Grigori Perelman

File:Grigori Perelman.jpg

Grigori Perelman.

Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман) (born 13 June 1966), who is known to his friends as 'Grisha', is a Russian mathematician who has made important contributions — possibly even landmark contributions — to Riemannian geometry. 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.

Overview

Perelman's early mathematical education was at the world-famous St. Petersburg School #239, which specializes in advanced mathematics and physics programs. As a high school student, he won a gold medal with a perfect score at the International Mathematical Olympiad in 1982. He earned his Ph.D. at the Mathematics & Mechanics Faculty of the St. Petersburg State University, one of the leading universities in the former Soviet Union. After graduation, Perelman began working at the highly renowned Petersburg Department of Steklov Institute of Mathematics of the Russian 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 worked at various universities in the United States. He returned to the Steklov Institute in 1996.

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 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 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 first half of 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 dimension four 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 three-manifolds, however, turns out to be the hardest of 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 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.

Interestingly enough, very broadly interpreted, there is a common thread between the geometrization conjecture and one way of thinking about FLT: both can be said to concern a kind of uniformitization. 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 delightfully intuitive: 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, 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 three-manifold. In principle the result is one of eight kinds of "normal form" or Thurston model geometry.

To adopt a slightly different metaphor, this is like formulating a kind of 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 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 been used by Hamilton. Singularities (including those at "infinite time") must occur in many cases for topological reasons, but here one would expect, given the truth of Thurston's geometrization conjecture, that a finite time singularity is essentially a kind of pinching along certain spheres corresponding to the prime decomposition of the 3-manifold and the infinite time singularities should result from certain collapsing pieces of the JSJ decomposition. Perelman's work shows this and proves the geometrization conjecture, and therefore the Poincaré 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, 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 [mathematicians] 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

In recent weeks, there has been growing speculation in the mathematical world that Perelman may be awarded a share of the next Fields Medal. This is the highest award in mathematics; it is awarded every four years at the International Congress of Mathematicians (ICM). The 2006 ICM will take place in Madrid (22 to 30 August).

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 in a peer-reviewed mathematics journal of his proof, as the rules for this prize currently require. Nonetheless, many mathematicians seem to feel that the scrutiny to which his eprints outlining his alleged proof have already been subjected far exceeds the "proof-checking" implicit in a normal peer review, and 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, may be uninterested in either honors or money. He has consistently been described by those who know him as shy and unworldly. In the 1990's, he turned down a prestigious prize from the European Mathematical Society. According to Overbye and other sources, Perelman is said to have resigned his position at the Steklov Institute in the spring of 2003, and since then his whereabouts have been unknown to the mathematical world. This reminds some observers of previous examples of "disappearances" of extremely talented mathematicians from the mathematical scene, including Alexander Grothendieck. As of 15 August 2006, it is by no means clear whether he will accept the Fields Medal if it is indeed offered to him.

See also

Related topics:

• Clay Mathematics Institute
• Fields Medal
• Geometrization conjecture
• Homology sphere
• Millennium Prize Problems
• Ricci curvature
• Ricci flow
• Poincaré conjecture
• Uniformization theorem

People:

• Michael Freedman
• Richard Hamilton
• Henri Poincaré
• Stephen Smale
• William Thurston
• Andrew Wiles

References

• Meet the cleverest man in the world (who's going to say no to a $1m prize) (James Randerson, The Guardian, Wed 16 Aug 2006)

External links

Perelman:

• 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
• International Mathematical Olympiad 1982 (Budapest, Hungary) Individual Scores

Commentary:

• Notes and commentary on Perelman's Ricci flow papers
• NY Times: Elusive Proof, Elusive Prover: A New Mathematical Mystery

References

• Mazur, Barry (1991). "Number Theory as Gadfly". American Mathematical Monthly. 98: 593–610. {{cite journal}}: External link in |title= (help)} This highly readable article (published before the appearance of the proof by Wiles) explains the connection between uniformatization and FLT.
• Schecter, Bruce (17 July, 2006). "Taming the fourth dimension". New Scientist. 183 (2456). {{cite journal}}: Check date values in: |date= (help)
• Anonymous, "Russian may have solved great math mystery". CNN. July 1, 2004. {{cite news}}: |access-date= requires |url= (help); Check date values in: |accessdate= and |date= (help); External link in |title= (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)
• Collins, Graham P. (2004). "The Shapes of Space". Scientific American (July): 94–103.
• 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)