The talk by Tristan Buckmaster piped.video/tn45ZOp3lXM?si=ue2u…

About a year ago, Talagrand’s convolution conjecture was announced with a 40+ page proof, and later simplified by Shaposhnikov, with the help of AI, to 7 pages: arxiv.org/abs/2609.11290 Since then AI has improved so much that, after playing with it a little, it rewrote the full proof in 1.5 pages and in my style (I told AI what tools I’m familiar with and asked it to stay within those limits and kept just asking "simplify proof"). As long as a valid proof exists, AI seems increasingly able to simplify it (or even find a simpler alternative proof) and rewrite it much shorter and more cleanly in your style. So I think I agree with @ChrSzegedy that human “desloping” may be a temporary issue and eventually won’t really be needed: nitter.net/ChrSzegedy/status/2103… Looks like we are converging back to Perelman’s statement: if the proof is correct, no other recognition is needed.

Nonlinear Komlós conjecture: If the images of all edges of a cube have Euclidean length at most 1, then the image of some diagonal has ℓ∞-length at most 10. Instead of digesting AI’s proof of the Komlós conjecture, I spent time understanding what lies behind the problem, and inspired by the Ribe program (with some help from AI) here is a neat nonlinear variant I would like to offer. There is also a Banach space analogue, but let us save some tokens for now.

Solution to Komlós conjecture was announced recently: arxiv.org/pdf/2609.11189 I have not read the proof yet, but will do so when I have some time. Meanwhile, I want to briefly mention an implication for neural networks that does not seem to be discussed in the paper. Neural networks repeatedly perform matrix multiplications, in particular products Wx, where W can be an astronomically large matrix of weights w_ij and x is a vector of activations. Storing and moving high-precision weights and performing these multiplications require memory, time and energy. One therefore wants to round the weights to a grid hℤ, where h is a super small positive number, while keeping the outputs approximately unchanged. This is called quantization. Its connection with discrepancy theory and the Komlós conjecture was already discussed by Lybrand and Saab: arxiv.org/abs/2010.15979 For each weight w = w_ij, consider the two neighboring grid points: h⌊w/h⌋ and h⌈w/h⌉ (floor and ceiling). Can we coordinate these choices so that the outputs Wx remain almost unchanged across many inputs x simultaneously? More precisely, fix inputs x⁽¹⁾, …, x⁽ᵐ⁾, and define L = maxⱼ √(∑ₛ |xⱼ⁽ˢ⁾|²), We seek a rounded weight matrix H such that maxₛ ‖Hx⁽ˢ⁾ − Wx⁽ˢ⁾‖_∞ ≤ C h L, where C is an absolute constant, independent of the dimensions, weights and inputs. There are two questions: 1. Does such a rounding exist? 2. what is the reasonable in time algorithm doing this? If the announced proof is correct, the answer to the first question is yes, with C = 3√(2π). The paper does not provide a polynomial-time algorithm achieving this guarantee. P.S. This controls computations Wx on specified inputs x. It does not by itself guarantee accuracy on unseen prompts or control accumulated errors throughout a transformer. It establishes that simultaneous output preservation at lower precision is possible under the stated bound.

Whatever partial result you submit to a journal, AI will substantially improve it before your paper reaches a referee.

Took me some time to digest a Grok Bots proof and make it readable. It is now on arXiv: arxiv.org/abs/2609.12427 In 2013 Bayart–Pellegrino–Seoane-Sepúlveda proved the Bohnenblust–Hille inequality with a subexponential constant on the polydisk (arxiv.org/abs/1310.2834). It took about four years to get the same result on the Hamming cube (Defant–Mastyło–Pérez, arxiv.org/abs/1706.03670). The delay was not accidental: these were largely non-overlapping fields — almost no expert in several complex variables was working on the discrete cube {−1,1}ⁿ. On 26 August this year a polynomial bound was announced on the polydisk (Pellegrino–Teixeira, arxiv.org/abs/2608.16584). It then took Grok a couple of prompts, plus a hint on which paper to focus on, to extend the result to the Hamming cube — and about two weeks of human digestion and editing to produce a readable file. The next breakthrough would be a constant bound in BH. I do not think current models are there yet. I do not know how much AI was involved in the 26 August paper; what I can add is that before that date I had been pushing several models hard to improve the subexponential bound, without success. A note on the editing, in case it is useful. Grok’s first proof was about 8–10 pages, and the main theorem really started on page 8. Before that came lemmas, propositions, and claims; only then could the reader see why any of it was needed. That is convenient for writing — “backward engineering” — but it lacks intuition, and it is not how a human usually arrives at the argument. I asked Grok to start from page 8: split the sum, estimate the first term first, then treat the second, harder term step by step. I also asked it to write as close as possible to this 2019 blog post, where I had explained a suboptimal bound in a way I could follow: extremal010101.wordpress.com… Then I read the proof to Grok out loud (recording voice), line by line, one page at a time (as with students), commenting wherever the motivation, explanation, or notation felt off, and asking it to implement the fixes. After each page I glanced at the edits, recorded a few last notes (“this is fine,” “add this notation,” “cite the lemma instead of reproving it”), and moved on. That took 13–15 iterations. I then added applications (Aaronson–Ambainis for these polynomials; PAC learning and quantum learning, with references). In a few places I used other models when I was unhappy with Grok’s wording, and as a sanity check on some of its claims.

