Grigori Perelman

File:Grigori Perelman.jpg

Grigori Perelman.

Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман), born 13 June 1966 in Leningrad, USSR (now St. Petersburg, Russia), sometimes known as Grisha Perelman, is a Russian mathematician of Jewish origin, 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, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics.

In August 2006, Perelman was awarded the Fields Medal, 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.

Early life and education

Grigori Perelman was born in Leningrad (now St. Petersburg) on June 13, 1966. 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. In the late 1980s, Perelman went on to earn a Candidate of Science degree (the Russian equivalent to the Ph.D.) at the Mathematics and 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). He was also a talented violinist and played table tennis.

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.

He has stated that he prefers to stay out of the limelight, saying that "I do not think anything that I say can be of the slightest public interest. I am not saying that because I value my privacy, or that I am doing anything I want to hide. There are no top-secret projects going on here. I just believe the public has no interest in me."

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 the proof of the Soul conjecture.

The problem

Main article: Poincaré conjecture

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, is the most famous open problem in topology. Loosely speaking, the conjecture surmises that if a closed three-dimensional manifold is sufficiently like a sphere in that each loop in the manifold can be tightened to a point, then it is really just a three-dimensional sphere. The analogous result has been known to be true in higher dimensions for some time, however the case of three-manifolds has turned out to be the hardest of them 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 the Millennium Prize Problems – a one million dollar prize for the proof of several conjectures, 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, but possibly even more far-reaching.

Perelman's proof

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

Perelman modifies 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 work its magic, 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 flavor of geometry, called Thurston model geometries.

This is similar to formulating a 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 process would not "hang up" 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.

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.

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 Harvard University, 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.

On 25 May 2006, Bruce Kleiner and John Lott, both of the University of Michigan, posted a paper on arXiv that claims to fill in the details of Perelman's proof of the Geometrization conjecture.

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, claiming to give a complete proof of the Poincaré and the geometrization conjectures. According to the Fields medalist Shing-Tung Yau this paper was aimed at "putting the finishing touches to the complete proof of the Poincaré Conjecture".

The true extent of the contribution of Zhu and Cao, as well as the ethics of Yau's involvement, has been controversial. Yau is both an editor-in-chief of the Asian Journal of Mathematics as well as Cao's doctoral advisor. Sylvia Nasar and David Gruber, writing for the New Yorker, has suggested that Yau was intent on being associated, directly or indirectly, with the proof of the conjecture and pressured the journal's editors to accept Zhu and Cao's paper on unusually short notice. Others have wondered if the "short time between the submission date...and the date when it was accepted for publication" by the journal was sufficient to let the paper be "refereed in a serious way". Yau has responded the paper had been refereed in the usual manner and that the journal "has very high standards." [1] Cao has stated, "Hamilton and Perelman have done the most important fundamental works. They are the giants and our heroes. In my mind there is no question at all that Perelman deserves the Fields Medal. We just follow the footsteps of Hamilton and Perelman and explain the details. I hope everyone who read our paper would agree that we have given a rather fair account." Cao also defended Yau by saying that Yau had remarked Perelman was deserving of the Fields Medal, including to reporters from the New Yorker. [2]

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 claim to provide a "detailed proof of the Poincaré Conjecture". On 24 Aug 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture.

The above work seems to demonstrate that Perelman's outline can indeed be expanded into a complete proof of the geometrization conjecture:

Dennis Overbye of the New York Times has said that "there is a growing feeling, a cautious optimism that [mathematicians] have finally achieved a landmark not just of mathematics, but of human thought." 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.

Sir John Ball, president of the International Mathematical Union, approached Perelman in St. Petersburg in June 2006 to persuade him to accept the prize. After 10 hours of persuading over two days, he gave up. Two weeks later, Perelman summed up the conversation as: "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." He went on to say that the prize "was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed."

On August 22, 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.

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.

