Research in Several Complex Variables

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract Boas/Staube This project has two main scientific components. The first goal is to advance the theory of the inhomogeneous Cauchy-Riemann equations on pseudoconvex domains in multidimensional complex space. Although global regularity of the d-bar Neumann problem holds on large classes of domains, it fails on the so-called worm domains. Currently there is no general theory that unifies the known positive results, much less one that also accounts for the negative results on the worm domains. Likewise, necessary and sufficient conditions for the stronger property of compactness of the d-bar Neumann operator are not known. A basic question to be addressed is how to unify Catlin's approach to global regularity (via compactness) with the investigators' vector field method. The investigators expect a new sufficient condition for global regularity to emerge from this study. They also hope to characterize compactness in the d-bar Neumann problem by some condition slightly weaker than Catlin's so-called property P. The second thrust of this project is to develop a new area in multi-dimensional complex analysis: namely, the study of how the theorems and estimates of the subject depend asymptotically on the dimension of the space as the dimension tends to infinity. Two particular problems of interest are the recently developed theory about extending Bohr's classical power series theorem to higher dimensions and the question of quantifying how zeroes of the Bergman kernel function depend on the dimension of the ambient complex space. During this project, the investigators will train graduate students, and they will supervise a young mathematician at the post-doctoral level (supported through a VIGRE grant at Texas A&M University) in his study of the d-bar Neumann problem on domains with corners. The study of analysis in several complex variables is motivated both by the centrality of the subject within mathematics and by its inherent usefulness. For example, one of the basic laws of nature, causality, when transcribed by a mathematical device called the Fourier transform, immediately gives rise to analytic functions of several (in this case four) variables. 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. For example, the Cauchy-Riemann equations form a model problem for a subject central to physics and engineering; Bohr's classical theorem has repercussions in operator theory; and the Bergman kernel function provides a concrete model for Berezin's quantization scheme in mathematical physics. In addition to advancing the frontiers of knowledge through basic research, this project will contribute significantly to the development of human resources through the scientific training of highly qualified personnel.