# James Maynard (mathematician)

James Maynard
Maynard in 2013
Born10 June 1987 (age 35)
Chelmsford, Essex, England
Alma mater
Known forWork on prime gaps
Awards
Scientific career
FieldsMathematics
Institutions
ThesisTopics in analytic number theory (2013)

James Alexander Maynard (born 10 June 1987) is an English mathematician working in analytic number theory and in particular the theory of prime numbers.[1] In 2017, he was appointed Research Professor at Oxford.[2] Maynard is a fellow[3] of St John's College, Oxford. He was awarded the Fields Medal in 2022.[4]

## Biography

Maynard attended King Edward VI Grammar School, Chelmsford in Chelmsford, England. After completing his bachelor's and master's degrees at Queens' College, University of Cambridge in 2009, Maynard obtained his D.Phil. from University of Oxford at Balliol College in 2013 under the supervision of Roger Heath-Brown.[5][1] He then became a Fellow by Examination at Magdalen College, Oxford.[6]

For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the University of Montreal.[7]

In November 2013, Maynard gave a different proof of Yitang Zhang's theorem[8] that there are bounded gaps between primes, and resolved a longstanding conjecture by showing that for any ${\displaystyle m}$ there are infinitely many intervals of bounded length containing ${\displaystyle m}$ prime numbers.[9] This work can be seen as progress on the Hardy–Littlewood ${\displaystyle m}$-tuples conjecture as it establishes that "a positive proportion of admissible ${\displaystyle m}$-tuples satisfy the prime ${\displaystyle m}$-tuples conjecture for every ${\displaystyle m}$."[10] Maynard's approach yielded the upper bound, with ${\displaystyle p_{n}}$ denoting the ${\displaystyle n}$'th prime number,

${\displaystyle \liminf _{n\to \infty }\left(p_{n+1}-p_{n}\right)\leq 600,}$

which improved significantly upon the best existing bounds due to the Polymath8 project.[11] (In other words, he showed that there are infinitely many prime gaps with size of at most 600.) Subsequently, Polymath8b was created,[12] whose collaborative efforts have reduced the gap size to 246, according to an announcement on 14 April 2014 by the Polymath project wiki.[11] Further, assuming the Elliott–Halberstam conjecture and, separately, its generalised form, the Polymath project wiki states that the gap size has been reduced to 12 and 6, respectively.[11]

In August 2014, Maynard (independently of Ford, Green, Konyagin and Tao) resolved a longstanding conjecture of Erdős on large gaps between primes, and received the largest Erdős prize (\$10,000) ever offered.[13][14]

In 2014, he was awarded the SASTRA Ramanujan Prize.[1][15] In 2015, he was awarded a Whitehead Prize[16] and in 2016 an EMS Prize.[17]

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.[18][19]

In 2019, together with Dimitris Koukoulopoulos, he proved the Duffin–Schaeffer conjecture.[20][21]

In 2020, in joint work with Thomas Bloom, he improved the best-known bound for square-difference-free sets, showing that a set ${\displaystyle A\subset [N]}$ with no square difference has size at most ${\displaystyle {\frac {N}{(\log N)^{c\log \log \log N}}}}$ for some ${\displaystyle c>0}$.[22][23]

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".[24]

## Personal life

Maynard was born on 10 June 1987 in Chelmsford, England.[1] His partner is Eleanor Grant, a medical doctor. They have a child.[4]

