In mathematics, the Poincaré conjecture (, ) is a theorem about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is ''homotopy equivalent'' to the 3-sphere, then it is necessarily ''homeomorphic'' to it.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily 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 Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have verified that Perelman's proof is correct.

The Poincaré conjecture, before being proven, was one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute 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 Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of U.S. mathematician Richard Hamilton (who first suggested a program for the solution). The Poincaré conjecture is the first and, as of July 2011, the only solved Millennium problem.

On December 22, 2006, the journal ''Science'' honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.

History

Poincaré's question

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere.

Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold 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 sphere, 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 group, and showed that the Poincaré sphere had a fundamental group 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 restated as "every loop can be shrunk to a point."

The original phrasing was as follows:

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:

Attempted solutions

This problem seems to have lain dormant for a time, until J. H. C. Whitehead 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 manifold.

In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos 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-reviewed form).

An exposition of attempts to prove this conjecture can be found in the non-technical book ''Poincaré's Prize'' by George Szpiro.

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? A 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 Smale 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 Freedman 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 Poincaré conjecture, in dimension four, remains open and is thought to be very difficult. Milnor's exotic spheres show that the smooth Poincaré 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 conjecture put it into a framework governing all 3-manifolds. John Morgan wrote:

Hamilton's program and Perelman's solution

Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow 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 was not found until Grigori Perelman published his papers using ideas from Hamilton's work.

In late 2002 and 2003 Perelman posted three papers on the arXiv. 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 Hamilton.

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 and John W. Lott 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 and Xi-Ping Zhu published a paper in the June 2006 issue of the ''Asian Journal of Mathematics'' giving a complete proof of the Poincaré and geometrization conjectures. They initially claimed the proof as their own achievement based on the "Hamilton-Perelman theory", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest "exposition of Hamilton–Perelman's proof". They were also forced to publish an erratum disclosing that they had failed to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the AJM editorial board issued an apology for what it called "incautions" in the Cao–Zhu paper. John Morgan and Gang Tian 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 ICM awarded Perelman the Fields Medal 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."

In December 2006, the journal ''Science'' honored the proof of Poincaré conjecture as the Breakthrough of the Year and featured it on its cover.

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 flow equations; :$\backslash partial\_t\; g\_\{ij\}=-2\; R\_\{ij\}$ 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.

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 ''S''²×R and ''S''³. 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 manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.

Notes

External links

The Poincaré conjecture described by the Clay Mathematics Institute.

The Poincaré Conjecture (video) Brief visual overview of the Poincaré Conjecture, background and solution.

The Geometry of 3-Manifolds(video) A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.

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 programme ''In Our Time'', 2 November 2006. Contributors June Barrow-Green, Lecturer in the History of Mathematics at the Open University, Ian Stewart, Professor of Mathematics at the University of Warwick, Marcus du Sautoy, Professor of Mathematics at the University of Oxford, and presenter Melvyn Bragg.

"Solving an Old Math Problem Nets Award, Trouble" – NPR segment, December 26, 2006.

Category:Geometric topology Category:3-manifolds Category:Mathematical theorems Category:Millennium Prize Problems

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Coordinates	40°26′30″N80°00′00″N
name	John Willard Milnor
birth date	February 20, 1931
birth place	Orange, New Jersey
Residence	United States
Nationality
field	Mathematics
work institution	Stony Brook University
alma mater	Princeton University
doctoral advisor	Ralph Fox
doctoral students	Tadatoshi Akiba Jon Folkman John MatherLaurent C. SiebenmannMichael Spivak
known for	Exotic spheresFary–Milnor theorem
prizes	Putnam Fellow (1949, 1950) Fields Medal (1962) National Medal of Science (1967) Leroy P Steele Prize (1982) Wolf Prize (1989) Abel Prize (2011)
footnotes	}}

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University. His wife, Dusa McDuff, is a professor of mathematics at Barnard College.

