Research and Education in Several Complex Variables

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Complex variables are used throughout mathematics and have important applications to engineering, physics and other scientific disciplines. This project will clarify and enhance the scientific understanding of phenomena observed in several complex variables. The results of this project will lead to significant advances related to commonly used tools, operators, and partial differential equations that originate in the study of multidimensional complex analysis. In addition, the project will aid the development of early career researchers at both the graduate and postdoctoral levels.The work to be carried out in the project initiates from a fundamental partial differential equation in the theory of several complex variables known as the d-bar Neumann problem. The problems to be considered interact with numerous areas of mathematics, and outcomes from this research will push the frontier of knowledge in partial differential equations, potential theory, differential geometry, and operator theory. The project will further the research of the Principal Investigator on the connection between global regularity of the d-bar Neumann operator and the Diederich-Fornaess index. In addition, the Principal Investigator and the co-Principal Investigator will collaborate on a study of the relationship between compactness of the d-bar Neumann operator and the local boundary structure at weakly pseudoconvex boundary points for a domain in complex Euclidean space. Finally, the Principal Investigator, in collaboration with his PhD student, will study operators between Bergman spaces of domains in complex space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.