Xena on Counterexamples

Posted on July 22, 2026 by Persiflage

There is an interesting post here on Xena’s blog.

I don’t centre “proof” in mathematics to anywhere near the extent that Kevin does, but I think emphasizing that distinction obscures the fact that our opinions about mathematics, mathematicians, and, I suspect, AI are quite similar.

It is fascinating to observe the progress of AI in mathematics (other people might use different adjectives). The result that a finite group scheme of order \(n\) need not be annihilated by \(n\) is the first AI-assisted result that I actually “care about.” I most closely associate this problem with a remark Hendrik Lenstra (the human equivalent of an LLM in the 90s) once made to me in the elevators of Evans Hall: namely, that there is a weaker result asserting that every finite flat group scheme \(G\) of order \(n\) is annihilated by \(n^{c(n)}\), for some integer \(c(n)\) independent of \(G\). The point is that there is, in an appropriate sense, a universal family of group schemes of order \(n\) over a finite-type base \(S\), where \(S\) is more or less the parameter space required to write down all the necessary Hopf-algebra data.

I absolutely agree with Kevin’s reaction to reading LLM-generated mathematics. Reading mathematics is already extremely hard; we are used to the fact that we might have to spend weeks understanding a few lines in a difficult paper. But what gives us the spirit to persevere is that we have been trained to give the author the benefit of the doubt: it is we who are missing the idea, and if only we think a little more about it, we will work it out. For better or worse, this is a cultural truth of mathematics. Once that is taken away (and it absolutely should be, for now, when reading LLM-generated mathematics) it becomes almost impossible to read mathematics. If there is one thing that LLMs have mastered, it is writing with the confidence and airs of someone possessing great authority.

Low-Hanging Fruit:

Some people have dismissed the recent AI-assisted breakthroughs because they were “obvious in retrospect” or because “not enough people tried to do them.” The first criticism seems clearly ridiculous, since “obvious in retrospect” is very far from “obvious.”

The second criticism reflects, in part, the way humans solve conjectures. In my experience, the process goes as follows. First, you learn about the conjecture and why it might be interesting. Second, you learn a little about why it is hard. You might think about it for a while and fail, and then file it away in the back of your mind. Later, when you encounter new ideas, you perform some pattern recognition and ask whether those ideas have any relation to previous problems you care about. If you are lucky, you “see a connection,” and then you can begin. Sometimes the first inspirational connection is most of the work; sometimes it is only the beginning, and there is much more work to do. In either case, that initial step is crucial.

An LLM, on the other hand (to anthropomorphize), can conduct an extremely thorough literature search and then throw a thousand different ideas at the problem to see whether anything sticks. If one of those ideas does seem relevant, and if the journey from that realization to the end of the proof is not a long one, then the LLM can solve the entire problem. As it becomes possible to chain together longer and longer stretches of reasoning, the only problems that can resist attack are those requiring a genuinely new idea (whatever that means).

Finally, there is the observation that these results are all counterexamples to conjectures that, as far as I know, were generally regarded as true. This phenomenon has a clear diagnosis, given by Deligne: “All problems in mathematics are psychological.” Well, to be precise, that quote was ascribed to Deligne* by Kisin in a lecture at Luminy, but it captures the essence of something clearly true. And the good news is that AI is an insane psychofreak with no hangups.

Addressing the more interesting—and controversial—question of what we, as a profession, are to do about all of this will have to wait until later. This week, I will be attending my first ICM in person. It is a much-maligned conference whose most exciting moment has already been blown by incompetent website design and which is being held in a city with slightly less appeal to me personally than St. Petersburg; but we shall see!

* Since I always check the original source, here is a slightly more nuanced version of that quote:


Dear Calegari,

I don’t remember the exact words, but I expect it is roughly accurate. Of course, it is not always true. The meaning was that we often have blocks which prevent us from seeing things which later will seem obvious to us.

