From First Principles - What OpenAI Actually Did to Navier-Stokes (EP 58)

Episode Date: September 24, 2026

What does it mean to solve an equation that describes almost every fluid around us, from the air over a wing to the water swirling down a drain?In Episode 58 of From First Principles, Lester Nare and Krishna Choudhary build the Navier-Stokes equations from the ground up before digging into OpenAI’s claimed breakthrough and the debate surrounding it.SummaryHow Newton’s laws become equations for a moving fluidVelocity fields, incompressibility, pressure and the nonlinear convective termWhy viscosity smooths a fluid while nonlinear motion can create finer structureWhat finite-time blowup means, and why simulation is different from proofHow forced and unforced equations differ, and why those assumptions matterEarlier work on Euler, Boussinesq and related fluid equationsOpenAI’s claimed result, Lean verification and the scope of the theoremThe dispute over scientific credit and the human research behind AI-assisted workThe METR investigation of the Hugging Face incidentEmergence World and long-running multi-agent experimentsAI-assisted biological discovery, oversight and recursive self-improvementSeparating demonstrated capabilities from claims and future scenariosChapters00:00 Can AI solve Navier-Stokes?00:34 Episode introduction02:20 Navier-Stokes: Mathematics Meets AI10:20 Building the Equations of Fluid Motion21:34 Velocity fields, divergence and incompressibility34:33 Acceleration and the convective term52:18 Why Fluid Motion Is Nonlinear1:02:08 Pressure, Euler and the Missing Physics1:14:50 How Viscosity Changes Everything1:33:36 Solving Equations vs. Simulating Fluids1:44:59 Can a Smooth Fluid Blow Up?2:13:52 The Road to the Claimed Breakthrough2:31:13 Inside the Claimed Navier-Stokes Proof2:48:26 The Dispute Over Scientific Credit3:07:23 From Chatbots to Agents3:08:57 The METR report and Hugging Face incident3:23:23 AI Risk, Oversight and the Race Ahead3:38:51 AI Discovery Beyond Mathematics3:52:47 Why “just turn it off” gets complicated4:06:47 Closing thoughts and what comes next4:08:46 OutroFeatured ResearchOpenAI’s Navier-Stokes announcementTristan Buckmaster’s statementMETR investigationEmergence WorldAI-assisted enzyme discoveryExplore FFPffppod.comffppod.com/fundingffppod.com/transfersffppod.com/America250Watch on YouTubeyoutu.be/NGfGw1tGxUYSupport the showffppod.com/donateFollow@FFPPod on X / Instagram / TikTok / Facebook

But in the Navier Stokes, and that's why it's so interesting. Because the Navier Stokes has this non-linearity, but it also has this smoothing term. It has this counterforce, which is the viscosity. And so you're asking, who's going to win? And are they always going to win? There are questions about credit, commercial incentives, accountability. Yes. But I do believe that the risks deserve real attention without being mired in those other things.

Yeah. Those other things can be true and the risk can still be real. Okay, so just turn it off. And what does just turn it off actually mean? Hello, Internet. This is your captain speaking. Lester Nare, joined as always by my co-host and our resident PhD, Krishna Chowdary. We are about to get into what will be our most requested episode in the history of From First Principles.

OpenAI announced a solution to one of the Millennium Problems. Many of you have either seen that announcement or all of the, the fallout and drama as it relates to AI that has arisen since that announcement. And so a quick overview of how we're going to structure this deep dive episode will be lucky to get it under three hours. So we're going to do something a little bit different than some of the mainstream coverage that starts with the drama. First, we're going to ground ourselves in the fundamental understanding of the mathematics as it relates to the open AI solution. What are the Navier-Stokes equations and what is the existence and smoothness problem? And how exactly did they solve it?

And then we will obviously move into some of the drama and tragedy that has arisen as a result of this and other progression in AI over the summer into the early fall. This is going to be a nuanced episode. We're going to talk about the science and some of the larger complexities if you want to understand science headlines, be sure to subscribe. As always, we are going to talk about the science from the ground up today because this is from first principles. It's not a question of if, but when. All of the people that have been paying attention to what AI has been doing in mathematics, that is what they are asking. When is AI going to solve one of the six open millennium problems? There's six of them that are left. They're kind of like the infinity stones. and I'd say AI is kind of like Thanos, you know, AI is coming. And the question was always, when is he going to get the first stone? Well, it seems he might have got it, okay?

