Yes, their parents are going around saying oof, the kids definitely didn’t get it from Roblox, or YouTube, or their peers.

really? you think they don't have friends/bros/tv/etc to imitate?

[flagged]

I'm a research mathematician. From what I can tell, the answer is roughly comparable to: if you posed 8,000 challenging open problems to the human math community, you might expect to see 372 of them solved within five years.

Probably some combination of: some of the 372 problems were easier than the rest; the AI got lucky on these 372; there were existing papers out there in the literature which proved especially helpful for these 372; and other similar factors.

If it took 3h for one of them, perhaps there was a time/compute budget cutoff along with a sorting based on some relevance.

My guess would be that these particular problems were vulnerable to an attack which built on recent advances and potentially tied in something unexpected from a distant area of mathematics. "Harder" is becoming harder to define. Harder for humans is probably not harder for LLMs.

There must be an element of luck, if they ran the remaining problems again with the same time constraints presumably a bunch would be solved

I've asked my students whether they still want to learn maths even if there will be a machine that will answer any question instantly and they will be homeless. They said yes.

(To my credit, I have warned them since more than a year ago that we will reach this point.)

To some extent this was discussed in the article, and in a way I think their goal is actually the same as it was: become the first human to understand something.

It's just that we lost one of the important ways to demonstrate understanding.

Pro AI blogging sponsored by ClosedAI, XTX markets and the Simons Foundation.

> what problems do these people reorient towards after this?

The same problem almost every person on earth is going to have to reorient to in the next decade, which is: how do we eat and stay housed when we have no real economic value?

They will continue to prove theorems and make discoveries, except now they will have AI to help them so hopefully progress will be faster. At the same time, new challenges will open up, for example how do you verify what the AI is doing and how do you explain it.

Math isn't about collecting random theorems, progress in math is about gaining understanding of new systems, and the theorems are guideposts to aid in that understanding.

You can prove 1000 theorems and not really increase any understanding about a subject, but gain knowledge of 1000 random facts. For example, I can write down some complicated equation and ask you "does this have a solution in the integers"? And if you do a maze of very complex and tedious algebra to show that there is a solution, you would have proved a theorem, but you would not have done much to move math forward at all.

On the other hand, if you introduce some completely new technique, say you take my equation and turn that into an algebraic surface, and then you count some special curves that live on this surface using geometric ideas, and then you show that if the number of such curves is odd, there must be a solution in the integers, and in this specific case, it is odd, so there is a solution -- well, then you have really pushed math forward and people will celebrate your proof, even though no one really cares if the equation I wrote down has a solution in the integers.

For example, there is a long history of failed attempts to prove Fermat's last theorem driving algebra and number theory forward by introducing the concept of ideals, for example, and this concept ended up much more important than whether Fermat's theorem is true or false, which is not too much more than a piece of trivia.

Or for example, the recent proof of the Poincare conjecture relies on the machinery of the Ricci flow introduced by Richard Hamilton, who then applied it to solve a number of open problems, but Perelman was able to take it even more forward to solve Poincare. So Ricci flow was massively important machinery.

For this reason, we celebrate people like Gromov, who didn't really prove that many theorems but introduced amazing machinery -- for example, the h-principle, or Gromov Compactness -- these were ideas and math is about the ideas. The ideas are then applied, using laws of logic, to form theorems.

So mathematicians will need to mine these proofs to see if there are any new techniques - new machinery - being introduced, or if the AI just used the existing machinery more efficiently. Here too, we are just looking at AI as a form of search, which it is really good at, since there are so many thousands of papers and so many ideas, that there might be a connection between two areas that lead to a solution and the human mathematician, not knowing all known results, can't make that connection. In the future, we may wonder how anyone did math without AI, much like we would wonder how anyone can be a writer without access to a dictionary or reference work. Is the AI just searching through a catalogue of known ideas and connecting them or is the AI coming up with genuinely new stuff like Ricci flow or the h-principle?

What is interesting is seeing whether we can get AI to actually discover new machinery for us. That would be huge.

And then we need to find efficient ways to detect these ideas and describe them.

Really this is very exciting and opens up whole new workstreams for mathematicians.

priesthood

Food and shelter /s

Have you learned something from every Fields medalist's research? If so you are a member of the extreme mathematical elite and you should probably just dig into the results yourself.

Often times the constant (2 in this example) is a conjectured minimum, so anything below that is a noteworthy result. Think of it as breaking through some theoretical limit.

for say FFT/integer multiplication or 3SUM, we have natural algorithms that have existed a long time with a given complexity (O(n \log n) and O(n^2), respectively). Given how long these natural algorithms have been the best algorithms we have, it is natural to conjecture they are optimal. Showing an O(n(\log n)^{.99999}) algorithm exists shows that these optimality conjectures are false.

Now, there are some critiques you can have of this. Namely, it is possible that these novel algorithms have significant trade-offs that make them almost never worthwhile in practice. "Fast" matrix multiplication algorithms are typically of this form. So perhaps this all points towards a deficiency in big O notation, which can be deceptive. But, for people who care about optimizing asymptotic complexity, it is still interesting.

Many people thought it could never be less than 2. They proved that it can. What is the true value? Nobody knows, now.

to get some intuition about why this is such a big deal, look up the history of strassen's algorithm, which solved matrix multiplication in less than O(n^3). this was a truly stunning result because it seemed intuitively obvious that the output matrix had n^2 cells each of which was calculated via an independent O(n) loop over a row/column of the input matrices, so how could you do better than n^3. but once strassen proved that you could do some clever tricks and reduce the overall time to something less than O(n^3) it started an entire cottage industry of people getting better and better algorithmic bounds. the initial breakthrough was a qualitative one, independent of how much it improved things in numerical terms.

https://hideoushumpbackfreak.com/algorithms/algorithms-stras...

Tell that to the humans working on matrix multiplication who spent years of their lives getting it from n^2.3728596 to n^2.371866, only for openai to blow it away at n^2.25

You just run it again and again and again

Yes, there was controversy around the Navier-Stokes result. But here are >300 problems and no corresponding >300 complaints of theft. Things are indeed moving really fast, and it's hard to keep up even as someone folrowing this with more obsession than would be healthy. Maybe somebody should keep a running short summary of The Situation...

Yes it was:

https://cepr.net/publications/ai-didnt-steal-the-mathematici...

Frontier theft is just faster.