QuesnayJr

You remember correctly. His result is a special case of Langlands.

Your reply isn't a response to the comment, but rather a general expression of antipathy towards mathematicians, who are apparently fatcats who have fancy lifestyles.

OpenAI sat on the results so they could drop 10 at a time. It's rumored that they are sitting more results: https://mathoverflow.net/questions/513818/a-serious-challeng...

So not only are we not going to get boatloads of results, we're only get as many results as necessary for OpenAI to market their models.

They are more than a strong student could achieve. I'm not equally familiar with the problems, but the ones I'm familiar with, if a student solved them people would be thinking "that's someone on track to win the Fields Medal one day".

If it works better here than for programming, then I would guess it's because you can give it a very precise prompt, so you either solve the problem or you don't. If you read the prompts people have shared for problems like this, then the instructions are basically "Solve this problem. Don't give up early. Don't solve a similar problem."

The ones I'm familiar with are big breakthroughs, but they are both counterexamples. Examples have an advantage in that once you have the example in hand and a sketch of the proof (which they have provided), then an expert can probably work out the details themselves.

The sofic groups question was the outstanding question about sofic groups. Almost everyone thought that non-sofic groups existed, and there were plausible candidates, but proving a group was non-sofic was out of reach. Now that we know how to do it once, we can probably do it a lot more.

The Connes rigidity conjecture I think people thought was false, but it was a provocative claim to make. The significance of conjectures is frequently not that the answer to the question is "yes", but that we don't know how to answer the question. And now, apparently, we do.

The Maxwell conjecture was a conjecture in theoretical physics (though not a particularly important one)

Mostly the significance of a conjecture is that we don't know the answer. A conjecture is successfully resolved if it proven or disproven. (In fact, of the three big conjectures to be resolved in the past couple of weeks, one was proven and two were disproven.) The Jacobian conjecture, which is the one that was resolved this week, was largely thought to be false. It was only the fact that no one could disprove that raised the tantalizing possibility that it was true.

The reason the author is upset is not that the conjectures were false, but that an AI settled them. That's why so much of his post is about the loss of the human element.

When that day comes, you think they won't? They spent their lives carefully honing a skill that is now worthless.

There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...

It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.

I remember back when Gemini looked like it was the best model that this comment section was full of confident predictions that Google had "won" and that no one would ever catch up with them again. The most embarassing part is that I kinda believed them.

If the poster's (is it Kevin Buzzard?) suggestion works out and AI finds a counterexample to the Hodge conjecture, that would be a really big deal. It's one of the Millenium problems, for example.

One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.

That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.

That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.

No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.

This seems like the next step, but the smallest counterexample in n=2 is degree greater than 100. (This is a paper of Moh. Wikipedia has details.)

The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true.

Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.

If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.

Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.

Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.

We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.

It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.

I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)

At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.

Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.

The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)

I haven't seen these claims, but all of the open problems that have been solved so far have been in the category of "humans could have solved them, but didn't". This doesn't diminish the significance of the results, but it does mean there's still work for mathematicians to do other than glorified prompt engineers.

This specific counterexample really is trivial. There's nothing to cite. People have wasted hours and hours on a question whose answer you could give as a homework problem in Calc II.

There's a certain (unfair) stereotype of Russian mathematicians that this fits.

If governments introduced a whataboutism sin tax, they could solve budget deficits overnight.

Both systems are at end-of-life. If anything, China is closer to the end. The total fertility rate in China is down to 1, while the US is 1.62.

At this point you're just openly shilling for the CCP. The CCP killed millions of people through incompetence, and then they had the brilliant idea of copying the exact development model of the Asian Tigers. It's their previous incompetence that's the reason that they haven't already caught up with South Korea.

Philosophy majors do surprisingly well in getting jobs outside academia, so I think the market demand for confusion is bigger than you might guess.

They get paid to produce confusion.

If it was the students, then students can have things they think are cool or uncool.

If it was the professor, then that would be very embarassing on his or her part.

It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts.

The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.

According to Wikipedia the term basically wasn't used by Marx at all, and was coined by other anti-capitalists.

There's an old paper about the mathematics of nomograms that I found interested when I stumbled across it: https://doi.org/10.1016/0001-8708(65)90042-3

The first time I saw the red squiggles is one of the few times I thought "Good job, Microsoft".

Someone said to me yesterday "I'm thinking about building my own cyberdeck" and I boggled, because the only meaning I knew was the Neuromancer one.