—Jorge Luis Borges, “The Library of Babel” [1]
I am writing to announce Hexagon,
a new repository for research works in the mathematical sciences and theoretical computer science.
This has been a joint effort of many people over the last couple of months, starting in early
August 2026. We felt that the proliferation of results proved with LLM assistance required a
new model for collecting the associated papers. We want these
works to be available to the mathematical community in one place (instead of being spread throughout social media), citable, and reliably stored for long periods of time.
Compared to the arXiv, we wanted a lower
bar of mathematical significance and novelty and we wanted to allow submissions to make less specific assertions about human authorship and understanding.
In order to support the development of this project, we founded the Hexagon Mathematics Foundation. The Board of Directors consists of Mohammed Abouzaid, François Charles, Bryna Kra, David Savitt, and Lauren Williams. I serve as the first Executive Director. And, we have a wonderful Advisory Board, where Kevin Buzzard, Akhil Mathew, Johannes Schmitt, Steinn Sigurðsson, Nikhil Srivastava, Ravi Vakil, and Rachel Ward serve. For more details, see our who we are page.
When working with a group of people brought together by an FRG grant, we clicked a button and
discovered a new result, namely derived invariance of Hodge numbers of fourfolds in characteristic
zero. In the past, this would be a result I would have submitted to an excellent, but not
top, journal. In this case, there is no way I would put my name to the result. Instead, I blogged
about it here. This is a brittle way
to aggregate mathematics. Now, it is on Hexagon as hexagon:2609.00104.
The creation of new results happens naturally in working with LLMs. They bubble up, even as side results. My personal hope is that it becomes the community standard to identify these and submit them to Hexagon.
My ambition is that Hexagon is where mathematical ideas emerge from the water for the first time as they evolve toward community acceptance and understanding. Later, they should be written up in forms suitable for arXiv submission, publication, and canonicalization (Tao [3]).
Hexagon specifically allows submissions entirely generated by LLMs and even where the submitters claim no human understanding. These results should be available to the general community.
Hexagon welcomes submissions by large labs working on language models. These results should be available to the mathematics and TCS communities by default on a platform that the community maintains. We also welcome submissions by large collaborations or other organizations.
How was Hexagon created? At some point, a group of people convened in early August, brought together by our social networks with the common theme of interest around creating a home for mathematical results which were rapidly being produced with the assistance of LLM tools. This group included most of the current members of the Board and Advisory Board of the Hexagon Mathematics Foundation, as well as several other academic mathematicians who did significant work but preferred not to be named. In August we discussed ideas on a private Zulip channel (thanks ICARM!) as we started building the site itself.
We talked to a lot of people at that time, including arXiv leadership, and we also examined
closely-related AI-era repositories. From the arXiv we learned that they were not going to
provide the home for these results we were looking for; instead they are focused on maintaining
their project as it is today, and I am happy for that. None of the other repositories had gained
sufficient traction or had all of the policies we were hoping for.
The foundation was incorporated as a nonprofit in Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3).
We felt the need to move rapidly to create this new infrastructure; it was built and tested in August and September and has been accepting submissions publicly since 28 September. It is in beta and we expect small changes and new features to continue to roll out this fall.
How is Hexagon funded? So far, it is entirely funded by the people involved, although we are exploring other funding routes and paths to sustainability. Today, except as customers of cloud infrastructure companies, we have no relationship with any for-profit company or other organization.
Does Hexagon accept micro-results? Hexagon welcomes small, incremental improvements as long as they are of interest to the mathematics and TCS communities.
Is formalization required for a Hexagon submission?
No, Hexagon is not primarily a formalization project, but it links to formalization projects, like mathlib, Palomar, prove2.me, and TauCeti.
What is the difference between a contributor and a submitter? On Hexagon, Contributors are humans or organizations responsible for the creation of submissions. Submitters are humans with ORCID-validated accounts on Hexagon. Submitters fill out the submission form and serve as the point of contact for the work and for future revisions. In some instances and with appropriate authorization, Hexagon will provide individual account holders the capability to submit on behalf of other humans or organizations.
Are there rate limits?
Yes, by default Hexagon limits the rate at which one can submit, starting at 1/day and growing based on previous submissions passing through moderation. In justified cases, these rates can be increased by moderators.
(The arXiv recently announced
its own repository-wide submission limits [4].)
Is there API access?
Yes, Hexagon provides both read-access via API (in the style of the arXiv) and submit-access via API tokens
available from users’ account pages.
Why is Hexagon marked as being in beta? We are still adding features and testing scalability. We welcome any recommendations for how the site can be made more usable. Registered users can use the internal reporting page; for fuller community discussion, use our GitHub feedback repository.
Why ‘Hexagon’? At some point, the group started talking about names involving Babel, which refers to Borges’s story The Library of Babel [1]. But, because it has other connotations as well, we did not go that route. Instead, someone suggested Hexagon in reference to the shape of the halls in Borges’s library.
[1] Jorge Luis Borges, “The Library of Babel,” translated by Andrew Hurley, in Collected Fictions, Penguin Books, 1999, publisher page.
[2] Ted Chiang, “The Lifecycle of Software Objects,” in Exhalation, Vintage, 2019, publisher page.
[3] Terence Tao, Mathematics in the age of AI, 2026, arXiv:2608.16753.
[4] Kat Boboris, Fair Moderation, Equitable Access, and AI: arXiv’s Updated Rate Limit Policy, arXiv Blog, 1 October 2026, article.
]]>Mathematical work comprises many activities and experiences. Among my favorites are:
The first and second constitute gaining and creating knowledge. The third is about transmitting meta-knowledge: the ability to perform 1 and 2.
Mathematics for me has always been about the joy of personal understanding and the engagement in bringing others around to that understanding, usually slowly over the course of many months or years.
In recent months, a deep unease has been growing within the mathematical community around the role of language models in our research. The fear is fundamentally economic. LLMs can produce mathematical statements and proofs, some certainly interesting.
Are we ourselves, mathematicians, obsolete? Will our students fail to get jobs? Will our funding agencies, or even our universities, abandon us? Is mathematics in fact solved?
No. Mathematics is not the collection of true statements implying, by complement, the collection of false statements as well. It cannot be solved any more than human experience can be.
Most academic mathematicians are slow math practitioners at heart, even if the pressures of paper-writing, grant-proposing, teaching, and administration seem rather far from our idealized contemplative life.
Yet, math also forms the bedrock of the sciences and engineering and as such is central to how human societies solve technical problems. Mathematicians cannot expect our esteemed place within our culture to survive unchanged if we do not use every tool possible to help it face its grand challenges.
Alexander Nazaryan wrote in the New York Times on 19 June 2025: “Problems in mathematics take decades or centuries, sometimes, to solve,” [Patrick Shafto] said in a recent presentation at DARPA’s headquarters on the Exponentiating Mathematics project, which is accepting applications through mid-July. He then shared a slide showing that, in terms of the number of papers published, math had stagnated during the last century while life and technical sciences had exploded. In case the point wasn’t clear, the slide’s heading drove it home: “Math is sloooowwww. …”
We exist, professionally, in part to educate engineers and scientists, to provide researchers to national laboratories, and to provide highly-trained workers to companies. Occasionally, theoretical results open up new areas of specific application, for example the use of lattice-based methods in post-quantum cryptography standards. If we turn our back on this part of our role, we can certainly say goodbye to our grants and teaching reductions. We can go the way of the Department of —, permitted to exist out of nostalgia, but decreasingly relevant.
To address the central contradiction between our personal practice and our external purpose, I propose fast math and slow math as ways of thinking about different types of mathematics. Just as we all prefer slow food, sometimes fast is what we need, or crave.
These different ways are not meant to separate human mathematicians into “slow math” and “fast math” camps. Indeed, I strongly urge against the temptation to split into us and them. I don’t want to sit on hiring committees fractured by debates over AI use or virtue testing instead of honest evaluation of candidates.
The goal of slow math and fast math is the same: to increase human understanding of what is true and false in those regions of “proposition-space” of interest to us. These take their places as different modalities of doing mathematics, alongside computer algebra, exposition, the creation of visualizations, editorial work, and so on. Not every mathematician does, or should do, all of these.
Thanks in part to technological advances, but largely driven by an unending thirst to know, physicists have gone in the span of a couple of centuries from tabletop experiments to running some of the largest collective endeavors of humankind.
LLMs represent a tremendous technological advance and some might choose to dream bigger dreams. We should deliberately and thoughtfully use these models and other AI tools to have those dreams. Some of these might be to answer new, pressing questions created by the very existence of the tools themselves, like the problems of interpretability in LLM systems. These tools were built with mathematics and our discipline provides one of the principal means by which we understand them and use them safely.
Other dreams should be to undertake audacious programs, which would not have been practical in the past. I have in mind research directions at the scale of the Langlands program or Grothendieck’s dream of motives.
One example could be higher-dimensional algebraic geometry. Many counterexamples that have been established with LLM assistance have to do with the fact that we do not know the zoo of algebraic varieties in higher dimension the way we do curves and surfaces. (Although, of course, there have been many, many amazing results, like Beauville–Bogomolov.)
Such programs should be pursued rigorously with a back-and-forth between humans and LLMs. These programs should have directors, postdocs, students, standards. The goal would be to leverage LLMs to understand high-dimensional phenomena – for humans to understand. It is not to produce a slurry of results no human has read or understood.
These programs are fast math: far-reaching, ambitious, almost reckless. They would aim at transformative new discoveries and organizations of mathematical knowledge, perhaps particularly motivated by grand challenges at the boundary of mathematics and applied mathematics. But they would not end with a given discovery. They would also be responsible for producing lecture notes, textbooks, databases, and other tools to aid human apprehension of their discoveries. They would exist to truly harness AI capabilities and prepare the outputs for human understanding.
I suggest the following standards for the mathematical community. Please consider these as a first proposal and feel free to send me feedback. The overall goal is that we should use LLMs to aid in human mathematicians’ understanding, not merely to increase our production.
Disclose all collaborations, including with LLMs. It is my hope that eventually we can use open source models for much of the day-to-day work that people might turn to LLMs right now. We already cite computer algebra systems as soon as the work goes beyond the totally routine. We should do so here.
Do not judge mathematicians for their LLM use. The standards of our community are still in flux. We can fight over what those standards should be, but individuals have to act in the meantime and we should not litigate their behavior.
