Microlocal Analysis and Monge-Ampere Type Equations in Geometry

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In this project the PI will continue to study problems mainly in differential geometry that can be formulated as highly nonlinear partial differential equations. One of the themes is the existence of canonical geometries or shapes on spaces. These originally grew out of Einstein's famous equation in general relativity. One example is the existence of Kahler-Einstein metrics with conic singularities. These beautiful structures turn out to be of central importance in mathematics and physics, and touch upon many fields and their theory involves progress relevant to algebraic, analysis, and geometry and topology. The analytic techniques developed in this proposal should be useful to researchers working in geometry, physics and elsewhere. Also, developing a better understanding for the polarity transform in convex geometry could be useful to solving a range of partial differential equations, and generalizes the known theory for the Legendre transform that is a classical tool in mathematics, mechanics and economics. The award also supports graduate students working on their dissertations in related topics.Understanding Kahler-Einstein metrics with conic singularities will deepen our understanding of smooth Kahler-Einstein metrics on both compact and non-compact Kahler 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. 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. Developing methods and techniques to construct and approximate such solutions and to study their regularity could have applications in other instances where these equations appear. Moreover, developing novel connections with algebraic geometry, convex geometry, and micro-local analysis will be an important goal of this project.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.