Timeline for Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?
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8 events
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| Apr 27, 2010 at 12:31 | comment | added | JSE | Yes, this is a good point. I am indeed talking about "Profinite Teichmuller theory." Which, as Andy says, doesn't address the Mumford conjecture; I brought it up just because it has this set-up which is meant to imitate the curve complex within algebraic geometry (and which set-up, as far as I understand, is correct.) | |
| Apr 27, 2010 at 5:20 | comment | added | Andy Putman | I think you all are conflating two unrelated false proofs of Boggi. In one sequence of two papers (on the arXiv and entitled "Relative pro-$\ell$ completions of Teichmüller groups" and "The Teichmüller group is good") he purports to prove the Mumford conjecture. I believe that this is what Jeffrey is talking about. In another paper (entitled "Profinite Teichmuller Theory"), he purports to prove the congruence subgroup property for the mapping class group. This is what JSE is talking about. As far as I can tell, the errors have nothing to do with one another. | |
| Apr 1, 2010 at 18:26 | comment | added | JSE | I'm not sure, but I know that Boggi's paper asserts that identity of profinite groups, that most people believe the identity is correct, and that people are definitely trying to write down a complete proof. | |
| Apr 1, 2010 at 16:26 | comment | added | Jeffrey Giansiracusa | As I understood it once, Boggi's work was meant to go further than simply understanding Harer stability in the algebro-geometric setting. It, together with some work of Looijenga, was meant to lead to a proof of the stable Mumford conjecture. I can't remember if this is related directly or not, but I recall someone once explaining to me that one (Looijenga and Boggi) could prove the Mumford conjecture in an algebraic way if one could first show that the mapping class group is 'good' in the sense that the profinite completion K(MCG,1)^ is equivalent to K(MCG^,1). Anyone know more of this? | |
| Jan 17, 2010 at 14:34 | history | edited | JSE | CC BY-SA 2.5 |
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| Jan 10, 2010 at 14:04 | comment | added | JSE | My understanding is that there are some mistakes (having to do with the passage between Hodge structures with Z-coefficients and Q-coefficients) which leave the proof of the main theorem incomplete, but that the "profinite teichmuller theory" setup is OK. | |
| Jan 10, 2010 at 13:26 | comment | added | Pete L. Clark | Could you elaborate on what you mean by controversial? | |
| Jan 10, 2010 at 13:13 | history | answered | JSE | CC BY-SA 2.5 |