Posted: March 1, 2007
Donal O'Shea, dean of faculty and vice president for academic affairs and Elizabeth T. Kennan Professor of Mathematics and Statistics, has written a new study chronicling efforts by mathematicians to solve one of the thorniest problems of their profession. The Poincaré Conjecture: In Search of the Shape of the Universewill hit bookstores March 6. O'Shea's book could not be more timely, as an obscure Russian mathematician has recently solved the Poincaré Conjecture.
O'Shea will read from his new book at 7 pm on Tuesday, March 13, at the Odyssey Bookstore in South Hadley.
Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth centuries. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting.
What is the conjecture? The Poincaré Conjecture lies at the heart of modern geometry and topology and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point.
According to an August 15, 2006, story in the New York Times: "Depending on who is talking, Poincare's Conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
"The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar, and a rabbit's head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere."
In 2000, the Clay Institute of Mathematics offered million-dollar prizes for solutions to seven important problems in mathematics, including the Poincaré Conjecture, which had thus far resisted solution.
Indeed, Poincaré's Conjecture was thought to be impossible to solve until Grigory Perelman, a reclusive Russian mathematician, solved it in several postings on the Internet in 2002. Since then some of the world's greatest mathematicians have been working to verify Perelman's work.
Although he worked in the United States during the 1990s, Perelman solved Poincaré's conjecture while working alone in St. Petersburg. After coming to the United States to give only a handful of lectures at several universities after his solution was made public, Perelman returned to Russia and says he is retired from the mathematic community.
O'Shea's book has already been hailed by a number of reviewers.
"Donal O'Shea has written a truly marvelous book. Not only does he explain the long-unsolved, beautiful Poincaré conjecture, he also makes clear how the Russian mathematician Grigory Perelman finally solved it. Around this drama O'Shea weaves a tapestry of elementary topology and astonishing concepts, such as the Ricci flow, that have contributed to Perelman's brilliant achievement. One can't read The Poincaré Conjecture without an overwhelming awe at the infinite depths and richness of a mathematical realm not made by us," observed Martin Gardner, writer on mathematics and author of The Annotated Alice and Aha! Insight.
The Poincaré Conjecture tells the story of the fascinating personalities, institutions, and scholarship behind centuries of mathematics that have led to Perelman's dramatic proof, and that have, in the process, broadened our understanding of how the universe works. O'Shea brings alive the achievements of Poincaré, Bernhard Riemann, William Thurston, Richard Hamilton, and others whose genius has transformed the field in the last century. He chronicles the dramatic events at the 2006 International Congress of Mathematicians in Madrid, where the eccentric Perelman was awarded a Field Medal--the mathematical equivalent of the Nobel Prize--only to turn it down, claiming he had no interest in the spotlight. The Poincaré Conjectureoffers a glimpse into our collective search for knowledge about the universe.