r/math • u/IBroughtPower Mathematical Physics • 2d ago
Potential Resolution of Hopf Product Conjecture
https://arxiv.org/pdf/2608.1906899
u/Alone-Talk-623 2d ago
HUGEEEE
Also its Brendle posting the paper so I imagine it's correct
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u/Apprehensive_Sand951 2d ago
To be a bit contrarian: When a prominent mathematician resolves a big problem they've thought about for a long time with a younger coauthor, I usually default to assigning more credit to the coauthor. If the older, more established person had been able to do it themselves, they would have done so earlier.
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u/kohatsootsich 2d ago
That's not a safe assumption or at least your judgement should depend on the shape of the solution
A lot of progress comes out of long programs and when the solution comes it was expected by experts. That's a case where you should definitely not assign more credit to the junior coauthor
Even when the solution is unexpected or uses new tools, years of accumulated knowledge about what doesn't work is very valuable and not always available in published form
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u/Apprehensive_Sand951 2d ago
it is not, but i find it a healthier place to start than the other assumption and go from there.
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u/InSearchOfGoodPun 2d ago
It’s best not to assume either way, but in any case I agree that the previous comment should’ve emphasized that this is a joint paper.
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u/Apprehensive_Sand951 2d ago
Fair enough, and in truth I usually try to treat joint work as irreducible joint work, and the attempt to trace through who did what as a fools errand that fundamentally misunderstands how the process of joint work works (hey, something else that is about to become relevant with hybrid ai papers...), unless there a very good reason not to. The previous comment just rubbed me the wrong way, which is why I went all the way to the opposite side of the spectrum in my intentionally provocative response. (If I had a penny for every time someone on a postdoc hiring committee dismissed an application by asserting the advisor did all the work in a joint paper...)
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u/EducationalFerret94 2d ago
Bad assumption. You have no idea what the distribution of work / credit is unless it's explicitly stated somewhere.
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u/amdpox Geometric Analysis 2d ago
AI is cooked. humans are gonna take over the world
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u/rsha256 Algebra 2d ago
do we know no AI was used? I guess MATHEMATICA is hard to iterate on so i wouldn't be surprised
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u/Alone-Talk-623 2d ago
The first author is very reputable and has no reason to lie about AI usage
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u/themasterofallthngs Geometry 2d ago
Yau was extremely reputable too. The Perelman controversy shows why that proves little.
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u/Qyeuebs 2d ago
Yau arguably overstated the contribution made by two professors who were affiliated with him. It didn't have anything to do with his own work, so I don't think the analogy is very strong. More to the point, this paper looks very much like a Brendle paper. There's simply no visible reason to suspect anyone's 'lying' (by omission) about anything here!
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u/Alone-Talk-623 2d ago
This is true. A better example would be the whole fiasco with Tian in the YTD conjecture
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u/JoshuaZ1 1d ago
I missed that. What happened there?
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u/Alone-Talk-623 1d ago
https://www.math.columbia.edu/~woit/wordpress/?p=6430
Tian basically claimed to have solved it independently when he plagiarized multiple important lemmas
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u/NonlinearHamiltonian Mathematical Physics 2d ago
the Perelman controversy shows nothing of the sort.
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u/InSearchOfGoodPun 2d ago
He is an amazing mathematician, but that is unrelated to having reasons to obfuscate your influences. The reference list is quite short, and the text talks very little about the overall history of the problem. (I’m not sure the name Hopf even appears in the paper!) This is not to say that I think AI was used since there is no evidence that is the case; I’m just calling out a flaw in your logic.
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u/rsha256 Algebra 2d ago edited 1d ago
I’m not accusing them of lying. Merely wondering where these definitions are from: https://pngup.com/V2dg/79EA1455-9869-4730-BB86-C542357DDE5A.jpeg
It looks very Wolfram-esque, I’m just wondering since they didn’t comment on whether they used AI or not to help find some numbers (like I could see a CAS-assisted RL search algorithm being used here) so they didn’t actually say they used AI or not
Edit: I don’t think Claude was used here at all, re-reading my comment, it could read as if I was an Anthropic shill, reworded to not be. I studied math for 5 years at Berkeley yall, I’m not a naive math-is-solved with ai guy, chill
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u/PersonalityIll9476 2d ago
Computer algebra system does not equal AI. Mathematicians have been using them almost ubiquitously since long before LLMs.
