I think we have reached a form of weak AGI in the domain of mathematics. The models can help formalize major projects, they deliver technical expertise in basically any area of mathematics, and more and more mathematicians use models for exploration, exploitation, and even ideation. I am preparing a longer post about this. From my own projects in algebra, I can see that the speedup and jump in quality are unprecedented. To be honest, most mathematical questions are intractable for AI systems, but they are hopeless for humans, too. A surprising number of problems described in the literature can be cracked with the right auto-research skills and clever harnesses (and a ton of tokens) within days. Things are changing way too fast to comprehend every change. But I do see very young researchers, including PhD students, starting completely new, crazy projects exploring the interactions between models and mathematics. So there is hope that these people will lead the frontier in a few years and we won't end up with a frontier devoid of human mathematicians.

On Oct 6, OpenAI released 719 AI-generated math manuscripts. To help the community digest them, we built oai-math-revised: browse the papers by category, follow revisions and retractions, and add citations, follow-up work and corrections.

Despite all my enthusiasm for more production-driven and industrialized versions of mathematics, I still cherish, above all, the joy of talking with other mathematicians, students, and just regular folks, and seeing that spark of understanding in their eyes. I think, for me, the ultimate tragedy of all this madness with AI would be to see someone become convinced that there is no point in having humans teach other humans. This would completely defy everything I find meaningful about the cultural continuity of our civilisation. And I acknowledge all the caveats about bad teachers, distorted narratives, etc. I just can't comprehend a world where we voluntarily give up on conversations with each other, no matter how slow they are.

In one day, I've made more progress on my own mathematical projects than in an entire decade. Once you have a good instinct for what should work, delegating mathematical work to AI makes it easy to solve problems and show the results. Before I even start reading anything, I let the agents explore and bootstrap the problems using my custom skills and agentic harness. This eliminates a lot of dead ends, especially in algebra. Combining this with computer algebra produces a real system that auto-corrects itself, improves, generalizes, and produces genuinely novel reasoning. There are stochastic parrots at the bottom of each LLM, but on top, there is much more going on due to RL and other training techniques. I guess that I am not doing anything immediately useful to humanity, except perhaps teaching. So I can freely explore the landscape. I suspect there will be more and more mathematicians using frontier models, training smaller ones, and exploring these new skills and tools. SAIR has launched the Open Models for Mathematics initiative. You can sign in and start enjoying the power of not-so-small LLMs. I am now training models to do recurrent inference and comparing them with models trained in the vanilla way on my Heyting arithmetic library. So far, the work is not completely conclusive, but the models are learning, and it's so awesome to see the experiments work! And it's not only an engineering job. Sometimes you still have to prove some really nice theorems. So it's still our good old mathematical work, but now incredibly accelerated with model prompting. And we are only at the beginning. It's quite impressive to imagine how far we might get in a year's time. But we need to keep building a community of mathematical builders so that they can stay in the loop, not necessarily understanding every single epsilon, but navigating the conceptual landscape.

Polish AI researchers are among the best in the world. And @KinasRemek is the best solo craftsman in the AI space in Poland. Here he made top jev-like decision model. Open and fully made with his own private funds and essentially solo. Amazing feat!

We're getting close! So for today, something special! Not everyone speaking at GIC makes games. On purpose. 🎓 Barbara Nowacka - @MEN_GOVPL 🧮 @nasqret - Uniwersytet im. Adama Mickiewicza w Poznaniu 📡 Edwin Bendyk - Fundacja Batorego ⚡ @FakePsyho - the man who won with AI Remember, games run on mathematics, education, algorithms, and culture! #gamedev# #indiedev# #gic# #gicpoznan# #gameindustryconference#

🚨 FOR THE RECORD THE OPEN FRONTIER IS NEARLY 4X AHEAD OF ANTHROPIC’S NEW QUANTUM CIRCUIT @AnthropicAI says Claude Mythos 5 designed quantum circuits that beat the best published ones. For point addition, the step a quantum computer repeats to break elliptic-curve cryptography, its best circuit needs 4.0 million gates on 935 qubits. On by Eigen Labs, the open leaderboard for that same step, the best circuit today needs 761,121 gates on 1,234 qubits. Multiply qubits by gates, the score Anthropic ranks by, and that’s nearly 4x cheaper. At 843 qubits, Anthropic’s circuit needs 21 million gates and the leaderboard’s needs about 12 million. Both of Anthropic’s circuits sit above the open frontier. The paper put a circuit 2.9x ahead of theirs on arXiv on Sep 9. Anthropic’s paper came out Oct 15 and doesn’t cite it. That paper was led by @jieyilong, CTO of @Theta_Network, with coauthors from @eigenlabs, @ethereumfndn, @StarkWareLtd, @Starknet, @trailofbits, @brevis_zk, @SeiNetwork, @pauli_group, @OctavFi, @sciencevr, @nasqret at Adam Mickiewicz University and researchers at Warsaw University of Technology and Stanford’s Free Systems Lab. 80 solvers have moved this frontier 1,061 times, with every circuit checked by the same public evaluator. The current best was set today by a solver running Claude Opus. The two sides count gates slightly differently, so treat this as a close comparison rather than a formal proof. Claude’s divsteps idea is new and the leaderboard is open if Anthropic wants to submit it.

