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

Keith Devlin has done a good job of explaining these seven extremely difficult problems. Granted, I am a mathematics professor, but I enjoyed the book and will rely on it when I teach my History of Mathematics course. Devlin is not able to make the Hodge Conjecture understandable in the last chapter, but that's a tall order; he frankly admits that the problem is difficult for even a professional mathematician to understand, and I doubt that many expositors could have done better with the Hodge Conjecture.

The goal of Keith Devlin's "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time" is "to provide the background to each problem, to describe how it arose, [to] explain what makes it particularly difficult, and [to] give you some sense of why mathematicians regard it as important." "In May 2000 ... the Clay Mathematical Institute (CMI) announced that seven $1 million prizes were being offered for the solutions to each of [the] seven unsolved problems of mathematics..." Devlin's book is a "general introductions to ... the official book on the problems..." "... readers ... wishes to ... solve one of the Clay Problems should read the definitive description ... in the CMI book." "The official CMI book consists primarily of detailed and accurate descriptions of the seven problems..." Keith Devlin was asked "to provide short introductory accounts of the problems to make the book more accessible to mathematicians...journalists...readers..." "To read my [Keith Devlin's] book, all you need...is...high school knowledge of mathematics...You will also need sufficient interest in the topic." The book has eight chapters. Chapter zero is the general introduction to the problems. Chapter one is about the Riemann Hypothesis. Riemann suggests that for Riemann's Zeta function to be zero, the roots have the form ½ + bi for some real number b. Chapter two is about Yang-Mills Theory and the Mass Gap Hypothesis. The Yang-Mills equations describe all of the forces of nature (electromagnetic force, the weak nuclear force, and the strong nuclear force) other than gravity. The hypothesis provides "an explanation of why electrons have mass." The problem asks for "missing mathematical development of the theory, starting from axioms." The third chapter is about computer (The P Versus NP Problem). "Computer scientists divide computational tasks into two main categories: Tasks of type P can be tackled effectively on a computer; tasks of type E could take millions of years to complete. Unfortunately, most of the big computational tasks that arise in industry and commerce fall into a third category, NP, which seems to be intermediate between P and E. But is it? Could NP be just a disguised version of P? ... no one has been able to prove whether or not NP and P are the same." Chapter four is about the Navier-Stokes Equations. The equations describe "the motion of fluids and gases--such as water around the hull of a boat or air over an aircraft wing." They are partial differential equations (PDE). "To date, no one has clue how to find a formula that solves these particular equations." Chapter five is about the Poincare Conjecture. "If you stretch a rubber band around the surface of an apple, you can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface...if you imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is

If you want to know about seven of the most difficult unsolved math problems for which the Clay Mathematics Institute offers 1 million dolars a piece to whoever can solve them, this is the right book. Actually, we might talk about six unsolved problems since Perelman apparently solved the famous Poincaré conjecture. A quite readable account for someone who has some training in math.

Want to make a fast million? It's just sitting out there, and all you have to do is solve a math problem. Only one of seven. What could be easier? Actually, as Keith Devlin points out in his smart little book, there are few things that could be more difficult. The math problems in question were chosen by the Clay Foundation and represent some of the significant ideas in math today. Even understanding the problems are a chore, let alone trying to solve them, but Devlin does a good job in explaining them on a layman's level. At least Fermat's Last Theorem could be easily understood; however, it is now proven, and thus not one of the problems in question. Instead we have things like the Riemann Hypothesis and the P vs. NP Problem and perhaps most exotic, the Hodge Conjecture, which Devlin says states: "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." I have a degree in math and I can barely understand that. Devlin is hard-pressed to sketch out even the bare ideas behind it. Some of the problems are easier to explain, but no less difficult to solve. As Devlin asserts, you can't go into this for the money. Chances are you would need a doctorate in math just to get started, and even then, you might work on one of these problems for decades without solving it. But the purpose of the award is to pique mathematical interest, and in that sense, it will work, drawing more people into the field. And in this generally math-averse and math-ignorant world, that can only be a good thing. And this book is a good thing too. Given the complex nature of the material discussed, Devlin does a good job of explaining with a light touch, even admitting occasionally that some of the material is beyond even him. This book may not be a fast ticket to a million dollars, but it is nonetheless rewarding in its own way.

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-dolla