Now back from vacation, here’s the latest on revolutionary developments in physics and mathematics:
- On the high energy physics front, the good news is that the LHC is performing remarkably well, with already over 13 inverse fb of luminosity, far above that expected at this time, on track to end up with a lot more than the targeted 25 inverse fb for the year. The bad news however is that new reliable rumors (together with the non-observation of any sign of a “special seminar” at CERN, see here) confirm non-existence of the 750 GeV state that would have killed the Standard Model and revolutionized the field. As far as I know, the plan is still to present these results publicly the first week of August at ICHEP, it looks like this will be on August 5.
For some interesting discussion of the statistical analysis issues that come up when trying to quantify how significant the 2015 evidence is for the supposed 750 GeV state, see the comment section of this blog entry. These subtleties it seems will be made irrelevant by the arrival of new data.
- In mathematics there has also been an unconfirmed claim of something revolutionary, but the problem is that there’s nothing analogous to new data coming in to help decide the issue. This is the claim first made four years ago by Mochizuki to have a proof of the abc conjecture, using new methods he calls “Inter-Universal Teichmuller Theory”. The current situation is an extremely unusual one, with experts still unable to understand and evaluate the purported proof. For the best summary of the situation, see Brian Conrad’s detailed explanation from last December here.
Not much seems to have happened since then, but one very recent development has been the appearance of a new survey of the theory. Unfortunately, my guess is that this is not likely to address the issues raised by Conrad and provide what he and other experts are looking for: precise checkable arguments. Instead the new survey is another attempt by Mochizuki to communicate his general high-level vision, often in very metaphorical terms. The last section of the survey is a remarkable attempt to position his ideas in the landscape of modern mathematics, which includes setting these ideas in opposition to those of the dominant research program (and thus of great value if they work out).
What’s really odd here is the way that usual mechanisms for transmitting understanding have failed. Mochizuki has worked to transmit understanding of his ideas to a small number of others, but the transmission has stopped there, with understanding of the abc proof not moving from them to others. For a while the hope was that Go Yamashita would be the one to move this forward, but he has not produced a promised document, or succeeded in communicating by his talks. More recently, last year Yuichiro Hoshi produced a document that is supposed to explain crucial ideas, but it is in Japanese, so inaccessible to most experts. Why this has not been translated remains very unclear.
Next week in Kyoto there will be another workshop trying to further understanding of the IUT theory. I hope this works out better than the last one. There’s a preparatory document here which to me seems to ignore the fundamental problem of figuring out what has gone wrong so far. In particular, its last point appears to be explicitly aimed at discouraging anyone in the audience from confronting speakers that are not successfully communicating ideas and insisting that they try to do better. It would be more fruitful to encourage this instead.
Latest Rumor : ATLAS shows gamma-gamma bump at 975 GEV, confirmed at 2.7 Sigma. Will update.
Elsewhere this had been crypto-rumoured for a while already:
https://twitter.com/dorigo/status/745528592927338498
https://twitter.com/dorigo/status/750266675526832128
These sigma significance levels need reevaluation. 2.5 sigma supposedly means a result is correct over 99% of the time – in other words, hardly ever wrong. But we know that’s nowhere near the reality.
Another Anon,
That’s the topic being discussed at Tommaso Dorigo’s blog (which would be a better place to discuss this than here). The subtleties of going from such a “local” number to a number reflecting the actual probability of something new are well-known in HEP physics (and the reason behind the very high “5 sigma” standard). This is nothing new, but has always been part of this subject.
Is this on anyone’s radar? http://www.ihes.fr/~lafforgue/math/NoriMotivesInformation.pdf
Re. Y. Hoshi, he gave ~4 hours of lectures in Paris on his mono-anabelian ideas, he has superb English so that is not a reason for lack of translation. One frustrating thing he does is give a new name to everything e.g. say A is an ‘isomorph’ of B if A is isomorphic to B; say that a morphism is ‘multiradial’ if it is not injective… etc etc. this sort of thing seemed to cause a lot of raised eyebrows in the theatre of IHP—this appears to be the m.o. in Mochizuki’s writings too.
On page 3 of the new overview, (“Alien Copies”), Mochizuki says, (slightly paraphrased):
“Let N be a fixed natural number > 1. Then the issue of bounding a given nonnegative real number h \ge 0 may be understood as the issue of showing that N h is roughly equal to h.”
This is an absurd statement, because h factors out of the approximate equality. It is not a typo, because he then repeats it in a formula.