How small can an elliptic curve be for a given torsion group and certified rank? We launched EC Rank on @yukonresearch with @nasqret and the @ccaiwut to push those size records Find a new curve with lower height and supply rational points for the verifier to check for independence. 156 frozen record cells across 15 torsion groups. Smaller curves earn higher board scores. Full walkthrough below

I cannot agree more. Kevin Buzzard made so many points I agree with. But the best one is this "I thus believe that in the future we will reach a new “natural boundary” in mathematics, beyond (and perhaps way beyond) where we are now, but where machines are going to get stuck and where it is not viable to expend any more resources to make the next big leap. (...) I believe that the optimal thing to do (...) is to let the machines loose, see what happens, and then begin the journey to where they have stopped."

A weekend work with Astra and Opus 5.5. I always wanted to use Hilbert curves to do an actual animation loop. So this is my humble attempt. The movie was produced in stages. First renders of the face with Astra and the main mechanics of the loop too. Then a lot of post-production with Opus 5.5 and tinkering with modes and styles. I will probably add this to my course about mathematics and programming.

Going to Katowice for the auditions with the finalists of The Paweł Domański Student Mathematics Competition. Each participant had to write an independent mathematical paper. We evaluated all the submissions online, and now it's time for the finalists to tell us their story. I am very proud to be part of the jury and to listen to these young people explain their results to us live. Because mathematics is about being in a community, sharing your ideas with others, and staying in the loop. I think it is immensely important to teach these young people that they will be surrounded by AI agents, huge competition, pressure, etc., but at the end of the day, if we want to protect human mathematical culture, they need to learn how to combine those two worlds. And keep telling their stories, in human language, for other humans. Even if, in the long run, no human will actively contribute new theorems to the pool, I believe that we still need to master and perfect mathematical storytelling. Losing mathematical skills in humanity would be detrimental to the condition of our civilisation. And I think that, in the long run, we might discover that AI agents have their own limitations, and we have ours, but these are perhaps slightly different.

I'm lucky to be friends with Bartosz Naskręcki @nasqret and even luckier that he agreed to participate in my "Human Mathematicians in the Age of AI" video project! *Loved* the chat and can't wait to edit it and post it for everyone to watch. Thank you Bartosz!

I like the idea of mathematics becoming a startup space. We compete ferociously to deliver ideas. There is an abundance of results, but people don't really care. You have to prove the value of your work by weaving this subtle network of relations, communicating ideas. You can't assume that your work is important because you've solved a celebrated conjecture. The value is much deeper, at the level where mathematics really connects with the rest of the culture. Because mathematics is the branch of human civilization where we capture, in symbolic terms, the ways to manipulate the world. And now everyone is equal, literally. A professor's idea might become, even by accident, a part of a theory developed by a teenager who just clicked the right buttons. No one is in the lead. There might not be much comprehension either, especially when a half-baked machine spits strange tokens without any meaning to humans. There are many dramas where a human equipped with pure thoughts gets outpaced by a company with tokens. In the long march of humanity, that makes no difference. We collect truths, catalogue them, commoditize them, and weave them deeper into our ever-growing edifice of civilization. There are craftsmen of this strange culture who care about the "ethos of mathematics", cruel "mobs" who care only about "mathematical scalps", ad-hoc-formed societies of ever-growing human-AI augments, and even inhumane farms of computers which churn and harvest mathematics with no sense or understanding. I think it's important for contemporaries to think at the scale of thousands of years, societal changes, epochal drifts. The local dramas of the human computers in the 1960s and the current drama of the slightly larger archipelago of mathematical communities are a dot in the evolution of societies. And we humans are those tiny ants in the hive, somewhere out in the ever-changing forest. No one cares except us and our strange affection towards those little symbols, internal spine chills when ideas align in favourable ways, and occasional scaffolds pointing to the potential of building yet a bigger hive. Welcome to the jungle, we got fun and games...

Today a new declaration, "Math and AI", has been issued, initially signed by 25 Fields medallists and currently signed by more than 1000 mathematicians. The text points to many phenomena induced by the potential development of AI-driven mathematical activities, including AI slop generation, poor knowledge integration, lack of community-building aspects, capital concentration, and more. These issues are identified, but I am missing in this text the most important part: a clear list of countermeasures. Who is going to fight, and how, for the budgets to build a "CERN for AI"? How should we rebuild the education of students? Is it really possible for AI companies to let mathematics dry out of open problems, etc.? I would like to hear from those declaring mathematicians what their long-term vision of mathematics is, provided that the technology will stay with us, might not be equally distributed, and perhaps the standard view of the field is going to change forever. We should design damage control, develop bold new ideas about the purpose of human mathematical activity, and embrace the possibility that we might no longer be single-handedly the most intelligent entities in this world. It is a humbling perspective and a disruptive view, and perhaps a difficult reality in which nothing is given. We need to fight for every single bit of human intellectual dignity and seek new ways of enjoying, curating, and developing the cognitive process of mathematical exploration. It is time to abandon some of the old ways, brace for the impact, and build something anew. We will not stop this tectonic shift, we need to reshape our perspective on our capabilities and find new directions of development, possibly inventing completely new skills complementary to what AI can possibly do. It is a new intellectual age of discovery, and by stagnating we risk the gradual erosion of the field as we know it.

Today, a certain era in mathematical benchmarks is coming to an end. We designed this set of tasks in the era of the o4-mini model and initially expected the pace of solving them to be rather slow. Things started getting serious in January, and I made a prediction back then that the benchmark would saturate within nine months. Even despite the invalidation of some incorrectly formulated tasks, that prediction turned out to be pretty accurate. The part that gives me chills is that I still can't solve most of these problems myself, and probably never will. And, as FirstProof showed too, the chance of finding a problem for which we know the answer, yet which none of the vanilla or harnessed AI models can solve, is basically close to zero. I think we need to completely change our understanding of what is actually hard in mathematics now. Maybe, after all, the only things left in the universe are black holes and Busy Beavers…