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| {{Millennium Problems}}
| | Greetings! I am Myrtle Shroyer. Managing individuals is his profession. Puerto Rico is exactly where he's always been living but she requirements to move because of her family members. Body building is 1 of the things I adore most.<br><br>Here is my weblog home std test ([http://www.megafilex.com/user/FKayser Internet Page]) |
| [[File:P1S2all.jpg|400px|thumb|right|For [[compact space|compact]] 2-dimensional surfaces without [[boundary (topology)|boundary]], if every loop can be continuously tightened to a point, then the surface is topologically [[Homeomorphism|homeomorphic]] to a 2-sphere (usually just called a sphere). The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.]]
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| [[File:Torus cycles.png|thumb|right|By contrast, neither of the two colored loops on this [[torus]] can be continuously tightened to a point. A torus is not homeomorphic to a sphere.]]
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| In [[mathematics]], the '''Poincaré conjecture''' ({{IPAc-en|p|w|ɛ|n|.|k|ɑ:|ˈ|r|eɪ}} {{respell|pwen-kar|AY|'}}; {{IPA-fr|pwɛ̃kaʁe|lang}})<ref>{{cite encyclopedia | encyclopedia=The American Heritage Dictionary of the English Language | title=Poincaré, Jules Henri | url=http://www.bartleby.com/61/3/P0400300.html | accessdate=2007-05-05 | edition=fourth | year=2000 | publisher=Houghton Mifflin Company | location=Boston | isbn=0-395-82517-2 }}.</ref> is a [[theorem]] about the [[Characterization (mathematics)|characterization]] of the [[3-sphere]], which is the hypersphere that bounds the [[unit ball]] in four-dimensional space. The conjecture states: {{quote|Every [[simply connected]], [[closed manifold|closed]] 3-[[manifold]] is [[homeomorphic]] to the 3-sphere.}} An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called [[homotopy equivalence]]: if a 3-manifold is ''homotopy equivalent'' to the 3-sphere, then it is necessarily ''homeomorphic'' to it.
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| Originally conjectured by [[Henri Poincaré]], the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a [[Closed manifold|closed]] [[3-manifold]]). The Poincaré conjecture claims that if such a space has the additional property that each [[path (topology)|loop]] in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An [[generalized Poincaré conjecture|analogous result]] has been known in higher dimensions for some time.
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| After nearly a century of effort by mathematicians, [[Grigori Perelman]] presented a proof of the conjecture in three papers made available in 2002 and 2003 on [[arXiv]]. The proof followed on from the program of [[Richard Hamilton (professor)|Richard Hamilton]] to use the [[Ricci flow]] to attack the problem. Perelman introduced a modification of the standard Ricci flow, called ''Ricci flow with surgery'' to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.
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| The Poincaré conjecture, before being proven, was one of the most important open questions in [[topology]]. It is one of the seven [[Millennium Prize Problems]], for which the [[Clay Mathematics Institute]] offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a [[Fields Medal]], which he declined. Perelman was awarded the Millennium Prize on March 18, 2010.<ref name="press-release-2010-03-18">{{cite press release|publisher=[[Clay Mathematics Institute]]| date=March 18, 2010 | format=PDF | title = Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman|url = http://www.claymath.org/poincare/millenniumPrizeFull.pdf | accessdate=March 18, 2010 | 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.}}</ref> On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of Hamilton's (who first suggested using the Ricci flow for the solution).<ref name="interfax">[http://www.interfax.ru/society/txt.asp?id=143603 ''Последнее "нет" доктора Перельмана''], [[Interfax]] 1 July 2010</ref><ref name="boston1">{{cite news| url=http://www.boston.com/news/science/articles/2010/07/01/russian_mathematician_rejects_1_million_prize/?p1=Well_MostPop_Emailed1 | work=The Boston Globe | first=Malcolm | last=Ritter | title=Russian mathematician rejects million prize | date=1 July 2010}}</ref> The Poincaré conjecture is the only solved [[Millennium problem]].
