Customer Reviews

The Poincare Conjecture: In Search of the Shape of the Universe

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4 star:		(5)
3 star:		(5)
2 star:		(0)
1 star:		(1)

 

 

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.

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.

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.

I myself am not a mathematician, but I am a fan of mathematics in general. I had been following Grigori Perelman's work in the news ever since he gave his lectures at MIT, and had been awaiting a book to cover this amazing story.

The book itself does quite a bit of leg work covering the history behind the Poincare Conjecture and the lives of the key contributors (Gauss, Riemann, Poincare, Klein, etc.). In the first few chapters, the author gives the reader a 'crash course' in topology (as well as talking about how the field of topology came to be), and in the last few chapters, talks about the failed attempts at proving the conjecture by various mathematicians, and finally, of course, the successful attempt by Perelman.

While making my way through this book, it felt like the author was attempting to do too much in too small a space. At exactly 200 pages of text (the last 90 pages or so are footnotes, appendices, and index), it's pretty much a featherweight when you consider the material the author is trying to cover. The book at once tries to be a history lesson, a treatise on the importance of mathematics (or learning in general depending on your interpretation), a short tutorial on topology, and a brief outline of the conjecture and it's proof. Each of these topics is covered in varying detail at the expense of brevity in others. I was shocked to see that talk about Perelman's proof occupied a scant 10 or so pages at the end of the book.

The history lesson, albeit very well researched and nicely written, shouldn't have been the main focus of the book. The author should have spent more time helping the reader gain a better understanding of topology, as well as helping to connect the dots between the various topics (how Perelman solved the problem of singularities in Ricci flows, how Ricci flows help prove Thurston's Geometrization Conjecture, and how Thurston's Geometrization Conjecture implies Poincare's Conjecture).

In all honesty, what I was hoping for was this book to be to Poincare's Conjecture what Simon's Singh's classic Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem was to Fermat's Last Theorem. Although a good try, it still falls short of my expectations.

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.

Donal O'Shea has written a wonderful book. He successfully conveys that we are living currently in the greatest age of mathematics ever seen. He does this by highlighting the recent solution of the Poincare Conjecture by Perelman. If you already know about the Poincare Conjecture and want to know the proof, this is not the book for you. O'Shea is aiming for those who have possibly never even heard of the Poincare Conjecture (in other words, most of humanity). He shows, primarily by recounting a lot of mathematical history, how mathematics is far more conceptual than computational. In particular I liked how he emphasized the profound influence of Riemann in shifting how people do mathematics. One word of warning for readers who were not math majors in college (a warning that should be applied to almost all mathematically oriented books): be willing to skim sections that are too complicated on first reading. This is what most of the pros do. This is a great book.

This book covers both the human side and the mathematical side of the Poincare Conjecture. On the human side, Donal O'Shea covers the historical environment from Euclid to Perelman which provide a background information for all layman readers who do not have any mathematical background. This book avoid all those controversy around Perelman is a good point since those controversy does not contribute anything that can help a layman to understand the soul and importance of Poincare Conjecture. On the mathematical side, this book tries to help a layman to get a "feeling" of Pointcare Conjecture, not an "understanding" of the Pointcare Conjecture. But this "feeling" is still too hard for layman without mathematical background. I suggest all readers who want to have a better "feeling" should read both Abbot's "Flatland" and Week's "The Shape of Space". These two books do not require much mathematical background.

chris tam
Hong Kong

I really enjoyed reading this book. Donal O'Shea is not always clear in explaining difficult mathematical ideas but he gives very good references to fill the gaps and does this in an appropriate manner. In spite of some lack of clarity, the book is really motivating: most important ideas underlying Poincaré conjecture have been explained. Geometrization theory (developed by Thurston) is very beautifully illustrated. After reading this book, one really wants to continue the subject. I recommend this book to any person with a mathematical taste. Everybody who likes mathematics should have this book in his bookshelf.

This is a delightful, passionate book, written in an easily readable style (I read it in one busy week). It does a wonderful job explaining the significance of Perelman's achievement amidst the background of mathematical history. O'Shea moves from Pythagoras and Euclid through Gauss, Riemann, and Klein and finally to Thurtson and Hamilton and the present day. He has wisely chosen to keep it to 200 pages plus extensive endnotes (which you should not skip). The reader should be forewarned that at most 20 of these 200 pages are specifically on Perelman's proof, so you get more of a sense of the accomplishment as embedded in history than you do of the specific techniques employed. However, one cannot help but learn a lot of (a) history, (b) geometry, and (c) Perelman-lore, and therefore the trip is most definitely worth it.

The book is excellent and worth rereading. But Amazon should fix those illustrations which are too small that enlarge into pixelated nothings. Is geometry spelled with acute first e intentionally to denote the structure rather than the area of study? If so it isn't clear and I never saw it elsewhere. Does the printed book have this? And fix the years which start with i instead of 1.