Mathematical Sciences: Geometric Function Theory in Several Complex Varaiables

这个项目继续研究多个复变量空间中的函数和域问题的数学工作。第一个涉及常微分方程和几个复变量的组合，其中一个人对实现为微分方程组的全纯向量场感兴趣。解向量提供了全n维空间的自同构单参数组的例子。当向量场是完备的时，自同构可以无限继续。这个项目的一个目标是确定每个全纯向量场可以被一个完整的向量场近似的程度。第二种研究集中在全纯嵌入下的二维空间的像。更准确地说，将在确定可以通过这种嵌入而省略的代数族的类型方面进行工作。众所周知，可以省略一条复数行，而不能省略三条复数行。为了理解Duval的例子如何允许拉格朗日圆盘嵌入到非多项式凸的二维空间中，还将进行额外的工作。然而，在适当的意义上，所有多项式壳总是可以用全纯圆盘来解释的。不同之处一定在全纯圆盘和更老的H无穷大圆盘的概念之间。这种类型的问题已经研究了几十年，因此进展预计会很缓慢。本世纪初出现了几个复变量，这是一个复变量函数研究的自然结果。很早就很明显，这一理论与其前身有很大不同。基本的几何学要难得多，而函数论与一阶偏微算符的亲和力要强得多。因此，它成长为一门混合学科，结合了微分几何和微分方程式的深刻特征。许多基本结构都是在过去30年里定义的。目前的研究仍然集中在理解这些基本的数学形式上。

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.