The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time Edition Unstated

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.

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 no way of shrinking it to a point without breaking either the rubber band or the doughnut...when you ask the same shrinking band idea distinguishes between four-dimensional analogues of apples and doughnuts...no one has been able to provide an answer." Chapter six is about the Birch and Swinnerton-Dyer Conjecture. The conjecture suggests that "there are infinitely many rational points on E [the elliptic curve] if and only if L(E,1)=0." Birch and Swinnerton-Dyer "creates" an counting device L(E,1) for rational points. Chapter seven is about the Hodge Conjecture. "The basic idea was to ask to what extent you can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension ... The Hodge conjecture asserts that for one important class of objects (called projective algebraic varieties), the piece called Hodge cycles are, nevertheless, combinations of geometric pieces (called algebraic cycles)."

Gives a very good description of complicated problems. Book was captivating.

As always, when I order math books at Amazon, I do that for 3 or 4 items together. When I receive them, it is often difficult to chose with which one to start. Since I had already read a lot about the millennium problems, I waited with this one, because I knew already at least something of the topics inside. However, when I finally started reading this book, I discovered that all the various problems had been written in not only a thorough way, but also in a pleasant, easily readable way. For instance, I had read a number of books about the Riemann Hypothesis already, but Keith Devlin even managed to explain some items in a new way. Also his description of the P versus NP problem was a pleasure to read. So, in fact of the 3 books I had ordered, this one appeared to be the best. And, in addition, I'm sure within a couple of weeks I will read it again. Just for fun.