In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the
Poincaré conjecture (French, ) is a
theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
about the
characterizationIn mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
of the
three-dimensional sphereIn mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
among
three-dimensional manifoldsIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three...
. It began as a popular, important
conjectureA conjecture is a proposition which is presumed to be real, true, or genuine, mostly based on inconclusive grounds. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
, but is now considered a theorem to the satisfaction of the awarders of the
Fields medalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
. The claim concerns a space that locally looks like ordinary three dimensional space but is connected, finite in size, and lacks any boundary (a
closedIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three...
). The Poincaré conjecture claims that if such a space has the additional property that each
loopIn mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
in the space can be continuously tightened to a point, then it is just a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.
After nearly a century of effort by mathematicians,
Grigori PerelmanGrigori Yakovlevich Perelman , born 13 June 1966 in Leningrad, USSR , sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture...
sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of
Richard HamiltonRichard Streit Hamilton is professor of mathematics at Columbia University.He received his Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...
. Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.
The Poincaré conjecture was, before being proven, one of the most important open questions in
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
. It is one of the seven
Millennium Prize ProblemsThe Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
, for which the
Clay Mathematics InstituteThe Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a
Fields MedalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
, which he declined. The Poincaré conjecture remains the only solved Millennium problem.
On December 22, 2006, the journal
ScienceScience is the academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious scientific journals. The peer-reviewed journal, first published in 1880 is circulated weekly and has a print subscriber base of around 130,000...
honored Perelman's proof of the Poincaré conjecture as the scientific "
Breakthrough of the YearThe Breakthrough of the Year is an annual award made by the journal Science for the most significant development in scientific research. Originating in 1989 as "the molecule of the year", inspired by Time's Man of the Year, it was renamed the "Breakthrough of the year" in 1996...
," the first time this had been bestowed in the area of mathematics.
Poincaré's question
At the beginning of the 20th century,
Henri PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
was working on the foundations of topology — what would later be called
combinatorial topologyIn mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
and then
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...
. He was particularly interested in what topological properties characterized a
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
.
Poincaré claimed in 1900 that
homologyIn mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
, a tool he had devised based on prior work by
Enrico BettiEnrico Betti was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations, giving early expositions of Galois theory. He also discovered Betti's theorem, a result in the...
, was sufficient to tell if a
3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three...
was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere. The Poincaré sphere was the first example of a
homology sphereIn algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
, a manifold that had the same homology as a sphere, of which many others have since been constructed. To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, the
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group or Poincaré group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
, and showed that the Poincaré sphere had a
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group or Poincaré group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of order 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different.
In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Poincaré's new condition - i.e., "trivial fundamental group" - can be re-phrased as "every loop can be shrunk to a point."
The original phrasing was as follows:
Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?
Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the
Poincaré conjecture. Here is the standard form of the conjecture:
Every simply connected, closedIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
3-manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
is homeomorphicIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
to the 3-sphere.
In other dimensions
The classification of closed surfaces gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the
Generalized Poincaré conjecture: is a homotopy
n-sphere homeomorphic to the
n-sphere? The stronger assumption is necessary; in dimensions four and higher there are simply-connected manifolds which are not homeomorphic to an
n-sphere.
Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961
Stephen SmaleStephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley . He entered the University of Michigan in 1948...
shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. In 1982
Michael FreedmanMichael Hartley Freedman is a mathematician at Microsoft Station Q. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born into a Jewish family in Los Angeles...
proved the Poincaré conjecture in dimension four. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not diffeomorphic to the four-sphere. This so-called
smooth Poincare conjecture, in dimension four, remains open and is thought to be very difficult. Milnor's
exotic sphereIn mathematics, an exotic sphere is a differentiable manifold that is homeomorphic to the standard Euclidean n-sphere, but not diffeomorphic. That means that such a manifold M is a sphere from a topological point of view, but not from the point of view of its differential structure...
s show that the smooth Poincare conjecture is false in dimension seven, for example.
These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the
geometrization conjectureThurston's geometrization conjecture states that compact 3-manifolds can be decomposed into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
put it into a framework governing all 3-manifolds.
John MorganJohn Willard Morgan is an American mathematician, well-known for his contributions to topology and geometry. He is currently the director of the Simons Center for Geometry and Physics at Stony Brook University.-Life:...
wrote:
"It is my view that before ThurstonWilliam Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics...
's work on hyperbolic 3-manifoldA hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s and . . . the Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the Geometrization conjecture) were true."
