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{{Short description|Russian mathematician (born 1966)}}
{{Use dmy dates|date=July 2022}}
{{Family name hatnote|Yakovlevich|Perelman|lang=Eastern Slavic}}
{{Infobox scientist
| name              = Grigori Perelman
| native_name       = {{nobold|Григорий Перельман}}
| native_name_lang  = ru
| image             = Grigori Perelman, 1993 (re-scanned) (cropped).jpg
| alt               = Headshot of Grigori Perelman. He is a white man with neck-length, brown, curly hair, a full beard and blue eyes, wearing a brown suit. 
| caption           = Perelman in 1993
| birth_name        = Grigori Yakovlevich Perelman
| birth_date        = {{Birth date and age|1966|06|13|df=yes}}
| birth_place       = [[Leningrad]], [[Soviet Union]]<br />(now Saint Petersburg, Russia)
| education         = [[Leningrad State University]] ([[PhD]])
| known_for         = Proof of the [[soul conjecture]], [[Poincaré conjecture]] and [[geometrization conjecture|geometrization]] of 3-manifolds
| awards            = {{ubl
  | [[Saint Petersburg Mathematical Society]] Prize (1991)
  | [[EMS Prize]] (1996, declined)
  | [[Fields Medal]] (2006, declined)
  | [[Millennium Prize Problems|Millennium Prize]] (2010, declined)
  }}
| fields            = {{ubl|
  | [[Differential geometry]]
  | [[Geometric analysis]]
  | [[Geometric topology]]
  }}
| workplaces        = {{ubl
  | [[St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences|PDMI]]
  | [[Courant Institute of Mathematical Sciences|New York University]]
  | [[University of California, Berkeley]]
  }}
| thesis_title      = Saddle Surfaces in Euclidean Spaces
| thesis_url        = https://www.mathgenealogy.org/id.php?id=84354
| thesis_year       = 1990
| doctoral_advisor  = {{ubl
  | [[Aleksandr Danilovich Aleksandrov|Aleksandr Aleksandrov]]
  | [[Yuri Burago]]
  }}
}}
'''Grigori Yakovlevich Perelman''' ({{langx|ru|Григорий Яковлевич Перельман}}, {{IPA|ru|ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman|pron|ru-Grigori Yakovlevich Perelman.oga}}; born 13{{nbsp}}June 1966) is a Russian [[mathematician]] and geometer who is known for his contributions to the fields of [[geometric analysis]], [[Riemannian geometry]], and [[geometric topology]]. In 2005, Perelman resigned from his research post in [[Steklov Institute of Mathematics]] 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.

In the 1990s, partly in collaboration with [[Yuri Burago]], [[Mikhail Leonidovich Gromov|Mikhael Gromov]], and Anton Petrunin, he made contributions to the study of [[Alexandrov space]]s. In 1994, he proved the [[Soul theorem|soul conjecture]] 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 [[Ricci flow]], and proved the [[Poincaré conjecture]] and [[geometrization conjecture|Thurston's geometrization conjecture]], the former of which had been a famous [[open problem]] 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.

In August 2006, Perelman was offered the [[Fields Medal]]<ref name=":0">{{Cite web |title=Fields Medals 2006 |url=http://www.mathunion.org/general/prizes/2006/ |url-status=dead |archive-url=https://web.archive.org/web/20130617222042/http://www.mathunion.org/General/Prizes/2006/ |archive-date=17 June 2013 |access-date=30 April 2006 |website=International Mathematical Union (IMU) – Prizes |df=mdy-all}}</ref> 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."<ref>{{Cite news |date=24 March 2010 |title=Russian maths genius Perelman urged to take $1m prize |url=https://news.bbc.co.uk/2/hi/8585407.stm |publisher=[[BBC News]]}}</ref> On 22 December 2006, the scientific journal ''[[Science (journal)|Science]]'' recognized Perelman's proof of the Poincaré conjecture as the scientific "[[Breakthrough of the Year]]", the first such recognition in the area of mathematics.<ref>{{Cite journal |last=Mackenzie |first=Dana |year=2006 |title=Breakthrough of the year. The Poincaré Conjecture{{snd}}Proved |journal=[[Science (journal)|Science]] |volume=314 |issue=5807 |pages=1848–1849 |doi=10.1126/science.314.5807.1848 |pmid=17185565 |doi-access=free}}</ref>

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay [[Millennium Prize Problems|Millennium Prize]]<ref>{{Cite web |title=The Poincaré Conjecture |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |url-status=dead |archive-url=https://web.archive.org/web/20140705132527/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-date=5 July 2014 |access-date=1 May 2014}}</ref> 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 [[Richard S. Hamilton]], the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture.<ref name=interfax/><ref name="PhysOrg1" /> He had previously rejected the prestigious prize of the [[European Mathematical Society]] in 1996.<ref name=bbc/>

==Early life and education==
Grigori Yakovlevich Perelman was born in [[Leningrad]], Soviet Union (now Saint Petersburg, Russia) on June 13, 1966, to [[History of the Jews in Russia|Jewish]] parents,<ref name="tdt">{{Cite news |last1=Osborn |first1=Andrew |last2=Krepysheva |first2=Olga |date=27 March 2010 |title=Russian maths genius may turn down $1m prize |url=https://www.telegraph.co.uk/news/worldnews/europe/russia/7530771/Russian-maths-genius-may-turn-down-1m-prize.html |url-status=live |archive-url=https://web.archive.org/web/20100330073600/http://www.telegraph.co.uk/news/worldnews/europe/russia/7530771/Russian-maths-genius-may-turn-down-1m-prize.html |archive-date=30 March 2010 |access-date=2 July 2010 |work=[[The Daily Telegraph]] |quote=He has suffered anti-Semitism (he is Jewish)....Grigory is pure Jewish and I never minded that but my bosses did}}</ref><ref>{{Cite web |last=McKie |first=Robin |date=27 March 2011 |title=Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen – review |url=https://www.theguardian.com/books/2011/mar/27/perfect-rigour-grigori-perelman-review |url-status=live |archive-url=https://web.archive.org/web/20131004063146/http://www.theguardian.com/books/2011/mar/27/perfect-rigour-grigori-perelman-review |archive-date=4 October 2013 |access-date=23 August 2013 |website=[[The Guardian]] |quote=Given that his parents were Jewish, Perelman, who was born in 1966, was fortunate in those who took up his cause.}}</ref><ref>{{harvtxt|Gessen|2009|p=48}}</ref> Yakov (who now lives in Israel)<ref name=tdt/> and Lyubov (who still lives in Saint Petersburg with Perelman).<ref name=tdt/> 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 after-school mathematics training program.<ref name="nyrb">{{Cite journal |last=Paulos |first=John Allen |author-link=John Allen Paulos |date=29 April 2010 |title=He Conquered the Conjecture |url=http://www.nybooks.com/articles/23863 |journal=[[The New York Review of Books]] |volume=57 |issue=7}}</ref>