Best,
Pierre Deligne

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The inverse Galois challenge

Posted on June 19, 2026 by Persiflage

I learnt a few days ago about the following AI challenge: find polynomials with Galois group \(G\) for each of the transitive subgroups of \(S_{24}\), with each of the possible signatures \((r_1,r_2)\) with \(r_1 + 2r_2 = 24\) where \(r_1\) is the number of fixed points for some (possibly trivial) involution \(c \in G\). (See the remark at the end about the formulation of this problem.)

I first heard about it at afternoon tea at SMRI. I had an interesting discussion (with Raymond van Bommel and others) on the merits of such a challenge. I was a bit confused to be honest. It seems as though the challenge has been “gamified” to some extent by awarding \(1\) point for each possible group and signature (with some additional points for small discriminants). The scoring system seems peculiar to me as well. I was challenged to commit to paper my thoughts on what the result would be, what examples I imagined were challenging and what would be easy.

The problem itself breaks up naturally into two subproblems: existence and construction. Now these are certainly different problems, and while the former is more interesting to me, the second can certainly also be challenging even when one has a positive answer to the first question. But that is also colouring my response here.

The case of solvable groups is more or less trivial and not so interesting. Certainly
existence is known, and the existence proof is more or less constructive, although there
is (potentially) going to be some issue of computational feasibility. That’s the aspect
of the problem on which I am the least informed. But I certainly don’t think the solvable
case is that interesting here from a theoretical point of view, unless one takes into account the additional problem of finding fields with small discriminants. From a computational point of view, I think one of the issues is identifying the most practical way of choosing the filtration of \(G\) in order to constructively set up the most efficient algorithm. As an example of what I mean, the efficient way to construct \(S_4\)-extensions is not to start with a quadratic extension! Of course, from a computational point of view, even constructing cyclic extensions of sufficiently high degree explicitly (say with some Galois structure to avoid trivialities with cyclotomic extensions) which might theoretically be trivial using class field theory becomes quite tricky, and Henri Cohen has written books about how to do this sort of thing. Perhaps these sort of questions are more to the heart of what this challenge is about. Another type of problem is to construct the \(S_n\) extension of \(\mathbf{Q}\) with smallest discriminant; obviously this is a much more subtle computational question than constructing a single example! Since John Jones (half of the team behind the database of local fields and the database of global fields now folded into the LMFDB) is one of the people behind this challenge this may indeed be more the spirit of this problem. That starts to touch on problems of Malle’s conjecture in very small ranges of discriminants which is definitely interesting, and certainly this blog post addresses quite different aspects of the problem related purely to the vanilla inverse Galois problem.

There are 25,000 transitive subgroups; 24193 are solvable and 807 are not.
In the remaining cases, you can bucket the groups by their (simple) composition
factors to get the following:

Non-solvable buckets by non-abelian composition factors:
{A_12}: 8
{A_12, A_12}: 4
{A_24}: 2
{A_5}: 267
{A_5, A_5}: 55
{A_5, A_5, A_5, A_5}: 45
{A_6}: 204
{A_6, A_6}: 64
{A_6, A_6, A_6, A_6}: 45
{A_8}: 20
{A_8, A_8, A_8}: 10
{LieA(1,11)}: 10
{LieA(1,11), LieA(1,11)}: 4
{LieA(1,23)}: 2
{LieA(1,7)}: 44
{LieA(1,7), LieA(1,7), LieA(1,7)}: 12
{M_11}: 3
{M_11, M_11}: 1
{M_12}: 5
{M_12, M_12}: 1
{M_24}: 1

While there may be subtleties, the examples involving \(A_n\) do not strike me as ones which would theoretically present that much difficulty. The general shape of these groups is presumably going to be something like \(A \subset H \subset G\), where \(\Gamma = H/A\) is the (direct product) of the non-simple groups in the bucket, and \(A\) and \(G/H\) will be solvable and quite small. So you are constructing \(\Gamma\) extensions over some small solvable field \(K\) with compatible \(\mathrm{Gal}(K/\mathbf{Q})\) action in a way that one can then solve some central extension problems, which is going to be a question of ensuring that the ramification in your \(\Gamma\) extensions is liftable. When you have an easy source of such \(\Gamma\) extensions, say when \(\Gamma = A_n\), this seems very manageable. When \(\Gamma\) is a product of some \(A_n\) then it is probably actually coming from \(A_n \wr C\) for some small group \(C\) which is equally easy.