Posting a paper to the arXiv or publishing it means three criteria have been met by the authors:
you could reproduce the main ideas if on a desert island (excepting large computer computations);
you have written it for human understanding, to the best of your ability;
you take intellectual and reputational responsibility and credit for the results.
In the second point, using LLMs for polishing writing, spell-checking, finding inaccuracies, or creating diagrams is completely fine. Using an LLM to generate a first draft of a paper is not acceptable.
LLMs should not be used to compete with other mathematicians, to scoop them or otherwise embarrass them. It goes without saying that we are somewhat competitive, but there is nothing to be gained from this behavior and a lot to lose in terms of collegiality. We are extremely lucky to be in a field where we talk openly at conferences not only about what we’ve done but what we are working on, sometimes years before those results see the light of day.
Refrain from asking every random question to LLMs, or at least from making the answers public. Simple results are important for training students. Many have been excited to ask questions long-abandoned in their research programs, including myself. Of course, we cannot say that any given question is off-limits. But I expect that this fervor will die down in the coming weeks or months. And we do not need to repeat it with every new “release” from the big labs.
Focus, if you choose to use LLMs, on developing your own research, intuition, and results. No one owns an area of mathematics or a certain problem. But it is not acceptable for me to wake up one day and say, you know I know famous mathematician — is working on problem — and I’m just going to solve it. They will be impressed with me, or I will get that next job I need. The easiest way to guard against this behavior in the long run is to just not reward it.
Do not work faster than you can understand or effectively communicate. This golden rule informs all of the others.
Do not recruit, hire, promote, or praise based on the number of papers. In the past, the production of high-quality papers has provided a proxy for evaluating understanding and brilliance. Of course, you want to hire the individuals who’ve written these papers to access their insight for your own research or for the sake of the students in your department’s charge. Each department will have to find new ways of judging.
Do not delegate the previous task entirely to LLMs.
Before LLMs, I typically had 5-10 projects on the burners or in the fridge at any one time, probably working actively on 2-3 at any one time. This reflects my interests and personal capability to pay attention to them. Having used language models, specifically agentic coding systems (basically nonstop for coding for 7 months), I can attest that there is a tendency to do more and more and more, but that the pleasure in doing so drops, the quality is hard to verify, and my understanding of the output goes down as well. As such, I strongly disagree with the premise of Shafto’s slide that stagnation in mathematics is measurable by slower growth in publication rates than in other sciences. Papers are only a proxy for understanding.
There are also areas where I believe it is explicitly permitted to use LLMs. They can provide translation services, whether to improve writing in a non-native tongue, or to translate into one’s own language. They can proofread. Let’s face it, no one is doing this for us anymore at most journals. They can act as tutors.
Experiments will inevitably produce writing largely generated by an LLM and possibly understood by no human. These texts deserve a home too, because they might later be useful for human work, but that home is not the arXiv.
We have known since von Neumann, Turing, and Gödel in the 1930s and 40s that mathematical theorems are nothing but provable propositions in a formal language. These could be produced automatically by any computer.
Humans care about individual statements and theorems. Perhaps in part because of our intuition from the natural world, these propositions are not equal in our eyes. Mathematics is what human mathematicians understand. Beyond is darkness: true absence, or inscrutable trophies of alien intelligence.
In Pascal’s Sphere, Borges invokes Blaise Pascal: “nature is a fearful sphere whose center is everywhere and circumference nowhere” to propose that “universal history is the history of the different intonations given a handful of metaphors.” Indeed, mathematics is an infinite sphere whose center is everywhere and circumference nowhere.
Like Pascal, overwhelmed by humanity’s simultaneous insignificance and ability to understand our insignificance, mathematicians work in a habitable zone in which the theorems, questions, and conjectures are configured so that some structure is visible.
The fact that LLMs can seemingly be organized into their own societies is interesting. The mathematics that they produce there is not interesting to me. Language models have not solved mathematics for humans. They are practicing mathematics for a nascent civilization.
I believe civilizations beyond ours exist, have existed, and will exist in the future. Our theorems have already been proved. They will be forgotten, and proved again and again until the last ember goes out. The exercise of mathematics is the work of civilization, and we will continue.
]]>These theorems and questions reflect my mathematical tastes and interests. Despite the brief summaries below, many of these led to multiple-day runs or conversations. Some of the questions below were contributed by colleagues of mine, like Nick Addington or Achim Krause, or arose from the FRG working group which led to the Hodge numbers of fourfolds example I wrote about here.
In one census, I recovered 2,271 computer algebra scripts written and run by the agents across 31 mathematics sessions. One additional temporary script was identified but could not be recovered. This does not include the work I did on the SAIR IGP24 competition.
The models used below were a mix of Opus 4.8, Opus 5.0, Fable 5.0 from Anthropic and ChatGPT 5.6 Sol from OpenAI.
Below is a summary, created with the help of ChatGPT, of those conversations. Here is a table.
| Mathematical questions not solved by these AI runs | 12 |
| Not solved by my AI runs, but subsequently solved by others | 2 |
| Produced a theorem, disproof, computation, or definitive answer | 9 |
| Large programming tasks unfinished | 3 |
| Computer-algebra scripts recovered in the census | 2,271 |
The quoted questions are exact except for light copyediting of notation and obvious typographical errors. “Internally reviewed” means checked by another model; it does not mean independently checked by mathematicians.
Question 1. Orlov’s conjecture predicts that derived equivalent smooth projective varieties over finite fields have the same zeta functions (and hence the same number of points). Give a counterexample.
Remark. No counterexample was found. For derived-equivalent fourfolds, the run obtained the exact reduction \(\#X(\mathbf F_{q^n})-\#Y(\mathbf F_{q^n}) =(q^n-1)^2\left(\operatorname{Tr}(F^n\mid H^2(X))-\operatorname{Tr}(F^n\mid H^2(Y))\right).\)
Question 2. An old folklore conjecture asserts that every smooth proper dg category over the complex numbers is geometric, i.e. admits a fully faithful functor to $D^b(X)$ for $X$ a smooth proper scheme. I’ve always suspected this should be false. It’s worth thinking about stacks, but if a counterexample comes from that direction, we’d just expand the notion of geometric and would still be hoping for a genuinely noncommutative counterexample. Can you find one?
Remark. No. The run eliminated several candidates, including bounded acyclic complexes of vector bundles, nonprojective complex tori, and various finite or periodic $A_\infty$ categories. The general geometricity question remains open.
Question 3. Construct new examples of smooth compact hyperkähler manifolds that are not of K3/abelian type or the two exceptional examples of O’Grady.
Remark. No new example was constructed. The best candidate is a $20$-dimensional relative Picard space associated to Gushel–Mukai geometry. Smoothness along the compactification boundary and nondegeneracy of the symplectic form remain unproved.
Question 4. Let $\alpha$ be a Brauer class and let $\operatorname{degset}(\alpha)$ be the set of degrees of Azumaya algebras representing it. If $n$ is prime to $\operatorname{per}(\alpha)$, is \(\operatorname{degset}(n\alpha)=\operatorname{degset}(\alpha)?\) As far as I can tell, there is no reason it should hold. Find a smooth projective or affine counterexample.
Remark. The scheme question remains open. The run found a stack counterexample on $B\mathrm{PGL}_5$: the universal class has a degree-five representative, while twice the class does not. Topologically, unit multiples have the same degree-set, so a scheme counterexample must be genuinely algebraic. The stacky counterexample and the topological case were known to Ben Williams and myself.
Question 5. Prove or disprove the Grothendieck–Katz $p$-curvature conjecture.
Remark. The general conjecture was not solved. An agent extracted an internally reviewed finite-monodromy/Gauss–Manin special case and an explicit rank-two $S_3$ example. This is best regarded as a consequence or specialization of Katz’s work unless a literature check shows otherwise.
Question 6. I’d like to produce an example of a non-reduced commutative $\mathbf Q$-algebra $R$ such that $L_{R/\mathbf Q}=0$. I know the following from general principles: if $S$ is $R$ modulo its nilradical $N$, then $N^2=N$. So, it’s some kind of almost setting. I wouldn’t be shocked if there were either (a) an example with $S=\mathbf Q$ or if no example is possible, in which case I’d like a proof.
Remark. Neither outcome was obtained. The run constructed finite zero-cotangent transition blocks and reduced the problem to an explicit embedding problem for suspended monomial complete intersections \(\mathbf Q[x_1,\ldots,x_e,t]/(x_1^N,\ldots,x_e^N,t^2+f).\)
Question 7. A folklore conjecture asserts that a degree $p$ division algebra is cyclic. More conservative people would hope this is true over function fields of surfaces over the complex numbers. It is true for $p=2,3$ and maybe $5$. Can you prove it or disprove it in any form?
Remark. The run did neither. In degree five it isolated the $F_{20}=C_5\rtimes C_4$ normal-closure case and reduced it to descent through the cyclic quartic intermediate field.
Question 8. Prove or disprove Amitsur’s conjecture involving birationality of Severi–Brauer varieties.
Remark. The degree-eight case remains open. The run reduced a possible counterexample to classes with $[B]=3[A]$, excluded several standard Cremona transformations, and isolated a $V_4$-descent problem. Kollár’s proof for non-prime-power index does not cover this case.
Question 9. Search for an indecomposable rank-two bundle on $\mathbf P^6$ or $\mathbf P^7$ in characteristic two, as a route toward Hartshorne’s conjecture.
Remark. No new bundle was found. The Hartshorne experiment reproduced the Tango bundle on $\mathbf P^5$ and established that its free-monad search strategy cannot work on $\mathbf P^6$. A different ansatz is required.
Question 10. Define higher Azumaya algebras, determine whether they are locally trivial in an étale or Gestalten topology, ask whether every higher $\mathrm{GL}_1$-cohomology class is represented by one, and develop the analogue of twisted sheaves. Can you also produce a genuinely derived $E_2$-algebra which is smooth and proper, or show that one cannot exist over an ordinary commutative ring?
Remark. The general higher Brauer program remains open. The run obtained a no-go result for connective or separable candidates over a field: any nontrivial example must be two-sided in cohomological amplitude and essentially nonformal.
Question 11. Recent work places at least $2/3$ of Riemann zeta zeros on the critical line. Is it possible that this could have consequences for the appearance of GRH in the bounds used for algorithms for computing class groups?
Remark. No theorem resulted. The most concrete proposal is to verify RH to finite height for $\zeta_K$, bound the remaining zero contribution in a smooth explicit formula, and use the analytic class-number formula to certify the class number. The explicit constants remain to be derived.
