- Millennium Prize Problems
Millennium Prize Problems P versus NP problem Hodge conjecture Poincaré conjecture Riemann hypothesis Yang–Mills existence and mass gap Navier–Stokes existence and smoothness Birch and Swinnerton-Dyer conjecture
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, 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, in 2010 by Grigori Perelman, who declined the award.
The seven problems are:
- P versus NP problem
- Hodge conjecture
- Poincaré conjecture (solved, see solution of the Poincaré conjecture)
- Riemann hypothesis
- Yang–Mills existence and mass gap
- Navier–Stokes existence and smoothness
- Birch and Swinnerton-Dyer conjecture
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, biology, philosophy and cryptography (see P versus NP problem proof consequences).
If the question of whether P=NP were to be answered affirmatively it would trivialise the rest of the Millennium Prize Problems (and indeed all but the unprovable propositions in mathematics) because they would all have direct solutions easily solvable by a formal system.
- "If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."
- — Scott Aaronson, MIT
Most mathematicians and computer scientists expect that P≠NP.
The official statement of the problem was given by Stephen Cook.
The Hodge conjecture
The official statement of the problem was given by Pierre Deligne.
The Poincaré conjecture (proven)
In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every two-dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.
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. Perelman was officially awarded the Millennium prize on March 18, 2010. On July 1, 2010, it was reported that Perelman declined the award and associated prize money from the Clay Mathematics Institute. In rejecting the Millennium Prize, Perelman stated that he believed the decisions by the organized mathematics community to be unjust and that his contribution to solving the Poincaré conjecture was no greater than that of Columbia University mathematician Richard Hamilton (who first suggested a program for the solution).
The Riemann hypothesis
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 far-reaching 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.
Yang–Mills existence and mass gap
In physics, classical Yang–Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic 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 Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.
The Navier–Stokes equations describe the motion of fluids. 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 insight into these equations.
The official statement of the problem was given by Charles Fefferman.
The Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer 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.
- Devlin, Keith J. (2003) . The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. New York: Basic Books. ISBN 0-465-01729-0.
- Carlson, James; Jaffe, Arthur; Wiles, Andrew, eds (2006). The Millennium Prize Problems. Providence, RI: American Mathematical Society and Clay Mathematics Institute. ISBN 978-0-8218-3679-8. http://www.claymath.org/library/.
- ^ "Maths genius declines top prize". BBC News. 22 August 2006. http://news.bbc.co.uk/2/hi/science/nature/5274040.stm. Retrieved 16 June 2011.
- ^ "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. http://www.claymath.org/poincare/millenniumPrizeFull.pdf. Retrieved March 18, 2010. "The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture."
- ^ Associated Press (1 July 2010). "Russian mathematician rejects $1 million prize". msnbc.com. http://www.msnbc.msn.com/id/38039068/ns/technology_and_science-science/?gt1=43001. Retrieved 16 June 2011.
- ^ Ritter, Malcolm (1 July 2010). "Russian mathematician rejects $1 million prize". The Boston Globe. http://www.boston.com/news/science/articles/2010/07/01/russian_mathematician_rejects_1_million_prize/.
- This article incorporates material from Millennium Problems on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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