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Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles | 
| Author: George G. Szpiro
Publisher: Plume
Category: Book
List Price: $15.00
Buy New: $4.95
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Avg. Customer Rating:  12 reviews
Sales Rank: 410800
Media: Paperback
Edition: Reprint
Number Of Items: 1
Pages: 320
Shipping Weight (lbs): 0.5
Dimensions (in): 8.1 x 5.5 x 0.7
ISBN: 0452289645
Dewey Decimal Number: 514
EAN: 9780452289642
ASIN: 0452289645
Publication Date: July 29, 2008
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Product Description
The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it
In the tradition of Fermats Enigma and Prime Obsession, George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri Poincare developed the Poincare Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldnt prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found.
Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.
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| Customer Reviews: Read 7 more reviews...
 Where are the Pictures? August 6, 2008
A story that I wanted to love. Unfortunately, the complete lack of illustrations left me increasingly in the dark. I'm very surprised that an editor would not have insisted on their inclusion in a book clearly marketed to the great unwashed.
 interesting book May 17, 2008
21 out of 23 found this review helpful
I am a mathematician/statistician and thoroughly enjoyed the book. The author George Szpiro writes a great story that is fascinating reading. Szpiro is a very well-qualified person to write this book as he holds a masters degree from Stanford and a PhD in mathematical economics from the Hebrew University. Dr. Grigori Perelman is generally created with solving a 100 year old problem that is eligible for the Clay Prize and actually had a great deal to do with his being awarded a Field's medal. Although this is about high level theoretical mathematics it is a historical account written for the general public and very understandable to general audiences.
As he usually does Dr. Lee Carlson has given a very detailed review on amazon for this book and discuss in length issues about whther or not Perelman's work really proves the conjecture. But Perelman is an odd character. He has divorced himself from the mathematical community and refuses to publish his work which is a requirement for th 1 million dollar Clay Prize! It is hard to understand why he won't do it. But then again it is also difficult to understand why he is the first and only recipient of the Field's Medal to refuse it! I believe that Szpiro believes as do most mathematicians that the Poincare conjecture is now a theorem and the Perelman is deserving of the Clay Prize. I think Dr. Carlson is a little too harsh in his assessment.
The story also tells of the life and works of Henri Poincare a mathematical genius who lived in the late nineteenth and early twentieth centuries. Poincare's accomplishments are impressive and his conjectures about the n body problem came out of his work that won him the first and only King Oscar award for his solution of the 3 body problem. Poincare's proof had a flaw in it that only he discovered. It was missed by the referee's of the entries in the competition. In the correcting his work and arriving at an interesting and different area, Poincare actually opened the door to Chaos theory and the mathematical subdiscipline of algebraic topology.
I also found very interesting the description of Poincare's earlier work as a mining engineer, a job he apparently like. His first work in that area was to determine the cause of a mining explosion that had cost several coal miners their lives. This was a field that Poincare was soon to abandon for his greater interest in mathematical research.
This is a beautifully written book that is hard to put down once you start it!
 good biographies and imaginative analogies April 3, 2008
This is a book about Poincare's Conjecture, the efforts to establish it as true
or demonstrate its falsity, and the mathematicians involved in those efforts.
The mathematical domain involved is called topology, previously analysis situs.
In two dimensions it is sometimes called rubber sheet geometry. It is about what
is true if the medium is bent, stretched, or compressed, but not torn or glued.
While early work in the field was concrete and easily visualized, such as walking
tours that satisfied various constraints, and relationships between the number
of surfaces, edges, and vertices of a solid, the subject quickly became very
abstract and dealt with things in more than 3 dimensions.
The book contains biographies of many mathematicians that worked on the problem.
Some are brief, and some are the size of magazine articles. Even if you are a
fan of mathematical history, you will probably meet many interesting people you
did not know about, or probably that you have not heard of. Many are noble and
many have feet of clay. Especially in recent decades there are many controversies.
I know none of the facts, but Szpiro seems to be an unbiased and accurate observer.
Many pairs of participants are linked by the PhD advisor to student relation.
This link seems to have led to some of the dubious behavior described. You can find
more such links on the web at the Mathematics Genealogy Project, a joint venture
of North Dakota State University and the American Mathematical Society.
The biographies are intertwined with a description of the problem and the techniques
used on it. This is not a math book. The mathematical descriptions are by
analogy. I did not enjoy the attempted explanations as much as the biographies,
perhaps because of my math degree. But you might disagree. In any case, I won't
blame the author for weak analogies. They are generally imaginative, and as
accurate as I can imagine. The problem is the problem domain. Most of us can not
imagine things like five dimensional bagels.
Overall, the book is good enough that I will try another of Szpiro's works,
and the chances that I'll try to learn more topology were increased by it.
