Topics in Analysis and Geometric Measure Theory

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

A basic principle in statistics and data science is that high-dimensional data can often be analyzed efficiently when it depends on only a few significant features. In analysis and geometric measure theory, an analogous question is whether an infinite set of points in space can be well-parameterized by a small number of variables—a property known as rectifiability. Determining whether a set is rectifiable is fundamental in analysis, and this project will develop new techniques to better understand rectifiability in non-Euclidean geometries relevant to physics and computer science. Another focus of the project is how to recover information from incomplete or noisy measurements—a challenge that arises in medical imaging, wireless communication, and the analysis of random systems. New forms of the uncertainty principle will be developed, with specific applications to these areas of science. Theoretical tools will be developed through a collaborative research environment that actively involves both undergraduate and graduate students. In doing so, the project will contribute to the advancement of mathematical understanding and the training of future scientists in disciplines that support developments in engineering, technology, and the natural sciences. A central question in geometric measure theory and harmonic analysis is whether the boundedness of a singular integral operator implies that the underlying measure must be rectifiable, or whether it can have purely fractal support. While this problem has been extensively studied in Euclidean spaces—where it connects to harmonic measure and free boundary regularity—much less is understood in non-Euclidean settings, such as the Heisenberg and parabolic groups. This project aims to extend the theory to these geometrically rich and analytically challenging contexts. A second major component addresses uncertainty principles and sampling theory. The focus is on characterizing the sets on which a function can be uniquely and stably reconstructed from partial information about its Fourier transform. These reconstruction results will be tailored for application to a range of problems, including mobile sampling, control theory for the wave and Schrödinger equations, and the long-term behavior of stationary Gaussian processes. 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.