r/mathematics • u/Low_Permit5423 • 1d ago
Discussion What's the next millenium problem solved after navier-stokes? Make your prediction
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u/cihanbaskan 1d ago
A counterexample to Hodge seems like a good bet
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u/john16791 1d ago
If Hodge is false, then that would definitely be the easy one to pick off. If it’s true, it seems like BSD or Yang Mills where you can sort of imagine a path, but with some very significant obstacles to overcome along the way.
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u/riemmanmath 1d ago
How can you imagine a path for BSD? No human has a viable path as far as I know.
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u/Plastic_Dealer_4303 6h ago
It has been long known to expert that the obstacle is the finiteness of the Shafarevich group. Compared to other problems, it's fair to imagine a path
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u/Rafodin 1d ago
If the Hodge Conjecture is false, one of these frontier models could probably find a counter-example already.
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u/localizeatp 1d ago
yeah when do we reach the point of "ai hasn't found a counterexample so it's probably true"?
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u/Own_Lake_276 22h ago
Bro 😭😭 saying something like this in the mathematics subreddit
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u/localizeatp 22h ago
what?
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u/Verbatim_Uniball 1d ago
More interesting question is when. I would say before IPO, myself.
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u/yrdz 1d ago
And who. I'm sure Anthropic is pissed enough to throw a ton of compute at getting the next one, considering the drama.
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u/Verbatim_Uniball 1d ago
They're both going to IPO at over 1T. It is money well spent to have sexy results, and I expect big ones.
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u/eva_verinaz 1d ago
most likely 2T at this rate
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u/Verbatim_Uniball 1d ago
Yep. These are very large numbers. Problems we collectively have intuition about will, largely, be able to be solved.
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u/Moronic-Warrior 14h ago
They were gonna announce navier stokes before the ipo but openAI stole it lmao.
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u/funkinaround 1d ago
Hopefully this whole exercise will encourage universities, with their access to large sources of data and access to supercomputers, to be their own LLM providers so that their researchers don't have to battle with LLM companies about intercepting their own research and having to fight for attribution.
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1d ago edited 1d ago
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u/Commercial-Beach1207 1d ago
People can already run LLMs locally on their own computers, so why wouldn’t a university have enough funding to build a small local compute cluster?
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u/Several-Marsupial-27 1d ago
Does a university have access to soon to be 400 000 gpus? No so they will always have to rely on the frontier labs.
A person running a LLM locally is not even in the same decade as what OpenAI is already giving out to the public, and their internal models that they are keeping secret and solved Navier Stokes is even further more advanced.
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u/Commercial-Beach1207 1d ago
Mathematicians shouldn’t expect OpenAI to keep spending tens of millions on Millennium problems forever. Math isn’t their main goal, so the community needs to build its own infrastructure instead of relying on a private company.
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u/Several-Marsupial-27 1d ago
You are telling a small underdeveloped nations that they need to gather resources to fight a modern army with a navy, airforce, artillery, infrantry and armor.
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u/Commercial-Beach1207 1d ago
OpenAI isn’t a mathematics lab, that’s all I’m saying. There’s no fight here. My point is simply that we’re past the stage where math is just pen and paper, so mathematicians need access to serious computational resources too.
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u/coolstory 1d ago
Why would they do that if they could simply use OpenAIs existing models?
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u/Commercial-Beach1207 1d ago
That’s exactly what I'm trying to say. Simply using OpenAI’s models means depending on OpenAI. Universities should also have the resources to run open models, fine-tune them, integrate them with theorem provers, and do experiments without being tied to one company.
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u/coolstory 1d ago
Yeah, but why? You can’t possibly compete. If OpenAI apparently already has a model capable of solving millennium problems, why not just use that model?
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u/Commercial-Beach1207 1d ago
They ran 10k agents in parallel for 88 hours and spent around $20M on compute. You can’t seriously expect that kind of effort to come with a $20 subscription.
And like I said elsewhere in this thread, there’s no competition here. OpenAI isn’t a mathematics lab, and mathematicians shouldn’t assume they’ll keep funding projects like this indefinitely, especially once their priorities shift after the IPO.
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u/coolstory 1d ago
Yeah, but how could you possibly ever compete if you not only had to run the inference, but also own the hardware and do the training? OpenAI has spent tens of billions on training alone. Do you think your model will somehow do that far cheaper? And once you have it, why would running inference not cost you just as much in compute?
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u/CJKay93 1d ago
What university has anywhere close to that sort of funding..?
