Mathematical Sciences: Complex Analysis and Geometry in Multidimensional Domains

Boas/Straube主要研究伪凸域边界无限型点集合的几何形状与d-bar-Neumann问题的正则性之间的相互作用。他们还将研究等价于d-bar-Neumann算子紧致性的几何条件。所使用的技术结合了函数理论、几何、偏微分方程和泛函分析的思想。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。

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.