Overview
My research is in algebraic number theory, specifically the explicit construction of units in number fields and points on abelian varieties. There are many classical conjectures regarding the relationships between these elements and special values of L-functions, such as the conjectures of Stark, Birch-Swinnerton-Dyer, and Beilinson. In my research I have made progress on these conjectures as well as stated and studied various generalizations and refinements that go beyond the world of L-functions. Much of my work uses the theory of modular forms and their associated Galois representations in order to shed light on these problems.
Current Appointments & Affiliations
James B. Duke Distinguished Professor of Mathematics
·
2025 - Present
Mathematics,
Trinity College of Arts & Sciences
Professor of Mathematics
·
2018 - Present
Mathematics,
Trinity College of Arts & Sciences
Recent Publications
On the equality of three formulas for Brumer–Stark units
Journal Article Journal of the London Mathematical Society · October 1, 2025 We prove the equality of three conjectural formulas for Brumer–Stark units. The first formula has essentially been proven, so this paper also verifies the validity of the other two formulas. ... Full text CiteBRUMER–STARK UNITS AND EXPLICIT CLASS FIELD THEORY
Journal Article Duke Mathematical Journal · January 1, 2024 Let F be a totally real field of degree n, and let p be an odd prime. We prove the p-part of the integral Gross–Stark conjecture for the Brumer–Stark p-units living in CM abelian extensions of F . In previous work, the first author showed that such a resul ... Full text CiteOn the Brumer-Stark conjecture
Journal Article Annals of Mathematics · January 1, 2023 Let H=F be a finite abelian extension of number fields with F totally real and H a CM field. Let S and T be disjoint finite sets of places of F satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element ΦH/F< ... Full text Open Access CiteRecent Grants
RTG: Linked via L-functions: training versatile researchers across number theory
Inst. Training Prgm or CMECo-Principal Investigator · Awarded by National Science Foundation · 2023 - 2028The Brumer-Stark Conjecture and its Refinements
ResearchPrincipal Investigator · Awarded by National Science Foundation · 2022 - 2027Beyond L-functions: the Eisenstein Cocycle and Hilbert's 12th Problem
ResearchPrincipal Investigator · Awarded by National Science Foundation · 2019 - 2022View All Grants
Education, Training & Certifications
University of California, Berkeley ·
2004
Ph.D.
Harvard University ·
1999
A.B.