Perelman is also due to receive a share of a Millennium Prize (probably to be shared with Hamilton). While he has not pursued formal publication in a peer-reviewed mathematics journal of his proof, as the rules for this prize require, many mathematicians feel that the scrutiny to which his eprints outlining his alleged proof have been subjected exceeds the "proof-checking" implicit in a normal peer review. 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 become eligible to receive a share of the prize. Perelman has stated that "I’m not going to decide whether to accept the prize until it is offered."

Withdrawal from mathematics

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, 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. According to a recent interview, Perelman is currently jobless, living with his mother in St Petersburg, and subsisting on her modest pension.

He has stated that he is disappointed with mathematics' ethical standards, in particular of Yau's effort to downplay his role in the proof and up-play the work of Cao and Zhu. He has said that "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" (a fuss about the mathematics community's lack of integrity) "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.”

Professor Marcus du Sautoy of Oxford University has said that "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."

Bibliography

• 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) Template:Ru icon

• Perelman, G. (1992). "Aleksandrov spaces with curvatures bounded below". Russian Math Surveys. 47 (2): 1–58. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Perelman, G. (1993). "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers" (PDF). Comparison Geometry. 30: 157–163. Retrieved 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. (1994). "Extremal subsets in Alexandrov spaces and the generalized Liberman theorem". St. Petersburg Math. J. 5 (1): 215–227. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Perelman's proof of the geometrization conjecture:

• Perelman, Grisha (November 11 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159. {{cite journal}}: Check date values in: |date= (help); Cite journal requires |journal= (help)
• Perelman, Grisha (March 10 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109. {{cite journal}}: Check date values in: |date= (help); Cite journal requires |journal= (help)
• Perelman, Grisha (July 17 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245. {{cite journal}}: Check date values in: |date= (help); Cite journal requires |journal= (help)

See also

• Clay Mathematics Institute
• Fields Medal
• Geometrization conjecture
• Homology sphere
• Manifold Destiny (On the New Yorker article)
• Poincaré conjecture
• Ricci curvature
• Ricci flow
• Soul theorem
• Uniformization theorem

References

• Robinson, Sara (2003-04-15). "Russian Reports He Has Solved a Celebrated Math Problem". The New York Times. {{cite news}}: |access-date= requires |url= (help); Check date values in: |date= (help); External link in |title= (help)
• Weisstein, Eric (2004-04-15). "Poincaré Conjecture Proved--This Time for Real". Mathworld. Retrieved 2006-08-22. {{cite web}}: Check date values in: |date= (help)
• Kusher, Rob. "Witnesses to Mathematical History Ricci Flow and Geometry" (PDF). Retrieved 2006-08-22. (an account of Perelman's talk on his proof at MIT; pdf file)
• The Associated Press, "Russian may have solved great math mystery". CNN. July 1, 2004. {{cite news}}: |access-date= requires |url= (help); Check date values in: |date= (help); External link in |title= (help)
• Schecter, Bruce (17 July, 2006). "Taming the fourth dimension". New Scientist. 183 (2456). {{cite journal}}: Check date values in: |date= (help)
• Begley, Sharon (July 21, 2006). "Major math problem is believed solved". Wall Street Journal. {{cite news}}: Check date values in: |date= (help); External link in |title= (help)
• Overbye, Dennis (2006-08-15). "An Elusive Proof and Its Elusive Prover". New York Times. {{cite news}}: |access-date= requires |url= (help); Check date values in: |date= (help); External link in |title= (help)
• Lobastova, Nadejda (2006-08-21). "Jobless maths whiz living with mother". The Age via The Telegraph. {{cite news}}: |access-date= requires |url= (help); Check date values in: |date= (help); External link in |title= (help)
• Weeks, Jeffrey R. (2002). The Shape of Space. New York: Marcel Dekker. ISBN 0-824-70709-5. (The author is a former Ph.D. student of Bill Thurston.)
• Collins, Graham P. (2004). "The Shapes of Space". Scientific American (July): 94–103.

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)
• Staff listing for Perelman at Petersburg Department of Steklov Institute of Mathematics
• Mathematics & Mechanics Faculty of St. 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) Individual Scores