There will be a documentary broadcast tomorrow in Japan on Mochizuki’s claimed proof of the abc conjecture. I was interviewed for this by the filmmakers last year, but don’t know anything about whether and how that footage will be used. I’d be curious to hear reports from any Japanese-speaking readers who see the documentary tomorrow.

Over the years there has been a detailed coverage of this story here on the blog. To make it more accessible, I’ve added an abc conjecture category. In case the documentary doesn’t make this clear, the current consensus of experts in the field is that there is no proof. Peter Scholze and Jacob Stix identified a problem with Mochizuki’s proof in 2018 (discussed in detail by Scholze and others here), and Mochizuki has not provided a convincing answer to their objections. No one else (including the journal editors who published the proof in PRIMS) has been able to provide a clear explanation of the problematic part of the proof.

Update: NHK has two web pages summarizing the content of the program, see here and here for English translations.

Taylor Dupuy is still making implausible claims that Scholze’s criticism of the proof is invalid. To judge for yourself, see here a long detailed discussion of the issue between them involving several other experts.

Reports I’m seeing from those who have watched the program say that it does correctly explain that the proof is not accepted by many experts.

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16 Responses to ABC on NHK

  1. Meata Konvo says:

    “Equivalent” theory by Kirti Joshi. No proof of ABC.

  2. David J. Littleboy says:

    I thought I had already missed it, but it’s tonight at 9:00 pm. I’ll record it and give a quick overview tomorrow.

  3. Random guy says:

    I saw parts of it. The main takeaway of the show, I believe, is that Mochizuki’s proof is a radical departure from conventional mathematics that continues to be the subject of fierce debate amongst mathematicians. Peter: there’s a scene where you say something to the effect that Peter Scholze’s objection was a huge deal for the community which allowed many to stop worrying about the legitimacy of the proof – I guess credit to the producers for including that..?

    Transcipt and screenshots at links below:

    First half:

    Second half:

  4. David J. Littleboy says:

    The NHK special on Mochizuki wasn’t as bad as it could have been.
    Long story short, it’s about how IUT is new and different and could revolutionize math. While it does present that there are mathematicians who don’t accept the proof, it glosses over that with claims that we don’t yet have ways to describe it yet; i.e. that the
    problem is more how to explain the ideas than that the proof doesn’t convince.

    Mochizuki himself refused to be interviewed, saying “How IUT differs from previous mathematics is an important question, but presenting it to a popular, or even mathematically sophisticated, audience in a TV program would be unreasonable.”

    Peter Woit is quoted at the beginning as saying (roughly)
    “It’s unlikely that the abc conjecture has been proven.”
    and David Roberts is quoted as saying “It most certainly has been.”
    At the end Roberts is back saying people who don’t get Mochizuki’s ideas
    are stuck in old mathematical thinking, and PW is back saying Sholze is the best
    number theorist we have, and if he’s not convinced, it’s not convincing.
    Then Peter is quoted saying “I’d guess most mathematicians are real happy that
    they don’t have to worry about it any more.”

    Much of the program was an attempt at an introduction to number theory. Attempt.

    Many mathematicians who were concerned with the abc conjecture or related mathematics prior to Mochizuki were interviewed.

    Taylor Dupuy is presented as a young and up and coming mathematician who thinks it worth betting his career on. (This may be a tad of an overstatement of his position, but he’s interviewed multiple times throughout the program and the point that there are young and upcoming mathematicians interested in it is one of the punch lines at the end of the program.)

    My SO watched it with me, and reports that to a non-technical audience, it reinforces the prejudice that math is a waste of time. Furthermore, she
    says that the folks at NHK should have listened to Mochizuki and not tried to explain it to a popular audience.

    OK, maybe it was as bad as it could have been…

  5. Z Y says:

    For me the program is an excellent exposition to the relevance of the abc conjecture, in some ways more understandable than the Notices AMS article of a few years ago, a good document to have for future historians and may influence young people to enter into mathematics

  6. David Roberts is quoted as saying “It most certainly has been.”
    At the end Roberts is back saying people who don’t get Mochizuki’s ideas

    Hmm, I’m pretty sure I didn’t say this. Or at least, didn’t mean to say this!

  7. David J. Littleboy says:

    David Roberts said:

    “Hmm, I’m pretty sure I didn’t say this. Or at least, didn’t mean to say this!”

    I was wondering if you’d say that. The program was structured (in those two segments) to put you in contrast with Peter, and gave the strong impression that that was what you thought. A careful translation of the Japanese might show that you were more careful than to say those things so clearly. Might.

    FWIW, the quote of yours on the web page referenced above is a good summary of the whole point of NHK’s presentation of Mochizuki’s theory, namely that he sets up two things as being the same and then handles them as being different. “What is strange about Mochizuki’s theory is the point that at the same time as saying two things are exactly the same, next, it handles them as being completely different. In mathematics, it is a basic principle that two things that are seen to be the same are the same. Is it possible for things to, while being the same at the same time to differ? I though about this really carefully. No, this is absolutely unreasonable.”

