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

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)."

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.

I'd read Dr Devlin's  Mathematics: The New Golden Age (Pelican)  and really enjoyed that book. I did not think his grasp of physics was that good in the chapter on the Yang- Mills mass gap problem, and came away not much wiser as to what the mass gap was about. Maybe a collaboration in a revised edition with a physicist would clarify the concept.Showing how fast mathematics can move, the chapter on the Poincare Theorem does not even mention Richard Hamilton's programme as a promising line of research, even though G. Perelman shortly afterwards solved the Conjecture using it.I found the other 5 chapters very illuminating and clear, albeit of increasing difficulty and think a second edition, describing Perelman's solution and the relevance of the Yang- Mills mass gap problem to the physics of the strong gluon force, would be marvellous. Mathematics: The New Golden Age (Pelican)

The author makes a determined attempt to describe the natrure of the Millenium Problems and their mathematical importance. The account is instructive an entertaining, and the difficulty of the task illustrated by the final chapter on the Hodge Conjecture, which is very much of the form "Well hardly anyone understands it so lets not waste paper" - an honest approach! Interesting for general mathematical readers. A view of Everest and its neighbours without the effort of the climb.

Great book

This book fills the gap between the Overviews and the Specialist Explanations of the Millenium Problems as described on the Cray site. Keith Devlin brings the reader from Level 0 to a good layman's understanding of the problems, their historical importance, why they are still important today and what might lie ahead for anyone who would attempt to solve any of them. If you have a mathematics background, you might find yourself skipping some of the descriptions, for example, the basics of differentiation, but this is not a fault. Overall, an absorbing, very well-written account, in my opinion.

Libro molto interessante, ben scritto.Ma NON contiene i quesiti dei 7 problemi. Traccia solo una storia e il possibile impatto qualora ciascun problema fosse risolto