Product Description

The definitive account of the Everests of mathematics--the seven unsolved problems that define the state of the art in contemporary math.

In 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.

Product Details

• Amazon Sales Rank: #300886 in Books
• Published on: 2003-10-15
• Released on: 2003-10-14
• Original language: English
• Number of items: 1
• Binding: Paperback
• 256 pages

From Publishers Weekly
The noble idea that advanced mathematics can be made comprehensible to laypeople is tested in this sometimes engaging but ultimately unsatisfying effort. Mathematician and NPR commentator Devlin (The Math Gene) bravely asserts that only "a good high-school knowledge of mathematics" is needed to understand these seven unsolved problems (each with a million-dollar price on its head from the Clay Mathematics Institute), but in truth a Ph.D. would find these thickets of equations daunting. Devlin does a good job with introductory material; his treatment of topology, elementary calculus and simple theorems about prime numbers, for example, are lucid and often fun. But when he works his way up to the eponymous problems he confronts the fact that they are too abstract, too encrusted with jargon, and just too hard. He finally throws in the towel on the Birch and Sinnerton-Dyer Conjecture ("Don't feel bad if you find yourself getting lost... the level of abstraction is simply too great for the nonexpert"), while the chapter on the Hodge Conjecture is so baffling that the second page finds him morosely conceding that "the wise strategy might be to give up." Nor does Devlin make a compelling case for the real-world importance of many of these problems, rarely going beyond vague assurances that solving them "would almost certainly involve new ideas that will... have other uses." Sadly, this quixotic book ends up proving that high-level mathematics is beyond the reach of all but the experts.
Copyright 2002 Reed Business Information, Inc.

From School Library Journal
Adult/High School-In May, 2000, the Clay Mathematics Institute posted a million-dollar prize to anyone able to solve any of what it considered the seven most important mathematical problems of the 21st century. They were chosen not for theoretical beauty alone, but because many of them deal with concepts in fields like physics, computer science, and engineering, and exist because practitioners in those fields are already using theoretical or practical design solutions that have not been mathematically proven. Devlin, "The Math Guy" from NPR's Weekend Edition, does a good job explaining the background of the problems and why theoretical mathematics as a discipline should matter to a general audience. Each problem has a chapter of its own and is given a treatment that, where applicable, extends back to the ancient Greeks. A passing knowledge of mathematics is important for taking in Devlin's work but a major in the subject is not, and this book should satisfy anyone looking for a layman's guide to modern theoretical mathematics. Or hoping to win a million dollars.
Sheryl Fowler, Chantilly Regional Library, VA
Copyright 2003 Reed Business Information, Inc.

From Booklist
Pledged by a wealthy amateur math enthusiast, $1 million per problem awaits whoever can solve the seven problems mathematician Devlin describes in this work. A similar proposition, minus the money, was made in 1900 by the German mathematician David Hilbert, who listed two dozen math mysteries he hoped would be dispelled in the coming century. All but one were, and that one, called the Riemann hypothesis, carries over to the new set of conundrums. The Riemann hypothesis is comprehensible to an advanced high-school math student, thanks to Devlin's clarity as well as his experience in popular exposition as the author of books such as The Math Gene (2000) and NPR's explainer of all things mathematical. As to the rest of the conjectures, Devlin directly states that no one without a doctorate could understand them, let alone crack them. But as a skilled guide pointing out the shape of the problems, and the practical implications of their solutions, Devlin's intriguing book will appeal to the lay reader curious about the abstract frontiers of math. Gilbert Taylor
Copyright © American Library Association. All rights reserved

Customer Reviews

Fails at an impossible task, BUT...
... but keep it in mind for that teenage nerd in your life.

To help you evaluate my evaluation, let me note up front that I have three long-ago years of graduate math courses under my belt, which made me familiar with four of the seven problems discussed here. I got bored with much of the account of those four, had fun with the discussion of the sixth problem (the Birch and Swinnerton-Dyer Conjecture, which has to do with rational points on elliptic curves), and obtained a vague picture of the remaining two.

My three-star rating is bound to be misleading. Keith Devlin has an enormous gift for mathematical explanation, but as he himself recognizes, in attempting to explain to the proverbial man on the street the seven Millenium Problems (for solving each of which the Clay Mathematical Institute, hoping to spur mathematical research in the 21st century somewhat as David Hilbert did with his famous set of 23 problems in the century just past, has put up a cool million American dollars), he has bitten off more than anyone could possibly chew. I don't mean to suggest it could have been done any better.

If you hanker to tackle the problems and win one of those millions for yourself, start hankering for some other pipe dream. These problems are tough. If you want to thoroughly understand what they consist of, you will need to go to the official technical description of the problems in the book jointly prepared by the Clay institute and the American Mathematical Society. If you want a light overview of them, there's no such thing, but this book is as good a compromise between ease and clarity as you will get. If you just want a feel for where mathematics in general stands at this point in history, the backward glance at Hilbert's problems given in "The Honors Class" is a better place to start.