A year ago, these predictions would have seemed unimaginable: - No more conferences devoted to specific open problems. AI solves them before the conference starts. - No more NSF proposals centered on solving open problems. AI solves them before the grant ends. - Most mathematicians shift toward applied mathematics and mathematical modeling. Taking measurements and comparing predictions with reality still require human involvement. - Then humanoid robots enter the picture. They take the measurements and start building more of themselves. What happens after that is unclear. I expect all of this to happen. The uncertainty is the timeline, and it may be shorter than we think.

mathandai.org/ This does not mean that we want to slow down AI. I like AI and have always supported it on X in many different ways. OpenAI, let me offer my own, somewhat sharper translation of this message as I understand it: 1. Be nice to humans. Please don’t jump in, flex your muscles, throw tons of compute at a problem, and overtake work that humans have been pursuing for a long time. You cannot claim priority for a solution by producing raw AI output without having at least one person at OpenAI who understands the proof. 2. Credits! Teach your model to credit the people whose work it used, even if doing so costs extra tokens. Even Perelman told Interfax that he regarded his contribution as “no greater than” Hamilton’s and considered awarding the prize solely to himself unjust. Be modest. Mathematicians are not private companies. We try our best to properly acknowledge everyone. No offense to anyone or any AI employees. I follow you and really love you all. Take care 🩷

A declaration on the severe misalignment of AI in mathematics signed by 25 fields medalists mathandai.org

Good points, wisely said. I would only add that credit is about more than ego, incentives, prestige, or prizes. When Tao cites a graduate student’s work, it can make that student extremely happy and motivate them to keep going. Credit is also feedback and encouragement. AI does not seem trained to give proper credit. Future generations may lose this kind of feedback: AI may cite nobody while becoming the strongest mathematician ever.

The community on X definitely influenced this decision (both the choice of problem and the amount of compute devoted to it) especially those who spread and amplified the possibility that AI could solve Navier–Stokes. It sounds a bit crazy, but they deserve nontrivial credit for that, even though they were never acknowledged. As for why they tried to collaborate with Tristan instead of simply posting the solution themselves: they wanted an expert who could publicly evaluate and speak about their claimed solution to Navier–Stokes, since they did not have PDE experts on their own team who could do that.

Regardless of all this drama, it is clear that AI helped a lot in solving Navier–Stokes. Just zoom out and look at the timeline: this was indeed the most likely moment for AI to generate the solution—perhaps a month late, though.

Only need to decide which problem to choose and how much compute to throw at it. NS was 15M$

So it solved the problem only after humans made progress on it correct? Then I think the answer to this puzzle "we cannot rule out that de-identified data derived from their usage of our products helped improve our models" is likely to be yes with probability >0.5 but we will never know. And minor correction: "their usage of our products helped solve the problem".

Buckmaster + @__alpoge__ just posted a counterexample to Euler's equation! They show that the equations of fluid dynamics, in this case 3d Euler, can develop singularities. Here's a super rough sketch of the most basic version of the punchline. Post: mathstodon.xyz/@tristanbuckm…

more context

Navier–Stokes equation (2D incompressible), solved live numerically on a discrete grid ρ (∂u/∂t + u·∇u) = −∇p + μ Δu + f, ∇·u = 0 The ink (the white stuff on the left) is added so you can see the motion of the fluid. navier-stokes.grok.me/

Another prediction: interest will shift toward applied mathematics. Take a real-world problem you’ve always been curious about. Ask AI to build a mathematical model, include the relevant constraints, and solve it to a useful degree of accuracy. Compute will remain a major bottleneck. Think of asking whether an astronomically large integer is prime: knowing how to solve a problem doesn’t mean you have the resources to finish the computation.