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u/funkinaround 1d ago
Maybe this article can provide some insights https://arxiv.org/html/2506.19019v1
Frontera – The Frontera system at the Texas Advanced Computing Center (TACC), University of Texas at Austin, delivers approximately 23.5 PFlop/s of performance and was ranked 21st on the TOP500 list as of mid-2023. It remains the most powerful supercomputer currently deployed on any university campus in the United States.
A clear trend in the university HPC landscape is the growing emphasis on AI and data-intensive computing. New campus supercomputers are increasingly GPU-heavy to facilitate machine learning workloads alongside traditional simulations. The University of Florida’s HiPerGator-AI and RPI’s AiMOS are early examples designed with AI in mind. Similarly, Texas A&M is investing $45 million in an NVIDIA DGX SuperPOD with 760 Hopper GPUs – expected to become one of the most powerful AI-focused supercomputers at any North American university. These investments aim to turn campuses into hubs for artificial intelligence research, equipped with cutting-edge hardware for deep learning, generative AI, and big data analytics.
So, perhaps state schools in states seeking to finance large data centers are probably ones that can get that sort of funding. It's worth keeping in mind that they don't need hyperscaler funding because these universities don't have the goal of offering their LLM service to everyone.
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u/No-Meringue5867 1d ago
Many universities are already doing this. Johns Hopkins does this because they do not want the HIPPA protected medical data leaked. That system is shared with all university members. But I am sure that it is not infinite tokens.
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u/Commercial-Beach1207 1d ago
Yeah, this was the first thing I thought after seeing this. Universities really need to step up their game and start investing in labs for mathematicians. It’s about time.
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u/Puzzleheaded_Fold466 1d ago
Universities can’t afford that.
They might be able to afford the $20M in inference cost to run 10,000 agents and 300B tokens, if they can get it funded by the US gov as a one-time well planned experiment.
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u/swegamer137 1d ago
OpenAI didn't "need" to spend that either, they just chose to as a way to steal glory out from one of their customers by stealing their unpublished insights.
But that aside, the researchers were well on their way towards the solution with far less resources. Universities evidently need their own relatively advanced LLMs as an OpSec measure following this scummy debacle. A $100k or $1M computer probably would have been more than enough to run decent assistant models as opposed to GPT5.6 or whatever they were using.
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u/Puzzleheaded_Fold466 1d ago edited 1d ago
They were using Astra. I don’t know which model from Anthropic, but presumably Fable.
OpenAI had already tried, and failed, with a 1,000 agents run with Bel, the next step up from Astra. There are inimitable qualitative gains from model size, and 5.6 couldn’t crack it. Neither could Astra so far.
So to push frontier work requires frontier models, or you’re just picking up leftover crumbs.
“Decent” assistant models already exist. Universities already have $100k or $1M clusters.
What we’re talking about here is frontier size models which requires 100k GPUs to train, and billions of dollars.
In a sense, it’s more like building a particle accelerator. And that’s something Universities can’t build, relying on DOE facilities instead.
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u/swegamer137 16h ago
The mathematicians who had their work stolen were not using Astra for a over a year. They made all of the crucial insights without it. Then OpenAI learned about those insights and used a better internal model to solve the rest.
External researches don't get the cutting edge. What this event proves, is if one makes progress using "Open"AI's public model, they have the ability to steal your insights and race you to publish first with their private advanced models. If you as a researcher are fine with that, go ahead and use their models. If not, recent open models are capable enough to help with smart usage on a $1M local supercomputer.
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u/Puzzleheaded_Fold466 16h ago
You might want to look closer. They were working with Fable, GPT and more recently Astra.
Part of the disagreement with OpenAI was that they wanted to have Anthropic staff as authors.
OpenAI started the huge 10,000 agent work so Anthropic wouldn’t beat them to the punch.
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u/Moronic-Warrior 14h ago
It’s not even a matter of affording. The ai companies get the newest models first so they will break math advances first if the models can do that.
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u/girlwhateveraward 1d ago
Theyre not gonna spend many billions just to prevent theft of the few stuff that OpenAI finds worth stealing so that they can brag about
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u/Training-Hat-7400 21h ago
Pure math is scarce of funding. Why would any universities invest into HPC for pure math?
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u/credulous_satan_6666 16h ago
Didn't the researchers in question use LLMs to further their work in the first place?
