Partial Differential Equations and Geometric Analysis in Several Complex Variables

多复变中的一个基本问题是理解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，主要研究者计划研究紧性和特征值谱的d-bar-Neumann算子，他将研究必要和充分条件的紧性的d-bar-Neumann问题的几何和potentialtheory术语。 他还将研究本征值谱的d-酒吧诺依曼问题的调查，除其他问题外，几个复杂的变量模拟马克卡茨的问题：``可以听到的形状鼓？''. 此外，主要作者计划研究不变度量，特别是Bergman和Kobayashi度量。他将考虑边界行为和零集的伯格曼核函数，以及完备性的小林度量。 另一个要研究的问题是自同构群理论及其与d-杆-Neumann问题的正则性的关系。复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、静电学和热传导中有重要的应用。 在这个项目中调查的问题不仅是内在有趣的，但也有影响的领域，如复杂的几何，算子理论，潜在的理论，数学物理和量子力学。 在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。调查员还将通过指导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.