Poincaré Conjecture

Grigory Perelman was awarded a Fields Medal at the Madrid meeting on the International Congress of Mathematicians for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow." A number of authors have written detailed expositions of Perelman's work. These papers, as well as other commentary and references, are listed on this page.

Articles by Grigory Perelman

The Entropy Formula for the Ricci Flow and its Geometric Applications, arXiv.org, November 11, 2002.

Ricci Flow with Surgery on Three-Manifolds, arXiv.org, March 10, 2003.

Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds, arXiv.org, July 17, 2003.

Web Resources

Notes and Commentary on Perelman's Ricci Flow Papers
(web page maintained by Bruce Kleiner and John Lott)

Groupe De Travail Sur Les Travaux De Perelman
(web page maintained by G. Besson et al)

Expository articles

Recent progress on the Poincaré conjecture and the classification of 3-manifolds, by John Morgan, Bull. AMS, 42 (2005), 57-78.

The Poincaré Conjecture, by John Milnor in "The Millennium Prize Problems," Clay Mathematics Institute and the American Mathematical Society, 2006.

Detailed Expositions

Articles are listed by date of first public availability.

Notes on Perelman's Papers, by Bruce Kleiner and John Lott, arXiv.org, May 25, 2006. Prior versions were available on the Kleiner-Lott web site since June, 2003 and September, 2004, for Perelman's first and second Ricci flow papers, resepectively.

Hamilton-Perelman's Proof of the Poincaré and Geometrization Conjecture by Huai-Dong Cao and Xi-Ping Zhu, arXiv.org, December 3, 2006. A revised version of the paper " A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow," published by the Asian Journal of Mathematics, June 2006.

Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian, arXiv.org, July 25, 2006. Manuscript received by CMI, May 19, 2006.

Additional commentary

Perelman's proof of the Poincaré conjecture — a nonlinear PDE perspective, by Terry Tao, ArXiv.org October 29, 2006