Partial Differential Equations and Geometric Analysis in Several Complex Variables

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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.