Donal O'Shea

The Poincare Conjecture

At a first glance at the Poincaré Conjecture it can be difficult to see what all the fuss is about - it seems to be saying something which is faily obvious. So it's useful to have a book such as The Poincaré Conjecture; : In Search of the Shape of the Universe, in which Donal O'Shea explains what the conjecture is really claiming, and why mathematicians have had such a hard time proving it. The book starts with a look at how people deduced the shape of the earth, pointing out that even after it was circumnavigated, they couldn't be sure what would happen if you kept going north - maybe you would go on for ever, or even reappear in the south.

O'Shea then looks at Euclidean geometry, and explains how non-euclidean alternatives were eventually discovered. He describes the work of Riemann on differential geometry, and then gets on to the work of Poincaré including his rivaly with Klein. This is followed by a chapter on the attempts to prove the conjecture in the 20^th century, culminating in the success of Grigory Perelman at the start of the 21^st

The use of equationsis avoided in this book, but I'm not sure that it's particularly suitable for those without some previous experience of the subject. O'Shea is keen to get across the nature of the work that has been done on the problem, and I would recommend this book to those who know a bit about topology and want to get a glimpse of the more advanced results in the field.

Amazon.com info

Paperback 304 pages
ISBN: 0802716547
Salesrank: 316942
Weight:0.6 lbs

Published: 2007 Walker & Company

Amazon price $10.85

Marketplace:New from $5.88:Used from $4.62

Buy from Amazon.com

Amazon.co.uk info

Hardcover 304 pages
ISBN: 1846140129
Salesrank: 244844
Weight:1.1 lbs

Published: 2007 Allen Lane

Amazon price £11.69

Marketplace:New from £10.00:Used from £13.37

Buy from Amazon.co.uk

Amazon.ca info

Paperback 304 pages
ISBN: 0802716547
Salesrank: 149959
Weight:0.6 lbs

Published: 2007 Bloomsbury US

Amazon price CDN$ 12.78

Marketplace:New from CDN$ 8.83:Used from CDN$ 5.88

Buy from Amazon.ca

Product Description
“O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman.”—Wall Street Journal In 1904, Henri Poincaré, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the Poincaré conjecture, a tantalizing puzzle that speaks to the possible shape of the universe. For more than a century, the conjecture resisted attempts to prove or disprove it. As Donal O’Shea reveals in his elegant narrative, Poincaré’s conjecture opens a door to the history of geometry, from the Pythagoreans of ancient Greece to the celebrated geniuses of the nineteenth-century German academy and, ultimately, to a fascinating array of personalities—Poincaré and Bernhard Riemann, William Thurston and Richard Hamilton, and the eccentric genius who appears to have solved it, Grigory Perelman. The solution seems certain to open up new corners of the mathematical universe.

Product Description

“O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman.”—Wall Street Journal
In 1904, Henri Poincaré, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the Poincaré conjecture, a tantalizing puzzle that speaks to the possible shape of the universe. For more than a century, the conjecture resisted attempts to prove or disprove it. As Donal O’Shea reveals in his elegant narrative, Poincaré’s conjecture opens a door to the history of geometry, from the Pythagoreans of ancient Greece to the celebrated geniuses of the nineteenth-century German academy and, ultimately, to a fascinating array of personalities—Poincaré and Bernhard Riemann, William Thurston and Richard Hamilton, and the eccentric genius who appears to have solved it, Grigory Perelman. The solution seems certain to open up new corners of the mathematical universe.

What is the Shape of the Universe? A Three-sphere? *****
Donal O'Shea's "The Poincare Conjecture: In Search of the Shape of the Universe" is about Henri Poincare's conjecture, which is "central to our understanding of ourselves and the universe in which we live." The book is written for "the curious individual who remembers a little high school geometry." The book traces "the history of geometry, the discovery of non-Euclidean geometry, and the birth of topology and differential geometry through five millennia..." What is the shape of the universe? With the proof of Poincare conjecture, we have a "method" to find out whether the universe is three-sphere or not. The method is "by using a complete atlas to check whether every closed loop could be shrunk to a point." "... Space and matter are intimately related, and the assertion that the universe has an infinite amount of matter causes serious theoretical problems ... The universe could have a boundary of some kind ... Regarding the size and shape of the universe, we are almost in precisely the same position that Columbus was in 1492 ... there was no complete atlas of the Earth in Columbus's time, there is no complete atlas of the universe today. If we left the Earth on a very fast spaceship, headed out in a fixed direction ... after a very long time, most cosmologists and mathematicians believe, we would come back close to where we started." "... a two-dimensional manifold is a mathematical object that shares a key property with the surface of our earth [... all regions can mapped onto on a piece of paper] ... The corresponding mathematical object that models our universe is a three-dimensional manifold, or thee-manifold. It is a set in which every point belongs to a region that can be mapped onto the points inside a clear aquarium or shoebox. In other words, the region around any point looks like space rather than a plane ... an atlas is a collection of maps that is complete in the sense that every point belongs to some region that is covered by one of the maps. A three-manifold is the object that is covered by all the maps in an atlas ... A three-dimensional manifold is called compact or finite if there is an atlas of it that is finite ... The very simplest finite three-manifold is the three-dimensional sphere, or three sphere." "Over the last century, many individuals have devoted their life's work to furthering our understanding of three-manifold. But ... all efforts ... [arrive] at an answer: Among all those three-manifolds, is there anyone that is different from the three-sphere and that has the property that every path can be shrunk to a point? If there is no such manifold, then we could say for sure whether our universe is a three-sphere by using a complete atlas to check whether every closed loop could be shrunk to a point. The Poincare conjecture states that there is no such manifold. ... the Poincare conjecture is the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (... homeomorphic to) the three-sphere..." "If the manifold is simply connected ( ... every loop can be shrunk to a point), ... Perelman proves that the Ricci flow [analogous to the diffusion of heat]... will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic the original manifold. Arguments that have been known for a long time show that a simply connected manifold with constant positive curvature is necessary the three-dimensional sphere. Therefore, Perelman's work proves the Poincare conjecture."

