Grigori Perelman - Knowledge

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the major contributors are unquestionably Hamilton and Perelman. In this paper, we shall give complete and detailed proofs especially of Perelman's work in his second paper in which many key ideas of the proofs are sketched or outlined but complete details of the proofs are often missing. 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.

4677: 990:, the second half of Perelman's second preprint is devoted to an analysis of Ricci flows with surgery, which may exist for infinite time. Perelman was unable to resolve Hamilton's 1999 conjecture on long-time behavior, which would make Thurston's conjecture another corollary of the existence of Ricci flow with surgery. Nonetheless, Perelman was able to adapt Hamilton's arguments to the precise conditions of his new Ricci flow with surgery. The end of Hamilton's argument made use of 5218: 890:, saying that with its publication it became clear that Ricci flow could be powerful enough to settle the Thurston conjecture. The key of Hamilton's analysis was a quantitative understanding of how singularities occur in his four-dimensional setting; the most outstanding difficulty was the quantitative understanding of how singularities occur in three-dimensional settings. Although Hamilton was unable to resolve this issue, in 1999 he published work on 5228: 866:, models how concentrations of extreme temperatures will spread out until a uniform temperature is achieved throughout an object. In three seminal articles published in the 1980s, Hamilton proved that his equation achieved analogous phenomena, spreading extreme curvatures and uniformizing a Riemannian metric, in certain geometric settings. As a byproduct, he was able to prove some new and striking theorems in the field of 4648: 1160:" Additionally, one of the pages of Cao and Zhu's article was essentially identical to one from Kleiner and Lott's 2003 posting. In a published erratum, Cao and Zhu attributed this to an oversight, saying that in 2003 they had taken down notes from the initial version of Kleiner and Lott's notes, and in their 2006 writeup had not realized the proper source of the notes. They posted a revised version to 1133:, giving a complete description of Perelman's proof of the Poincaré and the geometrization conjectures. Unlike Kleiner and Lott's article, which was structured as a collection of annotations to Perelman's papers, Cao and Zhu's article was aimed directly towards explaining the proofs of the Poincaré conjecture and geometrization conjecture. In their introduction, they explain 2862: 1079:, posted notes on Lott's website which, section by section, filled in details of Perelman's first preprint. In September 2004, their notes were updated to include Perelman's second preprint. Following further revisions and corrections, they posted a version to arXiv on 25 May 2006, a modified version of which was published in the academic journal 1341:"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 not people who break ethical standards who are regarded as aliens. It is people like me who are isolated." 1002:. In Perelman's adaptation, he required use of a new theorem characterizing manifolds in which collapsing is only assumed on a local level. In his preprint, he said the proof of his theorem would be established in another paper, but he did not then release any further details. Proofs were later published by Takashi Shioya and Takao Yamaguchi, 1140:

In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts,

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collapsing to its center. Perelman's proof of his canonical neighborhoods theorem is a highly technical achievement, based upon extensive arguments by contradiction in which Hamilton's compactness theorem (as facilitated by Perelman's noncollapsing theorem) is applied to construct self-contradictory

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pointed out a counterexample to one of Morgan and Tian's theorems, which was later fixed by Morgan and Tian and sourced to an incorrectly computed evolution equation. The error, introduced by Morgan and Tian, dealt with details not directly discussed in Perelman's original work. In 2008, Morgan and

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negative and bounded away from zero. Previous examples of such surfaces were known, but Perelman's was the first to exhibit the saddle property on nonexistence of locally strictly supporting hyperplanes. As such, his construction provided further obstruction to the extension of a well-known theorem

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Despite formal similarities, Hamilton's equations are significantly more complex and nonlinear than the heat equation, and it is impossible that such uniformization is achieved without contextual assumptions. In completely general settings, it is inevitable that "singularities" occur, meaning that

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proved the four-dimensional version in 1982. Despite their work, the case of three-dimensional spaces remained completely unresolved. Moreover, Smale and Freedman's methods have had no impact on the three-dimensional case, as their topological manipulations, moving "problematic regions" out of the

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Perelman's proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in ... Regarding the proofs, contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. (Some of the mistakes in were

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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. It was soon

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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." On 22 December 2006, the scientific

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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 ... was completely irrelevant for me. Everybody understood that if the proof is

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parents, Yakov (who now lives in Israel) and Lyubov (who still lives in Saint Petersburg with Perelman). Perelman's mother Lyubov gave up graduate work in mathematics to raise him. Perelman's mathematical talent became apparent at the age of 10, and his mother enrolled him in Sergei Rukshin's

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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." (''The New Yorker'' authors explained

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to give short series of lectures on his work, and to clarify some details for experts in the relevant fields. In the years afterwards, three detailed expositions appeared, discussed below. Since then, various parts of Perelman's work have also appeared in a number of textbooks and expository

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in three dimensions, showing that if a three-dimensional version of his surgery techniques could be developed, and if a certain conjecture on the long-time behavior of Ricci flow could be established, then Thurston's conjecture would be resolved. This became known as the Hamilton program.

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for the resulting Li−Yau length functional, Perelman established his celebrated "noncollapsing theorem" for Ricci flow, asserting that local control of the size of the curvature implies control of volumes. The significance of the noncollapsing theorem is that volume control is one of the

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for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of

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posted a paper on arXiv in which they provided a detailed presentation of Perelman's proof of the Poincaré conjecture. Unlike Kleiner-Lott and Cao-Zhu's expositions, Morgan and Tian's also deals with Perelman's third paper. On 24 August 2006, Morgan delivered a lecture at the

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of Thurston's conjecture. In a third paper posted in July 2003, Perelman outlined an additional argument, sufficient for proving the Poincaré conjecture (but not the Thurston conjecture), the point being to avoid the most technical work in his second preprint.

1458:. Zabrovsky says that in the interview, Perelman explained why he rejected the one million dollar prize. A number of journalists believe that Zabrovsky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman. 1107:

Since its 2008 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

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with surgery in three dimensions, systematically excising singular regions as they develop. As an immediate corollary of his construction, Perelman resolved a major conjecture on the topological classification in three dimensions of

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and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006.

1156:" from the abstract, some people interpreted Cao and Zhu to be taking credit from Perelman for themselves. When asked about the issue, Perelman said that he could not see any new contribution by Cao and Zhu and that they " 641:. In further unpublished work, Perelman studied DC functions (difference of concave functions) on Alexandrov spaces and established that the set of regular points has the structure of a manifold modeled on DC functions. 886:. As an application of his construction, Hamilton was able to settle a four-dimensional curvature-based analogue of the Poincaré conjecture. Yau has identified this article as one of the most important in the field of 834:

inside of the manifold which disconnect the space into separate pieces, each of which can be endowed with a uniform geometric structure. Thurston was able to prove his conjecture under some provisional assumptions. In

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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

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

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The "canonical neighborhoods theorem" is the second main result of Perelman's first preprint. In this theorem, Perelman achieved the quantitative understanding of singularities of three-dimensional

882:, Hamilton began a systematic analysis. Throughout the 1990s, he found a number of new technical results and methods, culminating in a 1997 publication constructing a "Ricci flow with surgery" for 696:. Cheeger and Gromoll conjectured that if the curvature is strictly positive somewhere, then the soul can be taken to be a single point, and hence that the original space must be diffeomorphic to 3689:"Erratum to "A complete proof of the Poincaré and geometrization conjectures – application of the Hamilton–Perelman theory of the Ricci flow", Asian J. Math., Vol. 10, No. 2, 165–492, 2006" 3284: 1199:

Tian posted a paper which covered the details of the proof of the geometrization conjecture. Morgan and Tian's two articles have been published in book form by the Clay Mathematics Institute.

