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

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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.