His mathematical education continued at the [[Saint Petersburg Lyceum 239|Leningrad Secondary School 239]], a [[specialized school]] with advanced mathematics and physics programs. Perelman excelled in all subjects except [[physical education]].<ref>{{Cite news |title=Eccentric 'Mathsputin' Rejects Million Dollar Prize |url=http://www.foxnews.com/scitech/2010/03/30/eccentric-mathsputin-solves-ancient-problem-rejects-million-dollar-prize/ |url-status=dead |archive-url=https://web.archive.org/web/20140715221607/http://www.foxnews.com/scitech/2010/03/30/eccentric-mathsputin-solves-ancient-problem-rejects-million-dollar-prize/ |archive-date=15 July 2014 |access-date=8 July 2014 |agency=[[Fox News]]}}</ref> In 1982, not long after his sixteenth birthday, he won a gold medal as a member of the Soviet team at the [[International Mathematical Olympiad]] hosted in Budapest, achieving a perfect score.<ref name="imo">{{Cite web |title=International Mathematical Olympiad |url=http://www.imo-official.org/participant_r.aspx?id=10481 |url-status=live |archive-url=https://web.archive.org/web/20121102120643/http://www.imo-official.org/participant_r.aspx?id=10481 |archive-date=2 November 2012 |access-date=25 December 2012 |publisher=Imo-official.org}}</ref> He continued as a student of the [[Saint Petersburg State University Mathematics and Mechanics Faculty|School of Mathematics and Mechanics]] (the so-called "матмех" i.e. "math-mech") at [[Saint-Petersburg State University|Leningrad State University]], without admission examinations, and enrolled at the university.{{Citation needed|date=May 2023}}

After completing his PhD in 1990, Perelman began work at the [[St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences|Leningrad Department of Steklov Institute of Mathematics]] of the [[USSR Academy of Sciences]], where his advisors were [[Aleksandr Danilovich Aleksandrov|Aleksandr Aleksandrov]] and [[Yuri Burago]]. In the late 1980s and early 1990s, with a strong recommendation from the geometer [[Mikhail Leonidovich Gromov|Mikhail Gromov]],<ref>{{harvtxt|Gessen|2009|p=45}}</ref> Perelman obtained research positions at several universities in the United States. In 1991, Perelman won the Young Mathematician Prize of the [[St. Petersburg Mathematical Society|Saint Petersburg Mathematical Society]] for his work on [[CAT(k) space|Aleksandrov's spaces]] of curvature bounded from below.<ref>{{Cite web |title=Young mathematician prize of the St. Petersburg Mathematical Society |url=http://www.mathsoc.spb.ru/mol_mat.html}}</ref> In 1992, he was invited to spend a semester each at the [[Courant Institute of Mathematical Sciences|Courant Institute]] in [[New York University]], where he began work on [[manifold]]s with lower bounds on [[Ricci curvature]]. From there, he accepted a two-year [[Miller Research Fellows]]hip at the [[University of California, Berkeley]], in 1993. After proving the [[Soul theorem|soul conjecture]] in 1994, he was offered jobs at several top universities in the US, including [[Princeton University|Princeton]] and [[Stanford University|Stanford]], but he rejected them all and returned to the [[St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences|Steklov Institute in Saint Petersburg]] in the summer of 1995 for a research-only position.<ref name=nyrb/>

==Early research==
===Convex geometry===
In his undergraduate studies, Perelman dealt with issues in the field of [[convex geometry]]. His first published article studied the [[Simplicial complex|combinatorial structures]] arising from intersections of [[convex polyhedra]].{{ran|P85}} With I. V. Polikanova, he established a [[measure theory|measure-theoretic]] formulation of [[Helly's theorem]].{{ran|PP86}} In 1987, the year he began graduate studies, he published an article controlling the size of [[Circumscribed sphere|circumscribed cylinders]] by that of [[inscribed sphere]]s.{{ran|P87}}

===Negatively curved hypersurfaces===
Surfaces of [[Curvature|negative curvature]] were the subject of Perelman's graduate studies. His first result was on the possibility of prescribing the structure of negatively-curved [[Polyhedron|polyhedral surfaces]] in three-dimensional [[Euclidean space]]. He proved that any such [[Metric space|metric]] on the plane which is complete can be continuously immersed as a polyhedral surface.{{ran|P88}} Later, he constructed an example of a smooth [[hypersurface]] of [[Four-dimensional space|four-dimensional Euclidean space]] which is complete and has [[Gaussian curvature]] 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.{{ran|P89}} As such, his construction provided further obstruction to the extension of a well-known theorem of [[Nikolai Efimov]] to higher dimensions.<ref>Efimov, N. V. Generation of singularites on surfaces of negative curvature. Mat. Sb. (N.S.) 64 (106) 1964 286–320.</ref>

===Alexandrov spaces===
Perelman's first works to have a major impact on the mathematical literature were in the field of [[Alexandrov space]]s, the concept of which dates back to the 1950s. In a very well-known paper coauthored with [[Yuri Burago]] and [[Mikhail Leonidovich Gromov|Mikhael Gromov]], Perelman established the modern foundations of this field, with the notion of [[Gromov–Hausdorff convergence]] as an organizing principle.{{ran|BGP92}} In a followup unpublished paper, Perelman proved his "stability theorem," asserting that in the collection of all [[Alexandrov space|Alexandrov spaces]] with a fixed curvature bound, all elements of any sufficiently small metric ball around a compact space are mutually [[homeomorphic]].{{ran|P91}} 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.

Perelman developed a version of [[Morse theory]] on Alexandrov spaces.{{ran|P93}} Despite the lack of smoothness in Alexandrov spaces, Perelman and Anton Petrunin were able to consider the [[Vector field|gradient flow]] of certain functions, in unpublished work.{{ran|PP95}} They also introduced the notion of an "extremal subset" of Alexandrov spaces, and showed that the interiors of certain extremal subsets define a [[Stratification (mathematics)|stratification]] of the space by [[topological manifold]]s.{{ran|PP93}} 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.{{ran|P95d}}

For his work on Alexandrov spaces, Perelman was recognized with an [[List of International Congresses of Mathematicians Plenary and Invited Speakers|invited lecture]] at the 1994 [[International Congress of Mathematicians]].{{ran|P95a}}

===Comparison geometry===
In 1972, [[Jeff Cheeger]] and [[Detlef Gromoll]] established their important [[soul theorem]]. It asserts that every complete [[Riemannian metric]] of nonnegative [[sectional curvature]] has a compact nonnegatively curved submanifold, called a ''soul'', whose normal bundle is [[diffeomorphic]] to the original space. From the perspective of [[homotopy theory]], this says in particular that every complete Riemannian metric of nonnegative sectional curvature may be taken to be [[closed manifold|closed]]. 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 [[Euclidean space]]. In 1994, Perelman gave a short proof of Cheeger and Gromoll's conjecture by establishing that, under the condition of nonnegative sectional curvature, [[Sharafutdinov's retraction]] is a [[submersion (mathematics)|submersion]].{{ran|P94b}} 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.