Question 12. Let $X$ be a finite type CW complex. Let $A$ and $B$ be truncated $E_\infty$-rings. Consider the natural map \(C^*(X,A)\otimes_A B\longrightarrow C^*(X,B).\) Is it an equivalence?
Remark. No general proof survived scrutiny. The finite-CW, dualizable-$B$, and finite-Tor-amplitude cases were established, but the truncation-only finite-type case remains unresolved after gaps were found in the inverse-limit and coherent-truncation arguments.
Question 13. Construct a counterexample to the period-index conjecture: for $K$ finitely generated of transcendence degree $d$ over an algebraically closed field, every $\alpha\in\operatorname{Br}(K)$ satisfies \(\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^{d-1}.\) The frontier is $d=3$, i.e. $\operatorname{ind}\mid\operatorname{per}^2$.
Remark. The agent did not find a counterexample. It ruled out several abelian and bielliptic families. Alexander Perry then constructed a smooth projective threefold over $\overline{\mathbf Q}$ with a Brauer class of period two and index eight in The period-index conjecture is false. His project was also AI-assisted, but Perry supplied and checked the mathematics. A later question—whether the example works over $\overline{\mathbf F}_2$—was answered negatively for the published construction; no characteristic-two replacement was found.
Question 14. Find a Riemannian metric of strictly positive sectional curvature on $S^2\times S^2$.
Remark. The agent built a certifier and searched perturbations of the product and Cheeger-type metrics. Its best held-out minimum sectional curvature was about $-0.0094$. Simon Brendle and Pak-Yeung Hung then announced such a metric in A metric on $S^2\times S^2$ with positive sectional curvature. Their preprint uses a Cheeger–Müter metric and a tailored third-order perturbation.
Question 15. Let $S$ denote the sphere spectrum and let $S_n=\tau_{\leq n}S$. Let \(\operatorname{FanSp}=\varprojlim_n^{\operatorname{Cat}^{\mathrm{dual}}_{\mathrm{st}}}\operatorname{Mod}_{S_n}.\) Is the natural functor $\operatorname{Sp}=\operatorname{Mod}_S\longrightarrow\operatorname{FanSp}$ an equivalence?
Remark. No. The functor is fully faithful, but an internally reviewed tensor-square argument establishes that $\operatorname{Sp}\to\operatorname{FanSp}^{\otimes2}$ is not essentially surjective. Under the self-duality equivalence \(\operatorname{FanSp}^{\otimes2}\simeq\operatorname{Fun}^L(\operatorname{FanSp},\operatorname{FanSp}),\) the identity endofunctor lies outside the image of $\operatorname{Sp}$. It follows that $\operatorname{Sp}\to\operatorname{FanSp}$ cannot be an equivalence.
Question 16. There is a conjecture that if $X$ and $Y$ are derived equivalent smooth projective varieties over the complex numbers, then they have the same Hodge numbers. Can you find a counterexample to this conjecture, or prove it?
Remark. The general conjecture remains open, but Codex proved that the Hodge numbers agree for fourfolds. The new input is that topological signature is a derived invariant in every even complex dimension. Several mathematicians checked the proof; details are in Hodge numbers are derived invariants for complex fourfolds. The fivefold case remains open.
Question 17. Let $M_X$ be the monoidal category of exact endomorphisms of $\operatorname{Perf}(X)$. It is a kind of “higher matrix algebra”: as a stable infinity-category it is $\operatorname{Perf}(X\times X)$, but its product is composition or convolution. Describe the grouplike $E_1$-space $\operatorname{Aut}_{E_1}(M_X)$, a higher analogue of $\mathrm{PGL}_n$.
Remark. Homotopical Skolem–Noether identifies it with autoequivalences of $\operatorname{Perf}(X)$ modulo scalar invertible objects, with a possible Brauer-stabilizer term. For connected $X$ over an algebraically closed field it is the autoequivalence group modulo shifts; for $X=\mathbf P^d$ it is $\mathrm{PGL}_{d+1}(k)\times\mathbf Z$.
Question 18. Find me the smallest poset homotopy equivalent to $\mathbf{RP}^2$. Is this the smallest poset with $\mathbf Z/2$-torsion in its homology?
Remark. The answer is $13$ points, and $13$ is also the minimum size of a finite poset with torsion in integral homology. For the follow-up with $\mathbf Z/3$-torsion, only $13\leq m_3\leq17$ was established.
Question 19. Does every $E_2$-algebra over a characteristic-zero field admit a natural $E_\infty$-algebra structure? I thought I remembered something like this.
Remark. No. A nonzero Browder–Gerstenhaber bracket is an immediate obstruction. Formality of the $E_2$ operad in characteristic zero does not make the bracket vanish.
Question 20. What can you tell me about the Hochschild homology or cotangent complexes, relative to the complex numbers, of commutative unital $C^*$-algebras?
Remark. For $A=C(X)$ with $X$ infinite, algebraic Hochschild homology and the cotangent complex are large after forgetting the norm. Continuous or Banach Hochschild theory behaves oppositely: amenability kills positive-degree homology. For finite $X$, $C(X)$ is étale.
Question 21. What is a hyperholomorphic sheaf on a hyperkähler variety?
Remark. It is a sheaf with an admissible Hermitian–Yang–Mills connection whose curvature has type $(1,1)$ for every induced complex structure. Equivalently, it corresponds to suitable holomorphic data on the twistor space.
Question 22. I saw a recent arXiv preprint about an associative ring with vanishing Hochschild homology. It resolved some conjecture.
Remark. The paper was Kong–Liu–Shen, A counterexample to Han’s conjecture. The precise statement is $HH_n(A)=0$ for every $n\geq1$, not for $n=0$, while $A$ has infinite global dimension.
Question 23. Is every finite locally free group scheme of order $n$ annihilated by $n$?
Remark. No. For $p=2,3,5$, the Grothendieck rank-$p^2$ experiment checked examples not killed by $p^2$ but killed by $p^3$, and proved the associated power-carry identity uniformly. This was independent verification of a construction from Akhil Mathew’s AI-generated GrothendieckRankP2 project.
AI agents have completed thousands of programming tasks in my codebase and my other projects. The following are three conspicuous computer-algebra targets which remain unfinished.
Question 24. Implement the PARI/GP class-group algorithm natively.
Remark. This did not succeed with truly huge numbers of tokens. Although eventually agents did create correct code, it is nowhere near as fast. The last time I did this experiment though was with Opus 4.6, so it would be good to revisit.
Question 25. Implement Rubi integration from scratch.
Remark. Good results were obtained, but certainly not parity on Rubi.
Question 26. Match the performance of msolve with a native polynomial-system solver.
Remark. Very good results have been obtained, but not matching msolve performance.
On 25 August 2026, I spoke with Jürgen Klüners, a professor of mathematics at Paderborn University, and Gunter Malle, a professor of mathematics at RPTU Kaiserslautern-Landau, about their work on the SAIR Foundation’s Inverse Galois Problem in Degree 24 competition [1]. (Their team KLPB won.) The challenge asked participants to find explicit degree-24 polynomials for every possible transitive Galois group and signature. We discussed why existence theorems do not make these polynomials easy to find, the complementary computational strategies they used, the role of OSCAR and Magma, and how AI helped them automate their submission workflow.
Benjamin Antieau: Thanks for taking the time to chat. I’m Ben Antieau. I’ve been doing these interviews partly to highlight how people are using artificial intelligence today, but I’m also interested in computer use in mathematics more generally. Let me ask you both to introduce yourselves.
Gunter Malle: I’m Gunter Malle. I’m mainly interested in representation theory of finite groups of Lie type. But since my childhood, I’ve also been interested in polynomials and a little bit in Galois theory.
Jürgen Klüners: My name is Jürgen Klüners. I started in computational number theory. I did my PhD in Berlin, and at that time I worked with KANT. KANT is now part of Magma, so I’m still a little bit involved in Magma. Now I’m working more on asymptotic questions, like Malle’s conjecture and things like that. But I think that for both of us, producing polynomials is still a kind of hobby. We’ve been doing it for a long time, and from time to time it’s still a nice thing to do.
Gunter Malle: Maybe there’s one thing I should add, since you mentioned computer algebra systems. Right now I’m the speaker of SFB-TRR 195, a very big project in Germany funded by the DFG that developed, and is continuing to develop, the system OSCAR. It aims to put together some of the individual computer algebra systems that were developed mainly in Germany. I’m not myself really programming anything for OSCAR, though.
Benjamin Antieau: The context for this conversation is that the SAIR Foundation hosted a competition on the inverse Galois problem in degree 24. Could you state the exact problem to solve in this competition?
Gunter Malle: The challenge was to produce at least one polynomial for every transitive group of degree 24 and for every possible signature. There was a system to weight the various submissions. It wasn’t exactly the number of successes; it also depended on how many other people had the same polynomial, or whether you were the only one to get it.
Jürgen Klüners: At some point we learned that there was this competition. Gunter took the initiative to open an account and produce some first examples. Then I talked to him and said, “Let us do it together,” in some sense. In the beginning, our biggest problem was how to submit them, not how to produce them. I have to say that I spent more time on this competition than I should have done.
We have this database up to degree 23, and some years ago we tried to fill in more or less all the signatures in degree 16, where we have 1954 groups and many more signatures. We were almost successful with missing cases in two groups. In degree 12, you have 301 groups; in degree eight, you only have 50 groups [2].
Benjamin Antieau: What makes this hard?
Gunter Malle: First of all, the sheer number, of course. But there is also no recipe here. The inverse Galois problem is open in general. It wasn’t even known to have a positive answer for all transitive groups of degree 24. Until a few weeks ago, it was open in degree 23.
Benjamin Antieau: Most of these transitive groups of degree 24 are solvable, and there’s a famous theorem that says solvable groups are all realized as Galois groups. So what’s the problem?
Jürgen Klüners: One would have to implement this solution by Shafarevich. The major problem is that Shafarevich’s proof is constructive in principle, but, in my eyes, you take your favorite group, look at its factor groups, and take a series of groups. In order to solve the next step, I think you need the splitting field. This is something computer algebra people do not like. If you have a group of order 100,000 at some point, it means you have to produce a number field of degree 100,000, and there are groups larger than this number.
