A Diophantine puzzle

I’ve had this piece of paper for a couple of years now. The picture shows some solutions to a relatively low-degree Diophantine equation (no more than degree 4, and probably even just degree 3), and the arrows show how a solution can be modified to create a new solution. The coefficients are integers, of course. What is the equation? I can’t remember what it is!

New solutions from old

Joshi’s quest

Kirti Joshi has been doing what seems to me, as an outsider, interesting work at trying to rehabilitate Mochizuki’s ideas captured in Inter-universal Teichmüller Theory (IUT), in terms of existing mathematics. At least, when I skim through his recent papers on arithmetic deformation theory (using “arithmetic Teichmüller spaces”), they read like normal mathematics, rather than something alien and mysterious. Additionally, there’s not (at least so far) hundreds of pages of somewhat unmotivated technical work whose sole purpose is, allegedly, the proof of the abc-conjecture. Personally, early on in the saga, I had hoped that Mochizuki’s work on Frobenioids and what-have-you would turn out to have independent interest, and spark investigations into side-topics, but that has not happened. With Joshi’s work, however, it’s mostly applying existing machinery of p-adic arithmetic geometry with some interesting innovations, and—as far as I can tell—trying to recreate what Mochizuki wanted to do in a much more efficient and understandable way. If this work turns out to not achieve the great heights it aspires to, it would be for the ordinary reasons, and not because the formalism obfuscates what’s going on and the author feels everything is obvious, and chooses to explain why critics are wrong using vastly oversimplified metaphors in non-standard language.

In any case, Joshi has just released a document (disclosure: I was given earlier copies to have a read of, and invited to give comments) that details the philosophy of what his programme of arithmetic deformation theory is trying to achieve, and how. I’m not expert enough to check the technical details, but I don’t find this in the same class of writing as Mochizuki’s “explanations” of IUT; I can follow along, and things look familiar and sensible, regardless of any downstream application to hard Diophantine problems like abc. It would be interesting to get expert comments on this high-level document on its own terms, to probe its robustness, as it currently stands, leaving aside the community’s issues with Mochizuki’s work. I appreciate that references to Mochizuki pervade the linked document, but I think it’s possible to just focus on the mathematics at hand, and treat the links to IUT as a kind of set dressing, providing a level of external motivation.

Bessis on “the secret mathematics”; teaching and research

I thought these (by David Bessis, sourced from this review of his book Mathematica by Michael Harris ) were wonderful, worth sharing, and worth keeping in mind when trying to convey mathematical ideas to students. At the very least, they resonate with me!

Understanding mathematics is seeing and feeling, it’s traveling along a secret path that brings us back to the mental plasticity we had as children

When a human being is faced with a mathematical text, the aim is not to read from the first line to the last, as a robot would. The aim is to grasp “the thoughts between the lines” … Mathematical texts are written by humans, for humans. Without our ability to give them meaning, without “the thoughts between the lines,” there would be no mathematical texts.

For Thurston and for all mathematicians, mathematics is a sensual, carnal experience situated upstream from language. Logical formalism is at the heart of the apparatus that makes this experience possible. Mathematics books are unreadable but we need them. They are a tool that allows us to share in writing the true mathematics, the only one that really counts: the secret mathematics, the one that is in our head.

One might say that it’s all very well to have this type of philosophical picture about what mathematics “really is”, but how does that help a student get past a blockage in their current understanding? I believe that being mindful that these type of unspoken, and often unconscious, processes are happening in mathematicians’ (eg the lecturer’s) heads can help us remember that mathematical understanding is a complex thing, not always contained in the lecture notes. We have the opportunity to reflect on and convey our own understanding, our own “secret mathematics”, hard-won through years of persistence, to the students.

Not every student is going to end up as a career mathematician, but for those who do, here is the most important thing:

More than publications or official works, the mathematician’s great creation is the intuition that is the accomplishment of an entire life.

