Award: DMS 1206284, Principal Investigator: Yanir Rubinstein
This project focuses on problems mainly in differential geometry and geometric analysis that can be formulated as real and complex Monge-Ampere type equations. These include (1) the existence and regularity of Kahler-Einstein metrics with conic singularities and their applications; (2) the existence and regularity of solutions to and well-posedness of the homogeneous Monge-Ampere equation; (3) new equations of Monge-Ampere type that arise in convex geometry and their relations to PDEs and the Legendre transform. One feature in key parts of this project is to combine tools of microlocal analysis to study these equations, in addition to the more traditional methods of PDEs, convex analysis, pluripotential theory, and several complex variables. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry.
In general terms, the analytic techniques developed in this proposal should be useful to researchers working in geometry, physics and elsewhere. On the one hand, deepening our understanding of canonical geometries on Kahler manifolds seems to be of interest to physicists trying to model the geometry of the universe. On the other hand, these canonical geometries have relations to a wide variety of established fields in mathematics. Moreover, Monge-Ampere type equations arise in a wide variety of problems in pure and applied mathematics and have a wide range of real-world applications, such as meteorology and optimal design of networks. Developing methods and techniques to construct and approximate solutions to such equations and to study their regularity could have applications in other instances where these equations appear. Finally, the Legendre transform is a classical tool in mathematics, mechanics and economics, and seeking generalizations of this theory to other settings, as in this project, could find a broad range of applications.