Chronicle of a Conjecture February 16, 2008
1 out of 1 found this review helpful
In 1904, Henri Poincare published a paper in which he asked: " Is it possible that the fundamental group of a manifold be trivial and yet the manifold not be homeomorphic to a sphere ? " and added that " this question would lead us too far astray."
For the next hundred years, mathematicians from different parts of the world chased a solution , sometimes even sacrificing their own careers.
The author begins with the International Congress of Mathematicians that took place in Madrid, Spain on August 22, 2006. It is an occasion when the Fields Medal (equivalent to the Nobel Prize) is awarded to selected brilliant mathematicians. Gregori Perelman, who was one of the medalists for his solution of the Poincare Conjecture, did not show up. The king of Spain had to wait in vain.
Perelman "spent the festive day hidden away in the modest apartment that he shared with his mother in a drab neighborhood of St. Petersburg."
We learn that Perelman is concerned about the ethics in the mathematics community. He says: " Even those who are more or less honest tolerate those who are not."
In the final chapter, the author tells about the million dollar prize by the Clay Institute for anyone who solves the Poincare Conjecture. Will Perelman be awarded ? will he accept ?
The second chapter is about the perception of dimensions. An ant crawling on a basketball thinks that the surface is completely flat. The sailors of Christopher Colombus were afraid they might fall off the edge of what they believed to be a flat world. A ball is a three-dimensional object and its surface is two dimensional. A gentle introduction.
In the next two chapters we get to know more about Poincare. He was trained as a mining engineer. His analytical mind came handy when he investigated a tragic accident in a coal mine, where sixteen people had been killed. Later, Poincare became a professor of math and won an Oscar Prize ( named after king Oscar II of Sweden ) for working on the three-body problem (the stability of the solar system is at stake !).
As I learned from other sources , Poincare was also a president of the Bureau of Longitudes and helped draw the world map for the colonies of the French empire. It is a puzzle that he did not come up with relativity theory after his intensive work on space, time and electrodynamics. One explanation is that he wanted to repair tradition and believed in such things as ether. The anti-authoritarian Einstein succeeded in defeating the Newtonian empire.
The next chapter "Geometry without Euclid" tells us about the origins and the purpose of topology. How to cross all the bridges (once each) of the town Koenigsburg ; how to classify objects according to their cavities , tunnels and twists. What are the betti numbers of pretzels, bagels and balls ?
The rest of the book is about the chronicle of the conjecture. The author tries to help the reader visualize the images of the objects. Manifolds can be imagined as flying carpets in the sky. As Poincare said : " Geometry is the art of reasoning well with badly made figures."
Two objects are topologically equivalent (homeomorphic), if they can be deformed to each other by pulling and creasing and crumpling , without tearing and gluing. A carpet is equivalent to a quilt but not to a poncho.
The Poincare Conjecture can help us figure out the shape of the universe. Are we living on a ball , a bagel or a pretzel ?
lively history; many math errors; where are the pictures? December 21, 2007
6 out of 6 found this review helpful
This book gives a nice account of the history of the various attempts to solve the Poincare conjecture, culminating with its recent proof by Perelman. Compared to the book by O'Shea, the history here seems more interesting and relevant. (Although here too the history is occasionally rambling and boring, at least we aren't subjected to a treatise on the rise of the German university system in the 19th century etc.) We get to meet lots of colorful characters and read many interesting stories about them. The author did an excellent job of interviewing all available people. The human side of mathematical research is very well presented here.
As for the math, although nice analogies are used to describe abstract concepts to the layman, the details are often garbled. Some of the basic mathematical statements made in the book are blatantly wrong. For example, the book states that the Poincare homology 3-sphere is the only homology 3-sphere other than the 3-sphere. (In fact, there are infinitely many different homology 3-spheres, and these comprise an intricate structure which is still being explored in present-day research.) We also learn in this book that the fundamental group of a genus 2 surface is Z^3, and the fundamental group of a genus 3 surface is Z^4. (Any student in an undergraduate topology class should know better.) The list goes on. I suppose that a layperson won't notice these mistakes, and will at least get an idea of what the math is like, modulo details. However there are other mathematical statements which, while not quite wrong, don't make any sense without more explanation. (Oh, and he keeps referring to three-dimensional manifolds as "floating in four-dimensional space", which really muddies the waters.) The description of Ricci flow at the end is quite a bit better than the math in the rest of the book; the acknowledgments indicate that the help of Christina Sormani played a big role here.
The most glaring omission in the book is the pictures. There aren't any. That's right, 300 pages of geometry without a single picture. Granted, professional mathematicians often write research articles in geometry with no pictures, only equations. But there the intended audience has enough knowledge to see the pictures in their mind. For a popular book on geometry to have no pictures is really disappointing. Maybe they were in a rush to get into print.
Conclusion: if you are a layperson who would like to learn about the Poincare conjecture, O'Shea's book is good for the math (which, while difficult to understand at times, is at least correct) and this book is good for the history.
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