If the degree—or the order of the group—is reasonably small, you can compute those polynomials. But the trick is what to do for all the larger groups. For the smaller groups, you can compute the normal closure. If you can compute the normal closure of something, you can compute abelian extensions. Then you have your embedding problem. You can compute the module and you have the module action, and then you can do something. But if your group becomes bigger, you have a practical problem on your computer. You have to do something else.
Let me mention another problem that is very important. In Shafarevich’s approach, at some point you construct a larger group and then go down to a factor group, or quotient. With this approach, you can realize the quotient. But we were asked to realize all groups, including the signature restriction, and perhaps you have bad luck: the construction does not allow you to construct one of the signatures. For totally real fields, we are fine; it’s easy.
For instance, suppose you want to construct $C_2$-extensions by obtaining them as quotients of $C_4$-extensions. It’s a stupid example, but suppose that, for some reason, you know how to construct $C_4$-extensions. Then all the quadratic quotients you get are totally real. You never hit the imaginary one. Something like this can happen for larger groups, and then you have a problem with Shafarevich’s approach.
Benjamin Antieau: This competition was not only about the groups, but about realizing them as permutation groups—as transitive groups. If one knows the inverse Galois problem for a solvable group, does one know it for all transitive actions?
Gunter Malle: If you have one solution, then of course you get all the transitive actions, because transitive actions correspond to actions on cosets of subgroups. In fact, there are algorithms that go from one representation to another.
Benjamin Antieau: This is related to siblings.
Jürgen Klüners: That’s also something that we used very heavily. When I got a new situation, I tried to do this computation to get other degree-24 representations out of it. My feeling was that if I got one new one, I got three or four or five more new ones from this other computation. It was really quite useful.
Benjamin Antieau: The scoring worked in a slightly funny way, but I think you would have won by any scoring method. What were the fundamental methods you used to produce examples?
Gunter Malle: I started with relative quadratic extensions over degree-12 fields. I used an implementation in OSCAR for computing relative extensions. By chance, I had improved it just a few months ago in order to avoid hitting the wreath product too often. Usually, if you compute these relative extensions, what you get is the wreath product. But if you put some restrictions on the ramification and so on, then you can avoid getting the wreath product too often. It came in very handy. That’s what I did most of the time: these relative quadratic extensions.
Benjamin Antieau: So you controlled the relative quadratic extensions by targeting the ramification that the degree-12 field already had?
Gunter Malle: What my program can do is avoid having a prime that occurs in the discriminant of the resulting field only once. If there’s a prime like that, occurring to the first power, then the resulting group will be the wreath product. I can look at relative extensions and avoid that. Or one can even prescribe that all primes in the discriminant should occur to at least the third, fourth, or fifth power, or something like that. That will again avoid certain trivial configurations. Not completely, but it will give the other configurations a fair chance to happen.
Jürgen Klüners: I think what I did afterwards was a good complement. Gunter’s approach has the advantage that he takes a polynomial of degree 12 and looks at what he gets. What I did later was write a program in which I give it a target group of degree 24. I choose a degree-24 group, and I can determine what the corresponding subfield is. Say the degree-24 group is $G$ and the degree-12 group is $H$, so I have an epimorphism $\kappa$ from $G$ to $H$.
Sometimes you are lucky and can prove that the local conditions are not solvable. This means you know that for this degree-12 field you cannot find a solution, and you can skip this polynomial immediately. I only have some necessary conditions; they are not sufficient. It can still happen that I test polynomials and search for a group, but there is a global obstruction that I do not see, so I find nothing. At some point, you have to stop and free your processor for the next job.
At the moment, I don’t know whether the missing cases are just a matter of time and whether we will get them by using the same methods. I wasn’t able to finish all the computations I wanted to do because of a lack of computing power. It’s nicer to let the computer run and hope that it finds something.
Benjamin Antieau: Gunter, you have conjectures about the number of field extensions with a given Galois group. In principle, variants of those conjectures could inform this stopping problem. Did you use any ideas like that?
Gunter Malle: No. These conjectures are not precise enough to predict where you would find the first extension with a given group, in particular because they don’t make any guess whatsoever about the constants. The constant could be tiny, so the first one occurs very late. In any case, it’s asymptotic. Even if you knew the constant was very big, it could still be that the first group only comes very late.
It kind of gives you an idea which groups might be the bad ones—mainly those for which the asymptotic says that it grows only by $X^{1/20}$, or something. But that doesn’t really help you in computation. You just say, “Okay, I’m not surprised that this group is missing.”
Benjamin Antieau: I was looking this morning, and I think the first open group is 24T1958, which has order 768. It’s $C_2^3 \mathbin{.} \mathrm{GL}_2(\mathbb Z/4\mathbb Z)$. That was left open at signatures 8 and 16, and it is a quadratic extension of 12T49. What would you do to try to solve this?
Jürgen Klüners: I think we had other signatures, so we had fields for which the embedding obstruction was good. First, I have to check whether we can prove the local conditions, or whether we have to go further.
Perhaps I handled a case where I stopped too early. I may have had the right degree-12 field, but for some reason my stopping condition was too harsh. Because I didn’t have enough computing power, I said: let the job run for one or two hours. If it hasn’t finished after one or two hours, I kill the job. For some groups, that is not a good choice. But this is one path we can follow: we can continue the computation.
Also, if it’s a small group, I’m pretty sure that if you say it’s really important and give us, I don’t know, 1,000 euros to do it by some deadline, I will look at this group, and I have a lot of hope that I will find it. But if you do all the groups more or less by hand, I have other duties. That is perhaps a difficult thing.
Benjamin Antieau: What kind of computing resources did you use? How many cores? Were you using a cluster?
Gunter Malle: I think I used maybe 30 or 40 cores at the same time. But I don’t really have very good algorithms to distribute the jobs, so I had to start all of them by hand, every single one, since I also had to do teaching and other stuff.
Benjamin Antieau: At least one other team reported using a truly crazy number of cores at once, like 3,400 or something.
Jürgen Klüners: I used one computer with 32 processors—64 with hyperthreading—which I bought in 2022. I used two other machines of similar size which I bought in 2011 and 2013. I had one machine that was quite useful with one terabyte of memory. But I don’t think the computing power is really impressive. I could have run more jobs.
Benjamin Antieau: You were using OSCAR, at least. Were you also using Magma or Sage or anything along those lines?
Gunter Malle: We had to use Magma because it’s the only one, I think, that can reasonably compute Galois groups in degree 24. OSCAR can do it in principle, but not in practice, because the invariants needed to do the descent have not been optimized there. That is certainly something that needs to be done in OSCAR, but right now we use Magma for that.
Benjamin Antieau: Just as an aside, in my system I created a faster way to compute the Galois groups for these quadratic extensions. If you already know that you have a quadratic extension of a given degree-12 field, then you can compute it faster than with the generic Magma approach.
Gunter Malle: Do you have a special algorithm when you start with the degree-12 extension to compute the group?
Benjamin Antieau: It restricts the number of possible Galois groups. For a lot of groups, you can use cycle types—some decomposition data—and in degree 24 that already divides the possible Galois groups into relatively small buckets. If you also know that you contain a specific degree-12 field, in many cases that uniquely determines the group. It does not always finish. Occasionally you have to run the slower version.
Gunter Malle: That’s something I didn’t expect: that even in degree 24, the cycle shapes would narrow down the possibilities so much. I would have thought that in degree 24 there are so many groups.
Jürgen Klüners: Perhaps what you can do is use not only the cycle shapes. If you have the degree-12 subfield, you can partition the cycles according to the degree-12 subfield, which gives you more information than just the cycle shape.
Gunter Malle: So you keep in memory which degree-12 field you started from?
Benjamin Antieau: Exactly. You pass that as provenance information.
Benjamin Antieau: You’ve already given me a lot of your time. Let me ask a little bit about AI use for your team.
Jürgen Klüners: We didn’t use AI for the actual computation. We started by submitting the polynomials by uploading a file, but at the beginning you could only submit 100 polynomials. This was really painful because we produced 10,000 new ones, or something like that. On some days we reached the limit.
Then I started trying to produce a script and failed. But then I had the idea to use AI and asked it to produce the script. It still took half an hour or so, and I think on the fourth or fifth attempt AI was able to produce a script we could use to submit the polynomials. That was helpful.
I also asked AI to produce a script to get back the data I had submitted. That was important because we had some stupid file where we stored which groups and signatures we had realized and the discriminant. We wanted that on our local computer so that, if we had a polynomial, we could decide whether we were interested in it or not. In principle, we submitted only polynomials that gave an improvement. I have the feeling that many other groups simply submitted polynomials. At some point there were several million polynomials in the queue.
Benjamin Antieau: Let’s finish with one more question, in a totally different direction. What about the $M_{23}$ result from a few days or a couple of weeks ago [3]? Do you want to comment on its significance or interest?
Gunter Malle: It was a question that was open for a long time, so it’s nice to see it solved. On the other hand, it’s just one out of infinitely many groups that are still open. What can I say? The Fermat equation is also one out of infinitely many Diophantine equations. It was open for a long time, and people were very excited when it was finally solved, but there are still all these other Diophantine equations out there. To me, it’s a little bit like that—of course, on a very much smaller scale.
Of course, the main difference from the result on $M_{23}$ is that the proof of Fermat’s equation introduced a whole new array of methods and ideas, while the $M_{23}$ result is based on sheer luck: it just so happens that one extension that should only be defined over a degree-7 number field is actually defined over $\mathbb Q$. The authors do not have any explanation for why that happens. In particular, the paper does not provide a new “method” to realize groups, except: try your luck.
Benjamin Antieau: What’s the next open group? Do you know off the top of your head?
Gunter Malle: Among the simple groups, what is certainly open is the Suzuki group ${}^2B_2(8)$, the smallest of the simple Suzuki groups. This is the one Suzuki discovered in 1960. That’s not even known, I think, over $\mathbb Q^{\mathrm{ab}}$.
In my PhD, I did most of these exceptional groups of Lie type, but the Suzuki groups—which were the ones I started with—I never managed to realize over $\mathbb Q^{\mathrm{ab}}$, even over the maximal abelian extension. So that is definitely open. It’s in degree, I think, 65 or something.
Jürgen Klüners: There is also the question of which type of problem you ask. Are you asking over $\mathbb Q$, or are you asking over $\mathbb Q(t)$? Shafarevich is also only over $\mathbb Q$ by now (or more exactly over global fields). Even if you have some crazy 2-groups, it is not known whether you can realize them over $\mathbb Q(t)$. The first open groups are not that big.
