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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. Currently, six of the problems remain unsolved. A correct solution to each problem results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture has been solved, but the solver Grigori Perelman has not pursued the conditions necessary to claim the prize. 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. 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 is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
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