Poincare Conjecture: In Search of the Shape of the Universe

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More About This Book

Overview

“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

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In everyday life, conjectures are cheap; in mathematics, they sometimes stimulate years of vigorous theorizing. In the case of Henri Poincaré (1854-1912), the supposition was simply stated. "Consider a compact 3-dimensional manifold V without boundary," he wrote. "Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?" At the time of his death, this pregnant thought was still in gestation. It would take almost a century for a mathematician to offer a persuasive proof of this major conundrum. Written for nonprofessional readers, The Poincaré Conjecture explains why the issue has loomed so large in scientific circles and also provides stimulating capsule biographies of the thinkers who prepared the way for its solution.
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 2006 Reed Business Information.
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Library Journal
In 1904, Henri Poincaré (1854–1912) challenged the world to solve one of the 20th century's most famous mathematical problems, collectively named the Poincaré Conjecture and involving configuring multidimensional space using algebraic topology. Poincaré asked a basic question of what it means for mathematical space to be curved. O'Shea (mathematics, Mount Holyoke Coll.) traces the footsteps of mathematicians like Euclid, Gauss, Riemann, and their contemporaries to prove one of mathematics' greatest puzzles, which eventually culminated with Grigori Perelman's eclectic, brilliant solution in 2003. O'Shea inspires readers to note the beauty, application, and humanity involved with this mathematical journey. Writing for readers with limited mathematical background, O'Shea successfully weaves mathematical proofs with curious insights to tell a great story, along with reams of valuable endnotes and figures. For all mathematicians and academic and larger public libraries.
—Ian D. Gordon

Product Details

  • ISBN-13: 9780802716545
  • Publisher: Walker & Company
  • Publication date: 12/26/2007
  • Edition description: First Edition
  • Pages: 304
  • Sales rank: 533,264
  • Product dimensions: 5.56 (w) x 8.33 (h) x 0.85 (d)

Meet the Author

Donal O’Shea is a professor of mathematics and the dean of faculty and vice president for academic affairs of Mount Holyoke College in Massachusetts. This is his first book for a general audience. He lives in South Hadley, Massachusetts.

Read an Excerpt

The Poincare Conjecture

In Search of the Shape of the Universe
By O'Shea, Donal

Walker & Company

Copyright © 2007 O'Shea, Donal
All right reserved.
ISBN: 080271532X
"Mathematical knowledge builds on the work of those who have gone before us. Any one of us with an elementary school education can solve arithmetic and algebraic problems that would have defeated the most learned Babylonian scribes. Any one of us with a few courses of calculus and linear algebra can solve problems that Pythagoras, Archimedes, or even Newton could not have touched. A mathematics graduate student today can handle topological calculations that Riemann and Poincaré could not have begun. We are not smarter than they. Rather, we are their beneficiaries."

Continues...
Excerpted from The Poincare Conjecture by O'Shea, Donal Copyright © 2007 by O'Shea, Donal. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents


Preface     ix
Cambridge, April 2003     1
The Shape of the Earth     6
Possible Worlds     21
The Shape of the Universe     32
Euclid's Geometry     46
The Non-Euclideans     57
Bernhard Riemann's Probationary Lecture     75
Riemann's Legacy     88
Klein and Poincare     106
Poincare's Topological Papers     122
The Great Savants     137
The Conjecture Takes Hold     151
Higher Dimensions     163
A Solution in the New Millennium     182
Madrid, August 2006     195
Notes     201
Glossary of Terms     241
Glossary of Names     247
Timeline     253
Bibliography     259
Further Reading     271
Art Credits     273
Acknowledgments     275
Index     279

Customer Reviews

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Sort by: Showing all of 2 Customer Reviews

  • Posted May 2, 2010

    Poincare's Conjecture

    Poincare's Conjecture is a mathematical guess about the ultimate shape of the physical universe. "Poincare's Conjecture", the book, is a fondly told story as much about the world of mathematicians and the role of mathematics, as it is the story of the Conjecture and its proof. I found it engrossing, but at the same time I have to say it is not light reading. The foot notes need to be read along with the text as they add much entry level information. I read it three times and found new insight with each reading.
    Henri Poincare lived in the last half of the 19th Century and the first half of the 20th. This was a time when European Universities were competing for prestige and the best mathematicians in a spirit of nationalism. It was a time of seminal mathematical progress. Carl Gauss laid down the laws of electromagnetism, the tools which make generators and electric engines as well as satellite TV possible. Bernhard Riemann developed the concept of space and geometry which revolutionized what 3 and higher dimensioned geometry might be like. Albert Einstein, a young scientist with some useful thoughts of his own about space curvature and the Universe, worked on his theories of relativity during this period. Poincare's conjecture in 1904 asked whether complicated convoluted 3-spaces might ultimately resolve to a simple 3-dimensioned sphere. In Poincare's words, "Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the 3-dimensional sphere?" Manifolds and Geometries having constant curvature are the only spaces which allow motion of rigid bodies whose lengths and angles do not change. The search for a proof also produced some notable people. John Milnor as an undergraduate at Princeton mistook a long standing problem on closed curves as a homework problem and solved it, reminiscent of Matt Damon's character in the movie, "Good Will Hunting". John Nash, the central character in "A Beautiful Mind", solved another longstanding problem, making him famous prior to his bouts with depression.
    Donal O'Shea wrote this book "for the curious individual who remembers a little high school geometry, but not much more". He also reveals his love of the topic. I enjoyed the tour. I would like to take one his classes some day.

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    Posted April 4, 2011

    No text was provided for this review.

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