That leaves the 83 remaining groups.

The group that first came to mind immediately upon hearing about this challenge as something that would be difficult was a totally real extension with Galois group \(G = \mathrm{PSL}_2(\mathbf{F}_{23})\). If I was to put money on one pair (group and involution) which would still be missing after this project, then this would be one. Moreover, I don’t really see how having every single other possible pair in the table computed would be helpful in understanding this last case. For example, if you choose the other conjugacy class in this case, even the computational problem is reduced to finding explicit Galois representations coming from mod \(p\) representations of modular forms.

Apparently the case of \(M_{23}\) as an explicit Galois group has been touted as a possible “super hard” problem for AIs to work on. I’m not super excited by this problem, since there are a number of places one could look and then randomly get lucky; my version of this problem would be \(\mathrm{SL}_2(\mathbf{F}_p)\) for all primes \(p\) (the case \(p=23\) related to the example above). The advantage of this example is that one clearly cannot “accidentally” find a solution for all \(p\) at once. (I’ve mentioned this example before at this blog.)

As for other difficult cases, it’s possible that some of the cases related to \(\mathrm{PSL}_2(\mathbf{F}_{11})\) could cause similar issues with difficult choices of involution. My impression is that one knows that \(M_{24}\) occurs as a regular extension, but I’m not sure which involutions one sees over this family, and that could also cause issues (e.g. my guess might be that the rigidity method produces/forces a particular choice of \(c\)).

Remark: Actually I’m not sure if the challenge requires one to find all pairs \((G,c)\) where \(c\) is a conjugacy class of involutions (which would be the most sensible choice) or all \(G\) with a possible pair \((r_1,r_2)\); while the former determines the latter the converse is not true.

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The moral panic over AI

Posted on May 26, 2026 by Persiflage

A number of journals appear to have frothed themselves into a moral panic over AI. There is certainly a lot of AI-generated crap appearing in multiple places, and the arXiv moderators are no doubt playing whack-a-mole trying to keep it down. On the other hand, other aspects of the profession continue on, unchanged.

I submitted a paper in January 2025 to TAMS. A number of my papers are long and technical, and I am not surprised that they take a long time to review. This paper, however, is both short and elementary. So it was somewhat surprising to me that it took over 16 months to receive a referee report. Out of curiosity, I also asked ChatGPT to produce a referee report. That took 30 minutes, and produced a far more substantial report than the one I had received. In addition to all of the more or less cosmetic issues addressed by the (presumably) human referee, ChatGPT also found non-trivial mathematical points that required addressing.

The most interesting correction, in my mind, was the following. At one point, we considered a lattice \(\Phi\) of rank \(d\) and chose successive minimal vectors \(\mathbf{v}_1, \ldots, \mathbf{v}_d\); that is, \(\mathbf{v}_1\) is a nonzero vector of shortest length, \(\mathbf{v}_2\) is a vector of shortest length not in the span of \(\mathbf{v}_1\), and so on. Then something happened in the paper which could be taken straight out of Serre’s talk on how “not to do mathematics”. Specifically, at some point it was assumed that these vectors generated the lattice \(\Phi\), but this was never stated. Instead; a claim was subsequently made that depended on this fact implicitly. The great thing about never mentioning something that you use is that you don’t have to prove it either, and in this case, when you are forced to actually think about it, it is easy to see that it is false! For example, \(\Phi\) could be the lattice \(\mathbf{Z}^n\) together with the vector \((1/2,1/2,\ldots,1/2)\). Such arguments are exactly a good way to slip something past a reviewer. To compound the issue, this was part of a section giving an alternate argument and was not used elsewhere in the paper. So the referee completely missed it, but ChatGPT did not.