  8. @David J. Littleboy

    my position is that I’m more cautious about how to interpret the clues we are getting from various parties. I very much respect Peter Scholze and I don’t think he would have made an elementary mistake. I take Scholze and Stix’s document at face value when it is worded cautiously (eg saying it isn’t a watertight argument), and of course Peter S will stand by it. It doesn’t hide the fact it’s not dealing with the full theory, but this is the sort of thing high-level people do work with in other sub-fields (Terry Tao has written things like this, for analytic number theory). Discussions privately with Peter S since have pointed to the exact fact he was never satisfied with in Mochizuki’s verbal explanations, in response to a specific question about passing to the full, un-simplified theory. Perhaps there is an explanation for this, but Mochizuki doesn’t have a good way to explain his intuition, or maybe there is a deeper flaw that is just surfacing like a wrinkle in a badly-fitted carpet. Taylor is more blunt, saying to Scholze and Stix “you can’t do such and such a step”, in the video (posted yesterday on YouTube) of him presenting his work from a year ago. This is of course, pointing to the actual contentious isomorphism (in the hexagon diagram), and is the clearest statement I’ve seen in pushback to Scholze and Stix’s argument, but it’s very light on details; given the medium, and time constraints, it was appropriate, but I’d love to hear more. Taylor is open in the video up front that there are things he thinks don’t necessarily join up in the argument, his complaint is that if there is a mistake, he doesn’t think it’s where Scholze and Stix are pointing.

    Taking Corollary 3.12 as a conjecture, as Taylor does, lots of interesting mathematics can be done. This is in my view akin to assuming GRH and then finding all kinds of qualitative estimates of prime distributions, etc. This is how I view the subsequent IUT work of Mochizuki et al, getting FLT etc. It’s all conditional on an interesting conjectured inequality in anabelian geometry/Arakelov theory. But, contra Mochizuki, we really can’t say that inequality (i.e. Corollary 3.12) is established. (More than one expert has described that step as a cliff, or a sudden leap etc. If experts are happy with everything before then, and then all struggle at that point, it’s not because the theory is just too hard for them.) I’d rather people take this view, than throw the lot in the bin. The glory of giving a new proof of FLT, or other hard number-theoretic results, is clearly lessened, but it’s at least a healthier way forward for the community (IMHO) to try to extract useful content here, being clear with what is accepted, and what is not (cf work of Kirti Joshi who is claiming no big result, but is adapting some IUT ideas into standard anabelian geometry, and getting non-zero mileage).

    It might be that my qualified optimism (conditional on assuming Cor 3.12 as a conjecture) happened to translate on screen into something less negative than others, including Peter W here. I can only go on what I think I would have said, and what is written on that NHK page for the documentary. I was quite complimentary towards Peter S in the interview, though it’s quite likely they didn’t use that, and I was fairly up-front that IUT doesn’t give a proof of abc, as far we currently understand it.

  9. Just to clarify, Taylor says in his video:

    I don’t claim to understand Mochizuki’s proof. I don’t see it as completely correct because there is a point of contact that I can’t see. And that’s all I really want to say about that.

    Taylor told me privately what that “point of contact” is, but not in any detail.

    Taylor’s discussion of what he sees as the weak point in the Scholze+Stix argument is at this time stamp, and goes for about 5 minutes. Make of it what you will.

  10. W says:

    @David Roberts:

    In addition to that video, the other place Taylor explains his point of view on Scholze-Stix is Remark 3.8.3 of but this explanation is even shorter. Both the talk and the paper are from 2020.

    He wrote a long Twitter thread ( responding to the text version of the documentary, where he expresses a lot of uncertainty on what the final outcome will be:

    “For me there is still the option that this was a big waste of time for everyone ”

    “The situation may end up more like Lame’s proof of Fermat or Kempe’s Proof of the Four Color Theorem. Lame’s proof was 175 years ago and we discuss it regularly when discussing the concept of a unique factorization domain. ”

    “There is also the possibility that we are in a Heegner-like situation where the proof is ultimately correct, but the author will pass away before the proof is finally accepted by the mathematical community (after Stark).”

    I will resist the temptation to say anything more about the content of the proof.

    @David J. Littleboy Your comment at the end about your SO makes me sad. I guess the point is mathematicians consider abstruse abstract problems and can’t even agree on those. But in fact this is the one thing that (some) mathematicians can’t agree on, and we agree on almost everything else, which is why it is apparently newsworthy.

  11. David J. Littleboy says:

    Hey, W! Don’t be too sad. My SO was irritated more with NHK for trying to make a story about something that can’t be made into a story than about math. Again, her main take away from this thing was that Mochizuki is a perfectly sensible bloke for not talking to NHK. I think I was able to get the point across that we don’t know what math is going to be useful 100 years from know, so we have to just keep at it.

    (We were both irritated that it didn’t have more on Mochizuki as a person. We’re both fascinated by the problems of bilingualism and biculturalism and the trials and travails of growing up being the smartest bloke in the room. There’s a human story there, too.)

  12. Z Y says:

    Is it only me that find remarkable that something as big as NHK took the decision to make a documentary about their countryman’s effort to prove ABC and that the end message is that the proof is not yet accepted by the majority of experts? No wonder why Mochizuki didn’t want to be interviewed

  13. Nayr says:

    Is there anywhere that we can rewatch the NHK documentary? It seems interesting.

  14. AG says:

    Recently ( on May 2) Prof. Mochizuki wrote a long comment about NHK documentary on his blog in Japanese. Its contents are very interesting.
    He himself says that the second half of what was broadcast was a complete flop.
    He is critical of the content of interviews and the way other researchers critique his theories. I know nothing about math and needless to say I can’t judge anything. I could understand just one thing: Mochizuki is a man of great integrity, and sincerely from the bottom of his heart thinks that his proof is complete and correct.
    It’s difficult for me to translate it into English. I hope someone translate correctly ( only if not illegal ) . To show the address of the blog here must be at least legal:

  15. Peter Woit says:

    I took a look at that (via Google Translate). It seems the part of the documentary he didn’t like was the part where people discuss problems with his proof. He doesn’t mention at all Scholze, (or me, I gather I mainly explain about Scholze’s objection and its significance). He complains in detail about Faltings, as well as about Taylor Dupuy.

    I do agree he surely believes in the correctness of his proof, but he is not following standard professional norms in dealing with criticism.

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