At a news conference in Tokyo today there evidently were various announcements made about IUT, the most dramatic of which was a 140 million yen (roughly one million dollar) prize for a paper showing a flaw in the claimed proof of the abc conjecture. It is generally accepted by experts in the field that the Scholze-Stix paper Why abc is still a conjecture conclusively shows that the claimed proof is flawed. For a detailed discussion with Scholze about the problems with the proof, see here. For extensive coverage of the IUT story on this blog, see here.

Between paywalls and the limitations of Google translate, I’m not sure exactly what the process is for Scholze and Stix to collect their million dollars. Perhaps they just need to publish their paper, but it seems that the decision may be up to the businessman who is contributing the funds, and it’s unclear what his process will be.

For a few sources I’ve found, see here, here and here. If others have reliable and more detailed sources they can point to (especially anything in English), please do so.

**Update**: Press release here. Rules for the million dollars are

Nobuo Kawakami makes his own judgment as an individual.

The review method will not be disclosed, but the papers to be reviewed must be papers in mathematics that have been published on MathSciNet and have published more than 10 papers on arithmetic geometry in the past 10 years. Only papers that have been peer-reviewed and published in journals.

Scholze and Stix may not want to take time to submit their paper to a journal (Scholze has a history of turning down large prizes…). It occurs to me that there are quite a few arithmetic geometers who understand well the problem with the proof, could write something up and possibly get it published. Maybe a collaboration could be formed to do this.

**Update**: New Scientist has a story here. The quotes from Fumiharu Kato aren’t especially encouraging for IUT, who “estimates that fewer than 10 people in the world comprehend the concept.”

Kato believes that the controversy stems from the fact that Mochizuki doesn’t want to promote his theory, talk to journalists or other mathematicians about it or present the idea in a more easily digestible format, believing his work speaks for itself. Kato says that his current and former students are also reticent to do the same because they see him “as a god” in mathematics and don’t want to go against his wishes.

Because of this, most mathematicians are “at a loss” for a way to understand IUT, says Kato, who concedes that, despite earlier optimism about the idea, it is possible that the theory will eventually be disproven.

Ivan Fesenko is much more of a believer:

He told New Scientist that there is no doubt about the correctness of IUT and that it all hinges on a deep understanding of an existing field called anabelian geometry.

“All negative public statements about the validity of IUT have been made by people who do not have proven expertise in anabelian geometry and who have zero research track record in anabelian geometry,” he says.

**Update**: Scientific American has a news story about this, which summarizes the situation with the proof as:

So despite Mochizuki’s latest publication, there is still doubt among experts about the state of the abc conjecture. Most number theorists cannot make up their own mind because they are unable to follow the proof. And because both Scholze and Mochizuki enjoy an excellent reputation in their field, it is unclear who is right.

This gets the story quite wrong, and misunderstands how mathematics works. The problem is that there is no proof that anyone (Mochizuki himself included) can explain to anyone else, this is not about “do I believe this guy or the other guy?”. Yes, most mathematicians don’t have the technical knowledge to evaluate this sort of proof, but there are plenty who do, and they are either saying there is no proof, or, for the few supporting the proof, unable to explain it to anyone else.

Coming after your previous post – from the sublime to the ridiculous.

So sad about Mochizuki. He was a first rate mathematician, and now he is promoted by shysters.

So some telecommunications founder put up the money for this prize it seems. Seems to linked to the Zen university institute.

We all know this prize is never going to be awarded, right?

Not allowing for ZBMath, only MathSciNet? Not proposing anything, but technically there’s already a review published on ZBMath that’s critical of the IUT papers…

Peter, you say “It is generally accepted by experts in the field that the Scholze-Stix paper … conclusively shows that the claimed proof is flawed.” I wanted to highlight one comment by Taylor Dupuy, one of the people who seems to have engaged with the papers at a technical level (comment from near the end of the “detailed discussion” PDF):

Arkadas,

Dupuy acknowledges that he does not understand Mochizuki’s proof and is not arguing that it is correct. I believe he would agree that it should not have been accepted by a referee and published. As for his argument with Scholze, that is available in detail in the discussion you refer to, and the idea that he got the better of that argument is just laughable.