Chris Austin,
In the text you mention Mochizuki is explicitly pointing to sections 2.3 and 2.4 for an explanation of what this is supposed to mean (i.e. what does “roughly equal”, or “=” mean), so you need to go there to interpret what he is writing. The problem is not that he is making trivial, easily identifiable mistakes.
This does however give some idea of the problem with this survey and with similar things he writes. All sorts of things are in quotes (or analogies, not equalities) and even when they’re not (e.g. roughly equal), it’s often hard to track down a precise definition, or figure out what precise claims are being made. This then makes it impossible not only to check for flaws in the argument, but even to understand exactly what the argument is.
The 2012 IUT papers are supposed to contain the precise statements, but these are 600 pages long and have resisted the efforts of experts to understand them. Mochizuki claims that four mathematicians have, with his help, gone through these and checked them. Normally what should have happened is that, armed with a detailed understanding, these mathematicians should have been able to transmit this to others, and to write up an independent version, one that would help others understand the papers. Besides the possible example of the Hoshi document in Japanese, this has not happened.
Chris Austin,
A clearer version of that statement appears at the bottom of page 11: The point is that ‘roughly equal’ is in the sense of the ‘absolute error’ not the ‘relative error’. For instance, if h and 101h differ by at most 1, then h can’t exceed 0.01. (By contrast, the relative error between Nh and h is fixed by N and, as you assert, would tell us nothing about h itself.)
Peter and Michael, thank you for the clarifications.
Regarding Hoshi’s document, Brian Conrad wrote in January in a comment on the same page that has his summary of the December workshop that Hoshi is hoping to make an English translation of his notes available around the time of summer workshop.
anon,
I’ve heard that such a translation is under way, but one of the mysteries of the subject is why this is taking so long. With the workshop less than a week away, it’s already pretty much too late to be useful to people preparing for the workshop.
Peter, you write that “Mochizuki has worked to transmit understanding of his ideas to a small number of others, but the transmission has stopped there, with understanding of the abc proof not moving from them to others.”
I would like to comment on that. There actually seems to be substantial “transmission” of IUT “to others” from Taylor Dupuy. Dupuy is the guy that Felipe Voloch “felt” “was the one that made the most progress learning the stuff” on the Oxford workshop (https://plus.google.com/106680226131440966362/posts/UHoetkZ7XXK).
You will find Dupuy’s has published several series of IUT-related videos on his vlog:
https://www.youtube.com/channel/UCHWnZ1NtJ4WvE5AHmNVXziw/playlists
The playlist “IUT strategy” seems to be a good starting point:
https://www.youtube.com/playlist?list=PLJmfLfPx1OednuaMHywSDgaDzRa0MiaOf
Then seems to be other relevant playlists too, like:
The geometry of Frobenioids, Anabelian Geometry, Etale Theta, Hodge Theatres, etc.
It seems that Taylor Dupuy is making an effort to “transmit” Mochizuki’s work to a wider audience.
Dupuy will be a speaker at the RIMS conference. It seems that the conference is going to be recorded on video, although maybe not streamed.
https://twitter.com/DupuyTaylor/status/744999080229765120
https://twitter.com/DupuyTaylor/status/744999335092436992
Otherwise, part of the confusion around this proof seems to come from the fact, that Mochizuki has worked on IUT for a long time and few professionals had done their homework before the Oxford workshop. Imagine you attend a class in some branch of maths you haven’t studied before and that you have ignored studying the required prequisite literature. I think that’s what happened at Oxford and that is really not very surprising.
I for sure experienced this quite some times during my math studies, when I had skipped my homework before class. The worst thing I could do in this situation would be to ask impertinent and not-so-very-clever questions.
The second issue worh mentioning is the fact that Mochizuki is Japanese and seems to be value that highly (someone wrote he is from a samuraj family).
Thus, when Mochizuki asks people to do their homework, to be polite and not to interrupt if it is not absolutely necessary, then this might be an other way to ask that the visitors to respect the “the Japanese language-based mathematical culture that exists” “at RIMS”.
http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.html
To quote Mochizuki in full:
Especially with regard to (1), the prospective applicant should be aware that Japanese
is the official language in which mathematical and administrative affairs are conducted here at RIMS. Some individuals may tend to regard this state of affairs as a sort of “unpleasant obstacle”. On the other hand, at the present time, there is no shortage of institutions throughout the world at which one may obtain a quality graduate education in mathematics in English. By contrast, the Japanese language-based mathematical culture that exists here at RIMS [and indeed at other Japanese universities] is, in my opinion, a precious cultural asset, both for Japan and for the world.
