Carlo Pagano

I have read the reaction of @marcusgnt and @ValerioCapraro to @ElliotGlazer astounding announcement. I strongly disagree with them and for practical reasons. I’m working on understanding and rewriting the OpenAI H10 paper: very far from done. But bringing Glazer’s work into the mix was actually helpful. It has shed light on some of the rewritings that a collaborator and I are trying to make. I’m still digesting things, and I’m not claiming to have verified his proposed proof, though I definitely hope eventually to understand it as well. I disagree with the idea that “slop on slop” is somehow additive. The effect need not be cumulative in the negative direction. There can be a nonlinearity here, where an attempt at a harder result can shed light on an earlier undigested argument. That is actually exactly what I’m seeing empirically in this case. Making more raw material available can actually help. It is the responsibility of the mathematical leadership to design new incentives that promote understanding. Asking people not to produce more raw material seems just not empirically useful and not an adequate response from academic leadership. Regarding Capraro's outrage. "The argument looks plausible, and plausibility takes the place of verification." Who said that this is so? The mathematical community will have to digest and verify these arguments of course. However the plausibility here is valuable for this process: this is more raw data that has a strong chance to be correct and be actually profitably used by the humans thinking about the proof. And I am not talking in generalities: this is exactly what is happening here.

Mark Shusterman and I have posted our resolution of the probabilistic version of Shafarevich's conjecture, posed by Liu--Wood and Sawin--Wood: arxiv.org/abs/2609.30200 The main idea of the proof was discovered independently by GPT5.5 pro and Aletheia, an internal agent at Google DeepMind. While digesting the proof we discovered several new applications. Among these, we settled a conjecture of Nikolov on the existence of universal perfect groups. This latter result came from an interesting experiment in letting the agent pattern-match problems to methods in reverse. I wanted to see the random group question settled since I studied Melanie and Yuan's paper back in 2021. Very happy this is finally done!

Classical constructions in explicit class field theory generate abelian extensions using division points on algebraic groups. A conjecture of Andrews–Petsche predicts that this is the only way to generate infinite abelian extensions by repeatedly taking preimages of points under polynomial maps. Today, two independent papers settle this conjecture: one by Zhuchao Ji, Jiarui Song and Junyi Xie, and one by Andrea Ferraguti and me! arxiv.org/abs/2609.24701 arxiv.org/abs/2609.24938 In a parallel project, appearing also today, Andrea Ferraguti, Patrick Ingram and I investigate the global function-field analogue. We formulate a precise conjecture incorporating new phenomena from Drinfeld modules and we prove it in degree smaller than the characteristic arxiv.org/abs/2609.24920. Still a lot to explore in the function field case! AI played a decisive role in the project with Andrea, while almost no role in the project with Andrea and Patrick.

Peter Koymans, Mark Shusterman and I have just posted our resolution of the probabilistic version of Chowla’s non-vanishing conjecture for imaginary quadratic characters over Fq(T), with q ≡ 1 mod 8: arxiv.org/abs/2609.11855 (Over function fields, one cannot expect deterministic non-vanishing: Wanlin Li constructed infinitely many examples of vanishing) We have been working on this for seven years, and it is very exciting to finally be able to share it!