Sadly, of course, the practicalities of life, academia etc mean that the former things are also necessary and not sufficient, but aside from having to ensure mouths and fed and bills and paid, it is indeed such a incorporeal thing that is the most valuable creation of a mathematician. People praise this among those mathematicians who are considered great: their feeling of rightness for new mathematics that leads to the most fruitful research programs (eg: Noether, Atiyah, Grothendieck, Erdős, Conway, Thurston, …. all in their own ways), even above their actual results sometimes.

A study in basepoints: guest post by Kirti Joshi

[The text below the divider is a response by Kirti Joshi in response to some comments at MathOverflow regarding his recent preprint, “Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups — Arithmetic Holomorphic Structures“. Kirti reached out to me regarding making a response, and I suggested that a blog posting would be better than an answer at MathOverflow, since it is not in the format of an answer to the question, to which he agreed. Regular readers of the blog will know that I follow developments around IUT with close interest, but I am not an expert in that area. My long-stated hope is that some interesting mathematics comes out of the whole affair, regardless of specifics about the correctness of Mochizuki’s proof or otherwise. –tHG]

I wish to clarify my work in the context of the discussion here. For this purpose suppose that X is a geometrically connected, smooth quasi-projective variety over a p-adic field which I will take to be {\mathbb{Q}}_p for simplicity. In Mochizuki’s context this X will additionally required to be an hyperbolic curve.

  1. First of all let me say this clearly: one cannot fix a basepoint for the tempered fundamental group of X in Mochizuki’s Theory [IUT1–IUT4]. The central role of (arbitrary) basepoints play in Mochizuki’s theory is discussed in (print version) [IUT1, Page 24], and notably the key operations of the theory, namely the log-link and theta-link, change or require arbitrary basepoints on either side of these operations [IUT2, Page 324] (and similar discussion in [IUT3]).
  2. This means one cannot naturally identify the tempered fundamental groups arising from distinct basepoints. [The groups arising from different basepoints are of course abstractly (and non-canonically) isomorphic. Mochizuki does not explicitly track basepoints while requiring them and so this makes his approach extremely complicated.]
  3. In the context of tempered fundamental groups, a basepoint for the tempered fundamental group of X is a morphism of Berkovich spaces \mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p}, where K is an algebraically closed complete valued field containing an isometrically embedded {\mathbb{Q}}_p. [Such fields are perfectoid.] As I have detailed in my paper, arbitrary basepoints requires arbitrary perfectoid fields K containing an isometrically embedded {\mathbb{Q}}_p. [For experts on Scholze’s Theory of Diamonds, let me say that the datum (X^{an}_{{\mathbb{Q}}_p}, \mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p}) required to define tempered fundamental group with basepoint \mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p} is related (by Huber’s work) to a similar datum for the diamond (X^{ad})^\diamond associated to the adic space for X/{\mathbb{Q}}_p.]
  4. In my approach I track basepoints explicitly (because of the (1) above) and I demonstrate how basepoints are affected by the key operations of the theory. [This is claimed in Mochizuki’s papers, but I think his proofs of this are quite difficult to discern (for me).]
  5. Because basepoints have to be tracked, and tempered fundamental groups arising from distinct basepoints cannot all be naturally identified, assertions which involve arbitrarily identifying fundamental groups arising from distinct basepoints cannot be used to arrive at any conclusion about Mochizuki’s Theory.
  6. In arithmetic geometry one typically works with isomorphism classes of Riemann surfaces i.e. with moduli of Riemann surfaces. Teichmuller space requires a different notion of equivalence and it is possible for distinct points of the classical Teichmuller space to have isomorphic moduli. This is also what happens in my p-adic theory.
  7. There is no linguistic trickery in my paper. I have developed my approach independently of Mochizuki’s group theoretic approach and my approach is geometric and completely parallels classical Teichmuller Theory. Nevertheless in its group theoretic aspect, my theory proceeds exactly as is described in [IUT1–IUT3] and arrives at all the principal landmarks with added precision because I bring to bear on the issues the formidable machinery of modern p-adic Hodge Theory due to Fargues-Fontaine, Kedlaya, Scholze and others. This precision allows me to give clear, transparent and geometric proofs of many of the principal assertions claimed in [IUT1–IUT3] without using Mochizuki’s machinery. [Notably my view is that Mochizuki’s Corollary 3.12 should be viewed as consequent to the existence of Arithmetic Teichmuller Spaces (at all primes) as detailed in my papers. One version of Corollary 3.12 is detailed (with proof) in my Constructions II paper. The cited version works with the standard complete Fargues-Fontaine curve {\mathscr{X}}_{{\mathbb{C}_p^\flat},{\mathbb{Q}}_p}, but there is a full version better adapted for Diophantine applications which works with the adic Fargues-Fontaine curve {\mathscr{Y}}_{{\mathbb{C}_p^\flat},{\mathbb{Q}}_p} and its finite étale covers and which exhibits the tensor packet structure of [IUT3, Section 3] which will appear in Constructions III (the two proofs are similar).] No claims are presently being made about the main result of [IUT4]. That is a work in progress.
  8. Apart from its intrinsic value, my original and independent work provides new evidence regarding Mochizuki’s work and I urge the mathematical community to reexamine his work using the emerging mathematical evidence.
  9. I am happy to talk to any mathematician who is interested in my work and I have written my papers as transparently as possible. [If there are any concrete mathematical objections to my papers, I will be happy to address them.]
  10. I thank David M. Roberts for a number of suggestions which have helped me improve this text.

