### CHARACTERS The Poincaré Conjecture In Search of the Shape of the Universe

Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century He revolutionized the field of topology which studies properties of geometric configurations that are unchanged by stretching or twisting The Poincaré conjecture lies at the heart of modern geometry and topology and even pertains to the possible shape of the universe The conjecture My meeting with this book fell considerably short of love at first sight Not saw it on sale yesterday at a Melbourne bookstore and asked if I thought it might be interesting I picked it up glanced at the less than brilliant cover and leafed through it for a minute or two the writing seemed lackluster and the first anecdote I found was one I d seen before I was about to put it back when I reconsidered It cost 10 and was evidently an easy read I d always wondered what the deal was with the mysterious Poincar conjecture Why not find outWell I couldn t have wrong this is a truly excellent book The bare bones of the story are easy to summarize The Poincar conjecture formulated in 1900 by Henri Poincar states cryptically that every simply connected closed 3 manifold is homeomorphic to the 3 sphere It remained an important unsolved problem for about a century until it was proved correct by the reclusive Russian mathematician Grigori Perelman Perelman was awarded two of the most prestigious prizes in mathematics but turned them downOn that description it doesn t sound very interesting but the author makes it come alive he s done a huge amount of background reading on both the mathematics and the history and when he puts it in its historical context you see how fascinating it is Well over half the book is a history of geometry starting from its foundations in antiuity with the Babylonians Pythagoras and Euclid O Shea a cultured mathematician with an intense interest in the history of his subject gives you plenty of material on the Greeks did you know there s a mistake in the proof of Euclid s Proposition 1 then traces how their work was passed through the Arabs to Renaissance Europe En route he finds a delightful way to explain to the non mathematicians what a 3 sphere is it turns out to be the shape of the universe as described in Dante s Divine Comedy two sets of concentric spheres mystically joined at their common surface He illustrates with a famous picture from Dor As he progresses towards the present day he finds opportunities to introduce the other terms that will eventually be used in the Conjecture and the narrative starts to focus in on the key concepts manifolds connectedness topology and above all non Euclidean geometry This is the clearest overview of the subject I ve ever seen and he has a whole bunch of stories and observations I hadn t come across before One thing I found particularly remarkable was the long guerilla war waged by the 19th century German mathematicians against Kant s conceptions of geometry I have had several discussions with philosophically knowledgeable people on this site about Einstein s claim to have refuted Kant What I didn t realize was that it was just the final battle in a campaign that had gone on for a century Gauss laid the groundwork but thought it was so controversial that he couldn t publish at least in Germany it wasn t possible to openly say that Kant was wrong and non Euclidean geometries made perfectly good sense But other great mathematicians Riemann Lobachevsky and Bolyai found the same ideas and they gradually came out in the open Einstein finished it off not only is it logically possible that the space we live in might be non Euclidean it actually happens to be true Another remarkable story from the end of this period is the intense rivalry between the German Klein who I learned married Hegel s granddaughter and the French Poincar a professional duel which so exhausted them that they both suffered nervous breakdowns as a result O Shea who knows both French and German includes lovely uotations from their correspondence By the time we reach 1900 and the formulation of the Conjecture it all makes perfect sense and it s obvious why the problem captivated several generations of top mathematicians I was worried that the last third would be anticlimactic but my fears again turned out to be groundless O Shea hardly loses momentum at all as he goes into the finishing stretch which involves explaining some horribly difficult mathematics once again he finds clever visual analogies to make the esoteric techniue of Ricci flow seem reasonable and intuitive It s obviously impossible to give us the details of Perelman s proof but he successfully conveys both its general outline and the process which led to its acceptance by the world mathematical communityAt the end there is the tantalizing mystery why did Perelman turn down the huge prizes he d won and what was the even larger discovery he hinted at which would make the Poincar conjecture no than a stepping stone If this had been a novel I would have groaned at the author s unsubtle attempt to set up a seuel but oddly enough it happens to be real life Stranger than fiction you know

