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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 Electing Judges 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

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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 So – the sh The White Nights of Ramadan 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 Beyond the Pale universe The conjecture So – the sh

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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 The fact is I

10 thoughts on “The Poincaré Conjecture In Search of the Shape of the Universe

  1. says:

    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 Poincaré conjecture Why not find out?Well I couldn't have wrong this is a truly excellent book The bare bones of the story are easy to summarize The Poincaré conjecture formulated in 1900 by Henri Poincaré 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 DoréAs 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 Poincaré 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 Poincaré 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

  2. says:

    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 Their universe is finite but has no boundaries There are no edges and if they start off from one point and keep going in the same direction they’ll come back to where they started Our universe is finite and without boundary in the same way If you get on a spaceship and keep going in the same direction eventually you’ll be back in the same neighborhood This one is harder to imagine isn’t it? In the case of two dimensional people living on a sphere we can see how it can be finite but without boundary because we can see how the sphere bends in a third dimension But how is it for our three dimensional universe? There’s no fourth dimension to bend in Reading this book didn’t make it any easier for me to really understand how the universe can be finite but without a boundary All I can do is uote the two dimensional analogy but I’m still a three dimensional earthling But even assuming that the universe is finite and without boundary – is it a three sphere? To go back to the two sphere analogy just because Magellan sailed in the same direction and came back to where he started doesn’t mean that the earth is a sphere It can also be doughnut shaped and the same would still happen No one really knows what the shape of the universe is There’s a lot of evidence for it being flat whatever that means And the Poincare Conjecture It says that a finite no boundary space that is “simply connected” is a three sphere This uestion is obviously of great interest both to mathematicians and to the physicists studying the geometry of the universe We still don’t know if the universe is “simply connected” or not A ball is simply connect but something like a doughnut is not simply connected Unlike Reimann’s Hypothesis the Poincare Conjecture was finally proved after much heartbreak and agony – by an eccentric Russian mathematician named Gregori Perelman who didn’t even accept the award for it The book tells the story of the conjecture and the man who proved it Good pop science and math history

  3. says:

    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 concepts

  4. says:

    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

  5. says:

    I've been interested in the Millennium problems since I first read about them several years ago It was exciting to read about the first one to be solved I never took topology in college though so I have to admit that much of this went right over my head If you wanted to know without reading all the math yes the Poincare conjecture turned out to be true Pretty cool stuff

  6. says:

    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

  7. says:

    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

  8. says:

    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 infinite learning If infinite books available I might not be able to start anew with the first book but having enough infinite time who knows? Life would be infinite even if memory would notI finished this book with a feeling of satisfaction with the great pleasure of having touched albeit with the points of my fingers the fascinating world of topology and geometry and while I want to learn I get the feeling I will not have enough time in this life to grasp this incredible world it's been opened to me to understand all the nuancies not even the most simple ones It is a very sad moment to realise this life is simply much too short to discover all the beauty hidden behind the walls of ignoranceThis is a fascinating book casting the search for a solution for an unsurmountable until Perelman arrived of course and difficult extremely difficult problem for performing the task of solving an open uestion well rather a conjecture posed by Poincaré one of the greatest mathematicians in history in the last page of his last work on topology 'analysis situs' the Poincaré's conjectureI knew little about topology and geometry before reading the book and after reading the book I want as a physicist I have the right to say I was ignorant before reading the book but I remain ignorant as well after the reading and this is uite disatisfyingSo many brilliant minds failed and then out of the blue well a blue which is not at all such looking at the brilliant background of the 'solver' and his career one clear mind Perelman of course came from the cold Russia with modesty and right attitude a bold mind who after solving such Conjecture went back to his cave from where he came from to never show up after saying something similar to 'hey world look what I leave for the future of maths Just some notes for you to read By the way I got solved the Poincaré's conjecture but please leave me alone I was just playing Sudoku Thank you very much'I loved and read with great pleasure the way the author presented the very difficult concepts and math topics to later give a sucint explanation of what was solved by Perelman actually I loved to read the historical overview surrounding the lifes and circumstances the diffculties and disappointments the many great mathematicians suffere the context and the background human and mathematical until a Russian mathematician came to fill in the void If this is not a fascinating story then you really don't have a sense for beauty and the misterious ways you may need to arrive at itAt times reading was difficult Mathematical concepts are not easy to explain for the layman but the author achieves when necessary almost always to use the correct explanation find the correct example or comparison to use the right words a clear mind would need to use and an average mind would need to understand Enlightning and absolutely recommendable

  9. says:

    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

  10. says:

    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 ring etc The highlights were the only occasional example of geometry in mathematical physics or when the author found time to elaborate a little on an interesting property of a certain metric or surface structure In fact the best part of the book is the final two sentences that state for about the 5th time but with the most clarity the thesis of the book as defined by its title Good grief