Besides traveling to Kyoto and extensively engaging with Mochizuki, Scholze is the most talented and knowledgeable mathematician in this field, Dupuy not so much. Anyone familiar with both of them would find it extremely unlikely that Dupuy would get the better of an argument between the two of them on this topic, and anyone who reads the discussion that happened can see that that expectation was fulfilled.

The problem I see with Scholze and Stix is certainly not that they’re not talented or knowledgeable enough but rather that, as far as I’ve seen, they haven’t been very forthcoming with their responses to the feedback they’ve gotten since their initial report and Scholze’s discussion with Dupuy. Mochizuki wrote a whole 100+ page paper supposedly analyzing and deconstructing the logical content of the version of IUT Mochizuki claims Scholze and Stix are oversimplifying his theory down to. Does anyone know what Scholze or Stix think of this paper? (We did get a response from Roberts, but I’m not sure if that lead to any discussion). In addition, Joshi’s work on so-called arithmetic spaces claiming to reformulate parts of IUT received a terse reply from Scholze on MathOverflow, but none further, even after Joshi’s lengthy response on Robert’s blog.

It’s possible there’s a lot of private discussion that’s going on, but if not, then I do get the impression that Scholze (and maybe Stix too) is not really interested in IUT any longer after their initial foray, which is fine since it’s not like they’ve run out of interesting mathematics. But it does mean the mathematical community should probably get a serious counter-response straight to Mochizuki’s response instead of relying on just two mathematicians. I think this challenge by IUGC presents a good opportunity for exactly this.

Soyoko U.,

Of the quite a few experts I’ve talked to, what they are saying is pretty consistent and uniform: Scholze identified the place (Corollary 3.12) where Mochizuki’s abc proof falls apart, he and Stix gave a serious argument that Mochizuki’s methods can’t fix this. The two of them put a lot of their time and effort into traveling to Kyoto and spending a week discussing this with Mochizuki. They left convinced he had no answer to the problem. In addition Scholze spent a fair amount of time during COVID lockdown going over this issue publicly with Dupuy and others, same result. Given this, there’s no good reason for anyone to spend any more time on this.

The only thing that can change this is for Mochizuki or someone else to come up with a new argument that successfully patches the flaw in the proof. Instead of revising his manuscript to do this, Mochiuki had his own journal publish the flawed version unchanged, and has a 156 page document explaining that Scholze is just too incompetent to understand an issue of elementary logic. I don’t see any reason Scholze should waste his time responding to it. If anyone who wants to can extract from it a serious argument that patches the proof, they could do so, but that has not happened. The latest circus with the million dollars and the wealthy donor who is going to decide the issue just goes further down a zero-credibility path.

I haven’t followed the details of Joshi’s work. My impression is that he believes that a significant variation on Mochizuki’s methods could work and prove abc. As far as I know, he hasn’t yet done what he needs to to get this taken seriously: write down a full proof of abc using his methods and have it refereed in the conventional way. I have no specific information about this, but I’m guessing Mochizuki is of no help, since he believes his own proof is correct, and that Scholze isn’t interested in spending more time on someone else’s research program that doesn’t look promising.

Peter,

Since both parties believe the other is incorrect with such conviction, it doesn’t seem like this circus will ever end. We agree that Scholze hasn’t wasted more of his time responding after the Dupuy discussion and probably won’t, and Mochizuki and his supporters are just going to stick to their guns and say Scholze and others oversimplified his theory, now going as far as to offer $1M for anyone who can provide a flaw. The dialogue ends, and Fesenko and Kato get funding for their own little institute with award ceremonies and high school brochures. Are we never going to see a resolution to this?

The prize seems to have necessary conditions, but no sufficient condition.