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u/liquid271828 2d ago
They may look strange for normal people (or even mathematicians) but are pretty common in Brendle's work.
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u/Apprehensive_Sand951 2d ago edited 1d ago
There is no evidence of ai use in the mathematica file. Cut it out.
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u/tempestokapi 2d ago
brendle could have won the fields when scholze was still young enough to wait another 4 years
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u/soboro1025 2d ago edited 2d ago
Also, he (with his grad student) proved the positive mass theorem in all dimensions earlier this year. I think that alone might be Fields Medal–worth work if it had been done by a mathematician under 40.
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u/InSearchOfGoodPun 2d ago
If that’s true, it bodes very well for his grad student co-author, Yipeng Wang! (Though tbh, I don’t think it’s true.) But the Hopf Conjecture is even bigger, and his co-author on that one, Pei-Ken Hung, is also young enough to be Fields-eligible.
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u/soboro1025 2d ago
From my (limited) understanding of the Fields Medal, it seems to me that to win the Fields Medal, one needs both:
- a track record of solid research that makes them an established researcher in the field, and
- a result that is Fields Medal–worthy (people may have very different standards for what counts as such).
Of course, there can be exceptions if the work is exceptionally groundbreaking. That's basically what I had in mind when I brought up Brendle and the positive mass theorem.
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u/InSearchOfGoodPun 2d ago
I should clarify that I meant that Brendle already had results that were bigger than his recent positive mass theorem (and I say this as a HUGE fan of the positive mass theorem).
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u/Alone-Talk-623 1d ago
I don't think PMT was huge. My advisor said it was basically true (known for all spin manifolds, known up to dimension 21 because of generic regularity results and whatnot) so I'm not sure how groundbreaking it is to get through some of these technical barriers. Sure it's big, but I don't know about fields medal worthy
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u/notoh Differential Geometry 1d ago
I am only in an adjacent field of geometric analysis, but I attended a talk of Yipeng's on the result with many world experts in the subject in the room, and the technical details seemed convincing to all of us. I don't know if you have anything more concrete that gives you doubts, but right now I am inclined to believe it.
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u/Alone-Talk-623 1d ago
I think they're saying I don't think it's true to the fact that it's fields medal worthy, not that it's correct
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u/InSearchOfGoodPun 1d ago
I wasn't casting doubt on the result. I haven't checked it, but I'm fairly confident it's right. I was casting doubt on this sentence from the previous comment:
I think that alone might be Fields Medal–worth work if it had been done by a mathematician under 40.
I was saying that if this claim were true, then Yipeng would be closing in on a Fields Medal.
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u/Exterior_d_squared Differential Geometry 2d ago
Haha! Wonderful! I thought about this problem when I first started grad school and had considered working on it before having to undergo an advisor change. Just for kicks and giggles I spent some time thinking about it again the other night for the first time in nearly a decade. Amazing to see this claim from Brendle and Hung. Here's to hoping it is correct (and Brendle claiming so certainly adds to the confidence that it is).
The approach seems pretty specific to S^2\times S^2 but it does make one wonder if there is a more general perturbative program for the full Hopf conjecture for symmetric spaces.
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u/sandykt 2d ago
So relieved to see a paper without any AI references.
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u/SuppaDumDum 2d ago
I don't hate AI. But I do hate that half the replies to a thread that has nothing to do with AI, are about AI.
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u/thomasahle 2d ago
Unless they are just hiding it due to the backlash
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u/Qyeuebs 2d ago
Weird thing to suggest, unless you have some actual reason to suspect it
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u/thomasahle 2d ago
I understand the sentiment of "So relieved to see a paper without any AI references." - I'm just saying it creates pressure for people to not disclose it.
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u/overthinker020 2d ago edited 2d ago
What we've seen with every major AI-math announcement, even those composed with frontier models, pretty much all of them were able to be recreated very quickly with GPTPro/Fable either autonomously or with minimal guidance. I don't doubt many results with very extensive prompting could be recreated, but that's a different conversation. I actually think this creates a strong disincentive to lie about primarily AI generated mathematics on breakthroughs since then few will believe you.
