Counteracting flatness with affine measures and related problems in harmonic analysis

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The field of harmonic analysis grew out of an effort to study and encode natural signals by breaking them into their fundamental parts. In this project, the principal investigator will endeavor to understand, in a quantitative way, the effects of curvature on such decompositions. For example, in medical imaging, X-rays are passed through a body, and one reconstructs the density function of the body by measuring how much of the X-radiation is absorbed or scattered. Mathematically, this process is represented by an operator, known as the X-ray transform, which averages functions along straight-line paths. The principal investigator is working on a project that seeks to understand what happens when averages are taken along curved paths. It is known that sufficient curvature of the paths leads to greater stability of the output, and the project aims to quantify the stabilizing effect in an intermediate case where the paths are curved, but may have flat regions. As another example, the Fourier transform expresses a natural signal as a superposition of constant velocity waves. If the velocities lie on a plane, then they are all aligned, and, like ocean waves formed by a steady breeze, the signal does not decay. But if the wave velocities lie on a curved surface, such as the surface of a ball, then there is some decay. Precisely measuring this decay is an important question in harmonic analysis, the answer to which is unknown. The principal investigator's research lies at the interface between these situations, when the velocities lie on a surface that is nearly planar in some regions and curved in others, and she seeks to precisely quantify the rate of decay. Such questions have potential implications to partial differential equations that arise in the study of quantum mechanics. The principal investigator will study curvature-related problems arising in Euclidean harmonic analysis, as well as some applications to dispersive partial differential equations. This work will encompass three directions. One is to prove new, curvature-independent bounds for the restriction of the Fourier transform to manifolds whose curvature vanishes along some nonempty set; another is to prove analogous results for averaging operators; finally, she will study extremizer problems and concentration compactness techniques, some having applications to partial differential equations. As part of this project, the principal investigator will organize conferences to facilitate the dissemination of mathematical knowledge and will make a dedicated effort to improve graduate training in harmonic analysis and to increase the participation of underrepresented groups, especially women.