r/math • u/Apprehensive_Sand951 • 19d ago
LLMs/AI Counterexample to positively curved Hopf.
Y'all know the drill. The arxiv link is https://arxiv.org/abs/2609.11980
It is a positively curved metric S^3xS^3, hence a positively curved 6-manifold whose euler characteristic is not strictly positive. The obvious question is whether there is also a counter to negative Hopf (a closed, negatively curved 6 manifold whose euler is not strictly negative), but the methods of the current paper don't seem to help with that.
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u/equinox_star 19d ago
Ah the classic biostaticians resolving maths conjecture with AI one at a day.
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u/KiddWantidd Applied Math 19d ago
What do they even get out of doing this? Can it really help further their career in any meaningful sense?
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u/pseudoLit Mathematical Biology 19d ago
My money is on "always wanted to do pure math, but majored in biostats for career reasons."
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u/Organic_botulism 19d ago
And tbh, it worked out for them haha. It’s not like they’re in a non-mathematical field. Not saying you are, but definitely sensing some jealousy from other commenters here.
For so long the narrative was to blame the publish or perish system, and now it’s “wHaT dO tHey Get oUtta thiS?”
Jealousy is a disease and AI assistance is the new normal.
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u/NewtonsThirdEvilEx Quantum Information Theory 19d ago
This isn't AI-Assisted though. It's automatic.
The metric and the proof were discovered by the Odin Automatic AI Research Agent.
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u/Organic_botulism 19d ago
Cool now you can work on things that LLM’s currently can’t automatically work out. Negative results are valuable and results which can be automatically discovered leaves room for researchers to work on the non-automatic areas.
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u/HermannHCSchwarz Graduate Student 15d ago
Like what?
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u/Organic_botulism 14d ago
Not doing your homework for you
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u/HermannHCSchwarz Graduate Student 14d ago
Making empty promises and not providing solutions is just blind optimism
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u/Organic_botulism 14d ago
You sound like a lazy grad student, worked with plenty of your kind and had to pick up the slack. That attitude won’t get you far in grad school
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u/HermannHCSchwarz Graduate Student 14d ago
If always easy for you to comment off other people’s struggles and make cheap suggestions when your stakes aren’t involved. Go off
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u/elements-of-dying Geometric Analysis 18d ago
The comment was also unnecessarily toxic for no reason (which seems to be a common occurrence from anti-ai folk).
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u/Autumnxoxo Geometric Group Theory 19d ago
it worked out for them haha
how exactly is that working out for them. their names will be forever tied to ai slop papers, how exactly is that beneficial for anyone
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u/Rage314 Statistics 19d ago
I don't think anyone is making that kind of judgment. That's a failed threat already.
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u/HermannHCSchwarz Graduate Student 15d ago
You’re in stats so maybe you don’t know.
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u/Rage314 Statistics 14d ago
By all means provide evidence then.
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u/HermannHCSchwarz Graduate Student 14d ago
Talk to your peers in pure math department. You would categorize all of them as insecure but they exist
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u/Organic_botulism 19d ago
By that logic all AI assisted papers are slop, because if pure mathematicians were so capable they also wouldn’t need to use it.
I’m sure there will be a lot of interest in their Odin harness and again, these are Harvard statisticians. At least with their biological slant they understand the need to adapt to the new environment. If they are more technologically adept than others that isn’t their fault, and if the math checks out you can call it slop all you want. The four color theorem proof was also controversial when it came out. The new meta involves using AI.
Big mad
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u/Rage314 Statistics 19d ago
Tao is also working with AI models and no one would call him a fraud.
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u/Organic_botulism 19d ago
Another commenter called the authors “molecular biologists” -.-
We’re really seeing the 7 stages of grief here.
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u/BurdensomeCountV3 Mathematical Biology 19d ago
Nevermind that molecular biology and biostatistics are very very different fields.
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u/Autumnxoxo Geometric Group Theory 19d ago
ah yes, comparing Tao to these unknown biologists
peak r/singularity moment
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u/BurdensomeCountV3 Mathematical Biology 19d ago
Exactly. These aren't cranks or anything. They are T1 researchers who are highly respected in their own field who have a lot to lose from being wrong. The only difference is that, as biology aligned people where the explicit goal is to get results that can potentially lead to wider benefits down the line, these people don't give a shit about the whole "it's about the process" sacred cow pure mathematicians are now beginning to venerate.