There has been more written about this:
“Despite mathematics being a universal language, culture clash could be getting in the way, says Kim. “In Japan people are pretty used to long, technical discussions by the lecturer that require a lot of concentration,” he says. “In America or England we expect much more interaction, pointed questions coming from the audience, at least some level of heated debate.”
https://www.newscientist.com/article/dn28682-mathematicians-left-baffled-after-three-year-struggle-over-proof/
I am certainly not an expert in Japanese culture, but I know so much, that they tend to put more emphasis on politeness and respect, than in the West.
Well in any case it seems that Dupuy has done his homework and is actually “transmitting” IUT to the West.
If Dupuy’s videos on IUT would get wider publicity, that might actually constructively contribute to transmitting IUT to a wider audience and thus to further mathematical knowledge and development :).
As a physics person, it’s hard to imagine that people are so invested in even trying to read Mochizuki’s paper. Given this kind of difficulty, it seems like a similar document in physics would have been abandoned and forgotten after a day or two, let alone years (Verlinde’s entropic gravity paper comes to mind). Maybe it just shows how different the goals are: here one can still hope to extract precise technical statements/proofs, whereas in a physics paper that would often be, at most, a secondary goal. Still, it seems strange… I guess Mochizuki must have some serious cred/social capital in his field?
Dan,
I don’t think there really could be a physics analog of this, since what’s at issue is whether a very complex set of new ideas is powerful enough to provide a proof, a kind of question physicists don’t deal with. Mochizuki is a well-known, accomplished and talented mathematician, and the fascination of this is that he clearly has new ideas, but the complexity of what he is doing and the way he has chosen to explain them has left experts unsure of how much can be gotten out of them.
The “entropic gravity” business is quite different. There was nothing at all complicated about that. It was a simple, but vague idea (not much different than ones other people had discussed in the past). Experts could immediately read Verlinde’s paper and see what he had. Different people would have different judgments about whether pursuing that kind of vague idea would ever lead anywhere, but that’s a different question.
I guess that’s part of my confusion- how vague/non-vague is Mochizuki’s work? Trying to parse the intro of his new paper is impossible for me; it sounds very vague, but obviously it’s above my pay grade. Maybe I can ask a sharper question: do people seem to agree that his paper is technically sound, but they just aren’t sure how it connects to standard literature? Or is there any possibility that he is “wrong”/any of this is jibberish?
Peter Zbornik,
I’m sorry, but I really don’t buy at all the “Japanese culture” explanation for the problem. Mochizuki grew up here in New York City (from age 5), went to school at Exeter, then got his undergraduate and graduate degrees at Princeton. His upbringing is more American than mine (I spent 5 years as a kid in France).
I also don’t buy the “didn’t do their homework” explanation for what happened in Oxford. Brian Conrad’s article I think does an excellent job of explaining the problems with the talks, and looking at the slides confirms what he has to say. He and others who were there are the best in the business, and claiming that the problem was that they were asking stupid uninformed questions is just ridiculous.
From what I can tell, Taylor Dupuy’s efforts are very enthusiastic (I’ve looked at some of them), but at a similar level of vagueness to Mochizuki’s surveys (or even much more so), and suffer from the same problems of being disconnected from the rest of solid mathematics, and being too imprecise to be checkable. Maybe I’m wrong, but as far as I know, he’s not claiming to fully understand the details and have checked them.
Dan,
The problem is twofold: the shorter “survey” papers are very vague, the 600 page 4-part IUT paper series more precise but bafflingly complex and invoking a completely new universe of mathematical objects. A lot of that complexity is likely irrelevant to the heart of the supposed proof. What’s needed is something in between: a detailed outline of the proof, with precise, checkable statements. The mystery here is why, four years out, such a thing seems to still not exist.
This kind of highly complex proof is inherently difficult to check: the mathematical objects involved can have subtle, often unintuitive behavior. To check such a proof, you need an expert, one who has some pretty deep insight into the behavior of the mathematical objects being manipulated. My impression is that we’re still unfortunately far from anyone (other than Mochizuki), being able to carry this out with confidence in the result.