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| On December 22, 2006, the journal ''[[Science (journal)|Science]]'' honored Perelman's proof of the Poincaré conjecture as the scientific "[[Breakthrough of the Year]]", the first time this had been bestowed in the area of mathematics.<ref name=science>{{cite journal | last = Mackenzie | first = Dana | authorlink = Dana Mackenzie | title = The Poincaré Conjecture--Proved | journal = Science | volume = 314 | issue = 5807 | pages = 1848–1849 | date = 2006-12-22 | publisher = American Association for the Advancement of Science | doi = 10.1126/science.314.5807.1848 | id = ISSN: 0036-8075 | url= http://www.sciencemag.org/cgi/content/full/314/5807/1848 | pmid=17185565}}</ref>
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| ==History==
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| ===Poincaré's question===
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| At the beginning of the 20th century, [[Henri Poincaré]] was working on the foundations of topology—what would later be called [[combinatorial topology]] and then [[algebraic topology]]. He was particularly interested in what topological properties characterized a [[sphere]].
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| Poincaré claimed in 1900 that [[homology (mathematics)|homology]], a tool he had devised based on prior work by [[Enrico Betti]], was sufficient to tell if a [[3-manifold]] was a [[3-sphere]]. However, in a 1904 paper he described a counterexample to this claim, a space now called the [[Poincaré homology sphere]]. The Poincaré sphere was the first example of a [[homology sphere]], a manifold that had the same homology as a sphere, of which many others have since been constructed. To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new [[topological invariant]], the [[fundamental group]], and showed that the Poincaré sphere had a [[fundamental group]] of order 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different.
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| In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Poincaré's new condition—i.e., "trivial fundamental group"—can be restated as "every loop can be shrunk to a point."
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| The original phrasing was as follows:
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| {{Quote|Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?}}
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| Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the '''Poincaré conjecture'''. Here is the standard form of the conjecture:
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| {{Quote|Every [[simply connected]], [[closed manifold|closed]] 3-[[manifold]] is [[homeomorphism|homeomorphic]] to the 3-sphere.}}
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| ===Attempted solutions===
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| This problem seems to have lain dormant for a time, until [[J. H. C. Whitehead]] revived interest in the conjecture, when in the 1930s he first claimed a proof, and then retracted it. In the process, he discovered some interesting examples of simply connected non-compact 3-manifolds not homeomorphic to '''R'''<sup>3</sup>, the prototype of which is now called the [[Whitehead manifold]].
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| In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influential mathematicians such as [[RH Bing|Bing]], [[Wolfgang Haken|Haken]], [[Edwin E. Moise|Moise]], and [[Christos Papakyriakopoulos|Papakyriakopoulos]] attacked the conjecture. In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere.<ref>{{cite journal | last = Bing | first = RH | authorlink = RH Bing | title = Necessary and sufficient conditions that a 3-manifold be S<sup>3</sup> | journal = Annals of Mathematics. Second Series | volume = 68 | issue = 1 | pages = 17–37 | year = 1958 | doi = 10.2307/1970041 | jstor=1970041}}</ref> Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.<ref>{{cite conference | last = Bing | first = RH | title = Some aspects of the topology of 3-manifolds related to the Poincaré conjecture | booktitle=Lectures on Modern Mathematics, Vol. II | pages = 93–128 | publisher = Wiley | year = 1964 | location = New York }}</ref>
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| Over time, the conjecture gained the reputation of being particularly tricky to tackle. [[J. W. Milnor|John Milnor]] commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect."<ref>{{cite web | url = http://www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf | title = The Poincaré Conjecture 99 Years Later: A Progress Report | accessdate=2007-05-05 | last = Milnor |first = John |authorlink = John Milnor | year = 2004 | format = PDF }}</ref> Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs, and tended to view any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in [[peer review|peer-reviewed]] form).<ref>{{cite journal | last = Taubes | first = Gary | title = What happens when hubris meets nemesis | journal = Discover | volume = 8 | pages = 66–77 | date = July 1987 }}</ref><ref>{{cite news | first = Robert | last = Matthews | title = $1 million mathematical mystery "solved" | url = http://www.newscientist.com/article.ns?id=dn2143 | work = NewScientist.com | date = 9 April 2002 |accessdate = 2007-05-05 }}</ref>
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| An exposition of attempts to prove this conjecture can be found in the non-technical book ''Poincaré's Prize'' by George Szpiro.<ref>{{cite book |last=Szpiro |first=George |title=Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles |date=July 29, 2008 |publisher=[[Plume (publisher)|Plume]] |isbn=978-0-452-28964-2}}</ref>
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| ===Dimensions===
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| {{Main|Generalized Poincaré conjecture}}
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| The [[Surface#Classification_of_closed_surfaces|classification of closed surfaces]] gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the '''Generalized Poincaré conjecture''': is a [[homotopy sphere|homotopy ''n''-sphere]] homeomorphic to the ''n''-sphere? A stronger assumption is necessary; in dimensions four and higher there are simply connected manifolds which are not homeomorphic to an ''n''-sphere.