Soyoko U.,

From reports, the motivation of the person funding this is exactly to resolve the question of the validity of the proof. But that question already is resolved, it’s just that Mochizuki/Fesenko/Kato don’t like the answer and have been engaging in a campaign to attract publicity and money based on claims that a flawed proof is actually valid. This doesn’t seem to be getting any traction outside of Japan, so the problem of how to deal with it is mainly a problem for Japanese mathematics in general, and RIMS in particular.

This article yesterday in the Asahi Shinbun:

https://www.asahi.com/ajw/articles/14951963

Maybe the controversy could be settled if, instead, the money was aimed at a rewarding a formal computer proof of Mizoguchi argument (akin to the Liquid Tensor Experiment)

fbrx,

My reading of the prize terms is that you could get the million dollars for producing a formal proof. The problem with formalization is that so far neither Mochizuki nor anyone else can produce an acceptable standard sort of proof, so there’s nothing to formalize. In the Liquid Tensor Experiment, Scholze started by producing a detailed conventional proof which seemed to be valid, but was complicated so hard to be 100% sure. Formalization could be carried out since the argument was there, and allowed one to be sure there was not a subtle problem with the argument.

That said, Mochizuki and those who claim that the argument is there could convince others by producing a formalized version. This seems unlikely to happen, since they don’t seem to have the necessary starting point. It would be a great idea for them to start a project to do a formalization, they likely would then start seeing clearly what the problem with the proof is.

“Dupuy acknowledges that he does not understand Mochizuki’s proof and is not arguing that it is correct. I believe he would agree that it should not have been accepted by a referee and published. As for his argument with Scholze, that is available in detail in the discussion you refer to, and the idea that he got the better of that argument is just laughable.”

This is entirely unfair to Dupuy. In fact, not only do I not think it’s “laughable” that Dupuy “got the better of that argument” (if you must put it in those terms): just the opposite is true.

It is not “generally accepted by experts in the field that the Scholze-Stix paper . . . conclusively shows that [Mochizuki’s] proof is flawed.” With very few exceptions (of which neither I, nor you, Peter, are one), no one can make heads or tails out of any of this. What there is, perhaps, is a somewhat common thought that Scholze and Stix are prominent and pedigreed mathematicians, and so if they say it it must be right. But that is atrocious reasoning under the best circumstances, and these circumstances are themselves atrocious, and have been since this stupid saga began a decade ago.

Dupuy (along with Dimitrov, Joshi, and others) has put in enormous effort to understand the Mochizuki papers. Not how I’d spend my time, but he understands these matters at least as well as Scholze and Stix do. The latter have, in their MS, SKETCHED an objection to Mochizuki, but I imagine they would be the first to admit that it is not fully rigorous. How could it be, given that they, like the rest of us, struggle to understand what Mochizuki is saying in the first place?

Anyways, now we have this $1M “prize”, which I find distasteful. Perhaps talk about who is right and who is mistaken, which claims are true and which false, etc., is not apposite. We are in the world of, well, “not even wrong”.

ILikeGeometry,

I don’t think I’m being at all unfair to Dupuy. The documented discussion between Scholze, Dupuy and others that took place on my blog is something that I followed very closely since I was moderating it, in consultation with experts I know in this field.

The question is not whether Scholze-Stix have rigorous proofs, but whether Mochizuki has a rigorous argument. All evidence is that there are now zero people in the world who can defend the claim of existence of such a rigorous argument, in the sense of stating the argument clearly and answering questions about it satisfactorily. Again, it’s very important to note that Dupuy himself acknowledges he cannot defend such a claim.

The current situation is actually very simple and not at all unprecedented. A mathematician with a serious attempt at a proof of a major conjecture has had a problem with his proof pointed out to him, is unable to fix the problem, and the community of relevant experts has lost interest in wasting more time on this.