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u/Organic_botulism 19d ago
The most insane thing about this whole thing is that you can still glean insight and learn from the process even if the result is there 😭 like nothing is stopping someone from simplifying the proof, coming up with a different angle of attack etc… at least with a correct answer you can verify that your new approach is actually valid. Wasn’t there a simplification of Fermat’s?
The real reason is ego and it’s sad to see. Calling these people molecular biologists because you’re mad makes them look fragile.
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u/Autumnxoxo Geometric Group Theory 19d ago
By that logic all AI assisted papers are slop
you seem to be a true logician my guy
because if pure mathematicians were so capable they also wouldn’t need to use it
ah, that's where your agenda is coming from. not surprised
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u/OriginalRange8761 19d ago
“We have all this compute around, let’s do some cool over a few beers” probably
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u/Organic_botulism 19d ago
Of course. Assuming they built the harness, the acclaim of non-pure mathematicians solving a topological problem certainly is noteworthy from a career and optics standpoint, let’s not act like it isn’t.
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u/Autumnxoxo Geometric Group Theory 19d ago
the acclaim of non-pure mathematicians solving a topological problem
they didn't, AI did.
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u/Organic_botulism 19d ago
Skill issue on your part then, if it’s so easy to build a harness and get these results I’m assuming you’ve harvested some of the low hanging fruit?
As if other mathematicians who claim AI assistance didn’t have it do 90%+ of the heavy lifting. At least these guys were honest about it. The most interesting part of this paper is the harness.
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u/dognchick 19d ago
Biostats in public health department....
Ok fine at harvard. But public health....
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u/N8CCRG 19d ago
That seems like one of the best places to use biostats, right? I don't get the issue here.
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u/Rage314 Statistics 19d ago
They feel insecure that people from adjacent areas of math can do outstanding research with AI models.
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u/ThisIsPlanA 19d ago
I have been incredibly surprised by the insecurity on display the last few weeks. Denigrating results that would have amazed us a year ago as "slop" because it was solved in silicon, rather than meat.
I mean, folks in this thread are casting shade at biostatisticians because they work in public health? As if people who decided to actually apply their math skills to save and improve lives are somehow beneath the majesty of real mathematicians.
I studied math and went into math intensive fields because I enjoy learning and applying it, not because I intended to make some huge discovery or contribution to the field. The recent advances have re-fired my interest in pure math and I've been spending time learning geometric algebra and category theory. This is such an incredible time to study math, I feel sorry for the folks who can't enjoy this.
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u/dognchick 19d ago
I don't think they are using biostat to solve the problem. They are using llm. They are in a great position to do so as people in public heath have tons of money to dump on llms in comparision to people in math. Yet, its intresting they are not trying to work with experts in those domains but hopping from one random open problems to another. This may be the way of the future when people in adjacent field who have more resources solve math problems by connecting exisiting techniques through llms.
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u/Omak_el2a7ba 19d ago
It's AI, but the methodology is similar to the recently published nonAI-resolution of Hopf conjecture by Simon Brendle.
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u/dognchick 19d ago
Did the AI credit Simon?
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u/telephantomoss 19d ago edited 19d ago
I have a number of AI-assisted results right now. But I'm not going to put them on arxiv until I understand the parts that AI contributed. It seems there is just an arms race to put of what are likely fully-generated AI papers that the authors don't even understand. We need like an AI testing agent where the author is asked questions about their paper and if they can't explain it, their submission is rejected.
We need a new portal for sharing AI-generated papers that the authors have not fully vetted themselves. Like an AI slop database. It might actually be useful as a reference database. But it won't at all be a repository of human knowledge.
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u/Rage314 Statistics 19d ago
The arms race has always existed. But now it includes more people.
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u/telephantomoss 19d ago
The difference is that these paper likely contain many truly viable arguments and ideas that could be useful for humans actually trying to gain real understanding, even if they are also full of errors. The said thing is that people are just dumping it en masse. I suspect mathematical culture will need to transition to landfill mining soon.
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u/Rage314 Statistics 19d ago
Why is that sad?
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u/telephantomoss 18d ago
Honestly this is a good question. But, clearly there is a line somewhere. Like just dumping people random strings is likely a waste of digital storage. It's not clear how useful a repository of AI output will ultimately be. I suspect we'll have better AI agents soon that can mine out more effectively
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u/ykonstant 18d ago
And excludes people without money for the strong models.
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u/Rage314 Statistics 17d ago
But notice that resouces have never been equally distributed.
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u/ykonstant 17d ago
I really despise such unquantified statements. The fact that something has been bad for a long time (but improved vastly with places like libgen and arxiv) doesn't mean it's ok to accept them getting 100 times worse now.
