Poincaré Conjecture

In mathematics, the Poincaré conjecture ( /pwɛn.kɑːˈreɪ/ pwen-kar-AY; ) is a theorem about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims 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. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard Hamilton to use the Ricci flow to attack the problem. Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.

The Poincaré conjecture, before being proven, was one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of Hamilton's (who first suggested using the Ricci flow for the solution). The Poincaré conjecture is the only solved Millennium problem.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.