Partial Differential Equations and Several Complex Variables (Mathematics)

在这项研究中，肖博士将继续她的线性偏微分方程的研究，这是由几个复杂变量的问题引起的。特别是，她将研究切向Cauchy-Riemann算子的可解性、正则性和可嵌入性。这些问题在复分析的函数理论中具有重要的基础意义。整体问题是将她关于b在弱伪凸边界上的L2存在性和估计的结果推广到Holder估计和Lp估计。局部问题是用核方法证明复向量场系统满足相容条件的局部解的存在性。抽象CR结构的可嵌入性也将被讨论。这个问题与b的局部可解性和b -同伦公式密切相关。本文还将研究b的范围和Szego投影的正则性等相关问题。这些问题的研究将受益于与威斯康星小组的互动，该小组也在积极研究相关问题。肖博士还将教授一个学期的研究生课程，内容是几个复杂变量。此外，她还将组织一系列由著名女数学家和顶尖研究人员参加的研讨会。该项目进一步推进了VPW计划的目标，即(1)为女性提供在工程和NSF支持的科学学科中发展职业的机会；(2)通过为在工业、政府和学术机构工作的女性科学家和工程师提供更大的知名度，鼓励女性在科学和工程领域追求职业。通过鼓励妇女参与科学，这是对国家未来科学活力的宝贵投资。

In this research, Dr. Shaw will continue her investigation of linear partial differential equations which arise from problems in several complex variables. In particular, she will study the solvability, regularity and embeddability of the tangential Cauchy-Riemann operators. These problems are of fundamental importance in the function theory of complex analysis. The global problem is an extension of her results on L2 existence and estimates for b on weakly pseudo-convex boundaries to Holder and Lp estimates. The local problem is to prove with kernel method the existence of a local solution for a system of complex vector fields satisfying the compatibility condition. The embeddability of abstract CR structures will also be addressed. This problem is closely related to the local solvability of b and the b - homotopy formula. Related problems on the range of b and the regularity of the Szego projection will also be studied. The study of these problems wil benefit from interactions with the Wisconsin group which is also actively working on related problems. Dr. Shaw will also teach a one-semester graduate course in Several Complex Variables. Additionally, she will organize a series of seminars by leading women mathematicians and top researchers in Several Complex Variables. This project furthers VPW program objectives which are (1) to provide opportunities for women to advance their careers in engineering and in the disciplines of science supported by NSF and (2) to encourage women to pursue careers in science and engineering by providing greater visibility for women scientists and engineers employed in industry, government, and academic institutions. By encouraging the participation of women in science, it is a valuable investment in the Nation's future scientific vitality.