I think is clear that the way of transmitting IUT has not been efficient. But on the other hand it may be that this theory is enough original and has been developed for such a long time that it would be as if Grothendieck presented to the world the EGA or SGA volumes in one stroke and thinking that in three years we could understand his proof of the first parts of the Weil conjectures.
Zoviyer,
Sure, that’s in principle possible. My guess though is that in such a hypothetical case it would take people a significant amount of time to understand the whole thing, but you would see progress taking place as this happened, and you wouldn’t see the odd phenomenon of people close to Grothendieck supposedly mastering the ideas, but unable to write up their own version and explain it to others. This is a really peculiar situation that I know of no historical analog for.
Zoviyer,
Thing is, while it took a bit for people to catch on to what Grothendieck was doing, they did, and in a few years. With the IUT the best people just have no clue. When Perelman proved Thurston’s geometrization conjecture, I was at a conference with many of the best geometric analysts in the world. Perelman had posted some work, and everyone was reading it, and within weeks they had a very good idea of what he had done. When Wiles proved Fermat, the same thing. The hole in his original argument was picked up quickly, took him a year to fix. What’s going on here is you have someone claiming to have essentially invented an entirely new branch of math, with little connection to other branches. It’s like Newton and Leibniz with calculus, more or less. People were using calculus almost immediately to do all sorts of amazing things, though of course it took more than a hundred years to put it on a good foundation. But people understood it, and used it, lots and lots of people. No one really seems to understand IUT, at least well enough to explain it in any fashion to very very smart people. No one has used it for anything. It could be correct, but I somehow doubt it. 600 pages of heavy duty math has probably tens of thousands of little things that could be wrong, and any one of them would likely blow the whole thing up.
Peter,
I agree with your points. From the outside looks as if both Hoshi and Yamashita had no internalized the crucial ideas and how they relate as steps of the proof, and that’s why it has not been communicated efficiently in the workshop or afterwards. The document that has not been translated to English is “just” an Introduction to IUT
With regards to the statistics issue – another point is the so called “look elsewhere” effect. The LHC experiments do hundreds of searches looking at a large number of final states. It is not just the probability of this particular search one has to consider but also what the entire experiment sees as a whole to properly compute the probability of having these results. This is why it often appears that results that taken on their own in isolation would indeed be perplexing if there were 3 sigma deviations coming and going all the time. However, what is neglected in this discussion is that people forget that there are hundreds of searches that land spot on the SM. Just like one expects that if one has a bunch of experimental points that you expect to fall on a line you don’t expect all of them to be within one error bar of the line. You expect a certain fraction of them to be 1 sigma, 2 sigma, 3 sigma, etc to be off just from the statistical nature of the measurement process. Unfortunately statistics and p-values are often misunderstood or misrepresented so people can easily get the wrong impression of what the real probability of seeing deviations is…
Peter, I agree with you, there is no proof published in a peer-reviewed journal.
But then we have a unique situation, where there is a claim, that Mochizuki has changed the foundations of mathematics.
Quoting the Nature article:
“Fesenko has studied Mochizuki’s work in detail over the past year, visited him at RIMS again in the autumn of 2014 and says that he has now verified the proof. (The other three mathematicians who say they have corroborated it have also spent considerable time working alongside Mochizuki in Japan.) The overarching theme of inter-universal geometry, as Fesenko describes it, is that one must look at whole numbers in a different light — leaving addition aside and seeing the multiplication structure as something malleable and deformable. Standard multiplication would then be just one particular case of a family of structures, just as a circle is a special case of an ellipse. Fesenko says that Mochizuki compares himself to the mathematical giant Grothendieck — and it is no immodest claim. “We had mathematics before Mochizuki’s work — and now we have mathematics after Mochizuki’s work,” Fesenko says.
http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509
Talking about the lack of examples: refering to Fesenko above, it would be interesting to get an example, where in “non-standard multiplication” 2*3=5
I would not agree with progress not being made at Oxford, quoting Conrad in his posed referred in your post:
“Despite the difficulties and general audience frustration that emerged towards the end of the week, overall the workshop was valuable for several reasons. It improved awareness of some of the key ingredients and notions. Moreover, in addition to providing an illuminating discussion of ideas around the vast pre-IUT background, it also gave a clearer sense of a more efficient route into IUT (i.e., how to navigate around a lot of unnecessary material in prior papers). The workshop also clarified the effectivity issues and highlighted a crucial cohomological construction and some relevant notions concerning Frobenioids.
