Grigori Perelman

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". The Fields Medal is widely considered to be the top honor a mathematician can receive. However, he declined to accept the award or appear at the congress.

On December 22, 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.^[2]

[edit] Early life and education

Grigori Perelman was born in Leningrad (now St. Petersburg) to a Jewish family on June 13, 1966. His early mathematical education occurred at the 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.^[3] 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". He is also a talented violinist and plays table tennis.^[4]

After graduation, Perelman began work at the renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences. 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. In 1992, he was invited to spend a semester each at New York University and Stony Brook University. From there, he accepted a two-year fellowship at the University of California, Berkeley in 1993. He was offered jobs at several top universities in US including Princeton and Stanford, but he rejected them all and returned to the Steklov Institute in the summer of 1995.

[edit] 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.

[edit] The problem

Main article: Poincaré conjecture

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was 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, but the case of three-manifolds had 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 – 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.

[edit] 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. See the Hamilton–Perelman solution of the Poincaré conjecture for a layman's description of the mathematics.

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.

[edit] 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.^[3]

As 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."

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.^[5]

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.^[6] According to the Fields medalist Shing-Tung Yau "Cao and Zhu put the finishing touches to the complete proof of the Poincaré Conjecture"^[7]. 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." ^[8]

On December 3, 2006, in response to plagiarism charges, Cao and Zhu retracted their original paper titled, “A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow” and renamed it more modestly, "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture." ^[9]. They also took the phrase "crowning achievement" out of the abstract.^[9]

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.^[10] On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture.^[11]

The above work demonstrates that Perelman's outline can indeed be expanded into a complete proof of the geometrization conjecture.

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."^[12]

[edit] 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.^[3] 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."^[3]

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".^[13] He did not attend the ceremony, and declined to accept the medal, making him the first person in history to decline this prestigious prize.^[14][15]

He had previously turned down a prestigious prize from the European Mathematical Society,^[15] allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.^[12]

Perelman may also be due to receive a share (or the totality) of a Millennium Prize. The rules for this prize - which can be changed, as stated by a member of the advisory board of the Clay Mathematics Institute - require his proof to be published in a peer-reviewed mathematics journal. While Perelman has not pursued publication himself, other mathematicians have published papers about the proof. This may make Perelman eligible to receive a share or the whole of a prize. Perelman has stated that "I’m not going to decide whether to accept the prize until it is offered."^[3]

Terence Tao spoke about Perelman's work on the Poincare Conjecture during the 2006 Fields Medal Event [1]:

[edit] Withdrawal from mathematics

As of the spring of 2003 Perelman no longer works in the Steklov Institute.^[4] 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.^[16] According to a recent interview, Perelman is currently jobless, living with his mother in St Petersburg.^[4]

Although Perelman says in a The New Yorker article that he is disappointed with the ethical standards of the field of mathematics, the article implies that Perelman refers particularly to Yau's efforts to downplay his role in the proof and play up the work of Cao and Zhu. Perelman 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."^[3] 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."^[3]

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.”^[3]

[edit] 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)

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

[edit] Notes

[edit] References

[edit] See also

• Homology sphere
• Manifold Destiny (On The New Yorker article)
• Soul theorem
• Uniformization theorem

[edit] External links

• Grigori Perelman at the Mathematics Genealogy Project
• 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)
• Maths solution tops science class (Link to BBC News page)
• BREAKTHROUGH OF THE YEAR: The Poincaré Conjecture--Proved