Grigori Perelman

Grigori Yakovlevich Perelman (Russian: Григо́рий Я́ковлевич Перельма́н; IPA: [ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman] /ˈpɛrᵻlmən/ PERR-il-mən; born 13 June 1966) is a Russian mathematician who made landmark contributions to Riemannian geometry and geometric topology before apparently withdrawing from mathematics.

In 1994, Perelman proved the soul conjecture. In 2003, 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^[1] 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, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."^[2] On 22 December 2006, the scientific 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.^[3]

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize^[4] for resolution of the Poincaré conjecture. On 1 July 2010, he turned down the prize of one million dollars, saying that he considered the award unfair and that his contribution to solving the Poincaré conjecture was no greater than that of Richard Hamilton, the mathematician who pioneered Ricci flow with the aim of attacking the conjecture.^[5][6] He also turned down the prestigious prize of the European Mathematical Society.^[7]

Early life and education

Grigori Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents^[8][9][10] Yakov (who now lives in Israel)^[8] and Lyubov. Grigori's mother Lyubov 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.^[11]

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.^[12] 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.^[13] He had been admitted to the School of Mathematics and Mechanics at the Leningrad State University without admission examinations for this endeavour and enrolled to the university, participated in numerous higher school student competitions, and was awarded the Lenin scholarship. In 1990, Perelman went on to earn a Candidate of Sciences degree (the Soviet equivalent to the PhD) at the same School of Mathematics and Mechanics of the Leningrad State University, one of the leading universities in the former Soviet Union, from where he had graduated. 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 Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below.^[14] In 1992, he was invited to spend a semester each at the Courant Institute in New York University and Stony Brook University 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.^[11]

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

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 3-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 3-sphere. The analogous result has been known to be true in dimensions greater than or equal to five since 1960 as in the 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 2000, 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.^[15][16][17]

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 then one should, in principle, eventually 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.

It was known 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 the Massachusetts Institute of Technology, Princeton University, Stony Brook University, Columbia University and New York University, where he gave a series of talks on his work.^[18]

Three independent groups of scholars have verified that Perelman's papers contain all the essentials for a complete proof of the geometrization conjecture:

1. 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.^[19] 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."
2. In June 2006, the Asian Journal of Mathematics published a paper by Zhu Xiping 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."^[20] (Asked about the paper, Perelman said the pair had not contributed anything original, and had simply reworked his proof because they "did not quite understand the argument".)^[18]
   1. 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.^[21]
   2. 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"^[22] and posted a revised version, renamed, more modestly, "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture".^[23] 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.
3. 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.^[24] On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture.^[25] This was followed up with the paper on the arXiv, "Completion of the Proof of the Geometrization Conjecture" on 24 September 2008.^[26]

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.^[18] 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 attempted 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."^[18] "'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.'"^[27] 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."^[28] He did not attend the ceremony, and declined to accept the medal, making him the only person to decline this prestigious prize.^[7][29]

He had previously turned down a prestigious prize from the European Mathematical Society.^[7]

On 18 March 2010, Perelman was awarded a Millennium Prize for solving the problem.^[30] On June 8, 2010, he did not attend a ceremony in his honor at the Institut Océanographique, Paris to accept his $1 million prize.^[31] 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,^[5] and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."^[6]

Perelman's proof was rated one of the top cited articles in Math-Physics in 2008.^[32]

Possible withdrawal from mathematics

Perelman quit his job at the Steklov Institute in December 2005.^[33] 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.^[34]

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."^[18] 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."^[18]

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).^[35]

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.^[36]

Perelman and media

Perelman has avoided journalists and other members of the media. Masha Gessen, the author of Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century, a book about him, was unable to meet him.^[37]

A Russian documentary about Perelman in which his work is discussed by several leading mathematicians including Mikhail Gromov was released in 2011 under the title "Иноходец. Урок Перельмана," "The Man Who Walks Differently: Perelman's Lesson."

In April 2011, Aleksandr Zabrovsky, producer of "President-Film" studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe.^[38] Zabrovsky says that in the interview,^[39] Perelman explained why he rejected the one million dollar prize.^[38] A number of journalists^[40][41][42] believe that Zabrovky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.

The writer Brett Forrest briefly interacted with Perelman in 2012.^[43][44]

Some Russian media wrote that Perelman had a job in Sweden since 2014.^[45][46][47]

Bibliography

Dissertation

• Перельман, Григорий Яковлевич (1990). Седловые поверхности в евклидовых пространствах (Saddle surfaces in Euclidean spaces): Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук (in Russian). Ленинградский государственный университет. 

Research papers

• Burago, Yu. D.; Gromov, M. L.; Perelman, G. Ya. (1992). "A. D. Aleksandrov spaces with curvatures bounded below". Russian Mathematical Surveys 47 (2): 1–58. doi:10.1070/RM1992v047n02ABEH000877. 
• Perelman, G. (1993). "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers" (PDF). Comparison Geometry 30: 157–163. Retrieved August 23, 2006. 
• Perelman, G. (1994). "Proof of the soul conjecture of Cheeger and Gromoll" (PDF). Journal of Differential Geometry 40 (1): 209–212. MR 1285534. Archived from the original (PDF) on March 1, 2012. 
• Perelman, G. (1993). "Collapsing with No Proper Extremal Subsets". Comparison Geometry 30: 157–163. 
• Perelman, G. (1994). "Elements of Morse theory on Aleksandrov spaces". Saint Petersburg Mathematical Journal 5 (1): 205–213. 
• Perelman, G. Ya.; Petrunin, A. M. (1994). "Extremal subsets in Alexandrov spaces and the generalized Liberman theorem". Saint Petersburg Mathematical Journal 5 (1): 215–227. 

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 [math.DG]. 
• Perelman, Grisha (March 10, 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109 [math.DG]. 
• Perelman, Grisha (July 17, 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245 [math.DG]. 

See also

• Homology sphere
• "Manifold Destiny" (On The New Yorker article)
• Uniformization theorem
• Asteroid 50033 Perelman

Notes

References

Further reading

• Gessen, Masha (2009). Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. Houghton Mifflin Harcourt. ISBN 978-0-15-101406-4. Retrieved December 12, 2012.  (Story of Grigory Perelman based on information from people who interacted with him.)

External links

• O'Connor, John J.; Robertson, Edmund F., "Grigori Perelman", MacTutor History of Mathematics archive, University of St Andrews .
• Grigori Perelman at the Mathematics Genealogy Project
• Grigori Perelman's results at the International Mathematical Olympiad