Mathematical Sciences: Boundary Behavior of Holomorphic Functions of Several Complex Variables

在这个项目中进行的数学研究遵循两个相关的主题，都是在几个复杂变量函数的背景下进行的。第一个是关于可移动奇点的问题。一旦从一个变量变为几个变量，奇点现象就会发生重大变化。全纯函数可以扩展为全纯函数的点或集称为可移动奇点。这项持续工作的目标之一是通过在奇点的邻域中定义的函数代数来表征可移动奇点。在已经算出的例子中，当奇异集相对于代数是凸的时候，它是可移动的。要讨论的一个相关问题是关于可移动域边界上光滑流形的表征。该项目的第二个重点是关于全纯函数的边界值。两种观点普遍存在。一个是确定边界上的光滑函数是否全纯地延伸到内部，另一个是研究内部上的全纯函数以某种合理的方式延伸到边界的程度问题。在前一种情况下，工作将集中在确定可以沿低维流形（例如复线）连续全纯的函数是否在大范围内是全纯的。边界考虑自然导致几何分析的应用，而可拓理论与求解偏微分方程组的理论密切相关。

Mathematical research undertaken in this project follows two related themes, both within the context of functions of several complex variables. The first is concerned with the question of removable singularities. There is a significant change in the phenomenon of singularities as soon as one goes from one to several variables. Points or sets where holomorphic functions may be extended as holomorphic functions are called removable singularities. One of the goals of this continuing work is to characterize removable singularities by means of the algebra of functions defined in neighborhoods of the singularities. In examples which have been worked out, when the singular set is convex with respect to the algebra, it is removable. A related question to be taken up concerns the characterization of smooth manifolds in the boundary of a domain which are removable. The second thrust of the project concerns the boundary values of holomorphic functions. Two points of view prevail. One is to determine whether a smooth function on the boundary continues holomorphically to the interior, the other is concerned with the question of the extent to which a holomorphic function on the interior extends to the boundary in some reasonable fashion. In the former case, work will concentrate on efforts to determine whether functions which can be continued holomorphically along lower dimensional manifolds, complex lines for example, are holomorphic in the large. Boundary considerations lead naturally to applications of geometric analysis, while the extension theory is closely tied with that of solving systems of partial differential equations.