CAREER: Structure and Interpolation in Number Theory and Beyond

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

The principal investigator (PI) will investigate connections between seemingly disparate data arising in number theory, a field of mathematics with deep ties to many areas in sciences and engineering. In particular, the PI will study structures associated to the whole numbers, fractions, and related sets called "number fields." The PI aims to elucidate how these structures vary across certain infinite collections of number fields, and furthermore, how their behavior explains currently mysterious phenomena in number theory, geometry, and beyond. In the course of the project the PI will organize workshops to educate graduate students about recent research developments and to promote diverse collaborations. To engage undergraduate students in topics motivating her research, the PI will develop an innovative, interactive course integrating approaches from the arts. Partly through a collaboration with museum curators, the PI will also organize public exhibits for the broader community. The research in this project focuses on L-functions, automorphic forms, and p-adic methods as tools to understand particular structures and how they vary in families. Anticipated consequences include progress toward instances of the Greenberg-Iwasawa main conjectures (connecting the structures of Galois groups with p-adic L-functions) and the Bloch-Kato conjectures (equating ranks of certain groups with the order of vanishing of associated L-functions at the central point, in analogy with the Birch and Swinnerton-Dyer conjecture). Key components, which are interconnected so that progress on one advances the others, include investigating the interplay between p-adic and archimedean properties of L-functions and automorphic forms, studying p-adic aspects of unitary Shimura varieties, and bridging different approaches to p-adic automorphic forms to gain insight into the behavior of associated data whose significance extends into number theory, homotopy theory, and arithmetic 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.