Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. 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.

Quotes

• The problems divided, very roughly, into two algebraic problems, two topological problems, two problems in mathematical physics, and one problem in the theory of computation.
  • Ian Hacking (30 January 2014). Why Is There Philosophy of Mathematics At All?. Cambridge University Press. p. 68. ISBN 978-1-107-72982-7. 

Solved problem

Poincaré conjecture

• Posed in 1904 by Henri Poincaré, the leading mathematician of his era and among the most gifted of all time, the Poincaré conjecture is a bold guess about nothing less than the potential shape of our own universe.
  • Donal O'Shea (30 October 2008). The Poincaré Conjecture: In Search of the Shape of the Universe. Penguin Books Limited. pp. 13. ISBN 978-0-14-190034-6. 

The official statement of the problem was given by John Milnor. A proof of this conjecture was given by Grigori Perelman in 2003.

Unsolved problems

P versus NP

• The P versus NP problem was first mentioned in a 1956 letter from Kurt Gödel to John von Neumann, two of the greatest mathematical minds of the twentieth century.
  • Lance Fortnow (2013). The Golden Ticket: P, NP, and the Search for the Impossible. Princeton University Press. p. 6. ISBN 0-691-15649-2. 

The official statement of the problem was given by Stephen Cook.

Hodge conjecture

• The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: algebra, topology, and analysis.
  • Ian Stewart (5 March 2013). Visions of Infinity: The Great Mathematical Problems. Basic Books. p. 211. ISBN 978-0-465-06599-8. 

The official statement of the problem was given by Pierre Deligne.

Riemann hypothesis

• At the beginning of the new millennium the most famous unsolved problem in Analytic number theory and complex analysis, if not in all of mathematics, is to determine whether the Riemann Hypothesis holds.

• • Theodore Gamelin (17 July 2003). Complex Analysis. Springer Science & Business Media. pp. 370. ISBN 978-0-387-95069-3. 

The official statement of the problem was given by Enrico Bombieri.

Yang–Mills existence and mass gap

• One should mention right at the start that one still does not understand whether quantum mechanics and special relativity are compatible at a fundamental level in our Minkowski four-space world. One generally assumes that this means finding a complete Yang-Mills gauge theory or the interaction of gauge fields with fermionic matter fields, the simplest form being quantum chromodynamics (QCD). Associated with this picture is the belief that the fundamental vector meson excitations are massive (as opposed to photons, which arise in the limiting case of an abelian gauge symmetry. The proof of the existence of a “mass gap” appears a necessary integral part of solving the entire puzzle. This question remains one of the deepest open issues in theoretical physics, as well as in mathematics.

Basically the question remains: can one give a mathematical foundation to the theory of fields in four-dimensions? In other words, when do quantum mechanics and special relativity lie on the same footing as the "classical physics" of Newton, Maxwell, Einstein, or Schrödinger—all of which fits into a mathematical framework that we describe as the language of physics. This glaring gap in our fundamental knowledge even dwarfs questions of whether there are other more complicated and sophisticated approaches to physics—those that incorporate gravity, strings, or branes—for understanding their fundamental significance lies far in the future. In fact, one believes that string proposals, if they can be fully implemented, have limiting cases that appear as relativistic quantum fields, just as relativistic quantum fields describe non-relativistic quantum theory and classical physics in various limiting cases.

• • Arthur Jaffe, Introduction to Quantum Field Theory, Spring 2005.

The official statement of the problem was given by Arthur Jaffe and Edward Witten.

Navier–Stokes existence and smoothness

The official statement of the problem was given by Charles Fefferman.

Birch and Swinnerton-Dyer conjecture

• The BSD Conjecture has its natural context within the larger scope of modern algebraic geometry and number theory.
  • Avner Ash; Robert Gross (2012). Elliptic Tales: Curves, Counting, and Number Theory. Princeton University Press. p. 245. ISBN 0-691-15119-9. 

The official statement of the problem was given by Andrew Wiles.