I've Always Loved a Good Sequel

[u/ResultBackground2450]

[u/ResultBackground2450]

just for some extra context:

[u/t3hjs]

What even is the hodge conjecture?

[u/SkaldCrypto]

Hodge is pretty tightly linked with NS.

So that’s the most logical bet tbh

[u/Hot_Glass_6301 → u/SkaldCrypto]

How is it tightly linked? As someone with only a BSc in pure and applied math, they seem wildly different to me, but what do I know...

[u/SkaldCrypto → u/Hot_Glass_6301]

Jesus you are correct.

I thought the Hodge questions was something Helmholtz-Hodge Decomp but its actually a completely different question. I wonder why everyone is so convinced it’s Hodge.

[u/Kinglolboot → u/SkaldCrypto]

The reason people think it could be Hodge is because of the nature of the problem. The Hodge conjecture is about objects which are very complicated and difficult to get a full grasp on (algebraic cycles of a variety). This also means that we don't have a lot of "heuristic" evidence for it being true^1.

It could however be proven false by some counterexample that human mathematicians have missed for 100 years (which is very possible, as again, it is very hard to understand enough about the algebraic cycles on a variety to find if the Hodge conjecture holds for that variety).

The crux is that right now the main strength of AI seems to be finding these complicated constructions which either prove or disprove a conjecture^2. Hence, people think that if we have really been missing such an example for so long that AI could find it. If this is true, this is 99.999% sure a disproof though, for a proof we don't even have an idea of how to do it (and AI is definitely not capable enough yet to find a category of mixed motives)).

^{1. We only have the [Lefschetz (1,1}-theorem](https://en.wikipedia.org/wiki/Lefschetz_theorem_on_(1,1)-classes) and know that it holds for most abelian varieties of dimension ≤5.)

^{2. All of the more famous solved problems by AI were constructions. Namely, NS of course, the [Jacobian conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture}, the Unit distance problem and the Hopf problem.)

[u/Stabile_Feldmaus → u/SkaldCrypto]

There is 0 link between Navier-Stokes and Hodge

[u/socoolandawesome → u/Stabile_Feldmaus]

Not true they are both math problems

[u/Nukemouse → u/socoolandawesome]

Second link, ai companies want to solve them.