The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time

Rating:
Summary: Fails at an impossible task, BUT...
Review: ... 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.

Rating:
Summary: Inspiring. And, it's from Keith Devlin.
Review: Great book. Although pushing the limits of "accessible", those who understand it will be intrigued by Devlin's discussion of the Millenium Problems, a set of puzzling mathematical problems that, if solved, could redefine and revolutionize mathematical science. If you thought cryptography was interesting, wait until you see this.

Also recommended: The Math Gene (also by Keith Devlin)

Rating:
Summary: Great book but sloppy physics history
Review: I've read several of Devlin's books and have loved them. So I was quite excited to see this book published. Overall, I enjoyed it immensely - he does a great job trying to give the reader the context of why a problem is important.

On fault I have with the book is his description of the Eddington's experiment to confirm general relativity. On page 80, he writes that Eddington made an accurate observation of the planet Mercury during a total solar eclipse. Not so - it was of a star. The orbit of Mercury rotates over time and it was general relativities first success to correctly calculate this effect. This is just very sloppy editing since this story is repeated in so many science popularizations.

Rating:
Summary: An honest attempt to explain deep mathematics
Review: 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.

Rating:
Summary: True million dollar problems
Review: 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.

Rating:
Summary: Oddly uneven exposition....
Review: Let's be frank: most people have a better chance of winning $1M at the state lottery than by proving any of these "millennium problems". Keith Devlin does a good job of explaining why. A little reverse psychology here and there ("...if you find the going too hard, then the wise strategy might be to give up.") just makes us want to push on toward the more difficult problems.

The going isn't too hard thanks to Devlin's expository ability, but alas, I think this will be true only for aficionados of mathematics and physics. In his columns for the Mathematical Association of America, Keith has always had in mind a varied audience of readers. But how can he hope to communicate to the non-mathematician when so much meaning resides in the equations that appear throughout the book? Still, his pedagogy prevents this from becoming "The Idiot's Guide to the Millennium Problems". (I suppose it'll appear real soon.)

Devlin hints at a disturbing idea. Will cutting edge problems become so abstruse some day that it will take the best minds all the fruitful years of their lives just to arrive at a position of comprehension? What then, mathematical AI?

There are some silly mistakes, perhaps caused by a looming deadline. One involves a mix-up between the relativistic precession of Mercury's orbit and the relativistic bending of light rays. A logical error appears in a footnote on pg.54, where the word "a" should replace "no". Another one appears in the caption of Fig. 5.5, where "Example" should replace "Proof". Would it be too much to ask that copy editors who are assigned technical books have a dim awareness of mathematical argumentation?

Rating:
Summary: It's Not An "Idiot's Guide to the Millenium Problems"
Review: Let's be frank: most people have a better chance of winning $1M at the state lottery than by proving any of these "millennium problems". Keith Devlin does a good job of explaining why. A little reverse psychology here and there ("...if you find the going too hard, then the wise strategy might be to give up.") just makes us want to push on toward the more difficult problems.

The going isn't too hard thanks to Devlin's expository ability, but alas, I think this will be true only for aficionados of mathematics and physics. In his columns for the Mathematical Association of America, Keith has always had in mind a varied audience of readers. But how can he hope to communicate to the non-mathematician when so much meaning resides in the equations that appear throughout the book? Still, his pedagogy prevents this from becoming "The Idiot's Guide to the Millennium Problems". (I suppose it'll appear real soon.)

Devlin hints at a disturbing idea. Will cutting edge problems become so abstruse some day that it will take the best minds all the fruitful years of their lives just to arrive at a position of comprehension? What then, mathematical AI?

There are some silly mistakes, perhaps caused by a looming deadline. One involves a mix-up between the relativistic precession of Mercury's orbit and the relativistic bending of light rays. A logical error appears in a footnote on pg.54, where the word "a" should replace "no". Another one appears in the caption of Fig. 5.5, where "Example" should replace "Proof". Would it be too much to ask that copy editors who are assigned technical books have a dim awareness of mathematical argumentation?

