Also adding where the proof falls apart:

- Peter Scholze,
*Global character of ABC/Szpiro inequalities*, MathOverflow (version: 2024-03-29).

Anonymouse

]]>added recent preprint

- Kirti Joshi,
*Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture*(arXiv:2403.10430)

Anonymouse

]]>Ah, now I see where my first edit went. It was deleted by Anonymous in rev 21.

If that article has problems, add a comment and let us know.

]]>I had re-organized the references-section, in order to disentangle general discussion from IUT stuff.

Then I also added this pointer:

Relating the abc conjecture to D=4 N=4 super Yang-Mills theory:

- Yang-Hui He, Zhi Hu, Malte Probst, James Read,
*Yang-Mills Theory and the ABC Conjecture*, International Journal of Modern Physics A Vol. 33, No. 13, 1850053 (2018) (arXiv:1602.01780, doi:10.1142/S0217751X18500537)

I can only hope that Mochizuki is patiently explaining his ideas in similar sorts of workshops.

As somebody in that blog discussion pointed out, on *Mochizuki’s Home page – What’s new‘?* is announced a workshop lecture that he is going to give next month, and the slides for that lecture he has here.
I gather the key idea is that alluded to on slide 4.

I haven’t been following the developments. But the discussion you linked to was of a very general sort and had little mathematical content.

Something in this situation reminds me of what Thurston wrote in his On Proofs and Progress in Mathematics (p. 15):

I also gave many presentations to groups of mathematicians about the ideas of studying 3-manifolds from the point of view of geometry, and about the proof of the geometrization conjecture for Haken manifolds. At the beginning, this subject was foreign to almost everyone. It was hard to communicate—the infrastructure was in my head, not in the mathematical community. There were several mathematical theories that fed into the cluster of ideas: three-manifold topology, Kleinian groups, dynamical systems, geometric topology, discrete subgroups of Lie groups, foliations, Teichmu ̈ller spaces, pseudo-Anosov diffeomorphisms, geometric group theory, as well as hyperbolic geometry.

We held an AMS summer workshop at Bowdoin in 1980, where many mathematicians in the subfields of low-dimensional topology, dynamical systems and Kleinian groups came. It was an interesting experience exchanging cultures. It became dramatically clear how much proofs depend on the audience. We prove things in a social context and address them to a certain audience. Parts of this proof I could communicate in two minutes to the topologists, but the analysts would need an hour lecture before they would begin to understand it. Similarly, there were some things that could be said in two minutes to the analysts that would take an hour before the topologists would begin to get it. And there were many other parts of the proof which should take two minutes in the abstract, but that none of the audience at the time had the mental infrastructure to get in less than an hour.

At that time, there was practically no infrastructure and practically no context for this theorem, so the expansion from how an idea was keyed in my head to what I had to say to get it across, not to mention how much energy the audience had to devote to understand it, was very dramatic.

I can only hope that Mochizuki is patiently explaining his ideas in similar sorts of workshops.

]]>Concerning #42: Zoran did you find time to check which theorems you were thinking of?

Generally: has anyone here further been following the developments? I see in discussion such as here that a certain scepticism is being felt in the number theory community.

]]>@hilbertthm90,

The Mordell conjecture completed a classification of the behavior of rational points on curves over Q. For genus 0 you get no points, or something isomorphic to P 1 and hence infinitely many. For genus 1, the Mordell-Weil theorem tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the case of finite can occur for higher genus. This is amazing!

This result also implies many non-trivial results. For example, fix a finite set of primes S, a dimension n, and a polarization degree d. There are only finitely many abelian varieties of dimension n and polarization degree d with bad reduction inside S. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3’s.

Thanks, that’s useful! I have moved that into the entry. Feel invited to further expand there.

Is the point that you want it to be interesting by relating to something that can be tied back to physics?

No. The point of my request since #2 is that I think an $n$Lab entry on something important/interesting should explain why that something is important/interesting. I am not an expert on this stuff, but I guess I know enough math that somebody who is can explain to somebody like me why he is fascinated by these conjectures. You just made a very good start. That helped, thanks.

