The d-bar-Neumann problem may be viewed as the study of the solution of the inhomogeneous Cauchy-Riemann equations having minimal L^2-norm. After Kohn's solution of the problem on strictly Pseudo-convex domains, important advances in the L^2-Sobolev theory included the work of Hormander on L^2-estimates, D'Angelo and Catlin on finite type and subellipticity, Boas and the PI on global regularity on domains of infinite type, and Kiselman, Barrett, and Christ on irregularity on worm domains. However, characterizing global regularity remains a fundamental open problem. Likewise, quite general sufficient conditions are known for compactness of the d-bar-Neumann operator, but a characterization on general pseudo-convex domains in terms of boundary data remains elusive. These long term goals represent two thrusts of this project. The PI recently observed intriguing links between sufficient conditions for regularity properties of the d-bar-Neumann operator and sufficient conditions for the existence of a Stein neighborhood basis for the closure of the domain. Investigating to what extent regularity properties of the d-bar-Neumann operator such as compactness imply the existence of a Stein neighborhood basis of the closure of the domain is a third thrust of this project. During this project, the PI will supervise several postdoctoral researchers and several graduate students. He will co-organize a special semester at the Erwin Schrodinger International Institute for Mathematical Physics dedicated to topics investigated in this project, and he will give a lecture course within the Institute's Senior Fellows program introducing these topics to graduate students and junior researchers.
The study of analysis in several complex variables can be motivated by the centrality of the subject within mathematics as well as through a direct appeal to its usefulness. For example, one of the basic laws of nature, causality, when transcribed via a mathematical device called Fourier transform, leads immediately to analytic functions of several (in this case, four) complex variables. The PI has shown that some of the problems to be studied in this project have intimate connections to issues emanating from quantum mechanics. The central object of study in this project, the Cauchy-Riemann equations, form a model problem for a subject central to physics and engineering (partial differential equations). Finally, the project will contribute significantly to the development of human resources through training of highly qualified personnel at the postdoctoral and graduate levels, respectively.