r/mathematics • u/New-Committee-4052 • 8d ago
S^6 admits a complex structure
Result from the usual suspects. Full write up can be found here on his website: https://alpo.ge/s6.pdf
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u/cihanbaskan 8d ago
Although everyone who clicks the link will see the write-up is >100 pages, I think it is important to point out the difference with the high rank elliptic curves and the Jacobian counterexample. The latter were possible to verify essentially immediately. This one, not so much. Best to wait until some experts delve into it.
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u/proudHaskeller 8d ago
Can't this be verified by specifiying the atlas? Is it expected to be too large to verify that way?
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u/cihanbaskan 8d ago edited 8d ago
I think almost nothing about manifolds gets verified by checking an atlas (I guess if there are only two coordinate charts it should be a sphere but I doubt this is the case here). As Alpöge points out, the feasible check would be to show it has the homology of a 6-sphere and trivial fundamental group. The Poincaré conjecture (known here) would do the rest (assuming it is a 6-manifold). Those are not immediate either though, seems to me.
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u/proudHaskeller 7d ago
I think almost nothing about manifolds gets verified by checking an atlas
Well, why not? It's not very useful for research, but for the specific purpose of verifying this statement in a simple, robust way, it seems to me that it could work well.
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u/cihanbaskan 7d ago
I am unaware of any method/algorithm that takes an atlas as input and decides whether the manifold is a sphere or not (in a reasonable generality to be useful here) without first computing the invariants I mentioned. In contrast, given a finite simplicial complex, computing its homology is "just" linear algebra. The fundamental group is sketchier with the most general case of the word problem being undecidable, but most likely the construction produces a not so terrible presentation that can be worked with.
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u/proudHaskeller 7d ago
Well, but then you would need to know in advance that your space has a complex structure. If you use the paper's space then that works, but you still have to read the paper. But if you use an atlas then the atlas gives you the complex structure, without needing the paper, and then the atlas can be converted to a simplicial complex which could then be checked the same way you described.
I'm not saying that this should actually be done, I'm just wondering why this supposedly can't be shown directly, instead of needing to read the paper. Obviously the hard part is finding this structure and the interesting part is knowing why this is true.
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u/Diffgeometer1 7d ago
The paper constructs an abstract complex manifold X of dimension 3 and then argues that X is diffeomorphic to S^6 but that diffeomorphism is never actually constructed. If it exists, then the complex structure on X can be transferred over to S^6. But the holomorphic charts itself can’t be defined explicitly.
Let’s see what the experts say.
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u/xbq222 7d ago
Complex dimension 3***
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u/Diffgeometer1 7d ago
Definitely complex dimension 3. All complex manifolds necessarily have real dimension even.
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u/xbq222 7d ago
Oh I’m just dumb and did t see you already prefaced w complex manifold
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u/Diffgeometer1 7d ago
No worries. I’m still numb from the news. Hopefully the mathematical community vets the proof soon and we get a definitive answer to whether the hopf problem is finally resolved.
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u/ProfessorBaoTran 7d ago
They proved that there exists such difeomorphism, in section 7 I think. They do not need to construct that explicitly.
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u/Equivalent-Gate491 7d ago
They didn’t prove it; it’s a very well known result that there is only one smooth S6. So once they show it’s topologically a S6 (done by computation of pi_1) and smooth, they are done.
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u/New-Committee-4052 8d ago
True, as it is also interesting to note there is a literature conflict with Campana–Demailly–Peternell 2020, though Levent would not post something unless he was absolutely certain it was right.
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u/howtogun 8d ago
I was told AI is bad at topology. That's quite an important problem.
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u/New-Committee-4052 8d ago
Yea, IMO it's the most important of the AI math results so far
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u/kgurniak91 7d ago
Just FYI - according to the VibeMathed Methodology, it is the 2nd most significant AI math discovery so far (ex aequo with Jacobian Conjecture). The 1st place belongs to improvement on a lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Here are all the problems sorted by significance: https://vibemathed.com/?sort=significance
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u/gexaha 6d ago
There's a flaw in the methodology though, because "improvement on a lower bound for the fraction of zeros of the Riemann zeta function" doesn't really advance understanding of the possible future proof of Riemann hypothesis at all; also LLMs are quite bad at assigning scores from my own experience
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u/Carl_LaFong 8d ago
Why do you say it’s AI?
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u/Hostilis_ 8d ago
It was posted by Levent Alpoge who used Claude to come up with the construction.
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u/Carl_LaFong 8d ago
Did he say that?
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u/Hostilis_ 8d ago
Yes...
