Editing Poincaré conjecture (section)

=== Hamilton's program and solution ===
[[File:Ricci flow.png|thumb|upright|200px|right|Several stages of the [[Ricci flow]] on a two-dimensional manifold]]

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 | author-link = Richard S. Hamilton | title = Three-manifolds with positive Ricci curvature | journal = Journal of Differential Geometry | volume = 17 | issue = 2 | pages = 255–306 | date = 1982 | mr = 0664497 | zbl = 0504.53034 | doi = 10.4310/jdg/1214436922 | doi-access = free }} Reprinted in: {{cite book | editor-last1 = Cao | editor-first1 = H. D. | editor1-link = Huai-Dong Cao | editor-last2 = Chow | editor-first2 = B. | editor-last3 = Chu | editor-first3 = S. C. | editor-last4 = Yau | editor-first4 = S.-T. | editor4-link = Shing-Tung Yau | title = Collected Papers on Ricci Flow | place = Somerville, MA | publisher = International Press | pages = 119–162 | series = Series in Geometry and Topology | volume = 37 | year = 2003 | isbn = 1-57146-110-8  }}</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.

In late 2002 and 2003, Perelman posted three papers on [[arXiv]].<ref>{{cite arXiv | last = Perelman | first = Grigori | author-link = Grigori Perelman | title = The entropy formula for the Ricci flow and its geometric applications | eprint = math.DG/0211159 | date = 2002 }}</ref><ref>{{cite arXiv | last = Perelman | first = Grigori | title = Ricci flow with surgery on three-manifolds | eprint = math.DG/0303109 | date = 2003 }}</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 | date = 2003 }}</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 S. Hamilton]].

From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:
* [[Bruce Kleiner]] and [[John Lott (mathematician)|John W. Lott]] posted a paper on arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture, following partial versions which had been publicly available since 2003.<ref>{{cite journal | first = Bruce | last = Kleiner | author-link = Bruce Kleiner |author2=John W. Lott | title = Notes on Perelman's Papers | year = 2008 | pages = 2587–2855 | volume = 12 | journal = Geometry and Topology | arxiv = math.DG/0605667 | doi=10.2140/gt.2008.12.2587 | issue = 5| s2cid = 119133773 }}</ref> Their manuscript was published in the journal ''[[Geometry and Topology]]'' in 2008. A small number of corrections were made in 2011 and 2013; for instance, the first version of their published paper made use of an incorrect version of Hamilton's compactness theorem for Ricci flow.
* [[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 | author-link = Huai-Dong Cao | author2 = Xi-Ping Zhu | author2-link = Xi-Ping Zhu | 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 | journal = Asian Journal of Mathematics | volume = 10 | date = June 2006 | issue = 2 | url-status = dead | 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 = 2012-05-14 }}</ref> The opening paragraph of their paper stated
{{Quote|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.}}
:Some observers interpreted Cao and Zhu as taking credit for Perelman's work. They later posted a revised version, with new wording, on arXiv.<ref>{{cite arXiv |author=Cao, Huai-Dong|author2=Zhu, Xi-Ping|name-list-style=amp |eprint=math.DG/0612069 |title=Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture |date=December 3, 2006 }}</ref> In addition, a page of their exposition was essentially identical to a page in one of Kleiner and Lott's early publicly available drafts; this was also amended in the revised version, together with an apology by the journal's editorial board.
* [[John Morgan (mathematician)|John Morgan]] and [[Gang Tian]] posted a paper on 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 | author-link = John Morgan (mathematician) | author2 = Gang Tian | author2-link = Gang Tian | title = Ricci Flow and the Poincaré Conjecture | eprint = math.DG/0607607 | date = 2006 }}</ref> and expanded this to a book.<ref>{{cite book | first = John | last = Morgan | author-link = John Morgan (mathematician) |author2=Gang Tian |author2-link=Gang Tian  | title = Ricci Flow and the Poincaré Conjecture |publisher= Clay Mathematics Institute |isbn = 978-0-8218-4328-4| date = 2007 }}</ref><ref>{{cite arXiv |last1=Morgan |first1=John |last2=Tian |first2=Gang |title=Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture |date=2015 |eprint=1512.00699 |class=math.DG }}</ref>

All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.

On August 22, 2006, the [[International Congress of Mathematicians|ICM]] awarded Perelman the [[Fields Medal]] for his work on the Ricci flow, but Perelman refused the medal.<ref>{{cite magazine | first = Sylvia | last = Nasar | author-link = Sylvia Nasar |author2=David Gruber | title = Manifold destiny | magazine = [[The New Yorker]] | pages = 44–57 | date = August 28, 2006 | title-link = Manifold destiny }} [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 = [[The New York Times]] | date = August 22, 2006 | url = https://www.nytimes.com/2006/08/22/science/22cnd-math.html?hp&ex=1156305600&en=aa3a9d418768062c&ei=5094&partner=homepage }}</ref> 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>

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/>