From Wikipedia, the free encyclopedia

Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман) (born 13 June 1966 in Leningrad, USSR) is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, it appears that he has proven Thurston's geometrization conjecture. If so, this solves in the affirmative the famous Poincaré conjecture, which has been regarded for one hundred years as one of the most important (and most difficult) open problems in mathematics.

In August 2006, Perelman was awarded the Fields Medal [1], which is widely considered to be the top honor a mathematician can receive. However, he declined to accept the award or appear at the congress.

Perelman is said to be a very private individual who has consistently tried to avoid the public eye. However, some facts about his career do appear to be verifiable.

Grigori Perelman was born on June 13, 1966. He comes from a Jewish family, and has a sister named Elena. ^[1] His early mathematical education occurred at the world-famous Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. In 1982, as a member of the USSR team competing in the International Mathematical Olympiad, an international competition for high school students, he won a gold medal, achieving a perfect score. He also achieved a perfect score at his Mensa International IQ entry test.^{[citation needed]} In the late 1980s, Perelman went on to earn a Candidate of Science degree (the Russian equivalent to the Ph.D.) at the Mathematics & Mechanics Faculty of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was entitled "Saddle surfaces in Euclidean spaces" (see citations below).

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences in St. Petersburg, Russia. His advisors at the Steklov Institute were Aleksandr Danilovich Aleksandrov and Yuri Dmitrievich Burago. In the late 80s and early 90s, Perelman held posts at several universities in the United States. He returned to the Steklov Institute in 1996.

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

In November 2002, Perelman astounded the mathematical world by posting to the arXiv the first of a series of eprints in which he claimed to have outlined a proof of Thurston's geometrization conjecture, a result that includes the Poincaré conjecture as a particular case.

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, has generally been considered to constitute the most famous open problem in topology. It states that any closed simply connected 3-dimensional manifold is homeomorphic to the standard 3-dimensional sphere. In the twentieth century, many leading mathematicians tried to prove the Poincaré conjecture--- beginning with Poincaré himself. All of them failed. However, an analogue of the Poincaré conjecture was finally proven for manifolds of dimension greater than four by Stephen Smale in 1960, and for manifolds of four dimensions by Michael Freedman in 1983. Both Smale and Freedman were awarded the highest honor in mathematics, the Fields Medal, for their work.

The case of three manifolds, however, turns out to be the hardest of all, roughly speaking because in topologically manipulating a three-manifold, there are too few dimensions to move "problematical regions" out of the way without interfering with something else.

In 1999, the Clay Mathematics Institute announced a one million dollar prize for the proof of several conjectures (these are known collectively as the Millennium Prize Problems), including the Poincaré conjecture. There is universal agreement that a successful proof would constitute a landmark event in the history of mathematics, fully comparable with the proof by Andrew Wiles of Fermat's Last Theorem (FLT), but possibly even more far-reaching.

Interestingly enough, very broadly interpreted, there is a common thread between the geometrization conjecture and one way of thinking about FLT: both can be said to concern a kind of uniformization. The geometrization conjecture is often considered to be the analogue for 3-manifolds of the uniformization theorem, which concerns two-manifolds, or surfaces. Similarly, according to Barry Mazur, FLT can be related to uniformization (see the article cited below).

Perelman's plan of attack on the geometrization conjecture involves a modification of 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 delightfully intuitive: 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 work its magic, eventually one should in principle obtain a kind of "normal form". According to Thurston, this normal form must take one of a small number of possibilities, each having a different flavor of geometry, called Thurston model geometries.

To adopt a somewhat different metaphor, this is like formulating a kind of dynamical process which gradually "perturbs" a given square matrix and which is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea had attracted a great deal of attention, but no-one could prove that the would not "hang up" by developing "singularities"--- at least, not 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.

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

Since 2003, Perelman's program has attracted increasing attention from the mathematical community. In the spring of 2003, he accepted an invitation to visit MIT and the State University of New York at Stony Brook, where he gave a series of talks on his work. However, after his return to Russia, he is said to have gradually stopped responding to emails from his colleagues.

As of August 2006, more formal versions of Perelman's purported proof are still being scrutinized. Several leading mathematicians have been involved, including Richard Hamilton, S. T. Yau, Michael Anderson, John Morgan (Columbia University), Robert Greene (UCLA) , Bruce Kleiner (Yale University), Gang Tian (Princeton University), John Lott (University of Michigan at Ann Arbor, MI), Huai-Dong Cao (Lehigh University), Xi-Ping Zhu (Zhongshan University), Gérard Besson and Laurent Bessières (both at Joseph Fourier University in Grenoble).

