Mathematical Sciences: Geometric Function Theory in Several Complex Varaiables

这个项目继续在几个复变量空间中的函数和域的研究问题上的数学工作。第一个涉及到常微分方程和几个复变量的组合，其中人们对全纯向量场感兴趣，实现为微分方程系统。解向量提供了全n维空间的自同构的单参数群的例子。当向量场完备时，自同构可以无限延续。这个项目的一个目标是确定在多大程度上每个全纯向量场可以被一个完全的向量场近似。第二方面的研究集中在全纯嵌入下二维空间的图像。更准确地说，工作将在确定代数变化的类型，可以通过这种嵌入省略。已知一条复行可以省略，三条复行不能省略。为了理解杜瓦尔的例子是如何允许拉格朗日圆盘嵌入到非多项式凸的二维空间中，我们将做额外的工作。然而，在适当的意义上，所有多项式壳总是可以用全纯盘来解释。这个差别一定存在于全纯圆盘的概念和h∞圆盘的旧概念之间。这类问题已经研究了几十年，所以预计进展会很慢。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 continues mathematical work on problems in the study of functions and domains in the space of several complex variables. The first involves a combination of ordinary differential equations and several complex variables where one is interested in holomorphic vector fields, realized as systems of differential equations. The solution vectors provide examples of one parameter groups of automorphisms of the full n-dimensional space. When the vector field is complete, the automorphisms can be continued indefinitely. One goal of this project is to determine the extent to which every holomorphic vector field can be approximated by a complete one. A second line of investigation focuses on the image of two-dimensional space under a holomorphic embedding. More precisely, work will be done in determining the type of algebraic variety which can be omitted by such an embedding. It is known that one complex line may be omitted and three complex lines cannot. Additional work will be done in an effort to understand how the example of Duval allows for a Lagrangian disk to be embedded in two-dimensional space which is not polynomially convex. Yet, in the appropriate sense, allpolynomial hulls can always be explained in terms of holomorphic disks. The difference must lie somewhere between the idea of a holomorphic disk and the older idea of an H-infinity disk. Questions of this type have been studied for decades, so progress is expected to be slow. 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 its 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.