Rating:
Summary: Another sloppy physics history
Review: Reviewer Ted Sung pointed out a sloppy remark of physics history made by the book. There is another serious mistake appeared on Page 91: The Yang-Mills gauge theory has never been awarded a Nobel Prize. C. N. Yang did get the Prize in 1957, but it was conferred to him and his another collaborator T. D. Lee for a different contribution to physics (postulating that parity is not conserved under weak interactions). Unfortunately the author did not perform a simple research on history but rather believed in his speculation. My four-star rating does not count in this unforgivable carelessness. Devlin has done a good job in popularizing these great mathematical problems. Perhaps someone could deliver even a finer exposition. However, it would definitely demand the readers another level of mathematical sophistication.

Rating:
Summary: Pedantic
Review: This book is more pedantic than I thought it would be. Being a smallish book and a smaller audience, it is understandable that the mathematical details are trimmed down (almost excised, you might say). Still, there is too much history and not enough details. Too often the author says that it may be above the reader's level. Overall, I was disappointed, but it was not a waste.

Rating:
Summary: The Everests of Mathematics
Review: Would you like to win a million dollars? Would you like to win it by solving a math problem? You have entered the right millennium to do so. There's a million bucks waiting for you if solve any one of seven problems, all of which are something like this: "Prove that every harmonic differential form (of a certain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles." Oh, dear. It is clear that you will be up against a task more daunting than just twisting a Rubik's cube. Keith Devlin has taken on the daunting task of explaining _The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time_ (Basic Books) to a lay audience. He knows it simply isn't going to be possible, and he has plenty of apologies. "Don't feel bad if you find yourself getting lost," he advises for one of the problems, "Most readers will. Like many parts of modern advanced mathematics, the level of abstraction is simply too great for the nonexpert to make much headway." And don't worry about the difficulty; even among experts, about one of the problems Devlin says, "... not only is there no 'smart money' on whether the conjecture will turn out to be true or false, there isn't even a consensus as to what it really says.")

So, why are these problems at all worth thinking about, by anyone other than eggheads who might have a crack at winning the million dollars for each one? First of all, the million dollar prizes are just a way of getting some publicity for higher math. The prizes were announced in May 2000. The money was committed by a mutual-fund magnate Landon Clay, who just finds math a fun hobby and likes to support mathematical research. One important reason to understand these problems, even just a little, is to try and understand what mathematicians do. Abstractions may be piled on abstractions, but these are not puzzles unconnected to the real world; the solutions will bear on communications, cryptography, nautical and aeronautical engineering, and much more. The announcement of these problems is similar to David Hilbert's announcement of 23 essential problems a century ago. Not only did he lay out mathematical effort for the twentieth century: it worked. All of the problems were solved, except for one. It is no surprise that that one, the Reimann Hypothesis, is one of the current seven. It has to do with the frequency and distribution of the mysterious prime numbers. Perhaps in the current century, building on all the previous attempts to solve it, someone will see the problem in a new way and it, too, will fall.

Devlin, in a very good-humored book, reminds us, "Still, it's important to remember that mathematicians belong to the same species as you. (Trust me on this.)" The prizes are out there to draw attention to the effort, but no mathematician is going to work on one of these problems in order to get rich. Anyone who without the prospect of a million-dollar prize would balk at the years of study required to grasp these problems "simply doesn't have the requisite commitment to mathematical research." It isn't hard to imagine that a boffin who works at this sort of problem is just the type to find much more joy in solving it than in adding many more digits to a bank account. On the other hand, it is hard to imagine that someone would pick up Devlin's book and consequently win one of the million dollar prizes (although there are a good many math kooks out there who are still engaged in follies such as trisecting the angle; this book cannot help but increase their efforts). Solving the problems can't be done on the basis of the descriptions here, as lucid as they might be. But that's not the point; trying to understand even in a superficial way the famous puzzles described here is not only a challenge in itself, but enables one to participate a little in the newest frontiers of the oldest intellectual endeavor our species has undertaken.