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    • CommentRowNumber101.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2018
    • (edited Jul 31st 2021)

    [ removed ]

    • CommentRowNumber102.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 6th 2018

    Yes, I saw that. I can’t understand Motl’s reasoning here.

    • CommentRowNumber103.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 29th 2020

    Added today’s

    • Taylor Dupuy, Anton Hilado, The Statement of Mochizuki’s Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies, arXiv:2004.13228

    • Taylor Dupuy, Anton Hilado, Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki’s Corollary 3.12, arXiv:2004.13108

    diff, v22, current

  1. Linking to a copy of Scholze and Stix’s article hosted on the nLab (fetched using the wayback machine), as it seems currently to not be openly available on Mochizuki’s website.

    diff, v24, current

    • CommentRowNumber105.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 28th 2021
    • (edited Mar 28th 2021)

    The links one and two are working for me, and I got to them from the page about the meeting.

  2. Adding a section explaining the logic of the final part of the proof, which is fairly straightforward in idea (i.e. if one ignores all the details!).

    diff, v25, current

    • CommentRowNumber107.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 28th 2021

    Added a note that the “theorem” is conditional, lest we give the wrong impression to the over-hasty reader.

    diff, v26, current

  3. Adding a few words on the key corollary, aiming to clarify the relevance of ’indeterminacies’.

    diff, v27, current

    • CommentRowNumber109.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 28th 2021

    I think that perhaps more pointers can be made to the work of Dupuy and Hilado, who teased out a “plain language” shorter path from a deconstructed Corollary 3.12 to weaker versions of Szpiro’s conjecture that are require a less-stringent inequality. The just-so story of pushing around elementary small inequalities between abstract quantities, I don’t find very helpful, because there is approximately ε\varepsilon mathematical content in them. This is no way manages to capture what is going on in and post Corollary 3.12, where there is a lot of various pieces of technology going on.

    • CommentRowNumber110.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 29th 2021
    • (edited Mar 29th 2021)

    The plan is to gradually try to go deeper into it; I have been studying the papers in occasional moments, and am starting to have some feeling for what I think is involved. Just ran out of time for the moment :-).

    Completely agree that what is on the page currently proves nothing! But one of the issues with understanding the papers is of course confusion about how the argument proceeds, so I think there is value in extracting the logic very clearly.

    • CommentRowNumber111.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2021

    Just looking at this page.

    Please let’s edit this part here:

    we shall assume that we have certain data 𝕄\mathbb{M}, which we shall treat as a black box.

    Given 𝕄\mathbb{M}, there are real numbers t 𝕄t_{\mathbb{M}} and q 𝕄q_{\mathbb{M}} such that t 𝕄q 𝕄-t_{\mathbb{M}} \geq - q_{\mathbb{M}}.

    lest we add to the general suspicion that somebody is trying to write a parody.

    If I understand well you are trying here to make tansparent the core structure of the argument, and that’s a great idea. But then this needs pointers to the details, lest it sounds like yet another absurdity.

    I suggest something like this:

    we have certain data 𝕄\mathbb{M} (that M calls … , see Def. xyz in IUT)

    there are real numbers t 𝕄t_{\mathbb{M}} (Def. abc in IUT) and q 𝕄q_{\mathbb{M}} (Def. lmn in IUT).

    Some minimum like this. Best some comment on what tt and qq stand for. Something the poor reader can hold on to.

    • CommentRowNumber112.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 29th 2021
    • (edited Mar 29th 2021)

    Hi Urs, I guess our posts crossed; as I wrote to David, I am planning to work up to this, just ran out of time.

    I do feel that there is little value in repeating Mochizuki’s terminology verbatim; the goal is to gradually drill down to greater detail in more familiar terms. I would say that I think someone trying to read the papers for the first time might well take some time to arrive at what is on the page even now, so I think there is already a little value in it.

    If I understand well you are trying here to make transparent the core structure of the argument

    Yes, exactly. More is coming :-).

