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Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман) (born 13 June 1966 in Leningrad, USSR) is a Russian Jewish mathematician who won (but declined to accept) a Fields Medal in 2006 for his important contributions to Ricci flow. In particular, an increasing number of knowledgeable mathematicians appear to believe that he has proven the Poincaré conjecture, which is universally held to be one of the most important open problems in mathematics.

Early life and education

Perelman's early mathematical education was at the world-famous Leningrad School #239, which specializes in advanced mathematics and physics programs. As a high school student, he was admitted to Mensa International, and won a gold medal with a perfect score at the International Mathematical Olympiad in 1982, representing the USSR team. He earned his Candidate of Science degree (Ph.D. equivalent in the USSR and Russia) at the Mathematics & Mechanics Faculty of the Leningrad State University, one of the leading universities in the former Soviet Union, in the late 1980s, his dissertation theme being "Saddle surfaces in Euclidean spaces" (Template:Lang-ru) . After graduation, Perelman began working at the highly renowned Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences in Leningrad. His advisors at the Steklov Institute were Aleksandr Danilovich Aleksandrov and Yuri Dmitrievich Burago. In the late 80s and early 90s, Perelman worked at various universities in the United States. He returned to Saint Petersburg, Russia in 1996 and continued work at Steklov Institute.

The Poincaré conjecture

Until the fall 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 geometrization conjecture can be considered an analogue for 3-manifolds of the uniformization theorem for surfaces.

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, is generally considered to be the most famous open problem in topology. It states that a certain condition suffices to ensure that a manifold is homeomorphic to a sphere. In the twentieth century, many leading mathematicians tried to prove the Poincaré conjecture—beginning with Poincaré himself. All of them failed. The 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 3-manifolds, however, turned out to require substantially differing techniques, 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.

Perelman's plan of attack on the geometrization conjecture involves significant 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 to formulate a "dynamical process" in which a given 3-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, and it ensures that "hot spots" of temperature will dissipate as the temperature becomes more evenly distributed, until a uniform temperature is achieved throughout an object with finite volume. The Ricci flow describes the behavior of a tensorial quantity, the Ricci curvature tensor, under the Ricci flow, barring singularities in the flow, concentrations of large curvature will spread out until the curvature is as uniform as possible over the entire 3-manifold. In principle the result is one of eight kinds of "normal form" or Thurston model geometry.

Hamilton's idea has attracted a great deal of attention, but the problem has been that despite much effort, no one has been able to show how to deal with singularities in the flow—at least, not until in his eprints Perelman sketched a program using Ricci flow with surgery. This modification of the standard Ricci flow enables him to remove the singular regions in a nice way and continue the flow until further singularities develop in which case the removal, or "surgery", is done again; this flow thus continues forever.

A similar process in four dimensions had previously been used by Hamilton. 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 (SUNY) at Stony Brook, NY, 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 (professor), 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) and Xi-Ping Zhu (Zhongshan University). A consensus now appears to be developing 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 only the parts needed to prove the Poincare conjecture. According to a recent news story:

There is a growing feeling, a cautious optimism that 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

The Fields Medal and Millennium Prize

On August 22, 2006, Perelman was awarded 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 received the award "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow".

However, Perelman did not turn up at the ceremony, and declined to accept the medal. He has consistently been described by those who know him as shy and unworldly. In the 1990s, he turned down a prestigious prize from the European Mathematical Society. According to Overbye and other sources, Perelman suffered a bitter split with the Steklov Institute (which failed to re-elect him as member) in the spring of 2003, and according to testimony to his friends currently finds mathematics a painful topic to talk about, even going so far as to say that it no longer interests him. He is currently jobless and living with his mother in St Petersburg, subsisting on her £30-a-month pension. This reminds some observers of previous examples of "disappearances" of extremely talented mathematicians from the mathematical scene, including Alexander Grothendieck.

Perelman is also due to receive a share of a Millennium Prize, should his proof become generally accepted. However, he has not pursued formal publication of his proof in a peer-reviewed mathematics journal, which the rules for this prize require - instead, he published the proof that he had been working on for 10 years on the internet. 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, appears to be uninterested in the money. Some have rumored that he is not uninterested in money; rather, that he is afraid the Russian mafia will kidnap his mother and hold her for ransom for the money they know is associated to the prize and to any other prize.

Footnotes

1. http://www.telegraph.co.uk/news/main.jhtml?xml=/news/2006/08/20/nmaths20.xml
2. Перельман, Григорий Яковлевич (1990). Седловые поверхности в евклидовых пространствах:Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук (in Russian). Ленинградский Государственный Университет.
3. http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf
4. http://www.newscientist.com/article/dn9813.html
5. http://news.bbc.co.uk/2/hi/science/nature/5274040.stm
6. http://www.smh.com.au/news/world/maths-genius-living-in-poverty/2006/08/20/1156012411120.html
7. http://top.rbc.ru/index.shtml?/news/society/2006/08/22/22132425_bod.shtml
8. http://www.theage.com.au/news/world/jobless-maths-whiz-living-with-mother/2006/08/20/1156012408003.html
9. http://www.telegraph.co.uk/news/main.jhtml?xml=/news/2006/08/20/nmaths20.xml

External links

• Perelman's eprints on the arXiv
• Grigori Perelman at the Mathematics Genealogy Project
• 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
• Maths genius living in poverty

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)
• 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)
• Collins, Graham P. (2004). "The Shapes of Space". Scientific American (July): 94–103.
• 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)
• Randerson, James (2006-08-16). "Meet the cleverest man in the world (who's going to say no to a $1m prize)". The Guardian. {{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)
• Nasar, Sylvia (2006-08-28). "Manifold Destiny: Who really solved the Poincaré conjecture?". The New Yorker. {{cite news}}: |access-date= requires |url= (help); Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

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