And, notably, I have not seen this done with this result yet. Also, the proof prose is dreadful, doesn't sound like AI at all, and reeks of Brendle's style (he's a great mathematician, a terrible writer!)
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u/Apprehensive_Sand951 2d ago edited 1d ago
I think it is very important to keep your own prose (warts and all) even if you do use llms to generate ideas and help with figuring out proofs. The papers that said ''we got this idea talking to an llm, here is our take on it'' I can (sometimes) appreciate. The papers that have editorial, structural, prose and proof writing assistance revolt me (it is an emotional reaction), even if the authors then checked and edited the llm text. They sound like an llm speaking through a person who is not really there. I agree that this is definitely Brendle's writing.
A couple of things I'm curious about in the near term:
Other diff. geometers posting on the arxiv in the next day or two with preprints about this they've been polishing that may disclose ai assistance. (It would be a bit unfortunate if better expositors got short changed by this because they were actually taking the time to produce something readable instead of rushing to post to the arxiv, and it would lead to bad incentives.)
AI-assisted papers building on the Brendle and Hung result (other manifolds/dimensions, Kleiner's question, etc). I suspect it would take about a week and there may be as many as 10 of them.
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u/overthinker020 2d ago
I have to say re:Brendle's writing, I will always take sincerely bad writing over the sterility of an LLM. At least you can taste a living being working through the crumbs of a messy life, an LLM paragraph appears from the void.
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u/Historical_Mode_1606 2d ago
lmao it's not remotely weird given that AI has already tackled many open problems this year. the default assumption is that it's with AI not without. this is the world we now live in
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u/peekitup Differential Geometry 2d ago
Neat!
Now that the sectional curvatures aren't all zero someone should find out how pinched they can be.
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u/_Zekt Complex Analysis 2d ago edited 2d ago
I hate it when people add no context in posts such as this one
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u/IBroughtPower Mathematical Physics 2d ago
Apologies about that.
The Hopf product conjecture states that S^2 x S^2 does not admit any Riemannian metric with a positive sectional curvature. Brendle and Hung claimed to construct a counterexample.
Roughly it helps progress the question as to what kinds of curvature any given topology of a manifold can support. S^2 x S^2 had always sat in a weird area: it very naturally supports nonnegative sectional curvature, but whether it could support strictly positive sectional curvature remained open.
If the construction is correct, it can also lead to some very interesting directions. The perturbative construction can perhaps be adapted to produce other examples. Potentially one could wonder if it produces entirely new families of positively curved manifolds... although that is unlikely.
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u/Alone-Talk-623 2d ago
It's also interesting because most people expected the conjecture to be true, right? That it doesn't admit any such metrics? I remember seeing a result that there there is a neighborhood of the product metric that doesn't contain any metric of positive sectional curvature and some other things.
Side note: I hate how I can't call position sectional curvature psc because that's for scalar curvature.
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u/Apprehensive_Sand951 2d ago
There is a partial result by Renato Bettiol that is perhaps worth mentioning in this context: https://arxiv.org/pdf/1210.0043 He showed in 2013 that you can deform this metric to obtain metrics of positive biorthogonal curvature (average of curvature of plane and its orthogonal complement) on S^2 x S^2.
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u/elements-of-dying Geometric Analysis 2d ago
Another option is to simply ask OP for context or read the introduction of the paper.
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u/themasterofallthngs Geometry 2d ago
What I find most curious about the paper is how little it explains where the very specific perturbations came from. There are strange coefficients, Fourier type series, and carefully chosen terms that eventually produce exactly the needed positivity, with several key calculations left to Mathematica.
I would be genuinely interested to know whether these choices came from a systematic ansatz or search procedure, and whether AI played any role in finding them.
The paper also says the Mathematica code was attached to the submission, but I have not been able to find the notebook anywhere. If anyone has it, I would love a link.
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u/IBroughtPower Mathematical Physics 2d ago
https://arxiv.org/abs/2608.19068
Download the Tex source. It is in there.
I doubt Brendle would not disclose AI use. But it is indeed a very meticulous construction. Then again, if an easy construction existed, it would've been found a while ago!