The challenge for Devlin (aside from gearing up to understand the two most abstruse problems himself) was to describe the problems without assuming any knowledge on the reader's part beyond high school algebra. So he has a humongous amount of ground to cover. With sprightly historical notes, he zips through complex numbers, complex functions, infinite sums and products, special relativity, quantum field theory, symmetry groups - and that's just the first two, easiest chapters. He does a particularly fine job, I felt, with the fifth chapter, on Poincare's conjecture. The mathematics needed for a precise statement of the conjecture is fairly daunting, but his informal description conveys the heart of it vividly and accurately.

All the above is subject to a major caveat. The real agenda for this volume is narrower than educating the general public. The main thing the Clay Institute wanted its prize offer to accomplish was to stir interest in math among students. Considered in those terms, I'd give it five stars, because the people who are going to lap the book up with relish are mathematically gifted high school students. If bits of each chapter go over their heads, it will only serve to whet their appetites. Because it's so ideally suited for them, I'd like to see (and I'm sure the Clay Institute would like to see) Devlin's opus in every high school library in the country.

An honest attempt to explain deep mathematics
In this book the author makes a sincere attempt to describe to a popular audience the content behind seven mathematical problems that were chosen by a private foundation called "The Clay Institute" as being deep enough to warrant a prize of $1,000,000 for their solution. The goal is realized in some parts of the book, but falls short in others, but it still is of value to those who are curious about the history and content behind these problems. The author is aware of the difficulty in describing the content of the problems to readers without substantial mathematical preparation, and he does a good job in general.

One can of course think of many other problems that fit the stature of the millennium problems, such as the invariant subspace conjecture, or developing a complete mathematical model of the cell, but these seven will no doubt spark the curiosity of a few young persons as they further their studies in mathematics. Some of the millennium problems, such as the Riemann hypothesis, the NP problem, the Poincare conjecture, and the Navier-Stokes equations, require only an undergraduate education. The others definitely require more background, just to understand even the statement of the problem. All of the them are fascinating, and will no doubt stimulate some incredibly interesting mathematical constructions.

Personal note for anyone interested (from someone who has worked on one of these problems for several years): For those readers who are thinking about attacking one of these problems, it is important to be really interested in solving it, for your own satisfaction, and not to be concerned about the financial reward or what the solution will bring you in terms of professional advancement. Large blocks of time will be needed to think about the problem, and therefore you will have to be concerned with your livelihood in the interim. Being a single person will definitely relieve you of the financial burden of having to support a family, but on the other hand a family will bring you personal warmth as you take the roller coaster ride of confidence and depression that goes with this kind of research. A traditional tenure-track position might be difficult to justify, since you will not be publishing and therefore your chances of obtaining tenure will be greatly diminished. It might also be wise in whatever job you work in to keep your ambitions to yourself, as colleagues and other mathematicians will typically not be encouraging in your decision to work on the problem. Therefore, you will definitely find yourself working on two problems in your life: the millennium problem and a constrained optimization problem, the latter being how to live your life in the interim, and whose solution possibly ranks in similar complexity. Your research in the millennium problem will probably take years, and as you see more lines appear on your face and your colleagues take the normal professional route, you might have doubts about your decisions. The more time spent on it without resolution of course will close the doors on a standard career in academia, and you will approach a critical point where there is no turning back. It is at this time that you will realize that it is you that has taken charge of yourself, your goals, and your attitudes about mathematics and life...and this of course is the best possible life anyone can have.

True million dollar problems
Is the solution of any mathematics problem worth one million dollars? Yes, in fact there are seven such problems. In 1999, Landon Clay established the Clay Mathematical Foundation and in 2000, the Clay Foundation announced seven separate prizes of one million U. S. dollars for the solution of each of seven mathematics problems. In keeping with the famous list of unsolved problems enunciated by David Hilbert at the turn of the previous century, this list can be considered the problems for the new century, which also happens to be a new millennium.
Make no mistake, these problems are very hard. Even with all his mathematical expertise. Devlin readily admits that he really does not understand them all and had a very difficult time writing about them at a level so that a general audience could understand the basics of the problems. The seven problems are

· The Riemann hypothesis
· Yang-Mills Theory and the Mass Gap Hypothesis
· The P vs. NP Problem
· The Navier-Stokes Equations
· The Poincare Conjecture
· The Birch and Swinnerton-Dyer Conjecture
· The Hodge Conjecture

and the Riemann hypothesis is distinguished in that it is the only one that was also on Hilbert's list at the turn of the previous century. In his descriptions of the last two problems, it is clear that Devlin is struggling to understand the fundamentals of the problems.
Nevertheless, he does manage to inform the reader about what the problems are about, as well as a taste of how difficult they are. Like the problems David Hilbert stated in 1900, this collection of problems forms a marker by which the mathematical progress of this century will be measured. For that reason, all mathematicians should learn something about them, and this book is an ideal initial step.

Published in Recreational Mathematics e-mail newsletter, reprinted with permission.