Grigori Perelman

Grigori Yakovlevich Perelman (Russian: Григо́рий Я́ковлевич Перельма́н , /ˈpɛrɨlmən/ perr -il-mən ; born 13 June 1966) is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.

In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology.

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, stating: "I'm not interested in money or fame, I don't want to be on display like an animal in a zoo." 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.

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On July 1, 2010, he turned down the prize, saying that he considers his contribution to proving the Poincaré conjecture to be no greater than that of U.S. mathematician Richard Hamilton, who introduced the theory of Ricci flow with the aim of attacking the geometrization conjecture.

Early life and education

Grigori Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lubov. Grigori's mother Lubov gave up graduate work in mathematics to raise him. Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school math training program.

His mathematical education continued at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In 1982, as a member of the Soviet Union 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 Sciences degree (the Soviet equivalent to the Ph.D.) at the School of Mathematics and Mechanics of the Leningrad State University, one of the leading universities in the former Soviet Union. His dissertation was titled "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 Aleksandrov and Yuri Burago. In the late 1980s and early 1990s, Perelman held research positions at several universities in the United States. In 1991 Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrov's spaces of curvature bounded from below . In 1992, he was invited to spend a semester each at the Courant Institute in New York University and State University of New York at Stony Brook where he began work on manifolds with lower bounds on Ricci curvature. From there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. After having proved the soul conjecture in 1994, 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 Saint Petersburg in the summer of 1995 for a research-only position.

He has a younger sister, Elena, who is also a scientist. She received a Ph.D. from Weizmann Institute of Science in Israel and is a biostatistician at Karolinska Institutet, in Stockholm, Sweden.

Perelman is also a talented violinist and a strong table tennis player.

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

Main article: Poincaré conjecture

The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was the most famous open problem in topology. Any loop on a three-dimensional sphere—as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted to a point. The Poincaré conjecture asserts that any closed three-dimensional manifold such that any loop can be contracted to a point is topologically a three-dimensional sphere. The analogous result has been known to be true in dimensions greater than or equal to five since 1960 (work of Stephen Smale). The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman. But the case of three-manifolds 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 : $1,000,000 prizes for the proof of any of seven conjectures, including the Poincaré conjecture. There was a wide agreement that a successful proof of any of these would constitute a landmark event in the history of mathematics.

Perelman's proof

Main article: Poincaré conjecture

In November 2002, Perelman posted the first of a series of eprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case.

Perelman modified 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 that gradually "perturbs" a given square matrix, and that is guaranteed to result after a finite time in its rational canonical form.

Hamilton's idea 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.

We know that singularities (including those that, roughly speaking, occur after the flow has continued for an infinite amount of time) must occur in many cases. However, any singularity that 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.

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 New York 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:

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. 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 Zhu Xiping 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. The June 2006 paper claimed: "This proof should be considered as the crowning achievement of the Hamilton-Per