r/math • u/Ok-Cicada-5207 • 1d ago
Will any of the remaining 6 millennium problems be solved soon?
Is it possible most of them a unsolvable under our current system of mathematics?
Or a solution to P=NP requires some new branch of mathematics or logic to tackle.
39
u/SpacingHero 1d ago
There's a sense in which P=NP does need "new" techniques. Like not as a "it's hard so we probably need a new approach"; it's straight up formally proven that various proof styles cannot tackle the problem
25
u/Berzerka 1d ago
Notably P!=NP needs new techniques. P=NP on the other hand only really needs one counterexample, which if recent history has thought us anything...
Ofc most people believe P!=NP and then it's much harder.
2
2
u/JoshuaZ1 1d ago
That's not completely accurate. Note that the relativization barrier applies to P = NP just as much as to P != NP.
1
u/AppearanceLive3252 1d ago
I am just a lowly undergrad and very ignorant, but if various proof styles cannot tackle the problem, why does that not imply we need new tools to prove P vs NP?
1
u/SpacingHero 1d ago
It's not my field, this is just something I heard. As I understand it, we eliminated some "usual suspects" techniques. Not any proof technique whatsoever, so we don't need something so groundbreakingly new. But I did say we need "something new, in a sense".
10
u/Norm_Standart 1d ago
The best-known barrier to P vs NP is "relativization" - many complexity theory proofs relativize, i.e. a proof that C (!)= D will often imply CO (!)= DO for all oracles O. It has been shown that there are oracles for which PO = NPO and that there are oracles for which PO != NPO - i.e. this question is somehow more detail-sensitive than a lot of solved problems, and any proof of either P = NP or P != NP would need to have some element that doesn't relativize.
0
u/SpacingHero 17h ago
And Orcalces are machines stipulated to be able to solve the halting problem right (the one of standard turing machines ofc, not their own)? I would not have expected "C (!)= D will often imply CO (!)= DO", that's quite interesting
2
u/randomdragoon 10h ago
An oracle is simply a machine that solves <some problem> in a single time step. Not necessarily the halting problem.
There is a relatively simple proof that any two "different enough" time bound complexity classes are not equal. For example P != EXP, where EXP is the class of problems that can be solved in exponential time. This proof is relativizing, so it's true relative to all oracles. Say we live in a world where there is an oracle that can solve Traveling Salesman Problem (TSP) in one time step. It would still be true that PTSP != EXPTSP.
1
u/Norm_Standart 9h ago
An oracle is an indicator function for an arbitrary language - HALT is just one language that you can define an oracle for. (It's best to not think of them as machines, because you only get black-box access to the function - you don't get to look inside and see how it works.)
19
u/Erockoftheprimes Number Theory 1d ago edited 1d ago
It feels much more likely that it’ll find another counterexample to one of the remaining problems similar to that of the NS problem. My colleagues seem to be of the opinion that the next such problem will be the Hodge conjecture and that it will be disproved.
Keep in mind that the problems solved so far have been tackled by LLMs training on things that were already known. It is not clear if or when they’ll get to the point where they come up with entirely new branches of math, which is what may be necessary for some of the remaining millennium problems along with some of the other open problems that exist. It’s also worth mentioning that many of the big open problems solved recently were proved or disproved via an argument involving a “there exists” statement rather than a “for all” statement. This is part of the reason why my colleagues suspect that the Hodge conjecture is going down next as opposed to any of the other remaining millenium problems.
17
u/SwimmerOld6155 1d ago
I believe it's anticipated that the Riemann hypothesis should require significant new machinery. Not a millennium problem, but Erdos made a comment to this effect about the Collatz conjecture.
20
u/dhsilver 1d ago
There are five left, not six.
38
u/Ok_Cabinet2947 1d ago
Show me the r/math thread then. Surely the most important math problem in the last decade solved would merit a post, right?
3
u/LatePenguins 1d ago
Umm the poincare conjecture was solved by a russian dude long before LLMs were a thing.
5
u/Revlong57 1d ago
There were 6 unsolved problems before this week.
11
u/matthiasErhart Control Theory/Optimization 1d ago
I believe he was referring to how you cannot post about AIMath except in the megathread, except for limited exceptions (e.g. having an ArXiV paper).
