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Ben Green (mathematician)
Ben Green (mathematician)
Ben Green (mathematician)
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Ben Joseph Green 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.

Key Information

Early life and education

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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.[1] He entered Trinity College, Cambridge in 1995 and completed his BA in mathematics in 1998, winning the Senior Wrangler title. He stayed on for 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 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

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The majority of Green's research is in the fields of analytic number theory and 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 progressions in the prime numbers: this is now known as the Green–Tao theorem.[2]

Amongst Green's early results in additive combinatorics are an improvement of a result of Jean Bourgain of the size of arithmetic progressions in sumsets,[3] as well as a proof of the Cameron–Erdős conjecture on sum-free sets of natural numbers.[4] He also proved an arithmetic regularity lemma[5] for functions defined on the first 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 norms with objects known as nilsequences. The theory derives its name from these nilsequences, which play an analogous role to the role that 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.[6] 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,[7] 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[8] for approximate groups, generalising the Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has worked, jointly with 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 .[9]

Green and Tao also have a paper[10] on algebraic combinatorial geometry, resolving the Dirac-Motzkin conjecture (see Sylvester–Gallai theorem). In particular they prove that, given any collection of points in the plane that are not all collinear, if is large enough then there must exist at least lines in the plane containing exactly two of the points.

Kevin Ford, Ben Green, Sergei Konyagin, 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 .[11] The form of the previously best-known bound, essentially due to Rankin, had not been improved for 76 years.

More recently Green has considered questions in arithmetic Ramsey theory. Together with 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 such that all have the same colour.[12]

Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the polynomial method to bound the size of subsets of a finite vector space without solutions to linear equations. He adapted these methods to prove, in function fields, a strong version of Sárközy's theorem.[13]

Awards and honours

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Green has been a Fellow of the Royal Society since 2010,[14] and a Fellow of the American Mathematical Society since 2012.[15] Green was chosen by the German Mathematical Society to deliver a Gauss Lectureship in 2013. He has received several awards:

References

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Ben Green (mathematician)

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Ben Green (born 27 February 1977) is a British mathematician specializing in additive combinatorics, number theory, and harmonic analysis. He serves as the Waynflete Professor of Pure Mathematics at the University of Oxford and as a Fellow of Magdalen College. Green is best known for his groundbreaking work on the distribution of prime numbers, particularly the Green–Tao theorem (proved jointly with Terence Tao in 2004), which demonstrates that the sequence of prime numbers contains arithmetic progressions of any finite length. Green grew up in Bristol, England, attending Bishop Road Primary School and Fairfield Grammar School before pursuing higher education at Trinity College, Cambridge. There, he earned a BA in Mathematics with first-class honors in 1998, a Certificate of Advanced Study in Mathematics with distinction in 1999, and a PhD in 2002 under the supervision of Timothy Gowers. His early career included a Junior Research Fellowship at Trinity College from 2001 to 2005, a professorship at the University of Bristol from 2005 to 2006, and the Herchel Smith Professorship of Pure Mathematics at the University of Cambridge from 2006 to 2013, where he was appointed at the age of 29. In 2013, he joined Oxford in his current role. Green has also held visiting positions at institutions such as MIT, Harvard, and the University of British Columbia. Green's research bridges multiple fields, including , , and , often applying analytic techniques to combinatorial problems. Beyond the , his notable contributions include advances in Freiman's theorem on arithmetic progressions in sumsets and generalizations of the Hardy–Littlewood method for primes in arithmetic progressions. He has authored over 100 publications and delivered more than 200 invited lectures worldwide. His achievements have been recognized with prestigious awards, including the Smith's Prize from in 2001, the in 2004 for his work on primes, the Salem Prize and Whitehead Prize in 2005, the Ostrowski Prize (jointly with ) in 2005, the SASTRA Ramanujan Prize in 2007, the European Mathematical Society Prize in 2008, the Sylvester Medal of the Royal Society in 2014, the Senior Whitehead Prize from Mathematical Society in 2019, and the I. Martin Isaacs Prize for Excellence in Mathematical Writing in 2024. Additionally, he was elected a in 2010.

