---
title: 'Breaking Maths News: what''s going on with millennium maths problems?'
source: 'https://youtube.com/watch?v=c79PsbkHw4A'
video_id: 'c79PsbkHw4A'
date: 2026-09-21
duration_sec: 1409
channel: 'Stand-up Maths'
---

# Breaking Maths News: what's going on with millennium maths problems?

> Source: [Breaking Maths News: what's going on with millennium maths problems?](https://youtube.com/watch?v=c79PsbkHw4A)

## Summary

OpenAI has announced a solution to the Navier-Stokes Millennium Problem, a major mathematical breakthrough achieved with AI. This video breaks down the equations, the solution, and the ensuing drama, including the race with human mathematicians and the cost of the AI effort.

### Key Points

- **Millennium Problem Solved by AI** [00:57] — OpenAI claims to have solved the Navier-Stokes Millennium Problem, one of seven major math problems set in 2000. Only one has been solved before.
- **Understanding the Equations** [01:23] — The Navier-Stokes equations describe fluid motion, with terms for acceleration, viscosity, pressure, and external forces. The left-hand side is essentially F=ma.
- **The Problem Statement** [05:42] — The problem asks to show that smooth, finite-energy solutions exist, or find a counterexample. OpenAI found a counterexample with a singularity in finite time.
- **OpenAI's Solution** [07:48] — OpenAI's proof is formalized in Lean, allowing machine verification. They found a smooth force that leads to a singularity, which is physically plausible but breaks the equations.
- **The Meme Image Explained** [09:04] — The viral image is a PR illustration, not a direct output of the proof. It's based on an exaggerated schematic from the paper.
- **The AI Effort and Cost** [15:10] — OpenAI used 10,000 AI agents, 3.7 million messages, and 130 billion tokens, costing an estimated $10-40 million, far exceeding the $1 million prize.
- **Drama with Human Mathematicians** [14:04] — Two human mathematicians, Levent Alpay and Tristan Buckmaster, were close to a solution for the Euler equations, leading to a race and accusations of OpenAI using their code.
- **Mathematicians' Reaction** [17:44] — Mathematicians are upset because the one-shot counterexample offers no insight and may discourage further work. The journey of discovery is valuable.
- **Practical Implications** [19:37] — The result is pure math with no practical application; real fluids would vaporize before the singularity. It raises questions about AI's role in pure math.

## Transcript

Breaking math news! A millennium problem has been solved by AI, and yes it's covered in drama, but the point is we now have a millennium math problem solution, and normally when these things
are announced, and I have to respond quickly, I'm on vacation at the beach. But this time, do not worry, I am not on vacation, at least not at the beach.
Yes, I am actually here at Disney World on vacation. Or am I? Wait for the end of the video.
But at the point, I'm hanging out with my mate. I thought I'd take some time off to talk about this because it is a big deal. The Millennium Prize problems were set up in 2000 as like the seven biggest, most important to some definition of that problems we have to try and solve.
And so far, only one of them have been solved. And now suddenly one, maybe two, that would be a different video. But the point is, OpenAI have declared they have a solution to the Navier-Stokes equation.
And whenever you see the reporting on this that's been happening this week, people go, oh, Navier-Stokes, something to do with like, you know, fluids moving around. But not here, this is standard math. We want to know what those equations actually are.
And what are the variables? Fluid movement is complicated. There's a lot going on, and the velocity of fluid it's moving at can change based on where it is in the fluid. It's ridiculous. We're going to need some complicated equations for this. It's not like it's just one particle bumping off another one.
Then we can use force equals mass times acceleration. Ah, oh, F equals MA. Forget that equation. We need the Navier-Stokes equation. Now you can formulate them different ways. I'm going to use the way the Clay Mathematics Institute writes them down on their Navier-Stokes equation page here, if you scroll down.
when they actually officially describe the problem. Thanks Charles. Ah, you can see right there at the top the Navier-Stokes equations, specifically three of them. And we can have a closer look at these. Here they are as written, or bear in mind,
technically within these ranges. So x actually represents, that is like all three coordinates, which are each a real value, and that's true for all of these, and then t starts at zero, and gets bigger. Pretty straightforward. Oh, and that n there should be three.