    • CommentRowNumber113.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 29th 2021

    I think there is little value in repeating Mochzuki’s cartoon attempts at “explanations” that laboriously illustrate elementary inequalities and the different between ’and’ and ’or’. Much better, as Urs say, to extract real mathematical content, and I feel that Dupuy–Hilado is much more fruitful a source (and Taylor Dupuy has had a reasonable amount of contact with Mochizuki, has a healthy skepticism about the claim, and also has managed, with Anton Hilado, to extract normal-looking mathematics that as far as I can tell is perfectly sound).

    Also, we should adopt Tim Campion’s version of a Frobenioid, if any, since he was able to fix some mistakes in Geometry of Frobenioids I (he might be prevailed on the supply some more details, but I can’t guarantee this).

    • CommentRowNumber114.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 29th 2021
    • (edited Mar 29th 2021)

    Much better, as Urs say, to extract real mathematical content

    Yes, I agree (and, again, this is the goal :-)).

    I feel that Dupuy–Hilado is much more fruitful a source

    Sounds great to add material from Dupuy-Hilado or elsewhere!

    My aim is to give an exposition of those parts of Mochizuki’s work that I have some understanding of. Personally, my interest comes principally from the fact that what Mochizuki does, or tries to do, can be thought of as a kind of algebraic geometry over 𝔽 1\mathbb{F}_1. These ideas are very interesting regardless of the application to the abc conjecture :-).

    At very least, this page has not been significantly edited in nearly 10 years, so what I am adding can hopefully be a spur for others to elaborate upon in any way they wish :-).

  4. Change a confusing notation (c_M didn’t depend on M)


    diff, v29, current

    • CommentRowNumber116.
    • CommentAuthorGuest
    • CommentTimeMar 29th 2021

    I agree it would be better to structure this as an outline of the IUTT papers. Ie, that X fact is being proven in Y page of IUTT. Example:


    • Corollary 3.12: Construction of real numbers t Mt_M and q Mq_M satisfying…


    • Theorem 1.10 : There exists a real number c Mc_M satisfying …

    • Corollary 2.2 : an inequality that implies abc conjecture…

    I didn’t read IUTT, just an example. The current formulation looks a bit ridiculous with the tautological 2.2, redundant clauses in 2.3, etc.

    • CommentRowNumber117.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2021
    • (edited Mar 29th 2021)

    Yes. In it’s current form this material is of the nature which, anywhere else, we would delete as spam. Currently Assumption 2.1 says almost verbatim: “Assuming a black box, there exists at least one real number. Maybe two.”

    Anyone just add a bare minimum: “where tt denotes…”. We don’t need to do it justice, just lift it to the level of sanity.

    • CommentRowNumber118.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 29th 2021

    Not to judge how people spend their time, but is this page really worth so much attention?

  5. Made the briefest of starts at working towards the notion of a pilot object. More to come; patience please :-)! Spent most of the time I had available today on the page field of moduli.

    diff, v30, current

    • CommentRowNumber120.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2021

    Thanks, Richard. No rush, I know it takes time to write decent entries.

    But here we are just lacking a handful of words that would make all the difference and that would just take 5 seconds of your time

    what does t 𝕄t_{\mathbb{M}} and q 𝕄q_{\mathbb{M}} refer to?

    Just say “where t 𝕄t_{\mathbb{M}} is the ABC of a given elliptic curve, while q 𝕄q_{\mathbb{M}} is its KLM”. Something like this.

    Or, if you feel that takes more than 5 seconds, then I suggest to remove the content currently in the entry and develop it in the Sanbox, until it looks more contentful. Because at the moment it looks nonsensical and thus makes a bad impression.

    • CommentRowNumber121.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 30th 2021
    • (edited Mar 30th 2021)

    Hi Urs, the problem is that what they are will be as good as meaningless if I just give the words (feel free to check yourself, the relevant references are given on the page). There are then two ways to proceed: bottom up (begin with familiar things which make sense in absolute terms), which is what Mochizuki does, and which very few have understood, because they get lost; or top down (begin with something which is clear, trivial indeed, logically, and then gradually add content), which I am trying. It is easier and more motivating to do this on the actual page rather than some kind of Sandbox, so I’d ask that an exception be made for the moment in this very unusual case. As you will have seen, I am adding content elsewhere as I work on this.

