## The Millennium Prize ProblemsJames Carlson, James A. Carlson, Arthur Jaffe, Andrew Wiles, Clay Mathematics Institute, American Mathematical Society On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an influential role in mathematical research. A century later, on May 24, 2000, at a meeting at the College de France, the Clay Mathematics Institute (CMI) announced the creation of a US$7 million prize fund for the solution of seven important classic problems which have resisted solution. The prize fund is divided equally among the seven problems. There is no time limit for their solution. The Millennium Prize Problems were selected by the founding Scientific Advisory Board of CMI--Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten--after consulting with other leading mathematicians. Their aim was somewhat different than that of Hilbert: not to define new challenges, but to record some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium; to recognize achievement in mathematics of historical dimension; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; and to emphasize the importance of working towards a solution of the deepest, most difficult problems. The present volume sets forth the official description of each of the seven problems and the rules governing the prizes. It also contains an essay by Jeremy Gray on the history of prize problems in mathematics. |

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### Contents

JEREMY GRAY | 3 |

ANDREW WILES | 31 |

PIERRE DELIGNE | 45 |

JOHN MILNOR | 71 |

The P versus NP Problem | 85 |

ENRICO BOMBIERI | 107 |

ARTHUR JAFFE AND EDWARD WITTEN | 129 |

Rules for the Millennium Prizes | 153 |

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3-manifolds abelian varieties Academy algebraic cycles algebraic varieties analytic Andrew Wiles arithmetic Arthur Jaffe awarded axioms Berlin Birch and Swinnerton-Dyer Cambridge century checking relation cl(Z classical Clay Mathematics Institute cohomology complexity theory computation construction counterexample Courtesy curvature defined Deligne dimensions Dirichlet divisor Edward Witten elliptic curves essay Euclidean example existence Fermat's Last Theorem finite fields formula four-dimensional gauge theory genus Geom geometry Hilbert Hodge conjecture homeomorphic input integral interaction known L-functions language linear lower bounds manifold mass gap Math mathematicians methods Millennium Prize Problems modular Navier-Stokes equations NP problem NP-complete number theory Paris Photo Phys physics Pierre Deligne Poincare Conjecture polynomial polynomial-time algorithm prime numbers Princeton proof properties prove quantum field theory question rational points renormalization Riemann hypothesis satisfies Sciences Smale smooth solved space space-time Springer-Verlag Tate three-dimensional topic topology Turing machine University variable vector weak solution Wightman Yang-Mills theory York zeros