That's the number of dimensions we're working on in this case. Now of the three, the last two are pretty boring. We'll deal with them first. This one here just says, at the beginning, the fluid is, you know, doing something, something nice.
Because u, the velocity, is a function of x, that's the position in 3D coordinates, and at time, time zero, that's just some, you know, u prime starting conditions.
So we don't have to worry about that too much. This just means the fluid cannot be compressed. It's an incompressible fluid. You try and squash it down, nah, not going to happen. So those two are just setting the thing.
Here's the real thing. This is the equation we worry about. This is telling us how the fluid as a whole, all of the particles together, are moving around. We'll look at the left-hand side first. And so you've still got U there.
All the U's are the velocity of one particular bit of fluid, which is what the eye is specifying. specifying, and this is its derivative with respect to time. You're thinking, well hang on Matt, that can't all be the derivative. Surely this,
there's your derivative of velocity with respect to time. Well, I don't know what this whole thing over here is doing. Well, the reason we need this is because it's not just like we're tracking one particle and its change of its velocity,
which is changing over time. The speed of a packet of fluid can change based on where it moves. It might move somewhere where the velocity of that location is different. We need all of this because the velocity u is a function of the location in 3D space as
well as being a function of time and actually the location in 3D space, I put this extra function of t in there, you'll often see it drawn this way. I want to emphasize the velocity is a function of both time and its position in 3D space
which itself is a function of time. It gets complicated. If you want to take the derivative of all of this with respect to time, it's chain rule time! There it is. So, that's why at the front here we've got the partial derivative with respect to t,
and then we've got to add in all the other ones. In fact, there's just one for each of the three different directions that our fluid can move in. And there's just one for each of the three different dimensions that we're operating these equations in. So, in short, that entire left-hand side, the derivative of
velocity, is just acceleration. And if we normalize mass, so just equals one, it turns out that is math times acceleration it is F equals MA the whole thing is F equals MA
yes it was just a more in bold version of force equals mass times acceleration this entire time well the other way around I guess technically it's MA MA equals F there we are okay so right the left
is MA on the right we've got some force so uh first of all this here is the viscosity nu, that little curvy V on the left, that's the viscosity measure, and this is the Laplacian
of the velocity. That's just where you sum up all the partial seconds or whatever it is. So that just means the fluid has some viscosity. And over here, this here introduces P, that's the pressure, so as well as the force due
to viscosity, we have the force due to pressure, and finally we have the force due to a force. This is a smooth force that you can apply to the fluid to move it around. And that's
going to be super important in a moment And now that we understand all the terms involved we can see what it means to solve or I guess disprove the Navier equation And Clay gave four ways it can be done Actually the first two are kind of the same
it just depends what numbers you're using for the coordinates, and the second two are kind of the same. But if we zoom in on C, which is what OpenAI say they've done, we can read it now. So on R3, so 3D coordinates, let's view there,
so that means it's got to have non-zero viscosity, fair enough, n equals 3, three dimensions, good. So you've got to show there exists a smooth, diversion-free vector field. Now, that's what I called U' before.
That's just the starting conditions. A nice, smooth, moving fluid at the beginning, and then a smooth, that's important, force. That's the force we saw at the very end, the last term a moment ago,
which acts on R3. And the conditions on statement C here is you've got to show that the fluid ends up with a singularity, and that's got to happen in finite time. So that's why there's a curved bracket on the end of our range here.
So time starts at zero, never quite reaches infinity. So you can't just be like, oh, asymptotically this will reach infinity. No, it's got to happen in finite time. It's got to use finite energy. And it's got to be smooth.
A smooth force moving the fluid around. You can't just like, you know, hit it with a hammer. So what does it mean that OpenAI has found a smooth force function that with finite energy and finite time can produce a singularity.
