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 any of the problems 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, by Grigori Perelman.
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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 farreaching 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.
The official statement of the problem was given by Pierre Deligne.
In topology, a sphere with a twodimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every 2dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with threedimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3manifolds.
The official statement of the problem was given by John Milnor.
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. Perelman declined that award.^{[1]} Perelman was officially awarded the Millennium prize on 18 March, 2010.^{[2]}
The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have farreaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert's eighth problem, and is still considered an important open problem a century later.
The official statement of the problem was given by Enrico Bombieri.
In physics, classical YangMills theory is a generalization of the Maxwell theory of electromagnetism where the chromoelectromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum YangMills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum YangMills theory and a mass gap.
The official statement of the problem was given by Arthur Jaffe and Edward Witten.
The Navier–Stokes equations describe the motion of liquids and gases. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations.
The official statement of the problem was given by Charles Fefferman.
The Birch and SwinnertonDyer conjecture deals with a certain type of equation, those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.
The official statement of the problem was given by Andrew Wiles.
This article incorporates material from Millennium Problems on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
