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Singularities and Smoothness in Geometric Partial Differential Equations

几何偏微分方程中的奇异性和光滑性

基本信息

批准号：
1809011
负责人：
Aaron Naber
金额：
$ 22.6万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1809011&HistoricalAwards=false
关键词：
Singularities Smoothness Geometric Partial Differential

项目摘要

The scientific goal of this project is to study various geometrically motivated equations and their applications. One area of investigation is energy-minimizing mappings between curved spaces (known as nonlinear harmonic maps) and their singular sets, i.e., points where the mapping ceases to be smooth. Another area of investigation is the structure of the spaces that arise as limits of smooth spaces with bounded curvature, and a third area is the regularity of solutions to the Yang-Mills equations which arise particle physics. The research project also touches on other mathematical subjects such as Reifenberg analysis, spectral analysis, stochastic analysis and metric-measure spaces. In each area, open problems which are at the forefront of the topic are being pursued. These are topics which have broad applications to many areas of mathematics and physics, and solutions to these problems would open doors to tackling the next generation of problems. Many parts of this project also involve the training of early-career researchers, as several lines of investigation involve collaboration with senior graduate students and postdoctoral scholars.The first part of this overall 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 a priori L1 estimates for their hessians. Valtorta and the PI have solved these conjectures for stationary Yang-Mills, but the methods do not work for harmonic maps, and the PI will solve the problems by other means. In another line of investigation, joint with Yu Wang, the PI will 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. Additionally, with Bob Haslhofer, the PI will study connections between bounded Ricci curvature and the analysis on path space. The PI will prove differential Harnack estimates for martingales on spaces with two sided Ricci curvature bounds. The solution to each of these problems involve the development of new techniques and ideas, which themselves one would expect to be even more interesting than the problems.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.

本项目的科学目标是研究各种几何动机方程及其应用。研究的一个领域是弯曲空间（称为非线性调和映射）与其奇异集之间的能量最小化映射，即，映射不再平滑的点。另一个领域的调查是结构的空间，出现的限制光滑空间有界曲率，第三个领域是正则性的解决方案，杨米尔斯方程产生粒子物理。该研究项目还涉及其他数学主题，如Reifenberg分析，谱分析，随机分析和度量空间。在每一个领域，都在探讨处于该专题前沿的未决问题。这些主题在数学和物理的许多领域都有广泛的应用，这些问题的解决方案将为解决下一代问题打开大门。这个项目的许多部分也涉及到早期职业研究人员的培训，因为有几条研究路线涉及到与高年级研究生和博士后学者的合作。这个整体项目的第一部分涉及非线性调和映射的能量恒等式和W2，1-猜想。粗略地说，能量恒等式是一幅图形，它给出了非线性调和映射序列爆破行为的一个显式公式。W2，1-猜想是关于平稳调和映射的Hessian有先验L1估计的简单猜想。Valtorta和PI已经解决了静态杨-米尔斯的这些问题，但这些方法不适用于调和映射，PI将通过其他方法解决问题。在另一条调查线，与王宇联合，PI将回答哈特和林在非线性调和映射上的一些开放性问题。粗略地说，目标是证明在强H1-映射类中可以求解稳定的调和映射。该提案的第二部分将解决涉及具有较低和有界Ricci曲率的空间的正则性问题。这部分的第一个项目是与姜文帅一起研究有界Ricci曲率流形极限的能量恒等式。虽然在精神上相似，但问题本身实际上与调和映射的情况非常不同。开始，能量在这个上下文中是L2曲率形式，直到最近人们才知道这是一个有界测度。其次，在这种情况下，缺陷测度的预测形式可以通过极限中的奇异性行为来计算。此外，与Bob Haslhofer一起，PI将研究有界Ricci曲率与路径空间分析之间的联系。PI将证明微分Harnack估计鞅的双边Ricci曲率界的空间。这些问题的解决方案都涉及到新技术和新想法的发展，人们会期望这些新技术和新想法本身比问题本身更有趣。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。