]]>Hopefully you are not deducing that something is the more interesting the more people talk about it! ;-)

Well, talking is not always communication. If you ”accept” an (incoming) communication (and you actively participate in it), you put the things being said in relation to what you already know and there is some resonance in your feeling and you will formulate an answer and thereby develop the subject in discussion… However the outcome of this communication may be that there is no strong relation between the discussed topic and the concepts you are interested in, but for the time the communication lasts - I think - you have an investigative interest in the topic, since you wish to put yourself into perspective to it.

But if we consider a group of people having no mathematical background and who just understand that a difficult problem has been solved, then they might have a discussion on difficult problems in general, but they won’t have a rich communication on the abc-conjecture.

]]>49 I find this link you found counterproductive to my interest, and you will likely agree. If somebody (Mordell, Faltings etc.) tells me that a striking and conceptual result of finiteness of number of points of ALL higher genus curves over rationals should be interesting because some particular series of diophantine equations (Fermat one) has no solution than I am really perplexed. Every statement in math (even the least interesting) can by Matiasevich theorem be reduced to solving a diophantine equation (the reduction is complicated and never illuminating to the very problem at hand). The Fermat equation may look nicer than average equation but one can easily write many diophantine equations which look nice to a person literate in elementary school algebra but which are of no importance in math and whose solution is often easy to diophantine experts. Truly, eventually it occurs that solving Fermat’s problem people found that it is related to some beautiful mathematics of elliptic curves and the problem is more known than Mordell among school boys and girls, and among mushroom hunters, but I would say the Mordell’s conjecture is appealing in its very statement (especially if you know the facts about genus 0 and 1, as 50 has kindly explained), while the Fermat problem becomes conceptually interesting only after a long chain of deductions resulting in deeper facts like modularity.

]]>That discussion reminded me of the following:

There is a paper: Moore and Seiberg equations, topological field theories and Galois theory, by Pascal Degiovanni. It attempts to show the links between TFTs and the questions in Grothendieck’s Esquisse. There are conjectured links between Rational Conformal Field Theories, and the Teichmuller tower, but from there to IUTT seems a very long way. (The paper is in the Dessins d’Enfants LMS lect. Notes 200, volume from 1994. There is a much longer (85 pages) version on the web in French. The short version is also on the web.

A Google search starting from there did give some other links, but not directly linked to the more purely number theoretic conjectures, more to Grothendieck’s starting point.

]]>“From time to time I keep googling for information on why these conjectures are interesting.” This must mean that you don’t find number theory in and of itself to be interesting, because I’d say most results in number theory are interesting because they relate to these ones not the other way around. They are quite fundamental.

Some examples: The Mordell conjecture completed a classification of the behavior of rational points on curves over Q. For genus 0 you get no points, or something isomorphic to $P^1$ and hence infinitely many. For genus 1, the Mordell-Weil theorem tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the case of finite can occur for higher genus. This is amazing!

This result also implies many non-trivial results. For example, fix a finite set of primes $S$, a dimension $n$, and a polarization degree $d$. There are only finitely many abelian varieties of dimension $n$ and polarization degree $d$ with bad reduction inside $S$. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3’s.

It proves the Isogeny Theorem which says that abelian varieties are isogenous if and only if they have isomorphic Tate modules. This one should fascinate anyone interested in geometry. It tells you exactly how much geometric information you can recover just by knowing a single cohomology group!

Is the point that you want it to be interesting by relating to something that can be tied back to physics? I think there are results in math that can be quite interesting just by relating them to more math. I dare say that this result is one of the foundations of modern number theory because of the use of Galois representations to prove something geometric.

]]>From time to time I keep googling for information on why these conjectures are interesting. I keep thinking that there will be some interesting implication of Mordell’s conjecture to something like Gromov-Witten theory or the like. But it seems to be tough.

Googling for

“abc conjecture is interesting”

yields 0 hits.

Googling for

“Mordell conjecture is interesting”

yields a single hit. That says that it is interesting because it says something about Fermat’s last theorem.

]]>a lot of people who can communicate with each other very well since they are doing similar things.