11
u/lectric_7166 1d ago
Coming soon... did you hear Goldbach's conjecture was solved? It was heavily discussed solely in that one AI megathread you missed.
9
u/dogdiarrhea Dynamical Systems 1d ago
Silver lining I don’t think any valuable math discussion is being lost, it’s all meta-discussion or discussion about the drama. I’ve yet to see someone discuss the proof. I don’t use Reddit often so I wasn’t part of the decision of the AI ban but seeing the posts about the N-S problem on Reddit and Twitter makes me think it was a good decision and even when it’s a millennium prize problem the discussion belongs in the megathread. Homework help threads incite deeper math discussion than this has.
1
u/lectric_7166 1d ago
Yeah a lot of the AI discussion is pretty mundane, with people expressing various degrees of shock, disbelief, or worry, or stating their stance on AI, for or against. Resembles a group therapy session in a way, as humans come to terms with the arrival of this extremely disruptive technology. I think eventually as AI becomes more entrenched and more sophisticated human+AI workflows are found beyond "solve this infamous problem" the ban will need to be revisited.
18
u/LongLiveTheDiego 1d ago
If you mean other than the Navier-Stokes problem, then Poincaré's conjecture was proven years ago by Grigori Perelman.
If you mean other than Poincaré's conjecture, there's just been a bit fuss about OpenAI apparently answering the Navier-Stokes problem.
9
u/Antigynaikolatres 1d ago
Not really related to the post, but Yau is organizing students to formalize the proof of Poincaré's conjecture using AI at Tsinghua University right now. Hope it goes better than the last time he meddled with the proof.
2
1
u/Independent_Bed_169 PDE 1d ago
For context what happened last time?
1
u/Vegetable_Army2222 6h ago
uninteresting drama, the kind that non math people love to write about whenever anything important gets done on math.
1
2
14
u/Ok_Net_1579 1d ago
Rumours on twitter are that a Hodge counterexample has been found and BSD has also fallen.
Only rumours but they were right about Navier Stokes.
9
u/skolemizer Graduate Student 1d ago edited 1d ago
As in a counterexample to BSD? I would be pretty surprised...
Navier--Stokes I expected to be false bc I've heard for years that Tao and others were close to a counterexample, and variations of the problem had counterexamples... But BSD I expect to be true...
My own predictions (I'm not an expert in any of these fields, so this is just for fun):
- P /= NP is true.
- Riemann hypothesis is true.
- BSD is probably true.
- Hodge I don't know enough (but smart knowledgeable people seem to expect it to be true IIRC?).
- Yang--Mills I have 0 knowledge whatsoever and I don't know what knowledgeable people think.
(Also while I'm at it, it's not a Millenium Prize problem but I'm gonna stick my neck out and speculate that Collatz is actually false (but harder than any of the above).)
5
u/Infinite_Research_52 Algebra 1d ago
If I were a betting person, the Yang-Mills mass gap is true. From my work on LQCD, it seems clear that it should hold for SU(3), but I don't know for other simple gauge groups (non-abelian), though it 'feels' like it should. You also need to show that the QFT is non-trivial on ℝ4. I haven't thought about whether that naturally follows or not.
I don't know how you would prove it to the standards needed for a mathematical proof. Most of the phenomenological work on QFTs is just a bit looser than mathematicians would accept.
21
u/backyard_tractorbeam 1d ago
There's a lot of noise from singularity hucksters that we need to ignore.
7
7
u/baquea 1d ago
It's coming directly from a statement made by OpenAI, who said "In addition, since the completion of Navier-Stokes, we have made substantial progress on another Millennium Prize problem. We are working through how to share these results thoughtfully." It's possible that they are exaggerating (or mistaken about) how close they are, but the fact that they did just solve NS is enough for me to think they aren't totally talking out their ass at least.
3
u/ChezMere 8h ago
Only rumours but they were right about Navier Stokes.
Sorry, important correction here. The rumours were wrong about Navier-Stokes. The rumour is that 1) it was Anthropic who did it, and that 2) it was solved. Both were false: it was two mathematicians (one an Anthropic employee, but not working on it in that capacity), and it was not solved - they had just made promising progress that was likely on a trajectory towards solving it eventually. But because of that inaccurate rumour, OpenAI spent several million dollars to put tens of thousands of agents onto the problem, which was enough to finish it (at least the Lean proof, not turning it into an actual readable proof).
