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，既在几个机构的职能范围内， 复杂变量 第一个问题是关于 可移除奇点 有一个显着的变化， 奇点的现象，只要一个人从一到 几个变量。 全纯函数 可以扩展为全纯函数称为可去函数 奇点 这项持续工作的目标之一是 用代数的方法刻画可去奇点 定义在奇点邻域中的函数。 在 已经计算出的例子，当奇异集是 凸相对于代数，它是可移动的。 一个相关 要讨论的问题涉及光滑的特性， 在一个域的边界上的流形是可移动的。 该项目的第二个重点是边界问题 全纯函数的值 有两种观点占上风。 一个是确定边界上的光滑函数是否 继续全纯的内部，另一个是关注 全纯函数在多大程度上 在内部延伸到边界，在一些合理的 时尚. 在前一种情况下，工作将集中于努力， 确定是否可以继续执行的功能 全纯沿着低维流形，复线 例如，在大范围内是全纯的。 边界的考虑自然会导致 几何分析，而可拓学理论是紧密联系在一起的 与求解偏微分方程组的方法相同。

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.