Millennium Prize problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. The problems are Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap. A correct solution to any of the problems results in a US $1 million prize being awarded by the institute to the discoverer(s).

At present, the only Millennium Prize problem to have been solved is the Poincaré conjecture, which was solved by the Russian mathematician Grigori Perelman in 2003.

In dimension 2, a sphere is characterized by the fact that it is the only closed and simply-connected surface. The Poincaré conjecture states that this is also true in dimension 3. It is central to the more general problem of classifying all 3-manifolds. The precise formulation of the conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

A proof of this conjecture was given by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was selected to receive the Fields Medal for his solution but he declined the award. Perelman was officially awarded the Millennium Prize on March 18, 2010, but he also declined that award and the associated prize money from the Clay Mathematics Institute. The Interfax news agency quoted Perelman as saying he believed the prize was unfair. Perelman told Interfax he considered his contribution to solving the Poincaré conjecture no greater than that of Columbia University mathematician Richard S. Hamilton.

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