The Poincaré Conjecture:,,In Search of the Shape,,of the Universe

http://www.amazon.com/Poincare-Conjecture-Search-Shape-Universe/dp/080271532X/ref=sr_1_1/104-0029617-0633535?ie=UTF8&s=books&qid=1175111307&sr=8-1
*The Poincare Conjecture: In Search of the Shape of the Universe (Hardcover) *
by Donal O'Shea <http://www.amazon.com/exec/obidos/search-handle-url/104-0029617-0633535?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Donal%20O%27Shea> (Author)
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.)
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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

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The shape we're in

IAN STEWART

The Poincaré Conjecture:

In Search of the Shape

of the Universe

By Donal O'Shea
Print Edition - Section Front



Walker, 292 pages, $34.95

We live in the golden age of mathematics. One by one, the great unsolved problems that have bedevilled the world's mathematicians for centuries have given up their secrets. Powerful new methods, entire theories, come into being so frequently that it is difficult to keep up.

Most of this activity goes on so far behind the scenes that outsiders have no idea that it is happening. But mathematicians are getting much better at telling the outside world of their triumphs. The latest, and arguably the greatest, of these breakthroughs is Grigori Perelman's amazing proof of the Poincaré Conjecture, which belongs to the area of mathematics known as topology: "rubber *** geometry," features that persist however a shape is bent and twisted. Features like holes, knots, boundaries, dimension.

Donal O'Shea, an accomplished topologist, tells the story of Perelman's proof, and what led up to it, for a general audience. Some passages show a distinct journalistic flair, fast-paced and easy on the brain. Others are more challenging, as the author makes a serious attempt to get below the surface and explain the big ideas. The overall framework is historical, a modern teaser to hook the reader's attention, leading into a history of human ideas about geometry from ancient Greece to the present day.

The mathematical problem emerged roughly a century ago from groundbreaking research of the great French mathematician Henri Poincaré (1854-1912). Before Poincaré, topology was a little-visited backwater; by the time he had finished, it was a major part of the mathematical mainstream. Topology is a flexible kind of geometry, a study of what shape things are. But unlike the rigid geometry of Euclid, with its straight lines and circles, topology allows shapes to be bent, stretched or otherwise distorted -- though tearing them is frowned upon.

Every popular science book has to tell a story, and O'Shea has framed his story in terms of the shape of the world. There is an illuminating analogy between Greek geometry and early attempts to deduce the shape of the Earth, and modern topology and today's investigations into the shape of the entire universe. This analogy offers so many opportunities to a writer that it is virtually irresistible, but it has its pitfalls. The main danger is that the framework takes over and distorts the history. The casual reader could be forgiven for thinking that the Poincaré Conjecture emerged from a wish to understand what shape our universe is. The truth is rather different: The main motivation was the internal structure of mathematics, not anything in the outside world. On the whole, the book makes this clear.

Poincaré built on what his predecessors had discovered about two-dimensional shapes: surfaces. The only possible topological surfaces are the sphere, or a sphere with several handles attached. The great Frenchman sought a similar understanding of three-dimensional shapes, but when he tried to adapt the methods to three dimensions he hit an annoying obstacle. He could easily define a three-dimensional analog of the sphere, now known as a 3-sphere. What he could not do, however, was to characterize that shape topologically.

In two dimensions, a sphere is the only surface in which every closed loop can be continuously shrunk until it dwindles to a single point. For all other surfaces, the loop can get caught on a handle, like your finger winding round the handle of a cup. (The cover of the book has a lovely illustration of an elastic band wrapped around an apple.) For a time, Poincaré thought that an analogous property characterized the 3-sphere. In fact, he thought this was obvious, and so made no attempt to prove it.

Later, he realized that an equally plausible characterization along slightly different lines was actually false, so he tried to prove that any three-dimensional shape, in which every loop shrinks to a point, must be a 3-sphere. He failed, and realized that the question is extremely difficult, despite its simple appearance.

Poincaré's question quickly became a conjecture, mathematical jargon for a statement that everyone believes must be true, but which lacks a proof. The Poincaré Conjecture is vitally important in mathematics, but not because it tells us what shape the universe is. It is important because our entire understanding of three-dimensional shapes depends on it. It is a major stumbling block in our methods, and until it is overcome, we can't really get started.

The history of the conjecture is complicated, with modern extensions to higher dimensions, failed attempts at proofs, all manner of related issues. O'Shea tells these stories clearly and well. The historical figures come over as real people. The basic mathematical concepts are explained in detail -- sometimes a little too much detail, perhaps. But the author should be applauded for not ducking the conceptual difficulties. This is not an easy area to write about honestly.

The climax, of course, is Perelman's dramatic proof, confirmed in the summer of 2006. Perelman did not submit his work to a journal. Instead, he posted it on the Web, at http://www.arXiv.org, a site for "preprints," drafts of papers prior to official publication. Ostensibly, he was working on the "Ricci flow," how a topological shape evolves as sharply curved regions try to flatten themselves out. But all the experts knew, thanks to an earlier insight of Richard Hamilton, that taming the Ricci flow would solve the Poincaré Conjecture -- and more, the Thurston Geometrization Conjecture, which describes all possible three-dimensional shapes, not just the humble 3-sphere. And this is what Perelman claimed to have done.

The image of the eccentric genius runs deep in the public perception of mathematicians. Mostly, it's nonsense, but occasionally not. Perelman seems reclusive rather than eccentric; press reports of him becoming disillusioned with mathematics are probably exaggerated. He has shunned the limelight, declining the prestigious Fields Medal and showing no interest in collecting a $1-million prize on offer from the Clay Mathematics Institute. To him, the only true reward is the mathematics itself. Arguably this shows wisdom, not eccentricity.

What about the shape of the universe, the book's subtitle? Poincaré may well have been interested in that question; he was interested in an awful lot of things. But it was far from his mind while he was exploring the landscape of topology. He was after something far more important: how to tell what shape anything is. The universe, or a doughnut, were just examples.

Mathematics creates general tools, which scientists and others use to solve specific problems, and tool-making itself is motivation enough for doing mathematics. And, in fact, mathematical physicists are currently using topological methods to understand the shape of the universe, as the book explains in its proper place.

It is always unfair to review a book by comparing it with another book which exists only in the reviewer's imagination, with the title What He Should Have Written Instead. In this ideal book, page lengths are so malleable that every conceivable side issue can be pursued, and expository difficulties miraculously vanish. But I would like to have seen a bit more about the beautiful geometrical ideas in Perelman's proof, and I would have been willing to forgo some of the earlier history to make room. Be that as it may, The Poincaré Conjecture makes one of the most important developments in today's mathematics accessible to a wide audience, and it deserves to be widely read.

Ian Stewart is professor of mathematics at the University of Warwick, in England. His recent books include Letters to a Young Mathematician and Why Beauty is Truth.