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| Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961 [[Stephen Smale]] shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental [[h-cobordism theorem]]. In 1982 [[Michael Freedman]] proved the Poincaré conjecture in dimension four. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not [[Diffeomorphism|diffeomorphic]] to the four-sphere. This so-called '''smooth Poincaré conjecture''', in dimension four, remains open and is thought to be very difficult. [[Milnor]]'s [[exotic sphere]]s show that the smooth Poincaré conjecture is false in dimension seven, for example.
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| These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the [[geometrization conjecture]] put it into a framework governing all 3-manifolds. [[John Morgan (mathematician)|John Morgan]] wrote:<ref>Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds.
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| Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78</ref>
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| {{Quote|It is my view that before [[William Thurston|Thurston]]'s work on [[hyperbolic 3-manifold]]s and . . . the Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the Geometrization conjecture) were true.}}
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| ===Hamilton's program and Perelman's solution===
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| [[File:Ricci flow.png|thumb|upright|200px|right|Several stages of the [[Ricci flow]] on a two-dimensional manifold]]
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| Hamilton's program was started in his 1982 paper in which he introduced the [[Ricci flow]] on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.<ref>{{cite journal | last = Hamilton | first = Richard | authorlink = Richard Hamilton (professor) | title = Three-manifolds with positive Ricci curvature | journal = Journal of Differential Geometry | volume = 17 | pages = 255–306 | year = 1982 }} Reprinted in: {{cite book | last = Cao | first = H.D. | coauthors = et al. (Editors) | title = Collected Papers on Ricci Flow | publisher = International Press | year = 2003 | isbn = 978-1-57146-110-0}}</ref> In the following years he extended this work, but was unable to prove the conjecture. The actual solution was not found until [[Grigori Perelman]] published his papers.
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| In late 2002 and 2003 Perelman posted three papers on the [[arXiv]].<ref>{{cite arxiv | last = Perelman | first = Grigori | authorlink = Grigori Perelman | title = The entropy formula for the Ricci flow and its geometric applications | eprint = math.DG/0211159 | year = 2002 | class = math.DG }}</ref><ref>{{cite arxiv | last = Perelman | first = Grigori | title = Ricci flow with surgery on three-manifolds | eprint = math.DG/0303109 | year = 2003 | class = math.DG }}</ref><ref>{{cite arxiv | last = Perelman | first = Grigori | title = Finite extinction time for the solutions to the Ricci flow on certain three-manifolds | eprint = math.DG/0307245 | year = 2003 | class = math.DG }}</ref> In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, [[Thurston's geometrization conjecture]], completing the Ricci flow program outlined earlier by [[Richard Hamilton (mathematician)|Richard Hamilton]].
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| From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
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| * [[Bruce Kleiner]] and [[John Lott (mathematician)|John W. Lott]] posted a paper on the arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture.<ref>{{cite journal | first = Bruce | last = Kleiner | authorlink = Bruce Kleiner | coauthors = John W. Lott | title = Notes on Perelman's Papers | year = 2006 | pages = 2587–2855 | volume = 12 | journal = Geometry and Topology | arxiv = math.DG/0605667 | doi=10.2140/gt.2008.12.2587 | issue = 5}}</ref>
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| * [[Huai-Dong Cao]] and [[Xi-Ping Zhu]] published a paper in the June 2006 issue of the ''[[Asian Journal of Mathematics]]'' with an exposition of the complete proof of the Poincaré and geometrization conjectures.<ref>{{cite journal | first = Huai-Dong | last = Cao | authorlink = Huai-Dong Cao | coauthors = [[Xi-Ping Zhu]] | title = A Complete Proof of the Poincaré and Geometrization Conjectures – 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 | format = [[PDF]] | journal = Asian Journal of Mathematics | volume = 10 |date=June 2006 | issue =2}}</ref> They initially implied the proof was their own achievement based on the "Hamilton-Perelman theory", but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest "exposition of Hamilton–Perelman's proof".<ref>{{cite arxiv |author=Cao, Huai-Dong and Zhu, Xi-Ping |eprint=math.DG/0612069 |title=Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture |date=December 3, 2006 |class=math.DG }}</ref> They also published an [[erratum]] disclosing that they had forgotten to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the AJM editorial board issued an apology for what it called "incautions" in the Cao–Zhu paper.