Some of Perelman's work dealt with the construction of various interesting [[Riemannian manifold]]s with positive [[Ricci curvature]]. He found Riemannian metrics on the [[connected sum]] of arbitrarily many [[complex projective plane]]s with positive Ricci curvature, bounded diameter, and volume bounded away from zero.{{ran|P97b}} Also, he found an explicit complete metric on four-dimensional [[Euclidean space]] with positive Ricci curvature and Euclidean volume growth, and such that the [[ultralimit|asymptotic cone]] is non-uniquely defined.{{ran|P97c}}

==Geometrization and Poincaré conjectures==
===The problems===
{{Main|Poincaré conjecture|Thurston's geometrization conjecture}}
The Poincaré [[conjecture]], proposed by mathematician [[Henri Poincaré]] in 1904, was throughout the 20th century regarded as a key problem in [[topology]]. On the [[3-sphere]], defined as the set of points at unit length from the origin in four-dimensional [[Euclidean space]], any [[path (topology)|loop]] can be contracted into a point. Poincaré suggested that a converse might be true: if a [[closed manifold|closed]] three-dimensional [[manifold]] has the property that any loop can be contracted into a point, then it must be [[Homeomorphism|topologically equivalent]] to a 3-sphere. [[Stephen Smale]] proved a [[Generalized Poincaré conjecture|high-dimensional analogue]] of Poincaré's conjecture in 1961, and [[Michael Freedman]] proved the four-dimensional version in 1982.<ref>Smale, Stephen. Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961), 391–406.</ref><ref>Freedman, Michael Hartley. The topology of four-dimensional manifolds. J. Differential Geometry 17 (1982), no. 3, 357–453.</ref> 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 way without interfering with other regions, seem to require [[Dimension|high dimensions]] in order to work.

In 1982, [[William Thurston]] developed a novel viewpoint, making the Poincaré conjecture into a small special case of a hypothetical systematic structure theory of [[topology]] in three dimensions. His proposal, known as the [[Thurston geometrization conjecture]], posited that given any closed three-dimensional [[manifold]] whatsoever, there is some collection of two-dimensional spheres and [[Torus|tori]] inside of the manifold which disconnect the space into separate pieces, each of which can be endowed with a uniform geometric structure.<ref>Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381.</ref> Thurston was able to prove his conjecture under some provisional assumptions. In [[John Morgan (mathematician)|John Morgan]]'s view, it was only with Thurston's systematic viewpoint that most topologists came to believe that the Poincaré conjecture would be true.<ref>John Morgan. "The Poincaré conjecture." Lecture at 2006 International Congress of Mathematicians.</ref>

At the same time that Thurston published his conjecture, [[Richard S. Hamilton|Richard Hamilton]] introduced his theory of the [[Ricci flow]]. Hamilton's Ricci flow is a prescription, defined by a [[partial differential equation]] formally analogous to the [[heat equation]], for how to deform a [[Riemannian metric]] on a manifold. The heat equation, such as when applied in the sciences to physical phenomena such as [[temperature]], 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.<ref>Hamilton, Richard S. Three-manifolds with positive Ricci curvature. J. Differential Geometry 17 (1982), no. 2, 255–306.</ref><ref>Hamilton, Richard S. Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153–179.</ref><ref>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.</ref> As a byproduct, he was able to prove some new and striking theorems in the field of [[Riemannian geometry]].

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 curvature accumulates to infinite levels after a finite amount of "time" has elapsed. Following [[Shing-Tung Yau]]'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 [[Geometrization conjecture|Thurston's conjecture]], Hamilton began a systematic analysis.<ref>{{Cite web |title=Autobiography of Richard S Hamilton &#124; the Shaw Prize |url=https://www.shawprize.org/prizes-and-laureates/mathematical-sciences/2011/autobiography-of-richard-s-hamilton}}</ref> Throughout the 1990s, he found a number of new technical results and methods,<ref>{{Cite journal |last=Hamilton |first=Richard S. |date=1995 |title=The formation of singularities in the Ricci flow |journal=Surveys in Differential Geometry |volume=II |pages=7–136}}</ref> culminating in a 1997 publication constructing a "Ricci flow with surgery" for [[Four-dimensional space|four-dimensional spaces]].<ref>{{Cite journal |last=Hamilton |first=Richard S. |date=1997 |title=Four-manifolds with positive isotropic curvature |journal=Comm. Anal. Geom. |volume=5 |issue=1 |pages=1–92 |doi=10.4310/CAG.1997.v5.n1.a1 |doi-access=free}}</ref> 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 [[geometric analysis]], saying that with its publication it became clear that Ricci flow could be powerful enough to settle the Thurston conjecture.<ref>Yau, Shing-Tung. Perspectives on geometric analysis. Surveys in differential geometry. Vol. X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.</ref> 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 [[Ricci flow]] 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.<ref>Hamilton, Richard S. Non-singular solutions of the Ricci flow on three-manifolds. Comm. Anal. Geom. 7 (1999), no. 4, 695–729.</ref> This became known as the Hamilton program.

===Perelman's work===
In November 2002 and March 2003, Perelman posted two [[preprint]]s to [[arXiv]], in which he claimed to have outlined a [[mathematical proof|proof]] of Thurston's conjecture.{{ran|P02}}{{ran|P03a}} 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.{{ran|P03b}}

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 [[Peter Li (mathematician)|Peter Li]] and [[Shing-Tung Yau]]'s differential Harnack inequalities to the setting of Ricci flow.<ref>Li, Peter; Yau, Shing-Tung. On the parabolic kernel of the Schrödinger operator. ''Acta Math.'' 156 (1986), no. 3-4, 153–201.</ref> By carrying out the proof of the [[Bishop–Gromov inequality]] 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 preconditions of Hamilton's ''compactness theorem''. As a consequence, Hamilton's compactness and the corresponding existence of subsequential limits could be applied somewhat freely.

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 [[Ricci flow]] which had eluded Hamilton. Roughly speaking, Perelman showed that on a microscopic level, every singularity looks either like a [[cylinder]] collapsing to its axis, or a [[sphere]] 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 manifolds.

Other results in Perelman's first preprint include the introduction of certain monotonic quantities and a "pseudolocality theorem" which relates curvature control and [[isoperimetry]]. However, despite being major results in the theory of Ricci flow, these results were not used in the rest of his work.

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 [[Ricci flow]] 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 [[closed manifold]]s which admit metrics of positive [[scalar curvature]]. His third preprint (or alternatively Colding and Minicozzi's work) showed that on any space satisfying the assumptions of the [[Poincaré conjecture]], the Ricci flow with surgery exists only for [[Glossary of mathematical jargon#finite|finite]] 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.

