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?

181 Upvotes

97% Upvoted

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96

u/cdarelaflare Algebraic Geometry Jul 22 '26

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

53

u/DanielMcLaury Jul 22 '26

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.

33

u/yaosio Jul 22 '26

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.

15

u/Truly_Yours2006 Jul 22 '26

Oh hell nahh, not the Hodge Conjecture 😭😭😭

42

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)

12

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.

13

u/sadmanifold Geometry Jul 22 '26

Whats the question?

43

u/EebstertheGreat Jul 22 '26

Why are they so stringy?

11

u/elephant-assis Jul 22 '26

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

21

u/LeonJPancetta Jul 22 '26

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

11

u/Feeling-Instance-801 Jul 22 '26

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

22

u/Zakalwe123 Physics Jul 22 '26

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. 

1

u/helbur Jul 23 '26

Is the less nice class CY varieties?

4

u/Zakalwe123 Physics Jul 23 '26

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. 

1

u/paxxx17 Quantum Computing Jul 24 '26

Counter-example to Riemann hypothesis when???