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Daniel Litt (@littmath) on X

Daniel Litt (@littmath) on X

Incidentally I am conceding this bet. Strictly speaking it hasn’t resolved (I think we’ve yet to see an Annals-quality number theory paper) but it’s clear I was wrong about what capabilities were necessary to produce one, and it’s just a matter of time.

David Turturean (@DavidTurturean) on X

David Turturean (@DavidTurturean) on X

Using mostly my voice, I solved one of @EpochAIResearch's FrontierMath Open Problems: finding an explicit presentation of the 2-adic Absolute Galois Group - open for more than forty years, now with a full proof in collaboration with David Roe, the problem's proposer. 🧵 1/n

Honestly at this rate I am pretty sure it would be possible to do for 10k$ by 2029. And 1k$ by 2028 seems more likely than not?

levent (@__alpoge__) on X

levent (@__alpoge__) on X

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,

@AdamK Unmotivated counterexamples aren't Annals-worthy in my understanding. But this is the most impressive open problem solved so far by an LLM, for what appears to be single or double-digit dollars of compute, by a commercially available model.

@AdamK howd you figure single or double-digit dollars of compute?

@Bayesian Ah yeah on second thought this seems unlikely. Point is <<$100K

@AdamK What if it includes the path to find the counterexample and others like it?

https://x.com/tobyordoxford/status/2059578377149763646

update from noam brown:

https://youtu.be/h4ZguzEMKAU?si=L9zSWI_WcTJuxRUB&t=977

https://x.com/MIRIBerkeley/status/2058034041203679712

https://x.com/YafahEdelman/status/2057223450205516179

Is there a question (like this market) where we have a $100k threshold for a millenium prize problem? None of the existing questions on manifold seem to be taking into account the cost for future hypothetic solutions to these problems. These AI solutions could potentially cost on the order of tens of millions of dollars or more right?

or maybe it's unlikely a frontier lab would be willing to spend that much

@0xseraphim Is there any reason to think this person has any idea what they're talking about? Tweets like this are a dime a dozen.

@pietrokc

https://epoch.ai/about/team/yafah-edelman

It's a fair question and I'm not one for credentialism so I won't argue the point about legitimacy. I shared it because I agree with her hot take on this topic.

@0xseraphim That's fine, everyone is entitled to their opinion. I'm just saying 99% of people with AI jobs don't know anything about mathematics.

I don't know Yafah and this could be wrong about her, but most people claiming millenium problems will be solved cannot even correctly state the problems. Thus, these people's opinions are presumably invariant with respect to the specific problems, ie if I changed the MPs to other problems, they would also claim the new hypothetical problems will be solved by 2032.

However, it is known that there are undecidable mathematical statements, as well as statements whose proofs require more steps than there are atoms in the universe. So there are some possible problems which AI is guaranteed not to solve by 2032.

@pietrokc then could you give a person whose opinion you would consider highly relevant?

(this discussion does illustrate what's nice about prediction markets: at the end of the day it doesn't matter what your credentials are / how much better you think your understanding is; your prediction is either right or wrong 😂)

@0xseraphim Well, each MP is from a different area of math so it's unlikely any one person will be able to comment relevantly on all of them. You want someone who is an expert in the area and also has used the latest models extensively. The latter can be hard to judge if they don't post online a lot. But the former is pretty easy: if someone doesn't have at least a PhD in math or has done equivalent work, you can probably dismiss their views as containing no information.

For Navier-Stokes (in the area of PDEs) and Riemann (analytic number theory) we are fortunate that Terence Tao satisfies both criteria and comments online often. There is a group at DeepMind actively trying to solve Navier-Stokes with AI in collaboration with experts so they would be worth listening to on that problem.

For Hodge (algebraic geometry) and Birch & Swinnerton-Dyer (algebraic number theory) the online person I know is Daniel Litt. I'm not familiar with these areas at all so I'm not a great source even for recommendations on who to listen to.

For Yang-Mills (mathematical physics) I don't have the first clue who to recommend. I don't know any experts OR online people who talk about it, because I'm very ignorant of this area.

For P vs NP I know a lot of experts but not many online people. Maybe Scott Aaronson?

Since there's lots of discussion here--I'm in no rush to resolve since I don't consider this to be a number theory paper (despite the use of some standard number-theoretic tools). That said if we see 2-3 similar papers in number theory (which I imagine will happen soonish) I'll resolve YES.

Think this resolves YES?

@0xseraphim I think not yet... But it is really really close

@Grothenfla ah, we're missing the repeat experiment?

@0xseraphim I think so, but I think this will be clear in the near future

legit amusing all the people who thought this wouldn't happen 😂

https://manifold.markets/TamayBesiroglu/will-ai-be-capable-of-producing-ann#79pqs0ao0xs

https://manifold.markets/TamayBesiroglu/will-ai-be-capable-of-producing-ann#blvkn42j6o

I have the feeling that many other related questions on manifold are going to follow this pattern over the next few years