Reviews elsewhere on the web:
American Scientist
Kevin Houston (pdf)
The Spectator
Mathematical Association of America
London Mathematical Society (pdf)

Donal O'Shea

The Poincare Conjecture

At a first glance at the Poincaré Conjecture it can be difficult to see what all the fuss is about - it seems to be saying something which is faily obvious. So it's useful to have a book such as The Poincaré Conjecture; : In Search of the Shape of the Universe, in which Donal O'Shea explains what the conjecture is really claiming, and why mathematicians have had such a hard time proving it. The book starts with a look at how people deduced the shape of the earth, pointing out that even after it was circumnavigated, they couldn't be sure what would happen if you kept going north - maybe you would go on for ever, or even reappear in the south.

O'Shea then looks at Euclidean geometry, and explains how non-euclidean alternatives were eventually discovered. He describes the work of Riemann on differential geometry, and then gets on to the work of Poincaré including his rivaly with Klein. This is followed by a chapter on the attempts to prove the conjecture in the 20th century, culminating in the success of Grigory Perelman at the start of the 21st

The use of equationsis avoided in this book, but I'm not sure that it's particularly suitable for those without some previous experience of the subject. O'Shea is keen to get across the nature of the work that has been done on the problem, and I would recommend this book to those who know a bit about topology and want to get a glimpse of the more advanced results in the field. info
Paperback 304 pages  
ISBN: 0802716547
Salesrank: 931607
Weight:0.6 lbs
Published: 2008 Walker Books
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Hardcover 304 pages  
ISBN: 1846140129
Salesrank: 1404071
Weight:1.1 lbs
Published: 2007 Allen Lane
Marketplace:New from £4.99:Used from £0.01
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Paperback 304 pages  
ISBN: 0802716547
Salesrank: 786359
Weight:0.6 lbs
Published: 2008 Walker Books
Marketplace:New from CDN$ 5.43:Used from CDN$ 5.43
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Product Description

"O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman."-Wall Street Journal
In 1904, Henri Poincaré, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the Poincaré conjecture, a tantalizing puzzle that speaks to the possible shape of the universe. For more than a century, the conjecture resisted attempts to prove or disprove it. As Donal O'Shea reveals in his elegant narrative, Poincaré's conjecture opens a door to the history of geometry, from the Pythagoreans of ancient Greece to the celebrated geniuses of the nineteenth-century German academy and, ultimately, to a fascinating array of personalities-Poincaré and Bernhard Riemann, William Thurston and Richard Hamilton, and the eccentric genius who appears to have solved it, Grigory Perelman. The solution seems certain to open up new corners of the mathematical universe.