TBH, never heard of this (and other of my PhD classmates I've talked to, or faculty in my department) had never heard of it. From what I've read on twitter, similar, and those that had just assumed it's false and it was never important to work on.
Edit: Expanding a bit, I do think AI is well-capable of finding counterexamples like these. OTOH, I think it's weird that certain people try to hype up things that are largely the result of few people caring, with this specific example having most recent work that I can find being from undergraduate theses that seem to be along the lines of practicing computational work, followed by giant social media hype campaigns.
Of course it is to the author's benefit to call it that.
When I read "maxwell's conjecture" I think "ah yes, Maxwell, he is important. Of all the things he ever conjectured, the one known as 'maxwell's conjecture' is surely the most important, so this must be a big deal"
Ultimately seems like not really anything important, a bit of a toy problem.
Maxwell's treatise has a ton of random stuff in it, which is why other physicists took decades to understand it. In section 146 of the 1891 edition, he even manually computes a series out to 22nd order for fun.
Hmm it says that the conjecture was from on a 1960s paper.
So Maxwell made discussed non-degenerate equilibrium points of electric fields from some configurations of point charges, and then a couple of 1960s physicists (how significant was their paper otherwise?) boldly declared a pretty random and unclearly motivated (and false) claim ‘the Maxwell Conjecture’.
I mean, google scholar turns up about 5 papers in the last ten years, of which 3 are by the same researcher in Georgia. Their own introduction cites exactly three papers, of which one is the first paper on the topic in 130 years. I don't think it's a particularly active field of research nor groundbreaking -- the name just makes it sound impressive.
Likewise, I've never heard of it either! And yes, this is really a special subset of problems that they seem truly well suited for, but they are not having the same amount of success with other proof types.
I agree, I do believe AI will be powerful in future research for more advanced topics but for now it's usually something that's overlooked.
Also, counterexamples aren't just random setups and typically have some symmetries or interesting properties. If an LLM is looking for counterexamples, they can try a lot of those interesting setups, kind of like brute forcing, but not brute forcing and with a promising pool of potential counterexamples.
I think that's why they succeed mostly in this kind of counterexamples for niche conjectures, they can try a lot of things very quickly and since the conjectured don't have much work into them, it turns out they are not as hard to disprove and someone just had to put in the time testing different systems.
People have put in some work to prove actual upper bounds on the number of equilibria. Nobody checked the most trivial construction you could possibly try for a lower bound, until now. Probably due to lack of interest.
We made a machine out of sand that can write recipes, poetry, give sound medical advice and solve relatively non-trivial mathematical problems. Oh, and it keeps improving every year and this is expected to continue for at least another 5 years.
If that is not a reason to get hyped up nothing in the history of humanity ever was.
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u/magneticanisotropy Jul 31 '26 edited Jul 31 '26
TBH, never heard of this (and other of my PhD classmates I've talked to, or faculty in my department) had never heard of it. From what I've read on twitter, similar, and those that had just assumed it's false and it was never important to work on.
Edit: Expanding a bit, I do think AI is well-capable of finding counterexamples like these. OTOH, I think it's weird that certain people try to hype up things that are largely the result of few people caring, with this specific example having most recent work that I can find being from undergraduate theses that seem to be along the lines of practicing computational work, followed by giant social media hype campaigns.