Life

Milnor was born in Orange, New Jersey. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fary–Milnor theorem. He continued on to graduate school at Princeton and wrote his thesis, entitled "Isotopy of Links", which concerned link groups (a generalization of the classical knot group) and their associated link structure. His advisor was Ralph Fox. Upon completing his doctorate he went on to work at Princeton.

In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He later went on to win the National Medal of Science (1967), the Leroy P Steele Prize for Seminal Contribution to Research (1982), the Wolf Prize in Mathematics (1989), the Leroy P Steele Prize for Mathematical Exposition (2004), and the Leroy P Steele Prize for Lifetime Achievement (2011). He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books which are remarkable for their easy, clear and precise style.

Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the ''New Scientist'' "It feels very good,", adding that, "One is always surprised by a call at 6 o'clock in the morning."

His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, and Michael Spivak.

Work

His most celebrated single result is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures (28 if one considers orientation). An ''n''-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to develop to this day.

In 1961 Milnor disproved the Hauptvermutung by exhibiting two simplicial complexes which are homeomorphic but combinatorially distinct.

Publications

;Books

;Journal articles

See also

Fary–Milnor theorem

Milnor conjecture in algebraic K-theory

Milnor conjecture in knot theory

Milnor conjecture concerning manifolds with nonnegative Ricci curvature

Milnor fibration

Milnor map

Milnor–Thurston kneading theory

Orbit portrait

References

External links

Home page at SUNYSB

Photo

Exotic spheres home page

Category:1931 births Category:20th-century mathematicians Category:Abel Prize laureates Category:American mathematicians Category:Fields Medalists Category:Institute for Advanced Study faculty Category:Living people Category:Members of the United States National Academy of Sciences Category:National Medal of Science laureates Category:People from Orange, New Jersey Category:Princeton University alumni Category:Princeton University faculty Category:Putnam Fellows Category:Sloan Research Fellowships Category:State University of New York at Stony Brook faculty Category:Topologists Category:Wolf Prize in Mathematics laureates

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Coordinates	40°26′30″N80°00′00″N
name	Grigori Perelman
birth date	June 13, 1966
birth place	Leningrad, Soviet Union
field	Mathematics
alma mater	Leningrad State University
doctoral advisor	Aleksandr AleksandrovYuri Burago
known for	Riemannian geometry and geometric topology
awards	EMS Prize (1996)Fields Medal (2006), declinedMillennium Prize (2010), declined
footnotes	}}

Grigori Yakovlevich Perelman (, ; born 13 June 1966) is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology.

In August 2006, Perelman was awarded the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." Perelman declined to accept the award or to appear at the congress. On 22 December 2006, the journal ''Science'' recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year," the first such recognition in the area of mathematics.

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On July 1, 2010, he turned down the prize, saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of U.S. mathematician Richard Hamilton, who introduced and developed the theory of Ricci flow with the aim of attacking the geometrization conjecture.

Early life and education

Grigori Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lubov. Grigori's mother Lubov gave up graduate work in mathematics to raise him. Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school math training program.

His mathematical education continued at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In 1982, as a member of the Soviet Union team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score. In the late 1980s, Perelman went on to earn a Candidate of Science degree (the Soviet equivalent to the Ph.D.) at the School of Mathematics and Mechanics of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was titled "Saddle surfaces in Euclidean spaces."

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences, where his advisors were Aleksandr Aleksandrov and Yuri Burago. In the late 1980s and early 1990s, Perelman held research positions at several universities in the United States. In 1991, he spoke at the Geometry Festival on Alexandrov spaces with curvature bounded from below. In 1992, he was invited to spend a semester each at the Courant Institute in New York University and State University of New York at Stony Brook where he began work on manifolds with lower bounds on Ricci curvature. From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. After having proved the soul conjecture in 1994, he was offered jobs at several top universities in the US, including Princeton and Stanford, but he rejected them all and returned to the Steklov Institute in Saint Petersburg in the summer of 1995 for a research-only position.

He has a younger sister, Elena, who is also a scientist. She received a Ph.D. from Weizmann Institute of Science in Israel and is a biostatistician at Karolinska Institutet, in Stockholm, Sweden.