## References

1. ^ a b c d Alladi, Krishnaswami. "James Maynard to Receive 2014 SASTRA Ramanujan Prize" (PDF). qseries.org. Archived (PDF) from the original on 1 February 2017. Retrieved 13 April 2017.
2. ^ "James Maynard appointed Research Professor and receives a Wolfson Merit Award from the Royal Society". 4 April 2018. Archived from the original on 4 April 2018. Retrieved 4 April 2018.
3. ^ "Professor James Maynard, St John's College, Oxford". Archived from the original on 22 April 2022. Retrieved 11 June 2022.
4. ^ a b Klarreich, Erica (June 2022). "A Solver of the Hardest Easy Problems About Prime Numbers". Quanta Magazine. Archived from the original on 5 July 2022. Retrieved 5 July 2022.
5. ^
6. ^ "James Maynard: Prime Numbers". Archived from the original on 16 April 2021. Retrieved 11 June 2022.
7. ^ "Dr James Maynard". Magdalen College, Oxford. Archived from the original on 20 May 2018. Retrieved 17 April 2014.
8. ^ Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. Princeton University and the Institute for Advanced Study. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. Archived from the original on 22 January 2014. Retrieved 16 August 2013.
9. ^ Klarreich, Erica (19 November 2013). "Together and Alone, Closing the Prime Gap". Quanta Magazine. Archived from the original on 5 December 2019. Retrieved 5 December 2019.
10. ^ Maynard, James (20 November 2013). "Small Gaps Between Primes". arXiv:1311.4600 [math.NT].
11. ^ a b c "Bounded gaps between primes". Polymath Project. Archived from the original on 28 February 2020. Retrieved 21 July 2013.
12. ^ Tao, Terence (19 November 2013). "Polymath8b: Bounded intervals with many primes, after Maynard". Archived from the original on 8 May 2021. Retrieved 17 April 2014.
13. ^ Klarreich, Erica (10 December 2014). "Prime Gap Grows After Decades-Long Lull". Quanta Magazine. Archived from the original on 15 July 2017. Retrieved 10 December 2014.
14. ^ Maynard, James (21 August 2014). "Large gaps between primes". arXiv:1408.5110 [math.NT].
15. ^ Alladi, Krishnaswami (December 2014), "Maynard Awarded 2014 SASTRA Ramanujan Prize" (PDF), Mathematics People, Notices of the AMS, 61 (11): 1361, ISSN 1088-9477, archived (PDF) from the original on 30 November 2020, retrieved 28 April 2021
16. ^ "2015 Whitehead Prize". Clay Mathematics Institute. 8 July 2015. Archived from the original on 20 August 2019. Retrieved 6 July 2022.
17. ^ "History of prizes awarded at European Congresses of Mathematics". European Mathematical Society. Archived from the original on 9 February 2015. Retrieved 6 July 2022.
18. ^ Grechuk, Bogdan (2021). Landscape of 21st Century Mathematics: Selected Advances, 2001–2020. Springer Nature. p. 14. ISBN 978-3-030-80627-9. Archived from the original on 7 July 2022. Retrieved 6 July 2022.
19. ^ Maynard, J.: Invent. math. (2019) 217: 127. https://doi.org/10.1007/s00222-019-00865-6 Archived 7 July 2022 at the Wayback Machine
20. ^ Sloman, Leila (16 September 2019). "New Proof Solves 80-Year-Old Irrational Number Problem". Scientific American. Archived from the original on 24 May 2022. Retrieved 6 July 2022.
21. ^ Koukoulopoulos, Dimitris; Maynard, James (1 July 2020). "On the Duffin-Schaeffer conjecture". Annals of Mathematics. 192 (1). arXiv:1907.04593. doi:10.4007/annals.2020.192.1.5. ISSN 0003-486X. S2CID 195874052. Archived from the original on 7 July 2022. Retrieved 6 July 2022.
22. ^ Bloom, T.; Maynard, J. (2020). "A new upper bound for sets with no square differences". arXiv:2011.13266 [math.NT].
23. ^ Doyle, John R.; Rice, Alex (5 September 2021). "Multivariate Polynomial Values in Difference Sets". ArXiv.org. p. 3. arXiv:2006.15400. Archived from the original on 17 August 2020. Retrieved 6 July 2022 – via Wayback Machine.
24. ^ "The Fields Medal 2022. James Maynard" (PDF). International Mathematical Union. Archived (PDF) from the original on 5 July 2022. Retrieved 6 July 2022.