Woit,

I don’t totally see this as a satisfactory resolution, not until at least some of the people in Mochizuki’s camp can agree. The ultimate burden of proof is still on them, and perhaps I’m being idealistic, but I think any mathematician at this calibre where they can understand IUT should know if there’s a problem when pointed out to them by Scholze. Why would Fesenko, Go Yamashita, and others put their own credibility on the line to vouch for the theory despite the claimed flaws? If they believe it’s correct and that Scholze and Stix are wrong, I feel like we’re missing their voices in the story, especially since Dupuy doesn’t actually claim it’s correct.

There may be some survivorship bias in the experts that land on this blog. Have you talked to mathematicians at RIMS? I’m not sure if Mochizuki would want to, but how about the other experts actually doing IUT, such as Fesenko, Hoshi, Joshi, Kato, Collas, Minamide, Tsujimura, Yang, or Porowski who can answer questions? There’s been some severe public dialogue missing, and I hope this situation will be rectified sooner than later.

Soyoko U.,

Yes, the burden of proof is on Mochizuki and anyone else claiming there is a proof, not on anyone else. Why should anyone else now put any time and effort into this?

One group of people who I think have a motivation to do something are the editors of the RIMS journal that accepted the papers. The reputation of their journal has taken a serious hit because of this. They have access to the referee reports, which either contain a convincing argument Scholze-Stix are wrong, or don’t. The fact that they have chosen to not release publicly such an argument and accept the damage to their reputation speaks volumes.

I’ve also followed the discussion between Scholze, Dupuy and others on this blog.

Even though most of it went over my head, of course, I’ve got an impression that Dupuy was making a more subtle statement than just saying that Scholze and Stix were wrong.

My understanding was that one of the mathematical objects involved was defined in Mochizuki’s work using some imprecise language. Scholze and Stix made a most obvious interpretation of that language, and with this interpretation they demonstrated that by the same logic that Mochizuki used one arrives to a statement that is clearly wrong.

Dupuy argued that it is possible that there might be another interpretation of these Mochizuki’s words, to which Scholze-Stix refutation would not apply. The objection seemed to be that Scholze and Stix did not rule out potential existence of such interpretation.

Dupuy however did not provide such interpretation. Neither did Mochizuki or anybody else, at least not in comprehensible form. Apparently Mochizuki is unwilling to clarify this point, and nobody else is able to.

My personal conclusion from the discussion is that even if Dupuy is right, Mochizuki’s proof as published is at best incomplete.

For a detailed, mathematical and evidence based discussion of the Mochizuki-Scholze-Stix issues, let me point out my two articles intended for a wider mathematical audience:

(1) Comments on Arithmetic Teichmuller Spaces (https://arxiv.org/pdf/2111.06771)

(2) Mochizuki’s Corollary 3.12 and my quest for its proof (https://www.math.arizona.edu/~kirti/joshi-teich-quest.pdf) [My mathematical conclusion is that Scholze-Stix claim is based on a flawed premise.]

(3) Links to my preprints related to Mochizuki’s work can be found at (https://ncatlab.org/nlab/show/Kirti+Joshi) or in the preprints section of my webpage (https://www.math.arizona.edu/~kirti/).

(1) Let me also add the following note which may be useful for many readers: https://www.math.arizona.edu/~kirti/Response-to-grouchy-expert.pdf

(2) My guest blog post on David M. Roberts blog https://thehighergeometer.wordpress.com/2022/11/25/a-study-in-basepoints-guest-post-by-kirti-joshi/ is also recommended.

Kirti Joshi,

Thanks for the detailed references. My personal opinion at this point, given the sad history of discussion of these issues, is that the best thing for everyone would be following the conventional process: credible refereeing by a reliable journal when you have a complete proof. The more I think about it, the source of the mess this subject is in is the lack of credibility in the way the IUT proof was handled by the RIMS journal editors. Doing this right this time is crucial (and might get you a million dollars…).

From the perspective of another scientific field, I think Woit’s argument is unfair.

The timeline is,

– Mochizuki claimed to have proved it, and a paper was published in a journal that might have problems.

– Scholze and Stix pointed out the problem and had a dialogue with Mochizuki, which ended in a disagreement.

