Mathematical Sciences: Complex Analysis and Geometry in Multidimensional Domains

DMS-9500916 Boas/Straube首席研究员将研究伪凸域边界上无穷类型点集的几何与d-bar-Neumann问题的正则性之间的相互作用。他们还将研究与d-bar-Neumann算子紧性等价的几何条件。这些技巧结合了函数论、几何学、偏微分方程式和泛函分析的思想。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。

DMS-9500916 Boas/Straube The principal investigators will study the interplay between the geometry of the set of infinite type points in the boundary of a pseudoconvex domain and the regularity of the d-bar-Neumann problem. They will also investigate geometric conditions equivalent to compactness of the d-bar-Neumann operator. The techniques to be used combine ideas from function theory, geometry, partial differential equations, and functional analysis. 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.