Geometric Variational Problems and Nonlinear Partial Differential Equations

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The interaction of geometry and analysis date back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world such as angles, distances and properties of certain shapes. Differential geometry was developed to use the tools of calculus to understand the geometry of curved spaces--for example, the curvature of space by matter as predicted by general relativity, or the properties of soap bubbles (which turn out to be related to the equations describing black holes). In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study. In addition to mathematical investigations, the PI will be organizing a summer residential STEM program in cooperation with the Chicago Pre-College Science and Engineering Program, and supported by the Notre Dame TRiO Program. This will be a two-week program run in the summers of 2019 and 2020 for high school students from 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 supported by this award. Poincare-Einstein manifolds are generalizations of the Poincare 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 supported by this award is the fundamental question of existence: given a manifold with boundary and a conformal class of metrics on the boundary, can one construct a Poincare-Einstein metric whose compactification induces the given conformal class on the boundary? In joint work with Q. Han and S. Stolz, the PI is developing new tools to detect obstructions to existence based on the topology and geometry of the boundary. On the other hand, in work with G. Szekelyhidi the PI was able to prove local existence of solutions (i.e., in a neighborhood of infinity). Another area of research with connections to physics is the PI's ongoing work with J. Streets and C. Kelleher on the variational properties of the Yang-Mills functional in four dimensions. Building on the recent work, which gave a new sharp lower bound for minimizing solutions, the PI will investigate the behavior of solutions with large Morse index. The PI will prove a lower bound for the energy depending (in a precise way) on the index of the solution and the geometry of the base manifold.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.