A consensus appears to have developed that Perelman's outline can indeed be expanded into a complete proof of the geometrization conjecture. Kleiner and Lott have written a long paper containing part of the expansion, Cao and Zhu have published a detailed paper in the Asian Journal of Mathematics, and Morgan and Tian have written a book manuscript focusing on the parts which are needed to prove the Poincare conjecture. Several recent news stories seem to establish that there is now a consensus that Perelman has indeed proven the geometrization conjecture:

There is a growing feeling, a cautious optimism that [mathematicians] have finally achieved a landmark not just of mathematics, but of human thought.

— Dennis Overbye, "An Elusive Proof and Its Elusive Prover", New York Times, 15 August 2006

Dr Perelman seems to have succeeded where so many failed. "I think for many months or even years now people have been saying they were convinced by the argument," said Nigel Hitchin, professor of mathematics at Oxford University. "I think it's a done deal."

— James Randerson, Meet the cleverest man in the world (who's going to say no to a $1m prize), The Guardian, August 16, 2006

On August 22, 2006, Perelman was offered in absentia a Fields Medal at the International Congress of Mathematicians in Madrid. The Fields Medal is the highest award in mathematics; two to four medals are awarded every four years. Perelman was offered the award "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow".^[2] However, he did not attend the ceremony,^[3] and he has declined to accept the medal.^[4]

Perelman is also due to receive a share of a Millennium Prize (probably to be shared with Hamilton). However, he has not pursued formal publication in a peer-reviewed mathematics journal of his proof, as the rules for this prize currently require. Nonetheless, many mathematicians seem to feel that the scrutiny to which his eprints outlining his alleged proof have already been subjected far exceeds the "proof-checking" implicit in a normal peer review, and the Clay Mathematics Institute has explicitly stated that the governing board which awards the prizes may change the formal requirements, in which case Perelman would presumably become eligible to receive a share of the prize.

Perelman, however, may be uninterested in either honors or money. He has consistently been described by those who know him as shy and unworldly. He has 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).

According to a recent story aired by BBC News:

John Ball, retiring president of the International Mathematical Union, said he had travelled to St Petersburg to meet Perelman in person to try to understand his reasons for declining the award. Professor Ball said he had spoken to Perelman of personal experiences with the mathematical community during his career that had caused him to remain at a distance… Manuel de Leon, chairman of the ICM said: "The reason Perelman gave me is that he feels isolated from the mathematical community and therefore has no wish to appear as one of its leaders."

— Maths genius declines top prize, BBC News, 22 August 2006

According to various sources, in the spring of 2003, Perelman suffered a bitter personal blow when the faculty of the Steklov Institute allegedly declined to re-elect him as a member,^[5] apparently in part out of continuing doubt over his claims regarding the geometrization conjecture. 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.^[6] According to a recent interview, Perelman is currently jobless, living with his mother in St Petersburg, and subsisting on her modest pension.^[7] A recent newspaper article quotes Marcus Du Sautoy (Oxford University) as saying:

"He has sort of alienated himself from the maths community… He has become disillusioned with mathematics, which is quite sad. He's not interested in money. The big prize for him is proving his theorem."

— James Randerson, Meet the cleverest man in the world (who's going to say no to a $1m prize), The Guardian, August 16, 2006

This circumstance has reminded some observers of previous examples of voluntary withdrawals of extremely talented mathematicians from the mathematical scene. In particular, in the 1960s, another Fields medalist, Alexander Grothendieck, resigned from a prestigious appointment and shortly thereafter withdrew entirely from formal participation in mathematics.

Перельман, Григорий Яковлевич (1990). Седловые поверхности в евклидовых пространствах:Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук (in Russian). Ленинградский Государственный Университет. (Perelman's dissertation) (Russian)

Perelman, G., Yu. Burago, M. Gromov (1992). "Aleksandrov spaces with curvatures bounded below". Russian Math Surveys 47 (2): 1-58.

Perelman, G. (1993). "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers". Comparison Geometry 30: 157-163. Retrieved on 2006-08-23.

Perelman, G. (1994). "Proof of the soul conjecture of Cheeger and Gromoll". J. Differential Geom. 40: 209-212.

Perelman, G. (1994). "Elements of Morse theory on Aleksandrov spaces". St. Petersbg. Math. J. 5 (1): 205-213.

Perelman, G.Ya., Petrunin, A.M. (1994). "Extremal subsets in Alexandrov spaces and the generalized Liberman theorem". St. Petersburg Math. J. 5 (1): 215-227.

Perelman, G., Buslaev, V. (1995). "On the stability of solitary waves for nonlinear Schrodinger equations". Am. Math. Soc. Transl. 164 (2): 75-98.

Perelman's proof of the geometrization conjecture:

Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications." November 11.

Perelman, Grisha (2003). "Ricci flow with surgery on three-manifolds." March 10.

Perelman, Grisha (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds." July 17.

Clay Mathematics Institute

Fields Medal

Geometrization conjecture

Homology sphere

Poincaré conjecture

Ricci curvature

Ricci flow

Soul theorem

Uniformization theorem