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Ricci Flow and the Poincare Conjecture
For over 100 years the Poincaré Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincaré Conjecture in the affirmative.
This book provides full details of a complete proof of the Poincaré Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincaré Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate.
The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincaré Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article.
The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem.
With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology.
The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
This book provides full details of a complete proof of the Poincaré Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincaré Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate.
The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincaré Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article.
The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem.
With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology.
The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
- GenresMathematicsScience
521 pages, Hardcover
First published August 14, 2007
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July 3, 2010Sure ....
This is not a review, as I have not started the book and have dubious qualifications to criticize it well, but a reflection on reading the popular books on this topic and similar topics. In fact, for all but the most gifted and dedicated a book such as this is unreadable or at best something close to a life's work to really understand in useful detail, and that is with my assumption that it is very well written. In any case, out of a minimal respect for reality and the thing itself I have not taken it out of the shrink wrap.
This is the real math behind the two popular accounts of the Poincare Conjecture (The Poincare Conjecture and Poincare's Prize). It has been a long time since even a math major could easily appreciate the solution to any long standing famous problem, what might ever have been described as low hanging fruit is long gone. Consider that this book is the product of three years of hard work by two specialists to fill out and validate to their satisfaction the terse but apparently sound work of Grigori Perelman that finally solved this enormously difficult problem. The good news is that as they worked it out, Perelman was very cooperative in answering questions despite being otherwise reclusive, a bit eccentric and at odds with the workings of the mathematical establishment as a whole [with admirable committment shown by his refusing a Fields medal, the closest math equivalent of a Nobel prize, and to submit his work to the formalities that might win him a million dollar Millenium prize (there is a huge separate argument about such prizes even existing and if accepted, how they should be split between one such as Perelman and others who made major recent contributions on which Perelman's work was built):] and that they found his work to be exemplary in style (after understanding a passage, one author notes (in Poincare's Prize) that given the same amount of space he would have written exactly the same, a notable achievement for a writer working alone without a collaborative review process who is not a native writer of English) as well as logically correct. A huge contrast to the many longer and more tortured attempts published through the conventional journals over the years that turned out to be fatally flawed. Even with the help of these experts I do not expect to get very far in this work even starting from a master's degree in math. I expect that there will a substantial following among those with the right doctorates.
There is now a problem with the general public (no condescending quibbling about the boobgeosie or couch potatoes allowed here) and famous unsolved math problems. The last "human readable" (that is, where the statement and to some extent the implications of its truth is readily grasped by a conscientious high school graduate with no special background in mathematics) famous math problem left, now that the Fermat "Last Theorem" and the Four Color Conjecture are considered solved, seems to be the Goldbach conjecture that every even number is the sum of two primes. The Poincare Conjecture is at the borderline of this standard being not too hard to describe (but still requires more description than the Fermat or Four Color problems) to a receptive person but every aspect of its solution rapidly departs from ordinary mathematical experience. The other great unsolved problem (surviving the Hilbert program and the Millenium prize bounty with no end in sight) that commands attention from popular math authors is the Riemann Hypothesis (I read Derbyshire's book). This is burst of publishing might be because it is mentioned in materials about John Forbes Nash (the Beautiful Mind, and don't get me started on the vast shortcomings of that overrated, over-awarded bit of glitzy cinematic distortion, read the book or watch the Nova documentary, A Brilliant Madness) as the problem that might have driven him over the edge in pursuit of a Fields medal, or because it ultimately relates to the deceptively accessible topic of prime numbers?. You can tell me how "human readable" this is: "All the non-trivial zeroes of the Riemann zeta function have real-part = one-half." Most people are not familiar with complex numbers let alone the zeta function, what a trivial zero might be, why anyone might have thought of it, why anyone cares about it, or what it means if the hypothesis is true. (Except for the part about complex numbers, that statement is probably true of most math majors with a bachelor's degree! A deep, focused treatment of number theory is not a typical requirement in today's degree programs. Personally, I think analytical number theory is amazing (that the prime number conjecture of Gauss was proven for one example) and that I should learn more about it. It is generally amazing how often distant seemingly unrelated areas of research come back around to provide critical insight to something central like number theory, as demonstrated by Andrew Wiles in cracking the Fermat problem.) This "problem" is nothing however compared to the problem of validating the massive proofs required for breakthroughs in present day mathematics and dealing with new methods, such as computer assisted proofs. The romantic era of math problems is nearly over, but not soon enough to prevent a reference to the eternal enigma of Fermat's Last Theorem to be uttered by the 24th Century Captain Picard, permanently recorded in an episode of Star Trek, The Next Generation.