Another memorable feature of the meeting was seeing the expertise of Y. Hoshi on full display. He could always immediately correct any errors by speakers and made sincere attempts to give answers to many audience questions (which were often passed over to him when a speaker did not know the answer or did not explain it to the satisfaction of the audience).”
The main problem for the participants in Oxford was, that they did not understand the big picture and thus it was not easy to dig into the details. Conrad wrote in the post you refered above, that: “Many are willing to work hard to understand what must be very deep and powerful ideas, but they need a clearer sense of the landscape before beginning their journey.” Dupuy and Mochizuki have now laid out the land whith their vlog resp. last paper. I believe this deserves some recognition.
There are several more papers with surveys ot IUT in the conference programme: https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-kyoto-pr.html
At the conference there seem to also be some “transmission made”.
The end of your post was not very polite nor correct:
“There’s a preparatory document here which to me seems to ignore the fundamental problem of figuring out what has gone wrong so far. In particular, its last point appears to be explicitly aimed at discouraging anyone in the audience from confronting speakers that are not successfully communicating ideas and insisting that they try to do better.”
The problem at the Oxie conference was that the interruptions made things worse, “party crashing-style”, quoting Conrad:
“The fact that the audience was interrupting with so many basic questions caused the lectures to fall behind schedule, which caused some talks to go even faster to try to catch up with the intended schedule, leading to a feedback loop of even more audience confusion, but it was the initial “too much information” problem that caused the many basic questions to arise in the first place. Lectures should be aimed at the audience that is present.”
As for the doing the homework before the conference at Oxie. It really seems that all these esteemed experts did not study the IUT and other papers before the conference.
Specifically: Voloch wrote in the last comment of his blog post, that:
“I prepared some but did not put 100s of hours as some had suggested but neither did most people I talked to.”
Conrad: “It was reasonable that participants with advanced expertise in arithmetic geometry should get something out of the meeting even without reading any IUT-related material in advance, as none of us were expecting to emerge as experts (just seeking basic enlightenment).”
Contrast this with Fesenko and Mochizuki in the nature article:
“Mochizuki has estimated that it would take a maths graduate student about 10 years to be able to understand his work, and Fesenko believes that it would take even an expert in arithmetic geometry some 500 hours. So far, only four mathematicians say that they have been able to read the entire proof.”
Contrast this with Conrads post, quote:
“At multiple times during the workshop we were shown lists of how many hours were invested by those who have already learned the theory”
Well, sad to say, “basic enlightenment” is seldom effortless. I think we can agree on that :o)
Yes, these guys were experts in their field, but they were faced with an other new field of maths and in this case, like it or not, you have to do your homework, because they cannot by definition be experts in this new field.
Now we have some Japanese and a Russian having spent considerable time with Mochizuki. Maybe some of the English-speaking “expert” community would also venture to do the same? That would certainly prove fruitful.
But then there definitely is a culture barrier here, undeniably, since Hoshi’s IUT introduction has not been translated to English and evidently the Japanese do not consider it to be a priority to translate it to the English-speaking mathematicians :o)
Please allow me a personal reflection. I believe we have been stuck with an outdated number system based addition and multiplication. At the same time there really has been almost no work on alternative construcions of the natural numbers or on generalizing multiplication (the only serious one I know of are the surreals) since Hilbert’s program. So from that point of view, for me, it is difficult not to see the work of Mochizuki as a possible “light in the tunnel”. There is no work on “un-ordered” “numbers” (whose cardinality exceed that of the largest possible ordered field). On the other hand we have all those wierd stuff with connections to quantum mechanicsa around the Riemann hypothesiss, which all point to the need to widen the notion of basic concepts like “number”, “addition”, “multiplication” and “order”.
Now let’s see if the new workshop will bring “order out of chaos” :0). It seems to be invitation-only, where the applicant has to state his or her “knowledge of IUT” (https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-rims16.html), so party-crashers may be stopped at the gates. It is organised at Mochizuki’s home turf (he has stopped travelling outside of Kyoto, last years) by Mochizuki, Fesenko and Taguchi. So it seems that the scene is set for some “transmission” of knowledge, provided that the participants have done their homework, will be polite and will not crash the lectures.
Somewhere I read, that Mochizuki estimated it would take like until, I believe, 2020 before IUT was understood and accepted.
Maybe in 2020 we will have the abc proof and some less rigid foundations of mathematics to go with it too, if Mochizuki doesn’t go Perelman playing ping pong with himself after all the hate thrown at him in the mean time.
