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.
One of the most fundamental ideas in geometry is to characterize each geometric object by a canonical metric with optimal curvature. The same idea is prevalent in theoretical physics where the optimal curvature condition is the equation of motion. These equations are notoriously difficult, but another layer of difficulty is added when complex structures play a central role, as in the case of K""ahler geometry and string theory. This project focuses on a range of such questions arising from complex geometry and string theory, including complex Monge-Ampere equations, K""ahler metrics of constant scalar curvature, the K""ahler-Ricci flow, and the holomorphic structure of string scattering amplitudes. Some of the main results obtained are the solvability and regularity of the complex Monge-Ampere equation describing geodesic rays in the space of K"ahler metrics; the construction of Green's functions with arbitrary local singularities; the solution of a boundary value problem inspired by the Ising model; and an algorithm for extracting a holomorphic representative from the cohomology class of a string amplitude. The project is also a vehicle for training graduate students and young postodoctoral researchers. A weekly seminar is held in the subject of complex geometry and partial differential equations which attracts participants from the whole tri-state area. Several postdoctoral participants have gone on to become distinguished mathematicians in their own right. Five students obtained their PhD's under the PI's supervision during the period of the grant, two more would obtain their PhD the following year, and three more are beginning work on their theses. Undergraduates have also been sponsored and supervised, and have produced original research.