As far as I understand the policy of TAMS, it would have been against the rules for the reviewer even to ask ChatGPT to look at the paper, let alone ask it to generate the report. But at least in this case — and I do stress this particular case — it would have been not only more time efficient by a factor of over 20,000, but also much more accurate and precise. I believe that literally the only comment made by the referee that was not made by ChatGPT was the recommendation to use the construction: Let \(k\) be an integer satisfying \((k,n)=1\) over the alternate Let \((k,n)=1\) be an integer.

There are many things in our profession which work quite well, and which AI threatens to make, if it hasn’t already, significantly worse. But there are many things in mathematics that are clearly broken as well. We should at the very least take the changes that will be forced on our profession by AI as a chance to finally address some of these lingering issues, many of which relate to what we publish and how we publish it, head on.

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arXiv endorsement requests

Posted on May 2, 2026 by Persiflage

With the arXiv’s new policies for posting, I am now getting inundated with requests from cranks for endorsements. Here is my suggestion for the arXiv: when someone requests an endorsement, there should be an option (rather than ignoring the email) of giving a negative endorsement, i.e. “here is a crank, please keep this person away from the arXiv”. That is certainly my reaction 100% of the time whenever I have received such a request. Perhaps even better, don’t allow anyone without an .edu account to post to the arXiv at all.

While we are making requests on the arXiv, perhaps the most useful one would be the ability to click on someone’s name and go to a list of their papers. Simply loading a search for “lastname, first initial” is close to useless when it comes to common Chinese surnames; it’s a strange choice of default setting.

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Dick Gross

Posted on January 15, 2026 by Persiflage

Here are some memories about Dick Gross, who sadly just passed away very recently.

Nothing was quite as reassuring as having Dick Gross in your audience. Inevitably, when your talk was done, he would both compliment you on it and have something very interesting mathematical to say. The last time this happened to me was at the Tate 100 conference in March. Dick was a student of Tate, and although he wasn’t able to come in person, he gave some prerecorded reminiscences. (Many of the very nice things Dick says about Tate can also be said about Dick.) But after my own talk (broadcast on Zoom), I still got an email from Dick titled “great talk,” which led to an interesting conversation between the relationship between Serre’s Conjecture and Artin’s Conjecture, as well as some analogs of these questions for \(\mathrm{GSp}_4\). I’m guessing I am not the only speaker at that conference to get such an email!

It’s hard to know where to start. Dick was a great mathematician — his collaboration with Don produced surely one of the greatest theorems in modern number theory (and many more great theorems besides; the paper on difference of singular moduli, for example). He was a great expositor — his Duke paper is a masterly exposition of quite a lot of the arithmetic theory of modular forms (and also a wonderful theorem). He was a great mathematician to chat with at afternoon tea or in the corridor, when you could learn all sorts of clever ideas that weren’t written down anywhere else. He was a pioneer in the arithmetic theory of automorphic forms on higher ranked groups. As Dick himself used to say, you start with \(\mathrm{GL}_2\), then remove the \(\mathrm{L}\), and then the \(2\).

In contrast, the first time I interacted with Dick, a little shy of 25 years ago, he was at the beginning of a second career beyond mathematics in administration, having become math department chair before his tenure as Dean. It was his job to let me know that the department was offering me a BP position. One line from that email was as follows:

For now, let me say how delighted I would be if you would join us next year, as a colleague.

This certainly made me feel pretty good at the time, and it is a line I have come back to and reused myself as a junior hiring chair. (Another line in that email, “I hear you are now an uncle. Behave accordingly”, is less versatile.) More generally, Dick was charming in the best possible way — combining not only the polish that this word suggests, but with an underlying spirit of someone who was attentive, personable, and conscientiously kind.

