Research and Education in Several Complex Variables

建议：DMS-9801539主要研究人员：哈罗德·P·博阿斯，埃米尔·施特劳贝摘要：L-2关于伪凸域上d-bar Neumann问题的索博列夫理论在20世纪80年代和90年代取得了巨大的进展，特别是通过Catlin和D‘Angelo关于次椭圆性和有限类型的工作，主要研究人员在大类(弱)伪凸域上的全局正则性，以及Barrett，Christian和Siu关于所谓的“蠕虫”域上的正则性失效的工作。然而，目前甚至还没有关于几何条件和位势理论条件的组合可以用来刻画d-bar Neumann问题的全局正则性的猜测。对这些条件的研究是本课题的研究方向之一。一个相关的(而且很可能更容易处理的)问题是用紧的d-bar Neumann算子来刻画区域。(这个问题在算子理论中也有影响。)此外，主要研究人员将研究有限类型区域上d-bar Neumann问题的具有增益的L-p估计。在这里，他们将建立在最近关于有限类型非光滑区域的工作中获得的新见解。从与域的Bergman核相关的Kaehler几何的角度来看，这些研究也是有意义的。(除了它们的内在意义，类Bergman核在一种量子化方案--Berezin量子化--中扮演着突出的角色，该方案最近引起了数学家和物理学家的极大关注)。在这个项目中，主要研究人员将培训三名研究生，他们将指导一名年轻的博士后水平的数学家(由德克萨斯农工大学的团体基础设施基金资助)研究与多项式壳有关的问题。(虽然这些问题具有内在的动机，但它们与控制工程有着密切的联系)。对几个复杂变量的分析的研究可以受到数学学科的中心地位的推动(因此，从长远来看，这对科学和技术企业的福祉是必不可少的)，或者通过更直接地呼吁其有用性。例如，自然界的基本定律之一，因果关系，当通过一种名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)复变量的解析函数。同样，如上所述，该项目的工作不仅将影响几个复变量和偏微分方程式的核心领域，还将通过与算子理论、数学物理和控制工程的联系影响其他科学领域。最后，该项目将对数学方面的人力资源开发作出重大贡献。

Proposal: DMS-9801539 Principal Investigators: Harold P. Boas, Emil Straube Abstract: The L-2 Sobolev theory of the d-bar Neumann problem on pseudoconvex domains has seen dramatic progress in the 1980s and 1990s, notably through the work of Catlin and D'Angelo on subellipticity and finite type, of the principal investigators on global regularity on large classes of (weakly) pseudoconvex domains, and of Barrett, Christ, and Siu on failure of regularity on the so called "worm" domains. However, there is currently not even a conjecture about what combination of geometric and potential theoretic conditions might be used to characterize global regularity in the d-bar Neumann problem. The study of such conditions is one of the directions of this project. A related (and quite possibly more tractable) question is that of characterizing domains with compact d-bar Neumann operators. (This question also has repercussions in operator theory.) In addition, the principal investigators will study L-p estimates with gain for the d-bar Neumann problem on domains of finite type. Here, they will build on new insight gained from recent work concerning nonsmooth domains of finite type. These investigations are also of interest from the point of view of the Kaehler geometry associated with the Bergman kernel of a domain. (In addition to their intrinsic significance, Bergman-like kernels play a prominent role in a quantization scheme --Berezin quantization -- that has recently attracted much attention from both mathematicians and physicists). During this project, the principal investigators will train three graduate students, and they will supervise a young mathematician at the post-doctoral level (supported through a Group Infrastructure Grant at Texas A&M University) in his study of problems concerning polynomial hulls. (While intrinsically motivated, these questions have intimate connections to control engineering). The study of analysis in several complex variables can be motivated by the centrality of the subject within mathematics (thus making it essential, in the long term, for the well-being of the scientific and technological enterprise) or through a more direct appeal to its usefulness. For example, one of the basic laws of nature, causality, when transcribed via a mathematical device called the Fourier transform, immediately gives rise to analytic functions of several (in this case four) complex variables. Likewise, as indicated above, 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 through connections to operator theory, to mathematical physics, and to control engineering. Finally, the project will contribute significantly to human resources development in mathematics.