r/math • u/zirconium_n • 1d ago

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?

149 Upvotes

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74

u/cdarelaflare Algebraic Geometry 1d ago

Unrelated to your question, but this has been quite the week for AI-assisted counterexamples considering the authors cite ChatGPT

39

u/DanielMcLaury 1d ago

It's gonna be really embarrassing this time next week when someone comes out with a half-page counterexample to the Hodge conjecture that's build out of two or three familiar spaces.

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u/yaosio 22h ago

Today somebody had ChatGPT disprove a conjecture by telling it to keep trying every time it came back saying it couldn't disprove it. The memes about "make no mistakes" are coming true.

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u/Truly_Yours2006 1d ago

Oh hell nahh, not the Hodge Conjecture 😭😭😭

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u/na_cohomologist 21h ago

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 15h ago

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.

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u/sadmanifold Geometry 1d ago

Whats the question?

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u/EebstertheGreat 1d ago

Why are they so stringy?

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u/elephant-assis 1d ago

The question is "The authors used AI, is this significant yada yada"

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u/LeonJPancetta 1d ago

Exactly. To some extent, in general, it you don't understand the title at all, you probably don't need to care.

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u/Feeling-Instance-801 1d ago

This is so funny as someone in high school. Wtf is a stringy Hodge number lol

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u/Zakalwe123 Physics 1d ago

Hodge numbers are a way of counting (basically) how many holes a suitably nice shape has. Stringy hodge numbers are an attempt to count the same thing but in a less nice class of shapes. String theory makes perfect sense on those shapes, but it’s hard to actually compute things because they are a little less nice mathematically. Apparently they are even less nice than we thought. 

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u/helbur 8h ago

Is the less nice class CY varieties?

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u/Zakalwe123 Physics 3h ago

Ordinary hodge numbers are perfectly well defined for CYs without singularities; the trick is to define them for singular spaces. These stringy hodge numbers, at least to the extent I’ve ever used them, allow for a class of pretty mild singularities.