chapter 1 The Poincaré Conjecture: In Search of the Shape of the Universe, meaning The Poincaré Conjecture: In Search of the Shape of the Universe, genre The Poincaré Conjecture: In Search of the Shape of the Universe, book cover The Poincaré Conjecture: In Search of the Shape of the Universe, flies The Poincaré Conjecture: In Search of the Shape of the Universe, The Poincaré Conjecture: In Search of the Shape of the Universe fdc355a550e1e Henri Poincar 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 Poincar Conjecture Lies At The Heart Of Modern Geometry And Topology, And Even Pertains To The Possible Shape Of The Universe The Conjecture States That There Is Only One Shape Possible For A Finite Universe In Which Every Loop Can Be Contracted To A Single PointPoincar 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 Equivalent Of A Nobel Prize, In August He Also Will Almost Certainly Share A Clay Institute Millennium AwardIn Telling The Vibrant Story Of The Poincar 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 Conjecture

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

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 out Well, 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 down.On 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 antiquity 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 logicallypossiblethat the space we live in might be non Euclidean, it actually happens to betrueAnother 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 quotations 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 technique 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 community.At 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 sequel, but oddly enough it happens to be real life Stranger than fiction, you know.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 question 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 quote 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 question 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.

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.

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 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

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.

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

Henri Poincar enuncio una congettura E possibile che il gruppo fondamentale di una varieta sia l identita , ma che la varieta in questione non sia omeomorfa alla sfera tridimensionale A chi non ha studiato matematica, tante delle parole presenti nella congettura di Poincar siamo a cavallo tra il 1800 e il 1900, non dicono niente Ma sfido chiunque a non rintracciare in esse un qualche fascino.Questa congettura e legata ad una domanda che, se possibile, e ancora piu affascinante qual e la forma dell universo Alla fine del libro si dice che la maggior parte degli studi attualmente esistenti dimostrerebbe che l universo e piatto, ovvero la sua curvatura e praticamente nulla di sicuro non e mai negativa.Il libro di O Shea e la storia del pensiero matematico da Euclide fino al russo Perelman che, nel 2002, ha dimostrato la congettura del matematico francese Una storia che scalda per il sapore romantico di qualche passo che appassiona per le sfide che racconta.Una cavalcata impressionante che coinvolge per i toni con cui l autore descrive le scoperte susseguitesi dall enunciato, attraverso i passi fatti nella topologia e nella geometria negli anni, fino alla recentissima dimostrazione.E una storia bella Che parla di come i matematici si siano lasciati nei secoli il testimone anche i piu eruditi matematici babilonesi non sarebbero stati in grado di risolvere problemi che oggi vengono affrontati con estrema facilita da chi ha imparato la matematica nelle scuole superiori Noi abbiamo adesso strumenti che prima erano impensabili e non si parla di computer o calcolatrici, ma di progressi fatti anche grazie ai primi passi mossi proprio da quei babilonesi Noi lasceremo alle generazioni future i mezzi per spingersi dove noi, ora, nemmeno possiamo immaginare.La matematica non e fredda I numeri sono davvero come una poesia perche , proprio come le poesie, sono scoperti e scritti dalla passione vedi Bruno D A in Matematica, stupore e poesia

Quando Grigori Perelman rifiut il milione di dollari che il Clay Institute gli aveva assegnato per la dimostrazione della Congettura di Poincar , la notizia raggiunse le prime pagine di tutti i giornali Non che la gente sapesse che diavolo fosse questa congettura, a dire il vero ma l idea di tutti quei soldi li stuzzicava Fortunatamente ci sono stati alcuni matematici che hanno pensato non tanto di raccontare la dimostrazione quanto di riuscire a dare uno sguardo generale sui temi trattati, per dare almeno un idea di quello di cui si stava parlando Donal O Shea ci riuscito benissimo con questo suo libro dopo l incipit molto americano ero un po prevenuto, ma lo stile del resto dell opera molto chiaro, e conduce man mano il lettore a capire il contesto in cui il problema nacque e fior , comprese le implicazioni con la relativit generale il tutto con un ampio apparato di note utili per chi volesse saperne di pi In fin dei conti la congettura di Poincar parla anche del nostro universo afferma infatti che se il nostro universo non infinito e si comporta come pensiamo faccia allora in un certo senso l equivalente quadridimensionale di una sfera Servir a qualcosa Probabilmente no, ma la matematica non si preoccupa certo della cosa La traduzione scorrevole, ma in qualche punto non matematico, a dire il vero mi ha dato l idea di essere stata tirata un po via, come nelle poesie in cinque versi che probabilmente sono limerick Troppa semplicit fa male

This was a decent book, but a bit of a hard read.Firstly, the book introduces many concepts by name, with some short descriptions, and then goes on to discuss them in some qualitative 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 language.Secondly, there are too many sections where names and dates and attempted proofs of such and such a conjecture theory etc are listed in these sections it very much feels like the only people who would be able to pull much meaning would be already quite familiar with the topics There is much of this in the last third or quarter of the book.The 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 family ancestry and the times and events their family lived through.