It is probably that, as the Beatles proclaimed, money can't buy love. But it sure can get you a lot of attention. In May 2000, the Clay Mathematics Institute announced that it was establishing one million dollar prizes for the solution of each of seven mathematical problems it considered of crucial importance. It didn't take long for the "Millennium Problems" to penetrate the conscience of mathematicians, and maybe even of the general public. Every news story about the solution of the Poincaré Conjecture, for example, mentions that it is worth a million dollars. The mere fact of the prize seems to make the problem more important.

The book under review is the official description of the seven problems. It includes the rules for the awarding of the prizes, but the core of the book are the seven essays describing the problems as precisely as possible. These are written by experts and aimed at mathematicians. (A description of the problems for the "general public" was attempted in Keith Devlin's The Millennium Problems.) The seven essays vary a lot in their accessibility. Milnor on the Poincaré Conjecture is very readable, while, for example, Deligne on the Hodge Conjecture will be very hard going for non-specialists. These essays have been available online at the Clay Mathematics Institute web site for some time, and they seem essentially unchanged.

An introductory essay by Jeremy Gray gives an overview of the history of prizes in mathematics. Gray highlights the difference between restrospective prizes, given for work already done, and prospective prizes, which promise rewards for the solution of specific problems. He points out that the latter were often unsuccessful: the problems were not solved within the set deadline, or turned out to be uninteresting. The Clay Millennium Prizes are, of course, prospective, and it is hard to foresee what their outcome might be. The good news is that one of the problems, the Poincaré Conjecture, seems to have been solved. The bad news is that the others still look very hard. The Abel Prize is not mentioned, and this suggests to me that the article was written some time ago, before that prize was announced, since it is today the most prominent of retrospective prizes.

Included in the book are a great number of photographs, a gallery including most of the people that are mentioned in the essays. Each of the authors of the essays gets a large photograph. These must have required a huge amount of work to collect, and they alone make the book worth getting. It's easy to get an image of, say, Riemann, but it's much rarer to find a good photograph of Enrico Bombieri or Hugh Montgomery.

Given the interest generated by the Millennium Problems, this book should be in every mathematics library, perhaps right next to Devlin's more popular account.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

Introduction vii

Landon T. Clay xi

Statement of the Directors and

the Scientific Advisory Board xv

A History of Prizes in Mathematics

Jeremy Gray 3

The Birch and Swinnerton-Dyer Conjecture

Andrew Wiles 31

The Hodge Conjecture

Pierre Deligne 45

Existence and Smoothness of the Navier–Stokes Equation

Charles L. Fefferman 57

The Poincar´e Conjecture

John Milnor 71

The P versus NP Problem

Stephen Cook 87

The Riemann Hypothesis

Enrico Bombieri 107

Quantum Yang–Mills Theory

Arthur Jaffe and Edward Witten 129

Rules for the Millennium Prizes 153

Authors’ Biographies 157

Picture Credits 161

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