What is the Shape of the Universe? A Three-sphere? *****

Donal O'Shea's "The Poincare Conjecture: In Search of the Shape of the Universe" is about Henri Poincare's conjecture, which is "central to our understanding of ourselves and the universe in which we live." The book is written for "the curious individual who remembers a little high school geometry." The book traces "the history of geometry, the discovery of non-Euclidean geometry, and the birth of topology and differential geometry through five millennia..."

What is the shape of the universe? With the proof of Poincare conjecture, we have a "method" to find out whether the universe is three-sphere or not. The method is "by using a complete atlas to check whether every closed loop could be shrunk to a point."

"... Space and matter are intimately related, and the assertion that the universe has an infinite amount of matter causes serious theoretical problems ... The universe could have a boundary of some kind ... Regarding the size and shape of the universe, we are almost in precisely the same position that Columbus was in 1492 ... there was no complete atlas of the Earth in Columbus's time, there is no complete atlas of the universe today. If we left the Earth on a very fast spaceship, headed out in a fixed direction ... after a very long time, most cosmologists and mathematicians believe, we would come back close to where we started."

"... a two-dimensional manifold is a mathematical object that shares a key property with the surface of our earth [... all regions can mapped onto on a piece of paper] ... The corresponding mathematical object that models our universe is a three-dimensional manifold, or thee-manifold. It is a set in which every point belongs to a region that can be mapped onto the points inside a clear aquarium or shoebox. In other words, the region around any point looks like space rather than a plane ... an atlas is a collection of maps that is complete in the sense that every point belongs to some region that is covered by one of the maps. A three-manifold is the object that is covered by all the maps in an atlas ... A three-dimensional manifold is called compact or finite if there is an atlas of it that is finite ... The very simplest finite three-manifold is the three-dimensional sphere, or three sphere."

"Over the last century, many individuals have devoted their life's work to furthering our understanding of three-manifold. But ... all efforts ... [arrive] at an answer: Among all those three-manifolds, is there anyone that is different from the three-sphere and that has the property that every path can be shrunk to a point? If there is no such manifold, then we could say for sure whether our universe is a three-sphere by using a complete atlas to check whether every closed loop could be shrunk to a point. The Poincare conjecture states that there is no such manifold. ... the Poincare conjecture is the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (... homeomorphic to) the three-sphere..."

"If the manifold is simply connected ( ... every loop can be shrunk to a point), ... Perelman proves that the Ricci flow [analogous to the diffusion of heat]... will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic the original manifold. Arguments that have been known for a long time show that a simply connected manifold with constant positive curvature is necessary the three-dimensional sphere. Therefore, Perelman's work proves the Poincare conjecture."

Proofread? ***
This book feels as if the author tried to edit it himself, complete with embarassingly frequent mistakes in grammar and punctuation, not to mention horribly botched illustrations. While several of the reviewers here have stated that they weren't satisfied with the mathematical "meatiness" of this book, I represent the lay side that found plenty of challenge following the concepts here (most of which I was seeing for the first time). As such, the histories were welcome asides to the often very long, hard to follow, and dubiously worded (AND poorly illustrated) technical paragraphs. Still, for someone who used this book as an introduction to topology, it was a fascinating read...in parts. If it ever sees another edition that allows for decent editing and proofreading, I imagine I would tack a fourth star onto the review.

Proofread? ***

This book feels as if the author tried to edit it himself, complete with embarassingly frequent mistakes in grammar and punctuation, not to mention horribly botched illustrations.