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| * [[John Morgan (mathematician)|John Morgan]] and [[Gang Tian]] posted a paper on the arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture)<ref>{{Cite arxiv | first = John | last = Morgan | authorlink = John Morgan (mathematician) | coauthors = [[Gang Tian]] | title = Ricci Flow and the Poincaré Conjecture | eprint = math.DG/0607607 | year = 2006 | class = math.DG }}</ref> and expanded this to a book.<ref>{{Cite book | first = John | last = Morgan | authorlink = John Morgan (mathematician) | coauthors = [[Gang Tian]] | title = Ricci Flow and the Poincaré Conjecture |publisher= Clay Mathematics Institute |isbn = 0-8218-4328-1| year = 2007 }}</ref>
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| All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.
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| On August 22, 2006, the [[International Congress of Mathematicians|ICM]] awarded Perelman the [[Fields Medal]] for his work on the conjecture, but Perelman refused the medal.<ref>{{Cite news | first = Sylvia | last = Nasar | authorlink = Sylvia Nasar | coauthors = David Gruber | title = [[Manifold destiny]] | work = [[The New Yorker]] | pages = 44–57 | date = August 28, 2006 }} [http://www.newyorker.com/archive/2006/08/28/060828fa_fact2 On-line version at the ''New Yorker'' website].</ref><ref>{{Cite news | first = Kenneth | last = Chang | title = Highest Honor in Mathematics Is Refused | work = [[New York Times]] | date = August 22, 2006 | url = http://www.nytimes.com/2006/08/22/science/22cnd-math.html?hp&ex=1156305600&en=aa3a9d418768062c&ei=5094&partner=homepage }}</ref><ref>{{Cite news | title = Reclusive Russian solves 100-year-old maths problem | work = [[China Daily]] | page = 7 | date = 23 August 2006 | url = http://www.chinadaily.com.cn/cndy/2006-08/23/content_671442.htm }}</ref>
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| John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."<ref>A Report on the Poincaré Conjecture. Special lecture by John Morgan.</ref>
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| In December 2006, the journal ''[[Science (journal)|Science]]'' honored the proof of Poincaré conjecture as the [[Breakthrough of the Year]] and featured it on its cover.<ref name=science/>
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| ==Ricci flow with surgery==
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| {{Main|Ricci flow}}
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| Hamilton's program for proving the Poincaré conjecture involves first putting a [[Riemannian metric]] on the unknown simply connected closed 3-manifold. The idea is to try to improve this metric; for example, if the metric can be improved enough so that it has constant curvature, then it must be the 3-sphere. The metric is improved using the [[Ricci flow]] equations;
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| :<math>\partial_t g_{ij}=-2 R_{ij}</math>
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| where ''g'' is the metric and ''R'' its Ricci curvature,
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| and one hopes that as the time ''t'' increases the manifold becomes easier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.
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| In some cases Hamilton was able to show that this works; for example, if the manifold has positive Ricci curvature everywhere he showed that the manifold becomes extinct in finite time under Ricci flow without any other singularities. (In other words, the manifold collapses to a point in finite time; it is easy to describe the structure just before the manifold collapses.) This easily implies the Poincaré conjecture in the case of positive Ricci curvature. However in general the Ricci flow equations lead to singularities of the metric after a finite time. Perelman showed how to continue past these singularities: very roughly, he cuts the manifold along the singularities, splitting the manifold into several pieces, and then continues with the Ricci flow on each of these pieces. This procedure is known as '''Ricci flow with surgery'''.
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| A special case of Perelman's theorems about Ricci flow with surgery is given as follows.