Russell’s paradox: the original letter to Frege

Russell’s Paradox is famous for having put a nail in the coffin of naive set theory, even if not directly, and it’s the type of thing that many people have been introduced to in varying ways: from the parable of the barber who only shaves those who don’t shave themselves, to the more “official” set of all sets that don’t contain themselves.

A version of Russell’s paradox, taken from Logicomix.

But, the original presents the result in two ways, only the second way is closer to what is presented. The first is in a much more Fregean style, not in a form of naive set theory. A translation of Russell’s letter (which was in German) is available in the excellent book From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, but that is not exactly filled with light reading. So here is a transcribed version of the one-page letter.

Note that Russell’s Paradox was not the first of the “antinomies” to be discovered: the Burali-Forti Paradox predates it by five years. And Ernst Zermelo also independently discovered Russell’s result a few years earlier, and communicated it privately to Hilbert, Husserl and possibly others. But it’s become the icon for the transition point between naive and axiomatic set theory, which was published by Zermelo in 1908.

The core of a Lie category is an open Lie subgroupoid

In Ehresmann’s first paper introducing what we today call Lie groupoids and topological groupoids, as well as categories internal to \mathbf{Top}, is Catégories topologiques et catégories différentiables, dating to 1959 and appearing in a rather obscure place. It is also available in his Œvres Complète, Part I, which are thankfully now online and available for free. Ehresmann in that paper also also talks about what might be called Lie categories, namely categories internal to \mathbf{Mfld}, with the added caveat that the source and target maps have locally constant rank—but I think it is also implicitly assumed that they are submersions. What I wanted to record was a write-up of a proof of the first result of that paper, which is that given a Lie category X, the maximal subgroupoid, the core \mathrm{Core}(X), of the underlying category, forms a Lie groupoid whose manifold of arrows is open in the manifold of arrows of X.

This is analogous to, and a vast generalisation of, the example where one has the Lie category with a single object, \mathbb{R}^n, and the manifold \mathrm{End}(\mathbb{R}^n) of linear maps for arrows; it is well known that in this instance GL(n) \subset \mathrm{End}(\mathbb{R}^n) is an open submanifold, given by the condition \det(A)\neq 0. It is not difficult to show that in fact the same is true for the Lie category \mathbf{Mat} where the objects are natural numbers, and \mathbf{Mat}(n,m) := \mathbf{Vect}_{\mathbb{R}}(\mathbb{R}^n,\mathbb{R}^m), the vector space of linear maps. Here the manifold of objects is zero-dimensional, and we have different dimensions among the connected components of the arrow manifold. So you can imagine that Ehresmann’s result generalises the case where the manifold of objects is positive-dimensional, and where one does not have a simple criterion measuring invertibility like determinant.

