Navier-Stokes Announcement

Though it would be funny if no one ever bothers publishing the result in an appropriate journal, and thus the prize technically can never be claimed.

Edit: Oh, didn't see the "qualifying outlet" condition. But Poincare was ever just put on arXiv, so arXiv must count as well.

A similar rule existed for the 100-year Wolfskehl prize established in 1906 for solving Fermat's last theorem; two years after publication.

Keyword: New Technologies

Possibly because of the ongoing debate about who actually deserves credit.

Very on brand with using AI for everything.

While the scandal is still unraveling, it seems that OpenAI did a rush job to steal other mathematicians' thunder and finish the proof first.

OpenAI released a statement that their work does not relate to the work of the other team, but it clearly does. They use the same niche smooth-forcing mechanism. Altman and Bubeck claim that because the proof used different scaling parameters and analytical steps, it's not related, but it seems that nobody else agrees. Oh, and OpenAI's Bubeck tried to threaten Buckmaster (mathematician working on the proof).

This brings nothing but shame for OpenAI.

The Poincaré conjecture guy also broke that rule. They wanted to give him the prize anyway but he refused. OpenAI announced they would also not claim the prize.

Looks like no one wants this prize lol

Looks like even a blog post is good enough, they just need to do the review by themselves.

OpenAI and Anthropic might have 3 millennium problems by December

I hope that pure mathematics research can retain a strongly human component forever. It would sadden me immensely for human understanding of our mathematical world to wither and die, and for us to become ignorant consumers of wonders beyond our understanding just because our robots can do it better than we can. As far as applied research goes, I hope we will always be able to understand what we want to, but I have less qualms about becoming more scalable and efficient.

Currently 0/2 Millenium problem solvers claimed the prize money so clearly money is not their motivation for tackling the problem.

OpenAI spent many multiples of the prize money in just a few days to get there and even if one solves a problem in the traditional way, that person is most likely already an accomplished professor at a reputable university where a million dollars doesn't mean as much as the eternal fame that comes with it.

If AI can do superhuman math that allows better medicines, cleaner energy etc that is great. But if AI replaces humans in all the creative and intellectual fields that is not only a loss of jobs but also a loss of deeply meaningful activities. This is waved away but I think that is mistaken.

What I fear is really the growing notion that "people shouldn't do math/art/music because machine do it better and cheaper".

Yes, you can make problems arbitrarily complex. But the prize problems were chosen not just because the solutions appear likely to be very complex (the problem statements aren't necessarily inherently complex--there is a way to restate the Riemann hypothesis that a junior high school student could easily understand, which I'll give below).

They were chosen because they were important problems that mathematicians really wanted solved, top people had worked on them for a long time and progress stalled a long time ago, and it seemed likely that solving them would require major breakthroughs.

Those kind of problems can be discouraging. Enough people who are probably better than you have spent enough time failing to solve them that realistically most researchers are going to focus all their efforts on something they are likely to make progress on.

A nice prize can get more people to at least work on them as side projects.

Here's that restatement of the Riemann hypothesis I mentioned.

The Riemann hypothesis is that the non-trivial zeros of the function ζ(s) occur on the line 1/2 + yi.

ζ(s) is 1/1^s + 1/2^2 + 1/3^s + ... when s is a complex number whose real part is greater than 1, and defined everywhere else except s = 1 by a process called analytic continuation. The trivial zeros are at s = -2, -4, -6, ... .

For a mathematician, or a non-mathematician who has taken complex analysis and hasn't forgotten much of that, that is not too complex a definition. For anyone else the first reaction is probably "Trivial zeros? How the heck does that thing even have zeros? And if it does how the heck can it have zeros at any negative integers! It is obviously infinity at every negative integer!!!".

Here's a different hypothesis that turns out to be exactly equivalent to the Riemann hypothesis. They are either both true of both false, so resolving one of them resolves the other.

Let H(n) = 1 + 1/2 + ... + 1/n for all positive integers n. These are called the harmonic numbers.

Let S(n) = the sum of the positive integer factors of n for all positive integers n. For example S(4) = 1 + 2 + 4, S(6) = 1 + 2 + 3 + 6, and S(17) = 1 + 17.

Hypothesis: S(n) <= H(n) + exp(H(n)) log(H(n)) with equality only when n = 1.

The proof that this is equivalent to the Riemann hypothesis is here [1].

[1] https://arxiv.org/pdf/math/0008177

...

>Now AI is changing that. I wonder what this means for the kind of mathematics that people have traditionally done.

Mathematics becomes engineering. I think it is great and long overdue. Saying that as a Math PhD dropout :) Of course like manual craftsmen had to adapt to Industrial Revolution, the same would need to be done by the mathematicians. And other scientists too.

Clearly the real-world cannot "blow-up" - real-world water vortices do not reach infinite velocity, etc.

The point of having Navier-Stokes as a Millennium prize was to hopefully generate new mathematics and techniques along the way, and auto-generating a sprawling AI-slop proof or millions of lines of Lean does not accomplish that result.

Clearly OpenAI has no interest in the math itself - to them this was just a trophy animal to shoot and stuff. I would be very surprised if they now helped analyze the proof and try to extract the mathematical value out of it, and this would obviously require outside help who are probably not inclined to help OpenAI math-wash their behavior.

At this point, how can we tell whether AI is improving or it's just reappropriating its users work? It's probably a bit of both. But still, thick milky.