As of July 2019, the only one of the Millennium Prize Problems to be solved and the only one to be refused.
In short, the proof builds upon the work of Hamilton (1982), who proved some special cases of the conjecture. Then, in 2002 and 2003, Russian mathematician Grigori Perelman published three papers (Perelman 2002, 2003a, 2003b) where he proved William Thurston’s Geometrization conjecture.
These papers were later commented and expanded by three independent groups, all of which admitted that the gaps they filled were relatively minor:
- In 2006, Kleiner and Lott (2006) published a paper with further «details that are missing in [Perelman’s papers], which contain Perelman’s arguments for the Geometrization Conjecture»;
- Cao and Zhu (2006) also published their own paper
- Morgan and Tan published a paper and book further detailing Perelman’s work (2006, 2007)
In 2006 Perelman was offered the Fields Medal but unlike his fellow nominee Terence Tao, declined it. Perelman said (Nasar and Gruber 2006):
[John M. Ball, president of the International Mathematical Union] 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. (…) Everybody understood that if the proof is correct then no other recognition is needed.
In 2010, Perelman also declined the Clay Mathematics Institute’s $1 million dollar award, considering it unjust for not being shared with Hamilton. Also (BBC News 2010):
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.
radical ← Andy’s Brevity Quest 2019 (284 words) → Gilgamesh
BBC News, ed. 2010. “Russian Maths Genius Grigory Perelman, Who Declined a Prestigious International Award Four Years Ago, Is Under New Pressure to Accept a Prize.” March 24, 2010. http://news.bbc.co.uk/2/hi/europe/8585407.stm.
Cao, Huai-Dong, and Xi-Ping Zhu. 2006. “Hamilton-Perelman’s Proof of the Poincaré Conjecture and the Geometrization Conjecture.” ArXiV Preprint, December. http://arxiv.org/abs/http://arxiv.org/abs/math/0612069v1.
Hamilton, Richard S. 1982. “Three-Manifolds with Positive Ricci Curvature.” Journal of Differential Geometry 17 (2): 255–306. https://doi.org/10.4310/jdg/1214436922.
Kleiner, Bruce, and John Lott. 2006. “Notes on Perelman’s Papers.” Geom. Topol. 12 (2008) 2587-2855, May. https://doi.org/10.2140/gt.2008.12.2587.
Morgan, John, and Gang Tian. 2007. Ricci Flow and the Poincare Conjecture (Clay Mathematics Monographs). American Mathematical Society. https://www.amazon.com/Ricci-Poincare-Conjecture-Mathematics-Monographs/dp/0821843281/ref=sr_1_1?keywords=Ricci+Flow+and+the+Poincare+Conjecture&qid=1563927214&s=books&sr=1-1.
Morgan, John W., and Gang Tian. 2006. “Ricci Flow and the Poincare Conjecture.” ArXiV Preprint, July. http://arxiv.org/abs/http://arxiv.org/abs/math/0607607v2.
Nasar, Sylvia, and David Gruber. 2006. “Manifold Destiny: A Legendary Problem and the Battle over Who Solved It.” The New Yorker, August. https://www.newyorker.com/magazine/2006/08/28/manifold-destiny.
Perelman, Grisha. 2002. “The Entropy Formula for the Ricci Flow and Its Geometric Applications.” ArXiV Preprint, November. http://arxiv.org/abs/http://arxiv.org/abs/math/0211159v1.
———. 2003a. “Ricci Flow with Surgery on Three-Manifolds.” ArXiV Preprint, March. http://arxiv.org/abs/http://arxiv.org/abs/math/0303109v1.
———. 2003b. “Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds.” ArXiV Preprint, July. http://arxiv.org/abs/http://arxiv.org/abs/math/0307245v1.