When I first found this result, just over a decade ago, it was utterly mysterious. My experience with the nitty-gritty of differential geometry was at that time relatively limited. Not to mention a) it was in French and b) it pre-dated standard modern treatments of differential geometry and c) it was by Ehresmann, who has a distinctive style. It seemed interesting, and perhaps a little bit important, but I didn’t know what to do with it. I asked on Math.Stackexchange about the analogous setup for monoids internal to schemes, and then on MathOverflow for the general case of categories in schemes, with some nice answers, but nothing I could use at that point.

Most of Ehresmann’s proof of the headline result, from his 1959 paper Catégories topologiques et catégories différentiables

Several times since then I’ve looked at it casually and tried to satisfy myself, and finally it clicked. So this blog post will explain the proof in my own words and notation, and with all the details that I felt were missing.

Continue reading “The core of a Lie category is an open Lie subgroupoid”

New paper — Rigid models for 2-gerbes I: Chern–Simons geometry

This is to finally announce the release of a slow-burn project joint with Raymond Vozzo …. at least, Part I of it.

  • DMR, Raymond F. Vozzo, Rigid models for 2-gerbes I: Chern–Simons geometry, arXiv:2209.05521, 63+5 pages.

Here’s the abstract:

Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make explicit calculations easier, for instance in applications to physics. To compare to the existing definition, we give a functorial construction of a bundle 2-gerbe as in the literature from our rigid model, including with connections. As an example we prove that the Chern–Simons bundle 2-gerbe from the literature, with its connective structure, can be rigidified—it arises, up to isomorphism in the strongest possible sense, from a rigid bundle 2-gerbe with connective structure via this construction. Further, our rigid version of 2-gerbe trivialisation (with connections) gives rise to trivialisations (with connections) of bundle 2-gerbes in the usual sense, and as such can be used to describe geometric string structures.

The entire point of this project is the drive to make calculations simpler. We started out gritting our teeth and working with bundle 2-gerbes as defined by Danny Stevenson (based on earlier work by Carey, Murray and Wang), but there was a combinatorial growth in the complexity of what was going on, even though it was still only in relatively small degrees. Ultimately, through a process of refining our approach we landed on what is in the paper. The length of the paper is due to two things: keeping detail from calculations (it’s done in eg analytic number theory papers, why not here? I dislike the trend in some areas of pure maths to hide the details to make the paper look slick and conceptual, when there’s real work to be done), and the big appendix. Oh that appendix. It was a useful exercise, I think, to actually work through the (functorial) construction of a bundle 2-gerbe as in the literature from one of our rigid bundle 2-gerbes. Relying on a cohomological classification result here feels a bit too weak for my liking (and, additionally, requires building up the classification theory, here we can build things by hand).

The introduction is intended to give an overview of the main ideas, so I will point you there, but perhaps this is the best place to outline the plan for the rest of the project. The main result of Part II will be to construct a universal, diffeological, Chern–Simons-style rigid bundle 2-gerbe, rigidifying the usual universal 2-gerbe on a suitable K(\mathbb{Z},4). Moreover, we build explicit classifying spans (that is, a span of maps the left of which is shrinkable) for the basic gerbes on arbitrary suitable G, taking the universal ‘basic gerbe’ to arise from a diffeological crossed module much as the basic gerbe is intimately linked to the crossed module underlying the string 2-group of G. This permits us to justify focussing on a such a rigid model, in that we will then know every bundle 2-gerbe should be stably isomorphic to a bundle 2-gerbe with a rigidification in our sense. Finally, Part III will give the intended first main application of this project, namely explicit geometric string structures for a wide class of examples. There are some spinoff applications, but they are further down the line.

Unifying ε-N and ε-δ arguments

I’m teaching an intro analysis topic at the moment, and so of course there’s the whole ordeal of introducing ε-δ arguments. However, when we say such a thing, we usually also have in mind the type of proofs that are used for convergence of sequences, which are not usually ε-δ, which is to do with continuity of a function, but involve finding some large N past which the sequence is close to a limit, or else some Cauchy-type condition: hence either | a_n - L| < \varepsilon or | a_n - a_m| < \varepsilon, for all n \geq N.

However, it is possible to present a convergence proof as a continuity proof, using \delta. This is not a massive secret, but it’s cute, so I thought I’d write it up.

