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

9322326 Stout该奖项支持在若干复变量全纯函数的边界行为理论和相关问题方面的数学研究。研究的一个主要方向是关于全纯函数的边值的可移奇点，即一个函数何时可以作为全纯函数扩展到一个点和一个邻域？这些点被称为可移动奇点。一个相关的研究方向是全纯函数的可移动集的研究。在这些集合中，全纯函数的自然边值与连续函数的自然边值一致。工作也将在不一定以光滑流形为界的领域进行，这是一个由几个复杂变量组成的领域，迄今为止还没有得到太多关注。最后，将继续研究柯西-黎曼函数在其基础域的一维切片方面的工作。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。＊＊＊

9322326 Stout This award supports mathematical research on aspects of the theory of boundary behavior of holomorphic functions of several complex variables and related questions. One major direction of the work concerns removable singularities for the boundary values of holomorphic functions, that is, when can a function be extended to a point and a neighborhood as an holomorphic function? These points are called removable singularities. A related direction of the work focuses on the study of removable sets for holomorphic functions. These are sets where the natural boundary values of an holomorphic function agree with those of a continuous function. Work will also be done on domains which are not necessarily bounded by smooth manifolds, an area of several complex variables which has not received much attention to date. Finally, work will continue on the study of Cauchy-Riemann functions in terms of one-dimensional slices of their underlying domains. 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. ***