Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles

This book is highly recommended for anyone who is interested to know how the proof of one of the most vexing mathematical conjectures of recent times came about. Mr. Szpiro easily walks the reader through complex and esoteric concepts in algebraic topology without assuming any previous background in the subject. Of course, if you have some familiarity with this field of mathematics, then the more you will get out of this book. What makes this book so engaging is that the author weaves into the story some details of the lives and the idiosyncrasies of the major mathematicians who were involved in the effort to prove the Poincare Conjecture. The writing style just right: it is concise and to the point, which is probably due to the fact that the author is a journalist and mathematician.

I read this book while enjoying my coffee and cinnamon roll at Borders. It describes the famous "Poincare puzzle" that is harder to explain than it is to imagine. To put it simply, is it possible to prove that a sphere is the only three-dimensional object without holes? Both the question and the solution relate to topology, interestingly a branch of mathematics developed almost single-handedly by Poincare. Well, the puzzle was solved by Gregory Perelman, a Russian mathematician but he went far beyond the mere proof of this one problem and actually provided an explanation for the more difficult Geometrisation Conjecture proposed by William Thurston (every 3-dimensional object can be divided into pieces, all of which have geometric structure). Strangely, he devised the explanation and then refused to acknowledge or accept the huge cash prize for his efforts. An excellect overview of mathematics and a wonderful (though brief) biography of the great Poincare and his unbelievable genius is provided as well as attending detail into the strange world of mathematical theory. I recomment this book wholeheartedly!

A delightful story of one of the major problems in mathematics and the numerous people, many Field medalists, that have intervened to solve it. Even if you are not an expert in topology you will get a feeling of the path to the proof via Thurston's geometrization conjecture and Hamilton's Ricci flow to the surgery of Perelman. The general educated reader will enjoy the stories of Smale in Copacabana and Hamilton's string of girlfriends which contrasts with the ascetism of Perelman and the political manouvering of Yau. In short, mathematics is a human endeavour and its practitioners are mortals which have similar passions, defects and excentricities as the rest of us, only they are extremely brilliant and passionate about the Queen of Sciences. Compared with a similar book by O'Shea this goes more directly to the point, whereas O'Shea introduces Poincaré only in page 111 after a very interesting but long detour from Babylon to Klein. Both books are worth reading and complement each other

I'm impressed and delighted at the way that Szpiro has been able to use analogy to provide appealing and memorable mental pictures of some of the deep and technical mathematical ideas. Here's a short excerpt from Chapter 12: "To prove Thurston's Geometrizaton Conjecture, Perelman described a process that would allow...surgery infinitely often for endless time. ... Let us consider the manifold to be the mythological multiheaded Hydra. ... Whenever he chops off a head, the Hydra keeps growing new ones. ... Had she just sprouted heads, Perelman would not have had a problem because spheres eventually go 'pop.' However, he really need to prevent the Hydra from sprouting extra bodies."