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The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture.

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of certain functions, in unpublished work. They also introduced the notion of an "extremal subset" of Alexandrov spaces, and showed that the interiors of certain extremal subsets define a

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In April 2011, 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

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in December 2005. His friends are said to have stated that he currently finds mathematics a painful topic to discuss; by 2010, some even said that he had entirely abandoned mathematics.

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The first half of Perelman's second preprint, in addition to fixing some incorrect statements and arguments from the first paper, used his canonical neighborhoods theorem to construct a

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in June 2006 to persuade him to accept the prize. After 10 hours of attempted persuasion over two days, Ball gave up. Two weeks later, Perelman summed up the conversation as follows:

1262:". He did not attend the ceremony and the presenter informed the congress that Perelman declined to accept the medal, which made him the only person to have ever declined the prize. 1324:

saying that he was disappointed with the ethical standards of the field of mathematics. The article implies that Perelman refers particularly to alleged efforts of Fields medalist

878:'s suggestion that a detailed understanding of these singularities could be topologically meaningful, and in particular that their locations might identify the spheres and tori in 708:. Perelman's theorem is significant in establishing a topological obstruction to deforming a nonnegatively curved metric to one which is positively curved, even at a single point. 622:. Vitali Kapovitch, who described Perelman's article as being "very hard to read," later wrote a detailed version of Perelman's proof, making use of some further simplifications. 1375:

mathematician, said that in 2007 Perelman confided to him that he was working on other things, but that it was too premature to discuss them. Perelman has shown interest in the

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Hamilton, Richard S. The Ricci flow on surfaces. Mathematics and general relativity (Santa Cruz, CA, 1986), 237–262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.

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Other results in Perelman's first preprint include the introduction of certain monotonic quantities and a "pseudolocality theorem" which relates curvature control and

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Yau, Shing-Tung. Perspectives on geometric analysis. Surveys in differential geometry. Vol. X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.

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hosted in Budapest, achieving a perfect score. He continued as a student of the School of Mathematics and Mechanics (the so-called "мехмат" i.e. "mech-math") at

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The Clay Institute subsequently used Perelman's prize money to fund the "Poincaré Chair", a temporary position for young promising mathematicians at the Paris

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time, so that the infinite-time analysis of Ricci flow is irrelevant. The construction of Ricci flow with surgery has the Poincaré conjecture as a corollary.

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Perelman's first preprint contained two primary results, both to do with Ricci flow. The first, valid in any dimension, was based on a novel adaptation of

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in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years.

393:, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture. He had previously rejected the prestigious prize of the 5327: 700:. In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, 3665: 2533:. Special Year in Differential Geometry held in Berkeley, CA, 1993–94. Mathematical Sciences Research Institute Publications. Vol. 30. Cambridge: 2470:. Special Year in Differential Geometry held in Berkeley, CA, 1993–94. Mathematical Sciences Research Institute Publications. Vol. 30. Cambridge: 2392:. Special Year in Differential Geometry held in Berkeley, CA, 1993–94. Mathematical Sciences Research Institute Publications. Vol. 30. Cambridge: 3276: 5307: 1408:. Russian media speculated he is periodically visiting his sister in Sweden, while living in Saint Petersburg and taking care of his elderly mother. 3162: 5337: 727:

with positive Ricci curvature, bounded diameter, and volume bounded away from zero. Also, he found an explicit complete metric on four-dimensional

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in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of

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Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381.

3036: 3430: 460:, Perelman obtained research positions at several universities in the United States. In 1991, Perelman won the Young Mathematician Prize of the 4436: 614:

as an organizing principle. In a followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all

5297: 3543:. Clay Mathematics Monographs, 5. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014. x+291 pp. 839:'s view, it was only with Thurston's systematic viewpoint that most topologists came to believe that the Poincaré conjecture would be true. 5262: 5257: 4478: 4453: 2064: 561:

were the subject of Perelman's graduate studies. His first result was on the possibility of prescribing the structure of negatively-curved

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Cao, Huai-Dong; Zhu, Xi-Ping (3 December 2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture".

4621:(an account of Perelman's talk on his proof at MIT; pdf file; also see Sugaku Seminar 2003–10 pp 4–7 for an extended version in Japanese) 1100:

corrected in .) We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman.

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developed a novel viewpoint, making the Poincaré conjecture into a small special case of a hypothetical systematic structure theory of

469: 224: 4454:"A Complete Proof of the Poincaré and Geometrization Conjectures – application of the Hamilton-Perelman theory of the Ricci flow" 4032: 3310: 936:. As a consequence, Hamilton's compactness and the corresponding existence of subsequential limits could be applied somewhat freely. 3142: 2378: 1294:

the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust.

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on the plane which is complete can be continuously immersed as a polyhedral surface. Later, he constructed an example of a smooth

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on Alexandrov spaces. Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the

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which had eluded Hamilton. Roughly speaking, Perelman showed that on a microscopic level, every singularity looks either like a

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for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow

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Hamilton, Richard S. Non-singular solutions of the Ricci flow on three-manifolds. Comm. Anal. Geom. 7 (1999), no. 4, 695–729.

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A Complete Proof of the Poincaré and Geometrization Conjectures – Application of the Hamilton-Perelman Theory of Ricci Flow

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Freedman, Michael Hartley. The topology of four-dimensional manifolds. J. Differential Geometry 17 (1982), no. 3, 357–453.

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Perelʹman, G. Ya.; Petrunin, A. M. (1994). "Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem".

1277: 4672: 4370: 959:. However, despite being major results in the theory of Ricci flow, these results were not used in the rest of his work. 461: 430: 3366:

Smale, Stephen. Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961), 391–406.

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Hamilton, Richard S. Three-manifolds with positive Ricci curvature. J. Differential Geometry 17 (1982), no. 2, 255–306.

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Hamilton, Richard S. Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153–179.

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Perelʹman, G. Ya. (1985). "Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in

799: 692:, this says in particular that every complete Riemannian metric of nonnegative sectional curvature may be taken to be 4652: 4266: 2916: 2542: 2479: 2401: 2319: 2210: 1493: 618:

with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually

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Efimov, N. V. Generation of singularites on surfaces of negative curvature. Mat. Sb. (N.S.) 64 (106) 1964 286–320.

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Perelman, Grisha (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds".

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A Russian documentary about Perelman in which his work is discussed by several leading mathematicians, including

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Perelman's reference to "some ugly thing" as "a fuss" on Perelman's part about the ethical breaches he perceived.)

