Mathematical Sciences: Partial Differential Equations and Several Complex Variables

DMS-9424122 Shaw 肖将继续她的工作偏微分方程的问题所产生的几个复杂的变量。 特别是，她将研究复杂和CR流形上的柯西-黎曼方程和切向柯西-黎曼方程。 她将工作的中心问题之一是了解当地和全球的可解性和规律性的切柯西-黎曼方程。 她还将研究伯格曼投影的正则性和非光滑伪凸域上d杆的边界正则性。 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-9424122 Shaw Shaw will continue her work on pde's arising from problems in several complex variables. In particular, she will study the Cauchy-Riemann equations and tangential Cauchy-Riemann equations on complex and CR manifolds. One of the central problems she will work on is to understand the local and global solvability and regularity of the tangential Cauchy-Riemann equations. She will also study the regularity of the Bergman projection and the boundary regularity of the d-bar on nonsmooth pseudoconvex 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.