Maybe it would be suitable to end this comment by quoting some “laws” from Fesenko’s website:
“Everything takes longer than you think [including learning IUT – comment by PZ].”
“When all else fails, read the instructions [of required reading for the conference – comment by PZ].”
“Abrams’s Advice: When eating an elephant, take one bite at a time [i.e. learning IUT takes time, there is no short-cuts – comment by PZ].”
https://www.maths.nottingham.ac.uk/personal/ibf/some.html
Peter Zbornik,
I think you’re selectively quoting Conrad, who was being exceedingly polite (despite being repeatedly insulted as a rude party-crasher who hadn’t done his homework). I hope the organizers of the Kyoto workshop think carefully about his analysis of what went wrong at Oxford and why Mochizuki’s work is not getting understood by experts. That they think the problem was experts asking questions and that they are trying to discourage questions is not confidence inspiring. When no one in the audience of a talk is understanding the speaker, it is much better for everyone if someone interrupts with questions than if the speaker is allowed to continue unimpeded.
Speaking as layperson, but with at least pretty rigorous undergraduate training, I also found it invaluable in understanding things well beyond just “drama + mystery proof.” Thanks for leading me to it.
IUT Kyoto Summit seems is progressing constructive discussions.
Christelle Vinsent (@ xl772) ‘s tweets is very convenient.
and this site is very nice. https://storify.com/xl772
Peter, for completeness, here is Fesenko’s take on the proceedings of the Oxford workshop and on the reasons for the relatively slower pace of knowledge transfer of IUT to a wider international audience: https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-i-rep.html
Peter Zbornik,
I wrote about that at the time here:
http://www.math.columbia.edu/~woit/wordpress/?p=8195
Fesenko kept changing the part of it blaming Brian Conrad (which I think is absurd), I see the latest version is
” Some participants who had ignored recommendations to study the theory before the workshop did not always behave considerately towards the speakers, interrupted the flow of talks by asking shallow questions and felt free to exhibit negative emotions without any concern to the other participants. ”
There’s an ongoing problem with how these ideas are being communicated, and instead of addressing it, Fesenko here tries to shoot the messenger. This isn’t helpful.
This whole saga just sounds unprofessional on the part of the principal actor. Blaming it on some inherent problem with an idea of Japanese culture derived from orientalist Hollywood minstrel shows seems insulting to Japanese culture.
@anon Don’t think it is fair to call anyone “unprofessional”. Simpler explanation is that it is *very very very hard* to simplify the work of a decade or two into digestible pieces. Not only is it hard work, but it is work that might require the surrounding community to help. Perhaps the struggle we’re observing is actually the only way to clarify things. Perhaps this really is the best of all possible worlds.
This might actually be the most efficient way to actually share a breakthrough, even if it is painful for all those directly involved?
Marshall Flax,
This is an unusually difficult situation, both for Mochizuki and for those trying to understand his ideas. We’ll see what emerges from the Kyoto workshop, but reports I’ve heard are not very encouraging, with the same obstructions to progress obvious at the Oxford workshop still there. I strongly disagree that this is the best of all possible worlds, in particular, better possible worlds exist in which
1. Hoshi’s survey exists in English
2. Fesenko took to heart criticisms of what happened in Oxford, encouraging people to demand better explanations of material no one is understanding, instead of attacking people for doing this.
3. Mochizuki paid attention to explanations from experts of what was needed for them to follow him, and wrote up a version of the proof addressing their concerns.
I agree with anon that blaming problems on Japanese culture makes no sense. From what I can see, part of the story here is something not that uncommon in mathematics: an author writes up his ideas in a way that is impossible for others to follow. What is supposed to happen then is that journals tell the author to rewrite. The author may not agree and be willing to do so, and then others need to step in, figure out what is going on, and write something readable themselves. What is highly unusual here is that we haven’t seen that happen, and the central question is why that is.
I think it’s pretty obvious what’s going on with Mochizuki. He wants to sell his theory, but others want to buy the proof for Vojta/Szpiro/ABC only. I think he sees the proof more as a motivational example for his theory, thus he has little interest to publish a more compact version that cuts out all unnecessary generality. I suspect in reading his papers in reverse chronology, concentrating only on the parts relevant to the proof, one would find more success than in trying to digest his published works of the last 10 years or so.
@Snowman … don’t think that’s precisely true either … at least some of the participants in the seminar are imagining that the theory might be extendable one day to address even deeper questions like BSD or Riemann.