In order to settle the [[Geometrization conjecture|Thurston conjecture]], 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 [[Jeff Cheeger]] and [[Mikhael Gromov (mathematician)|Mikhael Gromov]]'s theorem characterizing [[collapsing manifold]]s. 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,<ref>Shioya, Takashi; Yamaguchi, Takao. Volume collapsed three-manifolds with a lower curvature bound. ''Math. Ann.'' 333 (2005), no. 1, 131–155.</ref> [[John Morgan (mathematician)|John Morgan]] and [[Gang Tian]],<ref>Morgan, John; Tian, Gang. ''The geometrization conjecture''. Clay Mathematics Monographs, 5. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014. x+291 pp. 
{{ISBN|978-0-8218-5201-9}}</ref> Jianguo Cao and Jian Ge,<ref>Cao, Jianguo; Ge, Jian. A simple proof of Perelman's collapsing theorem for 3-manifolds. J. ''Geom. Anal.'' 21 (2011), no. 4, 807–869.</ref> and [[Bruce Kleiner]] and [[John Lott (mathematician)|John Lott]].<ref>Kleiner, Bruce; Lott, John. Locally collapsed 3-manifolds. ''Astérisque'' No. 365 (2014), 7–99. {{ISBN|978-2-85629-795-7}}</ref>

===Verification===
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 apparent that Perelman had made major contributions to the foundations of [[Ricci flow]], although it was not immediately clear to the mathematical community that these contributions were sufficient to prove the [[geometrization conjecture]] or the [[Poincaré conjecture]].

In April 2003, 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 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 articles.

* In June 2003, [[Bruce Kleiner]] and [[John Lott (mathematician)|John Lott]], both then of the [[University of Michigan]], 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 [[Geometry & Topology]] in 2008.<ref name="kleiner and lott">{{Cite journal |last1=Kleiner |first1=Bruce |last2=Lott |first2=John |year=2008 |title=Notes on Perelman's papers |journal=[[Geometry & Topology]] |volume=12 |issue=5 |pages=2587–2855 |arxiv=math/0605667 |doi=10.2140/gt.2008.12.2587 |s2cid=119133773}}</ref> At the 2006 [[International Congress of Mathematicians]], Lott said "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 correct." In the introduction to their article, Kleiner and Lott explained:
{{blockquote|text=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.}}
: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 2013.
* In June 2006, the [[Asian Journal of Mathematics]] published an article by [[Huai-Dong Cao]] of [[Lehigh University]] and [[Zhu Xiping]] of [[Sun Yat-sen University]], 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:
{{blockquote|text=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, 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.}}
:Based also upon the title "A Complete Proof of the Poincaré and Geometrization Conjectures – Application of the Hamilton-Perelman Theory of Ricci Flow" and the phrase "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow" from the abstract, some people interpreted Cao and Zhu to be taking credit from Perelman for themselves.<ref name="new yorker">{{Cite magazine |last1=Nasar |first1=Sylvia |last2=Gruber |first2=David |date=21 August 2006 |title=Manifold Destiny: A legendary problem and the battle over who solved it |url=http://www.newyorker.com/archive/2006/08/28/060828fa_fact2?currentPage=all |url-status=live |archive-url=https://web.archive.org/web/20110319235149/http://www.newyorker.com/archive/2006/08/28/060828fa_fact2?currentPage=all |archive-date=19 March 2011 |access-date=21 January 2011 |magazine=[[The New Yorker]] |authorlink1=Sylvia Nasar |authorlink2=David Gruber}}</ref> When asked about the issue, Perelman said that he could not see any new contribution by Cao and Zhu and that they "did not quite understand the argument and reworked it."<ref name="new yorker" /> 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,<ref>{{Cite journal |last1=Cao, Huai-Dong |last2=Zhu, Xi-Ping |year=2006 |title=Erratum to "A complete proof of the Poincaré and geometrization conjectures&nbsp;– application of the Hamilton–Perelman theory of the Ricci flow", Asian J. Math., Vol. 10, No. 2, 165–492, 2006 |journal=[[Asian Journal of Mathematics]] |volume=10 |issue=4 |pages=663–664 |doi=10.4310/ajm.2006.v10.n2.a2 |mr=2282358 |doi-access=free}}</ref> 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 [[ArXiv]]<ref>{{Cite arXiv |eprint=math.DG/0612069 |author=Cao, Huai-Dong |author2=Zhu, Xi-Ping |title=Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture |date=3 December 2006}}</ref> with revisions in their phrasing and in the relevant page of the proof.
* In July 2006, [[John Morgan (mathematician)|John Morgan]] of [[Columbia University]] and [[Gang Tian]] of the [[Massachusetts Institute of Technology]] posted a paper on arXiv in which they provided a detailed presentation of Perelman's proof of the Poincaré conjecture.<ref>Morgan, John W.; Tian, Gang ''Ricci Flow and the Poincaré Conjecture'' {{arxiv|math/0607607}}</ref> 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 [[International Congress of Mathematicians|ICM]] in [[Madrid]] on the Poincaré conjecture, in which he declared that Perelman's work had been "thoroughly checked."<ref>{{Cite web |title=Schedule of the scientific program of the ICM 2006 |url=http://icm2006.org/v_f/web_fr.php |url-status=dead |archive-url=https://web.archive.org/web/20100211112009/http://icm2006.org/v_f/web_fr.php |archive-date=11 February 2010 |access-date=21 March 2010 |publisher=Icm2006.org}}</ref> In 2015, [[Abbas Bahri]] 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.<ref>{{Cite journal |last=Bahri |first=Abbas |date=2015 |title=Five gaps in mathematics |journal=Adv. Nonlinear Stud. |volume=15 |issue=2 |pages=289–319 |doi=10.1515/ans-2015-0202 |s2cid=125566270 |doi-access=free}}</ref><ref>{{Citation |last1=Morgan |first1=John |title=Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture |date=2015 |arxiv=1512.00699 |bibcode=2015arXiv151200699M |last2=Tian |first2=Gang}}.</ref> The error, introduced by Morgan and Tian, dealt with details not directly discussed in Perelman's original work. In 2008, Morgan and Tian posted a paper which covered the details of the proof of the geometrization conjecture.<ref>Morgan, John W.; Tian, Gang ''Completion of the Proof of the Geometrization Conjecture'' {{arxiv|0809.4040}}</ref> Morgan and Tian's two articles have been published in book form by the Clay Mathematics Institute.