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Given that his parents were Jewish, Perelman, who was born in 1966, was fortunate in those who took up his cause.

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He has suffered anti-Semitism (he is Jewish)....Grigory is pure Jewish and I never minded that but my bosses did

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and others, was released in 2011 under the title "Иноходец. Урок Перельмана" ("Maverick: Perelman's Lesson").

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on the Poincaré conjecture, in which he declared that Perelman's work had been "thoroughly checked." In 2015,

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This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow

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on a manifold. The heat equation, such as when applied in the sciences to physical phenomena such as

385: 356: 173: 138: 4552:"Conjectures No More? Consensus Forming on the Proof of the Poincaré and Geometrization Conjectures" 4227:Интервью с математиком Григорием Перельманом: Зачем мне миллион долларов? Я могу управлять Вселенной 3199:"Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen – review" 2716: 927:'s differential Harnack inequalities to the setting of Ricci flow. By carrying out the proof of the 4698: 4428: 4008: 2456: 920: 724: 705: 445: 378: 4688: 3526:

Shioya, Takashi; Yamaguchi, Takao. Volume collapsed three-manifolds with a lower curvature bound.

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Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications".

1284:, Perelman refused to accept the Millennium Prize in July 2010. He considered the decision of the 598:

Perelman's first works to have a major impact on the mathematical literature were in the field of

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John Morgan. "The Poincaré conjecture." Lecture at 2006 International Congress of Mathematicians.

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curvature accumulates to infinite levels after a finite amount of "time" has elapsed. Following

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Cao, Jianguo; Ge, Jian. A simple proof of Perelman's collapsing theorem for 3-manifolds. J.

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for solving the problem. On 8 June 2010, he did not attend a ceremony in his honor at the

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On 18 March 2010, it was announced that he had met the criteria to receive the first Clay

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of curvature bounded from below. In 1992, he was invited to spend a semester each at the

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with advanced mathematics and physics programs. Perelman excelled in all subjects except

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can be contracted into a point. Poincaré suggested that a converse might be true: if a

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Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

4161:"Математика Григория Перельмана, уехавшего в Швецию, видели в купчинском супермаркете" 2078: 2059: 5227: 5145: 5113: 5059: 5032: 4978: 4912: 4790: 4625:

Morgan, John W.; Gang Tian (25 July 2006). "Ricci Flow and the Poincaré Conjecture".

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briefly interacted with Perelman in 2012. A reporter who had called him was told: "

1405: 1368: 1221: 972: 815: 803: 615: 599: 546: 530: 340: 295: 253: 95: 4001: 3973:"Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" 3013: 2994: 1864:; Perelʹman, G. (1992). "A. D. Aleksandrov spaces with curvatures bounded below". 774:, defined as the set of points at unit length from the origin in four-dimensional 5162: 5135: 5091: 5076: 4956: 4890: 3712: 3513:

Li, Peter; Yau, Shing-Tung. On the parabolic kernel of the Schrödinger operator.

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Hamilton, Richard S. (1995). "The formation of singularities in the Ricci flow".

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for his work on the Ricci flow. However, Perelman declined to accept the prize.

610:, Perelman established the modern foundations of this field, with the notion of 241: 5157: 5037: 4988: 4944: 4922: 4907: 4836: 4362: 3925:. International Congress of Mathematicians 2006. 22 August 2006. Archived from 3707: 3688: 3660: 2904: 2883: 2646: 1436: 1401: 1393: 1350:, led him to state that he had quit professional mathematics by 2006. He said: 1325: 1320: 924: 875: 790:

has the property that any loop can be contracted into a point, then it must be

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in 1994, he was offered jobs at several top universities in the US, including

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with positive Ricci curvature and Euclidean volume growth, and such that the

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Perelʹman, G. Ya. (1994). "Elements of Morse theory on Aleksandrov spaces".

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Some of Perelman's work dealt with the construction of various interesting

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In 2014, Russian media reported that Perelman was working in the field of

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In May 2006, a committee of nine mathematicians voted to award Perelman a

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apparent that Perelman had made major contributions to the foundations of

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recognized Perelman's proof of the Poincaré conjecture as the scientific "

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In his undergraduate studies, Perelman dealt with issues in the field of

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Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century

4482: 3736: 3252: 4841: 4802: 4189:Чтобы купить русского хлеба, Перельман пешком ходил через весь Нью-Йорк 2777:

Perelman, Grisha (2003). "Ricci flow with surgery on three-manifolds".

2332: 2268: 1786: 1763: 1692: 1634: 1417: 1333: 1164:

with revisions in their phrasing and in the relevant page of the proof.

1126: 1028: 963: 940: 891: 847: 759: 732: 562: 348: 26: 3345:"Young mathematician prize of the St. Petersburg Mathematical Society" 766:

in 1904, was throughout the 20th century regarded as a key problem in

4288: 4225: 3758: 3609: 2818: 2783: 2748: 2636: 2251:"A diameter sphere theorem for manifolds of positive Ricci curvature" 1444: 1178: 1007: 808: 558: 406: 161: 3839:

Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture

3774: 1367:

and subsequent seclusion Perelman stopped his mathematics research.

830:

whatsoever, there is some collection of two-dimensional spheres and

5189: 4704: 4366: 4187: 4111: 3893: 3847: 3280: 3069: 2979: 2880: 2191:

Proceedings of the International Congress of Mathematicians, Vol. 1

1385: 1281: 1092:" In the introduction to their article, Kleiner and Lott explained: 944: 904: 827: 819: 787: 771: 767: 644:

For his work on Alexandrov spaces, Perelman was recognized with an

477: 409:, Soviet Union (now Saint Petersburg, Russia) on June 13, 1966, to 3877: 2060:"Manifolds of positive Ricci curvature with almost maximal volume" 1400:. Shortly thereafter, however, he was spotted again in his native 437:, without admission examinations, and enrolled at the university. 4566:

Kleiner, Bruce; Lott, John (2008). "Notes on Perelman's papers".

3593:

Kleiner, Bruce; Lott, John (2008). "Notes on Perelman's papers".

3107: 1328:

to downplay Perelman's role in the proof and play up the work of

311:

and geometer who is known for his contributions to the fields of

34: 4647: 4291:[Interview with Perelman – fake?]. Versii. 5 May 2011. 1397: 1372: 1255: 1191: 948: 235: 2618:

Kapovitch, Vitali (2007). "Perelman's stability theorem". In

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After completing his PhD in 1990, Perelman began work at the

4143:""Komsomolskaya Pravda" found out where Perelman disappears" 807:

way without interfering with other regions, seem to require

339:, and Anton Petrunin, he made contributions to the study of 4607:"Witnesses to Mathematical History Ricci Flow and Geometry" 3572:

Kleiner, Bruce; Lott, John. Locally collapsed 3-manifolds.