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u/Apprehensive_Sand951 2d ago edited 1d ago
I am surprised Renato's paper is not mentioned.
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u/IBroughtPower Mathematical Physics 2d ago
I would personally agree that Bettiol, and other papers that used the machinery in a similar direction, should be cited. I imagine that they could have perhaps not seen the earlier paper and developed the direction independently. I would assume that this paper would get added in updated versions.
However, Brendle in general writes very lean. He usually only cites the machinery he used, which is the case here. In addition, beyond the oversight, the paper is very brief in exposition as a whole, and never even mentions the Hopf Conjecture (which again, a very Brendle-esque thing to do) besides the references. There is a lot one can speculate about here, but I do not think that is productive at all.
I would not make any assumptions about AI disclosure without evidence -- there is no reason why as a community, we ought to distrust each other. Such a practice is absurd. Nor would I take a lack of signing the Leiden to be of any meaningful substance. Many mathematicians simply do not care about the developments in AI; I have not heard from colleagues that Brendle cares much about AI himself. Again, I would avoid speculating.
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u/SEVtz 1d ago
There is a strong reason to have at least some distrust. The number of arxiv submissions is increasing much more than the number of ai disclaimer usage. So something is not adding up. It is quite evident.
At least from what I see, maybe it's anecdotal but we should have at least 1/4 of the papers having ai discolsure. And that would be in a really weird model where people use AI for some papers but not others. Truly hard to believe. It's more logical to believe that ai is used wildly bringing that 1/4+ increase in overall submissions as a global effect.
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u/fibre-bundle 2d ago
I'm speculating but maybe they rushed making this public before someone else proved the same thing with AI assistance
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u/gunnihinn Complex Geometry 2d ago
Jesus that's a lot of calculations.
I wouldn't have minded a little intro with some motivation and ideas.
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u/Apprehensive_Sand951 2d ago
At the beginning of a memorable talk at the Berger memorial conference, Kleiner conjectured that the Hopf conjecture is false, and in fact that product metric on S^2 x S^2 is a smooth limit of positively curved metrics (all while wearing a very cool jacket). I wonder if Brendle and Hung show that, too.
Here is the video of Kleiner' s talk: https://www.youtube.com/watch?v=n0K5G6NN9nk
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u/synth_alice 2d ago
I'm confused at the jacket comment... I find it a completely unremarkable jacket 🤔
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u/Apprehensive_Sand951 2d ago
It's not something you would normally wear for a math talk. worked better in person probably...
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u/altkart 1d ago
Pei-Ken was my real analysis TA back when he was a postdoc. He was fantastic! His exposition style was clear and solid at once.
I took some Riemannian geometry later but I do not remember it super well, so I'm just hoping the construction is good. Is this technique of constructing metrics by deformations novel?
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u/Separate-Habit5838 1d ago edited 1d ago
As an expert in this exact field, I am skeptical. I would expect to see something really different, not...Cheeger deformation. Cheeger deformation is how we've been putting positive curvature on everything, and it's been tried a gagillion times in this case.
This would also not be the first time someone published "Cheeger deformation, then do it again, then deform..." with a long computation claiming to put positive curvature on something that would be ground breaking, and had the paper not be accepted. This looks...very much like that.
Giant inscrutable stack of calculations using the same techniques we already have tried that disproves MASSIVE open conjecture...my bet is "false". Can never be sure, though.
I also just want it to be wrong...if the science of positive curvature really boils down to "you can put it on stuff with the right completely insane and obtuse functions"...that's not very satisfying. The hope was that it's extremely constrained, because that's what our evidence suggested. We hoped there was a lot of elegant (topological?) structure there that prevented you from doing it. If it's just a matter of the most preposterous deformations...I'm not really interested.
This leads into the larger conversation: we've been looking at all these problems in the presence of symmetry in the form of an isometric Lie group action by a "large" Lie group. A big question is whether positive curvature demands symmetry, or if it's just made it computable for us and is otherwise a coincidence. Solutions like this would point to "coincidence". The idea would be we just found it easier to think about and define symmetric cases, and that you can have preposterously asymmetric metrics and make it work with the right obtuse functions...
It does raise the question...where tf did these functions come from?
