Grigori Perelman의 정보

수상나이 40
출생년도 1966
사망년도
출생지 Russia
현재소속 St. Petersburg State University
관련출판물
시상내역
Other Award
연구단행본
연구실적 [자세히 보기]
Perelmans'proof

In November 2002, Perelman posted to the ? the first of a series of in which he claimed to have outlined a proof of the geometrization conjecture, a result that includes the Poincare conjecture as a particular case.

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 work its magic, 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 flavor 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 "hang up" 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. However, after his return to Russia, he is said to have gradually stopped responding to emails from his colleagues.

On 25 May 2006, Bruce Kleiner and John Lott, both of the University of Michigan, posted a paper on ?that claims to fill in the details of Perelman's proof of the Geometrization conjecture.

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, claiming to give a complete proof of the Poincare and the geometrization conjectures According to the Fields medalist Shing-Tung Yau this paper was aimed at "putting the finishing touches to the complete proof of the Poincare Conjecture".

The true extent of the contribution of Zhu and Cao, as well as the ethics of Yau's involvement, remain a matter of contention. Yau is both an editor-in-chief of the Asian Journal of Mathematics as well as Cao's doctoral advisor. It has been suggested that Yau was intent on being associated, directly or indirectly, with the proof of the conjecture and had pressured the journal's editors to accept Zhu and Cao's paper on unusually short notice. MIT mathematician Daniel Stroock has been quoted as saying, "I find it a little mean of [Yau] to seem to be trying to get a share of this as well."

In July 2006, John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology posted a paper on the ?titled, "Ricci Flow and the Poincare Conjecture." In this paper, they claim to provide a "detailed proof of the Poincare Conjecture". On 24 Aug 2006, Morgan delivered a lecture at the ICM in Madrid on the Poincare conjecture.

The above work seems to demonstrate that Perelman's outline can indeed be expanded into a complete proof of the geometrization conjecture:

Dennis Overbye of the New York Times has said that "there is a growing feeling, a cautious optimism that [mathematicians] have finally achieved a landmark not just of mathematics, but of human thought." 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."