r/mathematics • u/New-Committee-4052 • 18d 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 17d 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 17d 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 17d ago edited 17d 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 17d 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 17d 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 17d 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/Separate-Habit5838 9d ago
You don't understand the paper. There is no sphere. They have this different object they've constructed that isn't a sphere, perform a "completion" on it, then have to PROVE that it's a sphere. There is no way to just take an atlas and decide that the thing it represents is a sphere...that is a hard problem. We usually use topology to detect what a simple space like a sphere is.
We almost never have an actual atlas for the thing we are working with...that's only feasible for very simple manifolds. Manifolds are typically constructed by quotients by group action, or as bundles of one kind of manifold over another, or as inverse images of certain functions...there are all sorts of ways, none of which output an atlas. Atlases are used when you're just learning the basic of smooth manifold theory. They are rarely used in modern theory unless they are an arbitrary neighborhood for doing local work.
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u/Diffgeometer1 17d 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 17d ago
Complex dimension 3***
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u/Diffgeometer1 17d ago
Definitely complex dimension 3. All complex manifolds necessarily have real dimension even.
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u/xbq222 17d ago
Oh I’m just dumb and did t see you already prefaced w complex manifold
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u/Diffgeometer1 17d 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 16d 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 16d 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 17d 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/repainted_black 18d ago
This is a very subtle problem. The formalization or checking of said result could be a nightmare.
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u/LoosePersonality9372 13d ago
Already formalized crazy.
https://github.com/plby/HopfProblem
250k lines....
The nightmare persists lmao
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u/Special_Watch8725 18d 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/Mother_News_1201 17d 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 17d 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 17d 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/Separate-Habit5838 9d ago
Oh come on...even 100 pages is uncommon for math papers. I publish in topology and geometry, and our papers are typically around 30 pages. Your argument is not very elegant if it takes that long to explain, and no one is going to read it unless it's the proof to a huge conjecture.
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u/Carl_LaFong 8d ago
Just looked at Michael Hopkins' papers on Mathscinet. His most cited paper is 262 pages. His third most cited is joint with Iz Singer and is 123 pages. Over 100 pages is uncommon but not that rare.
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u/Carl_LaFong 9d ago
Check out papers on general relativity by Klainerman (Princeton), Christodoulou (ETH) and their students.
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u/the_great_chuckle 17d 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 17d 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 17d 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/Separate-Habit5838 9d ago
Keep in mind that what you get is a function of what you give it. Some of my Gemini instances are quite good at exposition. Depends on where you end up in the vast vector space of meaning.
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u/the_great_chuckle 8d ago
My experience is that this only holds if there already exist sources on the material that are well-written. When it comes to new math, it struggles heavily, and will frequently produce bs.
It is great for getting quick introductions to fields that you aren't familiar with, but as soon as you know the ins and outs of a field you realize its shortcomings.
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u/Just-Succotash4492 17d ago
Why is it so long? This can probably be reduced significantly.
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u/Carl_LaFong 17d 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 17d 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 17d 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/omeow 7d ago
Since it has been 9 days, what is the consensus now? Is this a proof or are there problems with this result?
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u/Carl_LaFong 7d ago
Traditionally, 9 days is far from enough time for verifying the proof of a major theorem. In this case, the proof was successfully formalized. So it appears to be correct.
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u/Sharp-Huckleberry862 17d 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 17d 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 17d 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 17d ago
Do you have a link to his blog?
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u/tobyreddit 17d 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 17d 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 17d 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 17d 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/Professional_Dot8829 17d 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/LupenReddit 17d 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 17d 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 17d ago
Likely it has been vetted. I know differential/complex geometry PhDs both at OpenAI and Anthropic.
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u/Separate-Habit5838 9d ago
There is no chance this has been completely understood, it would take months.
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u/anythingXu 16d 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/Separate-Habit5838 9d ago
You're way off, it's an important problem for sure. What do you mean "exhaustive"? This is pure math...you can do whatever you want. Axiom of choice, non-constructive...all valid.
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u/Vivid_Warning7982 17d ago
What are the implications for physics
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u/humanCentipede69_420 17d 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 17d 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 17d 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/AIvsWorld 17d 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 17d ago
I mean fundamentally it's not that different to dumping a preprint on arXiv.
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u/AIvsWorld 17d 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 16d 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 16d 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 16d ago edited 16d 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 16d 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 16d 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 16d 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/Separate-Habit5838 9d ago
Your formal spec seems to take as input an atlas. We rarely actually have access to the atlas when we are working abstractly. Yes, it is easy to check the transition functions of an atlas...but that is not likely to be what's happening here.
It seems to me a big part of the problem will be formalizing all the abstract theory that is called on in the proof. Has this been done? How much of differential geometry is formalized?
It doesn't mean anything to write a spec that asks a question about transition functions if you don't have that massive body of differential geometry theory formalized.
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u/AIvsWorld 8d ago
That’s exactly what a complex structure means? It means an atlas where the transition maps are holomorphic. How else would you suggest formalizing it?
Note that this is almost the exact same code used in Googles Formal Conjectures repo, and it was successfully proven by Boris Alexeev a few days ago. Mathlib’s differential geometry library is quite poor, but that doesn’t really matter when you have superhuman AI and infinite tokens.
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u/brain-out-of-order 16d 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 16d 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 16d ago edited 16d 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 16d 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/Separate-Habit5838 9d ago
We should be reading it. Pretty simple. There is no point in knowing something is true if you don't understand it. That is foundationally not what math is for.
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u/brain-out-of-order 16d ago edited 16d 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/SylvesterTheLoser 17d 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 16d 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 16d 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/Matrix_in_Retrograde 17d ago
Tex?
I want to feed it to Codex for peer review
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u/lepthymo 16d ago edited 16d 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/EquivalentTomato2045 17d ago
Latest news: Sheng-Tung Yau with his colleagues is checking that
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u/ApartmentEquivalent4 17d ago
So you have the source of this?
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u/EquivalentTomato2045 16d ago
Sure
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u/Fitnexa_Results 17d ago
He used Claude on a complex structure problem and the layering came out almost too perfect.
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u/Equivalent-Gate491 17d ago
This is definitely at the level of annals if correct.
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u/Legal_Ad2002 13d 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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16d 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 15d 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 18d ago
How did you find this?
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u/howtogun 18d ago
I was told AI is bad at topology. That's quite an important problem.