In the mathematical field of geometric topology, the Poincar conjecture (UK: , US: , French: [pwkae]), or Perelman's theorem, 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 Poincar in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincar 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 attempt 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 In unpublished arXiv preprints released in 2002 and 2003, Perelman presented his work proving the Poincar conjecture, along with the more powerful geometrization conjecture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 and the Leroy P. Steele Prize for Seminal Contribution to Research. The journal Science marked Perelman's proof of the Poincar conjecture as the scientific Breakthrough of the Year in 2006. The Clay Mathematics Institute, having included the Poincar conjecture in their well-known Millennium Prize Problem list, offered Perelman their prize of US$1 million for the conjecture's resolution. He declined the award, saying that Hamilton's contribution had been equal to his own.

Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман, IPA: [ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman] (listen); born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology.

In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made influential contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, thereby providing a detailed sketch of a proof of the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years.

In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." On 22 December 2006, the scientific journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first such recognition in the area of mathematics.On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture. He had previously rejected the prestigious prize of the European Mathematical Society in 1996.