Comments on: Perelman and the Poincare Conjecture http://www.math.columbia.edu/~woit/wordpress/?p=77 Tue, 24 Nov 2009 05:18:29 -0500 http://wordpress.org/?v=2.8.6 hourly 1 By: Not Even Wrong » Blog Archive » ICM 2006 http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-5142 Not Even Wrong » Blog Archive » ICM 2006 Fri, 30 Sep 2005 20:39:04 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-5142 [...] The International Congress of Mathematicians (ICM) takes place every four years and is the most prestigious international conference in mathematics. The 2006 ICM will take place next August in Madrid. One thing that happens at each ICM is the announcement of the winners of the Fields Medal. This has traditionally been considered the most prestigious award in mathematics, and the closest analog to a Nobel prize in math, although the recently instituted Abel prize may now compete for this honor. The Fields medal is awarded to between two and four people at each ICM, and recipients must be under the age of 40 on Jan. 1 of the year of the ICM. I have no inside information about who will will this year, but in gossip with mathematicians two names that tend to come up are those of Grigori Perelman (for his work on the Poincare conjecture), and Terence Tao. [...] [...] The International Congress of Mathematicians (ICM) takes place every four years and is the most prestigious international conference in mathematics. The 2006 ICM will take place next August in Madrid. One thing that happens at each ICM is the announcement of the winners of the Fields Medal. This has traditionally been considered the most prestigious award in mathematics, and the closest analog to a Nobel prize in math, although the recently instituted Abel prize may now compete for this honor. The Fields medal is awarded to between two and four people at each ICM, and recipients must be under the age of 40 on Jan. 1 of the year of the ICM. I have no inside information about who will will this year, but in gossip with mathematicians two names that tend to come up are those of Grigori Perelman (for his work on the Poincare conjecture), and Terence Tao. [...]

]]> By: Anonymous http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-866 Anonymous Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-866 Here is the link to the scores for that 1982 Olympiad http://www.srcf.ucam.org/~jsm28/imo-scores/1982/scores-order.html Here is the link to the scores for that 1982 Olympiad

http://www.srcf.ucam.org/~jsm28/imo-scores/1982/scores-order.html

]]> By: Anonymous http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-867 Anonymous Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-867 As a non-mathematician, I'd sure like to see this thread grow before it gets replaced by something else. These blogs (the good ones) are like gardens where the new blossoms crowd out the old ones, so you end up with only the topmost flowers. What a waste. This place needs a forum. As a non-mathematician, I’d sure like to see this thread grow before it gets replaced by something else. These blogs (the good ones) are like gardens where the new blossoms crowd out the old ones, so you end up with only the topmost flowers. What a waste.

This place needs a forum.

]]> By: Peter http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-868 Peter Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-868 Ooops. I just accidentally deleted two interesting comments. They pointed out that Perelman was the winner of the Math Olympiad competition in 1982, that if he was under 17 then he would be under 41 in 2006, and that Noam Elkies came in 4th behind Perelman that year. Ooops.

I just accidentally deleted two interesting comments. They pointed out that Perelman was the winner of the Math Olympiad competition in 1982, that if he was under 17 then he would be under 41 in 2006, and that Noam Elkies came in 4th behind Perelman that year.

]]> By: dolt http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-869 dolt Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-869 There is a nice, short article in Notices by John Milnor on the Poincare conjecture and Hamilton's work and a few words on Perelman's results, which I found helpful. http://www.ams.org/notices/200310/fea-milnor.pdf The nice property of Ricci flow equation is that is it like a heat equation (clear in the weak-field approximation, see Bakas). Just as heat flows from hot to cold so object gets uniform in temperature, the Ricci flow behaves similarly so "curvature tries to become more unoform", though there are several complications. My understanding is that Perelman showed how to take care of all that. The article also seems to suggest that the choice of the Ricci flow equation chosen by Hamilton was analogous to Einstein's derivation of his field equation: essentially R_{ij} is the unique 2-index tensor arising naturally from the first and second order derivatives of the metric. Persumably all other terms that can be written (terms of higher order in the Hilbert action in an effective field theory approach) will contain no additional geometric information. I am merely scratching the surface, I am sure, but very interesting stuff... There is a nice, short article in Notices by John Milnor on the Poincare conjecture and Hamilton’s work and a few words on Perelman’s results, which I found helpful.

http://www.ams.org/notices/200310/fea-milnor.pdf

The nice property of Ricci flow equation is that is it like a heat equation (clear in the weak-field approximation, see Bakas). Just as heat flows from hot to cold so object gets uniform in temperature, the Ricci flow behaves similarly so “curvature tries to become more unoform”, though there are several complications. My understanding is that Perelman showed how to take care of all that.

The article also seems to suggest that the choice of the Ricci flow equation chosen by Hamilton was analogous to Einstein’s derivation of his field equation: essentially R_{ij} is the unique 2-index
tensor arising naturally from the first and second order derivatives of the metric. Persumably all other terms that can be written (terms of higher order in the Hilbert action in an effective field theory approach) will contain no additional geometric information.