Personal background

Early life

Ben Green was born on 27 February 1977 in , . He attended Bishop Road Primary School from 1982 to 1988 and Fairfield Grammar School from 1988 to 1995. From a young age, Green displayed a strong interest in numbers, influenced by his family environment. His mother recalled that by age three, he was demanding "sums" to solve, reflecting an early fascination with numerical patterns such as car registration plates and distances on road signs. His father supported this curiosity by accompanying him on visits and sharing a mutual interest in folk and music, though no direct familial mathematical background is noted. Green independently learned subtraction around age five, a method his primary school later described as "the wrong way," but he did not encounter "real" until age thirteen. During his school years in , Green participated actively in mathematical competitions, supported by dedicated teachers including Julie Kirby and Frank Burke. At his relatively low-ranked —ranked around 2000th in the UK—he achieved the highest mark in a national competition for under-13s, highlighting his emerging talent. This success led to his involvement in the UK Olympiad movement, where he was mentored by figures such as Tony Gardiner, Christopher Bradley, and David Monk. Green represented the at the in both 1994 and 1995, earning silver medals each year for his performances. These experiences not only honed his skills but also fostered lasting friendships within the international mathematical community. Following these achievements, he transitioned to university studies at , in 1995.

Education

Green began his undergraduate studies in at , in 1995, earning a (BA) degree in 1998 with first-class honors in all three years and being named , the top undergraduate in his year. He continued at with the Master's program, completing the Certificate of Advanced Study in (Part III of the ), equivalent to an MMath, in 1999 with distinction. His early successes in mathematical olympiads had paved the way for his admission to . Green pursued his PhD at the from 1999 to 2002 under the supervision of , with a thesis titled Topics in Arithmetic Combinatorics focusing on arithmetic progressions in sets of positive density. During his doctoral studies, he spent a visiting graduate student period at from September 2000 to May 2001.

Professional career

Early appointments

Following the completion of his PhD at the in , Ben Green held a research fellowship under Title A at , from October 2001 to September 2005. During this period, he also undertook several postdoctoral and visiting positions, including an EU role at the Institute of Mathematics in from to April 2003, and a PIMS in at the from September 2003 to August 2004. In January 2005, Green was appointed Professor of Pure Mathematics at the , a position he held until September 2006. Concurrently, he began a two-year term as a Clay Research Fellow at the in July 2005. He also served as Visiting Professor at the from September 2005 to December 2006. Green's association with the University of Cambridge continued into his mid-career roles, where he was appointed Herchel Smith Professor of Pure Mathematics from September 2006 to July 2013. During this time, he held additional visiting positions, including a membership at the Institute for Advanced Study in Princeton from September to December 2007, and a Radcliffe Fellowship at Harvard University from September 2009 to June 2010.

Oxford professorship

In 2013, Ben Green was appointed the Waynflete Professor of at the , effective from 1 August, marking a significant step in his career following his position at the . This named chair, one of the oldest and most prestigious in at , underscores his expertise in areas intersecting , , and . At the Mathematical Institute of the , Green holds a central role, contributing to the institution's research environment as a Waynflete Professor. He is actively involved in the research group, focusing on additive and combinatorial aspects, and supports the broader efforts through his work at the interface of these fields. In addition to his professorial duties, he served as Associate Head of Department (Research) from 2020 to 2022, providing leadership in guiding the institute's research priorities and fostering interdisciplinary collaborations. Since his appointment, Green has maintained an active network of visiting and collaborative roles, including positions at institutions such as MIT, UCLA, the Institute for Advanced Study, the University of Geneva, and the École Normale Supérieure, which enable ongoing international partnerships in pure mathematics. These engagements complement his primary affiliation at Oxford and support the Mathematical Institute's global outreach in number theory and related areas.

