Research and Education in Several Complex Variables

D-bar-Neumann问题可以看作是研究具有最小^2范数的非齐次Cauchy-Riemann方程的解。在Kohn解决了严格伪凸域上的问题之后，L^2-Sobolv理论的重要进展包括Hormander关于L^2-估计的工作，D‘Angelo和Catlinon有限型与次椭圆性的工作，Boas和PI关于无限型域上的全局正则性的工作，以及Kiselman，Barrett和Christian关于蠕虫域上的不规则性的工作。然而，表征全球规律性仍然是一个基本的悬而未决的问题。同样地，THed-bar-Neumann算子紧性的充分条件是已知的，但是关于广义伪凸域上的边界数据的刻画仍然是难以捉摸的。这些长期目标代表了这个项目的两个推动力。PI最近观察到d-bar-Neumann算子正则性的充分条件与区域闭包的Stein邻域基存在的充分条件之间有趣的联系。研究d-bar-Neumann算子的紧性等正则性在多大程度上意味着域的闭包的Stein邻域基的存在是本项目的第三个重点。在这个项目中，PI将指导几名博士后研究人员和几名研究生。他将在Erwin Schrodinger国际数学物理研究所共同组织一个特别学期，致力于这个项目中研究的主题，他将在研究所的高级研究员中讲授一门课程，向研究生和初级研究人员介绍这些主题。对几个复杂变量的分析的研究可以受到该学科在数学中的中心地位以及对其有用性的直接呼吁的推动。例如，自然的基本法则之一，因果关系，当通过一种称为傅立叶变换的数学工具转录时，立即导致几个(在这种情况下，是四个)复变量的解析函数。PI已经表明，这个项目中要研究的一些问题与量子力学发出的问题有密切的联系。这个项目的中心研究对象，柯西-黎曼方程，形成了一个以物理和工程为中心的学科(偏微分方程)的模型问题。最后，该项目将通过培训高素质的博士后和研究生人员，大大促进人力资源的开发。

The d-bar-Neumann problem may be viewed as the study of the solutionof the inhomogeneous Cauchy-Riemann equations having minimalL^2-norm. After Kohn's solution of the problem on strictlyPseudo-convex domains, important advances in the L^2-Sobolev theoryincluded the work of Hormander on L^2-estimates, D'Angelo and Catlinon finite type and subellipticity, Boas and the PI on globalregularity on domains of infinite type, and Kiselman, Barrett, andChrist on irregularity on worm domains. However, characterizingglobal regularity remains a fundamental open problem. Likewise, quitegeneral sufficient conditions are known for compactness of thed-bar-Neumann operator, but a characterization on generalpseudo-convex domains in terms of boundary data remains elusive. Theselong term goals represent two thrusts of this project. The PIrecently observed intriguing links between sufficient conditions forregularity properties of the d-bar-Neumann operator and sufficientconditions for the existence of a Stein neighborhood basis for theclosure of the domain. Investigating to what extent regularityproperties of the d-bar-Neumann operator such as compactness implythe existence of a Stein neighborhood basis of the closure of thedomain is a third thrust of this project. During this project, the PIwill supervise several postdoctoral researchers and several graduatestudents. He will co-organize a special semester at the ErwinSchrodinger International Institute for Mathematical Physicsdedicated to topics investigated in this project, and he will give alecture course within the Institute's Senior Fellows programintroducing these topics to graduate students and junior researchers.The study of analysis in several complex variables can be motivated bythe centrality of the subject within mathematics as well as through adirect appeal to its usefulness. For example, one of the basic lawsof nature, causality, when transcribed via a mathematical devicecalled Fourier transform, leads immediately to analytic functions ofseveral (in this case, four) complex variables. The PI has shown thatsome of the problems to be studied in this project have intimateconnections to issues emanating from quantum mechanics. The centralobject of study in this project, the Cauchy-Riemann equations, form amodel problem for a subject central to physics and engineering(partial differential equations). Finally, the project willcontribute significantly to the development of human resourcesthrough training of highly qualified personnel at the postdoctoral andgraduate levels, respectively.