Hopefully you are not deducing that something is the more interesting the more people talk about it! ;-)

]]>“Can somebody say why it is interesting?”

Jürgen Habermas tried to detail the relation of knowledge and human interest (see also the enty in the german wikipedia web). In short he follows the pragmatism of Charles Sanders Peirce, who says that process of acquiring knowledge is bound to the collective of researchers who solve their task by communication. This process is conducted by knowledge-interests which need to be communicable. So Habermas would probably say that a necessary condition for the abc-conjecture to be interesting is to be communicable. Because it is hard to proof it can moreover constitute a long term research program involving a lot of people who can communicate with each other very well since they are doing similar things.

But I admit that some aspects of this have been said above. Also it gives only a necessary and no sufficient condition for hard problems in number theory to be interesting.

]]>I see. Actually reading those questions with hindsight it’s fairly obvious it wasn’t John.

To question Minhyong properly, it would probably be as well to understand his MO answer.

]]>@David C.,

re #8: it turns out that this interview is not actually *by* John Baez. It is by some reporter and John has only been forwarding it. John just pointed this out in the comment section. See at the very end currently.

Thanks. I have also copied it to and cross-linked with Mordell conjecture.

]]>I added a link to the ‘letter to Faltings’ at anabelian geometry.

]]>Can you say more concretely which ones?

Yes, if I go back to recheck my readings in last couple of days. But tomorrow and over the next week I am not likely to spend more time on this issue (our group collaborator is arriving on Saturday, among other things). I’ll try some evening though if time permits.

]]>some other deep theorems (related to progress partial results on Vojta’s conjecture and so on) are allegedly also used

Can you say more concretely which ones?

]]>Mochizuki wrote:

The notion of anabelian geometry dates back to a famous “letter to Faltings” [cf. [Groth]], written by Grothendieck in response to Faltings’ work on the Mordell Conjecture [cf. [Falt]].

So far as Mordell conjecture being not clearly interesting.

Urs said

Funny that you say that, given that the conjecture is now allegedly proven by way of anabelian geometry.

The fact that this is one of the main ingredients used is not in contradiction with what I am saying; some other deep theorems (related to progress partial results on Vojta’s conjecture and so on) are allegedly also used. This did not boil down to SOLELY anabelian geometry (except in the logical trivial sense that any theorem boils down to any single step in its proof while all other steps being done) nor the latter (anabelian geometry) boils down only to topos theoretic generalities (while it can be formulated in a specific instance of a such a topos theoretic setup).

The subject of number theory has its own conceptual ingredients DIFFERENT from topoi. The mathematics is of course intertwined and whenever the things COMBINE they become truly deep. The things which are just one kind of nonsense are more superficial than non-obvious combinations of things. As Manin once said, the deep problems in mathematics in their solution usually involve at least 5 different fields…

]]>some subjects have different conceptual ingredients, which do not boil down to “topos” nonsense but can freely combine with it.

Funny that you say that, given that the conjecture is now apparently proven by way of anabelian geometry.

]]>Mochizuki wrote:

The relationship between the classical “set-theoretic” approach to discussing mathematics — in which specific sets play a central role — and the “species-theoretic” approach considered here — in which the rules, given by set-theoretic formulas for constructing the sets of interest [i.e., not specific sets themselves!], play a central role — may be regarded as analogous to the relationship between classical approaches to algebraic varieties — in which specific sets of solutions of polynomial equations in an algebraically closed field play a central role — and scheme theory — in which the functor determined by a scheme, i.e., the polynomial equations, or “rules”, that determine solutions, as opposed to specific sets of solutions themselves, play a central role.

In fact this can be formalized by noting that the definable sets correspond to certain class of presheaves on the category of models and elementary monomorphisms. So if one works with appropriate notion of a scheme then I can make two remarks. First that the presheaf is typically here not representable. Second if it were representable the morphisms are typically different already in the underlying category: the elementary monomorphisms are not the typical kind of morphisms in mathematical practice. Rather one looks at natural transformations (morphisms of presheaves) and the naturality is here quite a constraint, while at the level of components one has also the usual restrictions of kind of morphisms in usual mathematics one desires for that kind of objects.

]]>