Ben Joseph Green
FRS (born 27 February 1977) is a British mathematician, specialising in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. He is the
Waynflete Professor of Pure Mathematics Waynflete could refer to:
*William Waynflete (1395–1486), English Lord Chancellor and bishop of Winchester
*Waynflete Professorships at Magdalen College, Oxford, named after William Waynflete
*Waynflete School, a private day school in Portland, M ...
at the
University of Oxford
, mottoeng = The Lord is my light
, established =
, endowment = £6.1 billion (including colleges) (2019)
, budget = £2.145 billion (2019–20)
, chancellor ...
.
Early life and education
Ben Green was born on 27 February 1977 in
Bristol
Bristol () is a city, ceremonial county and unitary authority in England. Situated on the River Avon, it is bordered by the ceremonial counties of Gloucestershire to the north and Somerset to the south. Bristol is the most populous city in ...
, England. He studied at local schools in Bristol,
Bishop Road Primary School
Bishop Road Primary School is a primary school in Bristol, England. It is on Bishop Road in the Bishopston area of Bristol. The school opened in 1896. It is the largest primary school in Bristol, notable for having educated Cary Grant and Paul ...
and
Fairfield Grammar School
Fairfield Grammar School was a secondary school in Bristol, England, founded in 1898 as Fairfield Secondary and Higher Grade School. It became a grammar school in 1945 and closed in 2000, to be replaced by a new comprehensive, Fairfield High Sc ...
, competing in the
International Mathematical Olympiad
The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...
in 1994 and 1995. He entered
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...
in 1995 and completed his
BA in mathematics in 1998, winning the
Senior Wrangler
The Senior Frog Wrangler is the top mathematics undergraduate at the University of Cambridge in England, a position which has been described as "the greatest intellectual achievement attainable in Britain."
Specifically, it is the person who a ...
title. He stayed on for
Part III
''Part III'' is the third studio album by American R&B group 112. It was released by Bad Boy Records on March 20, 2001 in the United States. Unlike the previous releases, the album is described as having edgier, techno-flavored jams, resulting ...
and earned his doctorate under the supervision of
Timothy Gowers
Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity Col ...
, with a thesis entitled ''Topics in arithmetic combinatorics'' (2003). During his PhD he spent a year as a visiting student at
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
. He was a research Fellow at Trinity College, Cambridge between 2001 and 2005, before becoming a Professor of Mathematics at the
University of Bristol
, mottoeng = earningpromotes one's innate power (from Horace, ''Ode 4.4'')
, established = 1595 – Merchant Venturers School1876 – University College, Bristol1909 – received royal charter
, type ...
from January 2005 to September 2006 and then the first
Herchel Smith Professor of Pure Mathematics The Herchel Smith Professorship of Pure Mathematics is a professorship in pure mathematics at the University of Cambridge. It was established in 2004 by a benefaction from Herchel Smith "of £14.315m, to be divided into five equal parts, to suppor ...
at the
University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
from September 2006 to August 2013. He became the
Waynflete Professor of Pure Mathematics at the
University of Oxford
, mottoeng = The Lord is my light
, established =
, endowment = £6.1 billion (including colleges) (2019)
, budget = £2.145 billion (2019–20)
, chancellor ...
on 1 August 2013. He was also a Research Fellow of the
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation (nonprofit), foundation dedicated to increasing and disseminating mathematics, mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address i ...
and held various positions at institutes such as
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
,
University of British Columbia
The University of British Columbia (UBC) is a public university, public research university with campuses near Vancouver and in Kelowna, British Columbia. Established in 1908, it is British Columbia's oldest university. The university ranks a ...
, and
Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
.
Mathematics
The majority of Green's research is in the fields of
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
and
additive combinatorics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A ...
, but he also has results in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. His best known theorem, proved jointly with his frequent collaborator
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, states that there exist arbitrarily long
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s in the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s: this is now known as the
Green–Tao theorem In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', there exist arith ...
.
Amongst Green's early results in additive combinatorics are an improvement of a result of
Jean Bourgain
Jean, Baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic t ...
of the size of arithmetic progressions in
sumset In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is,
:A + B = \.
The n-fo ...
s, as well as a proof of the
Cameron–Erdős conjecture In combinatorics, the Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in = \ is O\big(\big).
The sum of two odd numbers is even, so a set of odd numbers is always sum-free. There are \lceil ...
on sum-free sets of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s. He also proved an arithmetic regularity lemma for functions defined on the first
natural numbers, somewhat analogous to the
Szemerédi regularity lemma
Szemerédi's regularity lemma is one of the most powerful tools in extremal graph theory, particularly in the study of large dense graphs. It states that the vertices of every large enough graph can be partitioned into a bounded number of parts so ...
for graphs.
From 2004–2010, in joint work with
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
and
Tamar Ziegler
Tamar Debora Ziegler (; born 1971) is an Israeli mathematician known for her work in ergodic theory, combinatorics and number theory. She holds the Henry and Manya Noskwith Chair of Mathematics at the Einstein Institute of Mathematics at the Heb ...
