by Donal O'Shea
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“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.
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Average Customer Review:
0 of 2 people found the following review helpful:
 What is the Shape of the Universe? A Three-sphere?, 2008-05-02
Donal O'Shea's "The Poincare Conjecture: In Search of the Shape of the Universe" is about Henri Poincare's conjecture, which is "central to our understanding of ourselves and the universe in which we live." The book is written for "the curious individual who remembers a little high school geometry." The book traces "the history of geometry, the discovery of non-Euclidean geometry, and the birth of topology and differential geometry through five millennia..."
What is the shape of the universe? With the proof of Poincare conjecture, we have a "method" to find out whether the universe is three-sphere or not. The method is "by using a complete atlas to check whether every closed loop could be shrunk to a point."
"... Space and matter are intimately related, and the assertion that the universe has an infinite amount of matter causes serious theoretical problems ... The universe could have a boundary of some kind ... Regarding the size and shape of the universe, we are almost in precisely the same position that Columbus was in 1492 ... there was no complete atlas of the Earth in Columbus's time, there is no complete atlas of the universe today. If we left the Earth on a very fast spaceship, headed out in a fixed direction ... after a very long time, most cosmologists and mathematicians believe, we would come back close to where we started."
"... a two-dimensional manifold is a mathematical object that shares a key property with the surface of our earth [... all regions can mapped onto on a piece of paper] ... The corresponding mathematical object that models our universe is a three-dimensional manifold, or thee-manifold. It is a set in which every point belongs to a region that can be mapped onto the points inside a clear aquarium or shoebox. In other words, the region around any point looks like space rather than a plane ... an atlas is a collection of maps that is complete in the sense that every point belongs to some region that is covered by one of the maps. A three-manifold is the object that is covered by all the maps in an atlas ... A three-dimensional manifold is called compact or finite if there is an atlas of it that is finite ... The very simplest finite three-manifold is the three-dimensional sphere, or three sphere."
"Over the last century, many individuals have devoted their life's work to furthering our understanding of three-manifold. But ... all efforts ... [arrive] at an answer: Among all those three-manifolds, is there anyone that is different from the three-sphere and that has the property that every path can be shrunk to a point? If there is no such manifold, then we could say for sure whether our universe is a three-sphere by using a complete atlas to check whether every closed loop could be shrunk to a point. The Poincare conjecture states that there is no such manifold. ... the Poincare conjecture is the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (... homeomorphic to) the three-sphere..."
"If the manifold is simply connected ( ... every loop can be shrunk to a point), ... Perelman proves that the Ricci flow [analogous to the diffusion of heat]... will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic the original manifold. Arguments that have been known for a long time show that a simply connected manifold with constant positive curvature is necessary the three-dimensional sphere. Therefore, Perelman's work proves the Poincare conjecture."
1 of 1 people found the following review helpful:
 Proofread?, 2008-03-27
This book feels as if the author tried to edit it himself, complete with embarassingly frequent mistakes in grammar and punctuation, not to mention horribly botched illustrations.
While several of the reviewers here have stated that they weren't satisfied with the mathematical "meatiness" of this book, I represent the lay side that found plenty of challenge following the concepts here (most of which I was seeing for the first time). As such, the histories were welcome asides to the often very long, hard to follow, and dubiously worded (AND poorly illustrated) technical paragraphs.
Still, for someone who used this book as an introduction to topology, it was a fascinating read...in parts. If it ever sees another edition that allows for decent editing and proofreading, I imagine I would tack a fourth star onto the review.
1 of 1 people found the following review helpful:
Interesting and Enjoyable Read, 2008-01-28
I enjoyed reading this book very much and it really opens up my mind. For example, I did not know proving the Poincare conjecture has implications on finding the shape of our universe before I read this book. The author does a good job in introducing the ideas of topology, its history and origin, and of course the Poincare conjecture, to his readers. As a casual reader (I am not a mathematician), I found the level of details and explanation in the book just right to give its readers the basic and intuitive understanding of the mathematics behind. Now I am interested to read more about modern topology!
1 of 2 people found the following review helpful:
A must read!, 2008-01-22
A must read! One of the best book I have ever read! This book taught me more topology than the two university courses. A clear and precisely written history and exposition of one the most important ideas in science.
3 of 5 people found the following review helpful:
 Great beginning with a bad end, 2007-11-22
This book covers an interesting topic. In the beginning,author's analogies are really splendid, however in the middle and end part of the book author's imagination goes down and is very boring (only historical dates, facts, and people are provided).
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