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

by Donal O'Shea

Book Overview

The Poincar Conjecture tells the story behind one of the world's most confounding mathematical theories. Formulated in 1904 by Henri Poincar , his Conjecture promised to describe the very shape of the... This description may be from another edition of this product.

• Format: Paperback
• Language: English
• ISBN: 0802716547
• ISBN13: 9780802716545
• Release Date: January 2008
• Publisher: Walker & Company
• Length: 293 Pages
• Weight: 0.67 lbs.
• Dimensions: 0.8" x 5.8" x 8.3"

Customer Reviews

A great popular math book

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.

The first solution to one of the Millenium problems

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.

A wonderful book!

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.

A stunning book

This stunning book pushed me into new ways of thinking. In two hundred pages, the author tells the story of one of the most beautiful, complicated, and significant mathematical achievements of all time. The narrative blends social history, mathematical exposition and biography, and is supplemented by detailed notes, two glossaries and two indexes, all carefully done and full of unexpected delights. But it is the mathematics that takes center stage, and O'Shea writes in a way that allows you to understand the central mathematical ideas. And what ideas these are!! The universe can, like the surface of earth, have no walls and yet be finite. Angles on triangles need not sum to 180 degrees and only do on surfaces and in spaces that are flat. High school geometry is only a tiny piece of a larger, more satisfying story (no wonder it made so little sense for me). The same surface can carry different geometries and a donut can be flat. Geometric notions such as angles, parallel lines, and distance are separate from topological notions such as dimension, finiteness and how an object connects up. I have never felt more respected by an author. O'Shea clearly believes, and quietly demonstrates, that the Poincaré conjecture and the mathematics that underlies its statement lie within the grasp of every individual. There is no waffling and no purple prose. If he uses a word, he tells you what it means. By the end, you understand why individuals spent their life trying to prove the Poincare conjecture, and why the solution matters so much and the money so little. The mathematical ideas unfold against the main historical events of their time, and a reader who skipped all the mathematics would still be treated with to an unusually lively, and I think important, intellectual history. Passages in the text reflect on, among other things, the rise of the modern university, the American research establishment, nationalism and war, and the nature of mathematical knowledge and proof. The numerous biographical and historical sketches are first rate and the individuals come across as real people. My only caution is that the author's prose style is deceptive: many passages are lovely, but each page contains a *lot* of information. Read ten pages in a sitting, and you find yourself approaching information overload. The book is accessible, but more demanding than a quick perusal would suggest. Treat it as you would a very rich meal - carefully, without consuming too much too quickly. The force of the narrative and the sheer power and reach of the ideas leave one both sated and longing for more.

A Magnificent Story

O'Shea's beautiful and sweeping account of a thousand years of mathematics and mathematicians climaxing in Perelman's recent proof of the Poincaré conjecture gives a thrilling sense of the grandeur of humanity's progress and potential. It is a magnificent story, and that's how he tells it.