    Think of it like a Coq program: we are beginning with something that parses but where nearly everything is an axiom/assumption, which later needs to be replaced to say anything contentful.

    We can certainly add an ’Under construction’ note or similar though if desired.

    • CommentRowNumber122.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 30th 2021

    Even keywords that are broken links to entries yet-to-be written are better than an over-simplistic cartoon.

    • CommentRowNumber123.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2021

    Wait. Richard, David R., the entry aside, what is t 𝕄t_{\mathbb{M}} and q 𝕄q_{\mathbb{M}}??

    Surely you must know and be able to say within the number of keystrokes used above?

    But I am bowing out now. I just felt it’s my duty to provide the feedback that the entry as is does a disservice to the reputation of its author and its topic; and to suggest that the place to start drafting articles up to the point where they begin to make sense is the Sandbox, where reader’s won’t take issue with being offered such material.

    • CommentRowNumber124.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 30th 2021
    • (edited Mar 30th 2021)

    t 𝕄t_{\mathbb{M}} is:

    the procession-normalized mono-analytic log-volume of the holomorphic hull of the union of the possible images of a θ\theta-pilot object, relative to the relevant Kummer isomorphisms in the multiradial representation

    q 𝕄q_{\mathbb{M}} is:

    the procession-normalized mono-analytic log-volume of the image of a qq-pilot object, relative to the relevant Kummer isomorphisms in the multiradial representation

    My view is that adding this to the entry adds nothing currently. I am aiming to work down to explaining it, beginning with explaining what pilot objects are, which can be put in reasonably elementary terms.

    I cannot understand the antagonism. I am building up to adding content to a page that has not been edited for 10 years, which may in the best of cases help shed some light on an important work that few have been able to make sense of. I am adding content to other nLab pages in the process. If I am choosing to take the approach I am taking because I think it may ultimately be the most enlightening, can we not be patient and let things rest at that for a while? I will add an ’under construction’ type notice.

    If either of you can do better, you are very welcome to edit the page.

  6. I have now added the cautionary notice.

    • CommentRowNumber126.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 30th 2021

    @Richard I will send you an email.

    • CommentRowNumber127.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 31st 2021

    Clarification of the sense in which Dupuy–Hilado are surveys. They are very, very different to the articles by Mochizuki, Yamashita and Fesenko, in that they read like normal mathematics.

    diff, v32, current

  7. Beginning to add details of the construction of a category Mochizuki denotes F MOD *F_{MOD}^{*}, which is a concrete description of some kind of category of ’arithmetic line bundles’. Not finished yet, but have run out of time just now.

    diff, v33, current

    • CommentRowNumber129.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2021

    Had a look through Scholze’s new commentary.

    This is clear, with the following concise punchlines, in paraphrase:

    • §5: “Much of IUT is a laborious re-derivation of something weaker than Mochizuki himself proved 6 years ago.”

    • §6 “A central notion is a weird terminology for identifying two copies of \mathbb{N}.”

    • §7 “With the effect that the central steps in the main proof become unintelligible.”

    • §8 “The first three parts in the series boil down to a discussion of the group /2\mathbb{Z}/2.”

    • §9 “Previous critique of the proof has not been addressed.”

    There seems to be only one plausible conclusion to be drawn from §6-§9 and that hypothesis finds particularly clear support by §5.

    On this basis I suggest that, at this point, decency demands that however little we have on the nnLab regarding would-be mathematical content of IUT be essentially cleared, leaving only, if unavoidable, pointers to references and to the debate.

    Whatever IUT-inspired mathematical content contributors here feel should be on the nnLab, can easily be kept in other relevant entries.

    Particularly I feel that what the nnLab entry currently shows from “Assumption 2.1” to “Theorem 2.4” is unfortunate and does not help anyone. I understand that the idea is to give some broad-stroke outline of the argument. Such would certainly be good to have, but in its current form I find it does more harm than good.

