CAREER: Research and Training at the Intersection of Number Theory and Analysis

Duke University, Durham NC

Investigators

Abstract

Prime numbers have been a natural object of study for thousands of years and now play a foundational role in modern encryption systems for digital communications. Despite their long study, many questions about the distribution of prime numbers remain unsolved. Primes are also intertwined with the study of class numbers, which appear in many number-theoretic settings. Class numbers have been studied for 200 years, but still remain largely mysterious, although precise conjectures have been developed. Seemingly far away on the mathematical spectrum, Radon transforms quantify the distribution of the "mass" of functions along lower-dimensional surfaces; they are a critical part of the theory underlying Computed Tomography medical imaging. The study of Radon transforms is a central area in harmonic analysis with far-reaching connections both to the Carleson operator, which was instrumental in answering a historic question on Fourier series, and to the new world of discrete arithmetic operators, which blends harmonic analysis with number theory. This project, which is at the intersection of number theory and harmonic analysis, explores and connects all of these themes. During the course of the work, the project will contribute to the mathematical community through training postdocs, a graduate summer school, and mathematical outreach activities for children. The major aims of this research center on five projects at the intersection of number theory and harmonic analysis. First, new bounds relating to the divisibility of class numbers of number fields of arbitrary degree will be obtained. Second, new bounds will be obtained for short character sums, which historically provided an important subconvexity result for L-functions, and now in a multi-dimensional setting have the potential to impact problems involving counting integral solutions to Diophantine equations. Third, in the realm of Diophantine equations, new variations on the circle method, and closely related questions on multi-dimensional oscillatory integrals, will be developed. Fourth, new results for discrete operators will be proved by adapting number-theoretic methods to the setting of harmonic analysis. Finally, a systematic investigation of Carleson operators with polynomial phases and Radon-type behavior will be carried out.