Mathematical Sciences: Analytic and Geometric Problems in Several Complex Variables

本项目将集中于几个复杂变量领域的数学研究。将继续研究三维cr流形的可嵌入性问题，探索该问题与Kuranishi构造的曲面奇点通用变形之间的联系。这项研究的一个组成部分是计算机软件的发展，它有助于这些空间的计算。最终目的是在三维流形上构造cr -结构的标准形式和可嵌入结构的特性。此外，还将应用伪微分和爆破方法研究奇异度量域上的d-bar算子。其应用包括Bergman型度量的指标定理和龙格定理的无限小版本。本世纪初出现了几个复变量，作为一个复变量函数研究的自然产物。很明显，这个理论与它的前身有很大的不同。底层的几何结构更难掌握，而函数理论与一阶偏微分算子的关系要密切得多。因此，它成长为一门混合学科，结合了微分几何和微分方程的深层特征。许多基本结构是在过去三十年中确定的。目前的研究仍然集中在理解这些基本的数学形式。

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.