Timeline for Philosophy behind Mochizuki's work on the ABC conjecture
Current License: CC BY-SA 3.0
52 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2021 at 11:11 | answer | added | Olaf Teschke | timeline score: 7 | |
| Jan 15, 2015 at 1:49 | comment | added | Trent | what is the status of this in 2015? (in december, mochizuki posted a progress report on the verification of universal teichmüller theory / his alleged proof: kurims.kyoto-u.ac.jp/~motizuki/… ) | |
| May 2, 2014 at 17:03 | history | protected | Todd Trimble | ||
| Dec 1, 2013 at 20:59 | comment | added | vzn | more on the behind-the-scenes efforts to verify the proof: paradox of proof by chen on mochizuki attack | |
| Oct 2, 2013 at 13:33 | comment | added | Koushik | "The proof must be correct for if it was not he wouldn't have the ideas to invent them" just joking | |
| Jun 28, 2013 at 1:13 | history | edited | Kaveh |
adding exposition tag
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| Feb 22, 2013 at 18:21 | answer | added | user30304 | timeline score: 17 | |
| Sep 28, 2012 at 15:41 | comment | added | vzn | hi all fyi ABC also has significant implications in CS theory see eg cstheory.stackexchange.com/questions/12504/…, & just want to thank the math community & mathoverflow for keeping this question open to see extended engagement and analysis of the proof by professionals in the field, & hope mathoverflow will be open to further on-topic questions on the subj to facilitate further serious analysis. | |
| Sep 26, 2012 at 14:30 | history | reopened |
Pietro Majer Dmitri Pavlov algori Charles Rezk Gjergji Zaimi |
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| Sep 18, 2012 at 15:59 | history | closed |
Bugs Bunny Todd Trimble Felipe Voloch HJRW Ryan Budney |
not constructive | |
| Sep 17, 2012 at 14:34 | answer | added | Vesselin Dimitrov | timeline score: 37 | |
| Sep 16, 2012 at 15:52 | comment | added | Asaf Karagila♦ | John, the word "natural" has a precise mathematical definition. It does not mean "natural" like the natural world, which mathematically (and perhaps philosophically) is completely not well-defined. Mathematics cannot be done without precision, and the world is completely imprecise. | |
| Sep 15, 2012 at 18:49 | answer | added | Vesselin Dimitrov | timeline score: 91 | |
| Sep 13, 2012 at 20:24 | comment | added | John Sidles | As a striking example of the increasing prevalence of the notion of naturality in contemporary mathematics, Mochizuki’s four preprints employ the word "natural" and its derivatives on more than six hundred separate occasions (for details and related mathematical quotations, see this post on Gödel's Lost Letter and P=NP). | |
| Sep 12, 2012 at 11:02 | comment | added | Alexander Chervov | what-is-the-insight-of-quillens-proof-that-all-projective-modules-over-a-polynom mathoverflow.net/questions/19584/… | |
| Sep 10, 2012 at 2:11 | history | made wiki | Post Made Community Wiki by Kim Morrison | ||
| Sep 9, 2012 at 7:33 | comment | added | David Corwin | While this could hypothetically be a question that only Mochizuki can answer, it could instead be that there is a sketch of an "intuitive proof" known to experts for years before Mochizuki's work. In that scenario, no one knew how to put those intuitive (even wishful) ideas into a rigorous foundation, and Mochizuki developed his theory in part in order to create that foundation. But that proof sketch would be both obscure enough and well-known enough to put on MO. | |
| Sep 9, 2012 at 7:33 | comment | added | David Corwin | I really don't get why anyone thought to close this question. Given that Mochizuki thought his methods might be able to prove the ABC conjecture years before he came up with his (supposed) proof, it seems reasonable to think he might have an intuitive idea of a proof in his mind, and then the years of development of IUTeich were a means of putting those intuitive ideas into rigorous mathematical reality. | |
| Sep 8, 2012 at 18:01 | comment | added | Andy Putman | As discussed on the meta page, I just substantially edited the question to remove extraneous (and tendentious) material. | |
| Sep 8, 2012 at 17:59 | history | edited | Andy Putman | CC BY-SA 3.0 |