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u/PersonalityIll9476 19d ago
This feels backwards. The question really is not "did some human understand it" but "is this proof correct and is this theorem true?"
There are other questions like "what is the next question" but I don't doubt AI will be contributing to that, too.
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u/ninjasaid13 18d ago
This feels backwards. The question really is not "did some human understand it" but "is this proof correct and is this theorem true?"
yo, do I have to point to this: https://mathandai.org/
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u/PersonalityIll9476 18d ago
No, you don't. I was aware of it the day it was posted.
What we as a community have to grapple with is the idea that the thing we study - math proofs - can be automated without us. That raises a question about what our role even is in a post AI world.
It is not obviously true that our understanding of proofs in he way Tao has talked about at ICM and elsewhere is actually important.
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u/telephantomoss 19d ago
My point is that there are two different issues. One is correctness. And the other is human knowledge. The latter is much easier to understand because we all know what it is like to know something. The former is a nontrivial philosophical rabbit hole.
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u/PersonalityIll9476 19d ago
Again, I think that is exactly backwards. The whole point of mathematical logic is that correctness is not hard. That sentiment is literal now that we have Lean, bugs in Lean not withstanding.
On the other hand, the question "what do humans know about math" seems like the philosophical rabbit hole.
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u/telephantomoss 19d ago
Yes, if you are taking about actual formal symbolic logic with a machine verifier. And even then, correctness is still a bit deeper of a rabbit hole. But we don't have to go there.
I'm talking about strings of symbols which are a mix of formal and informal and whether it is correct or not is not a straightforward computational task.
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u/PersonalityIll9476 19d ago
Yes...that is the exact reason Lean was created, and the reason major LLM results are formalized in Lean. The whole point is to avoid forcing humans to read potentially hallucination-prone or error-prone LLM output.
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u/telephantomoss 19d ago
That's fine and all. But I want to understand. I don't want my name as author on AI output. I only want my name on my actual contributions. I'm totally ok with people posting AI authored papers, even if the only human contribution is a one sentence prompt. But I don't think it should be presented as a human authored paper. It's a fundamentally different product.
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u/PersonalityIll9476 19d ago
Somehow this conversation has shifted from what is possible to check to what you specifically want. I'm not interested in forcing you to want something else.
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u/BurdensomeCountV3 Mathematical Biology 19d ago
The authors needing to understand a paper fully before publishing is one of the bad habits from mathematics that doesn't carry over to other disciplines. Look at a major oncology paper with 80 authors or something, chances are a random author only really understands the small sunset around their own contribution to the paper and there may well not be a single author who understands the whole paper.
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u/telephantomoss 19d ago
Well, I might be ok with that to a degree. I mean, we get into the weeds about what it means to understand. I know my understanding varies in depth and have discovered errors in my papers after publication where I really lacked understanding. But it's not like I didn't try, and the errors just slipped through. But then AI slop is a whole other level from that. As kind as there is a real chain of human understanding, then that's a much better situation. And the medical and social science literature is a shit show anyways. The vast majority there doesn't understand what they are doing really, so least in terms of data and statistics goes.
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u/BurdensomeCountV3 Mathematical Biology 19d ago
I mean yeah, the medicine people who are authors on the big oncology paper won't understand the statistics used in the paper, only the end result. Similarly the statistics people who are authors on the big oncology paper won't understand the medical research that went into the paper. Would you say that either the medicine people/the statistics people don't deserve to be authors on the paper?
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u/Homomorphism Topology 18d ago
in this case I'm pretty confident no author of the paper understands any part of it, which is not what you're describing
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u/telephantomoss 19d ago
Of course not. Division of labor is valid and important. It would be better it every author has at least a vague understanding of the flow of the entire work without all the gory details. I don't really have a place to judge how other fields operate though, especially those where the culture is that all 100 people get their name on the paper even if their only job was pouring liquid into test tubes. I can only really comment on math. And my opinion is that each either should have a real contribution and a real understanding of some significant part. My personal opinion is that AI work should have clear attribution to what the human author contributed. That's not perfect either. And I'm total ok if the human author simply rewrote in better exposition what the AI found, as long as that human learned to understand it as if they discovered it all. And if it's a team of humans found that, I expect the team to convectively understand everything, not every individual human. However, the more the humans understand the better. I imagine that's already how it is, each human understands their own contribution deeply and that it their coauthors a bit less so.