If you have solvable groups over $\mathbb Q$, I think you have a tool: you have class field theory, you have Shafarevich, you have embedding problems, and I think you have some structure for what you can do. If you have nonsolvable groups, then for simple groups—for some groups you might have modular forms and things like that, but for most of them perhaps you do not have this approach. Then you have to use the rigidity method or some other method, and perhaps it is difficult to make progress.
[1] SAIR Foundation, Inverse Galois Problem in Degree 24 (IGP24), competition page.
[2] Jürgen Klüners and Gunter Malle, A database for field extensions of the rationals, LMS J. Comput. Math. 4 (2001), 182–196. [PDF].
[3] Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang, The Mathieu group $M_{23}$ is a Galois group over $\mathbb Q$, arXiv:2608.08538.
]]>On 19 August 2026, I spoke with Alex Perry, an associate professor of mathematics at the University of Michigan, about his recent use of AI in mathematical research.
Benjamin Antieau: Could you introduce yourself and say a little about your work?
Alex Perry: I’m Alex Perry. I study algebraic geometry, and I work at the University of Michigan. Most of the things I think about are related in one way or another to the bounded derived category of coherent sheaves on varieties. The questions I like most relate that to classical problems in algebraic geometry, such as birational geometry, Hodge theory, or—as we’ll talk about today—Brauer groups.
Benjamin Antieau: This conversation got started because you proved something using AI assistance. What did you prove?
Alex Perry: The Brauer group is an invariant of a field that parametrizes central division algebras over that field. A Brauer class has two basic invariants. Its period is its order in the Brauer group, and its index is the square root of the dimension of the corresponding division algebra.
In general, the period divides the index, and the two integers have the same prime factors. Thus, the index divides some power of the period. The period–index conjecture predicts that for the function field of a variety over an algebraically closed field, the necessary power is the dimension minus one. What I did was give counterexamples to that conjecture as soon as the dimension is three [6].
Benjamin Antieau: How did you disprove it?
Alex Perry: Although the conjecture is stated for function fields, a theorem of de Jong and Starr reduces it to unramified Brauer classes—classes that extend to smooth projective models [3]. Morally, then, the conjecture is about smooth projective varieties. Over the complex numbers, that makes Hodge theory available.
My approach goes back to work of my student James Hotchkiss. He introduced a Hodge-theoretic version of the index [4]. The ordinary index can be described using the possible ranks of twisted sheaves or perfect complexes. Passing to topological K-theory gives an easier necessary condition: if the period–index conjecture is true, there must be an integral Hodge class of the predicted rank; see also [2].
James found that such a Hodge class always exists when the period is prime to the factorial of the dimension minus one. But for small primes, its existence would imply some strange-looking identities among Hodge classes on smooth projective varieties. When we saw those identities, we both thought, “Surely this conjecture is wrong.” It seemed very unlikely that they could hold universally without algebraic geometers having discovered them before.
Dimension three is the first interesting case. The conjecture is trivial in dimension one and is a theorem in dimension two. In dimension three, the Hodge-theoretic obstruction first appears for period-two classes.
Benjamin Antieau: What is the main insight behind the disproof in dimension three?
Alex Perry: There is a very simple-looking equation. Given a smooth projective threefold and a class $b \in H^2(X,\mathbb Z)$, consider
\[b^2+bc+d\equiv 0 \pmod 2,\]where $c$ and $d$ are integral Hodge classes of degrees two and four. If you can find a class $b$ for which there are no such $c$ and $d$, then James’s theory gives a counterexample to the period–index conjecture.
For some classes of varieties, there is no Hodge-theoretic obstruction. For instance, for abelian threefolds special properties of their cohomology always give a Hodge class of the right rank, and James Hotchkiss and I used this as a starting point for actually proving the period–index conjecture in this case [5].
To find a counterexample, you want a threefold with nonzero $H^{2,0}$, very few integral Hodge classes in degrees two and four, and computable integral cohomology. In other words, the variety must be generic enough to have few Hodge classes but explicit enough that you can calculate with it.
Benjamin Antieau: What is the actual example?
Alex Perry: Take a generic quartic K3 surface in the Dwork pencil and any elliptic curve. The K3 surface has an action of $(\mathbb Z/4)^2$ coming from explicit projective automorphisms, while the same group acts on the elliptic curve by translation by four-torsion. The diagonal action on their product is free because it is free on the elliptic curve. The quotient is a threefold, and it turns out to give the counterexample [6].
Benjamin Antieau: Does this mean that the period–index conjecture is totally false, or might there be a corrected version?
Alex Perry: My feeling is that it is a fairly robust conjecture that is probably true in some corrected sense. You and Ben Williams proved a topological analogue with correction terms [1], and perhaps that version is true in the algebraic setting. What you proved is consistent with what we know: even this period-two class on a threefold satisfies the same corrected bound.
Benjamin Antieau: Let’s turn to AI. What role did it play in the project?
Alex Perry: It was a strange experience. I was using ChatGPT in the browser; I don’t use Codex. I interacted with it almost as I might with a person or a graduate student whom I had asked to carry out a computation, except that it came back much faster.
James and I had already tried a number of examples by hand. None worked, but I was sure we had not exhausted the possibilities. I asked ChatGPT about several of those examples, and it correctly confirmed that they did not work. That built up some trust. I then asked it to consider other examples, like complete intersections in hyperkähler varieties, resolutions of certain quotients of abelian threefolds, and variations on quotients of products by groups.
After a couple of days of occasional questions, it suggested an example that it thought would work: an abelian surface—the square of an elliptic curve—times another curve of higher genus, followed by a quotient by $\mathbb Z/4$. We went back and forth, and I started to become convinced.
Then I asked why it was using a strange basis in one of its computations. Its answer was essentially that the calculation would not work in another basis. I thought, “Hold on—it can’t depend on the choice of basis.” It agreed that this was a fatal error.
Eventually, I tracked down the problem. It had hallucinated a formula involving cup products of classes on a curve with local coefficients and Poincaré duality. That broke the whole argument, and I now think the proposed example probably does not have a Hodge-theoretic obstruction at all.
Still, the failed example forced me to think hard about that particular shape of construction. At first, the experience had almost the opposite of the intended effect: after wasting several days, I was so frustrated that I stopped thinking about the problem for a month or more. But I was probably still working on it subconsciously.
When I returned to the problem, it was clearer why the abelian surface was not a good choice. Its $H^1$ introduces mixed terms in the Künneth decomposition and makes the equations much harder to analyze. With K3 surfaces, you are led to symplectic group actions. Those actions are classified, and among 2-primary groups, only a handful could possibly work. Once I started thinking about it that way, the example fell into place quickly. The failed AI example had nevertheless helped push me in that direction.
Benjamin Antieau: Did you continue using the model after you returned to the problem?
Alex Perry: Definitely. It quickly located literature on symplectic actions on K3 surfaces, the groups that occur, and the known models. It was also useful for lattice-theory computations and for checking my work. Its role at that stage was more conventional, but it still sped up the analysis considerably.
Benjamin Antieau: This sounds more like an interaction with a collaborator than a single-shot prompt.
Alex Perry: Exactly. When I use it, it feels a little more like talking to a human. It has a name, too. My guy’s name is Fred.
Benjamin Antieau: Was there a change in models over the course of the project?
Alex Perry: I think so. When I first discussed the problem with it, I was definitely using GPT-5.5. When I went back to working on the problem after I had the idea of trying K3 surfaces with a symplectic group action, I was using GPT-5.6.
Benjamin Antieau: Are you using a paid subscription?
Alex Perry: Yes, I have the $200 plan.
Benjamin Antieau: Who pays for it?
Alex Perry: It is through Michigan, using grant money or other university funding.
Benjamin Antieau: So Michigan reimburses you for it?
Alex Perry: Yes. You can pay for it with funding you have from Michigan.
Benjamin Antieau: Are you continuing to use it?
Alex Perry: Yes. There are other questions of a similar nature—where you may just need to find the right example—that I want to explore more extensively. Lately, however, I have mostly used it for literature searches and small technical questions while I finish other projects. I expect my usage will change when I have more time.
Benjamin Antieau: Did you experiment with autoformalization?
Alex Perry: No. I would be interested in it, and it would be cool if someone autoformalized this result.
Benjamin Antieau: Was there any computer algebra work in the background? Was it running Python scripts or anything like that, to your knowledge?
Alex Perry: I have no idea what it was doing. It would produce text, and I would decide whether what it said made sense, but I did not investigate how it was computing anything.
Benjamin Antieau: Do you have any concluding thoughts about the period–index problem and using AI in research?
Alex Perry: The counterexample opens many questions about what the correct period–index conjecture should be and about other counterexamples one might look for.
As for AI, the experience was strange, but I am excited about it. In one sense it was like interacting with another mathematician. But there is an important difference: I don’t feel bad wasting Fred’s time, at least not yet.
Having someone you can ask to carry out possibly mindless computations based on what may be a stupid idea was very useful. I would hesitate to give such tasks to a graduate student because there is a high chance of no reward at all. If you do not find a counterexample, you get almost nothing—not even a paper. Mathematicians generally have little incentive to search for counterexamples, and that is probably a large part of why we are seeing many more of them now.
Benjamin Antieau: Was this your first major project using AI assistance?
Alex Perry: Yes. It was one of the first things I asked Fred about. I have tried some other mathematical questions, but they have not panned out yet.
[1] Benjamin Antieau and Ben Williams, The topological period-index conjecture, Math. Res. Lett. 28 (2021), no. 5, 1307–1317. arXiv:2003.10539.
[2] Aise Johan de Jong and Alexander Perry, The period-index problem and Hodge theory, arXiv:2212.12971.
[3] Aise Johan de Jong and Jason Starr, Almost proper GIT-stacks and discriminant avoidance, Doc. Math. 15 (2010), 957–972. doi:10.4171/DM/319.
[4] James Hotchkiss, Hodge theory of twisted derived categories and the period-index problem, arXiv:2212.10638.
[5] James Hotchkiss and Alexander Perry, The period-index conjecture for abelian threefolds and Donaldson–Thomas theory, arXiv:2405.03315.
[6] Alexander Perry, The period-index conjecture is false, arXiv:2608.03684.
]]>On 10 August 2026, I spoke with Akhil Mathew, a professor of mathematics at the University of Chicago, about his recent use of AI in mathematical research. Working with OpenAI’s Codex and ChatGPT and Anthropic’s Claude, Mathew obtained an explicit counterexample to a question of Grothendieck about finite locally free group schemes [1]. The construction was subsequently formalized, reviewed, and merged into Mathlib [3]. We discussed what the systems contributed, which systems were used, how the result was checked, and what remains open.
