The Poincare Conjecture: In Search of the Shape of the Universe [Paperback]

Donal O'Shea (Author)

Book Description

ISBN-10: 0802716547 | ISBN-13: 978-0802716545 | Publication Date: December 26, 2007 | Edition: 1st

“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.

Editorial Reviews

From Publishers Weekly

The reclusive Russian mathematician Grigory Perelman became a minor media celebrity last summer when he refused the prestigious Fields medal, awarded every four years to a mathematician under the age of 40. Perelman had succeeded in solving the Poincaré conjecture, named for 19th-century French mathematician Henri Poincaré, and which contemporary cosmologists believe has implications for our understanding of the shape of the universe. O'Shea, a professor of mathematics at Mount Holyoke College, begins his account of the long and contentious search for a solution to the puzzle by looking at how we came to understand the shape of the Earth, beginning with the Greeks, in particular Pythagoras and Plato. Writing for generalist science buffs, O'Shea gives a brief course in geometry and in topology and the topological structures called manifolds that are the basis of Poincaré's puzzle. Inexplicably, however, O'Shea doesn't give readers a formal statement of the conjecture itself until well into the book. O'Shea describes mind-bending structures in topology as clearly as most of us can describe a cube, but readers will need to do a little Wikipedia-ing first to find out just what it is they're reading about. Illus. (Mar.)
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. --This text refers to the Hardcover edition.

From Booklist

Euclid's Elements is historically the most popular mathematics book ever written, but one thing about it nagged its readers: its postulate that every line has exactly one line parallel to it. Doubt about the postulate's truth is O'Shea's starting point for this accessible if challenging presentation of a famous problem ultimately rooted in the parallel postulate. The great mathematician Henri Poincare (1854-1912) spent years investigating the implications of non-Euclidian space. Aided by diagrams and analogies, O'Shea, a professional mathematician, explains non-Euclidian spaces, populated by objects technically called manifolds and n-spheres (n means the number of dimensions), which leads to Poincare's conjecture, verbatim: "Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the three-dimensional sphere?" Readers defeated by such language, despite O'Shea's valiant nonnumerical clarity, can yet digest the author's connection of the conjecture to the shape of the universe, the biographical portraits that animate his text, and the drama of the conjecture's proof, announced in 2006. Gilbert Taylor
Copyright © American Library Association. All rights reserved --This text refers to the Hardcover edition.

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Product Details

• Paperback: 304 pages
• Publisher: Walker & Company; 1st edition (December 26, 2007)
• Language: English
• ISBN-10: 0802716547
• ISBN-13: 978-0802716545
• Product Dimensions: 8.3 x 5.8 x 0.8 inches
• Shipping Weight: 9.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (29 customer reviews)
• Amazon Best Sellers Rank: #169,886 in Books (See Top 100 in Books)

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72 of 74 people found the following review helpful:

4.0 out of 5 stars beautiful mathematics, May 8, 2007

By 

Nim Sudo - See all my reviews

This review is from: The Poincare Conjecture: In Search of the Shape of the Universe (Hardcover)

The Poincare conjecture was one of the most beautiful and important unsolved problems in mathematics for the last century. It has recently been solved, in a remarkable story, with the final breakthrough due to Perelman, who was awarded the Fields medal for his work but declined to accept it. The Poincare conjecture concerns the possible shapes of three-dimensional spaces, such as the universe that we live in. This book explains what the Poincare conjecture says, and tells the history of its formulation and proof. There are no equations in the main text (and only a couple in the endnotes), so in principle anyone can read this.

The book does a nice job of motivating the Poincare conjecture, by first discussing the possibilities for the shape of the two-dimensional surface of the earth (before we had explored the whole earth and figured out that it is a sphere), and then discussing the possibilities for the shape of the three-dimensional universe (which is currently unknown). The book also does a good job of explaining what modern geometry is about and how this has drastically changed since Euclid.

There were three things about the book that I didn't like. (Bear in mind that I do topology for a living so I am maybe being too critical here.) First, there is a lot of history, not only of the people who worked on the Poincare conjecture, but also of the institutions and political environment in which they worked. A lot of this seemed to me to have little relevance to the Poincare conjecture and didn't hold my interest. Second, in between these historical asides, the mathematical sections often rush through too much material, in not enough detail to be really understandable to a lay reader. Third, the pictures were subpar. Many of them looked like they were drawn with MacPaint, and are reproduced so small and dark that they are hard to make out. At least one picture is mathematically incorrect: it shows a disc of paper with a wedge cut out being folded to produce a spherical cap, but really one would get a cone instead. This mistake is unfortunate since it contradicts the whole point of the chapter in which it appears. In short, if I were writing this book, I would want to trim the history, remove some of the mathematics, explain the rest of the mathematics in more detail, and improve the pictures.

Perelman's papers finishing off the Poincare conjecture were sketchy, and a lot of work by other mathematicians was required to turn his papers into a detailed proof. There was some (in my opinion silly and unfortunate) controversy in the media regarding how much credit should go to various people for this. The book does not go into this controversy, which I think is a good thing (although it gives some hints without fully explaining the situation). There is also no discussion of why Perelman made the unusual decision to decline the Fields medal. Maybe no one really knows.