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| {{quote|The Ricci flow with surgery on a closed oriented 3-manifold is well defined for all time. If the fundamental group is a [[free product]] of [[finite group]]s and [[cyclic group]]s then the Ricci flow with surgery becomes extinct in finite time, and at all times all components of the manifold are connected sums of ''S''<sup>2</sup> bundles over ''S''<sup>1</sup> and quotients of ''S''<sup>3</sup>.}}
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| This result implies the Poincaré conjecture because it is easy to check it for the possible manifolds listed in the conclusion.
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| The condition on the fundamental group turns out to be necessary (and sufficient) for finite time extinction, and in particular includes the case of trivial fundamental group. It is equivalent to saying that the prime decomposition of the manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries ''S''<sup>2</sup>×'''R''' and ''S''<sup>3</sup>. By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a [[thick-thin decomposition]], whose thick piece has a hyperbolic structure, and whose thin piece is a [[graph manifold]], but this extra complication is not necessary for proving just the Poincaré conjecture.<ref>[[Terence Tao]] wrote an exposition of Ricci flow with surgery in: {{cite arxiv | first = Terence | last = Tao | title = Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective | eprint = math.DG/0610903 | year = 2006 | class = math.DG }}</ref>
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| ==Solution==
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| {{Refimprove|section|date=October 2013}}
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| [[File:Perelman, Grigori (1966).jpg|thumb|right|[[Grigori Perelman]]]]
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| In November 2002, Russian mathematician [[Grigori Perelman]] posted the first of a series of [[E-print|eprints]] on [[arXiv]] outlining a '''solution of the Poincaré conjecture'''. Perelman's proof uses a modified version of a [[Ricci flow]] program developed by [[Richard Hamilton (mathematician)|Richard Hamilton]]. In August 2006, Perelman was awarded, but declined, the [[Fields Medal]] for his proof. On March 18, 2010, the [[Clay Mathematics Institute]] awarded Perelman the $1 million [[Millennium Prize Problems|Millennium Prize]] in recognition of his proof.<ref>[http://www.claymath.org/poincare/ Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman]</ref>
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| Perelman rejected that [[prize]] as well.<ref name="interfax" /><ref name="PhysOrg1">{{cite web |url=http://www.physorg.com/news197209671.html |title=Russian mathematician rejects $1 million prize |publisher=[[Associated Press|AP]] on [[PhysOrg]] |author=Malcolm Ritter |date=2010-07-01 |accessdate=2011-05-15}}</ref>
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| Perelman proved the conjecture by deforming the manifold using the Ricci flow (which behaves similarly to the [[heat equation]] that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as [[Mathematical singularity|singularities]]. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.
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| The first step is to deform the manifold using the [[Ricci flow]]. The Ricci flow was defined by [[Richard Hamilton (mathematician)|Richard Hamilton]] as a way to deform manifolds. The formula for the Ricci flow is an imitation of the [[heat equation]] which describes the way heat flows in a solid. Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds. Hamilton used the Ricci flow to prove that some compact manifolds were [[diffeomorphic]] to spheres and he hoped to apply it to prove the Poincaré Conjecture. He needed to understand the singularities.
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| Hamilton created a list of possible singularities that could form but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line. Perelman proved this using something called the "Reduced Volume" which is closely related to an [[eigenvalue]] of a certain [[elliptic equation]].
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| Sometimes an otherwise complicated operation reduces to multiplication by a [[Scalar (mathematics)|scalar]] (a number). Such numbers are called eigenvalues of that operation. Eigenvalues are closely related to vibration frequencies and are used in analyzing a famous problem: [[can you hear the shape of a drum?]]. Essentially an eigenvalue is like a note being played by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the [[ricci_flow#cigar_soliton_solution|cigar soliton solution]], which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.
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| Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three-dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact homeomorphic to a sphere. This process is described in the fictional work by Tina S. Chang cited below.
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| One immediate question was how can one be sure there aren't infinitely many cuts necessary? Otherwise the cutting might progress forever. Perelman proved this can't happen by using [[minimal surfaces]] on the manifold. A minimal surface is essentially a soap film. Hamilton had shown that the area of
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| a minimal surface decreases as the manifold undergoes Ricci flow. Perelman verified what happened to the area of the minimal surface when the manifold was sliced. He proved that eventually the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra by Sormani in Szpiro's book cited below. This last part of the proof appeared in Perelman's third and final paper on the subject.