While several of the reviewers here have stated that they weren't satisfied with the mathematical "meatiness" of this book, I represent the lay side that found plenty of challenge following the concepts here (most of which I was seeing for the first time). As such, the histories were welcome asides to the often very long, hard to follow, and dubiously worded (AND poorly illustrated) technical paragraphs.

Still, for someone who used this book as an introduction to topology, it was a fascinating read...in parts. If it ever sees another edition that allows for decent editing and proofreading, I imagine I would tack a fourth star onto the review.

Interesting and Enjoyable Read *****
I enjoyed reading this book very much and it really opens up my mind. For example, I did not know proving the Poincare conjecture has implications on finding the shape of our universe before I read this book. The author does a good job in introducing the ideas of topology, its history and origin, and of course the Poincare conjecture, to his readers. As a casual reader (I am not a mathematician), I found the level of details and explanation in the book just right to give its readers the basic and intuitive understanding of the mathematics behind. Now I am interested to read more about modern topology!

Interesting and Enjoyable Read *****

I enjoyed reading this book very much and it really opens up my mind. For example, I did not know proving the Poincare conjecture has implications on finding the shape of our universe before I read this book. The author does a good job in introducing the ideas of topology, its history and origin, and of course the Poincare conjecture, to his readers. As a casual reader (I am not a mathematician), I found the level of details and explanation in the book just right to give its readers the basic and intuitive understanding of the mathematics behind. Now I am interested to read more about modern topology!

A must read! *****
A must read! One of the best book I have ever read! This book taught me more topology than the two university courses. A clear and precisely written history and exposition of one the most important ideas in science.

A must read! *****

A must read! One of the best book I have ever read! This book taught me more topology than the two university courses. A clear and precisely written history and exposition of one the most important ideas in science.

Great beginning with a bad end *
This book covers an interesting topic. In the beginning,author's analogies are really splendid, however in the middle and end part of the book author's imagination goes down and is very boring (only historical dates, facts, and people are provided).

Great beginning with a bad end *

This book covers an interesting topic. In the beginning,author's analogies are really splendid, however in the middle and end part of the book author's imagination goes down and is very boring (only historical dates, facts, and people are provided).

A bit too historic **
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out. I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel. This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough. A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture. A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'. In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly. This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

A bit too historic **

I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected... **
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall). This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does: about "betti numbers", and "homologies": -Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold; about the "fundamental group": -Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group. Sure, as I know from the beginning, that all this terms are associated with topology somehow! In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician. If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it! Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!! The book has one good thing though, has lots of references, articles, books and websites. For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Not quite what I expected... **

There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Interesting mix of mathematics and history ****
In The Poincaré Conjecture, Donal O'Shea explains a conjecture in topology from 1904 that had remained unsolved for nearly a century. Aside from its importance in topology, the conjecture also has implications on determining the shape of our own universe. It is also one of the seven Millennium Prize problems listed by the Clay Institute in 2000, with a one million dollar reward for a correct solution. It was finally solved in 2002 by Grigory Perelman and since then his solution has been accepted. He may be eligible for the Millennium Prize but does not appear to be interested. In 2006, he was awarded the Fields medal--the highest honor for mathematicians and which also carries a monetary reward--for his work but he declined the award. In this book, O'Shea takes us through the history of the conjecture and the attempts at solving it, and also takes some time to give us the historical context along the way by describing the social and political climate surrounding each mathematician that has sought to prove the conjecture. He does a good job of providing relatively clear and simple explanations of the complex ideas in topology and non-Euclidean geometry involved, but the book does move at a fairly brisk pace (minus the notes at the end the main text is only 200 pages long) so some work is still required to follow along, but I never felt completely lost. This book contains a nice mix of mathematical ideas and history for a general audience, and it managed to keep my interest throughout.

Interesting mix of mathematics and history ****

In The Poincaré Conjecture, Donal O'Shea explains a conjecture in topology from 1904 that had remained unsolved for nearly a century. Aside from its importance in topology, the conjecture also has implications on determining the shape of our own universe. It is also one of the seven Millennium Prize problems listed by the Clay Institute in 2000, with a one million dollar reward for a correct solution. It was finally solved in 2002 by Grigory Perelman and since then his solution has been accepted. He may be eligible for the Millennium Prize but does not appear to be interested. In 2006, he was awarded the Fields medal--the highest honor for mathematicians and which also carries a monetary reward--for his work but he declined the award.

In this book, O'Shea takes us through the history of the conjecture and the attempts at solving it, and also takes some time to give us the historical context along the way by describing the social and political climate surrounding each mathematician that has sought to prove the conjecture. He does a good job of providing relatively clear and simple explanations of the complex ideas in topology and non-Euclidean geometry involved, but the book does move at a fairly brisk pace (minus the notes at the end the main text is only 200 pages long) so some work is still required to follow along, but I never felt completely lost. This book contains a nice mix of mathematical ideas and history for a general audience, and it managed to keep my interest throughout.