Date: February 4, 2026

Time: 5:00–6:00 pm

Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA

Speaker: Barry Mazur, Harvard University

Title: About the Birch and Swinnerton–Dyer Conjecture

Abstract:

In the 1950s Bryan Birch and Peter Swinnerton–Dyer made computations that suggested a striking connection between a basic global invariant of an elliptic curve E over the field of rational numbers (namely, the rank of its group of rational points) and certain asymptotics of its local arithmetic invariants (i.e., the number of its rational points over finite fields).

This initial observation has evolved into their conjecture. My lecture will be an introduction to the general ideas behind its ever-expanding development.

Read more about the Birch and Swinnerton–Dyer Conjecture at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Barry Mazur joined the Harvard University faculty in 1959 as a Junior Fellow in the Society of Fellows and advanced through the ranks to become the Gerhard Gade University Professor of Mathematics, a position he has held since 1998. During his tenure at Harvard, he has mentored 60 doctoral students and served as a pivotal figure in bridging topology and number theory, notably through his classification of the possible torsion subgroups of elliptic curves over the rational numbers (Mazur’s torsion theorem), which identifies exactly 15 possible finite groups. This theorem, detailed in his 1977 paper “Modular curves and the Eisenstein ideal,” provided crucial insights into the Taniyama-Shimura conjecture and laid groundwork for Andrew Wiles’s 1994 proof of Fermat’s Last Theorem.

His broader research includes seminal works on étale homotopy theory (co-authored with Michael Artin in 1969), the arithmetic moduli of elliptic curves (with Nicholas M. Katz in 1985), and the Iwasawa main conjecture (proved with Andrew Wiles in 1984), as well as advancements in p-adic L-functions and the formulation of the Fontaine-Mazur conjecture on Galois representations. Mazur’s influence extends to public communication of mathematics; he has authored books like Imagining Numbers (2003), exploring historical perspectives on complex numbers.

Among his numerous honors, Mazur received the Cole Prize in Number Theory from the American Mathematical Society in 1982, the Chauvenet Prize in 1994 for expository writing, the Leroy P. Steele Prize for Lifetime Achievement in 2000, and election to the National Academy of Sciences in 1982. In 2011 (presented in 2013), he was awarded the National Medal of Science by President Barack Obama for his pioneering work in these fields.Most recently, in 2022, he received the Chern Medal from the International Mathematical Union, recognizing his profound discoveries and mentorship.

Millennium Prize Problems Lecture Series