Microlocal Analysis and Monge-Ampère Type Equations in Geometry

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This research project concerns questions in differential geometry that can be formulated in terms of nonlinear partial differential equations. One of the research themes is the existence of canonical geometries or shapes on spaces or manifolds, related to the original work of Riemann on curvature and Einstein's equations of general relativity. One example is the existence of Kähler-Einstein metrics with conic singularities; these structures turn out to be of central importance in mathematics and physics, and their theory involves developments in algebra, analysis, geometry, and topology. 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 the complex 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 Kähler-Einstein metrics with conic singularities 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.