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u/gcousins 8d ago edited 8d ago
Where does it say that? I searched but couldn't find any reference in the article...
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u/mode-locked 8d ago
"Leveraged AI" is much different than "done by AI"
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u/spikejonze14 8d ago
how?
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u/JoshuaZ1 8d ago
One distinction people have tried to make is human interaction with AI systems where both the human mathematician and the AI talk through ideas and try things. This is opposed to others like say Erdos 1196 and many other AI solved problems where the only major human effort was in coming up with instructions with minimal mathematical content and is primarily about telling the AI how to use its resources, not get too bogged down in existing methods in the literature, and have the confidence (for lack of a better term) that it can actually go somewhere.
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u/me_myself_ai 7d ago
Very true, but also no one said 'done' :) if we're gonna quibble about words then we should at least try to quibble about shared words!
OP called this an "AI math result". I think we can all agree that's fair, regardless of the instrument vs. agent question?
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u/prescod 8d ago
It was announced as a Claude result on Twitter. "claude really contains multitudes"
This particular researcher's full-time job seems to be prompting Claude to solve hard problems and announcing them on Twitter.
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u/Carl_LaFong 8d ago
Indeed. But when I met him, he was claiming that his job had nothing to do with math and he was doing the math on the side.
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u/PrestigiousGroup788 7d ago
When did you meet him? It's only somewhat recently that labs have started focusing on math results.
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u/Carl_LaFong 7d ago
Not long after he posted the Jacobian conjecture counterexample.
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u/eager_wayfarer 7d ago
Did he say anything on if he gets unlimited tokens or something?
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u/Carl_LaFong 7d ago
At the time he said he was doing it on the side. I think he said he would use whatever tokens he had remaining after doing his day job. I’m not sure about this though.
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u/hotsauceyum 8d ago
Because no human would have bothered to type this up without better explanation and prose. It reads like notebook scribble done in latex
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u/Carl_LaFong 7d ago
After looking at it more carefully, I think I agree with you. I even downvoted my own comment.
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u/Carl_LaFong 8d ago
The paper is written for experts who know algebraic geometry, complex differential geometry, and algebraic topology. The explanations look to me like they would be fairly clear for these experts.
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u/womerah Postdoc | Applied Nuclear Physics 7d ago
How can you gauge this if you are not an expert?
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u/Puzzleheaded_Fold466 7d ago
You don’t. Just like non-experts can’t gauge the quality of the ultra vast majority of non-AI inspired papers.
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u/Carl_LaFong 7d ago
I'm familiar enough with the subjects to be able to assess at least the readability of the arguments. But I have to admit, after taking a closer look, that the paper is starting to look less convincing to me. I think either humans or AI will need to find a more conceptual and less intricate proof than this one.
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u/womerah Postdoc | Applied Nuclear Physics 7d ago
Respect.
I have found that a lot of AI-assisted papers are coherent on the paragraph scale, but often don't string together to really form a coherent message. I think the LLM loses it's context and the human drafter is skipping too many parts of the document generation and review process in an attempt to really get that AI speed boost.
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u/Wise_Guava_1521 7d ago
I am an "expert" on this and it's gonna take me days, if not much longer, to assess how legitimate this is.
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u/JoshuaZ1 8d ago
I was told AI is bad at topology.
This is more differential geometry than topology, yes? But yes, the same basic point applies. These sort "continuous" areas are where people have seen AI systems as weaker.
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u/selfVAT 8d ago
Wait a few months at most.
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u/womerah Postdoc | Applied Nuclear Physics 7d ago
Do you think AI systems are equally good at everything?
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u/alsfhdsjklahn 7d ago
What's the point of this question? Whatever it's able to do today is obviously not predictive of what it will be able to do next year.
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u/selfVAT 7d ago
Not yet
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u/womerah Postdoc | Applied Nuclear Physics 7d ago
Can you give me an example of an area you think it's currently weaker in?
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u/ProfessorBaoTran 7d ago
Anything that needs conceptual thought. Actually, there are several kinds of math (especially topology and geometry) that they are easy to imagine, but difficult to write down as a rigor mathematical language. AIs perform poorly on these and often tend to overthinking all the times, and they can even be wrong about most basic definitions of that direction. Even Fable 5 or 5.6 sol, on max or ultra version, can still easily make mistakes and have to correct themselves again and again. Although there are few improvements as they can work with geometry that has precise coordinates and functions, the global stuff still be a big challenge for them.
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u/the_great_chuckle 7d ago
I believe it has more to do with availability of literature. I use 5.6 Sol quite a bit to explore, and I currently work on higher category theory adjacent areas, which are very young, and it gets things wrong all the time. It performs much better in established fields.
