Research and Education in Several Complex Variables

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The d-bar-Neumann problem may be viewed as the study of the solutionof the inhomogeneous Cauchy-Riemann equations having minimalL^2-norm. After Kohn's solution of the problem on strictlyPseudo-convex domains, important advances in the L^2-Sobolev theoryincluded the work of Hormander on L^2-estimates, D'Angelo and Catlinon finite type and subellipticity, Boas and the PI on globalregularity on domains of infinite type, and Kiselman, Barrett, andChrist on irregularity on worm domains. However, characterizingglobal regularity remains a fundamental open problem. Likewise, quitegeneral sufficient conditions are known for compactness of thed-bar-Neumann operator, but a characterization on generalpseudo-convex domains in terms of boundary data remains elusive. Theselong term goals represent two thrusts of this project. The PIrecently observed intriguing links between sufficient conditions forregularity properties of the d-bar-Neumann operator and sufficientconditions for the existence of a Stein neighborhood basis for theclosure of the domain. Investigating to what extent regularityproperties of the d-bar-Neumann operator such as compactness implythe existence of a Stein neighborhood basis of the closure of thedomain is a third thrust of this project. During this project, the PIwill supervise several postdoctoral researchers and several graduatestudents. He will co-organize a special semester at the ErwinSchrodinger International Institute for Mathematical Physicsdedicated to topics investigated in this project, and he will give alecture course within the Institute's Senior Fellows programintroducing these topics to graduate students and junior researchers.The study of analysis in several complex variables can be motivated bythe centrality of the subject within mathematics as well as through adirect appeal to its usefulness. For example, one of the basic lawsof nature, causality, when transcribed via a mathematical devicecalled Fourier transform, leads immediately to analytic functions ofseveral (in this case, four) complex variables. The PI has shown thatsome of the problems to be studied in this project have intimateconnections to issues emanating from quantum mechanics. The centralobject of study in this project, the Cauchy-Riemann equations, form amodel problem for a subject central to physics and engineering(partial differential equations). Finally, the project willcontribute significantly to the development of human resourcesthrough training of highly qualified personnel at the postdoctoral andgraduate levels, respectively.