 高维空间，定义为弱解， 某些退化的非线性偏微分方程的Levi曲率。 重点分析了伪凸函数的演化 超曲面和伪凹集，以了解 弱伪凸域和多项式壳的结构。 成功完成这项建议将解决， 其他的，关于全纯运动的重要问题， 对动力系统理论有重要的应用， Teichmuller理论。动力系统理论， 目前发展非常迅速， 许多应用问题。Teichmuller理论不仅 几何学的基本含义，但它也是一个重要的 弦理论的工具，最新的基本理论之一， 粒子物理学 兹比格涅夫·斯洛德科夫斯基

ABSTRACT Slodkowski Slodkowski will continue working on polynomial hulls improving on understanding the structure of hulls of families of Jordan arcs under low regularity assumptions. The first part of the proposal concentrates on a series of problems generated by the crucial question when is such a hull a topological hypersurface. Slodkowski plans to investigate them under regularity assumptions formulated in terms of quasiconformal geometry. Realization of this program will yield applications to holomorphic motions, among others. The second part of the proposal concerns the evolution of subsets in higher-dimensional space, defined in terms of weak solutions of certain degenerate nonlinear PDE's involving the Levi curvature. The main focus is on analyzing the evolution of pseudoconvex hypersufaces and of pseudoconcave sets, in order to understand the structure of weakly pseudoconvex domains and of polynomial hulls. A successful completion of this proposal would solve, among other things, important problems on holomorphic motions, which would have significant applications to the theory of dynamical systems and to Teichmuller theory. The theory of the dynamical systems, which is developing very rapidly at present, has important applications to to many applied problems. The Teichmuller theory has not only fundamental implications for geometry, but it is also an important tool for string theory, one of the newest theories of the elementary particles' physics. Zbigniew Slodkowski