Singularities and Smoothness in Geometric Partial Differential Equations

Project: Research project

Project Details

Description

Overview
The goal of the proposal is to study various geometrically motivated equations and their applications. The proposal mostly centers around ideas involving nonlinear harmonic maps, Yang-Mills, Ricci curvature, and Reifenberg analysis. However, the proposal will also touch on spectral analysis, stochastic analysis and metric-measure spaces. In all there are nine projects with eight coauthors. Each project will discuss first progress expected to be made over the next year, and then goals past that.
Intellectual Merit
The first part of the proposal is focused on the study of nonlinear harmonic maps and their singularities. In [64] new techniques were introduced in order to study the structure of the singular set nonlinear equations, and one would like to continue to push forward and build in new ways off this structure. The first project is concerned with the energy identity and the W2,1-conjecture for nonlinear harmonic maps. Roughly, the energy identity is a conjectural picture which gives an explicit formula for the blow up behavior of sequences of nonlinear harmonic maps. The W2,1-conjecture is the easily stated conjecture that stationary harmonic maps have apriori L1 estimates for their hessians. Valtorta and I have solved these conjectures for stationary Yang-Mills, but the methods do not work for harmonic maps, and we hope to solve the problems by other means. The second project is with Yu Wang and hopes to answer some open questions by Hardt and Lin on nonlinear harmonic maps. Roughly, the goal is show one can solve for stable harmonic maps in the class of strongly H1-mappings.
The second part of the proposal would address issues involving the regularity of spaces with lower and bounded Ricci curvature. Together with Wenshuai Jiang the first project of this part would study the energy identity for limits of manifolds with bounded Ricci curvature. Though similar in spirit, the problem itself is actually very different from the harmonic map case. To begin with, the energy in this context is the L2 curvature form, and it has only been very recently that one even knows this is a bounded measure. Secondly, the predicted form of the defect measure in this context can be computed by the singularity behavior in the limit. The second project of this part focuses on connections between bounded Ricci curvature and the analysis on path space. Building on work with Haslhofer, we would like to prove differential harnack estimates for martingales on spaces with two sided Ricci curvature bounds.
The third part of the proposal contains a variety of projects in areas including Reifenberg theory, Yang-Mills, and metric-measure space theory. In particular with Edelen and Valtorta we would like to prove Reifenberg results on Hilbert and Banach spaces which generalize those known in Euclidean space. We have made progress in this direction already, but there are many open problems left.
Broader Impact
The topics being investigated are directly related to many areas of research. The proposals include joint projects with eight coauthors, including five whom are either senior graduate students or in their early postdoc work (Yu Wang, Robin Neumayer, Eva Kopfer, Nick Edelen, Zahra Sinai). Two of my current postdoctoral students are minorities or females. The PI just finished organizing two yearlong programs in geometric analysis, one at Northwestern and one ongoing at Toronto, and the PI is currently organizing several conferences and summer schools, to take place over the next year. An important component of this proposal is to get the fundi
StatusActive
Effective start/end date7/1/186/30/21

Funding

  • National Science Foundation (DMS-1809011)

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