Poincaré conjecture
A compact 2-dimensional surface without boundary is topologically homeomorphic to a 2-sphere if every loop can be continuously tightened to a point. The Poincaré conjecture asserts that the same is true for 3-dimensional spaces. | |
| Field | Geometric topology |
|---|---|
| Conjectured by | Henri Poincaré |
| Conjectured in | 1904 |
| First proof by | Grigori Perelman |
| First proof in | 2002 |
| Implied by |
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| Generalizations | Generalized Poincaré conjecture |
| Millennium Prize Problems |
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In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/,[2] US: /ˌpwæ̃kɑːˈreɪ/,[3][4] French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century.
The eventual proof built upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow, Grigori Perelman was able to modify and complete Hamilton's program. In papers posted to the arXiv repository in 2002 and 2003, Perelman presented his work proving the Poincaré conjecture (and the more powerful geometrization conjecture of William Thurston). Over the next several years, several mathematicians studied his papers and produced detailed formulations of his work.
Hamilton and Perelman's work on the conjecture is widely recognized as a milestone of mathematical research. Hamilton was recognized with the Shaw Prize in 2011 and the Leroy P. Steele Prize for Seminal Contribution to Research in 2009. The journal Science marked Perelman's proof of the Poincaré conjecture as the scientific Breakthrough of the Year in 2006.[5] The Clay Mathematics Institute, having included the Poincaré conjecture in their well-known Millennium Prize Problem list, offered Perelman their prize of US$1 million in 2010 for the conjecture's resolution.[6] He declined the award, saying that Hamilton's contribution had been equal to his own.[7][8]
- ^ Matveev, Sergei (2007). "1.3.4 Zeeman's Collapsing Conjecture". Algorithmic Topology and Classification of 3-Manifolds. Algorithms and Computation in Mathematics. Vol. 9. Springer. pp. 46–58. ISBN 978-3540458999.
- ^ "Poincaré, Jules-Henri". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 2022-09-02.
- ^ "Poincaré". The American Heritage Dictionary of the English Language (5th ed.). HarperCollins. Retrieved 9 August 2019.
- ^ "Poincaré". Merriam-Webster.com Dictionary. Merriam-Webster. Retrieved 9 August 2019.
- ^ Mackenzie, Dana (2006-12-22). "The Poincaré Conjecture – Proved". Science. 314 (5807): 1848–1849. doi:10.1126/science.314.5807.1848. PMID 17185565. S2CID 121869167.
- ^ "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original (PDF) on March 22, 2010. Retrieved November 13, 2015.
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.
- ^ "Последнее 'нет' доктора Перельмана" [The last "no" Dr. Perelman]. Interfax (in Russian). July 1, 2010. Retrieved 5 April 2016. Google Translated archived link at [1] (archived 2014-04-20)
- ^ Ritter, Malcolm (1 July 2010). "Russian mathematician rejects million prize". The Boston Globe.