Research and Education in Several Complex Variables

这个项目解决了几个复变量和偏微分方程的交叉点的基本问题。两个重点关注的整体正则性和紧性的d-bar-Neumann算子在一个区域。首席研究员最近开发了一种理论，统一了已知的积极成果，预计全球的规律性，提供了一个重要的一步，对所需的表征这一属性的边界属性的域。目前还不知道距离已知的潜在的理论充分条件紧的d-巴诺依曼算子是必要的。基于他最近与一名研究生的合作（S。Munasinghe）提供了一种不同的方法，研究人员将寻求表征紧凑性。该项目的第三个推力启动了一个新的研究方向：现有的证据表明，正则性的d-酒吧诺依曼算子在一个域应该有积极的影响存在的斯坦邻域基础的封闭域。第四个推力来自研究者与博士后研究者A的合作。赖希，他和主要调查表明，经典的充分条件，紧凑的运营商在内部的一个域（与必要的修改）也产生紧凑的运营商的边界。目标是将他们的方法与研究者的早期方法联合收割机结合起来，以表明有限型在边界上隐含亚椭圆性。多复变分析的研究是由数学学科的中心地位及其实用性所激发的。例如，自然界的中心法则之一，因果关系，当通过称为傅立叶变换的数学装置转录时，立即导致几个（在这种情况下是四个）复变量的解析函数。另一方面，偏微分方程出现在处理随时间变化的系统的所有科学领域：物理学，工程学，经济学，生物学，气象学，环境科学等。因此，该项目位于基础研究的末端，处于科学和技术企业的核心数学两个领域的交叉点。它将通过资助研究生和调查员对博士后研究人员的监督直接影响人力资源开发，并通过调查员组织和参加讲习班/会议以及通过其论文写作间接影响人力资源开发。

This project addresses fundamental questions at the intersection of several complex variables and partial differential equations. Two thrusts concern global regularity and compactness of the d-bar-Neumann operator in a domain. The principal investigator has recently developed a theory that unifies the known positive results concerning global regularity that is expected to provide an important step towards a desired characterization of this property in terms of properties of the boundary of the domain. It is not known how far the known potential theoretic sufficient conditions for compactness of the d-bar-Neumann operator are from being necessary. Building on his recent work with a graduate student (S. Munasinghe) that provides a different approach, the investigator will seek to characterize compactness. A third thrust of the project initiates a new direction of research: the available evidence suggests that regularity properties of the d-bar-Neumann operator in a domain should have positive implications for the existence of a Stein neighborhood basis of the closure of the domain. The fourth thrust results from a collaboration of the investigator with the postdoctoral researcher A. Raich, in which he and the principal investigator showed that the classical sufficient conditions for compactness of the operators in the interior of a domain (with a necessary modification) also yield compactness of the operators on the boundary. The goal is to combine their methods with earlier methods of the investigator to show that finite type implies subellipticity on the boundary.The study of analysis in several complex variables is motivated both by the centrality of the subject to mathematics and by its usefulness. For example, one of the central laws of nature, causality, when transcribed via a mathematical device called the Fourier transform, leads immediately to analytic functions of several (in this case four) complex variables. Partial differential equations, on the other hand, arise in all areas of science that deal with systems that change over time: physics, engineering, economics, biology, meteorology, environmental sciences, and others. This project is thus located on the basic research end at the intersection of two areas of mathematics that are central to the scientific and technological enterprise. It will impact human resources development directly through funding for graduate students and through the investigator's supervision of postdoctoral researchers, and indirectly through the investigator's organization of and participation in workshops/conferences and through his expository writing.