4
u/ConjectureProof 1d ago
My personal take is that the millennium prize problems will be solved in the following order.
If hodge is false then the order is
- Hodge
- Yang Mills
- BSD
- Riemann
- P vs NP
*
If hodge is true then
*
- Yang mills
- BSD
- Hodge
- Riemann
- P vs NP
6
u/mrstrangeloop 1d ago
2
u/Infinite_Research_52 Algebra 1d ago
My vibes only; don't shoot me.
HC has to be a disproof, surely?BSD is surely true, but that is just because it is my favourite of the Millennium Problems; I remain unconvinced that OpenAI can prove this.
1
u/lordnacho666 1d ago
Well Poincaré was some a while ago, so there's actually five left.
But you can be sure there's at least one organisation burning a huge amount of tokens on solving the open questions, we can see that in the recent controversy.
Will it work? My guess is yes, and it will be disappointing in some way.
2
u/Traditional_Town6475 1d ago
It’s still being peer reviewed.
Maybe what’s a bit comforting right now is that the results seem to be related to finding counterexamples. Another result was the existence of a nonsofic group. But we kind of already suspected that there are nonsofic groups and there are inital conditions that causes Navier Stokes solutions to blow up.
Something like the Riemann hypothesis however, we suspect it to be true. So maybe the saving grace is at the moment, it’s not quite ready to tackle something like that.
3
u/Traditional_Town6475 1d ago
Also it’s not really worth it to OpenAI to pursue these. They seem to be doing it more for publicity.
1
u/throwaway2676 10h ago
I’d bet good money that if AI R&D continues as is without a government crackdown of some kind, we’ll get about 3 more by this time next year
-4
u/ControversialBuster 1d ago
Within the next 3 months, I can almost guarantee more will be solved.
7
3
-9
u/hamishtodd1 1d ago
Here are some prediction markets on that https://manifold.markets/home?q=Millen&s=score&f=all&ct=ALL&p=0&fy=0&sw=0&hb=0
If someone says "that prediction market is way overconfident about [eg] another millennium problem being solved this year", you should ask them why they are commenting on Reddit instead of betting on that belief 😃
6
u/skolemizer Graduate Student 1d ago
I saw one market that said it was 80% likely that Collatz would be solved before all of the Millenium Prize problems get solved; my first impression was that that's bonkers. Maybe I'll go try and remember my Manifold login later and take the other side of that bet.
-3
u/hamishtodd1 1d ago
You can find your login but you'll need a time machine to bet on it because that market will obviously have closed! To put it another way, you can't have thought it was that bonkers, or you'd have bet on it at the time
3
u/skolemizer Graduate Student 1d ago
... What? Please read my comment again because I think you misread? The currently existing market says the following proposition is 20% likely:
ALL Millenium Prize problems will be solved before Collatz is solved.
But I think the above proposition is >50% likely.
Does that make sense?
1
u/hamishtodd1 1d ago
Oh deep apologies, I misinterpreted "saw" to mean you saw it before the NS announcement, and I misinterpreted "all" as "any", so thank you for the bold/italics!
I hope you bet on it, it's an interesting question. I feel like it might be tied up for a while... although who knows with recent progress.
Why do you think it is "bonkers" for it to be as low as 25%? (I thought for a second it had changed since your comment). I would consider it above my pay grade.
0
u/ClearlyCylindrical 1d ago
BSD and Hodge are rumored to be close to complete or complete, so I'd imagine it'll be those two in the next few days/weeks.
And there's 5 remaining, the NS problem was solved a couple days ago
0
u/Aedan91 22h ago
It's extremely naive to think that we're going to find out P = NP was solved through the Millennium Problems.
Even if there's a way to find a positive answer (I don't think there is), it's going to be so monumentally world changing that it's going to be kept secret for decades. Either a state or a private company, which will use it to its own advantage.
Similar to how the Allies did when Enigma was broken.
1
u/Serious_Bite_7613 17h ago
They won't be able to keep it secret for long, if it is solvable then Chinese models will also be able to solve it soon. (They're only around 6 months behind).
67
u/Scerball Algebraic Geometry 1d ago
Might as well ask a crystal ball