Millennium Problems


The Millennium Problems are seven problems for the solution of which the Clay Mathematics Institute (CMI) is offering a prize of $7 million. Someone who can solve just one of these problems receives $1 million. For each problem, the CMI had a professional mathematician write up an official statement of the problem which will be the main standard by which a given solution will be measured against. The prize offer was announced in May 2004.

The seven problems are:

  1. 1.

    The Birch and Swinnerton-Dyer conjecture. It concerns elliptic curvesMathworldPlanetmath over rational numbersPlanetmathPlanetmathPlanetmath and the L-series attached to those curves. The official statement of the problem was given by Andrew Wiles.

  2. 2.

    The Hodge conjecture. It concerns Hodge cycles and their linear combinations. The official statement of the problem was given by Pierre Deligne.

  3. 3.

    “Existence and smoothness of the The Navier-Stokes equation.” The Navier-Stokes equation calculates momemtum in fluids (specifically, liquids and gases). The official statement of the problem was given by Charles Fefferman.

  4. 4.

    “P versus NP.” It concerns the differencePlanetmathPlanetmath between verification and computation in polynomial timeMathworldPlanetmath. The official statement of the problem was given by Stephen Cook.

  5. 5.

    The Poincaré conjecture. It is believed that every simply-connected compact 3-manifold is homeomorphic to S3. The official statement of the problem was given by John Milnor.

  6. 6.

    Quantum Yang-Mills theory. Just as Hilbert’s problems a hundred years before included a problem relating to the mathematics of physics, so does this 2000 collectionMathworldPlanetmath. The official statement of the problem was given by Arthur Jaffe and Edward Witten.

  7. 7.

    The Riemann hypothesisMathworldPlanetmath. It concerns the distribution of primes. This one was 8 of 23 in Hilbert’s problems. The official statement of the problem was given by Enrico Bombieri.

References

  • 1 Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time New York: Perseus Books Group (2002)

Appendix A External link

http://www.claymath.org/millennium/CMI page on Millennium Problems

Title Millennium Problems
Canonical name MillenniumProblems
Date of creation 2013-03-22 16:32:06
Last modified on 2013-03-22 16:32:06
Owner Mravinci (12996)
Last modified by Mravinci (12996)
Numerical id 8
Author Mravinci (12996)
Entry type Definition
Classification msc 01A65
Classification msc 01A61
Synonym Millenium Prize Problems
Related topic FamousAndInfamousOpenQuestionsInMathematics