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| ==References==
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| {{Reflist|colwidth=30em}}
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| ==Further reading==
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| * {{cite journal |author=Bruce Kleiner, John Lott |title=Notes on Perelman's papers |year=2008 |pages=2587–2855 |volume=12 |journal=Geometry and Topology |arxiv=math/0605667 |doi=10.2140/gt.2008.12.2587 |issue=5}}
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| * {{cite arxiv|author=Huai-Dong Cao, Xi-Ping Zhu|title=Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture | date = December 3, 2006|eprint=math.DG/0612069|class=math.DG}}
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| * {{cite arxiv |author=John W. Morgan, Gang Tian |title=Ricci Flow and the Poincaré Conjecture |eprint=math/0607607 |class=math.DG |year=2006}}: Detailed proof, expanding Perelman's papers.
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| * {{cite book |last=O'Shea |first=Donal |title=The Poincaré Conjecture: In Search of the Shape of the Universe |date=December 26, 2007 |publisher=[[Bloomsbury Publishing|Walker & Company]] |isbn=978-0-8027-1654-5}}
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| * {{cite arxiv|author=Perelman, Grisha|title=The entropy formula for the Ricci flow and its geometric applications | date = November 11, 2002|eprint=math.DG/0211159|class=math.DG}}
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| * {{cite arxiv|author=Perelman, Grisha|title=Ricci flow with surgery on three-manifolds | date = March 10, 2003|eprint=math.DG/0303109|class=math.DG}}
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| * {{cite arxiv|author=Perelman, Grisha|title=Finite extinction time for the solutions to the Ricci flow on certain three-manifolds|date = July 17, 2003|eprint=math.DG/0307245|class=math.DG}}
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| * {{cite book |last=Szpiro |first=George |title=Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles |date=July 29, 2008 |publisher=[[Plume (publisher)|Plume]] |isbn=978-0-452-28964-2}}
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| * {{cite journal |author=John Stillwell |title=Poincaré and the early history of 3-manifolds |year=2012 |pages=555–576 |volume=49 |journal=Bulletin of the American Mathematical Society |issue=4 |doi=10.1090/S0273-0979-2012-01385-X}}
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| ==External links==
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| * {{springer|title=Poincaré conjecture|id=p/p073000}}
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| * [http://www.claymath.org/sites/default/files/poincare.pdf The Poincaré conjecture described] by the Clay Mathematics Institute.
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| * [http://athome.harvard.edu/threemanifolds/ The Geometry of 3-Manifolds (video)] A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.
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| * Bruce Kleiner (Yale) and John W. Lott (University of Michigan): [http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html "Notes & commentary on Perelman's Ricci flow papers"].
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| *Stephen Ornes, [http://www.seedmagazine.com/news/2006/08/what_is_the_poincar_conjecture.php What is The Poincaré Conjecture?], ''Seed Magazine'', 25 August 2006.
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| *[http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20061102.shtml "The Poincaré Conjecture"] – [[BBC Radio 4]] programme ''[[In Our Time (BBC Radio 4)|In Our Time]]'', 2 November 2006. Contributors June Barrow-Green, Lecturer in the History of Mathematics at the [[Open University]], [[Ian Stewart (mathematician)|Ian Stewart]], Professor of Mathematics at the [[University of Warwick]], [[Marcus du Sautoy]], Professor of Mathematics at the [[University of Oxford]], and presenter [[Melvyn Bragg]].
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| *[http://www.npr.org/templates/story/story.php?storyId=6682439 "Solving an Old Math Problem Nets Award, Trouble"] – NPR segment, December 26, 2006.
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| *{{cite news | last = Nasar | first = Sylvia | coauthors = and Gruber, David | title = Manifold Destiny: A legendary problem and the battle over who solved it | work = [[The New Yorker]] | date = 21 August 2006 | url = http://www.newyorker.com/fact/content/articles/060828fa_fact2 | accessdate = 2006-08-24 }}
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| ===Articles===
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| * [http://www.newscientist.com/article.ns?id=mg18324565.000 Taming the fourth dimension], by [[B. Schechter]], [[New Scientist]], 17 July 2004, Vol 183 No 2456
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| * "Major math problem is believed solved", [[Wall Street Journal]], July 21, 2006 explains the current Millennium Prize situation.