3431:"Autobiography of Richard S Hamilton | the Shaw Prize" 3729: 3686: 738: 381:", the first such recognition in the area of mathematics. 4432: 2606:

Alexandrov's spaces with curvatures bounded from below II

1498:Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук. 842:

At the same time that Thurston published his conjecture,

680:

has a compact nonnegatively curved submanifold, called a

113: 4394:"Grigory Perelman, the maths genius who said no to $ 1m" 3965: 3871:

Completion of the Proof of the Geometrization Conjecture

2976:"Russian maths genius Perelman urged to take $ 1m prize" 1679:

Perelʹman, G. Ya. (1991). "Polyhedral saddle surfaces".

850:. Hamilton's Ricci flow is a prescription, defined by a 442:

Leningrad Department of Steklov Institute of Mathematics

4109:[The genius who has withdrawn from the world]. 2691:

Quasigeodesics and gradient curves in Alexandrov spaces

1525:

Geometric questions in the theory of functions and sets

1363:

It was unclear whether along with his resignation from

1346:

This, combined with the possibility of being awarded a

1306: 323:. In 2005, Perelman resigned from his research post in 2718:

DC structure on Alexandrov space (preliminary version)

1611:

Perelʹman, G. Ya. (1987). "k-radii of a convex body".

1158:

did not quite understand the argument and reworked it.

3277:"Eccentric 'Mathsputin' Rejects Million Dollar Prize" 2118:"Proof of the soul conjecture of Cheeger and Gromoll" 1856: 903:

In November 2002 and March 2003, Perelman posted two

4535:. Boston, Massachusetts: Houghton Mifflin Harcourt. 4009:"Clay Mathematics Institute 2010 Annual Report 2010" 3771:"Schedule of the scientific program of the ICM 2006" 2843: 1203: 508:

in the summer of 1995 for a research-only position.

