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

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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!

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.

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

George Szpiro has written a marvellous account of Poincaré conjecture (may be we should call it now Poincaré theorem). It is written for a general audience avoiding any mathematical formula or technicality. Szpiro highlights many important topolgic ideas underlying this problem and relates them to group theory with formidable lucidity. Unfortunately many aspects of this domaine of mathematics have been completely forgotten. Most importantly geometrical ideas underlying this theorem have been bypassed. There is no figures illustrating geometric or topologic details. Other important shortcoming is the lack of references. It would be nice if he gave some references to suitable sources in the course of developing each important idea. Anyway, I enjoyed reading this book but I think it must be complement by Donal O'Shea's book on the same subject.

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!

Written for the "general" audience and therefore essentially free of mathematical formalism, this book offers the reader a history of the work devoted to the resolution of the Poincare conjecture and the language of geometric topology that forms its context. For the more mathematically sophisticated reader, the reading is very fast, and the book can be finished in a few hours. The controversies behind the recent claims that the Poincare conjecture has been resolved by the Russian mathematician Grigory Perelman are discussed in some detail, but only from the standpoint of the author, who is relying on press reports and inside information from mathematicians who have talked to Perelman or knew him in some context. This part of the story is the most troubling, but also the most helpful, for it grants the reader insight into how some people, deemed to be highly intelligent, can behave when confronted with the possibility of eternal fame. There are many in the academy who act in ways that are similar to some of the individuals described by the author in this book: eager to claim credit for work done by others, or seduced by fame to such a degree as to make them unable to function without continuous praise from others.

The book's value thus goes beyond the helpful insights that it brings to the field of geometric topology and geometric analysis, even though these insights are purely descriptive. Several very important ethical issues are brought out and illustrated not by the thought experiments of philosophers but by people that are alive even now, and who occupy the lofty towers of the academy. When can speculate on the motivations of Perelman for acting as he did, and no doubt there will be a large variance in these speculations from each individual reader. But in the opinion of this reviewer without question Perelman acted appropriately when he turned down positions offered to him by a few of the "prestigious" universities in the United States. Perelman does not need them. He does not need the "publish or perish" mentality or any of the artificial "prestige" that they wanted to bestow upon him. Any prizes or faculty positions would not increase Perelman's talent in mathematics, nor would his refusal of them (and an absence of these offerings or prizes does not add to or subtract from one's mathematical ability). If Perelman did solve the Geometrization/Poincare conjecture, his proof is dependent on logic and clever mathematical constructions, intermixed with some of the vagueness of the English language. It does not depend at all on the thoughts or opinions of university administrators, private foundations, or funding agencies.

But did Perelman really resolve the Geometrization/Poincare conjecture? If one studies carefully his preprints, the answer would be no: there is not enough information in them to conclude that a proof has been obtained. This raises the question of whether the extensive notes of John Morgan and others actually serve to fill in the details of the proof. Without careful study of these notes one cannot say conclusively whether they represent a proof, and in this regard one should not take the word of the author, the press, the representatives of the Clay Institute, or indeed of anyone in the mathematical community that claims a proof has been given. Those interested in the validity of any proposed proof in mathematics should study it carefully for themselves, make their own conclusions, and never accept uncritically the opinions of experts who claim the proof is correct. Mathematical cognition is an intensely personal affair with private judgment always superseding the opinions of others. Therefore the truth of the proof of any mathematical result should remain open to those who have not examined it in thorough detail.

And even if it is a valid proof, it is still an open question as to whether the proof is unique, and why the Poincare conjecture in three dimensions requires techniques that are very different than in other dimensions. The author gives the reader some insight into the methodologies used in dimensions other than three. It would be interesting to find a general, unifying methodology that would apply to all dimensions, or find a framework that shows why this cannot happen. It would be fascinating to understand in more detail why the techniques have to be so different, with this understanding going beyond merely saying that there is more "room to maneuver" in higher dimensions. Perhaps those who spent a major portion of their lives in resolving the Poincare conjecture could find consolation in finding such a methodology or framework. Or, better yet, they could refrain from reading the proposed proofs of the conjecture, whether from Perelman or someone else, and consider it to be still open. They then could find their own proof and indulge themselves in the deeply personal and exhilarating journey that is the essence and ethos of mathematical research.

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 ?

This semester, I'm taking a course on topology, just for fun. Hence, I've grown more and more interested in stories about mathematicians who work in the field of topology, and no tale is grander than the race to prove the Poincare Conjecture!

Math enthusiasts will know that it is no longer a conjecture, but a theorem now. It was finally proven for n=3 by Grigori Perelman, an odd Russian genius who chose seclusion and anonymity over fame and glory. The book "Poincare's Prize" is full of stories about mathematicians who succeeded in proving the conjecture for certain dimensions (Smale proved n>4, and Freedman proved n=4), and others who spent the better part of their lives searching for one.

As if the writing wasn't enough, the lives of these mathematicians keep the reader engaged. The story of some of these mathematicians will bound to make the reader smile (people like Stephen Smale), some will evoke unlimited sympathy (all those who failed to find a proof), some will leave the reader angry (like Yau Shing Tung), and finally there is one person who will force the reader to imagine the unconstrained capabilities of the human mind, and that person is Grigori Perelman.

Perelman is the shining star of the book, his ultimate triumph and his withdrawal from mathematics and the media's attention not only makes for a potential Hollywood movie, but also forces the reader to think about the meaning of accolades and prizes, more importantly to ponder the underlying drive to find truth.

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."

This is one of the best books about mathematics that I have ever read.
It beautifully describes the (very sophisticated) math, in everyday, evocative
language, without using any equations, only very apt geometrical metaphors.
It also portrays the human side in vivid and gripping terms. A wonderful
combination of history, math, and drama.