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Millennium Prize Problems

The ** Millennium Prize Problems ** are seven problems in [mathematics] that were stated by the [Clay Mathematics Institute] in 2000. As of March 2010, six of the problems remain [unsolved] . A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a * Millennium Prize * ) being awarded by the institute. Only the [PoincarĂ© conjecture] has been solved, by [Grigori Perelman] .

** P versus NP **

The question is whether, for all problems for which a computer can * verify * a given solution quickly (that is, in [polynomial time] ), it can also * find * that solution quickly. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not all problems in NP are also in P. This is generally considered the most important open question in [theoretical computer science] as it has far-reaching consequences in [mathematics] , [philosophy] and [cryptography] (see [P=NP proof consequences] ).

The official statement of the problem was given by [Stephen Cook] .

** The Hodge conjecture **

The Hodge conjecture is that for [projective] [algebraic varieties] , [Hodge cycle] s are rational [linear combination] s of [algebraic cycle] s.

The official statement of the problem was given by [Pierre Deligne] .

** The PoincarĂ© conjecture (proven) **

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