==Fields Medal and Millennium Prize==
In May 2006, a committee of nine mathematicians voted to award Perelman a [[Fields Medal]] for his work on the Ricci flow.<ref name="new yorker" /> However, Perelman declined to accept the prize. [[John M. Ball|Sir John Ball]], president of the [[International Mathematical Union]], approached Perelman in [[Saint Petersburg]] 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:<ref name="new yorker" />

{{blockquote|text=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 ... [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed.}}

He was quoted as saying:<ref>{{Cite news |date=24 March 2010 |title=Maths genius urged to take prize |url=https://news.bbc.co.uk/2/hi/europe/8585407.stm |url-status=live |archive-url=https://web.archive.org/web/20100419061533/http://news.bbc.co.uk/2/hi/europe/8585407.stm |archive-date=19 April 2010 |access-date=25 March 2010 |publisher=[[BBC News]]}}</ref>

{{blockquote|text=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 me.}}

Nevertheless, on 22 August 2006, at the [[International Congress of Mathematicians]] in [[Madrid]], Perelman was offered the Fields Medal "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow".<ref name="icm2006">{{Cite news |date=22 August 2006 |title=Fields Medal&nbsp;– Grigory Perelman |url=http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf |url-status=dead |archive-url=https://web.archive.org/web/20121103230435/http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf |archive-date=3 November 2012 |access-date=22 August 2006 |publisher=International Congress of Mathematicians 2006}}</ref> 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.<ref name="bbc">{{Cite news |date=22 August 2006 |title=Maths genius declines top prize |url=https://news.bbc.co.uk/2/hi/science/nature/5274040.stm |url-status=live |archive-url=https://web.archive.org/web/20100815015937/http://news.bbc.co.uk/2/hi/science/nature/5274040.stm |archive-date=15 August 2010 |publisher=BBC News}}</ref><ref>{{Cite news |last=Mullins, Justin |date=22 August 2006 |title=Prestigious Fields Medals for mathematics awarded |url=https://www.newscientist.com/article/dn9813.html |work=[[New Scientist]]}}</ref>

He has also rejected a prestigious prize from the [[European Mathematical Society]].<ref name=bbc/>

On 18 March 2010, Perelman was awarded a [[Millennium Prize Problems|Millennium Prize]] for solving the problem.<ref name="press-release-2010-03-18">{{Cite press release |title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman |date=18 March 2010 |publisher=[[Clay Mathematics Institute]] |url=http://www.claymath.org/sites/default/files/millenniumprizefull.pdf |quote=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. |access-date=1 May 2014 |url-status=dead |archive-url=https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ |archive-date=22 March 2010}}</ref> On 8 June 2010, he did not attend a ceremony in his honor at the [[Institut océanographique de Paris|Institut Océanographique de Paris]] to accept his $1 million prize.<ref name="Clay Mathematics Institute 2010 Annual Report 2010">{{Cite web |title=Clay Mathematics Institute 2010 Annual Report 2010 |url=https://www.claymath.org/library/annual_report/ar2010/ar2010.pdf |access-date=21 April 2024}}</ref> According to [[Interfax]], Perelman refused to accept the Millennium Prize in July 2010. He considered the decision of the [[Clay Mathematics Institute|Clay Institute]] unfair for not sharing the prize with [[Richard S. Hamilton]],<ref name="interfax">{{Cite news |date=1 July 2010 |title=Последнее "нет" доктора Перельмана |url=http://www.interfax.ru/society/txt.asp?id=143603 |url-status=live |archive-url=https://web.archive.org/web/20100702155304/http://www.interfax.ru/society/txt.asp?id=143603 |archive-date=2 July 2010 |access-date=1 July 2010 |agency=[[Interfax]]}}</ref> and stated that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust."<ref name="PhysOrg1">{{Cite web |title=Russian mathematician rejects $1 million prize |url=http://www.physorg.com/news197209671.html |publisher=[[Associated Press|AP]] on [[PhysOrg]]|first=Malcolm|last=Ritter|date=1 July 2010|access-date=15 May 2011|archive-url=https://web.archive.org/web/20120117202129/http://www.physorg.com/news197209671.html|archive-date=17 January 2012|url-status=live}}</ref>

The Clay Institute subsequently used Perelman's prize money to fund the "Poincaré Chair", a temporary position for young promising mathematicians at the Paris [[Institut Henri Poincaré]].<ref>{{Cite web |date=4 March 2014 |title=Poincaré Chair |url=http://www.claymath.org/events/news/poincar%C3%A9-chair |url-status=dead |archive-url=https://web.archive.org/web/20230509214305/https://www.claymath.org/events/news/poincar%C3%A9-chair |archive-date=9 May 2023 |access-date=26 September 2016 |publisher=Clay Institute}}</ref>

==Possible withdrawal from mathematics==
Perelman quit his job at the [[Steklov Institute of Mathematics|Steklov Institute]] in December 2005.<ref>{{harvtxt|Gessen|2009|p=[https://archive.org/details/perfectrigorgeni00gess/page/185/mode/2up 185]}}</ref> 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.<ref>{{Cite news |date=22 August 2006 |script-title=ru:Главные новости |url=http://top.rbc.ru/society/22/08/2006/94703.shtml |url-status=live |archive-url=https://web.archive.org/web/20110716055141/http://top.rbc.ru/society/22/08/2006/94703.shtml |archive-date=16 July 2011 |access-date=21 March 2010 |work=Рбк |publisher=[[RBC Information Systems]] |language=ru}}</ref>

Perelman is quoted in a 2006 article in ''[[The New Yorker]]'' 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 [[Shing-Tung Yau]] to downplay Perelman's role in the proof and play up the work of [[Huai-Dong Cao|Cao]] and [[Zhu Xiping|Zhu]]. Perelman added:<ref name="new yorker" />
{{blockquote|text=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.}}

This, combined with the possibility of being awarded a [[Fields Medal]], led him to state that he had quit professional mathematics by 2006. He said:<ref name="new yorker" />
{{blockquote|text=As long as I was not conspicuous, I had a choice. Either to make some ugly thing{{efn|''The New Yorker'' authors explained Perelman's reference to "some ugly thing" as "a fuss" on Perelman's part about the ethical breaches he perceived.}} 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 was unclear whether along with his resignation from [[Steklov Institute of Mathematics|Steklov]] and subsequent seclusion Perelman stopped his mathematics research. [[Yakov Eliashberg]], another [[Russia|Russian]] 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 [[Navier–Stokes equations]] and the problem of their solutions' [[Navier–Stokes existence and smoothness|existence and smoothness]], according to ''[[Le Point]]''.<ref>{{Cite news |date=30 September 2010 |title=Le génie qui s'est retiré du monde |trans-title=The genius who has withdrawn from the world |url=http://www.lepoint.fr/monde/le-genie-qui-s-est-retire-du-monde-30-09-2010-1246189_24.php |url-status=live |archive-url=https://web.archive.org/web/20120721073747/http://www.lepoint.fr/monde/le-genie-qui-s-est-retire-du-monde-30-09-2010-1246189_24.php |archive-date=21 July 2012 |access-date=15 October 2010 |work=[[Le Point]] |pages=74–77 |language=fr}}</ref>