Attempted solutions
This problem seems to have lain dormant for a time, until
J. H. C. WhiteheadJohn Henry Constantine Whitehead , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....
revived interest in the conjecture, when in the 1930s he first claimed a proof, and then retracted it. In the process, he discovered some interesting examples of simply connected non-compact 3-manifolds not homeomorphic to
R3, the prototype of which is now called the
Whitehead manifoldIn mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture....
.
In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influential mathematicians such as
BingRH Bing was an influential American mathematician. He worked mainly in the area of topology, where he made many important contributions...
,
HakenWolfgang Haken is a mathematician who specializes in topology, in particular 3-manifolds.In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem...
,
MoiseEdwin Evariste Moise was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th century English poetry and had several notes published in that field.-Early life and education:Edwin E...
, and
PapakyriakopoulosChristos Dimitriou Papakyriakopoulos, commonly known as "Papa" was a Greek mathematician specializing in geometric topology...
attacked the conjecture. In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere. Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.
Over time, the conjecture gained the reputation of being particularly tricky to tackle. John Milnor commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect." Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs, and tended to view any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in
peer-reviewedPeer review is the process of subjecting an author's scholarly work, research, or ideas to the scrutiny of others who are experts in the same field. Peer review requires a community of experts in a given field, who are qualified and able to perform impartial review...
form).
An exposition of attempts to prove this conjecture can be found in the non-technical book "Poincaré's Prize" by George Szpiro.
Hamilton's program and Perelman's solution
Hamilton's program was started in his 1982 paper in which he introduced the
Ricci flowIn differential geometry, the Ricci flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold—in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric...
on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture. In the following years he extended this work, but was unable to prove the conjecture. The actual solution wasn't found until
Grigori PerelmanGrigori Yakovlevich Perelman , born 13 June 1966 in Leningrad, USSR , sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture...
of the
Steklov Institute of MathematicsSteklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics. It was established April 24 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in Leningrad...
,
Saint PetersburgSaint Petersburg is a city and a federal subject of Russia located on the Neva River at the head of the Gulf of Finland on the Baltic Sea. The city's other names were Petrograd and Leningrad...
published his papers using ideas from Hamilton's work.
In late 2002 and 2003 Perelman posted three papers on the
arXivThe arXiv is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, computer science, quantitative biology and statistics which can be accessed via the world wide web. In many fields of mathematics and physics, almost all scientific papers are placed on...
. In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by
Richard HamiltonRichard Streit Hamilton is professor of mathematics at Columbia University.He received his Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...
.
From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
- Bruce Kleiner
Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory.He received his Ph.D. in 1990 from the University of California, Berkeley. His advisor was Wu-Yi Hsiang. He is now Professor of Mathematics at New York University.Kleiner has...
and John W. LottJohn Lott is a Professor of Mathematics at the University of Michigan at Ann Arbor. He is working on Ricci flow.He and Bruce Kleiner of Yale University make up one of three teams formed for the purpose of verifying Grigori Perelman's proof of the Poincaré conjecture...
posted a paper on the arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture.
- Huai-Dong Cao
Huai-Dong Cao is A. Everett Pitcher Professor of Mathematics in Lehigh University. He collaborated with Xi-Ping Zhu of Zhongshan University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team is one of three teams formed for this purpose...
and Xi-Ping ZhuZhu Xiping is a Professor of Mathematics at Sun Yat-sen University. He collaborated with Cao Huaidong of Lehigh University in verifying Grigori Perelman's proof of the Poincaré conjecture. The Cao–Zhu team was one of three teams formed for this purpose...
published a paper in the June 2006 issue of the Asian Journal of Mathematics giving a complete proof of the Poincaré and geometrization conjectures, in which they used some earlier work by Kleiner and Lott.
- John Morgan
John Willard Morgan is an American mathematician, well-known for his contributions to topology and geometry. He is currently the director of the Simons Center for Geometry and Physics at Stony Brook University.-Life:...
and Gang TianTian Gang is a Chinese mathematician and an academician of the Chinese Academy of Sciences. He is known for his contributions to geometric analysis and quantum cohomology, among other fields...
posted a paper on the arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture) and expanded this to a book.
All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.
On August 22, 2006, the
ICMThe International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
awarded Perelman the
Fields MedalThe Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. The Fields Medal is often viewed as the top honor a mathematician can receive. It...
for his work on the conjecture, but Perelman refused the medal.
John Morgan spoke at the ICM on the Poincaré conjecture on August 24 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."
The August 2006 issue of
The New YorkerThe New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons, and poetry published by Condé Nast Publications...
contains an article, titled "
Manifold Destiny"Manifold Destiny" is an article in The New Yorker written by Sylvia Nasar and David Gruber and published in the August 28, 2006 issue of the magazine...
