Mathematical Sciences: Analytic and Geometric Problems in Several Complex Variables

该项目将集中在数学的几个领域 研究多个复变量。 工作将 继续讨论三维空间的可嵌入性问题 CR流形，探索这个问题和 曲面的非线性变形的Kuranishi构造 奇点这次调查的一个组成部分是 开发计算机软件，以促进 计算这些空间。 最终目标是 三阶CR-结构的正规形构造 维流形与可嵌入的特征 结构. 此外，还将开展工作， 拟微分和blow-up方法研究d-杆 域上的奇异度量算子 应用 包括Bergman型度量指标定理和无穷小指标定理 龙格定理的几个版本 几个复杂的变量出现在一开始， 世纪作为一个人的功能研究的自然产物 复变数 很明显，早期的理论 与其前身大不相同。 基本的几何形状是 更难掌握，函数理论也更难掌握， 一阶偏微分算子的仿射性 它 因此成为一门混合学科， 微分几何和微分方程。 许多 基本结构是在过去三年中定义的， 几十年目前的研究仍然集中在了解这些 基本数学形式。

This project will focus on several areas of mathematical research in the area of several complex variables. Work will continue on the problem of embeddability of three dimensional CR-manifolds, exploring connections between this problem and the Kuranishi construction of versal deformations for surface singularities. An integral part of this investigation is the development of computer software which facilitates the computation of these spaces. The ultimate goal is the construction of normal forms for CR-structures on three dimensional manifolds and the characteristics of embeddable structures. In addition, work will be done to apply pseudodifferential and blow-up methods to study the d-bar operator on domains with singular metrics. The applications include index theorems for Bergman type metrics and infinitesimal versions of the Runge theorem. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.