For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the
University of Montreal. In November 2013, Maynard gave a different
proof of
Yitang Zhang's theorem that there are bounded gaps between
primes, and resolved a longstanding
conjecture by showing that for any m there are infinitely many intervals of bounded length containing m prime numbers. This work can be seen as progress on the Hardy–Littlewood m-tuples conjecture as it establishes that "a positive proportion of admissible m-tuples satisfy the prime m-tuples conjecture for every m." Maynard's approach yielded the
upper bound, with p_n denoting the n-th prime number, :\liminf_{n\to\infty}\left(p_{n+1}-p_n\right)\leq 600, which improved significantly upon the best existing bounds due to the
Polymath8 project. (In other words, he showed that there are infinitely many prime gaps with size of at most 600.) Subsequently, Polymath8b was created, whose collaborative efforts have reduced the gap size to 246, according to an announcement on 14 April 2014 by the
Polymath project wiki. In 2014, he was awarded the
SASTRA Ramanujan Prize. In 2015, he was awarded a
Whitehead Prize and in 2016 an
EMS Prize. In 2016, he showed that, for any given decimal digit, there are infinitely many prime numbers that do not have that digit in their decimal expansion. In 2019, together with
Dimitris Koukoulopoulos, he proved the
Duffin–Schaeffer conjecture. In 2020, in joint work with
Thomas Bloom, he improved the best-known bound for
square-difference-free sets, showing that a set A \subset [N] with no square difference has size at most \frac{N}{(\log N)^{c\log \log\log N}} for some c > 0. Maynard was awarded the Fields Medal 2022 for "contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in
Diophantine approximation". Maynard was elected a
Fellow of the Royal Society (FRS) in 2023. ==Personal life==