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### Summary

### Summary

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) is likely to play a strong role in determining the course of mathematics in the current century. 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 Keith 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.

### Author Notes

Born in England in 1947 and living in America since 1987, Keith Devlin has written more than 20 books and numerous research articles on various elements of mathematics. From 1983 to 1989, he wrote a column on for the Manchester (England) Guardian. The collected columns are published in All the Math That's Fit to Print (1994) and cover a wide range of topics from calculating travel expenses to calculating pi. His book Logic and Information (1991) is an introduction to situation theory and situation semantics for mathematicians.

Co-author of the PBS Nova episode "A Mathematical Mystery Tour," he is also the author of Devlin's Angle, a column on the Mathematical Association of America's electronic journal.

Devlin lives in California, where he is dean of the school of science at Saint Mary's College in Morgana. He is currently studying the use of mathematics to analyze communication and information flow in the workplace.

(Bowker Author Biography)

### Reviews 4

### Booklist Review

Pledged by a wealthy amateur math enthusiast, $1 million per problem awaits whoever can solve the seven problems mathematician Devlin describes in this work. A similar proposition, minus the money, was made in 1900 by the German mathematician David Hilbert, who listed two dozen math mysteries he hoped would be dispelled in the coming century. All but one were, and that one, called the Riemann hypothesis, carries over to the new set of conundrums. The Riemann hypothesis is comprehensible to an advanced high-school math student, thanks to Devlin's clarity as well as his experience in popular exposition as the author of books such as The Math Gene (2000) and NPR's explainer of all things mathematical. As to the rest of the conjectures, Devlin directly states that no one without a doctorate could understand them, let alone crack them. But as a skilled guide pointing out the shape of the problems, and the practical implications of their solutions, Devlin's intriguing book will appeal to the lay reader curious about the abstract frontiers of math. --Gilbert Taylor

### Publisher's Weekly Review

The noble idea that advanced mathematics can be made comprehensible to laypeople is tested in this sometimes engaging but ultimately unsatisfying effort. Mathematician and NPR commentator Devlin (The Math Gene) bravely asserts that only "a good high-school knowledge of mathematics" is needed to understand these seven unsolved problems (each with a million-dollar price on its head from the Clay Mathematics Institute), but in truth a Ph.D. would find these thickets of equations daunting. Devlin does a good job with introductory material; his treatment of topology, elementary calculus and simple theorems about prime numbers, for example, are lucid and often fun. But when he works his way up to the eponymous problems he confronts the fact that they are too abstract, too encrusted with jargon, and just too hard. He finally throws in the towel on the Birch and Sinnerton-Dyer Conjecture ("Don't feel bad if you find yourself getting lost... the level of abstraction is simply too great for the nonexpert"), while the chapter on the Hodge Conjecture is so baffling that the second page finds him morosely conceding that "the wise strategy might be to give up." Nor does Devlin make a compelling case for the real-world importance of many of these problems, rarely going beyond vague assurances that solving them "would almost certainly involve new ideas that will... have other uses." Sadly, this quixotic book ends up proving that high-level mathematics is beyond the reach of all but the experts. (Nov.) (c) Copyright PWxyz, LLC. All rights reserved

### School Library Journal Review

Adult/High School-"The Math Guy" from NPR's Weekend Edition makes a case for why theoretical mathematics should matter to the general public. He devotes a chapter to each of the seven problems that the Clay Institute considers the most important mathematical problems of the century-and is, incidently, offering a million dollar prize to anyone who can solve one. (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

### Choice Review

Mathematicians generally wish they had more public understanding. The obstacle might seem mathematics' intrinsic difficulties, but physicians also must spend years mastering intricate technicalities and everyone appreciates the value of good doctoring. If long, rounded training must precede legitimate medical practice, any commitment to unwinding a mathematical mystery immediately makes one a mathematician; just looking closely initiates the merely curious into the fold. But few choose to look so close. Even so, mathematicians, some among the best, occasionally write books trying to tell the world about what they do. Mazur, a celebrated number theorist and (earlier) topologist, explores mathematical thinking through a topic whose tricky twists leave it just beyond the curriculum, namely, Bombelli's solution of the general cubic equation. Bombelli faced a conundrum: he had an equation, even knew its solutions, even had a formula that promised to yield those solutions, but the inner terms of the formula seemed nonsensical. Resolution required not solutions, already at hand, but rather imaginative interpretation of the intermediate process. Mazur's story concerns the formation of the universal modern image of imaginary numbers filling out the complex plane. Taking very seriously the designation "imaginary'' (otherwise seeming a mere historical artifact), Mazur creates a general meditation on imagination, meaning not mere fancy, but the inner work or play of forming of images, the sometimes slippery, sometimes sublime mental mirroring of worlds seen and unseen. He enlists poets to his cause as he weaves with the sensitivity of a fine literary critic an image of mathematics as a sort of poetry. So doing, he cultivates for mathematicians the sympathies of at least those who already sympathize with poets.A different public image for mathematicians casts them as mental sportsman or athletes, the best scaling previously impregnable mental summits or smashing through once-confining conceptual barriers. In May 2000, the Clay Institute announced seven $1 million prizes, one each for the first solution of seven well-known, important, and very difficult mathematical problems, challenges so difficult that many would view the Institute's money as very safe, at least in the short run. Just as difficult would seem explaining these questions to the public. Some hard problems (e.g., the Goldbach conjecture) will admit a crisp, elementary formulation, but not these. Devlin, a respected expositor, tries hard, but even the mighty fail at tasks. Professional mathematicians with another specialty will hardly glean anything from the chapters about the Hodge conjecture, or the Navier-Stokes equation, much less the general reader. The Riemann hypothesis, which holds pride of place, does admit elementary, if arcane, formulations (e.g., in terms of the Mobius function), which Devlin omits; he does not even show a graph of the Riemann zeta function, the key to the story as he tells it. Great experts can sometimes accomplish miracles of communication; this book needed (at least) seven such.Kappraff weaves a story around those ever-popular mathematical topics tied to attractive visuals: spirals, tessellations, Fibonacci numbers, phyllotaxis, chaos, and fractals. Pickover, for one, has made a whole career out of this genre. Unfortunately, only a small fraction of mathematics lends itself to this treatment, so one tends to see considerable recycling. But this book has a fresh feel, as Kappraff places particular emphasis on mathematical features emergent in cultural artifacts such as musical systems, architecture, and even quilts. He has also unearthed some rather unusual material; e.g., he devotes a chapter to the geometry of the Hebrew alphabet. Though the title reflects his conviction "that things in our lives that mean the most may be beyond measure,'' the material actually supports the ubiquity and effectiveness of mathematics. Where appropriate, the book offers ample calculations and formulae.Mathematical specialties in the eyes of other mathematicians can seem as arcane as mathematics itself viewed by the wider public. Even as mathematicians appeal to the public for more understanding, each mathematical discipline's survival depends on calling students to advanced work. Broadly speaking, a gap divides textbooks from journal literature; mathematical writers mostly presume a motivated reader, so they tend to explain "why'' better than they explain "why bother.'' Invitations to Geometry and Topology boldly pitches various research frontiers to the uncommitted student. Each essay starts with such basics as one might find in an expository monograph, yet advances to recent results. Eight chapters discuss perfect groups (important in K-theory), the word problem, the Fuller index, representations of the general linear group, Morse theory, the Dirac operator, Hermitian geometry, and vector fields on singular spaces. At least half the essays fall within the reach of a strong undergraduate mathematics major. ^BSumming Up: Devlin: Not recommended. All others: highly recommended; upper-division undergraduates. D. V. Feldman University of New Hampshire