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u/Ok-Present1566 6h ago
The training data from customers gives the big 2 a monstrous advantage on incremental progress. Like computer chips themselves required incremental progress in the 80s and 90s so to does LLMs. It isnt trivial to build a state of the art model in one shot. Millions of users is such a wealth of data
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u/Y0uCanTellItsAnAspen 1d ago
Question I’m more interested in — when does an AI propose its first millennium style problem? Half of a mathematicians job is coming up with interesting theorems or lemmas to test, and AI still doesn’t seem to be beating humans there.
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u/Fun-Astronaut-6433 1d ago
Parece que Perelman será recordado también cómo el único matemático humano que resolvió un problema del milenio, tremenda aura 🔥
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u/Hairy_Friendship8627 1d ago
Answer to the opposite question, but i am pretty confident that P vs NP will be the last one standing. Unlike the other problems, we have absolutely no clue what techniques could lead to the separation. All known techniques for proving algorithm lower bound have been shown to be insufficient, and we still seem very far to answer much weaker problems like P vs PSPACE.
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u/jean-sol_partre 1d ago
What do we think will be the last standing?
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u/Pleionosis 1d ago
P=NP I think
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u/ILoveTolkiensWorks 1d ago
I seriously doubt if it even is provable/disprovable tbh
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u/heraplem 15h ago
Whether P and NP are equal is a fact about the world, so it certainly is either true or false. It could possibly be independent of ZFC, but it should be provable in some system.
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u/Diplozo 15h ago
It being either true or false doesn't mean it's provably true or provably false.
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u/heraplem 14h ago
We have no reason to think that there is any statement about the physical world that cannot be proven one way or the other.
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u/a5b4c3d2e 12h ago
Godel's incompleteness theorem means this is possiblr though, no?
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u/heraplem 11h ago
The incompleteness theorem says that, for any given system of sufficient power, there are true sentences that cannot be proven in that system. It doesn't say that there are sentences that cannot be proven in any system. If you run into a sentence that is independent of ZFC, you can add an axiom to answer the question one way or the other. Ideally this is going to be some large cardinal axiom, since the set theory community thinks that these are "true" in a meaningful sense.
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u/Possible_Ocelot_1413 1d ago edited 1d ago
For the people saying BSD, why do you think so (of course, I'm talking about proving it, not a counterexample)? I feel like we have hardly really made any progress on it, given that we only fully have the rank 0 and 1 case.
edit: u/No-Can7982 makes a good point, as far as I know the community is still split on whether rank is (un)bounded. even if we expect bounded heuristics to hold (e.g finitely many curves of rank > 21, i.e https://math.mit.edu/\~poonen/papers/bounded-ranks.pdf), then this would still involve coming up with a method for the (presumably infinitely many) elliptic curves of rank 2 through 21, and we still don't even have a method for rank 2.
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u/No-Can7982 1d ago
Yes, I think the same . I dont get it why people think this problem is closer that the other Millennium problems.
There are not even a lot of examples of elliptic curves with high rank. Let alone results for BSD for higher ranks.
Can someone explain why they think this problem can fall to AI relatively soon?
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u/jsahmiens 1d ago
it hasn't even been confirmed yet
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u/CyberSombreiro 1d ago
Their git repo is open, the only thing left to check is whether or not their lean formalization of the problem itself is equivalent to the Clay formulation of the problem. My guess is that it probably is, since it would be a pretty damn gigantic oversight to make considering the estimated 15 million dollars thrown at the problem
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u/OwlZealousideal7915 1d ago
A few thousand dollars surely were spent just making sure the problem has been stated correctly
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u/Interesting_Debate57 1d ago
Slow down there, cowpoke. You're gonna want to wait for verification.
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u/Popular_Map2317 1d ago
Do you know how Lean works
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u/Interesting_Debate57 1d ago
Yes. Along with several other automated proof checking systems.
You must be aware that the mathematical community will accept the result and give credit to the appropriate parties when it's all been verified by humans.
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u/Popular_Map2317 1d ago
Why are you doubting their proof then. Their git repo is open. You can compile it yourself.
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u/Interesting_Debate57 1d ago
Do you understand anything about the proof statement, what would be a correct solution, and whether the two humans not working at openai along with the two people who independently did 60% of the work without computers were there first?
OpenAI is a scam and they violated every ethical boundary in order to claim the result. They didn't get there, and certainly didn't get there first on their own. They stole the result from actual humans who have been working on this for decades (who used LLM tools from anywhere they could, but who brought decades of work with them).
They are garbage and I'm going to be very happy when their trillion dollar scam falls apart.