3953:"Prestigious Fields Medals for mathematics awarded" 1564: 4528: 4488:Collins, Graham P. (2004). "The Shapes of Space". 4487: 3465:"Four-manifolds with positive isotropic curvature" 3161:Osborn, Andrew; Krepysheva, Olga (27 March 2010). 2193:. Zürich, Switzerland ( 3–11 August 1994). Basel: 1265:He has also rejected a prestigious prize from the 826:, posited that given any closed three-dimensional 552: 4682: 4565: 4423:. Encyclopedia of Mathematical Physics, Elsevier. 4315:"Grigori Perelman's interview full of mismatches" 3100:"Russian mathematician rejects $ 1 million prize" 1990: 5278:International Mathematical Olympiad participants 5244: 4115:(in French). 30 September 2010. pp. 74–77. 3586: 3160: 2811: 2776: 2741: 2687: 822:in three dimensions. His proposal, known as the 4624: 3950: 3163:"Russian maths genius may turn down $ 1m prize" 2950:International Mathematical Union (IMU) – Prizes 2803: 2768: 2733: 2706: 2679: 2594: 2509: 2446: 2368: 2304: 2240: 2173: 2107: 2049: 1982: 1915: 1848: 1770:with Gaussian curvature bounded away from zero" 1753: 1670: 1602: 1556: 1508: 1486:Седловые поверхности в евклидовых пространствах 688:to the original space. From the perspective of 504:, but he rejected them all and returned to the 296:[ɡrʲɪˈɡorʲɪjˈjakəvlʲɪvʲɪtɕpʲɪrʲɪlʲˈman] 4035:. Clay Institute. 4 March 2014. Archived from 1467:You are disturbing me. I am picking mushrooms. 1280:to accept his $ 1 million prize. According to 979:, the Ricci flow with surgery exists only for 4720: 3642: 3640: 3638: 3636: 1565:Polikanova, I. V.; Perelʹman, G. Ya. (1986). 1231:correct, then no other recognition is needed. 285: 4429:"Russian may have solved great math mystery" 4223: 4140: 2379:"Collapsing with no proper extremal subsets" 2065:Journal of the American Mathematical Society 1764:"An example of a complete saddle surface in 1472: 417:His mathematical education continued at the 4259:"6 странных ошибок в "интервью Перельмана"" 3646: 3592: 414:after-school mathematics training program. 400: 331:In the 1990s, partly in collaboration with 5328:University of California, Berkeley fellows 4727: 4713: 4675: 4452:Cao, Huai-Dong; Zhu, Xi-Ping (June 2006). 3836: 3633: 3093: 3091: 3060: 3058: 533:. With I. V. Polikanova, he established a 525:. His first published article studied the 63: 4630: 4579: 4219: 4217: 4185: 3846: 3813: 3735: 3706: 3608: 3480: 3012: 2992: 2817: 2782: 2747: 2645: 2635: 2617: 2135: 2077: 1923: 1785: 1761: 1678: 1610: 1516: 1494:Ленинградский государственный университет 1411: 1272:On 18 March 2010, Perelman was awarded a 1258:, Perelman was offered the Fields Medal " 911:, in which he claimed to have outlined a 405:Grigori Yakovlevich Perelman was born in 366:In August 2006, Perelman was offered the 5308:Saint Petersburg State University alumni 4136: 4134: 3462: 3443: 2714: 2602: 2517: 2454: 2376: 2312: 2248: 2181: 2115: 2057: 1424:, author of a biography about Perelman, 1318:Perelman is quoted in a 2006 article in 1252:International Congress of Mathematicians 1250:Nevertheless, on 22 August 2006, at the 1085:International Congress of Mathematicians 650:International Congress of Mathematicians 359:, the former of which had been a famous 5338:Members of the USSR Academy of Sciences 4694:MacTutor History of Mathematics Archive 4549: 4391: 3912: 3243: 3241: 3088: 3055: 2315:"Widths of nonnegatively curved spaces" 739:Geometrization and Poincaré conjectures 5245: 4526: 4451: 4256: 4214: 4202:from the original on 17 September 2012 4057: 3920:"Fields Medal – Grigory Perelman" 3752:Ricci Flow and the Poincaré Conjecture 3331: 3247: 3232: 3097: 1484:Перельман, Григорий Яковлевич (1990). 1288:unfair for not sharing the prize with 802:of Poincaré's conjecture in 1961, and 655: 4708: 4295:from the original on 26 December 2012 4238:from the original on 27 December 2012 4131: 3795: 3687:Cao, Huai-Dong; Zhu, Xi-Ping (2006). 3307:"International Mathematical Olympiad" 3196: 3156: 3154: 3152: 3133: 3131: 3129: 2184:"Spaces with curvature bounded below" 1183:Massachusetts Institute of Technology 1044:Massachusetts Institute of Technology 719:. He found Riemannian metrics on the 506:Steklov Institute in Saint Petersburg 484:. From there, he accepted a two-year 462:Saint Petersburg Mathematical Society 294: 156:Saint Petersburg Mathematical Society 5298:Mathematicians from Saint Petersburg 4734: 4321:from the original on 22 January 2013 4269:from the original on 17 October 2012 4224:Veligzhanina, Anna (28 April 2011). 4186:Gerasimov, Nikolai (27 March 2011). 4107:"Le génie qui s'est retiré du monde" 3313:from the original on 2 November 2012 3238: 3145:from the original on 15 August 2010. 3114:from the original on 17 January 2012 3066:"Последнее "нет" доктора Перельмана" 2940: 2938: 2936: 1379:and the problem of their solutions' 1307:Possible withdrawal from mathematics 1042:In April 2003, Perelman visited the 754:Thurston's geometrization conjecture 593: 357:Thurston's geometrization conjecture 5263:21st-century Russian mathematicians 5258:20th-century Russian mathematicians 4673:International Mathematical Olympiad 4481:. Revised version (December 2006): 4439:from the original on 13 August 2006 4141:Veligzhanina, Anna (23 July 2014). 3299: 3211:from the original on 4 October 2013 2688:Perelman, G.; Petrunin, A. (1995). 2036:: CS1 maint: untitled periodical ( 1993:St. Petersburg Mathematical Journal 1969:: CS1 maint: untitled periodical ( 1926:St. Petersburg Mathematical Journal 1835:: CS1 maint: untitled periodical ( 1740:: CS1 maint: untitled periodical ( 1529:Kalinin gosudarstvennyy universitet 1490:Saddle surfaces in Euclidean spaces 431:International Mathematical Olympiad 242:Saddle Surfaces in Euclidean Spaces 13: 4604: 4341:"Articles » Shattered Genius" 4317:. English Pravda.ru. 5 June 2011. 3951:Mullins, Justin (22 August 2006). 3900:from the original on 19 April 2010 3890:"Maths genius urged to take prize" 3668:from the original on 19 March 2011 3175:from the original on 30 March 2010 3149: 3126: 1811:Ukrainskiĭ Geometricheskiĭ Sbornik 1716:Ukrainskiĭ Geometricheskiĭ Sbornik 898: 824:Thurston geometrization conjecture 516: 490:University of California, Berkeley 307:; born 13 June 1966) is a Russian 229:University of California, Berkeley 14: 5349: 4641: 4550:Jackson, Allyn (September 2006). 4512:10.1038/scientificamerican0704-94 4373:from the original on 8 March 2013 4289:"Интервью Перельмана – подделка?" 