I am merely scratching the surface, I am sure, but very interesting stuff…

]]> By: dolt http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-870 dolt Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-870 Peter, Many thanks for your patient explanations; you are an invaluable resource! I will try to get a hold of the original references and try to understand it better. At least, now I have a pretty good chance of understanding it! Peter,

Many thanks for your patient explanations; you are an invaluable resource! I will try to get a hold of the original references and try to understand it better. At least, now I have a pretty good chance of understanding it!

]]> By: Peter http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-871 Peter Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-871 Just talked to Richard about this since he was upstairs at tea. He started work on this in the late seventies and didn't hear about the connection to the renormalization group and non-linear sigma models until many years later. He thinks the first time these equations occurred in physics were in Dan Friedan's thesis (1980), which he only heard about years later. Just talked to Richard about this since he was upstairs at tea. He started work on this in the late seventies and didn’t hear about the connection to the renormalization group and non-linear sigma models until many years later. He thinks the first time these equations occurred in physics were in Dan Friedan’s thesis (1980), which he only heard about years later.

]]> By: Peter http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-872 Peter Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-872 As far as I know, the way Hamilton and Perelman are using these equations has nothing to do with the fact that they are approximate renormalization group equations for a non-linear sigma model. (If I see Hamilton maybe I'll ask him if he knew about renormalization group eqs. when he started studying this equation). For them, the important point is just that the topology of the manifold doesn't change as you evolve the metric according to the Ricci flow (at least until you hit a singularity in the solution of the PDE). The idea of the proof of geometrization (drastically over-simplified...) is to show that, no matter what metric you start with, you end up at one of the finite number of possibilities on the list that Thurston conjectured were all the possible topologies of 3d manifolds. Solutions to the Ricci flow eq. certainly do develop singularities, which is one of the things that makes this very hard. As far as I know, the way Hamilton and Perelman are using these equations has nothing to do with the fact that they are approximate renormalization group equations for a non-linear sigma model. (If I see Hamilton maybe I’ll ask him if he knew about renormalization group eqs. when he started studying this equation). For them, the important point is just that the topology of the manifold doesn’t change as you evolve the metric according to the Ricci flow (at least until you hit a singularity in the solution of the PDE). The idea of the proof of geometrization (drastically over-simplified…) is to show that, no matter what metric you start with, you end up at one of the finite number of possibilities on the list that Thurston conjectured were all the possible topologies of 3d manifolds. Solutions to the Ricci flow eq. certainly do develop singularities, which is one of the things that makes this very hard.

]]> By: dolt http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-873 dolt Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-873 I read with great interest the talk of Bakas and tried to understand Perlman's papers. I have a basic question that perhaps somebody can answer. I understand that the Ricci flow equation is nothing but a 1-loop RG equation of the 2-dim (world-sheet) sigma-models with target space metric G_{\mu\nu}(X), i.e., \beta(G_{\mu\nu})= -(Ricci Tensor)_{\mu\nu}. The higher order terms are nonzero, but may be neglected for weak curvature. This Ricci flow equation is the basis for Perelman (and Hamilton's) analysis. How does this 1-loop approximate result become the foundation of studying geometry, especially global toopolgical questions like the Poincare conjecture. Issues like manifold surgery, Thurston geometrization conjecture are definitely way over my head. Thanks I read with great interest the talk of Bakas and tried to understand Perlman’s papers.

I have a basic question that perhaps somebody can answer.

I understand that the Ricci flow equation is nothing but a 1-loop RG equation of the 2-dim (world-sheet) sigma-models with target space metric G_{\mu\nu}(X), i.e., \beta(G_{\mu\nu})= -(Ricci Tensor)_{\mu\nu}. The higher order terms are nonzero, but may be neglected for weak curvature. This Ricci flow equation is the basis for Perelman (and Hamilton’s) analysis.

How does this 1-loop approximate result become the foundation of studying geometry, especially global toopolgical questions like the Poincare conjecture.

Issues like manifold surgery, Thurston geometrization conjecture are definitely way over my head.

Thanks

]]> By: Thomas Larsson http://www.math.columbia.edu/~woit/wordpress/?p=77&cpage=1#comment-874 Thomas Larsson Wed, 31 Dec 1969 19:00:00 +0000 http://www.math.columbia.edu/~woit/wordpress/?p=77#comment-874 Re age limits: I think Nobel's will stipulates that the Prize should go to the person who, within the appropriate field, has made the most important contribution to the benefit of mankind <em>during the past year</em>. One may argue whether academic research really benefits mankind the most - the Swedish inventors organization challenge that, arguing that society benefit more from inventions (like those of AN himself) than from basic research. Be that as it may, it is still hard to argue that your average Nobel laureate did his best work during the last year. Re age limits:
I think Nobel’s will stipulates that the Prize should go to the person who, within the appropriate field, has made the most important contribution to the benefit of mankind during the past year. One may argue whether academic research really benefits mankind the most – the Swedish inventors organization challenge that, arguing that society benefit more from inventions (like those of AN himself) than from basic research. Be that as it may, it is still hard to argue that your average Nobel laureate did his best work during the last year.

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