Let us fix some data: a sequence (a_n) in \mathbb{R}. Such a thing is a function a\colon \mathbb{N}\to \mathbb{R}. We say that (a_n) converges to L\in \mathbb{R} if:

\forall \varepsilon > 0,\ \exists N\in \mathbb{N},\ \forall n> N,\ |a_N - L| < \varepsilon

Another way to think about is is if we know that the function a extends to a function a'\colon \mathbb{N}_\infty = \mathbb{N} \cup \{\infty\}\to \mathbb{R}, satisfying a special property, then we have convergence. If we have convergence, we can clearly define such an extension where a'(\infty) = L. So what is this special property? It’s nothing other than continuity of a', where we have to put a particular topology on \mathbb{N}_\infty! From the point of view of pure point-set topology, we specify the neigbourhoods of \infty to be the cofinite subsets of \mathbb{N}_\infty containing \infty—that is, the subsets only missing finitely many elements of \mathbb{N}\subset \mathbb{N}_\infty—and otherwise for every n\in \mathbb{N}, \{n\} itself is a neighbourhood. Thus \mathbb{N} is a discrete subset, and only the point \infty “has nontrivial topology”, as it were. This at least means continuity of a' makes sense.

But the ε-δ definition of continuity is a metric definition, namely it’s treating \mathbb{R} as a metric space. So how do we make \mathbb{N}_\infty a metric space, so that the topology just defined is the metric topology? Here’s where we see the link we seek. Recall the fundamental sequence \frac{1}{n+1} \to 0 in the reals, whose convergence characterises Archimedean ordered fields. We can think of the set \mathcal{N}_0 := \{\frac{1}{n+1}\in \mathbb{R}\mid n\in \mathbb{N}\}\cup \{0\} = \mathcal{N}\cup \{0\} as inheriting the subspace topology from the reals, and in fact this gives something homeomorphic to \mathbb{N}_\infty. And, since \mathbb{R} is a metric space, we can give the subset a metric inducing this topology: we have d(\frac{1}{n},\frac{1}{m}) = |\frac{1}{n} - \frac{1}{m}| and d(0\frac{1}{n}) = \frac{1}{n}

However, this metric is slightly awkward, but it at least can inspire a slightly nicer metric d' on \mathbb{N}_\infty, namely d'(\frac{1}{n},\frac{1}{m}) = 1 and d(\infty,\frac{1}{n}) = \frac{1}{n}. There is a bijective short map \mathbb{N}_\infty\to \mathcal{N}_0 (n\mapsto \frac{1}{n},\ \infty \mapsto 0, but which is not invertible as as short map (nor even as a Lipschitz map, or a uniformly continuous function), but which is still a homeomorphism. As far as mere continuity goes, either version of the space is ok, but we will use \mathbb{N}_\infty with the metric d' and the resulting topology. So here we see where our very patient \delta is going to come in. A function \mathbb{N}_\infty\to \mathbb{R} (where both of these are considered as metric spaces) is continuous, precisely if

\forall \varepsilon > 0,\ \exists  \frac{1}{N},\ \forall \frac{1}{n}< \frac{1}{N},\ |a(n) - a(\infty)| < \varepsilon

So we could in principle dispense with the N, and only consider positive \delta, namely

\forall \varepsilon > 0,\ \exists \delta > 0,\ \forall \frac{1}{n}< \delta,\ |a_n - L| < \varepsilon

where we have set L:=a(\infty), and a_n = a(n), as usual. This is what happens if we know (a_n) has a limit. If we were to ask whether it has as limit, we should ask instead whether there is any continuous extension of \mathbb{N}\to \mathbb{R} along the inclusion \mathbb{N}\hookrightarrow \mathbb{N}_\infty. The usual proof of uniqueness of limits of sequences in metric space can be adapted to show that if such a continuous extension existed, then there is exactly one of them.

