Comment by oliculipolicula
16 days 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..
> complex structure in S^6 would mean an associative version of octonions
where can one read more about that?
In TFA itself!
The multiplication they call an "alternating [skew-symmetric?] invariant form": Q0 in symbols I think?
Basically a polynomial function in the components of the "superoctonions" that go into the product.that how I'd vibesplain to a fellow non-pro, anyways
P10, sec 2.4 Lemma 2.8
The complex structure itself is much more complicated.. or else alpoge would have posted a one line verification of the result? But a summary that excites everybody can only be done by a pro...
The next relevant thing in the paper though, that I might be able to attempt to vibesplain using my uh consumer knowhow, is "integrability" (starting in the section labelled "Context")
Or one could just ask Fable/ChatGPT to ELI10
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.
chat: https://tinyurl.com/4acu6uc3
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