And moreover, what on earth is this image that everyone keeps meaning? We'll start with the OpenAI page where they announced their solution. In the beginning, it's just a very general, you know, recap of the situation.
The problem is what we were saying before, just going through that, making sure everyone agrees on what's happening there. And here's their result. This is where it gets interesting. This first sentence is the main bit. So they're establishing first of all they've given an actual proof and a lean
Formalization so that's like basically code that people can verify that their proof is correct Doesn't give you any insight, but you can check it and what they found is an initially smooth fluid
So that was our starting conditions In fact, they've got it at rest and that could develop a singularity in finite time Now, they say singularity, that means that it blows up in mass term.
This to say the velocity is unbounded, it grows in a way where it can become infinitely fast. You think, well hang on, that's not physically possible. Well it can be if the region in which it's happening is getting smaller faster than the
velocity is getting bigger. And that's kind of the whole point, that's how the energy remains finite. So physically plausible in one sense, but it still blows up. which means that the equations don't work,
and that's why this whole thing is a counter-example to Navier-Stokes. And again, the smooth force is the whole thing that makes this so difficult. Now, you've already seen it poking out the bottom here. Let's check out the image. Here it is now.
For everything I did, producers themselves, people are just making it up. However, it is a PR image. There's no code that gives you this. It's not a 3D model. That was part of the proof.
It doesn't actually appear in the proof at all. this is just an illustration for the press release. So the teal bits are slow, and then the orange bits get faster and faster, but you can see the orange bits along this axial stretching,
which we'll get to in a moment, is orange all the way up and down. Now, if that was true, this would be producing a singularity that runs the entire kind of, you know, it'd be a linear one along the length of the axis. The singularity is just a single region right in the middle
at what they describe as the origin. Now that said, the diagram is based on something. If we go back up here and actually read the paper, which is always important to do, and you scroll down a bit, you can see, there it is.
This is the diagram that it was based on. Even this isn't super accurate. It's just a schematic to give you a rough sense of what's happening. And it's also, as it says here, it's exaggerated. So that meme image is a PR dress-up of an exaggerated schematic of the actual math OpenAI did.
So let's talk about the math. This is showing three steps in the evolution of this fluid as it approaches the singularity. And everything's been designed so the singularity happens at time t equals 1.
And then they define tau here as 1 minus t. So this means tau is like the countdown until the singularity happens. And the point is, as you get closer and closer to the singularity, the aspect ratio of this is changing.
It's stretching out, but oh my goodness, the reason I had to exaggerate it here is h. h is your value for, like, that's the exponent for the change in the aspect ratio, and that is always smaller than 1 100th.
So actually the aspect ratio changes so slowly as tau approaches zero, which is our big moment of singularity. To help me get my head around it, I made my own inaccurate diagram.
Check it out. So we are starting, so tau is 0.3, so this is closing in, we've got less than a third of the time left, it's a singularity moment, and over here I've got the max angular velocity, which is a proxy of how fast things are going in the middle, and you can see now, I've only got orange in the middle, because that's where the max speed is happening, it's not happening all the way up and down this axis.
So let's set it going. Now on the left I've tried to show that fluid is kind of spinning around the axis and coming in near the middle let's say, and as it gets closer and closer it spins faster.
