## The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our TimeIn 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader. |

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### Contents

The Gauntlet is Thrown | 1 |

The Music of the Primes The Riemann Hypothesis | 19 |

Appendix 1 Euclids Proof That There Are Infinitely Many Primes | 53 |

How Do Mathematicians Work Out Infinite Sums? | 55 |

How Euler Discovered the Zeta Function | 59 |

The Fields We Are Made Of YangMills Theory and the Mass Gap Hypothesis | 63 |

Group Theory The Mathematics of Symmetry | 97 |

When Computers Fail The P vs NP Problem | 105 |

Making Waves The NavierStokes Equations | 131 |

The Mathematics of Smooth Behavior The Poincare Conjecture | 159 |

Knowing When the Equation Cant Be Solved The Birch and SwinnertonDyer Conjecture | 191 |

Natation for Infinite Sums and Products | 211 |

Geometry Without Pictures The Hodge Conjecture | 215 |

231 | |

233 | |

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