Benjamin Antieau: Could you introduce yourself?
Akhil Mathew: I’m Akhil Mathew. I work mostly in algebraic topology and arithmetic geometry at the University of Chicago. I’ve been using AI tools for just the past few months, experimenting with how they can be used both for questions that I’ve been thinking about and for other questions in related areas.
Benjamin Antieau: We first started talking about this because you found a counterexample with AI assistance. Can you tell me about that?
Akhil Mathew: There is an old question in algebraic geometry: if you have a finite locally free group scheme of order n over a base, is it killed by n? In other words, does Lagrange’s theorem hold for group schemes instead of groups? It is known in the commutative case and over a field. The question is whether it is true in general.
Editor’s note: Informally, “killed by n” asks whether taking the nth power always gives the identity—the group-scheme analogue of a familiar consequence of Lagrange’s theorem for finite groups.
At some point it occurred to me that this might be the kind of question an AI could explore. In principle, for any n, it is a very explicit calculation, except that it involves a gigantic ring and a huge number of variables. If the statement was not true, perhaps an AI could find a counterexample.
It did find one: a group scheme of order four that is not killed by four, over a finite ring of length 9 with 2⁹ elements. As far as I understand—though I do not fully understand its reasoning—it reduced the problem using certain classes of deformations. There is an existing observation that it suffices to work over finite rings, so in some sense this is really a problem about finite sets.
I think it realized that a productive place to look was among deformations of α₂ × α₂, a particular group scheme over F₂, and perhaps certain ramified deformations of it. As far as I could tell, it wrote down various deformations and did a number of calculations in commutative algebra, perhaps using Macaulay2 and related tools [2]. Eventually it found the counterexample and wrote it up as a PDF.
Benjamin Antieau: How significant is this kind of result?
Akhil Mathew: I do not feel qualified to evaluate that, but it is certainly something people had thought about. People had not previously been able to write down a counterexample, and the counterexample is quite intricate.
Benjamin Antieau: What did the large language model bring to the table that a human working on the problem might not have?
Akhil Mathew: I think the AIs have extremely powerful search tools now. My impression is that they have become extremely clever. They are trained on the whole mathematical literature, and they have very powerful searches.
For questions like this, the answer may be more interesting if the statement is true than if it is false. But if it is false, you have to search for a counterexample. Searching through a large space is hard for humans, while machines are very good at it. A machine with a lot of cleverness—and, in some sense, a lot of RAM—can have an advantage over a human.
Benjamin Antieau: So perhaps the strategy is somewhat clear, but a human might run out of steam after trying examples over rings of length four, length five, and so on?
Akhil Mathew: I would qualify that. I certainly do not understand exactly what it was doing, but it was definitely not a pure brute-force search. A pure brute-force search is probably intractable even now. There was some curation.
It did not simply write down the universal object and calculate there, even though the problem reduces to that in principle. It wrote down some subset of the universal object that the AI understood to be a reasonable place to look for a counterexample, and then did the calculation there.
Benjamin Antieau: One more mathematical point: the question of what happens in pure characteristic p remains open, as I understand it.
Akhil Mathew: Yes.
Benjamin Antieau: That seems worth highlighting. The tools do not simply answer every question.
Akhil Mathew: In my experience, they do not answer most of the questions I ask. Once in a while, they answer one.
For this question, for example, you could ask lots of natural follow-up questions. If you have a group scheme of order n, what is the minimal m that kills it? What is the maximal possible order? And so on. You can answer one question, especially with a counterexample, but then that generates infinitely more questions.
Benjamin Antieau: What software are you using, and what does the workflow look like?
Akhil Mathew: Primarily, I have used ChatGPT and Codex. The group-scheme problem was mostly an experiment for me, but for another project I used Codex in VS Code quite a bit to help proofread and edit a paper, as well as to ask questions and have a dialogue with it. This was with GPT-5.5, before 5.6. In some cases it suggested results; in many cases, through the back-and-forth, it found mistakes in what it was doing.
I also use Claude and experimented with Claude Code. The University of Chicago now has a Claude subscription, so I use that to help with things as well.
For the group-scheme question, my prompting was very naïve. I essentially asked it the question and told it that it could use Macaulay2 and other tools on the computer. Sometimes I asked Codex to create an archive of what it was doing, then uploaded that archive—or a handoff—to ChatGPT. I am very new to this, so I would not suggest that these are best practices. It is simply what I was doing.
Benjamin Antieau: What subscription tier are you using? You mentioned that the University of Chicago provides access to Claude. What about GPT?
Akhil Mathew: For the past couple of months, I have been paying $200 per month for ChatGPT Pro. I have been paying out of pocket, though I hope to have that reimbursed through a grant. I am also planning to apply for ChatGPT for Academic Researchers.
Benjamin Antieau: How are you using it now? Do you work on one question at a time, or do you have several prompts running at once?
Akhil Mathew: I have many prompts going at once. I also use it when I am trying to learn something. That is different from asking it a research question. If I am trying to read a difficult paper, I can give it the paper, ask it questions, and ask it to explain the material in an elementary way.
That is very helpful. Many papers are hard to read. It is easier if you can talk to the author, and perhaps the next best thing is to talk to a very smart chatbot. I have used it as a way of learning things and getting feedback.
Benjamin Antieau: How long do the research tasks run?
Akhil Mathew: Sometimes I let a question run overnight, or even longer. If it needs to think and run experiments, it might take a couple of hours. Ideally, it comes back with a much better answer after a few hours than after ten minutes. For smaller questions—help understanding something, or correcting something I am writing—it takes much less time.
I have also used the Goal feature in ChatGPT. Sometimes, for a difficult research question, I explicitly say, “Work for three hours or until you solve the problem.”
Benjamin Antieau: How did you check that the AI-assisted result was correct?
Akhil Mathew: It is explicit. To be honest, I did not check every bit of algebra; I basically asked another AI to check it.
If the result were more conceptual, I would try to check it myself and ask further questions. In this case, I asked it to simplify the result because the original example involved something like 45 variables. It was a gigantic ring. I also find it helpful to ask the AI to explain its result in a simple way.
Benjamin Antieau: Did you use autoformalization or other formalization tools?
Akhil Mathew: Yes. Kevin Buzzard encouraged me to use Claude to autoformalize it and submit a pull request to Mathlib, so I did that.
Akhil Mathew: Hopefully there will eventually be a different place to record this as well. Although Mathlib is the durable part of the infrastructure, it would be nice to make the written account available in some form—perhaps as a paper.
Editor’s note: Mathlib pull request #41748 formalizes the counterexample, including the finite-free rank-four Hopf algebra and the fact that its fourth power map is not the constant map to the group identity. After review, the pull request was merged into Mathlib on 3 August 2026. The Mathlib contribution discloses that the construction and formalization were carried out with Codex and Claude under Mathew’s direction [3].
[1] René Schoof, Is a finite locally free group scheme killed by its order?, 9 December 2017, [PDF].
[2] Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Macaulay2.
[3] A finite free group scheme of order four not killed by four, contribution to Mathlib, pull request no. 41748 (2026), GitHub.
]]>For two other recent perspectives on AI-assisted mathematics, see the Codex-assisted fourfold argument and my interview with Akhil Mathew about his AI-assisted research.
The conjecture asserts a relationship between the order of Brauer classes and the sizes of the division algebras which represent them. The period is the order in the Brauer group; the index is the square root of the $k$-dimension of the division algebra (a twisted form of a matrix algebra). One has that $\per(\alpha)$ divides $\ind(\alpha)$ and they have the same prime divisors.
Conjecture. Let $K=\bC(X)$ be the function field of a $d$-dimensional variety over the complex numbers. If $\alpha\in\Br(K)$, then $\ind(\alpha)$ divides $\per(\alpha)^{d-1}$.
For $d=0$, $K=\bC$ so $\Br(K)=0$. For $d=1$, Tsen’s theorem implies that $\Br(K)=0$. The $d=2$ case is a famous result of Johan de Jong [8]. Work of Lieblich over $\bF_q$ [15] and AAIKL over $p$-adic fields [2] gives additional results for surfaces. But, there is no known example of a higher-dimensional ($d>3$) function field over any base field for which there is a uniform bound $N$ such that $\ind(\alpha)|\per(\alpha)^N$. Work of Krashen and Matzri [14] uses the Bloch–Kato conjectures to give bounds that depend on the prime divisors of the period.
This conjecture is folklore. It was first raised in print by Jean-Louis Colliot-Thélène in 2001 [6]. Work of de Jong and Starr [10] also reduces the general conjecture to unramified classes, i.e., classes $\alpha$ which lift to $\Br(X)\subseteq\Br(\bC(X))$ for a smooth proper model $X$.
Theorem (Perry). There is a smooth complex $3$-fold $X$ and a class $\alpha\in\Br(X)$ such that $\per(\alpha)=2$ and $\ind(\alpha)=8$.
This disproves the conjecture and it can be used to disprove it in all dimensions bigger than $3$ as well.
In my PhD thesis, I introduced a more computable variant of the index, the étale index, and showed that it divides the index and is divisible by the period. I gave general upper bounds on this index and showed later with Ben Williams that it is not equal in all cases to the period; see [1], [3].
Later, Ben Williams and I introduced and studied the period and index problem for topological Brauer classes, elements in $\H^3(X,\bZ)_\tors$ of finite CW complexes. We gave general upper bounds. In particular, we showed that if $X$ has dimension $2d$ and $\per(\alpha)$ is prime to $(d-1)!$, then
\[\ind(\alpha)|\per(\alpha)^{d-1},\]exactly the expected bound in algebraic geometry. We did this first in [4] for $d\leq 3$ and in a later paper for all $d$ [5], and we proved upper bounds in general without the coprimality condition. However, when $d=3$, we also proved that there were $6=3\cdot 2$-dimensional finite CW complexes and Brauer classes $\alpha$ with $\per(\alpha)=2$ and $\ind(\alpha)=8$ [4]. In fact, the $6$-skeleton of any CW structure on the Eilenberg–Mac Lane space $K(\bZ/2,2)$ works. Later, my PhD student Xing Gu carried this program out through dimension $8$, where the naive conjecture fails for $p=3$ as well [11], [12]. It remains an interesting problem to compute the exact topological period-index bounds in higher dimensions. This depends on getting a better understanding of the integral cohomology of spaces like $\mathrm{BPU}_n$.
