# 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.

## Contents

## Quotes[edit]

- 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.

- Ian Hacking (30 January 2014).

### Solved problem[edit]

#### Poincaré conjecture[edit]

- 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.

- Donal O'Shea (30 October 2008).

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[edit]

#### P versus NP[edit]

- 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.

- Lance Fortnow (2013).

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

#### Hodge conjecture[edit]

- 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.

- Ian Stewart (5 March 2013).

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

#### Riemann hypothesis[edit]

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

- Theodore Gamelin (17 July 2003).

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

#### Yang–Mills existence and mass gap[edit]

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

#### [edit]

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

#### Birch and Swinnerton-Dyer conjecture[edit]

- 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.

- Avner Ash; Robert Gross (2012).

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