    • CommentRowNumber130.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 1st 2021

    I totally agree that the nLab does not need to replicate more than the bare minimum here: IUT is aiming for some kind of F 1F_1 theory, and made an idiosyncratic use of category theory that seems to me to be really be out of kilter. At best pointing to the IUT papers, and maybe the early expository note discussing more concrete mathematics, and then the discussions of Scholze–Stix and the preliminary version of Scholze’s zbMath review definitely seems enough. If the discussion from Assumption 2.1 on was followed up at the time with concrete constructive details, and wasn’t trivial, content-free generic implications, then there might be an argument for at best keep a brief discussion. But Scholze’s review does a much better job of extracting what’s actually in there, and summarising it at a high level (and it’s coming from someone who has understood all the background arithmetic/anabelian geometry).

    • CommentRowNumber131.
    • CommentAuthorrongmin
    • CommentTimeAug 1st 2021

    Re. #130:

    IUT is aiming for some kind of 𝔽 1\mathbb{F}_1 theory, and made an idiosyncratic use of category theory that seems to me to be really be out of kilter.

    The above may seem like it’s coming out of left field for some people, so let me elaborate a bit.

    There is now some sort of consensus about what a reasonable theory of the field with one element 𝔽 1\mathbb{F}_1 could be, but back when things were still in flux, Mochizuki gave a talk at a 2004 Tokyo conference (conference website, Youtube video, Mochizuki’s notes) about a proposal that was motivated by the search for such a theory. This proposal later became IUT. Someone pointed this out to David on Twitter, especially the bit about “rings as 2-categories” on the first page of the notes, where Mochizuki sketched out his proposed approach to categorification.

    It’s unclear to me how much this misapprehension (as David saw it) has informed the development of IUT, but it seemed plausible to me that this may be behind the many peculiarities of IUT.

    • CommentRowNumber132.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 1st 2021
    • (edited Aug 1st 2021)

    Yes, I was being brief. There were statements in that 2004 talk by Mochizuki that made me seriously pause. If that was a talk in another part of the world, and the speaker not so respected, there would have been serious questions about some of the mathematical ideas. Instead it seems like the audience was nodding sagely along to statements like having to “solve” “equations” of the form x 1x 2x nx 1x_1\in x_2 \in \ldots \in x_n \in x_1, and how this is impossible in ZFC. There is no way in God’s green earth that statements about arithmetic geometry rely crucially on working around the Axiom of Foundation. There are talks where the speaker is motivating my metaphor, backed up by theorems that show the metaphor is true, and then …

    The event that sparked Rongmin to reminding me of this talk is the fact Mochizuki edited his most recent set of notes back in May to have a long section (1.10) on the legal theory of intellectual property, and how he would like to have some kind of recourse to the reputational damage to IUT based on what he sees as a false, poor imitation by Scholze (this is me paraphrasing, I can’t really bring myself to read it just yet). I find this section troubling for various reasons.

    • CommentRowNumber133.
    • CommentAuthorrongmin
    • CommentTimeAug 2nd 2021
    • (edited Aug 2nd 2021)

    Re. #132:

    Yes, I was being brief.

    That’s why I thought I’d fill in some of the details.

    There were statements in that 2004 talk by Mochizuki that made me seriously pause.

    Yup. That, and not Scholze-Stix, was what did it for me.

    While Scholze-Stix managed to derive a contradiction, that’s only a symptom of a problem. Everyone also seemed to agree that they used a simplified version of IUT, which unsurprisingly became a major point of dispute. With the 2004 talk, we get directly from Mochizuki some of the assumptions that he was working from, and that seems to point to something deep within IUT itself that makes it hard to survive contact with number-theoretic reality.

    it seems like the audience was nodding sagely along to statements

    Not really. There were many folded arms and bewildered looks.