– Dupty and Joshi, objective third parties, argued that Scholze and Stix’s points were problematic.

– Scholze and Stix offered no rebuttal to those points.

Here, Woit is telling us to submit complete proof to reliable journals.

However, to move this discussion forward, either we point out new problems with Mochizuki’s proof, we accept the proof, or the mathematical community should set up a committee to verify the proof. Anyway, I don’t think the ball is in Mochizuki’s court.

KK,

First, to correct your timeline: the journal published the papers in 2020, Scholze and Stix was 2018. This really is the source of the whole problem. These papers should never have been published over the strong objections of experts in the field and the lack of any significant revision of the papers to successfully meet the objections.

In 2020, see

https://www.math.columbia.edu/~woit/szpirostillaconjecture.pdf

Scholze extensively rebutted Dupuy’s points.

Joshi is not defending the argument in the Mochizuki papers, but believes he has his own somewhat different argument. He believes that he sees how to overcome the problem pointed out by Scholze-Stix, but as far as I know he has not yet produced his own version of what the Mochizuki lacks (a detailed and convincing proof of Corollary 3.12 and thus abc).

I’ve just re-read that discussion between Dupuy, Scholze, amd others.

Near the end Dupuy tries to summarize the discussion. While still maintaining that “the Stix-Scholze manuscript should not be held up as a reason to reject Mochizuki’s proof”, Dupuy adds that “what Mochizuki has written is not a proof in traditional terms”.

This is because in Mochizuki’s proof “many of the definitions use nonstandard

terminology and come off as ambiguous to many readers” and “Moreover, because of certain ambiguities in these definitions, readers are forced to search a large space of possible meanings”.

Finally, “Because of these difficulties I personally hold Mochizuki’s document as a program for proving ABC and consider Mochizuki’s formula a conjecture”.

Dr. Woit, while you state in your posts that you know nothing about my work, but you are free with your opinions regarding my work. So let me clarify. Between 2018–2023, I have written a number of papers on Mochizuki’s work, with complete proofs and examples. I had them vetted by experts before posting them on the arxiv for further comments. This is the normal process which I follow for my math pre-publications.

(1) Regarding the Scholze-Stix manuscript, I have established that the very premise it is founded upon is incorrect and hence it provides no mathematical conclusion regarding Mochizuki’s work at all.

(2) I have already posted a paper on the arxiv (Constructions II: Proof of a local prototype of Mochizuki’s Corollary 3.12). This has all the bells-and-whistles of Mochizuki’s Corollary 3.12. The general case will appear in (Constructions III). There has been extensive discussion on your blog about this corollary and many have asserted that this corollary is impossible.

(3) As regards to the timeline, let me point out the following important events:

(3A) (2000, 2001) Teichmuller Theoretic Proofs of the Geometric Szpiro Inequality due to Fedor Bogomolov (NYU) et. al published in (2000), Shouwu Zhang (Princeton) published in (2001).

(3B) (2012) Mochizuki’s papers I–IV posted online.

(3C) (2016) Mochizuki’s posted his discussion of Bogomolov et. al. and Zhang proofs and importantly he points out in his paper that these proofs are a `Rosetta Stone’ for his own proof.

(3D) (2018) Scholze-Stix post their report.

(3E) (2020) Earliest version of my Untilts paper was circulated (I have comments to it from Scholze, Stix, Mochizuki, Hoshi and Kedlaya (later F. Kato and I. Fesenko and other anabelian geometers received copies). This paper demonstrates that arithmetic data required for construction of tempered fundamental groups is not rigid (as asserted by Scholze-Stix).

(3F) (2021-2023) Number of my papers elaborating my ideas are posted on the arxiv. They demonstrate some of the key claims of Mochizuki’s papers I–III quite robustly.

(4) No mathematical challenges (privately or publicly) have been raised to my papers (on this topic) to date.