This is not a review, as I have not started the book and have dubious qualifications to criticize it well, but a reflection on reading the popular books on this topic and similar topics. In fact, for all but the most gifted and dedicated a book such as this is unreadable or at best something close to a life's work to really understand in useful detail, and that is with my assumption that it is very well written. In any case, out of a minimal respect for reality and the thing itself I have not taken it out of the shrink wrap.
This is the real math behind the two popular accounts of the Poincare Conjecture (The Poincare Conjecture and Poincare's Prize). It has been a long time since even a math major could easily appreciate the solution to any long standing famous problem, what might ever have been described as low hanging fruit is long gone. Consider that this book is the product of three years of hard work by two specialists to fill out and validate to their satisfaction the terse but apparently sound work of Grigori Perelman that finally solved this enormously difficult problem. The good news is that as they worked it out, Perelman was very cooperative in answering questions despite being otherwise reclusive, a bit eccentric and at odds with the workings of the mathematical establishment as a whole [with admirable committment shown by his refusing a Fields medal, the closest math equivalent of a Nobel prize, and to submit his work to the formalities that might win him a million dollar Millenium prize (there is a huge separate argument about such prizes even existing and if accepted, how they should be split between one such as Perelman and others who made major recent contributions on which Perelman's work was built):] and that they found his work to be exemplary in style (after understanding a passage, one author notes (in Poincare's Prize) that given the same amount of space he would have written exactly the same, a notable achievement for a writer working alone without a collaborative review process who is not a native writer of English) as well as logically correct. A huge contrast to the many longer and more tortured attempts published through the conventional journals over the years that turned out to be fatally flawed. Even with the help of these experts I do not expect to get very far in this work even starting from a master's degree in math. I expect that there will a substantial following among those with the right doctorates.
There is now a problem with the general public (no condescending quibbling about the boobgeosie or couch potatoes allowed here) and famous unsolved math problems. The last "human readable" (that is, where the statement and to some extent the implications of its truth is readily grasped by a conscientious high school graduate with no special background in mathematics) famous math problem left, now that the Fermat "Last Theorem" and the Four Color Conjecture are considered solved, seems to be the Goldbach conjecture that every even number is the sum of two primes. The Poincare Conjecture is at the borderline of this standard being not too hard to describe (but still requires more description than the Fermat or Four Color problems) to a receptive person but every aspect of its solution rapidly departs from ordinary mathematical experience. The other great unsolved problem (surviving the Hilbert program and the Millenium prize bounty with no end in sight) that commands attention from popular math authors is the Riemann Hypothesis (I read Derbyshire's book). This is burst of publishing might be because it is mentioned in materials about John Forbes Nash (the Beautiful Mind, and don't get me started on the vast shortcomings of that overrated, over-awarded bit of glitzy cinematic distortion, read the book or watch the Nova documentary, A Brilliant Madness) as the problem that might have driven him over the edge in pursuit of a Fields medal, or because it ultimately relates to the deceptively accessible topic of prime numbers?. You can tell me how "human readable" this is: "All the non-trivial zeroes of the Riemann zeta function have real-part = one-half." Most people are not familiar with complex numbers let alone the zeta function, what a trivial zero might be, why anyone might have thought of it, why anyone cares about it, or what it means if the hypothesis is true. (Except for the part about complex numbers, that statement is probably true of most math majors with a bachelor's degree! A deep, focused treatment of number theory is not a typical requirement in today's degree programs. Personally, I think analytical number theory is amazing (that the prime number conjecture of Gauss was proven for one example) and that I should learn more about it. It is generally amazing how often distant seemingly unrelated areas of research come back around to provide critical insight to something central like number theory, as demonstrated by Andrew Wiles in cracking the Fermat problem.) This "problem" is nothing however compared to the problem of validating the massive proofs required for breakthroughs in present day mathematics and dealing with new methods, such as computer assisted proofs. The romantic era of math problems is nearly over, but not soon enough to prevent a reference to the eternal enigma of Fermat's Last Theorem to be uttered by the 24th Century Captain Picard, permanently recorded in an episode of Star Trek, The Next Generation.
warily-considering
November 5, 2009It'd be fun to try this out occasionally, and see how far I can get with it.
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