Mathematical Sciences: Partial Differential Equations and Several Complex Variables

Shaw Shaw将继续研究由几个复杂变量的问题引起的pde。特别是，她将研究复流形和CR流形上的柯西-黎曼方程和切向柯西-黎曼方程。她将研究的核心问题之一是理解切向柯西-黎曼方程的局部和全局可解性和规律性。她还将研究非光滑伪凸域上Bergman投影的正则性和d杆的边界正则性。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-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.