CAREER: Degeneracies of Curvature in Harmonic Analysis

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The field of harmonic analysis attempts to understand physical signals and phenomena by decomposing mathematical objects into simpler parts. Initially, the tools in the field were optimized for the case when the fundamental components of these objects are linear: The Fourier Transform decomposes a signal (such as a sound) as a superposition of constant-frequency waves (notes); the X-ray transform is a way of understanding the density of a body by passing (straight line) beams of radiation through the body and measuring the strength of the beam emitted on the other side. In recent years, however, it has become increasingly apparent that harmonic analysis tools are also useful and important in situations where curvature plays a role. As an example, the Schrödinger equation describes the time evolution of a quantum system, and recent advances toward understanding solutions to this equation were facilitated by the fifty-year-old realization that all of the Fourier data of a solution to the Schrödinger equation lies on a parabolic object. Curvature has a localizing effect: since the waves all propagate in different directions, their superposition can only be large in a small region of spacetime; by contrast, if the Fourier data were to lie on a flat plane, the superposition would be constant in directions perpendicular to that plane. Other physical signals may have Fourier data constrained to different surfaces, and one aspect of the project is to precisely quantify the localization of these signals, in a way that depends only on the curvature of the underlying object. As another example, operators that average signals over curved surfaces arise in a variety of contexts, including three-dimensional geometric optics. Whereas averages over translates of a fixed plane are inherently unstable in directions perpendicular to the plane, averages over curved surfaces smooth out the original signal in all directions. The project will precisely quantify the degree of smoothing for surfaces that may have some curved parts and some flat parts. Finally, in recent years, there has been an explosion of interest in inverse problems wherein an attempt is made to characterize extreme cases consisting of mathematical signals for which the decay and smoothing effects described above are very weak; part of the project is to study some basic questions in this direction. These inverse problems are connected with potential applications in engineering, physics, and medical imaging. These scientific endeavors are inextricably linked with the investigator's efforts to help train the next generation of mathematicians. This workforce development encompasses three main directions: advising Ph.D. students in mathematics, creating summer research opportunities for undergraduate students, and organizing a series of symposia that foster interactions among mathematicians at all levels. During the past five decades, an important theme in harmonic analysis has been problems wherein the curvature of some underlying manifold causes operators to behave better than expected. The main part of this project considers Fourier restriction and averaging operators associated to manifolds with varying curvature. Curvature causes these operators to behave better than would be predicted by simply counting the dimension of the manifold, and the chief goal of the activity is to prove uniform bounds for these operators by equipping the manifold with a measure that gives small weight to regions where the curvature is small. These curvature-independent bounds are essentially the strongest possible for the operators considered. Moreover, these results would precisely quantify the role of curvature in the associated operators. The investigator will also work toward a characterization of functions that saturate the Lebesgue space inequalities for certain operators of this type. As an integral part of this project, the investigator will work to train junior mathematicians by serving as a dissertation advisor, by mentoring undergraduate researchers, and by organizing symposia that will include mathematicians at all levels.