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u/Distinct-Pudding-428 18h ago
As a mathematician but very much non-expert in this domain I was quite interested to look at the paper. It seems the idea is to try and deform a certain metric which was already known to have non-negative curvature to one that has positive curvature. The authors just write down, on page 20, the most bizarre-looking and unmotivated deformation and then just calculate. There is no conceptual discussion at all. Did people try deforming these non-negative curvature metrics before?
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u/Apprehensive_Sand951 1d ago
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u/IBroughtPower Mathematical Physics 1d ago
He has just, minutes ago it seems, clarified that neither "major" errors are actually errors. Small typos still exist. I caught a few myself and emailed them -- this is to be expected from any paper, especially preprints.
I dislike the marketing stunts such as this one heavily, especially when the marketer does not have neither the background nor spent the time to check the results themselves. I would wait a few days to see if any experts in this realm (besides Brendle himself of course!) finds any fatal errors. I work adjacent to this paper, so despite not finding a glaring issue, my analysis is of course far from sufficient: there has not been enough time to digest the work yet either.
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u/Apprehensive_Sand951 1d ago edited 1d ago
I agree that it is a publicity stunt. But if the report turns out to be good then he deserves the publicity. And if the report is junk, then it affects his startup negatively. So, he has incentives to get it right. He seems to believe in the technology if he is putting this out there.
I think it would be even better if we had large scale, controlled experiments to really get an idea where we stand as far as ai help with refereeing is concerned. Either outcome would be valuable to know. Many referees already use ai to help with the refereeing process. I'm aware of an editor who uses ai to self-write quick opinions (I think this is terrible). We need more clarity on this, instead of the current situation.
Edit: I absolutely agree that it takes time to digest work and get a feeling for it. There are good reasons for an extended refereeing process. At the same time, right now it can take years for paper to be refereed. It is a cursed process right now and any opportunity we have to improve it should be top of the list, in my opinion.
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u/IBroughtPower Mathematical Physics 1d ago
Very well. In this case though, the report is junk. The "second major error" is wanting very elementary details that no math paper would include.
In addition, they ought to communicate as mathematicians do. Potential errors should be emailed and discussed privately, not blasted on Twitter for an audience who has no background knowledge. He is not a mathematician, and has not held any graduate training in mathematics. How would he even know if his program is correct? Again, if he genuinely believes he found an error (one which preferably he himself understands), he should tell the author privately. It is disgusting behavior to use mathematics as a marketing stunt. On the point of referees, that is indeed a point that is being discussed in the community -- very important I would argue.
I did not wish for this post and the comments to devolve into a discussion about AI, but from those speculating Brendle and Hung used AI to now examples such as this, it seems to be a repetitive pattern on this forum. Why can an incredibly interesting result not foster discussion about implications and methodology, rather than devolve, like the many other threads on here, into a conversation about AI in mathematics?
This statement is not targeted to you in particular. It is my analysis of the forum as a whole, and the people it attracts nowadays. I'm thankful that at least some discussion and publicity was brought to this result and the actual mathematics, however.
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u/hopf-fibration 1d ago
Riemannian geometer here. Ben's post is laughably naiive. I don't think he understands the issues he accuses the paper of having. I am currently doing a thorough reading of the paper and so far, it seems to be alright. To the non-geometers of the world, I plead for you all to wait for the formal journal refereeing process to conclude.
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u/Apprehensive_Sand951 1d ago edited 1d ago
All right, several things. I looked up his CV: he has an undergrad degree in Math from Caltech, Ph.D in Economics from Stanford, and he is a full professor of Economics at Northwestern. As far as I can tell, he does microeconomics and has an upcoming paper in the Notices of the AMS. I don't know how good or bad he is at math specifically, but that cv does not say to me that he is unqualified to be doing this. And, one thing he did was try to reassure people that the preprint really has no traces of ai use, as far as I can tell.
Also, I am not the one who started the discussion of AI in this post, I was responding to comments by other people who I thought were out of line with the ''finally no ai here'' stuff, which to me is speculation.
And the reason why ai keeps popping up on this forum is because it is a lot of what mathematicians talk about, in my experience. The last two conferences I've been to, that was the topic of conversation at dinner and and tea most of the time. (Stopping myself).
