Post #3179051

2025-10-10 18:59 UTC

The AI performed well here too, basically working out that an elliptic coercivity estimate would indeed imply that the theorem was true if the curvatures were sufficiently close to 1 on average. Though it did point out unprompted that this was not actually a new result, because the hypothesis of curvatures being close to 1 actually implied star-shapedness! (It did slightly mishandle the estimation of a perturbative nonlinear term, but not in an unfixable fashion; it was comparable to a mistake that a human expert in nonlinear PDE might initially make.) To me, this resembled a "small data" result in PDE, leaving only the "large data" case remaining. The elliptic nature of the problem, combined with the bounded curvature, suggested to me that there was enough compactness to the problem that the problem could be reduced to a large finite computation using numerical PDEs. I suggested this approach to the AI, who agreed and gave an outline of how such a program might proceed. However, the approach would be extremely messy and unenlightnening, essentially a brute force enumeration of all possible shapes. (4/8)

Replies (1)

@tao@mathstodon.xyz 2025-10-10 18:59

So, at the "small scale" of fulfilling the requests asked of it, the AI performed very well, with only minor mistakes, and contributing several useful ideas that were in the literature, but which I was not aware of. At this point I felt though that further progress required input from actual differential geometry experts to avoid a very tedious brute force check, decided to write up the key results (in my own words, and checking step by step) and post them to the site. After doing so, I noticed in commentary for the original problem that the two-dimensional version of this problem was a classical result of Pestov and Ionin, which is famous enough to get its own Wikipedia page: https://en.wikipedia.org/wiki/Pestov%E2%80%93Ionin_theorem . But a glance at the picture on that page showed me that my intuition was quite wrong: in addition to "nearly round" sets, there were important cases where the set was very far from round, but instead resembled some roundish objects joined together by very long thin tubes. (5/8)

Open ##3179052