The $1 Million Math Problem

There are 7 Millennium Problems, and if you solve one, you get $1 million. Woot.

To date, there is only problem that's confirmed to be solved: the Poincaré Conjecture. Grigori Perelman solved it in the early 2000s and then just kind of disappeared. He rejected the fame and declined the money because he's a legend.

You may have seen a picture of him floating across the internet.

This man... is free.

Now OpenAI claims to have found a solution to one of the other Millennium Problems: the Navier-Stokes equations.

These equations describe the behavior of fluids. And fluids are interesting things. I spend a lot of my time reminding people that the sky is a fluid. Once you see it as a very thin ocean, planes and clouds make more sense.

Think of a fluid as anything that changes its shape based on the container that it's in. Coffee in a mug. Helium in a balloon.

If it fits, I sits.

This is important to note because the Navier-Stokes equations don't just affect how water moves, but the air too. That means the equations are behind the design of airplanes, rocket ships, and anything else that moves through air or sea.

Every mathematics student enters undergrad thinking they want to tackle the Millennium Problems - or find an equation to predict prime numbers. But by graduate school you learn that problems can be deceptive in their simpleness. And if there's anything mathematicians share in common with whoever designs economy seating: it's that looks aren't everything.

Grossly summarized, Navier-Stokes has been difficult to solved because nobody has been able to prove if it always produces finite answers, meaning that there isn't some weird exception where it "blows up," and spins off into infinity or something.

We really call it that: blowing up.

Mathematicians are just as bad at naming things as astrophysicists. Black hole. Blowing up. Big Bang Theory... terrible.

This is the image that OpenAI released to the press:

Imagine that it's water going down a drain. It spins faster and faster as it flows down toward the center. "Blowing up" would mean that maybe it spins infinitely into a single point. Obviously your sink doesn't do that. Water goes down a drain and into the sewage system.

OpenAI had around 10,000 AI agents working on it, and after about 4 days, they claim to have found a solution where it can "blow up" under very specific conditions.

Now this doesn't mean that it's going to happen in real life and your leftover rice is going to spin infinitely into oblivion. But it is an example of how the math we use to model the world has limits. And that's a whole other thing.

Mathematicians like to poke holes, that's kind of the job. We like to know why something works, and then prove that it's always going to work just like that. Meanwhile, someone else tries to prove that there are exceptions, and maybe it doesn't always work like that?

Anyway. The controversy here is that just before OpenAI's announcement, a few other mathematicians were using OpenAI to work on the problem as well. So the question is not only is the timing suspicious, but what is an AI company allowed to do with our data?

Not for the first time, we need to establish the laws that govern AI.

I'm sure we'll hear more about this in the coming months, but as far as your life goes and a possible solution having been discovered for the Navier-Stokes equations, nothing changes. A solution won't have much of an effect on engineering for a while. For mathematicians, however, this is a seismic shift.

Maybe AI really is coming for us all.