## Mathematics: Frontiers and PerspectivesThis remarkable book is a celebration of the state of mathematics at the end of the millennium. Produced under the auspices of the International Mathematical Union (IMU), the volume was born as part of the activities observing the World Mathematical Year 2000. The volume consists of 30 articles written by some of the most influential mathematicians of our time. Authors of 15 contributions were recognized in various years by the IMU as recipients of the Fields Medal, from K. F. Roth (Fields Medalist, 1958) to W. T. Gowers (Fields Medalist, 1998). The articles offer valuable reflections about the amazing mathematical progress we have witnessed in this century and insightful speculations about the possible development of mathematics over the next century.Some articles formulate important problems, challenging future mathematicians. Others pay explicit homage to the famous set of Hilbert Problems posed one hundred years ago, giving enlightening commentary. Yet other papers offer a deeply personal perspective, allowing singular insight into the minds and hearts of people doing mathematics today."" Mathematics: Frontiers and Perspectives"" is a unique volume that pertains to a broad mathematical audience of various backgrounds and levels of interest. It offers readers true and unequaled insight into the wonderful world of mathematics at this important juncture: the turn of the millennium. The work is one of those rare volumes that can be browsed, and if you do simply browse through it, you get a wonderful sense of mathematics today. Yet it also can be intensely studied on a detailed technical level for gaining insight into some of the great problems on which mathematicians are currently working. Editors Michael Atiyah and Peter Lax were winners of the famous Abel Prize awarded by The Norwegian Academy of Science and Letters for outstanding work in mathematics. |

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### Contents

II | 1 |

III | 13 |

IV | 33 |

V | 35 |

VI | 55 |

VII | 65 |

VIII | 79 |

IX | 93 |

XVII | 197 |

XVIII | 219 |

XIX | 235 |

XX | 251 |

XXI | 261 |

XXII | 271 |

XXIII | 295 |

XXIV | 321 |

X | 117 |

XI | 121 |

XII | 137 |

XIII | 153 |

XIV | 161 |

XV | 175 |

XVI | 189 |

XXV | 329 |

XXVI | 343 |

XXVII | 353 |

XXVIII | 403 |

XXIX | 417 |

XXX | 433 |

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### Common terms and phrases

abelian algebraic arithmetic automorphism bound bundle Calabi-Yau Calabi-Yau manifolds classical coefficients cohomology combinatorics compact complex conjecture connected construction defined denote differential equations dimension dimensional discrete dynamics eigenvalues elliptic curve Euler example existence finite formula Fourier fundamental Galois geometry given global graph Hilbert holomorphic ideas integer International Mathematical Union invariant irreducible isomorphism Kahler L-functions Lagrangian Lagrangian submanifolds lattice linear Majda manifolds Math mathematicians mathematics methods metric minimal submanifolds modular moduli space natural nonnegative nontrivial number theory p-adic p-motives Phys physicists physics points polynomial polytopes poset positive problem proof proved quadratic quantum cohomology quantum field theory question random rational curves real number representation result Riemann scalar curvature simplicial singular smooth solutions solved sphere stochastic string theory structure subgroup subset surfaces symmetric functions symplectic symplectic manifold theorem topology turbulent vanishing cycles variables vector zero zeta function