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

Slodkowski将继续研究多项式壳，以提高对低正则性假设下约旦弧族壳结构的理解。该提案的第一部分集中在由一个关键问题产生的一系列问题，即这种船体何时是拓扑超表面。Slodkowski计划在拟共形几何的正则性假设下研究它们。实现这个程序将产生应用到全纯运动，除其他外。该建议的第二部分涉及高维空间中子集的演化，定义为涉及Levi曲率的某些退化非线性PDE的弱解。本文主要分析伪凸超曲面和伪凹集的演化，以了解弱伪凸域和多项式壳的结构。这一建议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Teichmuller理论有重要的应用。动力系统理论是目前发展非常迅速的一门学科，它对解决许多实际问题具有重要的应用价值。Teichmuller理论不仅对几何具有基本意义，而且也是弦理论的重要工具，弦理论是基本粒子物理学的最新理论之一。兹比格涅夫•Slodkowski

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