Problems in complex analysis, complex geometry, and mathematical physics

This project focuses on a number of open problems about canonical metrics and stability in complex geometry, pluripotential theory and complex Monge-Ampere equations, Feynman rules in string theory, and integrable models related to gauge theories. These are fundamental problems in complex analysis and complex geometry that are also of great interest in algebraic geometry, partial differential equations, and mathematical physics. They are acknowledged to be difficult, but recent progress has revealed for them a very rich structure, as well as many unifying threads. The project builds upon previous research of the principal investigator, but it also branches out to explore relations with different concurrent lines of investigation. The principal investigator?s approaches to the various problems are tightly interwoven, so that progress on one could well lead to progress on others. Complex analysis and complex geometry are central fields in mathematics, whose role is essential in the very formulation and ultimate understanding of physical laws. Complex analytic methods are needed in every branch of both pure and applied mathematics. The geometric problems contemplated here either are rooted directly in attempts at understanding the laws of nature at their most fundamental level (as in the problems from string theory and gauge theories) or have strong analogies with basic equations from general relativity and other branches of science (as in the case of canonical metrics and Monge-Ampere equations). The proposed research will have an immediate beneficial effect on students and postdoctoral researchers at the principal investigator?s home institution. But it will also generate a lot of research problems and provide a fertile training ground for the many researchers in analysis and complex geometry nationwide. The principal investigator has actively encouraged junior people, irrespective of their affiliations, to participate in various components of this research. To this end, he plans to continue to disseminate the results of the research to a broad audience through lectures, survey papers, and graduate texts.