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

数学科学：多项式凸壳和列维曲率的伪凸集演化

• 批准号: 9706970
• 负责人: Zbigniew Slodkowski
• 金额: $ 7.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706970&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Polynomially; Convex; Hulls

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

摘要 Slodkowski Slodkowski 将继续研究多项式外壳，以提高对低正则性假设下约旦弧族外壳结构的理解。该提案的第一部分集中于由关键问题所产生的一系列问题，这样的船体何时是拓扑超曲面。斯洛德科夫斯基计划在用拟共形几何表述的正则性假设下研究它们。该程序的实现将产生全纯运动等方面的应用。 该提案的第二部分涉及高维空间中子集的演化，根据涉及列维曲率的某些简并非线性偏微分方程的弱解来定义。 主要重点是分析赝凸超曲面和赝凹集的演化，以了解弱赝凸域和多项式壳的结构。 该提案的成功完成将解决全纯运动的重要问题，这将对动力系统理论和泰希米勒理论具有重要的应用。目前发展非常迅速的动力系统理论对于许多应用问题有着重要的应用。泰希米勒理论不仅对几何学具有根本意义，而且还是弦理论（基本粒子物理学的最新理论之一）的重要工具。 兹比格涅夫·斯洛德科夫斯基