Perelman is also a talented violinist and a strong table tennis player.

Geometrization and Poincaré conjectures

Until the autumn of 2002, Perelman was best known for his work in comparison theorems in Riemannian geometry. Among his notable achievements was a short and elegant proof of the soul conjecture.

The problem

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was the most famous open problem in topology. Any loop on a three-dimensional sphere—as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted to a point. The Poincaré conjecture asserts that any closed three-dimensional manifold such that any loop can be contracted to a point is topologically a three-dimensional sphere. The analogous result has been known to be true in dimensions greater than or equal to five since 1960 (work of Stephen Smale). The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman. But the case of three-manifolds turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three-manifold, there are too few dimensions to move "problematic regions" out of the way without interfering with something else.

In 1999, the Clay Mathematics Institute announced the ''Millennium Prize Problems'': $1,000,000 prizes for the proof of any of seven conjectures, including the Poincaré conjecture. There was a wide agreement that a successful proof of any of these would constitute a landmark event in the history of mathematics.

Perelman's proof

In November 2002, Perelman posted the first of a series of eprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case.

Perelman modified Richard Hamilton's program for a proof of the conjecture, in which the central idea is the notion of the Ricci flow. Hamilton's basic idea is to formulate a "dynamical process" in which a given three-manifold is geometrically distorted, such that this distortion process is governed by a differential equation analogous to the heat equation. The heat equation describes the behavior of scalar quantities such as temperature; it ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved throughout an object. Similarly, the Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor. Hamilton's hope was that under the Ricci flow, concentrations of large curvature will spread out until a uniform curvature is achieved over the entire three-manifold. If so, if one starts with any three-manifold and lets the Ricci flow occur, eventually one should in principle obtain a kind of "normal form". According to William Thurston, this normal form must take one of a small number of possibilities, each having a different kind of geometry, called Thurston model geometries.

This is similar to formulating a dynamical process that gradually "perturbs" a given square matrix, and that is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea attracted a great deal of attention, but no one could prove that the process would not be impeded by developing "singularities", until Perelman's eprints sketched a program for overcoming these obstacles. According to Perelman, a modification of the standard Ricci flow, called ''Ricci flow with surgery'', can systematically excise singular regions as they develop, in a controlled way.

We know that singularities (including those that, roughly speaking, occur after the flow has continued for an infinite amount of time) must occur in many cases. However, any singularity that develops in a finite time is essentially a "pinching" along certain spheres corresponding to the prime decomposition of the 3-manifold. Furthermore, any "infinite time" singularities result from certain collapsing pieces of the JSJ decomposition. Perelman's work proves this claim and thus proves the geometrization conjecture.

Verification

Since 2003, Perelman's program has attracted increasing attention from the mathematical community. In April 2003, he accepted an invitation to visit Massachusetts Institute of Technology, Princeton University, State University of New York at Stony Brook, Columbia University and New York University, where he gave a series of talks on his work.

Three independent groups of scholars have verified that Perelman's papers contain all the essentials for a complete proof of the geometrization conjecture: #On 25 May 2006, Bruce Kleiner and John Lott, both of the University of Michigan, posted a paper on arXiv that fills in the details of Perelman's proof of the Geometrization conjecture. John Lott said in ICM2006, "It has taken us some time to examine Perelman's work. This is partly due to the originality of Perelman's work and partly to the technical sophistication of his arguments. All indications are that his arguments are correct." #In June 2006, the ''Asian Journal of Mathematics'' published a paper by Xi-Ping Zhu of Sun Yat-sen University in China and Huai-Dong Cao of Lehigh University in Pennsylvania, giving a complete description of Perelman's proof of the Poincaré and the geometrization conjectures. The June 2006 paper claimed: "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow." ##In November 2006, Cao and Zhu published an erratum disclosing that they had failed to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the ''AJM'' editorial board issued an apology for what it called "incautions" in the Cao–Zhu paper. ##On December 3, 2006, Cao and Zhu retracted the original version of their paper, which was titled “A Complete Proof of the Poincaré and Geometrization Conjectures — Application of the Hamilton–Perelman Theory of the Ricci Flow” and posted a revised version, renamed, more modestly, "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". Rather than the grand claim of the original abstract, "we give a complete proof", suggesting the proof is by the authors, the revised abstract states: "we give a detailed exposition of a complete proof". The authors also removed the phrase "crowning achievement" from the abstract. #In July 2006, John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology posted a paper on the arXiv titled, "Ricci Flow and the Poincaré Conjecture". In this paper, they provide a detailed version of Perelman's proof of the Poincaré conjecture. On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture. This was followed up with the paper on the arXiv, "Completion of the Proof of the Geometrization Conjecture" on 24 September 2008.

