Geometric Variational Problems and Nonlinear Partial Differential Equations

The interaction of geometry and analysis continues to be an important and active field of mathematical research. The classical subject of geometry grew out of our desire to understand properties of the physical world such as angles and distances. Differential geometry in turn was developed to use the tools of calculus to understand curved spaces.

For example, differential geometry can be used to understand the curvature of space by matter as predicted by general relativity. In the same way that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis. The project will investigate problems from geometry and mathematical physics that are united by the role played by mathematical analysis.

In addition to mathematical investigations, the PI will continue to organize a summer residential STEM program in cooperation with the Chicago Pre-College Science and Engineering Program. This is a program for high school students from the Chicago Public Schools, many of whom will be first-generation college students. The program will run for two weeks, and will include mathematics instruction and project-based learning

There are two main mathematical themes in this project. Poincaré-Einstein manifolds are generalizations of the Poincaré ball model of hyperbolic space. They are complete manifolds satisfying the Einstein condition (with negative Einstein constant) which can be compactified by conformally changing the metric that vanishes at an appropriate rate at infinity.

They arise in several areas of mathematics and theoretical physics; for example, in in the Fefferman-Graham theory of conformal invariants and in the AdS/CFT correspondence in quantum field theory. One area of investigation in this project is the fundamental question of existence: given a manifold with boundary and a conformal class of metrics on the boundary, can one construct a Poincaré-Einstein metric whose compactification induces the given conformal class on the boundary?

In joint work with S.Y.A Chang, PI will be studying the problem of using the geometry of the conformal boundary to solve a nonlinear PDE in the interior, and showing how the existence of solutions allows us to use Morse Theory to identify topological obstructions to existence. Another area of research with connections to physics is the PI’s ongoing work with S.

Perez-Ayala on extremizing eigenvalues of conformally covariant operators. In the case of surfaces, extremal eigenvalues of the Laplace-Beltrami operator have connections to minimal surfaces and harmonic maps. In recent work with Perez-Ayala PI showed that under certain natural conditions, one can extremize the low eigenvalues of the conformal Laplacian in higher dimensions, and there are examples of extremals that give rise to harmonic maps.

In ongoing work PI will study other operators and try to understand a kind of reverse construction; i.e., when a harmonic map gives rise to a maximal metric.

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.