4119:from the original on 21 July 2012 4087:from the original on 16 July 2011 3837:Morgan, John; Tian, Gang (2015), 3287:from the original on 15 July 2014 3139:"Maths genius declines top prize" 2933: 2917:Thurston elliptization conjecture 2320:Geometric and Functional Analysis 2079:10.1090/S0894-0347-1994-1231690-7 1278:Institut Océanographique de Paris 1204:Fields Medal and Millennium Prize 672:. It asserts that every complete 511: 16:Russian mathematician (born 1966) 5226: 5217: 5216: 4646: 4363:"Seven of the week's best reads" 3446:Surveys in Differential Geometry 3076:from the original on 2 July 2010 2860: 2846: 2123:Journal of Differential Geometry 1855: 1420:and other members of the media. 1218:International Mathematical Union 971:which admit metrics of positive 625:Perelman developed a version of 579:four-dimensional Euclidean space 325:Steklov Institute of Mathematics 4421:Singularities of the Ricci flow 4392:Harding, Luke (23 March 2010). 4385: 4355: 4333: 4307: 4281: 4257:Gessen, Masha (29 April 2011). 4250: 4179: 4153: 4099: 4067: 4051: 4025: 3979:. 18 March 2010. Archived from 3944: 3882: 3863: 3830: 3789: 3763: 3744: 3723: 3680: 3566: 3553: 3533: 3520: 3507: 3498: 3489: 3456: 3437: 3423: 3414: 3405: 3396: 3387: 3378: 3369: 3360: 3351: 3337: 3325: 3269: 3226: 3098:Ritter, Malcolm (1 July 2010). 2912:Spherical space form conjecture 2810: 2775: 2713: 2686: 2516: 2453: 2375: 2311: 2247: 2180: 2114: 2056: 1989: 1880:10.1070/RM1992v047n02ABEH000877 1571:Sibirskij Matematiceskij Zurnal 1563: 1021: 1010:, Jianguo Cao and Jian Ge, and 743: 553:Negatively curved hypersurfaces 4167:(in Russian). 20 December 2023 3197:McKie, Robin (27 March 2011). 3190: 3029: 2986: 2968: 2740: 2628:Metric and Comparison Geometry 2601: 1922: 1760: 1677: 1609: 1515: 529:arising from intersections of 419:Leningrad Secondary School 239 100:(now Saint Petersburg, Russia) 1: 4664:Mathematics Genealogy Project 4409: 3773:. Icm2006.org. Archived from 3541:The geometrization conjecture 3517:156 (1986), no. 3-4, 153–201. 3463:Hamilton, Richard S. (1997). 3253:"He Conquered the Conjecture" 3014:10.1126/science.314.5807.1848 2189:. In Chatterji, S. D. (ed.). 1774:Journal of Soviet Mathematics 1681:Journal of Soviet Mathematics 1614:Siberian Mathematical Journal 1567:"A remark on Helly's theorem" 1311:Perelman quit his job at the 1267:European Mathematical Society 947:collapsing to its axis, or a 852:partial differential equation 846:introduced his theory of the 492:, in 1993. After proving the 395:European Mathematical Society 23:Eastern Slavic naming customs 4462:Asian Journal of Mathematics 3869:Morgan, John W.; Tian, Gang 3750:Morgan, John W.; Tian, Gang 3694:Asian Journal of Mathematics 3258:The New York Review of Books 3141:. BBC News. 22 August 2006. 1867:Russian Mathematical Surveys 1115:Asian Journal of Mathematics 932:preconditions of Hamilton's 762:, proposed by mathematician 668:established their important 612:Gromov–Hausdorff convergence 287:Григорий Яковлевич Перельман 282:Grigori Yakovlevich Perelman 79:Grigori Yakovlevich Perelman 7: 3530:333 (2005), no. 1, 131–155. 2839: 1523:". In Ivanov, L. D. (ed.). 1456:The Formula of the Universe 1428:", was unable to meet him. 1148:Based also upon the title " 10: 5354: 4669:Grigori Perelman's results 3977:Clay Mathematics Institute 3798:"Five gaps in mathematics" 3708:10.4310/ajm.2006.v10.n2.a2 3563:21 (2011), no. 4, 807–869. 3539:Morgan, John; Tian, Gang. 2647:10.4310/SDG.2006.v11.n1.a5 2535:Cambridge University Press 2529:; Petersen, Peter (eds.). 2472:Cambridge University Press 2466:; Petersen, Peter (eds.). 2394:Cambridge University Press 2388:; Petersen, Peter (eds.). 1762:Perelʹman, G. Ya. (1992). 998:'s theorem characterizing 854:formally analogous to the 747: 702:Sharafutdinov's retraction 581:which is complete and has 569:. He proved that any such 486:Miller Research Fellowship 448:, where his advisors were 435:Leningrad State University 110:Leningrad State University 21:In this name that follows 20: 5333:21st-century Russian Jews 5293:New York University staff 5283:Jewish Russian scientists 5212: 4742: 3482:10.4310/CAG.1997.v5.n1.a1 3037:"The Poincaré Conjecture" 2806: 2771: 2736: 2709: 2682: 2597: 2512: 2449: 2371: 2307: 2256:Mathematische Zeitschrift 2243: 2203:10.1007/978-3-0348-9078-6 2176: 2110: 2052: 1985: 1918: 1851: 1756: 1673: 1605: 1559: 1511: 1473:Complete publication list 1237:He was quoted as saying: 1220:, approached Perelman in 1033:geometrization conjecture 800:high-dimensional analogue 735:is non-uniquely defined. 725:complex projective planes 684:, whose normal bundle is 476:, where he began work on 343:. In 1994, he proved the 286: 275: 252: 234: 216: 190: 183: 148: 120: 105: 74: 62: 53: 46: 4699:University of St Andrews 4226: 4188: 4075: 2993:Mackenzie, Dana (2006). 2927: 1381:existence and smoothness 1117:published an article by 929:Bishop–Gromov inequality 792:topologically equivalent 527:combinatorial structures 446:USSR Academy of Sciences 401:Early life and education 379:Breakthrough of the Year 54: 4590:10.2140/gt.2008.12.2587 4568:Geometry & Topology 4081:RBC Information Systems 3619:10.2140/gt.2008.12.2587 3596:Geometry & Topology 2009:English translation of 1942:English translation of 1808:English translation of 1713:English translation of 1377:Navier–Stokes equations 1301:Institut Henri Poincaré 1081:Geometry & Topology 986:In order to settle the 884:four-dimensional spaces 543:circumscribed cylinders 5268:Differential geometers 4527:Gessen, Masha (2009). 4426:The Associated Press, 3576:No. 365 (2014), 7–99. 2922:Uniformization theorem 2137:10.4310/jdg/1214455292 1412:Perelman and the media 1361: 1344: 1248: 1235: 1145: 1131:Sun Yat-sen University 1104: 1077:University of Michigan 1052:Stony Brook University 858:, for how to deform a 590:to higher dimensions. 5318:Soviet mathematicians 4655:at Wikimedia Commons 3815:10.1515/ans-2015-0202 3796:Bahri, Abbas (2015). 2715:Perelman, G. (1995). 2603:Perelman, G. (1991). 2518:Perelman, G. (1997). 2455:Perelman, G. (1997). 2377:Perelman, G. (1997). 2313:Perelman, G. (1995). 2249:Perelman, G. (1995). 2182:Perelman, G. (1995). 2116:Perelman, G. (1994). 2058:Perelman, G. (1994). 