In 2014, Russian media reported that Perelman was working in the field of [[nanotechnology]] in [[Sweden]].<ref name="kp.ru">{{Cite news |last=Veligzhanina |first=Anna |date=2014-07-23 |title="Komsomolskaya Pravda" found out where Perelman disappears |url=https://www.kp.ru/daily/26260.3/3138310 |work=Kp.ru -}}</ref> Shortly thereafter, however, he was spotted again in his native [[place of birth|hometown]] of [[Saint Petersburg]].<ref name="kp.ru" /> Russian media speculated that he periodically visits his sister in Sweden, while living in Saint Petersburg and taking care of his elderly mother.<ref>{{Cite web |date=2023-12-20 |title=Математика Григория Перельмана, уехавшего в Швецию, видели в купчинском супермаркете |url=https://www.rosbalt.ru/piter/2015/07/10/1417676.html |access-date=2023-12-20 |website=Росбалт |language=ru}}</ref>

==Perelman and the media==
Perelman has avoided [[Journalist|journalists]] and other members of the media. [[Masha Gessen]], author of a biography about Perelman, ''Perfect Rigour: A Genius and the Mathematical Breakthrough of the Century'', was unable to meet him.<ref>{{Cite news |last=Gerasimov |first=Nikolai |date=27 March 2011 |script-title=ru:Чтобы купить русского хлеба, Перельман пешком ходил через весь Нью-Йорк |trans-title=To buy Russian bread, Perelman walked through the whole New York |url=http://kp.ru/daily/25658/821159/ |url-status=live |archive-url=https://web.archive.org/web/20120917112720/http://kp.ru/daily/25658/821159/ |archive-date=17 September 2012 |access-date=25 December 2012 |work=[[Komsomolskaya Pravda]] |language=ru}}</ref> A reporter who had called him was told: "You are disturbing me. I am picking mushrooms."<ref>{{Cite news |last=Harding |first=Luke |date=23 March 2010 |title=Grigory Perelman, the maths genius who said no to $1m |url=https://www.theguardian.com/world/2010/mar/23/grigory-perelman-rejects-1m-dollars |work=[[The Guardian]]}}</ref>

A Russian documentary about Perelman in which his work is discussed by several leading mathematicians, including [[Mikhail Gromov (mathematician)|Mikhail Gromov]], [[Ludvig Faddeev|Ludwig Faddeev]], [[Anatoly Vershik]], [[Tian Gang|Gang Tian]], [[John Morgan (mathematician)|John Morgan]] and others was released in 2011 under the title "Иноходец. Урок Перельмана" ("Maverick: Perelman's Lesson").<ref>https://www.youtube.com/watch?v=Ng1W2KUHI2s</ref>

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 ''The Formula of the Universe''.<ref name="intrvu">{{Cite news |last=Veligzhanina |first=Anna |date=28 April 2011 |script-title=ru:Интервью с математиком Григорием Перельманом: Зачем мне миллион долларов? Я могу управлять Вселенной |trans-title=Interview with mathematician Grigori Perelman: Why do I need million dollars? I can control the world |url=http://kp.ru/daily/25677.3/836229/ |url-status=live |archive-url=https://web.archive.org/web/20121227090706/http://kp.ru/daily/25677.3/836229/ |archive-date=27 December 2012 |access-date=25 December 2012 |work=Komsomolskaya Pravda |language=ru}}</ref> Zabrovsky said that in the interview,{{citation needed|date=March 2023}} Perelman explained why he rejected the one million dollar prize.<ref name=intrvu/> A number of journalists believe that Zabrovsky's interview is most likely a fake, pointing to contradictions in statements supposedly made by Perelman.<ref>{{Cite web |last=Gessen |first=Masha |date=29 April 2011 |title=6 странных ошибок в "интервью Перельмана" |url=http://www.snob.ru/selected/entry/34911 |url-status=live |archive-url=https://web.archive.org/web/20121017224115/http://www.snob.ru/selected/entry/34911 |archive-date=17 October 2012 |access-date=8 May 2012 |website=Snob.ru}}</ref><ref>{{Cite news |date=5 May 2011 |title=Интервью Перельмана – подделка? |trans-title=Interview with Perelman – fake? |url=http://www.versii.com/news/231050/ |url-status=live |archive-url=https://web.archive.org/web/20121226235256/http://www.versii.com/news/231050/ |archive-date=26 December 2012 |access-date=25 December 2012 |publisher=Versii}}</ref><ref>{{Cite news |date=5 June 2011 |title=Grigori Perelman's interview full of mismatches |url=http://english.pravda.ru/society/stories/06-05-2011/117816-grigori_perelman-0/ |url-status=live |archive-url=https://web.archive.org/web/20130122115715/http://english.pravda.ru/society/stories/06-05-2011/117816-grigori_perelman-0/ |archive-date=22 January 2013 |access-date=25 December 2012 |publisher=English Pravda.ru}}</ref>

The writer Brett Forrest briefly interacted with Perelman in 2012.<ref>{{Cite web |title=Articles » Shattered Genius |url=http://brettforrest.com/shattered-genius/ |access-date=25 December 2012 |publisher=Brett Forrest}}</ref><ref>{{Cite web |date=1 September 2012 |title=Seven of the week's best reads |url=https://www.bbc.co.uk/news/magazine-19439860 |url-status=live |archive-url=https://web.archive.org/web/20130308151452/http://www.bbc.co.uk/news/magazine-19439860 |archive-date=8 March 2013 |access-date=25 December 2012 |publisher=[[BBC News]]}}</ref>

==Publications==

=== Dissertation ===
* {{Cite book |last=Перельман, Григорий Яковлевич |title=Седловые поверхности в евклидовых пространствах |publisher=[[Saint Petersburg State University|Ленинградский государственный университет]] |year=1990 |language=ru |trans-title=Saddle surfaces in Euclidean spaces}} Автореф. дис. на соиск. учен. степ. канд. физ.-мат. наук.

