I nearly made it through the first line, OK, sentence of what looks like the Intro, which doesn't bother introducing itself and isn't actually the introduction which occurs later.
I can only describe pages one and two as some sort of foreplay. If you manage to hang on until P108 you can dispense with a safe word in future.
100 pages of dense mathematics created by Claude. It doesn't even have an abstract or a summary. What are we looking at here? Has anyone read this? Please explain like I'm not a math PhD.
I’m pretty sure no one (except perhaps Anthropic insiders who had prior access, and probably not even them) has properly digested this paper yet, and some caution is warranted given the notoriety of this problem and the history of claimed solutions that did not stand up to scrutiny. But if it stands up, it’s a really big deal.
complex structure in S^6 would be an associative version of octonions (which are non associative "almost complex" versions of their cousins native to S^3, which you may know as (unit) "quaternions")
Big enough that Atiyah the godfather of this subfield, in 2016 claimed to have solved it (not sure if it was before or after the dementia)
This is not just any math PhD, it's the anthropic guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal if it were human
Hey very useful highlighting there, super, thanks. Then the result makes the normed division algebra family more in harmony, by curing the associativity defect of the lopsided member O. This can get the interest of physicists also. Also nice the reminder to think about spheres of magic number dimension for the algebras of the family. From a chat, unit norm elements form S^0, S^1, S^3 and S^7. Dimensions are the number one guess from the family name, minus one. nlab explains somewhere S^0 as two points, think -1 and +1. Quaternions (unit norm ones in S^3) act on S^2 by rotations but S^2 is the base space of a Hopf fibration with S^3 on top. One can ask about the relation between well known S^7 and the S^6 object.
yep I have feet wet with complex geometry/analysis but not to decipher lemma 2.8. But instead of asking ELI10 I asked about S^6 complex structure helping to cure O non-associativity and got open research directions for sibling goals more than clear cut "yes, if the paper pass validation we've got some new associative octonions". That may need some healthy challenging.
There's an abstract on page 3. But it's quite ... abstract.
By the end of the first sentence I'm pretty sure it would take me quite some time to understand what it's about (the first sentence of the abstract, not the whole paper)
I’ve been turning the handle on GPT for weeks, feeding in toy models of physics and seeing what pops out the other end.
It’s mostly this type of mathematics: classifying what high dimensional spaces can and can’t do.
Maths like this is where the AIs will be most useful in the next few years: tirelessly grinding through every possibility on the edge of human knowledge, filling in gaps and completing maps of no-go regions in the theory space.
This kind of papers go way over my head, but recently I was surprised that ChatGTP (free) found the proof for an expression had a square value for some variables. It dug up a relationship from a paper from 2009 and correctly applied it to the case.
Reducing the problem to a quadratic form is pretty natural, and the paper from 2009 (though i see 2015) is quite an overkill. The result is a fairly standard consequence of the theory of representation by quadratic forms, and follows from Legendre's three square's theorem after some massaging. There's a good couple of centuries of foundations behind it.
>Proofs which fewer and fewer people are able to understand
>Those who do understand, the rate at which models make discoveries exceed the amount of time those humans have in a day
>As a result, we will increasingly use other AI models to validate the proofs AI make for us
I don't see a future where this doesn't apply to everything. Let's say you're an evil CEO, you could ask an AI model to run for days cooking up every possible nefarious scheme to get out of a class-action lawsuit scott-free, to avoid taxes via complex financial engineering, etc. The plans will be far more complex than any human can understand. There simply aren't enough humans with the mental bandwidth to oppose you, so it's just machine vs machine, and if you have more money to pay for more compute, you win.
> > Proofs which fewer and fewer people are able to understand
This is the trend for proofs of famous old theorems in general, even pre-AI. Some particularly extreme examples are the proofs of Fermat's Last Theorem [0], the abc conjecture [1], the classification of finite simple groups [2], and the four colour theorem [3].
This is not any math PhD, this is the Anthropic guy who found the Jacobian conjecture counterexample, which was almost worth a field medal if one squints hard
complex structure in S^6 would be an associative version of octonions.
This is not just any math PhD, it's the guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal if it were human
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[ 2.3 ms ] story [ 21.4 ms ] threadI nearly made it through the first line, OK, sentence of what looks like the Intro, which doesn't bother introducing itself and isn't actually the introduction which occurs later.
I can only describe pages one and two as some sort of foreplay. If you manage to hang on until P108 you can dispense with a safe word in future.
Background: https://mathoverflow.net/questions/1973/is-there-a-complex-s...
complex structure in S^6 would be an associative version of octonions (which are non associative "almost complex" versions of their cousins native to S^3, which you may know as (unit) "quaternions")
Big enough that Atiyah the godfather of this subfield, in 2016 claimed to have solved it (not sure if it was before or after the dementia)
This is not just any math PhD, it's the anthropic guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal if it were human
where can one read more about that?
The multiplication they call an "alternating invariant form": Q0 I think?
P10, sec 2.4 Lemma 2.8
chat: https://tinyurl.com/4acu6uc3
You've got the tools, you've got the will
By the end of the first sentence I'm pretty sure it would take me quite some time to understand what it's about (the first sentence of the abstract, not the whole paper)
Announced by the same people, basically at the same time.
It’s mostly this type of mathematics: classifying what high dimensional spaces can and can’t do.
Maths like this is where the AIs will be most useful in the next few years: tirelessly grinding through every possibility on the edge of human knowledge, filling in gaps and completing maps of no-go regions in the theory space.
The chat: https://chatgpt.com/share/6a7f1b08-b9dc-83ea-b911-aaf8b65f1c...
>Proofs which fewer and fewer people are able to understand
>Those who do understand, the rate at which models make discoveries exceed the amount of time those humans have in a day
>As a result, we will increasingly use other AI models to validate the proofs AI make for us
I don't see a future where this doesn't apply to everything. Let's say you're an evil CEO, you could ask an AI model to run for days cooking up every possible nefarious scheme to get out of a class-action lawsuit scott-free, to avoid taxes via complex financial engineering, etc. The plans will be far more complex than any human can understand. There simply aren't enough humans with the mental bandwidth to oppose you, so it's just machine vs machine, and if you have more money to pay for more compute, you win.
> > Proofs which fewer and fewer people are able to understand
This is the trend for proofs of famous old theorems in general, even pre-AI. Some particularly extreme examples are the proofs of Fermat's Last Theorem [0], the abc conjecture [1], the classification of finite simple groups [2], and the four colour theorem [3].
[0]: https://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last...
[1]: https://mathoverflow.net/questions/232087/have-there-been-an...
[2]: https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
[3]: https://en.wikipedia.org/wiki/Four_color_theorem#Proof_by_co...
This is not just any math PhD, it's the guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal if it were human