Some assorted tidbits on this mesmerizing topic:
—
1) The publication of the proof might be on the way:
It seems, that Mochizuki’s ABC papers are being refereed, some say that probably by a Japanese journal:
“These papers are currently being refereed, and, although they have not yet been
officially accepted for publication, the refereeing process is proceeding in an orderly,
constructive, and positive manner.”
http://www.kurims.kyoto-u.ac.jp/~motizuki/Questions%20and%20Comments%20on%20IUT.pdf
—
2) Replacement for Japanese IUT introduction:
Maybe Hoshi’s Introduction to IUT, might in large be replaced by his lecture notes from the Oxie and RIMS workshops? Someone mentioned they were good.
—
3) Intellectual theft in the mathematical community:
Fear of stealing the proof “Perelman”-wise, and of confronting frustrated, impertinent mathematicions, might also play a role in the slower pace of knowledge transfer. After all, the Mathematical community is a jungle, according to Grothendieck, with whom Mochizuki compares himself (see the article in nature above) and who just couldn’t care less for his work to go through standard peer reviewing.
This is what Wikipedia has to say on Grothendieck: (https://en.wikipedia.org/wiki/Alexander_Grothendieck):
“While not publishing mathematical research in conventional ways during the 1980s, he [Grothendieck] produced several influential manuscripts with limited distribution, with both mathematical and biographical content.”
…
“In the 1,000-page autobiographical manuscript Récoltes et semailles (1986) Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner but which he progressively perceived to be governed by competition and status. He complains about what he saw as the “burial” of his work and betrayal by his former students and colleagues after he had left the community.[13] Récoltes et semailles work is now available on the internet in the French original,[36] and an English translation is underway. Parts of Récoltes et semailles have been translated into Spanish[37] and into Russian and published in Moscow.[38]”
…
“He […] criticized what he saw as the declining ethics of the scientific community, characterized by outright scientific theft that, according to him, had become commonplace and tolerated.”
https://en.wikipedia.org/wiki/Alexander_Grothendieck
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4) Speculation:
We might also speculate, that Mochizuki might considers other things more important, than making “PR” for his theory and fraternizing with the mathematical community, like for instance, using IUT to prove Riemann’s hypothesis. What then is a decade of slower acceptance of IUT and ABC compared to proving Riemann’s hypothesis?
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5) Cultural speculation
For some reason, I believe that the Japanese might not mind all that much, that IUT becomes a Japan-dominated field of study, centered at RIMS lead by Hoshi and guru Mochizuki :o) What these guys mind is not PR and marketing, but doing some fine new maths. Someone good at PR and marketing in the mathematics community should help them.
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6) My own take:
Personally I am more interested in the mathematics, that comes with the proof, than the proof itself, so therefore I see as the biggest problem the unvillingness of the experts to invest the 500 hours or more to study the problem.
Peter writes: “reports I’ve heard are not very encouraging, with the same obstructions to progress obvious at the Oxford workshop still there”.
Taylor Dupuy wrote on twitter: “Great conference, we got way way farther than Oxford”.
Following up on my last message re @DupuyTaylor ‘s comments, some more twitter feedback on the IUT Summit: Artur Jackson (@arturj) wrote “I am impressed with … the depth of the discussions. Better results than Oxford”. Ivan Fesenko wrote “A successful and rewarding workshop (see @math_jin feed; n.b. of course Ivan is an organizer) . Christelle Vincent (@xl772) has a number of tweets on the conference as well, see generally the hashtag #iutsummit . The tone of the feedback I have read appears generally, contra Peter’s comment, to be positive and encouraging than at Oxford.
e.ehrenweist,
I hope we’ll get some serious public reports from experts who were there, now that the workshop is over. The twitter traffic made clear that that’s not a good way of transmitting tricky arguments in arithmetic algebraic geometry. That things went better than Oxford (maybe due to Mochizuki himself playing a major role) is not inconsistent with the same problems still being there. Fesenko thought Oxford went great, so one might take that into consideration in reading his take on Kyoto.
Will likely write more about this once there’s more serious information available.
First report in by Nature: http://www.nature.com/news/monumental-proof-to-torment-mathematicians-for-years-to-come-1.20342
Christelle Vincent sums up the “mood at the conference” in a series of replies to a tweet here:
https://twitter.com/maths1Bsummer16/status/758179530817220608