Mathematical Sciences: Complex Analysis and Geometry in Multidimensional Domains

DMS-9500916 Boas/Straube 主要研究人员将研究在伪凸域的边界上的无穷型点的几何集合与d-杆-诺依曼问题的正则性之间的相互作用。 他们还将调查几何条件相当于紧的d-酒吧诺依曼算子。 将使用的技术结合联合收割机的想法，从功能理论，几何，偏微分方程和功能分析。 多复变出现在世纪初作为一个自然的产物研究的职能，一个复杂的变量。 很明显，这个理论与它的前身有很大的不同。 基本的几何是更难以掌握和函数理论有更多的亲和力与偏微分算子的一阶。 因此，它成长为一个混合学科结合了微分几何和微分方程的深刻特点。 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.