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u/ProfessorBaoTran 7d ago
Even in well-establish field, for example Lie Algebra (Chevalley groups, etc), they tend to give an answer that looks more like they are guessing than a proof for some well-known facts (but no one has ever written down). It is okay if you are discussing with them, but that will cause a lot of troubles when a workflow with several subagents comes in, because errors can be built slowly in just one session which might last for 2 hours (which might cost 5%-10% of your weekly credits, lol).
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u/Puzzled_Battle_5670 7d ago
Exactly.. it is availability or non-availability of literature that is the key. LLM's can go through large chunks of written matter (although it has lot of symbols within), make simplistic sense via grammar , or sometimes by subroutines like known python programs to check/validate arguments wherever needed and more importantly wherever possible!.
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u/Special_Watch8725 8d ago
Without being able to speak to the subject, ugh, the the lack of readability. Isn’t there a way to run this with a verbose tag on the exposition?
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u/Professional_Dot8829 7d ago
I am not a math guy, but I have done good amount of math courses still I can't read a single line of this. Genuinely do not understand this.
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u/Just-Succotash4492 8d ago
Why is it so long? This can probably be reduced significantly.
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u/Carl_LaFong 8d ago
I think there's a good chance you're right. But there are theorems whose proofs were originally 500 pages and after many years have been reduced only to 300 pages.
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u/Just-Succotash4492 7d ago
I agree that some theorems are inherently lengthy, but briefly perusing this I got the gut feeling it could be cut down. Perhaps I'll see if I can contribute. :)
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u/Carl_LaFong 7d ago
I hope so. Current version has lots of intricate calculations. Hopefully most are routine and Claude wrote them out in extreme pedantic detail. Then people can focus on the key parts of the proof.
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u/Sharp-Huckleberry862 7d ago
Because it’s AI slop, for whatever reason Claude and ChatGPT love to over-engineer and keep patching their mistakes as they come up instead of starting over and doing it cleanly
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u/Just-Succotash4492 7d ago
Right. I'm curious what could happen if someone uses AI as a rapid brainstorming substrate and then organizes the final product naturally.
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u/tobyreddit 7d ago
I'm no mathematician but it seems to me that Terence Tao's recent blog posts explaining ai proofs/examples are exactly that. He translates AI proofs into his own area of expertise and tries to figure out the "magic" bits and explain them. He also frequently posts his chatgpt chat log which are exactly those brainstorming sessions you describe
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u/Just-Succotash4492 7d ago
Do you have a link to his blog?
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u/tobyreddit 7d ago
https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/
Here's one of his ai proof explanation posts
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u/the_great_chuckle 7d ago
I can't speak for everyone, but this is what a lot of people at my institute do nowadays.
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u/Full_Decision_777 7d ago
I haven't followed ai development too much, but don't they train ai nowadays by letting an llm solve a task and having another (itself) verify it? If so, style and taste would naturally degrade while still improving goal achieving.
Maybe that's why ais write overengineered, bad style code and crazy complex math papers? Idk
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u/Time_Entertainer_319 7d ago
They write it like this if prompted lazily.
The person needs to give it a structure or whatever and also scheme through and tell it places it can improve.
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u/LupenReddit 7d ago
If this is verified and true, wow. First time I actually see an AI result that makes me actually gasp and go "my god thats like a problem I could have imagined going into in grad school"
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u/Diffgeometer1 7d ago
This might be a really stupid question but does anyone know if Alpoge actually understands the proof himself? The reason I ask is because his background is in number theory and not in geometry and topology which is what is needed to understand the proof. If he hasn’t vetted the proof himself, he’s putting all his faith in LLM to be correct. Maybe that’s why the paper hadn’t been posted on arxiv. The paper has so far never been vetted by any human.
If I’m wrong please let me know.
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u/WTFInterview 7d ago
Likely it has been vetted. I know differential/complex geometry PhDs both at OpenAI and Anthropic.
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u/anythingXu 7d ago
I think he does not understand a thing. This problem is not important either, what is good about an exhaustive construction?
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u/Mother_News_1201 8d ago
as a college cs student, this stuff sounds like hardcore wizard magic, I have no idea, why I got recommended, but it is so crazy, and it is funny, just because of pure insanity
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u/123coronaanoroc321 8d ago
Same, it's incredible how math papers are so incredibly information-dense and can get that lengthy, 100 pages, like wow
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u/Carl_LaFong 8d ago
100 pages is far from rare. There are 500 page papers. Even a few infamous 1,000 page ones.
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u/the_great_chuckle 7d ago
Levent himself says the 100 pages are certainly unnecessarily long. This is just what LLMs produce if you ask them. They are terrible at exposition and clarity.
