Partial Differential Equations and Geometric Analysis in Several Complex Variables

多复变量的一个基本问题是理解区域的解析性质和几何性质之间的相互作用。主要研究者计划通过研究d-bar-Neumann问题、不变度量和自同构群来解决这个问题。更具体地说，主要研究人员计划研究d-bar-Neumann算子的紧性和特征值谱。他将从几何和位势的角度研究d-bar-Neumann问题紧的充要条件。他还将通过研究与马克·卡克的问题类似的几个复变量来研究Thed-bar-Neumann问题的本征值谱。此外，首席研究者计划研究不变度量，特别是Bergman度量和Kobayashi度量。他将考虑边界行为和Bergman核函数的零集，以及Kobayashimical的完备性。另一个需要研究的问题是自同构群的理论及其与d-bar-Neumannan问题正则性的关系。复数分析是许多科学和工程领域的重要工具。例如，拉普拉斯变换在机械振动和电路的研究中是必不可少的。拉普拉斯方程在流体力学、静电学和热传导等领域有着重要的应用。这个项目研究的问题不仅本质上很有趣，而且还涉及到复几何、算符理论、位势理论、数学物理和量子力学等领域。这个项目中使用的许多工具来自数学和科学的其他分支。有些问题甚至可能依赖于计算机编程。调查员还将通过监督研究生，为人力资源的开发做出贡献。

ABSTRACTA basic problem in several complex variables is to understandthe interplay between the analytic and geometric natures of a domain.The principal investigator plans to address this problem by studying the d-bar-Neumann problem, invariant metrics, and automorphism groups. More specifically, the principal investigator plansto study compactness and eigenvalue spectrum of the d-bar-Neumann operator.He will investigate necessary and sufficient conditions for compactness of the d-bar-Neumann problem in geometric and potentialtheoretic terms. He will also study eigenvalue spectrum of thed-bar-Neumann problem by investigating, among other problems, the several complex variables analog of Mark Kac's question: ``Can one hear the shape of a drum?''. In addition, the principal investigatorplans to study invariant metrics, in particularly, the Bergman andKobayashi metrics. He will consider boundary behavior and the zero set ofthe Bergman kernel function, as well as completeness of the Kobayashimetric. Another problem to be studied is the theory of automorphismgroups and its relationship with the regularity of the d-bar-Neumannproblem. Complex analysis is a key tool in many areas of sciences and engineering.For example, the Laplace transform is essential in the study of mechanical vibrations and electric circuits. The Laplace equation has important applications to hydrodynamics, electrostatics, and heat conduction. The problems under investigation in this project are not only intrinsically interesting, but also have implications in areas such as complex geometry, operator theory, potential theory,mathematical physics, and quantum mechanics. Many tools to be used inthis project come from other branches of mathematics and sciences. Someproblems may even rely on computer programming. The investigator willalso contribute to the development of human resources by supervising agraduate student.