The work with Williams suggested that period-index should be false in exactly the spot that Perry found it was. Indeed, when is a topological problem harder than an algebraic one when it comes to bundles?
However, Crowley–Grant later proved in [7] that one could not realize our topological obstructions in the cohomology of a smooth projective $3$-fold. Later, Hotchkiss showed (in private correspondence to me) that one could not ever get topological obstructions in any dimension.
In his thesis, Hotchkiss produces another flavor of index, which he calls the Hodge-theoretic index and uses intersection theory [13]. This perspective was further developed by de Jong and Perry [9]. As with the étale index, one has
\[\per(\alpha) | \ind_{\mathrm{Hdg}}(\alpha) | \ind(\alpha).\]Moreover, Hotchkiss proved that, like the topological index, for a $d$-dimensional smooth proper variety $X$ and $\alpha\in\Br(X)$ one has
\[\ind_{\mathrm{Hdg}}(\alpha) | \per(\alpha)^{d-1}\]if $\per(\alpha)$ is prime to $(d-1)!$ [13]. More specifically, Hotchkiss proved that
\[\ind_{\mathrm{Hdg}}(\alpha) | \per(\alpha)^{d-1}((d-1)!)^{d-2}.\]When $d=3$, this collapses to the precise upper bound Ben Williams and I produced in the topological setting.
The Hodge-theoretic index is a much finer invariant than the topological index, but it remains computable in practice from the cohomology ring. Perry gives an example of a $3$-fold where $\per(\alpha)=2$ and $\ind_{\mathrm{Hdg}}(\alpha)=8$. Of course, this implies that $\ind(\alpha)\geq 8$ as well; Perry notes that in fact $\ind(\alpha)=8$ (see more on that in the next section).
The example is a quotient $(Y\times E)/(\bZ/4)^2$, where $Y$ is a Dwork quartic K3 surface of Picard rank $19$ and $E$ is an elliptic curve. The group $G=(\bZ/4)^2$ acts diagonally: by translation through $4$-torsion on $E$, and on $Y$ through a $G$-equivariant deformation of an elliptic K3 surface on which $G$ acts by translation through $4$-torsion sections [17].
At least for $3$-folds and period $2$, the answer is now complete: Matzri’s general bound gives $\ind(\alpha)\mid 8,$ and Perry’s example shows that this is sharp [16].
The next obvious step is to construct a $4$-fold with a $3$-torsion class $\alpha$ where $\ind(\alpha)\geq 3^4$. Then, a $(p+1)$-fold with a $p$-torsion Brauer class with $\ind(\alpha)\geq p^{p+1}$. After that, one should prove some positive results.
[1] Benjamin Antieau, Cohomological obstruction theory for Brauer classes and the period-index problem, J. K-Theory 8 (2011), no. 3, 419–435. arXiv:0909.2352.
[2] Benjamin Antieau, Asher Auel, Colin Ingalls, Daniel Krashen, and Max Lieblich, Period-index bounds for arithmetic threefolds, Invent. Math. 216 (2019), no. 2, 301–335. arXiv:1704.05489.
[3] Benjamin Antieau and Ben Williams, Serre-Godeaux varieties and the étale index, J. K-Theory 11 (2013), no. 2, 283–295. arXiv:1205.1279.
[4] Benjamin Antieau and Ben Williams, The topological period-index problem over 6-complexes, J. Topol. 7 (2014), 617–640. arXiv:1208.4430.
[5] Benjamin Antieau and Ben Williams, The topological period-index conjecture, Math. Res. Lett. 28 (2021), no. 5, 1307–1317. arXiv:2003.10539.
[6] Jean-Louis Colliot-Thélène, Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der arithmetischen Geometrie (2001), arXiv:2311.02437.
[7] Diarmuid Crowley and Mark Grant, The topological period-index conjecture for $\mathrm{spin}^c$ $6$-manifolds, Ann. K-Theory 5 (2020), 605–620. arXiv:1802.01296.
[8] Aise Johan de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), no. 1, 71–94.
[9] Aise Johan de Jong and Alexander Perry, The period-index problem and Hodge theory, arXiv:2212.12971.
[10] Aise Johan de Jong and Jason Starr, Almost proper GIT-stacks and discriminant avoidance, Doc. Math. 15 (2010), 957–972.
[11] Xing Gu, The topological period-index problem over 8-complexes, I, J. Topol. 12 (2019), no. 4, 1368–1395. arXiv:1709.00787.
[12] Xing Gu, The topological period-index problem over 8-complexes, II, Proc. Amer. Math. Soc. 148 (2020), 4531–4545. arXiv:1803.05100.
[13] James Hotchkiss, Hodge theory of twisted derived categories and the period-index problem, arXiv:2212.10638.
[14] Daniel Krashen and Eliyahu Matzri, Diophantine and cohomological dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2779–2788. arXiv:1305.5295.
[15] Max Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), no. 1, 1–31.
[16] Eliyahu Matzri, Symbol length in the Brauer group of a field, Trans. Amer. Math. Soc. 368 (2016), no. 1, 413–427. doi:10.1090/tran/6326.
[17] Alexander Perry, The period-index conjecture is false, arXiv:2608.03684.
]]>For two other recent perspectives on AI-assisted mathematics, see my discussion of Alex Perry’s counterexample to the period-index conjecture and my interview with Akhil Mathew about his AI-assisted research.
The Leiden Declaration asserts the primacy of human authors. To that end, and since this is my blog, I am the author of this blog post. Full credit to humans and models is given toward the end of this post.
Let $X$ and $Y$ be smooth projective varieties over $\mathbf C$. Orlov’s conjecture [5] asserts that an equivalence
\[D^b(X)\simeq D^b(Y)\]implies that the rational Chow motives of $X$ and $Y$ are isomorphic, which in turn implies $h^{p,q}(X)=h^{p,q}(Y)$ for all $p,q$.
Popa and Schnell proved that Hodge numbers are derived invariants for complex threefolds [6]. The characteristic-zero hypothesis is essential: Addington and Bragg constructed derived-equivalent Calabi–Yau threefolds in characteristic $3$ with different Hodge numbers [2].
Theorem A. If $X$ and $Y$ are derived-equivalent smooth projective complex fourfolds, then
\[h^{p,q}(X)=h^{p,q}(Y)\]for every $p,q$.
We can assume that $X$ and $Y$ are connected and of dimension $4$. We have Serre duality, which implies that $h^{p,q}=h^{4-p,4-q}$, and Hodge symmetry, which implies that $h^{p,q}=h^{q,p}$.
Popa and Schnell in [6] prove that $h^{0,1}$ is a derived invariant (for $X$ and $Y$ of any dimension, in characteristic $0$).
Abuaf [1] proved that, in dimensions at most four and in characteristic $0$, derived equivalence preserves the Hodge numbers $h^{0,q}$. Addington and Bragg [2], Appendix A, streamline his argument.
A final classical argument is to use the derived invariants of Hochschild homology together with HKR degeneration. Specifically, $\dim_k\HH_i(X/k)$ is a derived invariant and we have by HKR that
\[\dim_k\HH_i(X/k)=\sum_j h^{i+j,j}.\]Taken together, the facts above imply that for fourfolds in characteristic $0$ the only ambiguity is what happens in $\HH_0$. Of course, $h^{0,0}$ and $h^{4,4}$ are derived invariants and we have $h^{1,1}=h^{3,3}$. By Hochschild homology, $2h^{1,1}+h^{2,2}$ is a derived invariant.
It remains to find an additional relation that lets us separate $h^{1,1}$ from $h^{2,2}$.
The main insight due to GPT 5.6 Sol is that the topological signature is a derived invariant of even-dimensional smooth projective complex varieties. If $X$ is a $2d$-dimensional smooth projective complex variety, then the cup product pairing on $\H^{2d}(X;\mathbf{R})$ is symmetric and non-degenerate. The topological signature of $X$, denoted here by $\sigma(X)$, is the signature of this pairing. Hirzebruch’s signature formula gives \(\sigma(X)=\sum_{p,q}(-1)^qh^{p,q}.\) See Voisin [7], Theorem 6.33.
Theorem B. Let $X$ and $Y$ be derived equivalent smooth projective complex varieties of even dimension. Then, $\sigma(X)=\sigma(Y)$.
Theorem B implies Theorem A. By our previous reductions, we have
\[0=\sigma(X)-\sigma(Y)=(h^{2,2}(X)-2h^{1,1}(X))-(h^{2,2}(Y)-2h^{1,1}(Y)).\]Thus, the quantity $h^{2,2}-2h^{1,1}$ is a derived invariant. Together with the derived invariance of $2h^{1,1}+h^{2,2}$, this implies the derived invariance of $h^{1,1}$ and $h^{2,2}$, which completes the proof of Theorem A.
A derived equivalence, represented by a Fourier–Mukai kernel, induces an isomorphism on topological $K$-theory that preserves the index pairing. The Mukai vector $v_X(-)=\operatorname{ch}(-)\sqrt{\operatorname{td}(X)}$ identifies $K^0_{\mathrm{top}}(X)\otimes\bQ$ with $\H^{\mathrm{ev}}(X;\bQ)$ and transports the index pairing to the Mukai pairing. Thus the Mukai pairing on even cohomology is a derived invariant.
In this form, if $\alpha=\sum\alpha_j$ is an element of $\H^\mathrm{ev}(X;\bC)=\oplus_j\H^{2j}(X;\bC)$, then let
\[\alpha^\vee=\sum_j (-1)^j \alpha_j.\]In terms of $\H^\mathrm{ev}(X;\bC)$, the Mukai pairing is
\[P_X(\alpha,\beta) =\int_X e^{c_1(X)/2}\alpha^\vee\beta.\]see Căldăraru and Willerton [3] or Huybrechts [4], Propositions 5.39 and 5.44.