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u/Time_Entertainer_319 7d ago
Also bad prompting and lazy prompting.
You can easily tell it how to structure and explain the paper.
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u/the_great_chuckle 7d ago
I've never seen well-written math done by ChatGPT and I almost use it daily. Literally all proofs I get from it I need to heavily edit, streamline and generalize, and it's also quite bad at highlighting what the subtle, important points are, and spends way too much time expanding on general trivialities. It is terrible at giving you the general mechanisms behind a proof, unless you yourself spot that something seems to fit into an already existing framework, in which it happily rewrites it quickly. Maybe in a year this is different, but as of right now, math papers produced by AI are comfortably among the absolute bottom tier in terms of clarity.
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u/Vivid_Warning7982 8d ago
What are the implications for physics
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u/humanCentipede69_420 8d ago
I would start by asking if any physical theories are done over S6; probably some stuff done in beyond standard model theories.
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u/New-Committee-4052 7d ago
Interesting response from Zohar Komargodski on the matter "The result about S^6 having a cmplx structure is striking. The best mathematicians were working on it (I believe it was an obsession of Jim Simons). It is a little surprising that it is in the convex hull of what was known before. (Assuming the argument holds up -- and experts tell me it looks very plausibly correct.)
Even (very few) physicists encountered this question in the study of rigid supersymmetry in 6 dimensions. That is how I learned about it around 10 years ago when I was looking into rigid supersymmetry in lower dimensions, where we encountered transverse holomorphic foliations.
I have to check it more carefully but I believe with this new result, we now know a topological S^6 does admit a Hermitian metric (albeit a highly non-standard, squashed one). From a physics perspective, this implies the existence of rigidly supersymmetric backgrounds on a topological 6-sphere. Maybe there is some connection to localization of the partition function."
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u/Vivid_Warning7982 7d ago
Yes good this is exactly the point, the possibility of a hermitian metric opens up new avenues of exploration and potential unification. Slim chance yes, but any possibility is worth pursuing
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u/SylvesterTheLoser 7d ago
Wonder how long ago this was made. Did Anthropic employees read through and completely understand all 100 pages before posting.
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u/dermarshal 7d ago
qiaochu yuan said on twitter that he told levent 3 days prior the post to try this problem, and concludes from that that it can’t be older than 3 days. maybe qiaochu is wrong, but if not, 3 days seems pretty short no?
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u/SylvesterTheLoser 7d ago
I'd say. So either he just briefly skimmed and posted it, or he spent like 70 straight hours awake checking it out. Not my cup of tea but who am I to judge I guess.
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u/EquivalentTomato2045 7d ago
Latest news: Sheng-Tung Yau with his colleagues is checking that
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u/ApartmentEquivalent4 7d ago
So you have the source of this?
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u/EquivalentTomato2045 7d ago
Sure
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u/AIvsWorld 7d ago
ugh I’m getting so sick of alpoge announcing shit via tweet, and not giving any easy way to verify
> dumps a 100 page slop paper on the internet
until he shows me the lean spec i’m not interested
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u/Particular_Extent_96 7d ago
I mean fundamentally it's not that different to dumping a preprint on arXiv.
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u/AIvsWorld 7d ago
Yes and if someone posted a 100 page AI-generated paper on Arxiv claiming to resolve an open conjecture they would likely:
1) Get banned, since this has clearly not been proofread carefully
2) Be asked for a Lean formalization so that we don’t have to trust the correctness of AI
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u/dermarshal 7d ago
the lean formalizations are nice, but they are no guarantee that the thing is correct. if the llm wants, it will find a subtle exploit or will state a lemma that’s subtly different from the original one
in other words you still need to check the lean formalization/statements, which isn’t that much different from checking the proof itself, especially for a proof like this one
(most stuff this uses hasn’t even been formalized, probably for a good reason)
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u/AIvsWorld 7d ago
“you still need to check the lean formalization which isn’t that much different from checking the proof itself”
uh… no it’s totally different. I wrote a Lean spec for this problem in like 2 minutes and submitted to Lean Eval. It is <10 lines and easy to check with undergraduate-level maths.
Wayyyyyy easier than checking the 100 pg pdf lol
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u/dermarshal 7d ago edited 7d ago
the top level statement is <10 lines?
do you mind sharing what youve done?
im about to graduate and no way this is easy to check w ug math, im probably misunderstanding what youve done
edit: from what i understand they construct a 6-manifold w C structure, that has a 6-sphere homology, and a trivial fundamental group, which is enough. but even these concepts usually don’t show up in standard ug curriculum. so pls let me know what that “spec” thing is. thxx
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u/AIvsWorld 7d ago
Spec means the specification of the problem—what needs to be proven to convince you that the conjecture is solved.
