View source for Ben Green (mathematician)

{{Short description|British mathematician (born 1977)}}
{{other people|Ben Green}}
{{BLP sources|date=January 2022}}
{{EngvarB|date=June 2017}}
{{Use dmy dates|date=November 2022}}
{{Infobox scientist
| name              = Ben Green
| birth_name        = Ben Joseph Green
| image             = Ben Green.jpg
| image_size        = 
| caption           = Green in 2010
| birth_date        = {{birth date and age|df=yes|1977|2|27}}
| birth_place       = [[Bristol]], England
| death_date        = 
| death_place       = 
| fields            = Mathematics
| workplaces        = [[University of Bristol]]<br>[[University of Cambridge]]<br>[[University of Oxford]]<br>[[Princeton University]]<br>[[University of British Columbia]]<br>[[Massachusetts Institute of Technology]]
| alma_mater        = [[Trinity College, Cambridge]]<br>([[Bachelor of Arts|BA]], [[Part III of the Mathematical Tripos|MMath]], PhD)
| doctoral_advisor  = [[Timothy Gowers]]
| thesis_title      = Topics in Arithmetic Combinatorics
| thesis_year       = 2003
| doctoral_students = {{plainlist|1=
*[[Adam Harper]]
*[[Vicky Neale]]
*[[Lilian Matthiesen]]
}}
| known_for         = 
| influences        = 
| influenced        = 
| awards            = {{nowrap|[[Clay Research Award]] (2004)<br>[[Salem Prize]] (2005)<br>[[Whitehead Prize]] (2005)<br>[[SASTRA Ramanujan Prize]] (2007)<br> [[European Mathematical Society|EMS Prize]] (2008)<br> [[Fellow of the Royal Society]] (2010)<br>[[Sylvester Medal]] (2014)<br> [[Senior Whitehead Prize]] (2019)}}
| religion          = 
| signature         = 
| footnotes         = 
}}
'''Ben Joseph Green''' [[Fellow of the Royal Society|FRS]] (born 27 February 1977) is a British mathematician, specialising in [[combinatorics]] and [[number theory]]. He is the [[Waynflete Professor of Pure Mathematics]] at the [[University of Oxford]].

== Early life and education ==
Ben Green was born on 27 February 1977 in [[Bristol]], England. He studied at local schools in Bristol, [[Bishop Road Primary School]] and [[Fairfield Grammar School]], competing in the [[International Mathematical Olympiad]] in 1994 and 1995.<ref>{{IMO results |id=1137}}</ref> He entered [[Trinity College, Cambridge]] in 1995 and completed his [[Bachelor of Arts|BA]] in mathematics in 1998, winning the [[Senior Wrangler (University of Cambridge)|Senior Wrangler]] title. He stayed on for [[Part III of the Mathematical Tripos|Part III]] and earned his doctorate under the supervision of [[Timothy Gowers]], with a thesis entitled ''Topics in arithmetic combinatorics'' (2003). During his PhD he spent a year as a visiting student at [[Princeton University]]. He was a research Fellow at Trinity College, Cambridge between 2001 and 2005, before becoming a Professor of Mathematics at the [[University of Bristol]] from January 2005 to September 2006 and then the first [[Herchel Smith Professor of Pure Mathematics]] at the [[University of Cambridge]] from September 2006 to August 2013. He became the [[Waynflete Professorship|Waynflete Professor]] of Pure Mathematics at the [[University of Oxford]] on 1 August 2013. He was also a Research Fellow of the [[Clay Mathematics Institute]] and held various positions at institutes such as [[Princeton University]], [[University of British Columbia]], and [[Massachusetts Institute of Technology]].

== Mathematics ==

The majority of Green's research is in the fields of [[analytic number theory]] and [[Additive Combinatorics|additive combinatorics]], but he also has results in [[harmonic analysis]] and in [[group theory]]. His best known theorem, proved jointly with his frequent collaborator [[Terence Tao]], states that there exist arbitrarily long [[arithmetic progression]]s in the [[prime number]]s: this is now known as the [[Green–Tao theorem]].<ref>{{Cite journal|last1=Green|first1=Ben|last2=Tao|first2=Terence|date=2008|title=The Primes Contain Arbitrarily Long Arithmetic Progressions|jstor=40345354|journal=Annals of Mathematics|volume=167|issue=2|pages=481–547|arxiv=math/0404188|doi=10.4007/annals.2008.167.481|s2cid=1883951}}</ref>