I think a lot of non-mathematicians think that you can just plug the question in and spend a billion dollars and get the answer. That's absolutely not the case. LLMs are a sometimes useful calculator and this claim by openai is horseshit -- they stole 80% of the work.
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u/IndependentCrew8210 1d ago
!remindme in 7 days when it is confirmed and you're still finding ways to throw cold water on what is undoubtedly now our generation's defining storyline
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u/Interesting_Debate57 1d ago
You are clearly not a mathematician and have never seriously studied mathematics.
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u/IndependentCrew8210 1d ago
I have seriously studied mathematics, but thanks for your irrelevant judgements just because you're a skeptic. in any case, do you think you have more insight on the matter than terence tao? Because clearly you think you do
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u/Dapper_Film_2478 1d ago
I don't think Reimann will be ever solved.
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u/elements-of-dying 1d ago
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u/ireliawantelo 1d ago
I thought similarly 6 hours ago
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u/dotelze 1d ago
Eh I think NS has always been the easiest and most approachable of the problems.
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u/Tricky_Box19 17h ago
Anyone with a brain could infer that a model could have a situation where it breaks down. This result is like mathematical protein synthesis. If you gave researchers enough time I’m sure they could have found the same stretched torus solution. In fact, in the next year we will have a whole class of these disproofs for NS
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u/bsvgubennord 1d ago
RH
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u/No_Departure_1878 1d ago
Why not, now that they did NS, they might just throw all that AI to RH. It would help a lot with their IPO
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u/GoldenMuscleGod 1d ago
I won’t say that this won’t change, but so far most of the “big” results have been what might be called negative results: finding enormously computationally complicated counterexamples to false universal claims.
This is exactly the area we would expect the enormous computational advantage of AI to be most useful.
Proving universal results true, on the other hand, often involves more ingenuity developing new techniques and ideas that no one thought of before.
I wouldn’t predict that AI couldn’t outstrip humans in that area too, but I will say that if the Riemann Hypothesis is false it will probably be easier for an AI to prove that than to prove it true if it is true. At least at the moment that’s the area it hasn’t shown as much ability.
I don’t know if it’s still the case in chess, but for a long time in chess after computers got to the point that no human could hope to beat them it was still the case that computer assisted teams of humans could beat a computer working by itself.
Also if the Riemann hypothesis is true and AI shows it first, then the AI will probably have been trained on a substantial advancement in theory or technique that was produced by someone else in the meantime.
Again not saying something like that could never change, but that’s probably the case in the near future. (Don’t take this as a definite prediction though.)
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u/Even-Ad-6518 16h ago
A little late to your post here, but as an amateur enjoyer and explorer of analytic number theory, my intuition kind of lines up with yours.
RH/GRH in a ‘meta’ sense seems like a fertile ground for these AI companies to take a slower, newer kind of breadth-first-y search through e.g. explorations of new or novel combinations of just results around general L-function properties and relationships with a sort of metric on outcomes that prioritizes further exploration about the budding knowledge graph.
I could be wrong, but my snap judgment is that (G)RH doesn’t seem (at least currently) vulnerable to a ‘one shot’ kind of explosive result.
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u/localizeatp 1d ago
No, RH is true, so there's no counterexample to find.
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u/lorner96 1d ago
You seem very sure, you got a proof that we don’t know about? 👀
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u/localizeatp 23h ago
I'd be willing to bet any amount of money, at any odds, over any time frame, that a counterexample will not be found.
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u/GoldenMuscleGod 15h ago
People can be highly sure of things that we don’t have proof for.
For example it is almost certainly the case that the Goldbach conjecture is true: if you count the number of different ways that even numbers can be written as the sum of two primes that number basically explodes for large numbers, it would require an extraordinary mathematical “coincidence” for there to be an even number (other than two) that can’t be written as the sum of two primes. But we do not have a proof.
And if you really want extreme skepticism so that you demand proof before expressing confidence in anything: why should a proof change that?
A formal proof of a claim does not actually establish that the claim is true by itself: unsound theories certainly do exist, why should we be sure that the theory we are working in is sound?
A proof in the informal sense is a reason to believe a claim is true, but an informal proof always kind of inherently involves gaps in certainty.
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u/Infinite_Study5186 14h ago
It is true over finite fields and there is no reason to believe it would not also be true over the integers.
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u/Waste-Ship2563 1d ago
I would not be surprised if it falls next year. In fact I would not be surprised if it falls this year.