", that details some of the issues surrounding Perelman's accomplishment, particularly some disagreements that arose between the mathematicians responsible for verifying his proof.
The proof was called the "Breakthrough of the year" by
ScienceScience is the academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious scientific journals. The peer-reviewed journal, first published in 1880 is circulated weekly and has a print subscriber base of around 130,000...
magazine.
Ricci flow with surgery
Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold. The idea is to try to improve this metric; for example, if the metric can be improved enough so that it has constant curvature, then it must be the 3-sphere. The metric is improved using the
Ricci flowIn differential geometry, the Ricci flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold—in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric...
equations;
where
g is the metric and
R its Ricci curvature,
and one hopes that as the time
t increases the manifold becomes easier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.
In some cases Hamilton was able to show that this works; for example, if the manifold has positive Ricci curvature everywhere he showed that the manifold becomes extinct in finite time under Ricci flow without any other singularities. (In other words, the manifold collapses to a point in finite time; it is easy to describe the structure just before the manifold collapses.) This easily implies the Poincaré conjecture in the case of positive Ricci curvature. However in general the Ricci flow equations lead to singularities of the metric after a finite time. Perelman showed how to continue past these singularities: very roughly, he cuts the manifold along the singularities, splitting the manifold into several pieces, and then continues with the Ricci flow on each of these pieces. This procedure is known as
Ricci flow with surgery.
A special case of Perelman's theorems about Ricci flow with surgery is given as follows.
The Ricci flow with surgery on a closed oriented 3-manifold is well defined for all time. If the fundamental group is a free productIn mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
of finite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s and cyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s then the Ricci flow with surgery becomes extinct in finite time, and at all times all components of the manifold are connected sums of S2 bundles over S1 and quotients of S3.
This result implies the Poincaré conjecture because it is easy to check it for the possible manifolds listed in the conclusion.
The condition on the fundamental group turns out to be necessary (and sufficient) for finite time extinction, and in particular includes the case of trivial fundamental group. It is equivalent to saying that the prime decomposition of the manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries
S2×
R and
S3. By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a
graph manifoldIn topology, a graph manifold is a 3-manifold which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen in 1967...
, but this extra complication is not necessary for proving just the Poincaré conjecture.
External links
- The Poincaré conjecture described by the Clay Mathematics Institute.
- Bruce Kleiner (Yale) and John W. Lott (University of Michigan): "Notes & commentary on Perelman's Ricci flow papers".
- Stephen Ornes, What is The Poincaré Conjecture?, Seed Magazine, 25 August 2006.
- The slides used by Yau in a popular talk on the Poincaré conjecture.
- "The Poincaré Conjecture" - BBC Radio 4
BBC Radio 4 is a domestic UK radio station that broadcasts a wide variety of spoken-word programmes, including news, drama, comedy, science and history. It replaced the BBC Home Service in 1967.-Outline:...
programme In Our TimeIn Our Time is a live BBC radio discussion programme hosted by Melvyn Bragg. Each week, three guest speakers cover a specific historical, philosophical, religious, artistic or scientific topic...
, 2 November, 2006. Contributors June Barrow-Green, Lecturer in the History of Mathematics at the Open UniversityThe Open University is the distance learning university founded and funded by the UK Government. It is notable for having an open entry policy, i.e. students' previous academic achievements are not taken into account for entry to most undergraduate courses...
, Ian StewartIan Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...
, Professor of Mathematics at the University of WarwickThe University of Warwick is a British campus university located on the outskirts of Coventry, West Midlands, England. It was established in 1965 as part of a government initiative to expand access to higher education, and in 2000 Warwick Medical School was opened as part of an initiative to train...
, Marcus du SautoyMarcus Peter Francis du Sautoy is a Professor of Mathematics at the University of Oxford. Formerly a Fellow of All Souls College, and Wadham College, he is now a Fellow of New College. He is currently an EPSRC Senior Media Fellow and was previously a Royal Society University Research Fellow. His...
, Professor of Mathematics at the University of OxfordThe University of Oxford , located in the UK city of Oxford, is the oldest surviving university in the English-speaking world and is regarded as one of the world's leading academic institutions. Although the exact date of foundation remains unclear, there is evidence of teaching there as far back...
, and presenter Melvyn BraggMelvyn Bragg, Baron Bragg, FRSL, FRTS is an English author, broadcaster and media personality who, aside from his many literary endeavours, is perhaps most recognised for his work on The South Bank Show.-Biography:...
.
- "Solving an Old Math Problem Nets Award, Trouble" - NPR segment, December 26, 2006.