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| * [http://www.scientificamericandigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=1550EBB0-2B35-221B-6A293BA37A12BAFF The Shapes of Space], by [[G. P. Collins]], [[Scientific American]], 2004 July, pp. 94–103
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| * [http://www.sciencenews.org/articles/20030614/bob10.asp If it looks like a sphere...]{{Dead link|date=February 2011}}, by [[E. Klarreich]], [[Science News]] Online, June 14, 2003, Vol 163, No. 24, p 376.
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| * [http://www.nytimes.com/2006/08/15/science/15math.html?pagewanted=1&_r=1 Elusive Proof, Elusive Prover: A New Mathematical Mystery], by [[Dennis Overbye]], [[New York Times]], Science, August 15, 2006.
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| * [http://www.ams.org/notices/200402/fea-anderson.pdf Geometrization of Three Manifolds via the Ricci Flow], by [[Michael Anderson (differential geometer)|Mike Anderson]] ([[SUNY Stony Brook]]), Notices of the AMS, Vol 51, Number 2, (written for mathematicians)
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| * [http://arxiv.org/abs/math.DG/0610903 Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective] by [[Terence Tao]], unpublished [[arxiv.org]] preprint (written for mathematicians).
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| ===Fiction===
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| * [http://www.jstor.org/pss/25678738 Perelman's Song], by [[Tina S. Chang]], listed on [http://math.cofc.edu/kasman/MATHFICT/default.html Kasman's Mathematical Fiction website], appeared ''[[Math Horizons]]''.[http://pulse.yahoo.com/_LYTXAIQ3BIDXHCARFNHZUUTF4Q/blog/articles/45709?listPage=index]
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| ===Lectures===
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| * [http://arxiv.org/abs/math.DG/0607821 Structures of Three-Manifolds], for the scientifically inclined audience by [[Shing-Tung Yau]] ([[Harvard]]), June 20, 2006.
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| * [http://www.math.lsa.umich.edu/~lott/seminarnotes.html The Work of Grigori Perelman]{{Dead link|date=March 2011}}, talk by [[John Lott (mathematician)|John Lott]] ([[University of Michigan]]) [[International Congress of Mathematicians]] 2006 Presentation, for mathematicians in all areas, excellent graphics
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| * [http://comet.lehman.cuny.edu/sormani/others/perelman/introperelman.html Perelman and the Poincaré Conjecture], talk by [[Christina Sormani]] ([[CUNY Graduate Center]] and [[Lehman College]]) presented at [[Williams College]], [[Wellesley College]], [[Lehman College]] and [[Tufts University]]. Transparencies are posted for public use (same as the graphics above) and a guide for math professors interested in giving a similar talk (recommends studying the resources posted here).
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| ===Websites===
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| * [http://www.claymath.org/millennium/Poincare_Conjecture/ Clay Mathematics Institute] has a description of the Poincaré Conjecture as a Millennium Problem by [[John Milnor]] ([[SUNY]] [[Stony Brook, New York|Stony Brook]]) as well as a press release about the proof.
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| * [http://comet.lehman.cuny.edu/sormani/others/perelman/introperelman.html Intro Perelman Website] by [[Christina Sormani]], ([[CUNY Graduate Center]] and [[Lehman College]]) was used as a framework for this article and a resource for the initial set of links.
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| * [http://www.slate.com/id/2147954 Who cares about Poincaré?], by [[Jordan Ellenberg]], [http://www.slate.com Slate], August 18, 2006, is for the layman
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| * [http://www.gang.umass.edu/~kusner/other/3mfd.html A Bit of Cosmic Background], by [[Robert Kusner]], UMass Math Dept Newsletter 2007.
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| * [http://terrytao.wordpress.com/category/teaching/285g-poincare-conjecture/ Lectures on Perelman's proof] by T. Tao.
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| ===Videos===
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| *{{Youtube|AUoaTrQTM5o|The Poincaré Conjecture}}, Brief visual overview of the Poincaré Conjecture, background and solution.
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| {{Breakthrough of the Year}}
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| {{DEFAULTSORT:Poincare conjecture}}
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| [[Category:Geometric topology]]
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| [[Category:3-manifolds]]
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| [[Category:Theorems in topology]]
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| [[Category:Millennium Prize Problems]]
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