, he developed so-called
higher order Fourier analysis
Higher may refer to:
Music
* The Higher, a 2002–2012 American pop rock band
Albums
* ''Higher'' (Ala Boratyn album) or the title song, 2007
* ''Higher'' (Ezio album) or the title song, 2000
* ''Higher'' (Harem Scarem album) or the title song ...
. This theory relates
Gowers norm
In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomne ...
s with objects known as
nilsequences. The theory derives its name from these nilsequences, which play an analogous role to the role that
characters
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
play in classical
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
. 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. This generalises the classical approach using
Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually at ...
. Many aspects of this theory, including the quantitative aspects of the inverse theorem for the Gowers norms, are still the subject of ongoing research.
Green has also collaborated with
Emmanuel Breuillard
Emmanuel Breuillard (born 25 June 1977) is a French mathematician. He was the Sadleirian Professor of Pure Mathematics in the
Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge, and is now Professor ...
on topics in group theory. In particular, jointly with
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, they proved a structure theorem for
approximate group In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of ...
s, generalising the
Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has work, joint with
Kevin Ford and
Sean Eberhard, on the theory of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
, in particular on what proportion of its elements fix a set of size
.
Green and Tao also have a paper on algebraic
combinatorial geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geome ...
, resolving the Dirac-Motzkin conjecture (see
Sylvester–Gallai theorem
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester ...
). In particular they prove that, given any collection of
points in the plane that are not all
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, 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
Sergei Vladimirovich Konyagin (russian: Серге́й Владимирович Конягин; born 25 April 1957) is a Russian mathematician. He is a professor of mathematics at the Moscow State University.
Konyagin participated in the Internat ...
,
James Maynard and
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, 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
. 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
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a ...
. Together with
Tom Sanders he proved that, if a sufficiently large
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of prime order is coloured with a fixed number of colours, then the field has elements
such that
all have the same colour.
Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the
polynomial method
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
to bound the size of subsets of a finite
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
without solutions to
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s. He adapted these methods to prove, in function fields, a strong version of
Sárközy's theorem.
Awards and honours
Green has been a Fellow of the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
since 2010, and a Fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
since 2012. Green was chosen by the
German Mathematical Society
The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathe ...
to deliver a
Gauss Lectureship
The Gauss Lectureship (''Gauß-Vorlesung'') is an annually awarded mathematical distinction, named in honor of Carl Friedrich Gauss. It was established in 2001 by the German Mathematical Society with a series of lectures for a broad audience.
Eac ...
in 2013. He has received several awards:
* 2004:
Clay Research Award __NOTOC__
The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research. The following mathematicians have received the award:
{, class=" ...
* 2005:
Salem Prize
The Salem Prize, in memory of Raphael Salem, is awarded each year to young researchers for outstanding contributions to the field of analysis. It is awarded by the School of Mathematics at the Institute for Advanced Study in Princeton and was fo ...
* 2005:
Whitehead Prize
The Whitehead Prize is awarded yearly by the London Mathematical Society to multiple mathematicians working in the United Kingdom who are at an early stage of their career. The prize is named in memory of homotopy theory pioneer J. H. C. Whiteh ...
* 2007:
SASTRA Ramanujan Prize
The SASTRA Ramanujan Prize, founded by Shanmugha Arts, Science, Technology & Research Academy (SASTRA) located near Kumbakonam, India, Srinivasa Ramanujan's hometown, is awarded every year to a young mathematician judged to have done outstanding ...
* 2008:
European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...
prize recipient
* 2014:
Sylvester Medal
The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...
, awarded by the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
.
* 2019:
Senior Whitehead Prize
The Senior Whitehead Prize of the London Mathematical Society (LMS) is now awarded in odd numbered years in memory of John Henry Constantine Whitehead, president of the LMS between 1953 and 1955. The Prize is awarded to mathematicians normally ...
of the
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
References
External links
Ben Green personal homepage at OxfordBen Green faculty page at OxfordBen Green Homepage at Trinity College, CambridgeClay Research Award 2004 announcement*
math.NT/0404188 – Preprint on arbitrarily long arithmetic progressions on primes
{{DEFAULTSORT:Green, Ben
1977 births
Living people
20th-century English mathematicians
21st-century English mathematicians
Academics of the University of Bristol
Alumni of Trinity College, Cambridge
Cambridge mathematicians
Clay Research Award recipients
Combinatorialists
Fellows of Magdalen College, Oxford
Fellows of the American Mathematical Society
Fellows of the Royal Society
Fellows of Trinity College, Cambridge
International Mathematical Olympiad participants
Number theorists
Scientists from Bristol
Recipients of the SASTRA Ramanujan Prize
Senior Wranglers
Simons Investigator
Waynflete Professors of Pure Mathematics
Whitehead Prize winners
Professors of the University of Cambridge