1416:Perelman has avoided 1352: 1338: 1239: 1226: 1136: 1095: 1083:in 2008. At the 2006 880:Thurston's conjecture 639:topological manifolds 565:in three-dimensional 480:with lower bounds on 450:Aleksandr Aleksandrov 262:Aleksandr Aleksandrov 198:Differential geometry 4685:Robertson, Edmund F. 4369:. 1 September 2012. 4232:Komsomolskaya Pravda 4195:Komsomolskaya Pravda 3309:. Imo-official.org. 2946:"Fields Medals 2006" 2537:. pp. 165–166. 2474:. pp. 157–163. 2396:. pp. 149–155. 2197:. pp. 517–525. 2018:(1): 242–256. 1993. 1951:(1): 232–241. 1993. 1531:. pp. 129–131. 1492:] (in Russian). 1048:Princeton University 1000:collapsing manifolds 723:of arbitrarily many 713:Riemannian manifolds 466:Aleksandrov's spaces 4683:O'Connor, John J.; 4504:2004SciAm.291a..94C 4491:Scientific American 4469:(2). Archived from 3857:2015arXiv151200699M 3802:Adv. Nonlinear Stud 3777:on 11 February 2010 3168:The Daily Telegraph 3007:(5807): 1848–1849. 2894:Hyperbolic manifold 2531:Comparison geometry 2468:Comparison geometry 2390:Comparison geometry 1627:1987SibMJ..28..665P 1292:, and stated that " 1290:Richard S. Hamilton 1216:, president of the 1175:Columbia University 1075:, both then of the 1060:New York University 1056:Columbia University 1037:Poincaré conjecture 988:Thurston conjecture 977:Poincaré conjecture 934:compactness theorem 868:Riemannian geometry 750:Poincaré conjecture 678:sectional curvature 656:Comparison geometry 563:polyhedral surfaces 474:New York University 391:Richard S. Hamilton 353:Poincaré conjecture 317:Riemannian geometry 225:New York University 135:Poincaré conjecture 5303:Russian scientists 5232:Mathematics portal 4689:"Grigori Perelman" 4559:Notices of the AMS 4083:. 22 August 2006. 3932:on 3 November 2012 3654:(21 August 2006). 3249:Paulos, John Allen 2854:Mathematics portal 2333:10.1007/BF01895675 2269:10.1007/BF02571925 1787:10.1007/BF01097177 1693:10.1007/BF01097421 1635:10.1007/BF00973857 1336:. Perelman added: 1152:" and the phrase " 1113:In June 2006, the 888:geometric analysis 811:in order to work. 786:three-dimensional 583:Gaussian curvature 559:negative curvature 427:physical education 423:specialized school 321:geometric topology 313:geometric analysis 208:Geometric topology 203:Geometric analysis 56:Григорий Перельман 5240: 5239: 4651:Media related to 3975:(Press release). 3896:. 24 March 2010. 3582:978-2-85629-795-7 3549:978-0-8218-5201-9 3251:(29 April 2010). 2835: 2834: 2800: 2799: 2765: 2764: 2730: 2729: 2703: 2702: 2676: 2675: 2657:978-1-57146-117-9 2584: 2583: 2570:on 27 August 2021 2506: 2505: 2443: 2442: 2429:on 25 August 2021 2365: 2364: 2301: 2300: 2237: 2236: 2170: 2169: 2104: 2103: 2046: 2045: 1979: 1978: 1912: 1911: 1845: 1844: 1750: 1749: 1722:: 100–108. 1988. 1667: 1666: 1599: 1598: 1553: 1552: 1313:Steklov Institute 1123:Lehigh University 860:Riemannian metric 674:Riemannian metric 616:Alexandrov spaces 600:Alexandrov spaces 594:Alexandrov spaces 547:inscribed spheres 535:measure-theoretic 470:Courant Institute 351:, and proved the 341:Alexandrov spaces 279: 278: 185:Scientific career 5345: 5273:Fields Medalists 5230: 5220: 5219: 4736:Fields Medalists 4729: 4722: 4715: 4706: 4705: 4701: 4679: 4660:Grigori Perelman 4653:Grigori Perelman 4650: 4636: 4634: 4620: 4618: 4616: 4611: 4601: 4583: 4574:(5): 2587–2855. 4562: 4556: 4546: 4534: 4523: 4498:(July): 94–103. 4477: 4475: 4458: 4448: 4446: 4444: 4404: 4403: 4389: 4383: 4382: 4380: 4378: 4359: 4353: 4352: 4350: 4348: 4337: 4331: 4330: 4328: 4326: 4311: 4305: 4304: 4302: 4300: 4285: 4279: 4278: 4276: 4274: 4254: 4248: 4247: 4245: 4243: 4221: 4212: 4211: 4209: 4207: 4183: 4177: 4176: 4174: 4172: 4157: 4151: 4150: 4138: 4129: 4128: 4126: 4124: 4103: 4097: 4096: 4094: 4092: 4071: 4065: 4055: 4049: 4048: 4046: 4044: 4033:"Poincaré Chair" 4029: 4023: 4022: 4020: 4018: 4013: 4005: 3999: 3998: 3993: 3991: 3986:on 22 March 2010 3985: 3969: 3963: 3962: 3948: 3942: 3941: 3939: 3937: 3931: 3924: 3916: 3910: 3909: 3907: 3905: 3886: 3880: 3867: 3861: 3859: 3850: 3834: 3828: 3827: 3817: 3793: 3787: 3786: 3784: 3782: 3767: 3761: 3748: 3742: 3741: 3739: 3727: 3721: 3720: 3710: 3684: 3678: 3677: 3675: 3673: 3644: 3631: 3630: 3612: 3603:(5): 2587–2855. 3590: 3584: 3570: 3564: 3557: 3551: 3537: 3531: 3524: 3518: 3511: 3505: 3502: 3496: 3493: 3487: 3486: 3484: 3469:Comm. Anal. Geom 3460: 3454: 3453: 3441: 3435: 3434: 3427: 3421: 3418: 3412: 3409: 3403: 3400: 3394: 3391: 3385: 3382: 3376: 3373: 3367: 3364: 3358: 3355: 3349: 3348: 3341: 3335: 3329: 3323: 3322: 3320: 3318: 3303: 3297: 3296: 3294: 3292: 3273: 3267: 3266: 3245: 3236: 3230: 3224: 3223: 3218: 3216: 3194: 3188: 3187: 3182: 3180: 3158: 3147: 3146: 3135: 3124: 3123: 3121: 3119: 3095: 3086: 3085: 3083: 3081: 3062: 3053: 3052: 3050: 3048: 3039:. Archived from 3033: 3027: 3026: 3016: 2990: 2984: 2983: 2982:. 24 March 2010. 2972: 2966: 2965: 2963: 2961: 2956:on June 17, 2013 2952:. Archived from 2942: 2900:Manifold Destiny 2876:Ancient solution 2870: 2868:Biography portal 2865: 2864: 2863: 2856: 2851: 2850: 2823: 2821: 2804: 2788: 2786: 2769: 2753: 2751: 2734: 2725: 2723: 2707: 2698: 2696: 2680: 2669: 2649: 2639: 2620:Cheeger, Jeffrey 2613: 2611: 2595: 2589:Unpublished work 2579: 2577: 2575: 2569: 2563:. Archived from 2524: 2510: 2501: 2461: 2447: 2438: 2436: 2434: 2428: 2422:. Archived from 2383: 2369: 2360: 2305: 2296: 2241: 2232: 2188: 2174: 2165: 2139: 2108: 2099: 2081: 2050: 2041: 2035: 2027: 2012:Algebra i Analiz 2008: 1983: 1974: 1968: 1960: 1945:Algebra i Analiz 1941: 1916: 1907: 1849: 1840: 1834: 1826: 1817:: 99–102. 1989. 1807: 1789: 1769: 1754: 1745: 1739: 1731: 1712: 1671: 1662: 1603: 1594: 1557: 1548: 1522: 1509: 1497: 1406:Saint Petersburg 1369:Yakov Eliashberg 1274:Millennium Prize 1222:Saint Petersburg 973:scalar curvature 969:closed manifolds 844:Richard Hamilton 816:William Thurston 804:Michael Freedman 637:of the space by 531:convex polyhedra 464:for his work on 386:Millennium Prize 306: 305: 304: 298: 293: 289: 288: 254:Doctoral advisor 248: 176:(2010), declined 174:Millennium Prize 170:(2006), declined 164:(1996), declined 92: 88: 86: 69:Perelman in 1993 67: 57: 48:Grigori Perelman 44: 43: 5353: 5352: 5348: 5347: 5346: 5344: 5343: 5342: 5243: 5242: 5241: 5236: 5208: 5207: 5204: 5177: 5150: 5123: 5096: 5069: 5052: 5025: 4998: 4971: 4949: 4927: 4900: 4883: 4856: 4829: 4812: 4795: 4778: 4761: 4738: 4733: 4644: 4639: 4632:math.DG/0607607 4614: 4612: 4609: 4581:math.DG/0605667 4554: 4543: 4476:on 14 May 2012. 