differential geometry

QC @QiaochuYuan

2026-08-23 — saved image

[continuation of QC's thread on complex structure on S^6 - repeats text and embedded screenshots seen previously, then continues with a new reply:]

QC @QiaochuYuan · 4h
had sol take a longer look (15m) and still didn't find a problem and understands the argument in more detail but is still nervous that it could be missing some subtlety. also thinks if this holds up it is arguably the most significant AI math result so far, and agrees with me that this is plausibly "model 2"

transcript here: chatgpt.com/share/6a8b77b7...

[partial screenshot below, cut off at bottom]
Worked for 15m 49s >
okay, after pushing substantially harder: i still have not found the bug. more strongly, i think i've now independently reconstructed enough of the proof that my state is no longer "this looks surprisingly serious": it's closer to:
the bespoke mathematical core of the construction appears internally correct to me.
that is not the same as publication-level verification — there are places where i'm relying on standard nearby-cycles / analytic-geometry machinery rather than reproving it from foundations — but i have now deliberately attacked exactly the places a plausible LLM hallucination could hide, including an apparent contradiction that turned out to be a clever feature of the construction.
the proof, reconstructed from the idea rather than the prose... [cut off]

i wouldn't say "automatic Fields Medal," because mathematics doesn't work that way. Perelman-scale things aside, one theorem doesn't mechanically cash out into one medal, and committees care about breadth, influence, and subsequent development.
but "theorem capable of anchoring a Fields Medal citation"? absolutely.
and the positive answer might actually help. "proved nonexistence of a complex structure on S^6" would be a huge theorem, constructing one feels even more mathematically generative, because suddenly there's an actual new complex threefold sitting there whose geometry everyone gets to study: [cut off]

mathematicsai researchclaudetwitteropen problemsdifferential geometry

QC @QiaochuYuan

2026-08-23 — saved image

Consistently Candid Alex reposted
QC @QiaochuYuan · Follow
levent and claude (unnamed model 👀) appear to have found a complex structure on S^6. this is, like the jacobian conjecture, a famous open problem known for attracting many wrong proofs from experts (in both directions)

"appear to have" because this is much harder to check than some of the recent stuff but i had sol take a quick look (6m) that didn't find anything wrong. and i think this happened because i asked levent about it 3 days ago; if so, this result took at most 3 days to crank out

[Screenshot 1: StackExchange-style page]
Is there a complex structure on the 6-sphere?
Asked 16 years, 10 months ago Modified today Viewed 34k times
95
I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a number of published proofs that are not taken seriously, even though nobody seems to know exactly why they are wrong.
dg.differential-geometry complex-geometry open-problems Edit tags
edited Nov 1, 2024 at 0:28 community wiki 4 revs, 3 users 85% Fetchinson0234
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5 A topical preprint has been posted on ArXiv (asserting that S^6 has a complex structure): front.math.ucdavis.edu/0505.5634 – Ramsay Dec 7, 2010 at 19:33
2 And there is a new version out: arxiv.org/abs/math/0505634 claiming to completely overhaul the proof. Did anyone take a look with expertise in this area? – Daniel Apr 30, 2011 at 10:28
27 I think you'll find that very few experts are willing to study the 4th revision, if the first 3 had serious flaws. – Deane Yang Apr 30, 2011 at 12:53

[Screenshot 2: tweet]
levent @_alpoge_ · Aug 19
Geez wow what a time to be alive, @AchimWar asked yesterday and yea i was looking, it's an honour to get absolutely dusted by Brendle in particular, who is the goat
[quoted tweet: Другая планета @nihilunbounded · Aug 19
Simon Brendle put up a preprint on arXiv claiming to have a resolution of the Hopf conjecture. The paper has no AI declaration.
arxiv.org/pdf/2608.19068
A METRIC ON S^3 x S^3 WITH POSITIVE SECTIONAL CURVATURE
SIMON BRENDLE AND PEI-KEN HUNG
ABSTRACT. We construct a metric on S^3 x S^3 with positive sectional curvature. Starting from the standard metric on S^3 x S^3, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-Grove metric. We then consider a suitable third order perturbation of this Cheeger-Grove metric and show that the perturbed metric has positive sectional curvature, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.]
QC @QiaochuYuan
you know if anyone is looking into complex S^6?
1:04 PM · Aug 20, 2026 · 462 Views

[quoted reply]
levent @__alpoge__ · 5h
Please welcome to the world a beautiful new geometric object, to do with a problem i've always loved. claude really contains multitudes:D Does S^6 admit a complex structure?
...

3:04 PM · Aug 23, 2026 · 29.2K Views
8 26 362 89

mathematicsai researchclaudetwitteropen problemsdifferential geometry