Amongst Green's early results in additive combinatorics are an improvement of a result of [[Jean Bourgain]] of the size of arithmetic progressions in [[sumset]]s,<ref>{{Cite journal|last=Green|first=B.|date=1 August 2002|title=Arithmetic progressions in sumsets|journal=Geometric & Functional Analysis|language=en|volume=12|issue=3|pages=584–597|doi=10.1007/s00039-002-8258-4|s2cid=120755105|issn=1016-443X}}</ref> as well as a proof of the [[Cameron–Erdős conjecture]] on sum-free sets of [[natural number]]s.<ref>{{Cite journal|last=GREEN|first=BEN|date=19 October 2004|journal=Bulletin of the London Mathematical Society|language=en|volume=36|issue=6|pages=769–778|doi=10.1112/s0024609304003650|issn=0024-6093|arxiv=math/0304058|title=The Cameron–Erdos Conjecture|s2cid=119615076}}</ref> He also proved an arithmetic regularity lemma<ref>{{Cite journal|last=Green|first=B.|date=1 April 2005|title=A Szemerédi-type regularity lemma in abelian groups, with applications|journal=Geometric & Functional Analysis|language=en|volume=15|issue=2|pages=340–376|doi=10.1007/s00039-005-0509-8|issn=1016-443X|arxiv=math/0310476|s2cid=17451915}}</ref> for functions defined on the first <math>N</math> natural numbers, somewhat analogous to the [[Szemerédi regularity lemma]] for graphs.

From 2004–2010, in joint work with [[Terence Tao]] and [[Tamar Ziegler]], he developed so-called [[higher order Fourier analysis]]. This theory relates [[Gowers norm]]s with objects known as [[nilsequences]]. The theory derives its name from these nilsequences, which play an analogous role to the role that [[Character group|characters]] play in classical [[Fourier analysis]]. Green and Tao used higher order Fourier analysis to present a new method for counting the number of solutions to simultaneous equations in certain sets of integers, including in the primes.<ref>{{Cite journal|last1=Green|first1=Benjamin|last2=Tao|first2=Terence|date=2010|title=Linear equations in primes|jstor=20752252|journal=Annals of Mathematics|volume=171|issue=3|pages=1753–1850|doi=10.4007/annals.2010.171.1753|doi-access=free|arxiv=math/0606088}}</ref> This generalises the classical approach using [[Hardy–Littlewood circle method]]. Many aspects of this theory, including the quantitative aspects of the inverse theorem for the Gowers norms,<ref>{{Cite journal|last1=Green|first1=Ben|last2=Tao|first2=Terence|last3=Ziegler|first3=Tamar|date=2012|title=An inverse theorem for the Gowers U s+1 [N]-norm|jstor=23350588|journal=Annals of Mathematics|volume=176|issue=2|pages=1231–1372|doi=10.4007/annals.2012.176.2.11|doi-access=free|arxiv=1006.0205}}</ref> are still the subject of ongoing research.

Green has also collaborated with [[Emmanuel Breuillard]] on topics in group theory. In particular, jointly with [[Terence Tao]], they proved a structure theorem<ref>{{Cite journal|last1=Breuillard|first1=Emmanuel|last2=Green|first2=Ben|last3=Tao|first3=Terence|date=1 November 2012|title=The structure of approximate groups|journal=Publications Mathématiques de l'IHÉS|language=en|volume=116|issue=1|pages=115–221|doi=10.1007/s10240-012-0043-9|issn=0073-8301|arxiv=1110.5008|s2cid=119603959}}</ref> for [[approximate group]]s, generalising the [[Freiman's theorem|Freiman-Ruzsa]] theorem on sets of integers with small doubling. Green also has worked, jointly with [[Kevin Ford (mathematician)|Kevin Ford]] and [[Sean Eberhard]], on the theory of the [[symmetric group]], in particular on what proportion of its elements fix a set of size <math>k</math>.<ref>{{Cite journal|last1=Eberhard|first1=Sean|last2=Ford|first2=Kevin|last3=Green|first3=Ben|date=23 December 2015|title=Permutations Fixing a k-set|journal=International Mathematics Research Notices|language=en|volume=2016|issue=21|pages=6713–6731|doi=10.1093/imrn/rnv371|issn=1073-7928|arxiv=1507.04465|bibcode=2015arXiv150704465E|s2cid=15188628}}</ref>

Green and Tao also have a paper<ref>{{Cite journal|last1=Green|first1=Ben|last2=Tao|first2=Terence|date=1 September 2013|title=On Sets Defining Few Ordinary Lines|journal=Discrete & Computational Geometry|language=en|volume=50|issue=2|pages=409–468|doi=10.1007/s00454-013-9518-9|issn=0179-5376|arxiv=1208.4714|s2cid=15813230}}</ref> on algebraic [[Discrete geometry|combinatorial geometry]], resolving the Dirac-Motzkin conjecture (see [[Sylvester–Gallai theorem#The number of ordinary lines|Sylvester–Gallai theorem]]). In particular they prove that, given any collection of <math>n</math> points in the plane that are not all [[collinear]], if <math>n</math> is large enough then there must exist at least <math>n/2</math> lines in the plane containing exactly two of the points.