4473: 4456: 4442: 4440: 4435:. 1 July 2004. 4427: 4412: 4407: 4390: 4386: 4376: 4374: 4361: 4360: 4356: 4346: 4344: 4343:. 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3214: 3212: 3195: 3191: 3178: 3176: 3159: 3150: 3137: 3136: 3127: 3117: 3115: 3096: 3089: 3079: 3077: 3072:. 1 July 2010. 3064: 3063: 3056: 3046: 3044: 3035: 3034: 3030: 2991: 2987: 2974: 2973: 2969: 2959: 2957: 2944: 2943: 2934: 2930: 2889:Homology sphere 2866: 2861: 2859: 2852: 2845: 2842: 2837: 2836: 2831: 2801: 2796: 2766: 2761: 2731: 2726: 2721: 2704: 2699: 2694: 2677: 2672: 2658: 2609: 2586: 2585: 2580: 2573: 2571: 2567: 2545: 2522: 2507: 2502: 2482: 2459: 2444: 2439: 2432: 2430: 2426: 2404: 2381: 2366: 2361: 2302: 2297: 2238: 2233: 2213: 2186: 2171: 2166: 2105: 2100: 2047: 2042: 2029: 2028: 2010: 1980: 1975: 1962: 1961: 1943: 1913: 1908: 1846: 1841: 1828: 1827: 1809: 1765: 1751: 1746: 1733: 1732: 1714: 1668: 1663: 1600: 1595: 1554: 1549: 1518: 1503:Research papers 1475: 1441:Anatoly Vershik 1414: 1383:, according to 1309: 1206: 1024: 901: 899:Perelman's work 809:high dimensions 794:to a 3-sphere. 776:Euclidean space 756: 748:Main articles: 746: 741: 733:asymptotic cone 729:Euclidean space 717:Ricci curvature 698:Euclidean space 690:homotopy theory 676:of nonnegative 658: 646:invited lecture 596: 567:Euclidean space 555: 539:Helly's theorem 537:formulation of 523:convex geometry 519: 517:Convex geometry 514: 494:soul conjecture 482:Ricci curvature 403: 345:soul conjecture 300: 299: 291: 271: 246: 227: 223: 212: 179: 144: 129:soul conjecture 101: 99: 93: 90: 84: 82: 81: 80: 70: 58: 55: 49: 42: 17: 12: 11: 5: 5351: 5341: 5340: 5335: 5330: 5325: 5320: 5315: 5310: 5305: 5300: 5295: 5290: 5285: 5280: 5275: 5270: 5265: 5260: 5255: 5238: 5237: 5235: 5234: 5224: 5213: 5210: 5209: 5206: 5205: 5203: 5202: 5197: 5192: 5187: 5180: 5178: 5176: 5175: 5170: 5165: 5160: 5153: 5151: 5149: 5148: 5143: 5138: 5133: 5126: 5124: 5122: 5121: 5116: 5111: 5106: 5099: 5097: 5095: 5094: 5089: 5084: 5079: 5072: 5070: 5068: 5067: 5062: 5055: 5053: 5051: 5050: 5045: 5040: 5035: 5028: 5026: 5024: 5023: 5018: 5013: 5008: 5001: 4999: 4997: 4996: 4991: 4986: 4981: 4974: 4972: 4970: 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1752: 1748: 1747: 1718:(in Russian). 1687:(1): 735–740. 1675: 1669: 1665: 1664: 1621:(4): 665–666. 1607: 1601: 1597: 1596: 1577:(5): 191–194. 1561: 1555: 1551: 1550: 1513: 1507: 1506: 1500: 1499: 1474: 1471: 1437:Ludwig Faddeev 1433:Mikhail Gromov 1413: 1410: 1394:nanotechnology 1326:Shing-Tung Yau 1321:The New Yorker 1308: 1305: 1286:Clay Institute 1205: 1202: 1201: 1200: 1169:In July 2006, 1166: 1165: 1135: 1134: 1110: 1109: 1094: 1093: 1067:In June 2003, 1023: 1020: 996:Mikhael Gromov 925:Shing-Tung Yau 900: 897: 876:Shing-Tung Yau 764:Henri Poincaré 745: 742: 740: 737: 715:with positive 666:Detlef Gromoll 657: 654: 635:stratification 608:Mikhael Gromov 595: 592: 588:Nikolai Efimov 554: 551: 518: 515: 513: 512:Early research 510: 458:Mikhail Gromov 402: 399: 337:Mikhael Gromov 284:(Russian: 277: 276: 273: 272: 270: 269: 264: 258: 256: 250: 249: 238: 232: 231: 218: 214: 213: 211: 210: 205: 200: 194: 192: 188: 187: 181: 180: 178: 177: 171: 165: 159: 152: 150: 146: 145: 143: 142: 141:of 3-manifolds 139:geometrization 131: 124: 122: 121:Known for 118: 117: 107: 103: 102: 98:, Soviet Union 94: 78: 76: 72: 71: 68: 60: 59: 51: 50: 47: 15: 9: 6: 4: 3: 2: 5350: 5339: 5336: 5334: 5331: 5329: 5326: 5324: 5321: 5319: 5316: 5314: 5311: 5309: 5306: 5304: 5301: 5299: 5296: 5294: 5291: 5289: 5288:Living people 5286: 5284: 5281: 5279: 5276: 5274: 5271: 5269: 5266: 5264: 5261: 5259: 5256: 5254: 5251: 5250: 5248: 5233: 5229: 5225: 5223: 5215: 5214: 5211: 5201: 5198: 5196: 5193: 5191: 5188: 5186: 5185:Duminil-Copin 5182: 5181: 5179: 5174: 5171: 5169: 5166: 5164: 5161: 5159: 5155: 5154: 5152: 5147: 5144: 5142: 5139: 5137: 5134: 5132: 5128: 5127: 5125: 5120: 5117: 5115: 5112: 5110: 5107: 5105: 5104:Lindenstrauss 5101: 5100: 5098: 5093: 5090: 5088: 5085: 5083: 5080: 5078: 5074: 5073: 5071: 5066: 5063: 5061: 5057: 5056: 5054: 5049: 5046: 5044: 5041: 5039: 5036: 5034: 5030: 5029: 5027: 5022: 5019: 5017: 5014: 5012: 5009: 5007: 5003: 5002: 5000: 4995: 4992: 4990: 4987: 4985: 4982: 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3408: 3399: 3390: 3381: 3372: 3363: 3354: 3346: 3340: 3334:, p. 45) 3333: 3328: 3312: 3308: 3302: 3286: 3282: 3278: 3272: 3264: 3260: 3259: 3254: 3250: 3244: 3242: 3235:, p. 48) 3234: 3229: 3222: 3210: 3206: 3205: 3200: 3193: 3186: 3174: 3170: 3169: 3164: 3157: 3155: 3153: 3144: 3140: 3134: 3132: 3130: 3113: 3109: 3105: 3101: 3094: 3092: 3075: 3071: 3067: 3061: 3059: 3042: 3038: 3032: 3024: 3020: 3015: 3010: 3006: 3002: 3001: 2996: 2989: 2981: 2977: 2971: 2955: 2951: 2947: 2941: 2939: 2937: 2932: 2923: 2920: 2918: 2915: 2913: 2910: 2907: 2906: 2901: 2897: 2895: 2892: 2890: 2887: 2885: 2882: 2879: 2877: 2874: 2873: 2869: 2858: 2855: 2849: 2844: 2830: 2826: 2820: 2815: 2809: 2805: 2795: 2791: 2785: 2780: 2774: 2770: 2760: 2756: 2750: 2745: 2739: 2735: 2720: 2719: 2712: 2708: 2693: 2692: 2685: 2681: 2667: 2663: 2659: 2653: 2648: 2643: 2638: 2633: 2629: 2625: 2621: 2615: 2614: 2608: 2607: 2600: 2596: 2591: 2590: 2566: 2562: 2558: 2554: 2550: 2546: 2544:0-521-59222-4 2540: 2536: 2532: 2528: 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1069:Bruce Kleiner 1066: 1065: 1064: 1061: 1057: 1053: 1049: 1045: 1040: 1038: 1034: 1030: 1019: 1017: 1013: 1012:Bruce Kleiner 1009: 1005: 1001: 997: 993: 989: 984: 982: 978: 974: 970: 965: 960: 958: 953: 950: 946: 942: 937: 935: 930: 926: 922: 917: 914: 910: 906: 896: 893: 889: 885: 881: 877: 871: 869: 865: 861: 857: 856:heat equation 853: 849: 845: 840: 838: 833: 829: 825: 821: 817: 812: 810: 805: 801: 797: 796:Stephen Smale 793: 789: 785: 781: 777: 773: 769: 765: 761: 758:The Poincaré 755: 751: 736: 734: 730: 726: 722: 721:connected sum 718: 714: 709: 707: 703: 699: 695: 691: 687: 686:diffeomorphic 683: 679: 675: 671: 667: 663: 653: 651: 647: 642: 640: 636: 632: 631:gradient flow 628: 623: 621: 617: 613: 609: 605: 601: 591: 589: 584: 580: 576: 572: 568: 564: 560: 550: 548: 544: 540: 536: 532: 528: 524: 509: 507: 503: 499: 495: 491: 487: 483: 479: 475: 471: 467: 463: 459: 455: 451: 447: 443: 438: 436: 432: 428: 424: 420: 415: 412: 408: 398: 396: 392: 387: 382: 380: 376: 375: 369: 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