Partial Differential Equations and Geometric Analysis in Several Complex Variables

University Of Wyoming, Laramie WY

Investigators

Abstract

ABSTRACT A basic problem in several complex variables is to understand the 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 plans to 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 potential theoretic terms. He will also study eigenvalue spectrum of the d-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 investigator plans to study invariant metrics, in particularly, the Bergman and Kobayashi metrics. He will consider boundary behavior and the zero set of the Bergman kernel function, as well as completeness of the Kobayashi metric. Another problem to be studied is the theory of automorphism groups and its relationship with the regularity of the d-bar-Neumann problem. 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 in this project come from other branches of mathematics and sciences. Some problems may even rely on computer programming. The investigator will also contribute to the development of human resources by supervising a graduate student.