My interactions with Dick at Harvard mostly continued through his capacity as chair. At one point, I realized that I had been slightly overpaid (I was getting a mix of money both from Harvard and from AIM). His remark at the time, which I can only paraphrase due to the passage of time — said entirely deadpan — was something like, “I have two pieces of advice in life: avoid paying your taxes as much as possible, and don’t tell anyone if you are overpaid.”

There was only one moment where I saw him anywhere approaching being exasperated (though presumably that must have happened quite often as Dean). DeBacker and I had been put in charge of the colloquium committee. The speakers had already been invited by the time it started, so the main task was simply organizing the dinners for the speakers. This was all done by DeBacker, who paid for everything himself and was later reimbursed. On one occasion, Richard Borcherds gave the colloquium, and we ended up going to a quite fancy restaurant (Harvest) in Harvard Square. I don’t think a single senior faculty member came, but lots of graduate students did. We were, I think, quite liberal with the purchase of some nice bottles of Chablis. As you can imagine, the price of the dinner (fully paid by the department) was on the higher side, and at some point it must have gone to Dick’s desk to be approved. I believe Dick’s remark to Stephen was along the lines of, “I don’t want to hear the words ‘colloquium dinner,’ ‘graduate students,’ and ‘$2000’ in the same sentence ever again.”

My best mathematical interactions with Dick mostly came through casual conversations and emails (or even comments on this blog!). I did once answer an actual mathematical question raised by Dick in a joint Inventiones paper with Lubin from 1986. They asked whether a certain Hecke algebra of level \(\Gamma_0(p^2)\) localized at an Eisenstein ideal above \(p\) was always a discrete valuation ring, and I found this could be answered in the positive using ideas of Chenevier and Bellaiche.

Of course, that particular question and answer are no more than mathematical ephemera. But Dick’s legacy — as a mathematician and as a person — will live on.

More from other sources:
An article on Dick in Celebratio Mathematica
Faculty Spotlight Harvard Interview

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Arbeitsgemeinschaft 2026

Posted on June 4, 2025 by Persiflage

The April 2026 Oberwolfach Arbeitsgemeinschaft will be on Arithmetic Holonomy Bounds and Applications to Irrationality, and in particular will discuss some of the results of this paper.

For those who don’t know, the Arbeitsgemeinschaft (“study group”) is different from usual Oberwolfach workshops (or workshops more generally) — the idea is that the participants learn the material and then teach it to each other. I have never actually been to one, or rather I almost did but it was first cancelled due to Covid and then went online due to Covid. That workshop asked a lot of participants in terms of background which made it particularly tough to be online rather than in the Black Forest. Hence I’m not sure that I can describe what a usual Arbeitsgemeinschaft is like any better than reproducing the official blurb here:

The Arbeitsgemeinschaften mainly address to non-specialists who want to broaden their outlook on mathematics and to junior researchers who wish to enter a field for future research. Experts are also welcome. The idea is “learning by doing” – similar to the Seminaire Bourbaki. Participants have to volunteer for one of the lectures described in the program of the Arbeitsgemeinschaft. After the deadline for application the organizers choose the actual speakers to give them enough time to understand the subject and to prepare for their lectures.

