The PI plans to investigate projects in three areas, related by the theme of the complex Monge-Ampere equation. The first project will build on the PI's work with Tosatti which gave an analogue of Yau's Theorem for compact Hermitian manifolds. The PI proposes to generalize these estimates and apply them to obtain new results on the Bott-Chern space of a complex manifold. The second project deals with the Kahler-Ricci flow - which can be regarded as the parabolic complex Monge-Ampere equation - on algebraic varieties. As part of a program to understand the analytic minimal model program, the PI and Song investigated the Kahler-Ricci flow on algebraic varieties and gave necessary conditions under which the flow contracts an exceptional divisor. The PI proposes to extend these results to deal with more general singularities and also to study the behavior of the curvature tensor along the Kahler-Ricci flow. The final project concerns the Calabi-Yau equation on symplectic manifolds. Donaldson conjectured that the complex Monge-Ampere equation solved by Yau has an analogue for symplectic 4-manifolds with compatible almost complex structures. He gave applications of his conjecture to symplectic topology. Special cases of Donaldson's conjecture were established by the PI and his co-authors. The PI will undertake an analysis of the structure of the blow-up set for the Calabi-Yau equation, with the ultimate goal of proving Donaldson's conjecture.
An important problem in mathematics, and physics, is to understand the interaction between geometry and differential equations. This proposal concerns a well-known and centuries-old mathematical object called the Monge-Ampere equation. This equation arises naturally in the study of geometry and is closely related to Einstein's equations in physics. A main goal of this project is to find applications of the Monge-Ampere equation to more general and commonly occuring geometric objects where it was not previously known that a connection exists. By finding new relationships between this classical differential equation and geometry, the PI aims to further our understanding of what kind of geometric structures can exist. In addition, the Monge-Ampere equation is deeply related to geometry via an associated heat flow. It is expected that this heat flow will help us understand some old and difficult problems concerning solutions to algebraic equations. Indeed, the solutions of algebraic equations define geometric objects, and the heat flow deforms these objects and can extract information from them. The PI will investigate the precise behavior of the Monge-Ampere heat flow on such geometric objects.