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…

The paper is led by @jieyilong, CTO of @Theta_Network, with coauthors from @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. On the comparison, the 2 efforts use different interfaces and accounting conventions, so the paper treats this as a numerical comparison rather than a formal claim. Craig Gidney and Tanuj Khattar of @GoogleQuantumAI reviewed the manuscript before publication. Read the full paper on @arxiv:

Even in a world of superintelligent AI in math, we will need people like Bartosz @nasqret to inspire the next generation of mathematicians augmenting their work with these tools. I wish I’d had a math teacher like him growing up.

I’m excited about the possibility of seeing finite-time Navier–Stokes blow-up reproduced in a tank of water. In this turbulent moment let's pause and contemplate how far humanity has pushed their understanding of physical reality. And congratulations to all those scientists who were involved in the process! And this is a long list of contributors.

I got very serious recently about using formal languages in mathematics, and I am trying (like I did once with Magma) to internalize how they function and how I can think in them naturally and basically keep up with a formal proof like I can grasp the flow of regular mathematical text in my field. Obviously, due to the popularity, structure of the type theory, and expressive power (+Mathlib), I decided to explore in depth Lean and one other language which I've built with agents for Peano arithmetic. So far, the main obstacle I can see is that the architecture of many tactics makes it super unfriendly to follow the proof. This is one of those gaps + strange syntax in Lean which still makes me very confused when I try to read such a text. I don't have such problems with Magma, where the ideas are encoded in a much more natural way. This is probably one of those directions in which I want to develop: how to make formal languages which are easy to follow, have a particular deduction style, or are simply expressive enough to make the argument look very compact, yet understandable. In hindsight, I can see that these were some of the problems which Georg Cantor had when he tried to formalize the mathematical work of the day. I had a look at Anthropic's formalization of FLT and... there is so much work :) I can definitely sympathize with Kevin Buzzard that his quest is different. We can appreciate some proof artifact, but the composition of the code, high-level engineering, and difficult choices about how things should be formalized to make them work better (filters for limits...) are equally important, or sometimes even more important than a particular proof artifact itself. Overall, mathematics is an art of exploring new avenues, and Lean and other formalized languages give us so much space for beautiful discoveries, new engineering, and can simply help us organize human knowledge better. So yes, learn formalization as something entirely new, a higher, deeper insight into thoughts. You won't be disappointed, maybe only with how badly AI is still doing this autonomously. We should keep improving this too. Going back to work on elliptic curves in Lean :)

I tested GPT-Astra on mathematics. It's a quantum leap. You can talk with the model and prove the statements live in Lean. The feeling is absolutely stunning. You can verify your ideas, compile truth. For a mathematician it feels like finally we arrived in the era where we can focus entirely on the ideation and exploration. Each lemma flows once the logic is set. Before the verification was lagging behind but Astra is very fast and for many tasks the formalization happens as you write your argument in Codex. If you tell the model to use literate programming + LaTeX you end up with your proof combined with the Lean code, everything explained as you wrote, mixed with small chunks of Lean which are digestible. I don't want to go back to the era where the only confirmation of the proof was "aha". Now the "aha" is followed by a green tick that indeed tells you that you have captured the essence. Imagine how cool it will be to have all the lemmas of the world combined in one giant database, pointing to people and models who found them. You compose and mix your ideas and build on the shoulders of the giants. But you see much further now and build much faster. And we are just at the beginning of those changes. So much work to do, so much fun!