Classic. And actually more is going out the top than the bottom, opening eye are clear that the asymmetry there is quite key However on the left I keeping it full sized On the right is what actually happening While it spinning and evolving it getting dramatically smaller which I just showing by zooming out and you can see tails are
going at a linear rate. All the action happens right at the very end. That's when the aspect ratio changes significantly and we get exactly at t equals one the singularity that breaks the Navier-Stokes equations. Please keep in
This is just my ridiculous diagram and it's super inaccurate as well. I just wanted to try and give some kind of dynamic sense of the math situation that OpenAI were describing in their paper. If you want to go a tiny bit deeper, what's interesting is OpenAI didn't find the smooth
force required to make this happen. They found a setup like a velocity field, and so that's what the diagram is trying to show you here. And what they realized was, if we come up to page 2, they point out for any incompressible
flow U, that's our speed and pressure P, we're going to define the external force F to be the residue, which means you've got Navier-Stokes explaining all the standard fluid mechanics,
so you take the velocity field you need for the singularity, you run it backwards through Navier-Stokes and take out all the fluid mechanics stuff, and what is left over has to be the additional force required to get that velocity, but that force is so often not smooth, and if you
just set up this system here to get the singularity, the force wouldn't be smooth. What OpenAI did, their real big breakthrough down here on page 6, they have a sequence of pulses in the velocity, and they had to tune a whole bunch of these to make that spiral
plausible, but in such a way that the force comes out smooth. And so this is what they actually designed was a very clever way to pulse the velocity field
to get the singularity without making the force no longer smooth. So have OpenAI actually solved the Millennium Problem? The consensus seems to be yes, they actually have. That said, their proof, their result, hasn't been verified independently yet,
and they're a long way off actually getting the prize awarded if indeed they even claim it. And that brings us to the drama. The drama all kicked off on Tuesday 8th of September,
and the first move was humans, because OpenAI were the only people working on this. Two human mathematicians, Levent Elphaga, sorry I pronounced that wrong, and Tristan Buckmaster, were nearing a solution themselves.
Tristan actually put out the post on Mastodon, linking to three papers, where they had solved not the Navier-Stokes, but the very similar Euler equations, which is the same thing but with the viscosity removed
so it's a little bit easier and they found a way to make the Euler equation break which means they would be very close to doing the same thing for Navier-Stokes. The non-math drama then unfolded in a state that Tristan published at the same time
and it kind of falls into two different flavors. One is that OpenAI heard the rumors and realized some humans were close to claiming this millennium prize and so they just pulled out all the stops
and powered up, as they say, 10,000 AI agents to all work on it at the same time. Now, this is over, because it worked. They got there first. But OpenAI do say that those agents sent 3.7 million messages
and used about 130 billion output tokens. A lot of people online have tried to work out how much that would have cost it if you were actually paying for all those tokens. and they figure that OpenAI spent between $10 to $40 million
getting the Navier-Stokes solution. That's discounting wasted tokens on other Millennium Prizes. They made another stole, but at the end of the day, everyone used to be, wow, Millennium Prizes, they're like a million dollar prize each.
And now, someone has spent an order of magnitude more money to win one. The second type of drama is just messy drama. So Tristan and Levin did use ChatGPT's codecs to help write some of their code
and they're like, well, does that end up in the training data that OpenAI used? OpenAI were like, well, we can't rule it out, so that's a bit of a mess. And then there's all these accusations that OpenAI got in touch with Tristan and said, oh, you can publish alongside us, so you've got to get rid of Levin,
who just coincidentally happens to work at Anthropic and different AI companies, even though they were doing this work on their own time. And so there was a lot of back and forth unpleasantness. It's just not how academics ideally should be behaving.
Drama aside, mathematicians were also pretty upset at how this one-shot AI result was just dropped. And I've seen that sold online as mathematicians just being emotional because they're being replaced by AI.
Which is a little bit true. But more importantly, for the most part, mathematicians have been pro good use of AI to help support mathematics research. If you remember my previous video, which was like months ago now, where we looked at the Euler distance problem,
that was discovered by AI thinking for like 80 minutes and then just dumping a result. But it had techniques and methods within that, and they've already been repurposed by humans to solve other things.
So mathematicians are not just like blanket AI is bad, they're like the way we're using AI now is bad. Specifically in this case, mathematicians were really annoyed because it was a counterexample drop in a way that's not human understandable.
There's no insight we can get from this other than we know there's a counterexample. And moreover that means it less likely mathematicians will work on this problem because we kind of already know the answer which means we going to skip everything in between
from wondering about the problem to eventually solving it. And sure, we could just say, look, reverse engineer the one-shot counterexample and see what you learn, but humans are humans. We're not going to have the same motivation if we already know the ending.
