A complex structure on S^6 [pdf]

57 pointsposted 12 hours ago
by robinhouston
(alpo.ge)

28 Comments

mr_mitm

11 hours ago

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.

robinhouston

11 hours ago

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.

Background: https://mathoverflow.net/questions/1973/is-there-a-complex-s...

oliculipolicula

6 hours ago

for maybe undergrads,

complex structure in S^6 would mean 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").

Also fancy cousins of unit vectors (that you can do a nonzero triple cross product with), ie "non associative vectors" living on S^2, the 2D surface of a sphere "with radius 1"

If you haven't quite got the PhD https://ncatlab.org/nlab/show/quaternionic+structure

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 (for impact if not notoriety or complexity) if it were human

Disclaimer: not a pure maths PhD myself, just someone who likes this stuff so..

cmenge

11 hours ago

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)

user

6 hours ago

[deleted]

linzhangrun

2 hours ago

Obviously a Fields Medal–level achievement... Even the most optimistic person would not have expected it to happen so quickly six months ago

atleastoptimal

8 hours ago

It seems the trend for AI math is:

>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.

gucci-on-fleek

3 hours ago

> It seems the trend for AI math is:

> > 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...

HedonicEscal8r

8 hours ago

Tax law is not arbitrarily complicated, and unexpected edge cases are settled by human judges, not by mathematical proofs.

juleiie

3 hours ago

Yeah people forget that law is not mathematics.

At the end of the day judge either believes you or not and you are ducked

And if your defense is so complicated then that’s worse for… you

user

7 hours ago

[deleted]

dinga

11 hours ago

I skimmed through it. Looks fine. Had a bit of trouble following step 5 of theroem 10.5 (page 89), but after a second look it became all too obvious.

cmenge

11 hours ago

I was more confused by the application of Nakayama’s lemma in the following paragraph, but then it suddenly became clear to me.

scuppernong

11 hours ago

christ

gerdesj

11 hours ago

I'm with you. That is quite a dense paper.

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.

user

6 hours ago

[deleted]

user

6 hours ago

[deleted]

esafak

11 hours ago

Is this a proof a human could have come up with? Is it elegant? How significant was the human contribution?

jiggawatts

11 hours ago

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.

user

6 hours ago

[deleted]

akomtu

4 hours ago

This is a big pile of AI slop. I doubt anyone will ever read it.

Davidzheng

3 hours ago

someone surely will. It's a breakthrough.