Microlocal analysis and Monge-Ampere type equations in Geometry

This research project concerns questions in geometry, analysis, and algebra that can be formulated in terms of nonlinear partial differential equations. One of the research themes is the existence of canonical shapes and structures on manifolds or geometric spaces, related to the original work of Riemann on curvature and Einstein's equations of general relativity. The analytic techniques to be developed in this project are expected to be useful to researchers working in geometry, physics, and related areas. Additionally, the project aims to develop better understanding of relations between complex geometry and convex geometry, with applications to algebraic geometry of thresholds. Some of these relations involve novel generalizations of the Legendre transform, which could be useful in solving a range of partial differential equations, generalizing the theory for the Legendre transform that is a classical tool in mathematics, mechanics, and economics. The project involves research training of graduate students in related topics. Understanding Special Lagrangian submanifolds of Calabi-Yau will deepen understanding of minimal submanifold theory as well as the Lagrangian mean curvature flow and its singularities. Understanding singular Kähler-Einstein metrics and spaces will deepen understanding of smooth Kähler-Einstein metrics on both compact and non-compact Kähler manifolds, including Fano and Calabi-Yau spaces. These spaces are central in a wide variety of fields, ranging from algebraic geometry and number theory to theoretical physics where the Eguchi-Hanson metric appears. Monge-Ampère type equations arise in a wide variety of questions in pure and applied mathematics and have a wide range of practical applications. This project aims to develop methods to construct and approximate such solutions and to study their regularity, which will have applications in other instances where these equations appear. The project also intends to develop novel connections with algebraic geometry, convex geometry, and micro-local analysis. 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.