[[Kevin Ford (mathematician)|Kevin Ford]], Ben Green, [[Sergei Konyagin]], [[James Maynard (mathematician)|James Maynard]] and [[Terence Tao]], initially in two separate research groups and then in combination, improved the lower bound for the size of the longest gap between two consecutive primes of size at most <math>X</math>.<ref>{{cite arXiv|last1=Ford|first1=Kevin|last2=Green|first2=Ben|last3=Konyagin|first3=Sergei|last4=Maynard|first4=James|last5=Tao|first5=Terence|date=16 December 2014|title=Long gaps between primes|eprint=1412.5029|class=math.NT}}</ref> The form of the previously best-known bound, essentially due to [[Robert Alexander Rankin|Rankin]], had not been improved for 76 years.

More recently Green has considered questions in arithmetic [[Ramsey theory]]. Together with [[Tom Sanders (mathematician)|Tom Sanders]] he proved that, if a sufficiently large [[finite field]] of prime order is coloured with a fixed number of colours, then the field has elements <math>x,y</math> such that <math>x, y, x{+}y, xy</math> all have the same colour.<ref>{{Cite journal|last1=Green|first1=Ben|last2=Sanders|first2=Tom|date=1 March 2016|title=Monochromatic sums and products|journal=Discrete Analysis|language=en|volume=5202016|issue=1|doi=10.19086/da.613|issn=2397-3129|arxiv=1510.08733|s2cid=119140038}}</ref>

Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the [[polynomial method in combinatorics|polynomial method]] to bound the size of subsets of a finite [[vector space]] without solutions to [[linear equation]]s. He adapted these methods to prove, in function fields, a strong version of [[Furstenberg–Sárközy theorem|Sárközy's theorem]].<ref>{{Cite journal|last=Green|first=Ben|date=23 November 2016|title=Sárközy's Theorem in Function Fields|journal=The Quarterly Journal of Mathematics|language=en|volume=68|issue=1|pages=237–242|doi=10.1093/qmath/haw044|issn=0033-5606|arxiv=1605.07263|s2cid=119150134}}</ref>

== Awards and honours ==
Green has been a Fellow of the [[Royal Society]] since 2010,<ref>{{cite web|url=http://royalsociety.org/people/ben-green/|title=- Royal Society}}</ref> and a Fellow of the [[American Mathematical Society]] since 2012.<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society]. Retrieved 19 January 2013.</ref> Green was chosen by the [[German Mathematical Society]] to deliver a [[Gauss Lectureship]] in 2013. He has received several awards:

* 2004: [[Clay Research Award]]
* 2005: [[Salem Prize]]
* 2005: [[Whitehead Prize]]<ref>{{cite web|url=http://www.lms.ac.uk/content/list-lms-prize-winners#Whitehead_Prize|title=List of LMS prize winners – London Mathematical Society}}</ref>
* 2007: [[SASTRA Ramanujan Prize]]
* 2008: [[European Mathematical Society]] prize recipient
* 2014: [[Sylvester Medal]], awarded by the [[Royal Society]].
* 2019: [[Senior Whitehead Prize]] of the [[London Mathematical Society]]

== References ==
{{reflist|30em}}

== External links ==
*[http://people.maths.ox.ac.uk/greenbj/ Ben Green personal homepage at Oxford]
*[https://www.maths.ox.ac.uk/contact/details/greenbj Ben Green faculty page at Oxford] {{Webarchive|url=https://web.archive.org/web/20140426113144/https://www.maths.ox.ac.uk/contact/details/greenbj |date=26 April 2014 }}
*[http://www.trin.cam.ac.uk/index.php?pageid=176&conid=155 Ben Green Homepage at Trinity College, Cambridge]
*[http://www.claymath.org/research Clay Research Award 2004 announcement]
*{{MathGenealogy |id=76979 }}
*[https://arxiv.org/abs/math.NT/0404188 math.NT/0404188 – Preprint on arbitrarily long arithmetic progressions on primes]

{{FRS 2010}}
{{Recipients of SASTRA Ramanujan Prize winners|state=collapsed}}
{{Authority control}}

{{DEFAULTSORT:Green, Ben}}
[[Category:1977 births]]
[[Category:Living people]]
[[Category:20th-century English mathematicians]]
[[Category:21st-century English mathematicians]]
[[Category:Academics of the University of Bristol]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:Cambridge mathematicians]]
[[Category:Clay Research Award recipients]]
[[Category:Combinatorialists]]
[[Category:Fellows of Magdalen College, Oxford]]
[[Category:Fellows of the American Mathematical Society]]
[[Category:Fellows of the Royal Society]]
[[Category:Fellows of Trinity College, Cambridge]]
[[Category:International Mathematical Olympiad participants]]
[[Category:Number theorists]]
[[Category:Scientists from Bristol]]
[[Category:Recipients of the SASTRA Ramanujan Prize]]
[[Category:Senior Wranglers]]
[[Category:Simons Investigator]]
[[Category:Waynflete Professors of Pure Mathematics]]
[[Category:Whitehead Prize winners]]
[[Category:Professors of the University of Cambridge]]