    Mochizuki edited his most recent set of notes back in May to have a long section (1.10) on the legal theory of intellectual property

    This was discussed on Reddit, where functor7 quoted the salient bits.

    how he would like to have some kind of recourse to the reputational damage to IUT based on what he sees as a false, poor imitation by Scholze

    That is why I also agree with Urs in #129 that

    decency demands that however little we have on the nLab regarding would-be mathematical content of IUT be essentially cleared, leaving only, if unavoidable, pointers to references and to the debate.

  8. Removing the experimental sketched outline of the proof logic.

    diff, v35, current

    • CommentRowNumber135.
    • CommentAuthorRichard Williamson
    • CommentTimeAug 2nd 2021
    • (edited Aug 2nd 2021)

    My thoughts on IUTT (and this page) as expressed here and elsewhere on the nForum remain unchanged, but I have removed the material under discussion above now as I do not currently have the time/energy to work on it further, or to defend trying to proceed in such a way.

    • CommentRowNumber136.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 2nd 2021

    Thanks, Richard, I appreciate it.

    • CommentRowNumber137.
    • CommentAuthorrongmin
    • CommentTimeAug 4th 2021
    • (edited Aug 4th 2021)

    Apologies for bumping this up, but there is now a new development re. #129:

    Had a look through Scholze’s new commentary.

    There is now an official version of Scholze’s review:

    The explanation by zbMATH:

    Technically, the main modification from the preliminary version erroneously available is that we suggested to have a stable version of “Why ABC is still a conjecture” linked at the homepage (to avoid the situation of the now broken link - Peter Scholze kindly provided this, but needed of course also the agreement of Jakob Stix, hence the delay. (Somewhat ironically, our mistake lead to an temporarily unstable version of the review itself - once more, apologies!)

    • CommentRowNumber138.
    • CommentAuthorRichard Williamson
    • CommentTimeAug 4th 2021
    • (edited Aug 4th 2021)

    Slightly amusing that a link to Scholze’s homepage is considered more stable than a link to the nLab, which is what was present in the original review (given that it is the link on Mochizuki’s homepage that is now unavailable, which is why I archived it on the nLab)!

    • CommentRowNumber139.
    • CommentAuthorrongmin
    • CommentTimeAug 4th 2021
    • (edited Aug 4th 2021)

    Re. #138:

    Slightly amusing that a link to Scholze’s homepage is considered more stable than a link to the nLab

    Presumably, Scholze isn’t going anywhere anytime soon, and the server space is provided by an institution that also employs him. Should he decide to move to another institution, a redirect can probably be set up easily. The situation with the nLab is, as I understand it, a little bit more fluid. Perhaps that’s one of the factors that was considered.

    it is the link on Mochizuki’s homepage that is now unavailable, which is why I archived it on the nLab

    And it’s a good thing you did, so thank you, Richard. I think there’s value in having multiple archived copies.

  9. Hehe, yes, I have no problem with the link being to Scholze’s homepage, I was just amused by the sociology of it :-). Links to things on university home pages very frequently go down too.

    • CommentRowNumber141.
    • CommentAuthortomr
    • CommentTimeSep 10th 2021
    Just note Mochizuki just received this year Breakthrough prize. But, obviously, this not an argument here.
    • CommentRowNumber142.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 10th 2021

    Different Mochizuki :-)

  10. added links to KJ; this page needs a complete overhaul.


    diff, v38, current

    • CommentRowNumber144.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2024

    Yes. If it helps, I think you can feel free to clear the page and start afresh.

  11. This page is being redone. But it will take some time before it is finished.


    diff, v39, current

  12. Typo fixes


    diff, v39, current

  13. Text added for clarity.


    diff, v40, current

    • CommentRowNumber148.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 11th 2024

    I’m not sure using the nLab to make value judgements about your own work (” a clearest proof of this Teichmuller aspect has been provided (in all dimensions) in the works of Kirti Joshi.”) is really the best way to go about things. By all means point out that your work provides a description/explanation of this aspect, and that IUT I-IV do not explicitly address why it’s supposed to be somehow a Teichmüller theory, but please tread a fine line here. If you want to make value judgements, point out that it’s your judgement, not coming somehow from the nLab (and by extension its core contributor base) itself.