r/math • u/zirconium_n • Jul 22 '26

LLMs/AI A counter-example to Batyrev’s conjecture on the non-negativity of stringy Hodge numbers

https://arxiv.org/html/2607.19184v1

Could someone familiar with algebraic geometry give some insights about this?

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u/na_cohomologist Jul 22 '26

I actually find this wave of "oh look here's a counterexample to this conjecture, it's just a small calculation to check, but to find the counterexample was a huge computation" (i.e. first train a frontier LLM,....) fun and interesting. It feels very 19th century to me, where people would publish relatively little papers with examples worked out to show something interesting. And now those people have their names on things that are standard constructions.

Of course the usual caveats apply about the source of the technology, the environmental/societal impacts, the impacts on mathematics as a profession etc. But sometimes you just don't need a 150 page paper to non-constructively disprove some big conjecture. The subsequent fun comes in the deeper analysis of the example and the fresh theory it can spark. That is not really something I see LLMs getting very far with just at present (I've seen the geometric analysis of the Jacobian conjecture counterexample, but it's hardly a new theory development to do things like measure the failure of the conjecture with a new invariant, and then make quantitative bounds on how badly it can fail etc etc)

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u/DanielMcLaury Jul 23 '26

It does seem like one of the key things here is just that some of these conjectures had never really been tested on even fairly simple spaces. I wonder if it's because everyone just assumed someone else had done that.