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.
- 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.
- 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.
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.
- 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.
- At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds.
The official statement of the problem was given by Enrico Bombieri.
Yang–Mills existence and mass gap
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.