CAREER: Analysis of Operators on Rough Sets

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Fractals and self-similar structures permeate the physical word, from statistical mechanics to material sciences. This project seeks to analyze how some fundamental physical objects interact with fractal structures. For instance, suppose we know that a field associated to a body in space (for instance its gravitational field) is well-behaved (say, has bounded magnitude), then what can we infer about the geometry of the body? Does the condition upon the field preclude the surface of the body from having a fractal type structure? A second topic considered concerns understanding signals whose frequencies have a self-similar or fractal behavior. More precisely, one is interested in the extent to which the signal can be reconstructed uniquely from a "small" set of values. The principal investigator (PI) will incorporate both graduate and undergraduate students into the research program, training them for future careers in STEM fields. The PI will organize annual undergraduate research symposia that will bring together undergraduate students interested in mathematical research across southeastern states to present their undergraduate research, build a network with others interested in pursuing research in the mathematical sciences, and provide information about research opportunities at the graduate level. This project concerns the analysis of operators when the geometry underlying the problem is rough or fractal in nature. The first question considered is: What can be deduced about a measure from the knowledge that a Calderon-Zygmund operator associated to it is bounded? Under these circumstances, can the measure have a fractal structure, or must its support be contained in (a countable number of) Lipschitz submanifolds of appropriate dimension? This basic question in analysis has found applications in the calculus of variations, the study of free boundaries, and the geometry of harmonic measure. Despite intense study over the last thirty years, the tools that serve as a bridge from the analytic condition on the field to the geometric structure of the measure are currently underdeveloped, something that this proposal aims to rectify. The second question is: How sparse must the support of the Fourier transform of a function be to ensure that the function can be uniquely reconstructed by its values on a "thick" subset? Although this question is a classical form of the uncertainty principle, it is not yet well understood, especially in several dimensions. 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.