Geometric Harmonic Analysis: Advances in Radon-like Transforms and Related Topics

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The mathematics of geometric averages known as Radon-like operators is of fundamental importance in a host of technological applications related to imaging and data analysis: CT, SPECT, and NMR, as well as RADAR and SONAR applications, all depend on a deep understanding of the Radon transform, and related ideas appear in optical-acoustic tomography, scattering theory, and even some motion-detection algorithms. Somewhat surprisingly, there are many basic theoretical problems in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. This project studies a family of questions in the area of geometric averages which, for example, correspond to quantifying the relationship between small changes in the imaged objects and the expected changes in measured data (which in practice would be processed computationally to recover an approximate picture of the original object). The theoretical challenge in a problem such as this is to precisely quantify the notion of change and to establish essentially exact relationships between the magnitude of input and output changes. Thanks to recent advances in the PI's work to understand these objects, the project is well-positioned to yield important results. Achieving the main goals of this project would lead to advances in a number of related areas of mathematics and may influence future imaging technologies. The project furthermore provides unique opportunities for the advanced mathematical training of both undergraduate and PhD students, who can transfer these skills to other areas of critical need once in the workforce. The PI studies topics in mathematical analysis related to the development of new geometric approaches to Radon-like transforms, oscillatory integrals, and Fourier restriction problems. This work includes various special cases of both sublevel set and oscillatory integral problems. Major special cases deserving mention include multiparameter sublevel set estimates, maximal curvature for Radon-like transforms of intermediate dimension, degenerate Radon transforms in low codimension, Fourier restriction and related generalized determinant functionals, and multilinear oscillatory integrals of convolution and related types. The PI's approach to these involves a variety of new tools developed within the last 5 years which incorporate techniques from Geometric Invariant Theory, geometric measure theory, decoupling theory, and other areas. Among these new tools is a recent result of the PI which provides an entirely new way to estimate norms of Radon-Brascamp-Lieb inequalities in terms of geometric quantities which can be understood as analogous to Lieb's formula for the Brascamp-Lieb constant. A major goal of this project is to understand the local geometric criteria which implicitly govern the finiteness of the nonlocal integrals appearing in the Radon-Brascamp-Lieb condition. The project has numerous potential applications to other problems of interest at the intersection of harmonic analysis, geometric measure theory, and incidence geometry. 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.