He was particularly interested in what topological properties characterized a sphere. In this way he was able to conclude that these two spaces were, indeed, different. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere? Here is the standard form of the conjecture: Every simply connected , closed 3- manifold is homeomorphic to the 3-sphere. Note that "closed" here means, as customary in this area, the condition of being compact in terms of set topology, and also without boundary 3-dimensional Euclidean space is an example of a simply connected 3-manifold not homeomorphic to the 3-sphere; but it is not compact and therefore not a counter-example. Attempted solutions[ edit ] This problem seemed to lie dormant until J.

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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.

After having proved the soul conjecture in , 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 for a research-only position.

Then the soul of M is a point; equivalently M is diffeomorphic to Rn. 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.

The central idea is the notion of the Ricci flow. 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 quantity , the Ricci curvature tensor.

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. 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.

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 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. All indications are that his arguments are correct.

The June paper claimed: "This proof should be considered as the crowning achievement of the Hamilton—Perelman theory of Ricci flow.

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. 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. 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. 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.

He has said that "As long as I was not conspicuous, I had a choice. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit. Fellow countryman and mathematician Yakov Eliashberg said that, in , Perelman 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 problem of their existence and smoothness.

The writer Brett Forrest briefly interacted with Perelman in One who managed to reach him on his mobile was told: "You are disturbing me. I am picking mushrooms. Седловые поверхности в евклидовых пространствах [Saddle surfaces in Euclidean spaces] in Russian. Ленинградский государственный университет. Research papers.

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## Grigori Perelman

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## Poincaré conjecture

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## Conjectura de Poincaré

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