Millennium Prize Problems

This article is about the math prizes. For the technology prize, see Millennium Technology Prize.

Millennium Prize Problems
P versus NP problem Hodge conjecture Poincaré conjecture (solved) 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 November 2015, 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. The Poincaré conjecture was solved by Grigori Perelman, but he declined the award in 2010.

Solved problem

Poincaré conjecture

Main article: Poincaré conjecture

In topology, a sphere with a two-dimensional surface is characterized by the fact that it is compact and simply connected. The Poincaré conjecture is that this is also true in one higher dimension. The problem is to establish the truth value for this conjecture. The truth value had been established for the analog conjecture for all other dimensionalities. The conjecture 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 but he declined that award. Perelman was officially awarded the Millennium Prize on March 18, 2010, but he also declined the 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 Hamilton.

Unsolved problems

P versus NP

Main article: P versus NP problem

The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, and to biology, philosophy and cryptography (see P versus NP problem proof consequences).

“ 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.

Hodge conjecture

Main article: Hodge conjecture

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.

Riemann hypothesis

Main article: 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

Main article: 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 official statement of the problem was given by Arthur Jaffe and Edward Witten.

Navier–Stokes existence and smoothness

Main article: Navier–Stokes existence and smoothness

The Navier–Stokes equations describe the motion of fluids. Although they were first stated in the 19th century, they are still 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.

Birch and Swinnerton-Dyer conjecture

Main article: Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with certain type of equations; 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.

See also

• Hilbert's problems
• List of unsolved problems in mathematics
• Paul Wolfskehl (offered a cash prize for the solution to Fermat's Last Theorem)
• Smale's problems

References

• This article incorporates material from Millennium Problems on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Further reading

• Devlin, Keith J. (2003) [2002]. 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. 

External links

• The Millennium Grand Challenge in Mathematics
• The Millennium Prize Problems