=== Research papers ===
{{Refbegin}}
{{longitem|{{rma|P85|tw=4em|{{Cite conference |last=Perelʹman |first=G. Ya. |year=1985 |editor-last=Ivanov |editor-first=L. D. |title=Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in {{math|'''R'''<sup>2k − 1</sup>}} |location=Kalinin |publisher=[[Tver State University|Kalinin gosudarstvennyy universitet]] |pages=129–131 |mr=0829936 |zbl=0621.52003 |book-title=Geometric questions in the theory of functions and sets}}}}}}

{{longitem|{{rma|PP86|tw=4em|{{Cite journal |last1=Polikanova |first1=I. V. |last2=Perelʹman |first2=G. Ya. |year=1986 |title=A remark on Helly's theorem |url=http://eudml.org/doc/62756 |journal=Sibirskij Matematiceskij Zurnal |volume=27 |issue=5 |pages=191–194 |mr=0873724 |zbl=0615.52009}}}}}}

{{longitem|{{rma|P87|tw=4em|{{Cite journal |last=Perelʹman |first=G. Ya. |year=1987 |title=k-radii of a convex body |journal=[[Siberian Mathematical Journal]] |volume=28 |issue=4 |pages=665–666 |bibcode=1987SibMJ..28..665P |doi=10.1007/BF00973857 |mr=0906047 |s2cid=122265141 |zbl=0637.52009}}}}}}

{{longitem|{{rma|P88|tw=4em|{{Cite journal |last=Perel'man |first=G. Ya. |date=1991 |title=Polyhedral saddle surfaces |journal=Journal of Soviet Mathematics |volume=54 |issue=1 |pages=735–740 |doi=10.1007/BF01097421 |issn=0090-4104 |mr=0971977 |s2cid=121040191}}}}}}

{{longitem|{{rma|P89|tw=4em|{{Cite journal |last=Perel'man |first=G. Ya. |date=1992 |title=Example of a complete saddle surface in R4 with Gaussian curvature bounded away from zero |url=https://link.springer.com/content/pdf/10.1007/BF01097177.pdf |journal=Journal of Soviet Mathematics |volume=59 |issue=2 |pages=760–762 |doi=10.1007/BF01097177 |issn=0090-4104 |mr=1049373 |s2cid=121011846 |access-date=13 January 2025 |doi-access=free}}}}}}

{{longitem|{{rma|BGP92|tw=4em|{{Cite journal |last1=Burago |first1=Yu |last2=Gromov |first2=M |last3=Perel'man |first3=G |date=30 April 1992 |title=A.D. Alexandrov spaces with curvature bounded below |journal=Russian Mathematical Surveys |volume=47 |issue=2 |pages=1–58 |doi=10.1070/RM1992v047n02ABEH000877 |issn=0036-0279 |mr=1185284 |s2cid=250908096 |zbl=0802.53018}}}}}}

{{longitem|{{rma|P93|tw=4em|{{Cite web |title=G. Ya. Perel'man, "Elements of Morse theory on Aleksandrov spaces", Algebra i Analiz, 5:1 (1993), 232–241; St. Petersburg Math. J., 5:1 (1994), 205–213 |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=374&option_lang=eng |access-date=14 January 2025 |website=Math-Net.Ru |language=ru |ref={{sfnref|Math-Net.Ru}}}}}}}}

{{longitem|{{rma|PP93|tw=4em|{{Cite web |title=G. Ya. Perel'man, Petrunin, "Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem", Algebra i Analiz, 5:1 (1993), 242–256; St. Petersburg Math. J., 5:1 (1993), 215–227 |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=375&option_lang=eng |access-date=14 January 2025 |website=Math-Net.Ru |language=ru |mr=1220498 |zbl=0815.53072 |ref={{sfnref|Math-Net.Ru}}}}}}}}

{{longitem|{{rma|P94a|tw=4em|{{Cite journal |last=Perelman |first=G. |year=1994 |title=Manifolds of positive Ricci curvature with almost maximal volume |journal=[[Journal of the American Mathematical Society]] |volume=7 |issue=2 |pages=299–305 |doi=10.1090/S0894-0347-1994-1231690-7 |mr=1231690 |zbl=0799.53050 |doi-access=free}}}}}}

{{longitem|{{rma|P94b|tw=4em|{{Cite journal |last=Perelman |first=G. |year=1994 |title=Proof of the soul conjecture of Cheeger and Gromoll |journal=[[Journal of Differential Geometry]] |volume=40 |issue=1 |pages=209–212 |doi=10.4310/jdg/1214455292 |mr=1285534 |s2cid=118147865 |zbl=0818.53056 |doi-access=free}}}}}}

{{longitem|{{rma|P95a|tw=4em|{{Cite conference |last=Perelman |first=G. |year=1995 |editor-last=Chatterji |editor-first=S. D. |title=Spaces with curvature bounded below |url=https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1994.1/ICM1994.1.ocr.pdf |conference=Zürich, Switzerland ( 3–11 August 1994) |location=Basel |publisher=[[Birkhäuser]] |pages=517–525 |doi=10.1007/978-3-0348-9078-6 |isbn=3-7643-5153-5 |mr=1403952 |zbl=0838.53033 |book-title=Proceedings of the International Congress of Mathematicians, Vol. 1}}}}}}

{{longitem|{{rma|P95b|tw=4em|{{Cite journal |last=Perelman |first=G. |year=1995 |title=A diameter sphere theorem for manifolds of positive Ricci curvature |url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002443910 |journal=[[Mathematische Zeitschrift]] |volume=218 |issue=4 |pages=595–596 |doi=10.1007/BF02571925 |mr=1326988 |s2cid=122333596 |zbl=0831.53033}}}}}}

{{longitem|{{rma|P95c|tw=4em|{{Cite journal |last=Perelman |first=G. |year=1995 |title=Widths of nonnegatively curved spaces |url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN000462241 |journal=[[Geometric and Functional Analysis]] |volume=5 |issue=2 |pages=445–463 |doi=10.1007/BF01895675 |mr=1334875 |s2cid=120415759 |zbl=0845.53031}}}}}}

{{longitem|{{rma|P97a|tw=4em|{{Cite conference |last=Perelman |first=G. |year=1997 |editor-last=Grove |editor-first=Karsten |editor-link=Karsten Grove |editor2-last=Petersen |editor2-first=Peter |title=Collapsing with no proper extremal subsets |url=http://library.msri.org/books/Book30/files/percol.pdf |conference=Special Year in Differential Geometry held in Berkeley, CA, 1993–94 |series=Mathematical Sciences Research Institute Publications |location=Cambridge |publisher=[[Cambridge University Press]] |volume=30 |pages=149–155 |isbn=0-521-59222-4 |mr=1452871 |zbl=0887.53049 |archive-url=https://web.archive.org/web/20210825212623/http://library.msri.org/books/Book30/files/percol.pdf |archive-date=25 August 2021 |access-date=29 July 2020 |book-title=Comparison geometry |url-status=dead}}}}}}

{{longitem|{{rma|P97b|tw=4em|{{Cite conference |last=Perelman |first=G. |year=1997 |editor-last=Grove |editor-first=Karsten |editor-link=Karsten Grove |editor2-last=Petersen |editor2-first=Peter |title=Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers |url=http://www.msri.org/publications/books/Book30/files/perricci.pdf |conference=Special Year in Differential Geometry held in Berkeley, CA, 1993–94 |series=Mathematical Sciences Research Institute Publications |location=Cambridge |publisher=[[Cambridge University Press]] |volume=30 |pages=157–163 |isbn=0-521-59222-4 |mr=1452872 |zbl=0890.53038 |book-title=Comparison geometry}}}}}}