And the journey is the whole point, much like a rollercoaster. at a theme park, see, tied it in, also yes it is the next day, right, the journey is the experience and we're going to miss out a lot of math if we don't do that journey. I mean in the past if an AI
had one shot and said you can't square the circle would we have bothered inventing transcendental numbers? If AI had said there are no solutions with the quintet do we still get Galois theory?
That we have to still be doing the journey and actually my close personal friend Grant Sanderson of 3Blue1Brown fame is doing a blog post on Terry Tao's blog and their own video
which is coming out about now due to the nature of breaking news. I don't know if this video will be first or there, you should absolutely check that out. It's got really good insight into why we do the journey of mathematics. All of that said, for the longest time mathematicians have been
saying keep funding us because the maths we're doing might one day be useful. Just trust us bro. And mathematicians we may have applied maths ourselves into the
corner and now society is saying yeah if mathematics is as useful as you say and maybe it's more like a monorail ride where the destination is the point and surely we should just fire up the AI math machine and spit out as many results
we can to see if they're useful and uh yeah i kind of got a point there in one sense navier stokes was the worst millennium problem to be ai'd first because it sounds so applied we care about
fluids we want to know how they move but in reality for the longest time anyone doing actual fluid mechanics already knew the limitations of navier soaks and how to calculate the movement of fluid
in any practical situation. The Millennium Prize version was very much pure math dressed up in fluid mechanics. I quite enjoyed on the fluid mechanics subreddit, someone actually ran the numbers for a real fluid to see what would happen,
and it would vaporize long before that singularity. So for all practical purposes, this discovery is actually not helping humans at all. And look, if OpenAI had found a new way to, I don't know, multiply matrices,
everyone would be very excited. No one would be complaining, ooh, you didn't show us you're working out. We'd be like, great, new way to multiply matrices, which is what happened back in 2025, when I did the video about DeepMind,
coming up with a better 4x4 complex number matrix form of multiplication. There are definitely new applied bits of mathematics, techniques and algorithms that AI can help us find and implement, and we should all celebrate that.
but just picking off pure bits of mathematics with no immediate application is just going to undermine both the education and the motivation to discover and learn more pure mathematics.
And that is a very different subject. That's the one where the journey is the point, because what we learn along the way are little detours we notice that we hadn't previously considered taking. So what happens now? Well, I think in some regards, a lot of people thought something like this was inevitable.
I would have guessed some kind of neural network, LLM, linked up to some kind of proof-bearing flying lean system. But it would have been like a niche thing that mathematicians had.
I did not see it becoming this kind of, you know, Silicon Valley startup culture, move fast, break things situation, where now math specifically has become this tech pro appendage measuring competition to just dump big answers out there.
Although that said, given the response to Navier-Stokes, I believe the AI companies are sitting on a war chest of big math results, and they're going to try to work out how to release them.
And I say, please, just do it nicely. But all of that aside, one day this will all settle down. We'll have a magical math box over here, finding better ways to multiply matrices. and over here we'll have AI helping mathematicians better understand and
discover new bits of mathematics. I just hope we get there get there quickly but it's unavoidable. The AI tide is coming in. I think it's important we have a kind of society-wide nuanced conversation about this. Surely that's
that's what's gonna happen. Anyway that's it this time. I won't do another one of these for a while unless there's some ridiculous breaking mask news, we'll see what happens. One day I'll just have a regular vacation. Oh no,
that's dead. I have actually been here doing some work. The video will be out probably in November. Subscribe if you want to see it. But this, this actual video, this has nothing to do with Disney or anyone else. This is literally me on
vacation and thought it would be fun to put together. In fact, I've accidentally bought two overqualified filmmakers to help me put this video together. Now, much like the poor, crying children at the end of the day out of Epcot, we're going to wrap this up.
You can say thanks to my poor friends who just want to go back to their holidays.