Class Groups, Character Sums, and Oscillatory Integrals

Duke University, Durham NC

Investigators

Abstract

This project will investigate central questions and conjectures in the mathematical fields of number theory and harmonic analysis. The research will elucidate properties of fundamental mathematical objects, and will develop new techniques that can be applied in the future by other mathematicians. In addition, the principal investigator, Dr. Lillian Pierce, will enhance the transmission of scientific knowledge and broaden participation in STEM research and learning. Dr. Pierce will fund and mentor trainees in the process of performing original research and obtaining PhD’s in mathematics. This award will support two conferences that aid the full participation of women and non-binary students in US-based math PhD programs, a summer school in harmonic analysis, and the innovative new journal “Essential Number Theory.” Dr. Pierce’s investigations will focus on open questions in analytic number theory, as well as projects in analysis that are motivated by an arithmetic viewpoint. These questions center on three themes. The first project will investigate a well-known conjecture on class groups of number fields, using methods involving character sums, sieves, and L-functions. The second project will investigate short mixed character sums, with applications to understanding L-functions and improving sieve methods for counting integral solutions to Diophantine equations. The third project will investigate square functions, convergence questions in partial differential equations, and maximally modulated oscillatory integral operators. 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.