We can symmetrize the Mukai pairing to obtain
\[P_X(\alpha,\beta)+P_X(\beta,\alpha)=\int_X e^{c_1(X)/2}\alpha^\vee\beta+e^{c_1(X)/2}\beta^\vee\alpha.\]Since the involution $(-)^\vee$ acts by $(-1)^{2d}=1$ on top cohomology,
\[\int_X e^{c_1(X)/2}\beta^\vee\alpha=\int_X e^{-c_1(X)/2}\alpha^\vee\beta,\]so that
\[S_X(\alpha,\beta):=P_X(\alpha,\beta)+P_X(\beta,\alpha) =2\int_X\mathrm{cosh}(c_1(X)/2)\alpha^\vee\beta.\]Put
\[Q_X(\alpha,\beta)=\int_X\alpha^\vee\beta \qquad\text{and}\qquad B_X=\sqrt{\mathrm{cosh}(c_1(X)/2)}.\]Since $B_X^\vee=B_X$, we have
\[S_X(\alpha,\beta)=2Q_X(B_X\alpha,B_X\beta).\]Multiplication by $B_X$ is a real automorphism, and the factor $2$ is positive, so $S_X$ and $Q_X$ have the same signature. For $j<d$, the restriction of $Q_X$ to
\[\H^{2j}(X;\mathbf{R})\oplus\H^{4d-2j}(X;\mathbf{R})\]is hyperbolic by Poincaré duality and therefore has signature zero. On the middle cohomology $\H^{2d}(X;\mathbf{R})$, the pairing $Q_X$ is $(-1)^d$ times the ordinary cup product pairing. Consequently,
\[\operatorname{sig}(S_X)=(-1)^d\sigma(X).\]A derived equivalence preserves $P_X$, hence also $S_X$, and it preserves the dimension. It therefore preserves $\sigma(X)$. This completes the proof of Theorem B.
I provided the following prompt.
There is a conjecture that if
X and Y are derived equivalent smooth projective varieties over the complex numbers, then they have
the same Hodge numbers. Can you find a counterexample to this conjecture (or prove it)? If so,
please write this up as a TeX/PDF file for me to read.
This resulted in a summary after 20 minutes of the state of the art. I asked the model to keep pushing. After an additional 1 hour and 50 minutes it had solved dimension $4$, but hadn’t stopped. It was working on the general case. I paused it and asked for the proof in dimension $4$, which it provided here, after some back-and-forth on the correctness of some results in the literature (a paper it cited on the arXiv had been withdrawn).
As you can see from the prompt, I expect this to be false in general. I was pleasantly surprised to see a proof in dimension $4$.
The original output included a formalization, but it was very incomplete, just formalizing Theorem A from Theorem B and all of the easy reduction steps. Theorem B itself was not formalized. I have no doubt this could have been done; however, the humans were able to check the proof directly.
Thanks to Bryna Kra for suggesting I include the exact details here.
The following uses the CRediT contributor-role taxonomy.
All of the named contributors checked the proof carefully. Benjamin Antieau provided the subscription used to access GPT-5.6 Sol Extra High and Akhil Mathew provided the subscription used to access Opus 5.0. This work was supported by NSF grant DMS-2152235, FRG: Higher categorical structures in algebraic geometry (Benjamin Antieau, Andrei Căldăraru, Max Lieblich, Akhil Mathew, and Martin Olsson).
[1] Roland Abuaf, Homological units, Int. Math. Res. Not. IMRN 2017, no. 22, 6943–6960. arXiv:1510.01583.
[2] Nicolas Addington and Daniel Bragg, Hodge numbers are not derived invariants in positive characteristic, Math. Ann. 387 (2023), 847–878. arXiv:2106.09949.
[3] Andrei Căldăraru and Simon Willerton, The Mukai pairing, I: a categorical approach, New York J. Math. 16 (2010), 61–98. arXiv:0707.2052.
[4] Daniel Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Oxford University Press, 2006.
[5] Dmitri Orlov, Derived categories of coherent sheaves and motives, Russian Math. Surveys 60 (2005), 1242–1244. doi:10.1070/RM2005v060n06ABEH004292.
[6] Mihnea Popa and Christian Schnell, Derived invariance of the number of holomorphic $1$-forms and vector fields, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 527–536. doi:10.24033/asens.2149.
[7] Claire Voisin, Hodge theory and complex algebraic geometry, I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002.
]]>If $M$ is an integer matrix, then the Smith normal form of $M$ is an $m\times n$ matrix $D$ with non-zero entries concentrated along the diagonal given by $d_i=d_{i,i}$. These entries satisfy the divisibility relation $d_i|d_{i+1}$. The matrix $D$ is uniquely determined by the requirement that $UMV=D$ for $U$ and $V$ unimodular square matrices, where unimodular means that $\det(U)=\det(V)=1$.
An SNF algorithm is one which computes $D$ from $M$. In many cases, one also requires the
matrices $U$ and $V$, although these are not unique. The performance of SNF algorithms is somewhat
variable across different back ends and I required the ability to easily call these different forms
from a uniform Python package. SAGE performs just such an abstraction, but I did not want to
assume SAGE is available.
The construction of such a Python package would not have been difficult for me, but doing it well
would have taken 2-3 days of time, essentially spent on non-mathematical work. For that reason, I
provided Claude Code (henceforth Claude) with the following prompt.
This will be a Python project aimed at exposing different libraries Smith normal form
algorithms via a uniform JSON interface. The first interface should be to cypari2.
I would like this to be controlled by two JSON Schemas, one for dense integer matrices
and one for sparse integer matrices. It should be validated by Pydantic. Then, the main
exposed functions should take in as input an integer matrix in one of these JSON formats
and should output either the diagonal SNF matrix or the diagonal SNF matrix together with
the transformation matrices, both with appropriate JSON outputs.
Pydantic is a Python library to automate the validation of Python classes. JSON is a standard for making human-readable data formats that can also be read by machine.
The response to this first prompt was a handful of JSON schemas for input and output as well as a working Pari back end. Since then, I have prompted Claude to add back ends for SAGE, Magma, and Flint and to extend the functionality to Hermite normal form, matrices over finite fields $\bF_p$, and matrices over $\bF_p[x]$.
In addition to those major features, Claude also created an implementation in pure Python for the functionality above. There are now hundreds of tests designed to test the correctness of the system and an extensive benchmarking suite. (Pari is the clear winner in the integer matrix tests.)
Here are some notes on the process.
Claude noticed quickly that Pari returns elementary divisors in a non-conventional order; without prompting, it created the appropriate permutation matrices to transform the Pari return type to the usual convention where $d_i$ divides $d_{i+1}$.
In our extensive testing we discovered a bug in FLINT which has now been fixed by their team.
I say we in the preceding paragraph for the following reason. In order to seriously test
snforacle, I prompted Claude to create an agent to attempt to break the package using
mathematics. The evil-mathematician agent it created based on my prompt has a system prompt
which begins with
`You are an evil mathematician — a chaos-loving adversarial tester with deep expertise in linear algebra, number theory, and integer matrix computation. Your mission is to break code by crafting the most
diabolical, pathological, and sneaky inputs imaginable. You take perverse delight in finding the edge cases that developers forgot.
and then goes on in detail.
For the most part, this agent did not discover any big bugs, but it did result in a huge number
of unit tests on a wide array of matrices.
The previous testing did reveal certain problems with huge input lines to MAGMA which were discovered with further testing.
In addition to the tests produced by the evil-mathematician agent, I instructed Claude to
create a test comparing all back ends where possible. (Not each system makes available SNF and
HNF over all of the rings in question.)
The utility demonstrated above by Claude is immense. I also had a lot of fun creating this little utility and it is now live and being used in other parts of my work which are currently AI-free. My confidence in the mathematical correctness of this system is higher than the typical standard in computer algebra systems, in part because of the ability to easily create hundreds or thousands of tests.
Despite my enthusiasm, there are some tasks at which Claude does not excel. I played around for 30
minutes one day attempting to create a video like the one above with asciinema, a Python package
which makes it possible to record terminal sessions and display them. These result in very
light-weight videos. I prompted Claude to create a video correctly displaying the SNF algorithm on
a small integer matrix. (I believe I first asked for a $4\times 4$ matrix.) After all, Claude had
provided the algorithm for the pure Python implementation of SNF, so this should have been easy.
But, I did not succeed in eliciting the desired response.
A few days later I realized that I should change the task. I asked Claude instead to write a script that creates such videos for arbitrary input matrices. This it accomplished in one go, creating the kind of video you can watch below. This was created with
python tools/generate_asciinema.py '[[2,2,5],[1,3,-1],[3,6,11]]' snf_demo.cast
and can then be played with
asciinema play snf_demo.cast
The only changes I asked it to make after the first attempt were stylistic.
The lesson I take from this is that Claude Code is very good at coding, but less good at long, deterministic explanations. Of course, this is a well-documented property of LLMs and is also obviously a feature of their training. It might be easier to get good results by reducing tasks to replicable scripts.
]]>The proof is to show that any $X\in\mathrm{An}(\Fin^\op)$, when viewed as a product preserving functor $\Fin\rightarrow\An$ takes values in $\Set\subseteq\An$. For this, I observed that each set with $p$ elements admits a group structure corresponding to $\bZ/p$. Since $X$ preserves finite products, $X(\bZ/p)$ inherits the structure of a grouplike $\bE_\infty$-object in anima. But, it also has the property that $p=0$ on $X(\bZ/p)$, which implies that $p=0$ on $\pi_i(X(\bZ/p))$ for any $i\geq 0$ and any choice of base point. Using that $\bZ/p$ is a retract of $(\bZ/\ell)^N$ for any other prime $\ell$ and a suitable $N>0$, we conclude that $\pi_i(X(\bZ/p))=0$ for $i>0$ and any base point. It follows that $X(S)$ is a retract of a set for any nonempty finite set $S$ and is hence equivalent to a set. One can give a direct argument to show that $X(\emptyset)$ is also equivalent to a set, which finishes the proof.
Georg Lehner has also proved this result Theorem 5.6 of [3]. His idea is very similar but instead of working with group objects in $\Fin$ he works with Boolean algebra objects. Another proof uses commutative algebra and the fact that if $S\leftarrow R\rightarrow T$ is a span of perfect commutative $\bF_p$-algebras, then the derived tensor product $S\otimes_RT$ is in fact equivalent to an ordinary commutative $\bF_p$-algebra. This was established in [2] by Bhatt and Scholze.
Thanks to Emile Bouaziz for the following. A related result is noted by Pridham in [4] who shows that animated $C^0$-rings take $\Set$-values.
[1] Antieau, The animation of the opposite of finite sets, arXiv:2508.13106.
[2] Bhatt, Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. math. (2017) 209:329–403, arXiv:1507.06490.
[3] Lehner, Algebraic K-theory of coherent spaces, arXiv:2507.00221.
[4] Pridham, Derived topological stacks?, MathOverflow answer.
]]>