Research contributions

Additive combinatorics

Ben Green has made foundational contributions to additive combinatorics, a field concerned with the additive structure of subsets of integers or abelian groups, often using tools from , , and . His work emphasizes capturing and quantifying additive patterns, such as arithmetic progressions and sumsets, in dense sets. Early in his career, Green resolved longstanding conjectures and developed structural lemmas that have influenced subsequent research in the area. One of Green's key innovations is the arithmetic regularity lemma, developed in collaboration with , which provides a way to decompose functions on the integers into structured components that reveal their additive behavior. Specifically, the lemma states that for a f:Z/NZ→Cf: \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}, there exists a α(n)\alpha(n) (a structured object arising from ) and a small error term such that ff is well-approximated by α\alpha in a suitable sense, allowing the extraction of additive patterns like arithmetic progressions. This analogue to Szemerédi's graph regularity lemma has applications in counting linear configurations and has been extended to higher degrees and non-abelian settings. The associated counting lemma further enables precise estimates for the number of solutions to linear equations in sets with controlled additive energy. Green also proved the Cameron–Erdős conjecture, which posits that the number of sum-free subsets of {1,…,N}\{1, \dots, N\}—sets containing no three terms x,y,zx, y, z with x+y=zx + y = z—is asymptotically O(2N/2)O(2^{N/2}). His proof shows that nearly all such subsets are contained in one of two canonical examples: the odd integers up to NN or the integers from ⌈N/2⌉+1\lceil N/2 \rceil + 1 to NN. This result, obtained using Fourier-analytic methods and energy estimates, resolved a problem from 1983 and highlighted the scarcity of sum-free sets relative to the total size 2N2^N. Independently proved around the same time by Sapozhenko, Green's approach has been generalized to other groups and sum-free variants. In collaboration with from 2004 to 2010, Green advanced higher-order , introducing tools to study uniformity norms beyond the classical , with applications to combinatorial problems like on arithmetic progressions. Their joint work established inverse theorems for Gowers uniformity norms, which measure the additive of functions. For instance, the U3U^3 inverse theorem asserts that if a function f:Z/NZ→Cf: \mathbb{Z}/N\mathbb{Z} \to \mathbb{C} has small U3U^3 norm (indicating high uniformity), then ff correlates with a quadratic nilsequence, providing a structural description that facilitates proofs of density theorems in additive bases. This framework, extended to higher Us+1U^{s+1} norms in later work with Tamar Ziegler, has become central to modern additive , enabling breakthroughs in pattern avoidance and . Green's collaboration with Emmanuel Breuillard on approximate groups addressed the growth properties of sets that nearly form . An approximate group is a AA in a group GG such that A−1AAA^{-1}A A is covered by KK left translates of AA for some constant K≥1K \geq 1. Their theorem proves that any KK-approximate subgroup of an arbitrary group is essentially a of a of bounded step, controlled by KK, resolving a of Harald Helfgott and providing a classification akin to the solvable word problem in group theory. This result, obtained via product set estimates and non-commutative Balog-Wojciechowski techniques, has implications for expander graphs and the classification of progression-free sets. Together with , Green resolved the Dirac–Motzkin conjecture in combinatorial geometry, which concerns the minimal number of ordinary lines—lines passing through exactly two points—determined by nn points in the plane in . They proved that for sufficiently large nn, any such configuration determines at least n/2n/2 ordinary lines, confirming the conjecture's asymptotic threshold and identifying extremal configurations like points on a cubic curve or a perturbed grid. Although rooted in , the proof employs additive combinatorial tools such as sum-product inequalities to control collinearities, bridging with sumset growth. More recently, in 2023, Green collaborated with Kevin Ford and Dimitris Koukoulopoulos to study equal sums in random sets and the concentration of divisors, providing new insights into the distribution of divisors and additive structures in random subsets. In 2025, with Freddie Manners and , he revisited sumsets and entropy, developing entropic methods to bound the size of sumsets and advancing understanding of additive energy in structured sets.