63 of 67 people found the following review helpful:

3.0 out of 5 stars I'm Not Sure Who the Real Audience Is for This Book, July 21, 2007

By 

Steve Koss (New York, NY United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: The Poincare Conjecture: In Search of the Shape of the Universe (Hardcover)

Describing some of Henri Poincare's work, author Donal O'Shea writes on page 132 that Poincare "produced an infinite family of closed three-dimensional manifolds that are not homeomorphic to one another and showed that one can have nonhomeomorphic manifolds with the same Betti numbers and, in fact, with the same Betti numbers as a sphere." Oh, so you don't know what a Betti number is? Well, O'Shea's Glossary of Terms describes it as "an integer counting that number of inequivalent manifolds of a given dimension in a manifold that do not bound a submanifold of one dimension higher." If this is not quite your cup of mathematical tea (as it is not quite mine, despite my B.S. in mathematics from a highly-regarded engineering school), then THE POINCARE CONJECTURE might just be a full teapot that you want to skip.

A fundamental rule of nonfiction is to identify and write for your intended audience, and it is difficult to imagine who O'Shea saw here as his audience. THE POINCARE CONJECTURE addresses a mathematically famous but publicly obscure hypothesis from the general field of topology, the study of shapes and curvature. Filled to overflowing with historical background, dating back to Euclid's original five postulates for plane geometry, the main body of O'Shea's book is exclusively textual. Even the hypercritical Ricci flow equation, the main vehicle through which Grigory Perelman achieved his landmark proof, is relegated in abbreviated form to the footnotes. All of this, combined with O'Shea's opening chapters attempting to explain topological manifolds, suggests a book targeted at nonmathematicians. Yet as the excerpt above demonstrates, the author seems too often not to have found the necessary nonmathematical explication to reach successfully a nonprofessional mathematical audience. Conversely, those who are sufficiently mathematically versed to follow O'Shea's elucidation will likely find it far too light in pure mathematical content.

In point of fact, author O'Shea defers even stating or describing the Poincare Conjecture until page 45: "...the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (that is, homeomorphic to) the three-sphere." What follows is 136 pages of interesting mathematical history tracing from Euclid through Gauss, Bolyai, and Lobachevsky to Riemann, Felix Klein, Poincare himself, and then on to David Hilbert, Smale, Thurston, Richard Hamilton, and finally, Perelman. The final eighteen pages attempt in some small way to capture the excitement behind Perelman's revelation of the solution to one of the Clay Institute's seven millennium problems (each with an accompanying million dollar prize), but the conclusion is disappointingly anticlimactic. Perelman being the professional hermit and cipher to which he apparently aspires, O'Shea gives us little sense of the man and his background and virtually no sense of his labors (and, presumably, travails) over this infamous hypothesis. Perelman appears in this book as he appeared before the mathematical community in 2003 - on the stage for too precious few moments and then whisked away, backed to self-selected obscurity in Russia. Compare this to Simon Singh's brilliant treatment of Andrew Wiles and his search for the proof of the so-called Fermat's Last Theorem.

In that same final eighteen pages, O'Shea returns to the "real world" question that his book's subtitle suggests would be answered as a result of proving Poincare's Conjecture: the shape of the universe. He devotes slightly over one page to revealing his answer -- that "the question...is still very much open." The teaser subtitle is just that, a teaser for the scientifically interested and inclined. Gee, thanks.

THE POINCARE CONJECTURE is long on the history of non-Euclidian geometry and the various subdisciplines of topology and unsatisfying short on the actual proof of its eponymous theorem, the peculiar man behind that proof, and the real world implications of the result. The text itself runs 200 pages but offers just short of 100 more space-filling pages larded with footnotes, glossary of terms, glossary of names, timeline, bibliography, further reading, art credits, acknowledgments, and an index. Frankly, the New Yorker's 14-page article about Perelman in August 2006 by Nasar and Gruber was more informative. For this book, a gentleman's 3 stars is the most I can muster.

10 of 11 people found the following review helpful:

5.0 out of 5 stars The first solution to one of the Millenium problems, August 1, 2007

By 

Jaume Puigbo Vila "cosmos fan" (Barcelona, Spain) - See all my reviews
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This review is from: The Poincare Conjecture: In Search of the Shape of the Universe (Hardcover)

Lee Carlson's review casts some doubt about the validity of Perelman's proof, but this is not what the mathematical community of experts is saying. Even the people who have filled in the details of Perelman's proof agree that all the merit is his. As this book shows, Morgan clearly states in his address in the ICM in Madrid that Perelman proved the Poincaré's conjecture and much more (Thurston's conjecture) and introduced new methods that will be used by many mathematicians in the coming years.

O'Shea's book is a good complement to Szpiro's. O'Shea is more encompassing and starts the history of the conjecture going back as far as Babylonic mathematics. It only gives the biography of Poincaré in page 111 and misses some of the details of the controversy provoked by Yau and explained in detail in an article in New Yorker and also in the book by Szpiro. It also has some more technical details, but both books are good reading for a mathematically educated reader.