The Navier-Stokes' existence and smoothness problem might have been solved. And naturally, we were on vacation. I was in beautiful Lake Geneva in Wisconsin, and somebody forwarded me a tweet, and this is what I was feeling. I made a meme about it. You know, we're also anticipating GTA-6. Yes. And this is the, ah, here we go again.

Because honestly, I didn't want to keep talking about AI and math. We'd already done the Jacobian conjecture. We'd already done the Riemann hypothesis or a problem related to that, both by anthropic. But this is history, if it's true. You know, this is a big deal. And it's not anthropic this time. It's open AI. And the announcement was met with a different kind of reaction than the other two.

And the other two, there was a reaction of, oh my gosh, AI is so good at mathematics. This is incredible. There's a bit of an existential crisis in mathematics and so on and so forth. But here, there was also something else. There was a fight. There were allegations of academic misconduct. There were questions about who deserved credit. There's questions about what open AI proved themselves and what they maybe stole from other

people's work. that's someone else being the NYU mathematician, Tristan Buckmaster, and Anthropic researcher Levant Alpogue, who was the hero behind the Jacobian conjecture a few episodes back. So there's a lot to unpack. But before we get into any of that, as you said, we're going to do something very deliberately. We're going to talk about the mathematics.

Because at the end of this two-page document that Buckmaster published, which really kicked off the controversy, this document where he talked about all the shenan. nanigans that had happened behind the scenes. He makes the argument that if open AI has really solved Navier Stokes, then they should just say so, and he'd rather be discussing the mathematics than anything else. And honestly, same. For me, same. So that's where we're going to begin. We're going to do the math first. We're going to do the drama second. So here's the roadmap for the episode. We're going to start from first principles and try to build up the Navier Stokes equations.

There's set of two equations. And at the end of that segment, I want you to have an intuitive. understanding of every single term in that equation. So it's not just a bunch of words and fancy symbols. At the end of that, we'll confront the actual millennium problem, this question of existence and smoothness. There is going to be some calculus involved. There's going to be some equations, but I have a lot of visuals. So hopefully it's not going to be that bad. And we'll describe it for our audio listeners. But I do suggest this is a very visual heavy episode, as most of our episodes are. Then we'll take a tour through the mathematics over the last several decades and get to the main event, which is OpenAI's 165 page proof. The claim is that they have

solved it. Now, I am not a fluid dynamist, and I'm certainly not an expert in the kinds of linear partial differential equations that the Navier-Stokes problem is. This is a problem that has attracted some of the greatest mathematicians of our time, including Terrence Tao, among others. So I'm approaching this from a perspective of a physicist who's trying to understand fluid mechanics. And so I went back to our trusty old Landau Lifshitz course in theoretical physics, Volume 6, fluid mechanics. Landau Lifshitz goes through some amazing, amazing mathematics here. And Navier Stokes is in the first, in the early chapters, because there's so much more to fluids than just Navier-Stokes. Shout out to my professor, Robin Bruinsma, who taught me this course.

It was an amazing course where we got to apply some of my favorite mathematics, i.e. complex analysis, to calculate the wing, the lift on a wing of an airplane using, like, a complex integral, which was one of the most mind-blowing things that you could use complex mechanics to, like, talk about, like, real things, you know, not just prime numbers, as we've been known. So at the end of all that, we're going to be understanding the solution, and then we'll talk about the drama, and then all of the existential stuff about what all of this means, because we're going to have to zoom out. This is not happening in a vacuum. And there is a lot of crazy developments that are honestly happening daily. It is really happening daily. And once Krishna has kind of walked us through the mathematics, I kind of want to look at the larger world that this announcement happened in. there is legitimate disputes over the credit and how we think about acknowledgement in scientific discovery and understanding. And this is also happening in the context of an already existing and accelerating battle between multiple stakeholders as it relates to this issue.