Nigel Hitchin, professor of mathematics at Oxford University, has said that "I think for many months or even years now people have been saying they were convinced by the argument. I think it's a done deal."

The Fields Medal and Millennium Prize

In May 2006, a committee of nine mathematicians voted to award Perelman a Fields Medal for his work on the Poincaré conjecture. The Fields Medal is the highest award in mathematics; two to four medals are awarded every four years. However, Perelman declined to accept the prize. Sir John Ball, president of the International Mathematical Union, approached Perelman in Saint Petersburg in June 2006 to persuade him to accept the prize. After 10 hours of persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: "He proposed to me three alternatives: accept and come; accept and don't come, and we will send you the medal later; third, I don't accept the prize. From the very beginning, I told him I have chosen the third one... [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed." "'I'm not interested in money or fame,' he is quoted to have said at the time. 'I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me.'" Nevertheless, on 22 August 2006, Perelman was publicly offered the medal at the International Congress of Mathematicians in Madrid "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." He did not attend the ceremony, and declined to accept the medal, making him the first person to decline this prestigious prize.

He had previously turned down a prestigious prize from the European Mathematical Society, allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.

On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem. On June 8, 2010, he did not attend a ceremony in his honor at the Institut Océanographique, Paris to accept his $1 million prize. According to Interfax, Perelman refused to accept the Millennium prize in July 2010. He considered the decision of Clay Institute unfair for not sharing the prize with Richard Hamilton, and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."

Terence Tao spoke about Perelman's work on the Poincaré Conjecture during the 2006 Fields Medal Event:

Perelman's proof was rated one of the top cited articles in Math-Physics in 2008.

Possible withdrawal from mathematics

As of the spring of 2003, Perelman, no longer worked at the Steklov Institute. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; some even say that he has abandoned mathematics entirely. According to a 2006 interview, Perelman was then unemployed, living with his mother in Saint Petersburg.

Perelman is quoted in an article in ''The New Yorker'' saying that he is disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman's role in the proof and play up the work of Cao and Zhu. Perelman added, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." He has also said that "It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated."

This, combined with the possibility of being awarded a Fields medal, led him to quit professional mathematics. He has said that "As long as I was not conspicuous, I had a choice. Either to make some ugly thing or, if I didn't do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit." (The ''New Yorker'' authors explained Perelman's reference to "some ugly thing" as "a fuss" on Perelman's part about the ethical breaches he perceived).

It is uncertain whether his resignation from Steklov and subsequent seclusion mean that he has ceased to practice mathematics. Fellow countryman and mathematician Yakov Eliashberg said that, in 2007, Perelman confided to him that he was working on other things but it was too premature to talk about it. He is said to have been interested in the past in the Navier–Stokes equations and the set of problems related to them that also constitutes a Millennium Prize, and there has been speculation that he may be working on them now.

Perelman and media

Perelman avoided journalists and other media people. Masha Gessen, the author of a book about him, found it challenging to write about a person she never met.

