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

9322326 Stout该奖项支持对多个复变量的全纯函数的边界行为理论和相关问题方面的数学研究。这项工作的一个主要方向是关于全纯函数边值的可去奇性，即函数什么时候可以作为全纯函数扩展到点和邻域？这些点称为可移去奇点。这项工作的一个相关方向是研究全纯函数的可去集。这些集合是全纯函数的自然边值与连续函数的自然边值一致的集合。也将在不一定以光滑流形为边界的区域上进行工作，光滑流形是一个由几个复变量组成的区域，到目前为止还没有得到太多的关注。最后，将继续研究柯西-黎曼函数在其基本区域的一维切片方面的工作。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。***

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. ***