deleted 1295 characters in body; edited title
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| Sep 8, 2012 at 8:31 | answer | added | Minhyong Kim | timeline score: 214 | |
| Sep 8, 2012 at 6:22 | answer | added | Pasten | timeline score: 35 | |
| Sep 8, 2012 at 5:01 | history | reopened |
Tom Leinster JBorger Theo Johnson-Freyd Marty Andy Putman |
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| Sep 8, 2012 at 2:04 | vote | accept | James D. Taylor | ||
| Sep 8, 2012 at 0:17 | history | closed |
user631 Benjamin Steinberg Bill Johnson Suvrit Andy Putman |
not a real question | |
| Sep 7, 2012 at 22:31 | comment | added | Emerton | Dear James, Just to echo grp's answer, it is hard to overstate the extent to which most of the number theory community has not engaged with Mochizuki's work before now, and now people are desperately trying to catch up. It will take time before anyone can explain what is really going on. Regards, | |
| Sep 7, 2012 at 22:26 | comment | added | Emerton | ... which ultimately allows him to (in some vague sense, at least) mimic the function field argument. But perhaps this intuition is off, and in any case, it hasn't helped me much in penetrating what he is actually doing. That is going to take hard work! | |
| Sep 7, 2012 at 22:24 | comment | added | Emerton | Not being active on MO anymore, I don't much care if this question survives or not, but I am very interested in understanding more about Mochizuki's argument, so to the extent that insight appear here, I'll happily take advantage of it, and I'm sure others will too. My own sense was that Mochizuki's program has been motivated by trying to get around Faltings's "no go" theorem on arithmetic KS, by constructing a new, non-linear (or perhaps anabelian) interpretation of Hodge theory (both classical and p-adic) and related ideas, leading to a construction of some sort of arithmetic KS map, ... | |
| Sep 7, 2012 at 20:21 | comment | added | Gerhard Paseman | James, I think your question is a reasonable one to ask. However "reasonable" and "appropriate for MathOverflow" are not the same thing. If you wish, we can discuss this further on meta.mathoverflow. Gerhard "Ask Me About Appropriate Asking" Paseman, 2012.09.07 | |
| Sep 7, 2012 at 18:27 | comment | added | Todd Trimble | I strongly agree with Benjamin's comment. I share the concerns of quid, grp, and others, that but for Mochizuki, no one is currently able to answer definitively, and therefore CW it should be (if it even remains open). | |
| Sep 7, 2012 at 18:11 | answer | added | Marty | timeline score: 184 | |
| Sep 7, 2012 at 15:53 | comment | added | James D. Taylor | David, your comments are precisely the type of answer I'm looking for. It is okay that it's 20 years old. | |
| Sep 7, 2012 at 14:26 | comment | added | Benjamin Steinberg | It would seem that only Mochizuki could actually give a correct answer. Anything else would be speculation. Therefore, IMHO this should be a CW question since it cannot have a single correct answer (barring Mochizuki responding). That grp's popular answer was made CW by grp further substantiates this. | |
| Sep 7, 2012 at 13:28 | answer | added | grp | timeline score: 45 | |
| Sep 7, 2012 at 12:37 | comment | added | David E Speyer | @grp I absolutely agree that everything I am talking about is 20 years old, and it cannot be "what is new" in Mochizuki. I do not know what is new; someone else would need to write that and I hope they do. That's why I asked whether this sort of context is useful. I'll try to get together a reply re whether the function field analogy is relevant at some point later. | |
| Sep 7, 2012 at 12:14 | comment | added | grp | @David: The link between ABC and Szpiro's Conjecture (which is the content of the application of the Frey-like construction) long predates Mochizuki's work, and the "function field case" of ABC seems to have nothing to do with the ideas relevant in Mochizuki's work in the number field case much as the "function field case" of FLT is totally irrelevant to the actual proof of FLT. So although each aspect is very interesting for someone who has never heard of the ABC Conjecture, neither of them sheds light on anything that has happened since the time Mochizuki began his work on these matters. | |