In April 2011 Perelman gave his first interview to Aleksandr Zabrovsky, producer of "President-Film" studio and agreed to shoot a film about him, under the tentative title ''The Formula of the Universe''. A number of news outlets wrote about this interview, mentioning that Perelman finally explained why he rejected the one million prize:

I've learned how to calculate the voids; along with my colleagues we are getting to know the mechanisms for filling in the social and economic "voids". Voids are everywhere. They can be calculated, and this gives us great opportunities ... I know how to control the Universe. So tell me — why should I chase a million?

A number of journalists claimed that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.

Bibliography

; Dissertation

; Research papers

; Proof of the geometrization conjecture

; Books about Perelman Story of Grigory Perelman based on information from people who interacted with him.

See also

Homology sphere

"Manifold Destiny" (On ''The New Yorker'' article)

Soul theorem

Uniformization theorem

Asteroid 50033 Perelman

Notes

References

Anderson, M.T. 2005. Singularities of the Ricci flow. Encyclopedia of Mathematical Physics, Elsevier. (''Comprehensive exposition of Perelman's insights that lead to complete classification of 3-manifolds'')

The Associated Press, Erratum. Revised version (December 2006): Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (an account of Perelman's talk on his proof at MIT; pdf file; also see Sugaku Seminar 2003-10 pp 4–7 for an extended version in Japanese) (The author is a former Ph.D. student of Bill Thurston.)

External links

Fields Medals awarded at the 2006 IMU

International Congress of Mathematicians 2006

Perelman's arXiv eprints (link to APS mirror due to server strain on arxiv.org re Perelman)

Mathematics & Mechanics Faculty of Saint Petersburg State University

Petersburg Department of Steklov Institute of Mathematics

Notes and commentary on Perelman's Ricci flow papers

International Mathematical Olympiad 1982 (Budapest, Hungary)

Maths solution tops science class (Link to BBC News page)

Breakthrough Of The Year: The Poincaré Conjecture--Proved

Perelman refuses a million dollars to live in complete poverty

ICM 2006 Video Presentation

He Conquered the Conjecture essay on Perelman by John Allen Paulos from ''The New York Review of Books''

The man who refused a million dollars

Category:1966 births Category:Living people Category:Russian mathematicians Category:Russian Jews Category:20th-century mathematicians Category:21st-century mathematicians Category:Fields Medalists Category:Differential geometers Category:New York University staff Category:Fellows at the University of California, Berkeley Category:Topologists Category:Jewish scientists Category:People from Saint Petersburg Category:Saint Petersburg State University alumni

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Coordinates	40°26′30″N80°00′00″N
Name	William Thurston
Birth date	October 30, 1946
Nationality	American
Field	Mathematics
Work institutions	Cornell UniversityUniversity of California, DavisUniversity of California, BerkeleyPrinceton University
Alma mater	New College of FloridaUniversity of California, Berkeley
Doctoral advisor	Morris Hirsch
Doctoral students	Richard CanaryBenson FarbDavid GabaiWilliam GoldmanSteven KerckhoffYair Minsky
Prizes	Fields Medal (1982)Oswald Veblen Prize in Geometry (1976)National Academy of Sciences (1983) }}

William Paul Thurston (born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

Mathematical contributions

Foliations

His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. His more significant results include:

The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).

The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.

With John Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology.

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).

The geometrization conjecture

His later work, starting around the late 1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. This was the first example of a hyperbolic knot.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement. He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next revolutionary theorem.

Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem.

To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.

The geometrization theorem has been called ''Thurston's Monster Theorem,'' due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. A proof to that conjecture follows from the recent work of Grigori Perelman (2002–2003).

Orbifold theorem

In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Hamilton's work on the Ricci flow.

Education and career

Thurston was born in Washington, D.C. to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida) in 1967. For his undergraduate thesis he developed an intuitionist foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley, in 1972. His Ph.D. advisor was Morris W. Hirsch and his dissertation was on ''Foliations of Three-Manifolds which are Circle Bundles''.

After completing his Ph.D., he spent a year at the Institute for Advanced Study, then another year at MIT as Assistant Professor. In 1974, he was appointed Professor of Mathematics at Princeton University. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, he moved to University of California, Davis. In 2003, he moved again to become Professor of Mathematics at Cornell University.