(5) Importantly: Scholze-Stix make no reference to Bogomolov et. al., Zhang proofs and nor are they mentioned in any public discussion led by Scholze, and they are not mention in Scholze’s ZbMath review (which is also flawed for the same reasons as above). Mochizuki’s discussion of these proofs was available on his website (released in 2016). Scholze-Stix report was released in 2018. Omission is quite striking given the importance of Mochizuki’s arithmetic claim and its relationship to the geometric case (normal procedure in arithmetic is to understand the geometric case first (e.g. geometric Langlands and the Langlands correspondence resp.) [My discussion of the Bogomolov et. al., Zhang proofs including both, my point of view and Mochizuki’s appears in my (Constructions II(1/2)) paper.]

(6) For the lay-readers, in mathematical logic, a false proposition can be used to arrive at any conclusion one wants. So Scholze-Stix report provides no useful conclusion regarding Mochizuki’s work.

(7) A links to all my paper referred above can be found at https://ncatlab.org/nlab/show/Kirti+Joshi.

Kirti Joshi,

First of all, the only reason I’ve said anything about your work is that it is repeatedly used to justify claims for the validity of the Mochizuki proof. It would be helpful if you would state clearly whether the Mochizuki papers contain a valid proof of Corollary 3.12 and thus abc.

I’ve expressed no opinion and have no opinion about the validity of your argument that the Scholze-Stix objection can be overcome, allowing a proof of Corollary 3.12 and thus abc. Here I have just been stating the obvious: the way to conclusively resolve this is to produce such a proof and have it carefully checked by experts.

I am certainly no expert in any of this but it appears that Kirti Joshi has anted up a very respectable attempt at doing just what you have asked for.

Jack Morava,

My understanding is that Joshi has given a proof of a “local prototype” of Corollary 3.12, which is not strong enough to give abc, with a proof of the full result still to come in a “Constructions III” paper.

Thanks, looking forward hopefully to clearer skies…

Not directly related to this, but the comments to the original post are closed now: the big $1M+ prizes at the International Congress of Basic Science in Beijing went to Mumford and Adi Shamir. So, surprisingly, no Witten… A bunch of “best paper” prizes were also given to works published in the last ~five years. From a quick look, the selections (in fields I know about at least) seemed reasonable, though obviously there would have been many other equally deserving candidates.

There was a discussion about Kirti Yoshi’s preprint on MO with a direct refutation from Peter Scholze all happening in November 2022. See https://mathoverflow.net/a/435112

mnmltype,

Thanks, I had somehow missed that.

(1) @mnmltype, @Peter Woit: Both of you have evidently missed my responses to the claims made by Scholze and Sawin on MathOverFlow (in 2022). These claims are addressed in detail in my guest blogpost (2022) on David M. Roberts blog (https://thehighergeometer.wordpress.com/2022/11/25/a-study-in-basepoints-guest-post-by-kirti-joshi/) and in my Quest Paper (https://www.math.arizona.edu/~kirti/joshi-teich-quest.pdf). [These links are already posted in the list of links I have given above.] You may find my response to “Grouchy Expert” (also listed above) useful as well.

(2) It is clear that some people on both the sides of this debate are still unwilling to come to terms with incontrovertible mathematical facts uncovered by my investigations. I urge them to please read my mathematical writings on this matter.

(3) How many times does it have to be said that Scholze-Stix argument is based on a flawed mathematical premise and therefore it provides no valid mathematical conclusions regarding Mochizuki’s work? [Repeating an incorrect argument ad nauseam does not make it true.]

(4) To put it in Richard Feynman’s memorable words: “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.” I performed a mathematical equivalent of an experiment (see my Untilts Paper 2020-version) by finding a counter example and hence recognized that Scholze-Stix argument is wrong.

(5) For detailed mathematical proofs see my papers, links provided above or at https://ncatlab.org/nlab/show/Kirti+Joshi.

@Kirti Joshi

It is not possible to falsify a mathematical claim by introducing a new definition for terms used in the claim so that, under the new definition, the claim is false. The definition of “arithmetic holomorphic structures” you use in your work is – as you point out in the linked Quest paper, for example – different from Mochizuki’s.