Analytic number theory

Ben Green's most celebrated contribution to analytic number theory is the Green–Tao theorem, which he proved in collaboration with in 2004 and published in 2008. The theorem establishes that the sequence of prime numbers contains arithmetic progressions of any finite length, resolving a longstanding in . This result extends on arithmetic progressions in dense sets to the primes, which have zero , by employing a transference principle that embeds a positive proportion of primes into a pseudorandom subset of almost-primes amenable to Szemerédi's methods. In 2014, Green collaborated with Kevin Ford, Sergei Konyagin, James Maynard, and to prove significant bounds on the gaps between consecutive primes. Their work demonstrates that there exist gaps between consecutive primes that are larger than any fixed multiple of the logarithm of the primes' magnitude, providing the first proof of such unbounded gaps relative to logarithmic scales. This advances earlier conjectures and partial results by quantifying how primes can cluster while leaving arbitrarily large intervals devoid of them, with implications for the distribution of primes in short intervals. Central to Green's approaches in these prime-related problems is the integration of and to model the irregular distribution of primes. Pseudorandom measures allow the primes to be analyzed as behaving like random sets in certain structural senses, enabling the application of combinatorial theorems despite the primes' sparsity, while underpins the handling of correlations and densities in arithmetic progressions. These techniques, often leveraging Green's earlier arithmetic regularity lemma as a foundational tool for decomposing sets into structured and pseudorandom components, have been instrumental in bounding prime gaps and exploring their arithmetic structure. In 2024, Green proved a significant result on Sárközy's theorem for shifted primes, establishing that there are infinitely many primes pp such that p+n2p + n^2 is also prime for certain shifts, advancing the understanding of prime differences by squares.

Other areas

Green's applications of to additive emphasize as a tool for analyzing structure in dense sets. In particular, stability concepts from help classify approximate subgroups and identify stable patterns in subsets of abelian groups or s with significant additive energy. His survey on finite field models highlights how these ideas, inspired by Hrushovski's work on stable structures, enable the decomposition of dense sets into stable and unstable components, facilitating proofs of structural theorems without relying solely on . This approach has influenced subsequent developments in understanding approximate homomorphisms and controlled approximate groups in dense settings. In , Green has advanced the study of multiple recurrence in dynamical systems, leveraging measure-preserving transformations to establish combinatorial correspondences. His joint work with develops quantitative bounds on polynomial orbits in nilmanifolds, which underpin multiple recurrence results for actions and provide equidistribution estimates essential for dynamical proofs of arithmetic phenomena. For instance, these ergodic techniques demonstrate that certain configurations recur with positive upper in amenable dynamical systems, offering a bridge between topological dynamics and additive structures. Such methods extend Furstenberg's multiple recurrence theorem to more general settings, emphasizing the role of invariant measures in capturing recurrent behaviors. Green's early research addressed uniform distribution and discrepancy theory, focusing on the asymptotic behavior of sequences and sets in arithmetic progressions. In his 2001 paper, he quantified the number of squares within sets defined by bounded doubling constants, using discrepancy estimates to bound deviations from expected uniform distribution in additive bases. This work connects discrepancy measures to the irregularity of distribution for quadratic residues, providing bounds that control how closely such sets approximate uniform measures modulo primes. Later extensions in joint papers explored uniform distribution properties of multivariate polynomials over finite fields, with applications to norms measuring and low-discrepancy configurations.

Recognition

Major awards

In 2001, Green received the Smith's Prize from the for his outstanding performance in the final year of the . This award, shared annually among the top students, marked his early excellence during his PhD studies. In 2004, Green received the from the for his outstanding contributions to and , including his seminal joint work with establishing the existence of arbitrarily long arithmetic progressions of prime numbers. This award, one of the highest honors for early-career , recognized the profound impact of his results on additive shortly after his PhD. In 2005, Green and jointly received the Ostrowski Prize for their exceptional contributions to mathematics, particularly the on primes in arithmetic progressions. Awarded biennially for outstanding achievements in , it highlighted their groundbreaking proof. The following year, in 2005, Green was awarded the by the School of Mathematics at the Institute for Advanced Study for his innovative contributions to and . The prize, named after Raphaël Salem and given annually to young analysts, highlighted Green's breakthroughs in understanding structure in sets of integers, building on his recent ergodic and combinatorial advances. Also in 2005, the London Mathematical Society bestowed upon Green its Whitehead Prize, acknowledging his exceptional early achievements in combinatorics and combinatorial number theory. This prestigious award for mathematicians under 40 underscored the rapid recognition of his work during a pivotal phase of his career at the University of Bristol. In 2007, Green was honored with the SASTRA Ramanujan Prize by Shanmugha Arts, Science, Technology & Research Academy for his distinguished contributions to additive number theory. Established to celebrate Ramanujan's legacy, the prize emphasized Green's role in advancing the field through tools from ergodic theory and Fourier analysis. At the 2008 European Congress of Mathematics in , Green received the Prize, one of ten awarded to young European researchers for significant achievements in mathematics. The award celebrated his broad influence across and , coinciding with his growing international prominence. In 2014, the Royal Society awarded Green the Sylvester Medal for his landmark result on primes in and subsequent breakthroughs in related areas. This medal, the Society's highest honor in , marked the culmination of over a decade of transformative research. In 2019, the London Mathematical Society granted Green the Senior Whitehead Prize for his groundbreaking results in additive combinatorics and . As a senior-level distinction, it affirmed his enduring leadership in the field, reflecting the lasting significance of his contributions. In 2025, Green received the inaugural I. Martin Isaacs Prize for Excellence in Mathematical Writing from the for his article “On Sárközy's theorem for shifted primes”. This new award recognizes clarity and impact in mathematical exposition.