| Sep 7, 2012 at 7:00 | comment | added | David E Speyer | The second is that I think the Mochizuki is thinking of the target $\mathbb{P}^1$ as the $j$-line, so that maps from $\mathrm{Spec} \mathbb{Z}$ to it correspond (roughly) to elliptic curves over $\mathbb{Q}$. This is very analogous to the way that introducing an elliptic curve made FLT provable. Are these things you already understand, or would it be useful for me to write them up in more detail? Again, this is all with the caveat that I haven't looked at anything beyond the introductions, and I understand only a little bit of them. | |
| Sep 7, 2012 at 6:57 | comment | added | David E Speyer | @James Taylor I have not made any serious attempt to read the papers. However, I can point you to two things which I think are relevant, based on hints from the introductions. The first is the very easy proof of function field ABC, which turns into an analysis of the possible branching behavior of maps $\mathbb{CP}^1 \to \mathbb{CP}^1$. For number field ABC, the source $\mathbb{CP}^1$ should turn into $\mathrm{Spec}(\mathbb{Z})$ and the target should still be $\mathbb{P}^1$ (continued). | |
| Sep 7, 2012 at 4:55 | history | reopened |
Steven Landsburg Douglas Zare David Corwin Will Sawin Alexander Chervov |
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| Sep 7, 2012 at 2:44 | history | closed |
user9072 Will Jagy Andrés E. Caicedo JSE Daniel Litt |
not a real question | |
| Sep 7, 2012 at 2:32 | comment | added | user9072 | I reply on meta. | |
| Sep 7, 2012 at 2:21 | comment | added | James D. Taylor | @quid: you're being stubborn. Is it not legitimate to ask questions about mathematics that is available but difficult to read and understand? @Kevin: thanks! | |
| Sep 7, 2012 at 2:18 | comment | added | Will Jagy | META tea.mathoverflow.net/discussion/1438/mochizuki-proof-of-abc | |
| Sep 7, 2012 at 2:15 | comment | added | Kevin Ventullo | @James: Have you looked at Remark 1.10.1 in IUTeich Theory IV? He actually states that the computations in the proof of the previous theorem, which seems to be the main theorem from which ABC is derived, were known to him as early as 2000, and actually compares this to the proof of the Weil Conjectures. So it might be worth trying to study the proof of Theorem 1.10 (obviously much easier said than done). | |
| Sep 7, 2012 at 2:13 | comment | added | user9072 | Well, then, read more! And if you do not care enough or lack the appropriate background to do so, I do not see why you need to know this so urgently. If experts become optimistic and understand the thing well enough, expositions will be all around. Just wait | |
| Sep 7, 2012 at 2:05 | history | edited | James D. Taylor | CC BY-SA 3.0 |
deleted 4 characters in body; deleted 21 characters in body
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| Sep 7, 2012 at 2:00 | comment | added | James D. Taylor | @quid: the expositions I've seen (such as kurims.kyoto-u.ac.jp/~motizuki/2010-10-abstract.pdf) are mostly teasers to make people read more. My question is about the sketch underlying the proof of the ABC conjecture, which I don't see evident there. If you have an exposition that you would recommend, I suggest that you write it as an answer. | |
| Sep 7, 2012 at 1:55 | comment | added | James D. Taylor | Correction: "an enthusiastic report". Sorry, Jordan! | |
| Sep 7, 2012 at 1:51 | comment | added | JSE | The suggestion that what's on my blog constitutes even "an extremely vague glimpse" of what Mochizuki is trying to get at is false advertising of the most extravagant kind! | |
| Sep 7, 2012 at 1:17 | comment | added | user9072 | There is some expository material on Mochizuki's website. Did you try to read it already? If not, please, do so first. For lack of any indication of having done so, vote close (as by FAQs 'homework' ought to be done before asking). | |
| Sep 7, 2012 at 1:17 | comment | added | Will Jagy | I'm afraid we do not permit the word "behooves." | |
| Sep 7, 2012 at 1:06 | history | asked | James D. Taylor | CC BY-SA 3.0 |