If you are interested in learning this material, please consider applying! A number of people seemed keen on knowing the details of our paper up until the point they learnt it was 220 pages long. But I truly think that many of the ideas can be broken down into bite size chunks which I think makes this topic ideally suited to the intended format. The background for the lectures doesn’t involve much beyond the complex theory of modular forms as well as some complex analysis. To see what we have in mind, you can click here to see an outline of how we have conceived the breakdown of lectures might be. When you apply, you can choose which lecture you (might be) prepared to give, assuming you are given enough advanced warning! All the links you might need (for applying and other information) can be found here:

2026 Arbeitsgemeinschaft: Arithmetic Holonomy Bounds and Applications to Irrationality

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En Passant: Mailbox and Tate 100

Posted on April 22, 2025 by Persiflage

I lost the key to my office mailbox about a year ago (probably more), and just had it replaced. So today I got to enjoy the bounty of new mail, which consists of:

  • A request from the AMS for money, dated Sep 14, 2024,
  • A seasons greetings card from the dean of the college, year unknown,
  • A brochure from Nasco education selling math toys,
  • A poster for the 2025 Arizona Winter Schoo, dated Sep 25, 2024,
  • The Fall 2024 University of Chicago library magazine,
  • The Fall 2024 Berkeley Mathematics magazine.

I might make another visit in another year! Leafing through these, I see that Martin Olsson is asking me for money, but also that Alex Paulin was voted the best professor at Berkeley for the second year in a row by the Daily Cal:

I actually really liked the answer to the question about his motivations, which resonated with me.

Recently there was a conference to celebrate the legacy of John Tate (who would have turned 100 this year). I was only able to slip in and out of Cambridge for the day to give my talk (with the one evening I was there fortuitously coinciding with the banquet), but I was also very happy that the entire conference was live streamed on video. For those of us who limit our travel, it’s really nice to be able to follow along. As with Alex, I care about doing a good job when giving a talk, although I suspect I am not quite as successful. You can watch my talk here, which is ostensibly about my recent work with Boxer, Gee, and Pilloni, but perhaps with fewer details than any of them would give in such a talk.

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It wasn’t me!

Posted on March 13, 2025 by Persiflage

I have a gentleman’s bet with another number theorist that I will be able to write a nonsense paper and get it published in respectable journal. To be honest, I barely have enough time to write actual papers let alone a nonsense paper, but it has crossed my mind from time to time.

The respective mathematician recently inquired as to whether I was the author of a certain preprint, presumably written in an attempt to collect on our bet; I just want to say that it wasn’t me! First of all, I wouldn’t try to claim a major conjecture, the aim would be to write a paper that zero people actually read (including the referee). My impression from a quick browse is that the preprint in question is AI generated nonsense. Apart from the irritation of having AI generated nonsense on the arXiv, it is not very interesting, and so I won’t say too much more about it.

It actually points to something about AI and mathematics that sticks out to me like a sore thumb whenever I try it out. Namely, AI does a remarkably good job of phrasing things with the full confidence of someone who knows what they are talking about, even when (which is mostly the case) they are saying utter nonsense. In other words, it does a great job of capturing the tone but not any of the actual structure of mathematical discourse. I hope that, at least, is something that will change as AI improves! Maybe also that’s why this fake paper is annoying; as a community, we give each other the benefit of the doubt that we are trying our best both to give correct proofs and honest exposition. That makes it especially annoying when someone deliberately tries to exploit or undermine that generosity. When the work generated by AI remains this obviously stupid it is less of a problem. But if AI improves AND people try to use it maliciously to publish fake proofs, that could potentially be a problem.

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A new blog

Posted on March 3, 2025 by Persiflage

For various hosting reasons, I have moved my old blog back to galoisrepresentations.org rather than galoisrepresentations.com. The forwarding link will die at some point so please update your links accordingly! Hopefully the previous content is still operational but let me know if you see anything broken.

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The Poincaré homology sphere

Posted on January 5, 2025 by Persiflage

This illusion (from the Chicago museum of illusions, and duplicated, I believe, in other similar museums in other cities) “almost” appears to give a tiling of \(\mathbf{R}^3\) by regular dodecahedra, which for a number of reasons is not possible. (It also looks remarkably like knot not.) Moving one’s point of view slightly, one can observe that the interior faces don’t quite match up. But can this be remedied with curved mirrors? That is, is there a way to shape these mirrors so that one can is looking inside the Poincaré dodecahedral space?

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