Fellowships and lectureships

Green was elected a in 2010, recognizing his outstanding contributions to . In 2012, he was selected as a in its inaugural class, honoring his significant work in the field. Green received the Philip Leverhulme Prize in 2006, which provided research funding to support his ongoing investigations in . He was awarded the Gauss Lectureship by the German Mathematical Society in 2013, delivering a series of lectures at their annual meeting in on topics in additive combinatorics. At the in in 2006, Green presented a section lecture on , highlighting advancements in arithmetic progressions. He delivered a plenary lecture at the 2014 in , addressing approximate algebraic structures and their applications.

References

  1. https://royalsociety.org/people/ben-green-11540/
  2. https://www.maths.ox.ac.uk/system/files/users/cvs/bencv_0.pdf
  3. https://www.magd.ox.ac.uk/people/professor-ben-green/
  4. https://www.claymath.org/library/annual_report/ar2006/06report_green.pdf
  5. https://www.maths.ox.ac.uk/node/69941
  6. https://people.maths.ox.ac.uk/greenbj/papers/bencv.pdf
  7. https://www.imo-official.org/participant_r.aspx?id=1137
  8. https://genealogy.math.ndsu.nodak.edu/id.php?id=76979
  9. https://www.bristol.ac.uk/news/2004/564.html
  10. https://www.claymath.org/people/ben-green/
  11. https://www.magd.ox.ac.uk/news/new-waynflete-professor-of-pure-mathematics/
  12. https://www.maths.ox.ac.uk/groups/number-theory/information-prospective-students
  13. https://www.maths.ox.ac.uk/people/ben.green
  14. https://www.lms.ac.uk/sites/default/files/files/b_green.pdf
  15. https://arxiv.org/abs/1002.2028
  16. https://arxiv.org/abs/math/0304058
  17. https://arxiv.org/abs/math/0503014
  18. https://arxiv.org/abs/1110.5008
  19. https://arxiv.org/abs/1208.4714
  20. https://arxiv.org/abs/1908.00378
  21. https://arxiv.org/abs/2306.13403
  22. https://annals.math.princeton.edu/wp-content/uploads/annals-v167-n2-p03.pdf
  23. https://arxiv.org/pdf/1412.5029.pdf
  24. https://arxiv.org/abs/2206.08001
  25. https://terrytao.wordpress.com/2009/12/06/non-commutative-freiman-theorems-and-model-theory/
  26. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/multiple-correlation-sequences-not-approximable-by-nilsequences/CC22C323285837211E1304FECD3DEB82
  27. https://www.claymath.org/library/annual_report/ar2004/04report_clayaward.pdf
  28. https://www.ostrowski.ch/index_e.php?ifile=preis
  29. https://www.ias.edu/math/salem-prize
  30. https://www.lms.ac.uk/sites/lms.ac.uk/files/files/339%2520-%2520July%25202005.pdf
  31. https://www.sastra.edu/downloads/newsletters/vol7/volume%2520VII-No.4-2007.pdf
  32. https://royalsociety.org/news/2014/awards-and-medals-2014/
  33. https://www.lms.ac.uk/news-entry/28062019-1621/lms-prize-winners-2019
  34. https://www.ams.org/isaacs-prize
  35. https://www.ams.org/profession/fellows/inaugural-class
  36. https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2006.2/ICM2006.2.ocr.pdf
  37. http://www.icm2014.org/en/program/scientific/plenary.html
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