{{longitem|{{rma|P97c|tw=4em|{{Cite conference |last=Perelman |first=G. |year=1997 |editor-last=Grove |editor-first=Karsten |editor-link=Karsten Grove |editor2-last=Petersen |editor2-first=Peter |title=A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone |url=http://library.msri.org/books/Book30/files/perex.pdf |conference=Special Year in Differential Geometry held in Berkeley, CA, 1993–94 |series=Mathematical Sciences Research Institute Publications |location=Cambridge |publisher=[[Cambridge University Press]] |volume=30 |pages=165–166 |isbn=0-521-59222-4 |mr=1452873 |zbl=0887.53038 |archive-url=https://web.archive.org/web/20210827005607/http://library.msri.org/books/Book30/files/perex.pdf |archive-date=27 August 2021 |access-date=29 July 2020 |book-title=Comparison geometry |url-status=dead}}}}}}
{{Refend}}

=== Unpublished work ===
{{Refbegin}}
{{longitem|{{rma|P91|tw=4em|{{Cite tech report |url=https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCBFB2+.pdf |title=Alexandrov's spaces with curvatures bounded from below II |last=Perelman |first=G. |year=1991 |type=Preprint}}
* See also: {{Cite conference |last=Kapovitch |first=Vitali |year=2007 |editor-last=Cheeger |editor-first=Jeffrey |editor-link=Jeff Cheeger |editor2-last=Grove |editor2-first=Karsten |editor2-link=Karsten Grove |title=Perelman's stability theorem |series=Surveys in Differential Geometry |location=Somerville, MA |publisher=International Press |volume=11 |pages=103–136 |arxiv=math/0703002 |doi=10.4310/SDG.2006.v11.n1.a5 |isbn=978-1-57146-117-9 |mr=2408265 |doi-access=free |book-title=Metric and Comparison Geometry}}}}}}
{{longitem|{{rma|PP95|tw=4em|{{Cite tech report |url=https://anton-petrunin.github.io/papers/qg_ams.pdf |title=Quasigeodesics and gradient curves in Alexandrov spaces |last=Perelman |first=G. |last2=Petrunin |first2=A. |year=1995 |type=Preprint}}}}}}
{{longitem|{{rma|P95d|tw=4em|{{Cite tech report |url=https://anton-petrunin.github.io/papers/alexandrov/Cstructure.pdf |title=DC structure on Alexandrov space (preliminary version) |last=Perelman |first=G. |year=1995 |type=Preprint}}}}}}
{{longitem|{{rma|P02|tw=4em|{{Cite arXiv |eprint=math/0211159 |author=Perelman, Grisha |title=The entropy formula for the Ricci flow and its geometric applications |year=2002 }} {{zbl|1130.53001}}}}}}
{{longitem|{{rma|P03a|tw=4em|{{Cite arXiv |eprint=math/0303109 |author=Perelman, Grisha |title=Ricci flow with surgery on three-manifolds |year=2003 }} {{zbl|1130.53002}}}}}}
{{longitem|{{rma|P03b|tw=4em|{{Cite arXiv |eprint=math/0307245 |author=Perelman, Grisha |title=Finite extinction time for the solutions to the Ricci flow on certain three-manifolds |year=2003 }} {{zbl|1130.53003}}}}}}
{{Refend}}

==See also==
* [[50033 Perelman]]
* [[Ancient solution]]
* [[Homology sphere]]
* [[Hyperbolic manifold]]
* "[[Manifold Destiny]]"
* [[Spherical space form conjecture]]
* [[Thurston elliptization conjecture]]
* [[Uniformization theorem]]

{{clear}}

==Notes==
{{Notelist}}

==References==
{{Reflist|30em}}

==Sources==
{{refbegin|30em}}
*[[Michael T. Anderson|Anderson, M.T.]] 2005. [http://www.math.sunysb.edu/%7Eanderson/papers.html Singularities of the Ricci flow]. Encyclopedia of Mathematical Physics, Elsevier.
*The Associated Press, {{Cite news |date=1 July 2004 |title=Russian may have solved great math mystery |url=http://www.cnn.com/2004/US/West/01/07/math.mystery.ap/index.html |url-status=live |archive-url=https://web.archive.org/web/20060813185225/http://www.cnn.com/2004/US/West/01/07/math.mystery.ap/index.html |archive-date=13 August 2006 |access-date=15 August 2006 |agency=[[CNN]]}}
* {{Cite journal |last1=Cao |first1=Huai-Dong |last2=Zhu |first2=Xi-Ping |date=June 2006 |title=A Complete Proof of the Poincaré and Geometrization Conjectures&nbsp;– application of the Hamilton-Perelman theory of the Ricci flow |url=http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf |url-status=dead |journal=[[Asian Journal of Mathematics]] |volume=10 |issue=2 |archive-url=https://web.archive.org/web/20120514194949/http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf |archive-date=14 May 2012}} [https://web.archive.org/web/20120514194801/http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-Erratum.pdf Erratum]. Revised version (December 2006): [https://arxiv.org/abs/math.DG/0612069 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture]
* {{Cite journal |last=Collins, Graham P. |year=2004 |title=The Shapes of Space |journal=[[Scientific American]] |volume=291 |issue=July |pages=94–103 |bibcode=2004SciAm.291a..94C |doi=10.1038/scientificamerican0704-94 |pmid=15255593}}
* {{Cite book |last=Gessen |first=Masha |url=https://archive.org/details/perfectrigorgeni00gess |title=Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century |publisher=Houghton Mifflin Harcourt |year=2009 |isbn=978-0151014064 |location=Boston, Massachusetts |url-access=registration}}
* {{Cite journal |last=Jackson |first=Allyn |date=September 2006 |title=Conjectures No More? Consensus Forming on the Proof of the Poincaré and Geometrization Conjectures |url=https://www.ams.org/notices/200608/comm-perelman.pdf |journal=Notices of the AMS}}
* {{Cite journal |last1=Kleiner |first1=Bruce |last2=Lott, John |year=2008 |title=Notes on Perelman's papers |journal=Geometry & Topology |volume=12 |issue=5 |pages=2587–2855 |arxiv=math.DG/0605667 |doi=10.2140/gt.2008.12.2587 |s2cid=119133773}}
* {{Cite web |last=Kusner |first=Rob |title=Witnesses to Mathematical History Ricci Flow and Geometry |url=http://www.gang.umass.edu/~kusner/other/new-perelman.pdf |access-date=22 August 2006}} (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)
* {{Cite arXiv |eprint=math.DG/0607607 |first=John W. |last=Morgan |author2=Gang Tian |title=Ricci Flow and the Poincaré Conjecture |date=25 July 2006}}
{{refend}}

==External links==
{{Sister project auto}}

* {{MathGenealogy}}
* {{IMO results}}
* {{MacTutor Biography|id=Perelman}}

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