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: 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: Looks like we are converging back to Perelman’s statement: if the proof is correct, no other recognition is needed.

Grok 4.6 (build) solved the “Greedy is least speedy” conjecture of Holmes–Holroyd–Ramírez by proving their stronger Conjecture 5 (shown in the figure), which immediately implies it. chat transcript: polished paper: I was chatting with my colleague (one of the strongest mathematicians I know) about AI’s capabilities in mathematics, and he seemed a bit skeptical. So I proposed a test: give me a problem that seems within reach for an expert but beyond current AI. He suggested Conjecture 5 from the Holmes–Holroyd–Ramírez paper. After some experimenting I told him, “Look, I think AI solved it.” He was a little surprised and, out of curiosity, gave me two other questions that he believes he can solve, to see whether AI could solve them as well. So far, AI has not solved either of them. This conjecture (now theorem) has an interesting application: imagine a particle on the integer line ℤ. At each position x, it moves one step right with probability pₓ in (0,1) and one step left with probability 1 − pₓ. The environment is m-periodic, so p₀, …, pₘ₋₁ repeat forever. Starting from 0, let Xₙ be its position after n steps. It is known that lim Xₙ/n exists almost surely, call it velocity v. Now fix the probabilities p₀, …, pₘ₋₁, but allow yourself to rearrange them within one period. How should they be arranged to make the particle escape from its starting point as slowly as possible? It turns out that an optimal arrangement is the pendulum (greedy) arrangement z* shown in the figure (except the degenerate case ∏ⱼ (1 − pⱼ) = ∏ⱼ pⱼ when the random walk is recurrent so every arrangement has v=0).

AI’s ranking of open problems solved in today’s arXiv list. AI put problems connected to Gromov’s work in the top two spots. For #3#, I remember attending a talk by one of the authors. #4# is, to me, one of the cutest problems in complex analysis, I first learned about it in Chapter I of Garnett–Marshall’s Harmonic Measure book about 15 years ago.