CAREER: Harmonic Analysis and the Stability of Singularities in the Calculus of Variations

这个项目研究变分和偏微分方程中几个重要的物理模型中的奇点。变分最初是在力学的背景下发展起来的，它是一个数学研究领域，研究使能量最小的形状或函数。例如，肥皂泡之所以呈圆形，是因为在给定固定封闭体积的情况下，它可以最大限度地降低表面张力。在某些情况下，这些自然能量的极小化子会表现出奇点--在那里解是不顺利的。这个项目研究了这种奇点的形成和结构。该项目的综合教育部分支持高中生的数学暑期计划，一个旨在增加当地研究研讨会机会的学习研讨会，以及一门新的跨学科研究生课程。该项目研究变分的三个领域的奇点形成：(静止的)自由边界问题，抛物型偏微分方程解的节点集，以及流形上的能量临界演化。每个主题都代表了奇点形成研究中的一个中心问题。奇点什么时候存在？它们什么时候是稳定的或通用的？什么时候才能准确地描述奇点形成的时空邻域中的解的行为？该项目将开发技术来区分极小化过程中的奇点形成和稳定溶液中的奇点形成。该项目的第二部分涉及流中奇点的摄动，而不使用单调性。该项目的第三部分研究了解析性在气泡和孤子分辨的唯一性中的作用。该项目的教育部分寻求通过高中、本科生和研究生水平的新的和更容易获得的内容来增加对分析和微分方程式感兴趣的受训人员的供应。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates singularities in several physically important models arising in the calculus of variations and partial differential equations. Developed initially in the context of mechanics, the calculus of variations is a mathematical field of study, which investigates shapes or functions which minimize energy. For example, a soap bubble takes its round shape because it minimizes surface tension given a fixed enclosed volume. In some situations, minimizers to these natural energies exhibit singularities – places where the solution is not smooth. This project investigates the formation and structure of such singularities. The integrated educational component of the project supports a mathematical summer program for high school students, a learning seminar designed to increase access to local research seminars, and a new and interdisciplinary graduate course.The project studies singularity formation in three areas of the calculus of variations: (stationary) free boundary problems, nodal sets of solutions to parabolic partial differential equations, and energy critical evolution on manifolds. Each topic represents a central question in the study of singularity formation. When do singularities exist? When are they stable or generic? When can one precisely describe the behavior of a solution in a space-time neighborhood of the singularity formation? The project will develop techniques to distinguish singularity formation in minimizers from singularity formation in stable solutions. A second component of the project involves the perturbation of singularities in flows without the use of monotonicity. The third component of the project investigates the role of analyticity in the uniqueness of bubbling and soliton resolution. The educational component of the project seeks to increase the supply of trainees interested in analysis and differential equations via new and more accessible content at the high school, undergraduate and graduate levels.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.