His Ph.D. students include Richard Canary, Renaud Dreyer, David Gabai, William Goldman, Benson Farb, Detlef Hardorp, Craig Hodgson, Richard Kenyon, Steven Kerckhoff, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm, Richard Schwartz, Martin Bridgeman, William Floyd and Jeffrey Weeks. His son, Dylan Thurston, is an assistant professor of mathematics at Barnard College, Columbia University.

Thurston has turned his attention in recent years to mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and as head of The Geometry Center. As director of Mathematical Sciences Research Institute from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public.

In 2005 Thurston won the first AMS Book Prize, for ''Three-dimensional Geometry and Topology''. The prize ''recognizes an outstanding research book that makes a seminal contribution to the research literature''.

Thurston has an Erdős number of 2.

Selected works

William Thurston, ''The geometry and topology of 3-manifolds'', Princeton lecture notes (1978–1981).

William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5

William Thurston, ''Hyperbolic structures on 3-manifolds''. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246.

William Thurston, ''Three-dimensional manifolds, Kleinian groups and hyperbolic geometry'', Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.

William Thurston. ''On the geometry and dynamics of diffeomorphisms of surfaces''. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431

Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. ''Word processing in groups''. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. ISBN 0-86720-244-0

Eliashberg, Yakov M.; Thurston, William P. ''Confoliations''. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5

See also

References

External links

Thurston's page at Cornell

Category:Topologists Category:Members of the United States National Academy of Sciences Category:American mathematicians Category:Fields Medalists Category:University of California, Berkeley faculty Category:Cornell University faculty Category:University of California, Davis Category:New College of Florida alumni Category:1946 births Category:Living people Category:Sloan Research Fellowships Category:University of California, Berkeley alumni

zh-min-nan:William Thurston de:William Thurston es:William Thurston fr:William Thurston ko:윌리엄 서스턴 it:William Thurston ht:William Thurston nl:William Thurston ja:ウィリアム・サーストン no:William Thurston pnb:ولیم تھرسٹن pl:William Thurston pt:William Thurston ru:Тёрстон, Уильям Пол sk:William Thurston sl:William Thurston fi:William Thurston uk:Вільям Терстон zh:威廉·瑟斯顿

Tian Gang (; born November 1958) 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. He was born in Nanjing, China, was a professor of mathematics at MIT from 1995–2006 (holding the chair of Simons Professor of Mathematics from 1996), but now divides his time between Princeton University and Peking University. His employment at Princeton started from 2003, and now he is entitled Higgins Professor of Mathematics; starting 2005, he has been the director of Beijing International Center for Mathematical Research (BICMR).

Biography

Tian graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, after having studied under Shing-Tung Yau. This work was so exceptional he was invited to present it at the Geometry Festival that year. In 1998, he was appointed as a Cheung Kong Scholar professor at the School of Mathematical Sciences at Peking University, under the "Cheung Kong Scholars Programme" (长江计划) of the Ministry of Education. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award in 1994, and the Veblen Prize in 1996. In 2004 Tian was inducted into the American Academy of Arts and Sciences.

Mathematical contributions

Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds under the direct of Yau. In particular he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern class, and showed that hypersurfaces with a Kähler–Einstein metric are stable in the sense of geometric invariant theory. He proved that a Kähler manifold with trivial canonical bundle has trivial obstruction space, known as the Bogomolov–Tian–Todorov theorem.

He (jointly with Jun Li) constructed the moduli spaces of maps from curves in both algebraic geometry and symplectic geometry and studied the obstruction theory on these moduli spaces. He also (jointly with Y. Ruan) showed that the quantum cohomology ring of a symplectic manifold is associative.

In 2006, together with John Morgan of Columbia University, amongst others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman.

Publications

(Selected)

Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629--646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.

Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with $C\sb 1(M)>0$. Invent. Math. 89 (1987), no. 2, 225--246.

Tian, G.; Yau, Shing-Tung. Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), no. 3, 579--609.

Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101--172.

Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32 (1990), no. 1, 99--130.

Ruan, Yongbin; Tian, Gang. A mathematical theory of quantum cohomology. J. Differential Geom. 42 (1995), no. 2, 259--367.

Tian, Gang. Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), no. 1, 1--37.

Ruan, Yongbin; Tian, Gang. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130 (1997), no. 3, 455--516.

Li, Jun; Tian, Gang. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc. 11 (1998), no. 1, 119--174.

Liu, Gang; Tian, Gang. Floer homology and Arnold conjecture. J. Differential Geom. 49 (1998), no. 1, 1--74.

Liu, Xiaobo; Tian, Gang. Virasoro constraints for quantum cohomology. J. Differential Geom. 50 (1998), no. 3, 537--590.

Tian, Gang. Gauge theory and calibrated geometry. I. Ann. of Math. (2) 151 (2000), no. 1, 193--268.

Tian, Gang; Zhu, Xiaohua. Uniqueness of Kähler-Ricci solitons. Acta Math. 184 (2000), no. 2, 271--305.

Cheeger, J.; Colding, T. H.; Tian, G. On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12 (2002), no. 5, 873--914.

Tao, Terence; Tian, Gang. A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Amer. Math. Soc. 17 (2004), no. 3, 557--593.

Tian, Gang; Viaclovsky, Jeff. Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160 (2005), no. 2, 357--415.

Cheeger, Jeff; Tian, Gang. Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. Vol. 19, No. 2 (2006), 487--525.

Morgan, John; Tian, Gang. Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007, 525pp.

Song, Jian; Tian, Gang. The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170 (2007), no. 3, 609--653.

Chen, X. X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1--107.

Kołodziej, Sławomir; Tian, Gang A uniform $L^\infty$L∞ estimate for complex Monge-Ampère equations. Math. Ann. 342 (2008), no. 4, 773–787.

Mundet i Riera, I.; Tian, G. A compactification of the moduli space of twisted holomorphic maps. Adv. Math. 222 (2009), no. 4, 1117–1196.

Rivière, Tristan; Tian, Gang The singular set of 1-1 integral currents. Ann. of Math. (2) 169 (2009), no. 3, 741–794.

Tian, Gang Finite-time singularity of Kähler-Ricci flow. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1137–1150.

Students

Vladimir Bozin, MIT, 2004;

Xiaodong Cao, MIT, 2002;

Sandra Francisco, MIT, 2005;

Zuoliang Hou, MIT, 2004;

Ljudmila Kamenova, MIT, 2006;

Peng Lu State, University of New York at Stony Brook, 1996;

Zhiqin Lu, New York University, 1997;

Dragos Oprea, MIT, 2005;

Yanir Rubinstein, MIT, 2008;

Sema Salur, Michigan State University, 2000;

Bianca Santoro, MIT, 2006;

Natasa Sesum, MIT, 2004;

Jake Solomon, MIT, 2006;

Michael Usher, MIT, 2004;

Lijing Wang, MIT, 2003;

Hao Wu, MIT, 2004;

Zhiyu Wu, Columbia University, 1998;

Baozhong Yang, MIT, 2000;

Zhou Zhang, MIT, 2006;

Hans-Joachim Hein, Princeton, 2010;

Richard Bamler, Princeton, 2011;

Chi Li, Princeton, 2011;

Mohammad Farajzadeh Tehrani, Princeton 2011;

Giulia Saccà, Princeton, 2012;

Guangbo Xu, Princeton 2012;

References

External links

Veblen prize citation

M.I.T. home page for Gang Tian (probably out of date).

Category:1958 births Category:20th-century mathematicians Category:21st-century mathematicians Category:Chinese mathematicians Category:Harvard University alumni Category:Members of the Chinese Academy of Sciences Category:Peking University alumni Category:Peking University faculty Category:Fellows of the American Academy of Arts and Sciences Category:Princeton University faculty Category:Living people

de:Gang Tian ja:田剛 pt:Gang Tian zh:田刚