Research in Several Complex Variables

AbstractBoas/Staube该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中拟凸域上的非齐次Cauchy-Riemann方程的理论。虽然d-bar Neumann问题的全局正则性在大类域上成立，但它在所谓的蠕虫域上失败。目前还没有一个通用的理论来统一已知的积极结果，更不用说也解释蠕虫域上的消极结果了。同样，d-bar Neumann算子具有较强紧性的充要条件也是未知的。一个需要解决的基本问题是如何将Catlin的全局正则性方法(通过紧致性)与研究人员的矢量场方法统一起来。研究人员预计，这项研究将为全球规律性提供一个新的充分条件。他们还希望用一些比Catlin所谓的性质P稍弱的条件来刻画d-bar Neumann问题的紧性。这个项目的第二个目的是在多维复分析中发展一个新的领域：即研究当空间的维度趋于无穷时，主题的定理和估计如何渐近地依赖于空间的维度。两个特别令人感兴趣的问题是最近发展起来的关于将玻尔经典的幂级数定理推广到更高维的理论，以及量化Bergman核函数的零点如何依赖于环境复空间的维度的问题。在这个项目中，研究人员将培训研究生，他们将指导一位年轻的博士后水平的数学家(由德克萨斯A&M大学的Vigre资助)研究带角域上的d-bar Neumann问题。对几个复杂变量的分析的研究既受到数学学科的中心地位的推动，也受到其内在实用性的推动。例如，自然界的基本定律之一，因果关系，当被称为傅立叶变换的数学工具转录时，立即产生几个(在这种情况下是四个)变量的解析函数。该项目的工作不仅将影响几个复变量和偏微分方程的核心领域，而且还将影响科学的其他领域。例如，柯西-黎曼方程形成了物理和工程中心学科的模型问题；玻尔的经典定理在算子论中产生了影响；伯格曼核函数为贝雷津的数学物理量子化方案提供了一个具体的模型。除了通过基础研究推进知识前沿之外，该项目还将通过科学培训高素质人员，大大促进人力资源的开发。

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.