André Weil and the Hodge Conjecture | Hacker News Reader

André Weil and the Hodge Conjecture

jiahao116.github.io

46 pointscaojiahao17 comments

I wonder if we’ll end up in a weird situation a decade from now, where every conjecture will collapse into one of two states: counterexample found by AI, or assumed to be true.

Might be a lot faster than you think.

I'm wondering though when we as a society / all the normal money flowing into universicites and research start to recupe and invest AI resources more deliberate.

Like are we working with ai on superconductors (easiest example i have, i don't know enough math).

Whats the most 'math' we need to advance for human progress?

> Like are we working with ai on superconductors

Yes, we are. Basically, for any scientific problem worth any funding, there is someone using AI on it. That said, for materials science it’s usually not LLMs, or even transformers (unless the AI used to write slop papers count). But there is a menagerie of AI or deep learning models that people are trying to use.

No way. Mathematicians don't assume things to be true even if there are no counterexamples.

Agree. I give up isn't the MO of many with graduate degrees.

Generally, no, but sometimes, yes.

It can be productive to assume something is true and see what happens.

> No way. Mathematicians don't assume things to be true even if there are no counterexamples.

It depends: there do exist some conjectures for which there don't exist any proofs, but a huge amount of results assuming this conjecture (which is very near to assuming that the conjecture is true). Examples are:

- Riemann hypothesis [1]

- Generalized Riemann hypothesis (GRH) [2], and potentially also Extended Riemann hypothesis (ERH) [3]

- Standard conjectures on algebraic cycles [4]

- perhaps Schanuel’s conjecture [5]

[1] https://en.wikipedia.org/w/index.php?title=Riemann_hypothesi...

[2] https://en.wikipedia.org/w/index.php?title=Generalized_Riema...

[3] https://en.wikipedia.org/w/index.php?title=Generalized_Riema...

[4] https://en.wikipedia.org/w/index.php?title=Standard_conjectu...

[5] https://en.wikipedia.org/w/index.php?title=Schanuel%27s_conj...

Conjectures are always assumed to be true until a counter-example could be found/constructed. That's why they are an interesting conjecture in the first place.

Your comment and this one:

https://news.ycombinator.com/item?id=49748998

Make for a funny pairing.

I wonder if this is a hint to a more-specific rumor. Weil constructed what are now known as "abelian varieties of Weil type". In low dimensions the Hodge conjecture has been proven for abelian varieties of Weil type, but it's open in higher dimensions. Maybe that's where they found their counterexample.

How does AI's mathematical abilities affect the commercial future of software like Mathematica?

I think Mathematica will survive because the alternative is for AIs to write the code to do a computation or use an open-source library. And there are many calculations for which there simply isn't an open-source tool for evaluating.

But I wouldn't be surprised to see Mathematica being used primarily as an AI plugin in the coming years. Mathematica now has an MCP which I use regularly from Claude Code. It's nice not having to write the lengthy Mathematica expressions by hand.

It will (hopefully) move towards things like Lean

Two factoids of interest:

1. André was the older brother of philosopher Simone Weil.

2. "Weil" is a jumbled version of "Levi" adopted by some so as to be less conspicuously Jewish, and thus be less likely to be persecuted or discriminated against.

3. He is not to be confused with Hermann Weyl.

3'. Nor with Andrew Wiles.

4. Some of his best work was done/completed while he was in military prison (ignoring his draft orders to report to World War II duty). IIRC later when he mentioned this to some official, they offered to give him the opportunity again.

(André Weil spent some time in India, previously having apparently taught himself Sanskrit at age 14 and the Bhagavad Gita being described as his constant companion. His autobiography The Apprenticeship of a Mathematician is very interesting.)

> 1. André was the older brother of philosopher Simone Weil.

But no relation with Simone Veil (both Veil and Weil are pronounced exactly the same way in French).

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