### REVIEW ✓ PDF, eBook or Kindle ePUB Ò Donal O'Shea

T 2006 He also will almost certainly share a Clay Institute millennium awardIn telling the vibrant story of The Poincaré Conjecture Donal O'Shea makes accessible to general readers for the first time the meaning of the conjecture and brings alive the field of mathematics and the achievements of generations of mathematicians whose work have led to Perelman's proof of this famous conjectu This was a decent book but a bit of a hard readFirstly the book introduces many concepts by name with some short descriptions and then goes on to discuss them in some ualitative detail how one concept leads to another how concepts fail to connect For me at least this was difficult to follow Granted in order to truly understand what is being discussed you would need to understand the mathematics perhaps this is just an insurmountable problem in trying to translate high level and difficult mathematics into lay languageSecondly there are too many sections where names and dates and attempted proofs of such and such a conjecturetheoryetc are listed in these sections it very much feels like the only people who would be able to pull much meaning would be already uite familiar with the topics There is much of this in the last third or uarter of the bookThe middle 85% of the book isn t about the Poincare Conjecture per se In this I would describe the book as the history of mathematicians and mathematics from ancient times to today as told from the point of view of the Poincare Conjecture An analogy might be something like a book that details the life of some famous figure by telling the history of their familyancestry and the times and events their family lived through The Black Ice Score (Parker, ualitative detail how one concept leads to another how concepts fail to connect For me at least this was difficult to follow Granted in order to truly The Black Painting understand what is being discussed you would need to Fire and Desire understand the mathematics perhaps this is just an insurmountable problem in trying to translate high level and difficult mathematics into lay languageSecondly there are too many sections where names and dates and attempted proofs of such and such a conjecturetheoryetc are listed in these sections it very much feels like the only people who would be able to pull much meaning would be already The Forgotten Memoir of John Knox uite familiar with the topics There is much of this in the last third or The Illusionists uarter of the bookThe middle 85% of the book isn t about the Poincare Conjecture per se In this I would describe the book as the history of mathematicians and mathematics from ancient times to today as told from the point of view of the Poincare Conjecture An analogy might be something like a book that details the life of some famous figure by telling the history of their familyancestry and the times and events their family lived through

### Donal O'Shea Ò 6 FREE DOWNLOAD

States that there is only one shape possible for a finite universe in which every loop can be contracted to a single pointPoincaré's conjecture is one of the seven millennium problems that bring a one million dollar award for a solution Grigory Perelman a Russian mathematician has offered a proof that is likely to win the Fields Medal the mathematical euivalent of a Nobel prize in Augus This book was in the mathematics section in the library and I was expecting something mathematics focused Hence I was disappointed by the history lesson this book turned out to be Except for the initial confusion it was a nice read

My meeting with this book fell considerably short of love at first sight Not saw it on sale yesterday at a Melbourne bookstore and asked if I thought it might be interesting I picked it up glanced at the less than brilliant cover and leafed through it for a minute or two; the writing seemed lackluster and the first anecdote I found was one I'd seen before I was about to put it back when I reconsidered It cost 10 and was evident

So – the shape of the universe It’s a giant ball right Especially when you think of its beginning in a big bang But that brings up the awkward uestion of what’s outside the ball Space universe is not infinite It’s believed to be finite but without a boundary It becomes easier to understand this if you consider two dimensional beings living in a spherical the two dimensional surface of a ball universe Th

There was some explanation earlier in the book but later explanation was poor I came away with little understanding of how the Poincare conjecture was solved The book was a disappointment but did provide a reference to book by Jeffrey Weeks that might offer better layman level explanations of topological conce

Why is this book not widely read It's at least as good as books like Fermat's Last Theorem with far mathematical content If any layman wants a glimpse into the world of top level mathematics I cannot recommend a better book

I've been interested in the Millennium problems since I first read about them several years ago It was exciting to

This book was in the 'mathematics' section in the library and I was expecting something mathematics focused Hence

As a recent grad student in mathematics I found this book incredibly interesting It made me want to go on and get my PhD in manifold theory

The fact is I would need infinitive sets of lifes to read all the books I want and another set of infinitive lifes to put into practice everything I read in all the books I would achieve to read in those other infinite sets of lifes certainly an infinite number of books And yet I would need an infinite memory to recall all the things I learn from them and correct maybe all the infinite sets of mistakes I would make during my inf

This was a decent book but a bit of a hard readFirstly the book introduces many concepts by name with some short descriptions and then goes on to discuss them in some ualitative detail; how one concept leads to another; how concepts fail to connect For me at least this was difficult to follow Granted in order to truly understand what is bein

This book was about as painful as reading the book of Genesis its pages mostly comprise a chronological list of mathematicians and so and so's work begot so and so's thesis interspersed with definitions sans explanation or example a group a