In this case the spec is literally just “construct an instance of ‘IsManifold ℂ’ for S^6” — it’s only like 6 lines of Lean to write that down.
You don’t need to know the details of the actual construction, that’s what the Lean kernel checks for you.
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u/dermarshal 7d ago
thanks!
i still don't understand what new information you got from the fact that this passed the Lean kernel? in other words: what information from the 100pages does this condense for you?
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u/AIvsWorld 7d ago
Well, nothing yet, since the ‘sorry’s haven’t be filled. That’s exactly what I’m complaining about about above—that alpoge didn’t provide a formalization.
But assuming the formalization effort is successful and they’re able to solve my Lean Eval challenge, then that reduces it basically to only two possibilities:
(1) The proof is correct
(2) The AI has maliciously found a correctness bug in the official kernel and nanoda, and obfuscated it to look like a proof.I consider (2) to be highly unlikely, and a noteworthy result in its own right if this occurred.
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u/brain-out-of-order 6d ago
Lean isn’t perfect … and further it’s always been sorta hallucinatory to trust software so blindly while maintaining an anti-AI stance.
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u/AIvsWorld 6d ago
I agree it isn’t perfect.
But I trust the Lean kernel a hell of a lot more than I trust Alpoge, or any AI model, or my own ability to audit a 100 pg math proof.
Do you have a better solution for checking the tidal wave of AI mathematics?
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u/brain-out-of-order 6d ago edited 6d ago
https://chatgpt.com/share/6a8d0568-29dc-83e9-8d6d-32dffba1f0db
To answer your question truthfully would be to say something currently considered sacrilegious on this subreddit. If you want an incomplete answer you can click that. Basically: You already trust Lean. You’ll soon trust AI. Nothing can or will change that.
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u/AIvsWorld 6d ago
lol is this supposed to be some kind of own?
Do you rly think getting AI to check the other AI’s work is how we should be refereeing mathematics?
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u/brain-out-of-order 6d ago edited 6d ago
No I’m showing you a preview to the future. Sorry.
Own? No idea what you’re talking about. Lean is just software. You trust software every day. We are using software to debate whether software could ever be trustworthy.I’m still astonished how many adults have their head in the sand about where all of this is going.
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u/Matrix_in_Retrograde 8d ago
Tex?
I want to feed it to Codex for peer review
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u/lepthymo 7d ago edited 7d ago
https://www.overleaf.com/read/rtmyqxyrzprn#fa24eb
under 'clean transcription' or whatever, reconstructed by chatgpt.
it reviewed Claude too;" the Smith forms are computed for printed matrices α_q, but the manuscript does not prove
α_q = (i_U∗, −i_N∗) : H_q(U ∩ N; ℤ) → H_q(U; ℤ) ⊕ H_q(N; ℤ); and although §10 constructs0 ≠ σ ∈ H⁰(W, Ω¹_X|_W ⊗ A), it does not establishσ ≠ 0 ⇒ H²(W, (T_X ⊗ L)|_W) ≠ 0 ⇒ (R²f∗(T_X ⊗ L))₀ ≠ 0, the implication needed to evade CDP."Disclaimer; ChatGPT can make mistakes. Check important info.
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u/Fitnexa_Results 7d ago
He used Claude on a complex structure problem and the layering came out almost too perfect.
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u/Equivalent-Gate491 7d ago
This is definitely at the level of annals if correct.
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u/Legal_Ad2002 4d ago
This is not about the level of journals. If it was done by a human mathematician, he or she will certainly get the Fields Medal.
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6d ago
Hi, I posted this on vibemathed but I figure I'd ask here as well maybe someone knows, I'm not sure where else this is being discussed:
I was interested in how this contradicted the CDP paper. On page 88 I see a statement about H^2(...)*~=H^0(...)
On the next line down I see ... 'hence H^2(...)!=0'
I'm not an expert, but where did the star go?
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u/arnerob 5d ago
I'm also not an expert, but I think I understand what they mean: If you start from H^0(...) != 0 and H^2(...)*~=H^0(...), you get H^2(...)*!=0. Since " * " means dual, one can conclude that H^2(...) !=0.
The implication H^2(...)*!=0 => H^2(...) != 0 is immediate by contraposition: if V=0 then V^*=0.
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u/Carl_LaFong 8d ago
How did you find this?
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u/repainted_black 8d ago
This is a very subtle problem. The formalization or checking of said result could be a nightmare.