In 2000 the Clay Institute proposed the seven current mathematical problems that they hoped would guide mathematical research in our current era. The millenium problems are a respectful nod to a similar set of mathematical problems compiled by David Hilbert in 1900; one of the original Hilbert problems, the Riemann Hypothesis, has found its way into the Clay Millenium problems list.
The millenium problems are unimaginably (to most of us) abstract and intractable, and to even attempt to explain them to the lay-person is an impossible task. Nevertheless Devlin has made a brave and worthy attempt. Each of the problems: The Riemann Hypothesis, The Yang-Mill Theory and Mass Gap Hypothesis, The P v NP problem, The Navier-Stokes Equation, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and the Hodge Conjecture, has its own chapter. Each chapter gives some historical background to the problem, a mathematical overview, the possible implications of its proof (or disproof) and as lucid an explanation as is possible of the problem itself.
The cover reviews state that you'll come away feeling much the wiser after reading this book and whilst this is true, wisdom in this instance is a double-edged sword, the insight gained from Devlin's explanations bring home the realisation of just how difficult and obscure modern day mathematical research is.
A very good book, although taxing in parts; if you're at all interested in mathematics, and unless you're here by accident I assume you must be, then I would recommend reading it.
In May 2000, the Clay Mathematics Institute announced that seven $1 million prizes were being offered for the solution to each of seven of the most difficult unsolved problems in mathematics. These were 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. Keith Devlin's excellent book aims not to give a detailed description of these difficult problems but provides the background to each problem and explains why mathematicians regard it as important. 'The Millenium Problems' is an accessible and illuminating book that should be of interest not only to mathematicians but also the general reader curious about the frontiers of mathematics.
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.
I had not read any Keith Devlin before reading The Millennium Problems and I have to say I regret not having done so. He has a very engaging and clear writing style ( not to mention a humorous style too ) and has managed to present some very difficult mathematics as well as I think anybody could.
This book does require some mathematically background - though not to a great extent - which I guess anyone with enough interest in mathematics to be considering this book will already have. The mathematics presented are obviously difficult but that doesn't mean the book itself is difficult to read because it isn't. Overall this is a very enjoyable and accomplished book and I highly recommend it.
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.