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 Duke 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 Scholarly Works
Ranks of matrices of logarithms of algebraic numbers II: The matrix coefficient conjecture
Journal article Advances in Mathematics · March 1, 2026 Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p -adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant, then after a ratio ... Full text CiteOn Constructing Extensions of Residually Isomorphic Characters
Conference Springer Proceedings in Mathematics and Statistics · January 1, 2026 This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet’s Lemma for GL2 in the residually indistinguishable case. We suppose we are given a Galois representation taking values in the total rin ... Full text CiteOn 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 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 - 2029The 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
University of California, Berkeley ·
2004
Ph.D.
Harvard University ·
1999
A.B.