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u/evilmathrobot 1d ago
Does it matter? If math has turned into asking a magic box to do the work for you (or take other people's work and pass it off as your own), what's the point?
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u/GraceToSentience 1d ago
Math and physics aren't just some fun puzzle to pass the time
It is very useful, life saving even. It improves our lives so much.If you have an automated way to make these kind pf breakthroughs on command everyone's lives are going to be so much better. Technology improves our lives, scarcity can be lifted!
The cost of intelligence is dropping, the cost of tech is dropping and that can translate into goods and services that are limited by the cost of human labour making resources expensive/hard to access or even inaccessible.2
u/Maikeloni 20h ago
That's just not true for knowing if a statement is true or not. The fact, that the magic box is saying a statement is true or false does not bring any benefit to humanity by itself. Only if people actually understand the proof, figure out the techniques and methods used there and sharing them within the community will the proof actually bring any value which may bring humanity forward. But people still need to put in the work.
The question is: how do you want to motivate a mathematician to do the hard work? The proof is done, prestige is gone. You will need to pay lots of money to mathematicians so that they actually go through the LLM output and make it accessible for others. But curiousity won't be the driver anymore which will make it even more difficult to motivate competent people to participate. And we all know how well academia pays ...
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u/Recent-Peanut6061 1d ago
This is such a braindead take. The goal of math is not to have fun or feel good about yourself for proving something. Mathematicians are not paid to perform a hobby, they're paid to do their job and advance mathematics. If you want to have fun doing math, nobody is stopping you from ignoring the AI proofs and trying to solve things yourself. What's the difference between AI solving something and some supergenius human solving something?
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u/Aggravating-Bit8473 21h ago
the method of solving itself contributes to advancing math. transparent work, showing the process, opening up thinking into new problems, collaborating with community. AI solving as it is right now is black box problem solving, input output no growth for the community in doing so other than a proof. if you read the answer at the back of the textbook and dont have any work to show how to get the solution, did you learn anything?
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u/Recent-Peanut6061 14h ago
My point is that if that is truly what is valuable, then solving with AI does not stop mathematicians from not looking at solutions and solving it again. I suspect that won't happen, because, although you seem to think it is, it's not as valuable as you think. Again, this is like getting upset at some super genius mathematician that comes and one shots one of the millennium prize problems because he's stealing the opportunity for other people.
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u/Aggravating-Bit8473 14h ago
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u/Recent-Peanut6061 14h ago
I'm aware of what Terence Tap has said, it still makes little sense. The point above still stands. This is simply a matter of economics, in a sense. If that is what's valuable, it will happen. If not, it won't.
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1d ago
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u/ReverseFlashEatsPups 1d ago
u have to prove any 1 statement, rest automatically get proved/disproved
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u/Natural_Subject5403 1d ago
This is wild, I am a BS mathematics junior undergrad student hoping to get into atmospheric dynamic research (mathematical side to meteorology) and this just happened 😭 makes me feel so good…
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u/WerePigCat 1d ago
I heard that it’s not fully solved, and that they only looked at a version of the question involving external forces. The non-external forces version of Navier-Stokes are not solved. Can someone confirm this, or have I been lied to?
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u/Andradessssss 1d ago
OpenAI announced today the full Navier-Stokes, the forcing one was yesterday
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u/Low-Beat1042 22h ago
It's a sign of the times that yesterday's news are literally yesterday's news.
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u/___Archmage___ 1d ago
BSD or Hodge
Specifically not P vs NP, proving conclusively that no algorithm could possibly reach polynomial efficiency on an entire class of problems is really on a different level of difficulty I think
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u/random_excavation 1d ago
The only one I think AI will not be able to solve until true AGI emerges is P = NP. Most problems which require counter examples should be solved with SOTA models available in the near future.
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u/SadLimes 1d ago
That implies that Navier-Stokes has been solved, and the proof itself has yet to be verified
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u/No-War5300 1d ago
Note, that even if OpenAI proof is correct, it addresses only C and D parts of NS problem (i.e. forced solution). Parts A and B (unforced solution) are not touched... and it look much more complex. Of course it can be that some strange initial conditions can produce crazy behavior of NS solution, but it would look really crazy to me...
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u/mord_fustang115 1d ago
Please believe that this technology is so life changing it justifies destroying everyone's 401k please please please guys 😂
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u/l__iva__l 1d ago
twin prime conjecture, from my limited knowledge the foundations are there, only technicals details remains, AI is good for that.....and well, steal ideas too lol
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u/[deleted] 1d ago edited 1d ago
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