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"In 2000, the Clay Foundation of Cambridge, Massachusetts, announced a historic competition: Whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 David Hilbert, one of the greatest mathematicians of his day, proposed twenty-three problems, now known as the Hilbert Problems, that set much of the agenda for mathematics in the twentieth century. The Millennium Problems are likely to acquire similar stature, and their solution (or lack of one) will play a strong role in determining the course of mathematics in the current century. They encompass many of the most fascinating areas of pure and applied mathematics, from topology and number theory to particle physics, cryptography, computing and even aircraft design. Keith Devlin, renowned expositor of mathematics, tells here what the seven problems are, how they came about, and what they mean for math and science." "These problems are the brass rings held out to today's mathematicians, glittering and just out of reach. In the hands of Devlin, "the Math Guy" from NPR's Weekend Edition, each Millennium Problem becomes a fascinating window onto the deepest and toughest questions in the field. For mathematicians, physicists, engineers, and everyone else with an interest in mathematics' cutting edge, The Millennium Problems is the definitive account of a subject that will have a very long shelf life."--Jacket.… (more)
User reviews
Devlin organizes the book from most comprehensible to least, beginning with the relatively straightforward Riemann Hypothesis dealing with the distribution of prime numbers (though ease in understanding is not ease in solving; this is the longest-standing of the Problems, first proposed in 1859.) He works his way through the classic algorithmic question of P vs NP and the Poincaré Conjecture (proved in 2006, after the publication of this book) to the utterly opaque Hodge conjecture, where he effectively throws up his hands in despair at even attempting to explain the problem.
The book assumes an interest in math (why else would anyone read it, after all?), but not much knowledge of the field. This was actually rather baffling; Devlin has no qualms about introducing the fundamentals of complex analysis or group theory, but doesn't assume the reader knows or remembers differential calculus. I thus found myself skimming impatiently at some points while being baffled at others. It's probably safe to assume that people who didn't take any math at all in college wouldn't touch this book, so why the coyness about high school calculus?
As a physicist and astronomer by training, it was difficult for me to understand why some of these problems matter. P vs NP is obviously tremendously important to modern computing, and the Navier-Stokes equations of fluid mechanics have clear real-world applications (and their inclusion in the list of problems validated my longstanding opinion that quantum mechanics and general relativity are childs' play compared to the terrors of fluid dynamics), but what will it actually mean, even to mathematicians, whether the Hodge conjecture is true?
This review may seem fairly critical, but overall The Millennium Problems is a very interesting read; the task it tackles may just be nearly as difficult as some of the problems it describes.
The final chapter was intriguing in that it hinted at connnections between differential forms (which I care about) and algebraic geometry (which I've tried to avoid