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Meanwhile the federal government seems to be nibbling away at its own credibility, putting part of itself in suspended animation and toying with the idea of not fulfilling its financial promises. In the s, partly in collaboration with Yuri Burago , Mikhael Gromov , and Anton Petrunin , he made influential contributions to the study of Alexandrov spaces. In , he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. The full details of Perelman's work were filled in and explained by various authors over the following several years.

In August , Perelman 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: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo. Hamilton , the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture.

Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program. His mathematical education continued at the Leningrad Secondary School , a 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 his work on Aleksandrov's spaces of curvature bounded from below. 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.

Perelman's most notable work in this period was in the field of Alexandrov spaces , the concept of which dates back to the s. In a well-known paper coauthored with Yuri Burago and Mikhail Gromov , Perelman laid out the modern foundations of this field, with the notion of Gromov-Hausdorff convergence as an organizing principle. In , Perelman developed a notion of Morse theory on these non-smooth spaces. For his work on Alexandrov spaces, Perelman was invited to lecture at the International Congress of Mathematicians.

This was of interest since Cheeger and Gromoll had established the result under the stronger assumption that all sectional curvatures are positive. As the deformation from nonnegative to positive curvature is not well understood, the soul conjecture was proposed. Three notable papers of Perelman's from to deal with the construction of various interesting Riemannian manifolds with positive Ricci curvature. 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. This was followed by the two other preprints in Perelman modified Richard S. Hamilton 's 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, with the distortion process 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 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, 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.

However, it was widely expected that the process would be impeded by developing "singularities. Perelman's articles sketched a solution. According to Perelman, every singularity looks either like a cylinder collapsing to its axis, or a sphere collapsing to its center. With this understanding, he was able to construct a modification of the standard Ricci flow, called Ricci flow with surgery , which can systematically excise singular regions as they develop, in a controlled way.

The idea for Ricci flow with surgery had been present since a article of Hamilton, [19] who had successfully carried it out in in the setting of higher-dimensional spaces subject to certain restricted geometric conditions. Perelman showed that 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. Their argument, given the prerequisite of some sophisticated geometric measure theory arguments as developed in the s , is particularly simple. Perelman's preprints quickly gained the attention of the mathematical community, although they were widely seen as hard to understand since they had been written somewhat tersely.

Against the usual style in academic mathematical publications, many technical details had been omitted. In April , Perelman visited the Massachusetts Institute of Technology , Princeton University , Stony Brook University , Columbia University and New York University to give short series of lectures on his work, and to clarify some details for experts in the relevant fields.

In June , Bruce Kleiner and John Lott , both then of the University of Michigan , posted notes on Lott's website which, section by section, filled in many of the details in Perelman's first preprint. In September , their notes were updated to include Perelman's second preprint. 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. Perelman's proofs are concise and, at times, sketchy.

The purpose of these notes is to provide the details that are missing in [Perelman's first two preprints] Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. Some of the mistakes in [Perelman's first paper] were corrected in [Perelman's second paper]. We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman.

In their introduction, they explain. In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. As we pointed out before, we have to substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program.

All three of the expositions above have been revised after publication. Kleiner-Lott and Morgan-Tian's expositions were found to have errors which didn't affect the large scope , while Cao-Zhu's exposition attracted criticism for their phrasing and for an attributional error. Since publication, Kleiner and Lott's article has subsequently been revised twice for corrections, such as for an incorrect statement of Hamilton's important "compactness theorem" for Ricci flow. The latest revision to their article was in In , Abbas Bahri pointed out an error in Morgan and Tian's exposition, which was later fixed by Morgan and Tian and sourced to a basic computational mistake.

Cao and Zhu's paper underwent criticism from some parts of the mathematical community for their word choices, which some observers interpreted as claiming too much credit for themselves. When asked about the issue, Perelman said that Cao and Zhu had not contributed anything original, and had simply reworked his proof because they "did not quite understand the argument". In a published erratum, [28] Cao and Zhu attributed this to an oversight, saying that in they had taken down notes from the initial version of Kleiner and Lott's notes, and in their writeup had not realized the proper source of the notes.

They posted a revised version to the arXiv [29] with revisions in their phrasing and in the relevant page of the proof. For these observers, the troublesome parts of the proof are in the second half of Perelman's second preprint. For instance, Fields medalist Shing-Tung Yau said in that [30]. Although it may be heresy for me to say this, I am not certain that the proof is totally nailed down. I am convinced, as I've said many times before, that Perelman did brilliant work regarding the formation and structure of singularities in three-dimensional spaces—work that was indeed worthy of the Fields Medal he was awarded.

About this I have no doubts [ This suggests to me that other mathematicians don't yet have full command of this work and its methodologies either. After 10 hours of attempted persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows: "He proposed to me three alternatives: accept and come; accept and don't come, and we will send you the medal later; third, I don't accept the prize.

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 March , Perelman 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 alleged 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. 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. In , Russian media reported that Perelman was working in the field of nanotechnology in Sweden. Perelman has avoided journalists and other members of the media. In April , Aleksandr 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 I am picking mushrooms. Media related to Grigori Perelman at Wikimedia Commons. From Wikipedia, the free encyclopedia. Russian mathematician. In this Eastern Slavic name , the patronymic is Yakovlevich and the family name is Perelman. Leningrad , Soviet Union. Aleksandr Aleksandrov Yuri Burago.

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