Research in Several Complex Variables

【摘要】本项目有两个主要的科学组成部分。第一个目标是提出多维复空间伪凸域上非齐次柯西-黎曼方程的理论。尽管d-bar Neumann问题的全局正则性在大范围域中成立，但它在所谓的蠕虫域中失效。目前还没有统一已知的积极结果的通用理论，更不用说解释蠕虫领域的消极结果了。同样，d-bar Neumann算子的紧性较强的充要条件也不知道。要解决的一个基本问题是如何将Catlin的全局正则性方法（通过紧致性）与研究者的向量场方法统一起来。研究人员期望从这项研究中出现一个新的全球规律性的充分条件。他们还希望在d-bar Neumann问题中通过一些比Catlin所谓的性质p稍弱的条件来描述紧性。这个项目的第二个重点是在多维复分析中开发一个新的领域：即研究当维度趋于无穷大时，该主题的定理和估计如何渐近地依赖于空间的维度。两个特别有趣的问题是最近发展的关于将玻尔经典幂级数定理扩展到高维的理论，以及量化Bergman核函数的零如何依赖于周围复空间的维数的问题。在这个项目中，研究人员将培训研究生，他们将指导一名博士后水平的年轻数学家（由德克萨斯农工大学VIGRE资助）研究角域上的d-bar诺伊曼问题。在几个复杂变量的分析研究的动机是由中心的主题在数学和其固有的有用性。例如，自然的基本定律之一，因果关系，当被称为傅里叶变换的数学装置转录时，立即产生几个（在这种情况下是四个）变量的解析函数。本项目的工作不仅会影响几个复杂变量和偏微分方程的核心领域，而且会影响其他科学领域。例如，柯西-黎曼方程为物理和工程的核心学科形成了一个模型问题；玻尔经典定理对算子理论产生了影响；Bergman核函数为数学物理中Berezin的量化方案提供了具体的模型。除了通过基础研究推进知识前沿之外，该项目还将通过对高素质人才的科学培训，为人力资源的发展做出重大贡献。

AbstractBoas/StaubeThis project has two main scientific components. The first goal is to advance the theory of the inhomogeneous Cauchy-Riemann equations on pseudoconvex domains in multidimensional complex space. Although global regularity of the d-bar Neumann problem holds on large classes of domains, it fails on the so-called worm domains. Currently there is no general theory that unifies the known positive results, much less one that also accounts for the negative results on the worm domains. Likewise, necessary and sufficient conditions for the stronger property of compactness of the d-bar Neumann operator are not known. A basic question to be addressed is how to unify Catlin's approach to global regularity (via compactness) with the investigators' vector field method. The investigators expect a new sufficient condition for global regularity to emerge from this study. They also hope to characterize compactness in the d-bar Neumann problem by some condition slightly weaker than Catlin's so-called property P. The second thrust of this project is to develop a new area in multi-dimensional complex analysis: namely, the study of how the theorems and estimates of the subject depend asymptotically on the dimension of the space as the dimension tends to infinity. Two particular problems of interest are the recently developed theory about extending Bohr's classical power series theorem to higher dimensions and the question of quantifying how zeroes of the Bergman kernel function depend on the dimension of the ambient complex space. During this project, the investigators will train graduate students, and they will supervise a young mathematician at the post-doctoral level (supported through a VIGRE grant at Texas A&M University) in his study of the d-bar Neumann problem on domains with corners.The study of analysis in several complex variables is motivated both by the centrality of the subject within mathematics and by its inherent usefulness. For example, one of the basic laws of nature, causality, when transcribed by a mathematical device called the Fourier transform, immediately gives rise to analytic functions of several (in this case four) variables. The work in this project will impact not only the core areas of several complex variables and partial differential equations, but also other areas of science. For example, the Cauchy-Riemann equations form a model problem for a subject central to physics and engineering; Bohr's classical theorem has repercussions in operator theory; and the Bergman kernel function provides a concrete model for Berezin's quantization scheme in mathematical physics. In addition to advancing the frontiers of knowledge through basic research, this project will contribute significantly to the development of human resources through the scientific training of highly qualified personnel.