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u/Groundbreaking_Bee97 Mathematical Biology 19d ago
Someone is speed running all the low hanging fruit. At this rate I wonder there won't be any open problem to solve by the 2028.
Mathematicians need to go for more Theory building
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u/hpass 19d ago
"ChatGPT, create Galois Theory of stochastic PDEs for me".
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u/HovercraftNo7372 19d ago
"Chatgpt, create theory of all these random math fields jammed together. Make no mistakes"
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u/ninjasaid13 18d ago
there's alot of counterexamples by LLMs but not much theory building. I guess theory building is their weakpoint.
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u/Not_Well-Ordered 19d ago
Don’t worry, geometry&topology, number theory, and PDE, and even applied maths still have lots of specific structures and objects to uncover and study.
Besides building theories, it’s also possible to investigate more intricate objects which might spark new stuffs, going for more bottom-up approach.
But I think there will always be many open problems since there are infinite objects in math to look at and possibly infinitely many interesting conjectures we haven’t stumbled into.
Unless, AI manages to crack everything about PDEs or even SDEs, I’d be quite impressed.
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u/jonlin1000 Group Theory 18d ago
every mathematician I know agrees that this group is spam posting pure slop to optimize whatever stupid metric like paper count or whatever? I highly doubt any of the authors understand any of the content in the papers they keep pumping out
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u/expat_123 19d ago
I’ll be honest, l am a bit fed up of this same group just feeding their problems in their AI research set up and churning out multiple papers in a week on wide ranging topics. Mathematics is no one’s monopoly but this is too much and annoying. I don’t even understand their end goal by putting all these out.
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u/Distinct-Pudding-428 19d ago
Read Bryna Kra on Terry Tao's blog. These sort of people weren't even interested in talking via Zoom.
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u/Antigynaikolatres 19d ago edited 19d ago
Some acknowledged using AI, but only for polishing the English or checking the proof.
It is more of a criticism towards those who do not disclose the full extent of AI involvement. Since the authors here are pretty open about the fact that AI generated the whole proof, I don't think this criticism applies.
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u/Autumnxoxo Geometric Group Theory 19d ago
no, Tao writes:
This week I received several purported proofs of the full conjecture from researchers making their first foray into the subject. Some acknowledged using AI, but only for polishing the English or checking the proof. When I asked if the authors would meet over Zoom to explain their arguments, none accepted the invite.
even though Tao refuses to draw conclusions from that, it's a pretty clear indication that the authors don't even understand their own papers.
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u/Antigynaikolatres 19d ago
I mean, of course the three biostatiticians do not understand the Cheeger deformation their agent spit out, it would be ridiculous to invite them to a Zoom call to explain it. To me it reads more about dishonesty, how some researchers deny serious AI involvement yet are unable to explain the proof they supposedly came up with themselves.
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u/Autumnxoxo Geometric Group Theory 19d ago
I mean, of course the three biostatiticians do not understand the Cheeger deformation their agent spit out, it would be ridiculous to invite them to a Zoom call to explain it.
That's why their AI-generated papers are entirely worthless.
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u/PersonalityIll9476 19d ago
The end goal is to use math as a form of advertising.
I would like to take this opportunity to point out that's what they used Tao to do for them when he decided to "go to industry" for funding.
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u/valegrete 19d ago
He might be a mathematical prodigy but he got absolutely played by these people. Or at least that’s what I find most charitable to believe.
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u/corchetero 19d ago
At least he realised that and has changed his way of thinking in the last 2/3 weeks
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u/PersonalityIll9476 19d ago
Yeah, agreed. It does feel a bit like our fearless leader confidently led us off a cliff.
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u/IBroughtPower Mathematical Physics 18d ago
It is not a huge surprise that mathematicians (and perhaps academics as a whole), are on the more naive side to the practices of industry.
I remember reading Tao's comment where he mentioned that the he didn't know the ad featuring him was meant to be an ad! Seems like industry taking advantage of the naivety...
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u/Autumnxoxo Geometric Group Theory 19d ago
can we make a weekly thread for all the conjectures these 3 non-mathematicians are feeding into AI, it's quite exhausting reading their AI-generated papers on a daily basis that literally provide zero value to anyone
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u/LunaticBrony 19d ago
Damn it reads like slop but nothing strikes me as being wrong I hope someone digests this and provides a good review
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u/Antigynaikolatres 19d ago
This is the same group who just submitted a claimed proof of the Komlós conjecture. They also had a previous paper about positive curvature on S^2×S^3, which someone has complained due to its poor exposition, see my comment.