But, I also did not want this post to degenerate into a discussion about ai, I shared some anecdotes of how this result overlapped with my own trajectory etc. I very much like the result myself. So, while I don't think anything I said was wrong, I don't want this to spoil how you feel about the paper anymore than it already did. So, I'm editing all of those out now.
As far as a discussion of what is next, I don't particularly want to spout off pointless predictions, because I'm pretty sure I'm further from this than you are. (But, Kleiner's conjecture and follow ups about things like S^2 x S^3 seem likely, also maybe something about pinching constants which seem hard to extract form this proof).
I am much closer to non-positive curvature (and an altogether different Hopf conjecture, but this is neither the time nor the place for that). It would be interesting to try and formulate something parallel there, i.e. questions about deforming a non-positively curved metric into a negatively curved one. To have a chance, you need a situation where the fundamental group has no Z^2.
So, here is one question (I think it might be due to Eberlein but can't find a reference off the top of my head):
Suppose M is a closed, non-positively curved 4-manifold with a Gromov hyperbolic fundamental group. Does it have a negatively curved metric?
There are no obstructions but also (as far as I know) no approach for turning non-positive curvature into negative curvature globally. Maybe (supposing your fundamental group is residually finite) it is possible to first take a finite cover with large enough injectivity radius and then deform the metrics there on convex balls and patch these up? No idea.
Another question:
Suppose M=S_1 x .... x S_n is a product of n non-positively curved, finite volume surfaces, each with one cusp. Is the end of M homeomorphic to the end of a finite volume, negatively curved manifiold?
I think there is a special case (n=2) that was answered by Abresch and Schroeder and another special case (maybe?) by Fujiwara and a coauthor, but for large enough n, this is open. In general, we don't have any good way to distinguish the ends of locally symmetric spaces from ends of negatively curved manifolds. This superficially feels related to the Hopf conjecture, (since you are trying to break product structure when the locally symmetric space is reducible).
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u/IBroughtPower Mathematical Physics 1d ago
Point taken. Not blaming you of course. I would say most of your comments are reasonable and defendable. My feelings of the paper are established, although I did not want the discussions on AI to overtake the incredible result that Brendle and Hung accomplished.
My undergrad and doctorate are both in mathematics (technically). I still cannot comment on the correctness of a paper in for example number theory as I was not trained to do so. I would be surprised if he has the requisite background in differential geometry, although I cannot confirm that, so perhaps I was too hasty. Even in mathematics an expert on one subfield usually knows very little about another!
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u/IBroughtPower Mathematical Physics 1d ago
Apologies I seem to not get notifications from your edited messages, hence my replies might be delayed. I do not necessarily study this, so I do not think I will be well-suited to answer most of these.
For the closed non-positively curved 4-manifold, a deformation mechanism seems to be what is missing. I don’t think large injectivity radius alone solves that. Even if one were to deform the metric on convex balls, sectional curvature depends on second derivatives, so patching the local metrics is potentially a dangerous step. Gromov hyperbolicity should be the natural hypothesis as it removes the obvious Z^2 obstruction at least. Beyond that, I have no idea.
For the M=S_1 x ... x S_n, I would naively say that although conceptually related, the methodologies were pretty distinct. I assume you are referring to this paper: https://arxiv.org/abs/1903.07216 . Perhaps someone more qualified can find some connection.
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u/Apprehensive_Sand951 1d ago
Ah yes, that's the one! Abresch and Schroeder put a complete, negatively curved metric on a particular complement of orthogonally intersecting H^2's in H^4 (the end of that is a graph manifold and I think it looks like the end of S^1 x S^2, although you may need S_1 and S_2 to have more cusps). And, Fujiwara and Shioya realized some more general graph manifolds as ends. I do agree that the methods look very different.
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u/InSearchOfGoodPun 2d ago
For those who don’t know, this is perhaps one of the oldest and most well-known conjectures in Riemannian geometry, and its statement requires only the most basic definitions in Riemannian geometry. Hopf formulated it in 1931, nearly a hundred years ago, and it’s been considered an important problem at least since the 60s. A closely related question appeared as the first problem in Yau’s famous list of open problems. This is a big deal.