Surely Scholze and Stix were thinking about Mochizuki’s definitions when they made their claim that distinct arithmetic holomorphic structures do not exist. The fact that, under your definition, they do exist, is not a falsification.

Mathematical works that consist in large part of new definitions, and proofs of theorems whose statements depend on these new definitions, can never be evaluated solely on a mathematical basis of “Are the proofs of the claimed theorems correct”? To evaluate the importance of the work one needs to estimate the importance and relevance of the new definitions. This will always be at least somewhat subjective and never incontrovertible.

@Will Sawin: I don’t really understand the logic of your above comments: are they meant to be a serious mathematical discussion or just musings for social media–I can’t tell. But I will address them all the same.

To refresh your memory, and for the record, let me say that in early November, 2022, I spent considerable amount of time explaining my work to you by email. It was clear to me from our correspondence, that you did not wish to be persuaded on this matter at all (as evidence I cite your subsequent Nov 2022 posts on Mathoverflow–to which I gave a detailed response in my guest blogpost on David Robert’s blog and I provided further details in my Quest Paper).

Now let me address your comments posted above:

(1) This time your defense of Scholze-Stix is that (I quote) “Surely Scholze and Stix were thinking about Mochizuki’s definitions when they made their claim that distinct arithmetic holomorphic structures do not exist.” I am sorry, I can only address their written mathematics and not what they may have been thinking.

(2) You assert that “The definition of “arithmetic holomorphic structures” you use in your work is – as you point out in the linked Quest paper, for example – different from Mochizuki’s.”

As I have demonstrated in my Untilts paper, my definition of arithmetic holomorphic structures provides arithmetic holomorphic structures in Mochizuki’s sense.

Let me add that both Mochizuki’s theory and my theory of these structures is based on the structure of tempered fundamental groupoids. Mochizuki and I read the information in the tempered fundamental groupoids differently, and these readings lead to our respective definitions, but there is no change in the essential content.

In fact as I point out in my Untilts paper, the structure of the tempered fundamental groupoid (of a variety over a p-adic field) is quite close to that of the Diamond associated to the said variety by Scholze.

(3) Whether you like it or not, and regardless of whether Mochizuki has said it or not, my observation is that the key premise of the Scholze-Stix report is fundamentally flawed. My reasoning is carefully and meticulously documented in my papers, with proofs, examples and all details. As noted in the Quest paper (see \S 1.8), amongst other things, the Scholze-Stix argument is flawed because of their claim that Teichmuller Theory cannot possibly exist because moduli theory exists!

(4) You state that “…This will always be at least somewhat subjective…”–nothing in this context or in mathematics in general is ever “somewhat subjective.”

(5) If you still have any questions about my papers and its relationship to Mochizuki’s work, I will be happy to visit your math department for a seminar and defend my work.

@Kirthi Joshi: You seem to be actively promoting your writings via various blog comments, blog posts, pdf documentations of your “quest”, etc., but it is not clear to me at all what *interesting theorems* you are claiming. Your papers – which I have read – seem to consist largely of definitions, constructions, and rhetoric. But I see no reason to care about the constructions you’ve cooked up so far, because you haven’t done anything *non self-referential* with them. Calling something an “arithmetic holomorphic structure” doesn’t automatically make it interesting!

All,

I think the argument with Kirti Joshi has gone well past the point of shedding light on anything and needs to end.

Nothing I’ve seen here or heard from others has changed the basic facts of the matter: no one has a proof of abc. Joshi believes he has a viable way to get a proof, other experts are skeptical. To convince the skeptics, he just needs to produce a proof, experts will look at it and see if it works. If he’s right, hew will richly deserve the $1 million.

I don’t think this is about the million dollars…

Alex Kontorovich has a great reaction to the SciAm article, see this Twitter chain: https://twitter.com/alexkontorovich/status/1685461590429081600?s=61&t=U5PW80qvOnhPbSXZ3SpfUQ

A formalization challenge!