A similar descriptions can be made for the Cauchy property, except in this instance, one really does need to use the metric space (\mathcal{N},d) (not the space \mathcal{N}_0!), so that the function \mathbb{N}\to \mathcal{N} is a Cauchy sequence. Notice here that we do not have the limit point, since Cauchyness doesn’t refer to any actual limiting value, even assuming one exists. If we include \{0\}, hence consider the metric space \mathcal{N}_0, then we are dealing with a Cauchy sequence known to be convergent (in the reals, this is all Cauchy sequences, but one can consider all this in the rationals, for instance, where only some Cauchy sequences have a limit).

Thus there are four different things happening here. And this is where I put on my category-theorists hat: we have 1) a generic sequence, 2) a generic convergent sequence, 3) a generic Cauchy sequence, and 4) a generic convergent Cauchy sequence. There are maps from the generic sequence to the generic Cauchy sequence, from the generic sequence to the generic convergent sequence, from the generic convergent sequence to the generic convergent Cauchy sequence, and from the generic Cauchy sequence to the generic convergent Cauchy sequence. Since a X-sequence (say in \mathbb{R}, but it works in any metric space) is given by a continuous function from the generic X-sequence, precomposing with these maps just described forget properties of the sequence (for instance, take a convergent sequence together with its limit, and then forget what the limit is, or take a Cauchy sequence and forget this fact). I’ve been a bit sloppy as to what category all this is happening in, it should at least be metric spaces and continuous maps, but one could see if the screws could be tightened, and something like this work in eg uniformly continuous maps. I leave this as an exercise for the reader.

Convergence of an infinite sum in the rationals

I’m teaching an intro to analysis course this semester, and we are starting with the usual axiomatic treatment of numbers. I made a small emphasis on the rationals as a Archimedean field, and we can actually start with the analysis before we even get to talking about the real numbers. Moreover, since everything here is so close to the metal, we can be proving results at the level of using induction.

I wanted to use this blog post to record the proof using no more than the rationals (i.e. no embedding things into the real numbers), that the geometric series \sum_{n=0}^\infty x^n converges in \mathbb{Q} for 0 < x < 1 and rational. One can perform the usual manipulation of partial sums (possible in \mathbb{Q}) to get

\sum_{n=0}^\infty x^n = \dfrac{1}{1-x} - \dfrac{1}{1-x}\lim_{n\to \infty} x^n

(assuming the RHS exists) and hence it suffices to prove that \lim_{n\to \infty} x^n = 0. It is easy to prove (say with induction) that x^{n+k} < x^n for all k > 1.

Then the limit is zero when for all N\in \mathbb{N}, we can find some n\in \mathbb{N} such that x^n < \dfrac{1}{N}. This is close to being dual to the statement of the Archimedean property, which is of the form \exists N,\ \dfrac{1}{N} < y, for each positive rational y. Initially I thought of trying to leverage the Archimedean property for the positive rational numbers, in a multiplicative sense (Archimedeanness makes sense for any ordered group), but I didn’t end up making this work (more on this below). Ultimately I found an argument in Kenneth Ross’ book Elementary analysis, which I simplified further to the following.

We write x = \dfrac{1}{1+a/b} for positive integers a, b, since x < 1. One can prove (by induction) the estimate (1+a/b)^n > na/b (Ross had this as a corollary of the binomial theorem, but there is a simple direct proof). Then we have x^n =\dfrac{1}{(1+a/b)^n} < \dfrac{1}{na/b} = \dfrac{b}{a}\cdot\dfrac{1}{n}. Given N\in \mathbb{N}, we can choose n = bN, so that x^n < \dfrac{b}{abN} \leq \dfrac{1}{N}, as needed.

What I like about this argument is that it uses nothing other than the ordered field axioms on \mathbb{Q}, together with two very easy applications of induction. It’s a lovely proof to present to an undergrad class.

Returning my failed first idea at a proof, I had reduced the problem to that of showing that for every rational x > 1, there is an n such that x^n > 2 (challenge: can you leverage this fact to conclude the convergence as desired? It’s the base case of an induction proving the multiplicative Archimedean property). User @Rafi3AK on Twitter supplied an explicit estimate of the required n, using the binomial theorem, namely n = \lceil 1/(x-1)\rceil (i.e. round up to the ceiling).