Singularities in Harmonic Analysis and the Calculus of Variations

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Harmonic analysis and the calculus of variations are two of the oldest branches of mathematical analysis. Harmonic analysis is the study of the scale invariant properties of functions such as the oscillation of violin strings; it underpins much of modern image and signal processing. The calculus of variations is the study of energy minimizers; for example, a soap film hanging off a wire forms a shape which minimizes the surface tension. The purpose of this project is twofold: first to use harmonic analysis to study the behavior of energy minimizers under perturbations (like a soap film in a mild wind). Second, to use techniques from the calculus of variations to study a central problem in harmonic analysis, namely how does diffusion (i.e. heat spreading through a room) detect geometry. This synergy should not only lead to mathematical breakthroughs but also refine our ability to use these mathematical ideas to predict physical phenomena. The project provides research training opportunities for graduate students.The investigator is interested in singularities of minimizers (places where the minimizer fails to be smooth, like the corners soap bubbles form when they touch each other). In particular, the investigator wants to develop tools to study these singularities which are persistent under perturbations (either of the energy or the initial conditions). To develop these tools, new ideas from harmonic analysis and geometric measure theory are necessary. The investigator is also interested in the problem of how solutions to elliptic differential equations (which model diffusion) detect the geometry of domains in Euclidean space. Recently, there have been major breakthroughs showing that homogeneous diffusion can detect some geometric features. The investigator believes that ideas from the calculus of variations can extend this breakthrough to understand what finer geometric details inhomogeneous diffusion can detect.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.