Grigori Perelman

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Grigori Yakovlevich Perelman
Born June 13, 1966 (1966-06-13) (age 42)
Leningrad, USSR
Fields Mathematician
Alma mater Leningrad State University
Known for Riemannian geometry and geometric topology
Notable awards Fields Medal (2006), declined

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 who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston's geometrization conjecture. 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 until it was solved.

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.

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.[2]


[edit] Early life and education

Grigori Perelman was born in Leningrad (now Saint Petersburg) on 13 June 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 Soviet 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".

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 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 the Courant Institute in New York University and Stony Brook University. From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. 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 the summer of 1995.

He has a younger sister, Elena, who is also a mathematician. She received a PhD from Weizmann Institute of Science and is a biostatician at Karolinska Institutet.

Perelman is a talented violinist and also plays table tennis.[4]

[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 a short and elegant proof of the soul conjecture.

[edit] 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 sphere in three dimensions can be contracted to a point; the Poincaré conjecture surmises that any closed three-dimensional manifold where any loop can be contracted to a point, 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 Solution of the Poincaré conjecture for a layman's description of the mathematics.

Perelman modified Richard Hamilton's program for a proof of the conjecture, in which the central idea is the notion of the Ricci flow (named after the Italian mathematician Gregorio Ricci-Curbastro). 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 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 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 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, any singularity which 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.

[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]

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.[5] 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."
  2. 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]
  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.[7] On 24 August 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincaré conjecture.[8] This was followed up with the paper on the arXiv, "Completion of the Proof of the Geometrization Conjecture" on 24 September 2008.[9].

A separate check was also performed by Richard Hamilton with Gerhard Huisken and Tom Ilmanen. Giving the first plenary lecture at the ICM2006, he said, "I am enormously grateful with Grisha for doing this".

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."[10]

[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 Saint 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… [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed."[3]

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".[11] He did not attend the ceremony, and declined to accept the medal, making him the first person in history to decline this prestigious prize.[12][13]

He had previously turned down a prestigious prize from the European Mathematical Society,[13] allegedly saying that he felt the prize committee was unqualified to assess his work, even positively.[10]

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 Poincaré Conjecture during the 2006 Fields Medal Event [1]:

They [the Millennium Prize Problems] are like these huge cliff walls, with no obvious hand holds. I have no idea how to get to the top. [Perelman's proof of the Poincaré Conjecture] is a fantastic achievement, the most deserving of all of us here in my opinion. Most of the time in mathematics you look at something that's already been done, take a problem and focus on that. But here, the sheer number of breakthroughs...well it's amazing.

[edit] Withdrawal from mathematics

As of the spring of 2003, Perelman no longer works at 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.[14] According to a 2006 interview, Perelman is currently jobless, living with his mother in Saint 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) (in Russian). Седловые поверхности в евклидовых пространствах:Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук. 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". Saint Petersburg Math. J. 5 (1): 215–227. 

Perelman's proof of the geometrization conjecture:

  • Perelman, Grisha (11 November 2002). The entropy formula for the Ricci flow and its geometric applications. arΧiv:math.DG/0211159. 
  • Perelman, Grisha (10 March 2003). Ricci flow with surgery on three-manifolds. arΧiv:math.DG/0303109. 
  • Perelman, Grisha (17 July 2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arΧiv:math.DG/0307245. 

[edit] See also

[edit] Notes

  1. ^ "Fields Medals 2006". International Mathematical Union (IMU) - Prizes. Retrieved on 2006-04-30. 
  2. ^ "The Poincaré Conjecture--Proved". BREAKTHROUGH OF THE YEAR. 22 December 2006. Retrieved on 2006-12-29. 
  3. ^ a b c d e f g h Nasar and Gruber.
  4. ^ a b c Lobastova and Hirsh
  5. ^ Bruce Kleiner, John Lott Notes on Perelman's papers Geometry & Topology 12 (2008) 2587–2855, doi:10.2140/gt.2008.12.2587, arΧiv:math/0605667
  6. ^ Cao and Zhu.
  7. ^ John W. Morgan, Gang Tian Ricci Flow and the Poincaré Conjecture arΧiv:math/0607607
  8. ^ Schedule of the scientific program of the ICM 2006
  9. ^ John W. Morgan, Gang Tian Completion of the Proof of the Geometrization Conjecture arΧiv:0809.4040v1
  10. ^ a b Randerson.
  11. ^ "Fields Medal - Grigory Perelman" (PDF). International Congress of Mathematicians 2006. 22 August 2006. 
  12. ^ Mullins.
  13. ^ a b "Maths genius declines top prize". BBC News. 22 August 2006. 
  14. ^ Главные новости ::

[edit] References

[edit] External links

NAME Perelman, Grigori Yakovlevich
DATE OF BIRTH 13 June 1966
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