In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere Several teams of mathematicians verified that Perelman’s proof was correct. The Poincaré conjecture, before . The actual solution was not found until Grigori Perelman published his papers. In late and Perelman. Grigori Perelman was attributed a Fields Medal for «his history of the conjecture and the mathematical path that led to Perelman’s proof is. Perelman became infamous for not only declining the Fields medal, and the Millenium prize, but also denouncing the mathematical community in general for .

Author: | Zolotilar Voran |

Country: | Suriname |

Language: | English (Spanish) |

Genre: | Spiritual |

Published (Last): | 22 April 2013 |

Pages: | 237 |

PDF File Size: | 11.80 Mb |

ePub File Size: | 20.5 Mb |

ISBN: | 440-2-67202-724-1 |

Downloads: | 28823 |

Price: | Free* [*Free Regsitration Required] |

Uploader: | Zulkigor |

Grigori Yakovlevich Perelman Russian: He has made contributions to Riemannian geometry and geometric topology. InPfrelman proved the soul conjecture. Inhe proved Thurston’s geometrization conjecture. The proof was confirmed in In AugustPerelman was offered the Fields Medal [1] for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow “, but he declined the award, stating: Hamiltonthe mathematician who pioneered the Perelmqn flow with the aim of attacking the conjecture.

Grigori’s mother Lyubov 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.

## Poincaré conjecture

His mathematical education continued at the Leningrad Secondary Schoola specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In the late s and early s, with a strong recommendation from the geometer Mikhail Gromov[14] Perelman obtained research positions at several universities in the United States. Petersburg Mathematical Society for porof work on Aleksandrov’s spaces of curvature bounded from below.

### [math/] Perelman’s proof of the Poincar\’e conjecture: a nonlinear PDE perspective

After having proved the soul conjecture inhe was offered jobs at several top universities in the US, including Grigoei and Stanfordbut he rejected them all and returned ptoof the Steklov Institute in Saint Petersburg in the summer of for a research-only position.

Perelnan latePerelman 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. Any loop on a 3-sphere —as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted into a point.

The analogous result has been known to be true in dimensions greater than or equal to five since as in the work of Stephen Smale. The four-dimensional case resisted longer, finally being solved in 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.

The most fundamental contribution to the three-dimensional case had been produced by Richard S. The role of Perelman was to complete the Hamilton program. Pperelman modified Richard S.

### Perelman’s Solution | Clay Mathematics Institute

Hamilton pperelman program for a proof of the conjecture. The central idea is the notion of the Ricci flow. Hamilton’s fundamental 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 which much earlier motivated Riemann to state his Riemann hypothesis on the zeros of the zeta function 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 quantitythe 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, then one should, in principle, eventually 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.

## Mathematics > Differential Geometry

This is similar to formulating a dynamical process that gradually “perturbs” a given square matrix and that is guaranteed to result after a finite pperelman in its rational canonical form. Perelamn 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 simple procedure for overcoming these obstacles.

According to Perelman, a modification of the standard Ricci flow, called Ricci flow with surgerycan systematically excise singular regions as they develop, in a controlled way. It was known 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. Perelman’s work was checked relatively quickly. On 25 MayBruce Kleiner and John Lottboth of the University of Michiganposted a paper on arXiv that fills in the details of Perelman’s proof of the Geometrization conjecture.

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.

The June paper claimed: In NovemberCao and Zhu published an erratum disclosing that they had failed to cite properly the previous work of Kleiner and Lott published in ptoof In the same issue, the AJM editorial board issued an apology for what it called “incautions” in the Cao—Zhu paper. The authors also removed the phrase “crowning achievement” from the abstract.

After 10 hours of attempted persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: From the very beginning, I told him I have chosen the third one Everybody understood that if the proof is correct, then no other recognition is needed.

I’m not a hero of mathematics.

I’m not even that successful; that is why I don’t want to have everybody looking at me. He had previously rejected a prestigious prize from the European Mathematical Society.

On 18 MarchPerelman was awarded a Millennium Prize for solving the problem. He considered the decision of the Clay Institute unfair for not sharing the prize with Richard S. Hamilton[5] and stated that “the main reason is my disagreement with the organized mathematical community. I don’t like their decisions, I consider them unjust. Perelman quit his job at the Steklov Institute in December Perelman is quoted in an article in The New Yorker saying that he is disappointed with the ethical standards of the field of mathematics.

The article implies that Perelman refers particularly to the efforts of Fields medalist Shing-Tung Yau to downplay Perelman’s role in the proof and play up the work of Cao and Zhu. Perelman added, “I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.

It is people like me who are isolated. This, combined with the possibility of being awarded a Fields medal, led him to quit professional mathematics.

He has said that “As long as I was not conspicuous, I had a choice. Either to make some ugly thing or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit. It is uncertain whether his resignation from Steklov and subsequent seclusion mean that he has ceased to practice mathematics. Grigoori countryman and mathematician Yakov Eliashberg said that, inPerelman confided to him that he was working on other things but it was too premature to talk about it.

He is said to have been interested in the past in the Navier—Stokes equations and the set of problems related to them that also constitutes a Millennium Prize, and there has been speculation that he may be working on them now. InRussian media reported prpof Perelman was working in the field of nanotechnology in Sweden. Perelman has avoided pereljan and other members of the media. Masha Gessenthe author of Perfect Rigour: A Genius and the Mathematical Breakthrough of the Centurya book about him, was unable to meet him.

In AprilAleksandr Zabrovsky, producer of “President-Film” studio, claimed to have held an interview with Perelman and agreed to shoot a film about him, under the tentative title The Formula of the Universe. The writer Brett Forrest briefly interacted with Perelman in One who managed to reach him on his mobile was told: I am picking mushrooms. From Wikipedia, the free encyclopedia. This name uses Eastern Slavic naming customs ; the patronymic is Yakovlevich and the family name is Perelman.

Saint Petersburg Mathematical Society Prize Aleksandr Aleksandrov Yuri Burago. Geometry portal Topology portal Mathematics portal.

Archived from the original on June 17, Retrieved April 30, Archived from the original on July 5, Retrieved May 1, Archived from the proif on July 2, Retrieved July 1, Archived from the original on January 17, Retrieved May 15, Archived from the original on August 15, Archived from the original on March 30,