James Maynard

fl. 1987, Chelmsford, England, United Kingdom

Arithmetic and number theoryFields Medal 2022

English number theorist who gave a simpler, more flexible proof of bounded gaps between primes (2013) and, with Dimitris Koukoulopoulos, resolved the Duffin–Schaeffer conjecture (2019); awarded the Fields Medal in 2022.

James Maynard was born in Chelmsford, England, and studied at Queens' College, Cambridge, before completing a DPhil at Balliol College, Oxford, in 2013 under Roger Heath-Brown. After postdoctoral positions in Montreal and visits to Berkeley and Princeton, he returned to Oxford as a research professor in 2017.

Shortly after his doctorate, Maynard developed a multidimensional sieve method that gave a far simpler and more flexible proof of bounded gaps between primes than Yitang Zhang's 2013 breakthrough, and moreover extended to show bounded gaps between clusters of $m$ primes for any $m$ ; a similar independent method was found by the Polymath8 collaborative project. In 2014 he resolved a related conjecture of Paul Erdős on the existence of arbitrarily large gaps between consecutive primes, and in 2016 proved there are infinitely many primes missing any chosen digit from their decimal expansion. With Dimitris Koukoulopoulos he gave, in 2019, a complete proof of the Duffin–Schaeffer conjecture in metric number theory, a decades-old problem about approximating irrational numbers by fractions with prescribed denominators.

Maynard received the Fields Medal in 2022 "for his spectacular contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation." He continues to work at the University of Oxford, where he is a fellow of St John's College.

Workplaces: University of Oxford

United Kingdom

Award-winning work

Fields Medal 2022Awarded for contributions to analytic number theory, notably on the distribution of prime numbers and Diophantine approximation. In 2013 Maynard introduced a new and more flexible sieve method — now called the Maynard–Tao sieve — that improved Zhang's bound on gaps between primes and established bounded gaps for tuples of primes of arbitrary length. With Dimitris Koukoulopoulos he proved the Duffin–Schaeffer conjecture (2019), a central statement on how well almost all real numbers can be approximated by rationals. His methods have reshaped modern sieve theory and driven a string of results on primes in restricted sets.

Contributions, linked to the library

Great problemTwin prime conjectureGave an independent, more flexible sieve proof of bounded gaps between primes (2013).TheoremBounded gaps between primes (Zhang, Maynard)Independent 2013 proof using a multidimensional sieve, extended to clusters of m primes.