Grigori Perelman

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 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.

Early life and education

Grigori Perelman was born in Leningrad (now Saint Petersburg) to a Jewish family 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. 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 and is a biostatician at Karolinska Institute.

Perelman is a talented violinist and also plays table tennis.

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 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.

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 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 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, 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.

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.

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

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

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."

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. 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