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The Millennium Prize problems. (English) Zbl 1155.00001

Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 0-8218-3679-X/hbk). xvii, 165 p. (2006).

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Publisher’s description: “Guided by the premise that solving some of the world’s most important mathematical problems will advance the field, this book offers a fascinating look at the seven unsolved Millennium Prize problems. This work takes the unprecedented approach of describing these important and difficult problems at the professional level.
In announcing the seven problems and a US$7 million prize fund in 2000, the Clay Mathematics Institute emphasized that mathematics still constitutes an open frontier with important unsolved problems. The descriptions in this book serve the Institute’s mission to “further the beauty, power and universality of mathematical thinking.”
Separate chapters are devoted to each of the seven problems: the Birch and Swinnerton-Dyer Conjecture (Andrew Wiles), the Hodge Conjecture (Pierre Deligne), the Navier-Stokes Equation (Charles L. Fefferman), the Poincaré Conjecture (John Milnor), the \(\mathbf P\) versus \(\mathbf{NP}\) Problem (Stephen Cook), the Riemann Hypothesis (Enrico Bombieri), and Quantum Yang-Mills Theory (Arthur Jaffe and Edward Witten).
An essay by Jeremy Gray, a well-known expert in the history of mathematics, outlines the history of prize problems in mathematics and shows how some of mathematics’ most important discoveries were first revealed in papers submitted for prizes. Numerous photographs of mathematicians who shaped mathematics as it is known today give the text a broad historical appeal.
Anyone interested in mathematicians’ continued efforts to solve important problems will be fascinated with this text, which places into context the historical dimension of important achievements.”
This book really deserves its place in any mathematics library or even on almost all mathematician’s desk.
The Problem concerning the Poincaré conjecture has meanwhile been solved by Grisha Perelman (see the reviews in Zbl 1130.53001, Zbl 1130.53002 and Zbl 1130.53003) who, however, decided not to accept the Fields Medal (the Nobel Prize equivalent for Mathematics) awarded at the ICM 2006 in Madrid.

MSC:

00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general
00A05 Mathematics in general
00A07 Problem books
01A67 Future perspectives in mathematics
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