You have the frontier labs, you have the investor and venture capital community, you have politicians beginning to finally weigh in on this. Obviously, you have the general public who, for many people, don't really see real world impact on this in their everyday life. And, you know, for context around some of these AI tools, I personally have been a user of them just as a technologist and someone who's interested in these subjects in the way that we cover on the show since GPT2. And I've seen what the progress looks like from a very direct personal context. And I do think that there is.

is a challenge in discussing this issue right now because you have to fit into one of two boxes, which is, this is all hype and this is going to cause human extinction. And in my view, there's a lot of space in between this is all hype and this is going to cause human extinction. And I would love for us to be better able to have conversations within that gap. And so we're going to talk through some of the larger context to help frame this because this will be a, technology, a transformation that will be with us for the foreseeable future. And I think it's going to be important to look at these two things. And we'll look at some concrete examples of other things that have happened like the Hugging Face incident where there were some issues with OpenAI's

models as it relates to the company Hugging Face because I think that will help really ground us around this shift from this is fancy auto-complete or this is just a better way to do search. and this agentic world that has been actually a driver behind how Open AI got to this solution. So we're going to get into all of that, a real brief moment of housekeeping. For those of you who may be joining us for the first time, welcome to the best science show on the planet. You can find us on all platforms where you listen to your favorite podcasts as well as social for clips. Our website has all of the research papers, background material, and other ways. to look into what's happening in frontier science research at ffppod.com.

Any engagement that you do on our videos on all platforms, help us reach more people with what we think are some of the most fun and informative conversations about science, and we have some great merch. So with that, we're going to jump right in to what is the meat and bones of what's going to be, I think, a very fantastic episode. Yes, and we will start with the mathematics.

So these are the Navier-Stokes equations. There's two of them. And our goal is to understand these equations. These equations are an equation of motion that describes fluids, how fluids behave, what are the forces that they exert? If I know what the fluid is doing right now, can I predict what the fluid will be doing at some later time? These are the equations that allow that to happen. And when I say they're an equation of motion, that's what I mean. They're like, they're telling you how the equation, how the fluid is going to move around in some space. Now, in classical systems, the starting point for all equations of motion is always Newton's laws. Okay? Newton's first law is an object at rest, stays at rest, and an object in motion will keep going unless it's acted on

by another force. So, you know, if you're in empty space and you just have a satellite that's moving in one direction, it's going to keep moving. What happens when there's a force? Well, when there's a force, there is an acceleration. It's not a velocity. It's not like you need a force to keep it going. If it's going, it'll just keep going. If there's a force, it'll change how it's moving. So that's called an acceleration. It's a change in velocity. And the relationship between the amount of force that I put on an object and how fast it's going to accelerate is dependent on the mass. It's a mass is a kind of inertia. It's like how much resistance to a force do I have is literally what mass is.

And there's a pretty common demonstration that you see in physics classrooms in high school where you've got two rubber bands. They're the same rubber band. But one of them is attached to a block of wood and the other one is attached to two of the same blocks of wood. And you can see that the single block of wood accelerates a lot faster than the one with two blocks of wood. That's because the two blocks of wood has twice the mass. So it's going to have half the acceleration. There's Newton's second law.

And Newton's third law is something about conservation of momentum, which is like, you know, if I'm pushing on an object, then that object pushes back on me. So those are the three laws. And if we want to understand how something moves around, all we have to do is figure out what is the nature of the forces on this object. And if we're good enough at the math, we can figure out how this object is going to move forever.

This is partly what enables us, for example, to land on the moon, everything, or at Mars, all of these kinds of, obviously much more surface area than that. But this is why we can make predictions over such long periods of time. Yeah, really, in order to get to the moon, you just need Newton's laws. I guess you need electricity to, like, work the capsule or whatever. But, like, you know, if there was some analog version where, like, you had fluids that did computation or something, you could land on the moon with just Newton's laws. So let's take, for example, the mass on a spring.

I want to go through a singular example of a very simple problem where I can have an equation of motion derived from Newton's laws, and I can then figure out what that thing is going to do forever. A common example is a mass on a spring. A spring is a substance that obeys Hook's law, meaning if I pull on the thing, it's going to resist and it's going to pull back. And how much it's going to pull back is dependent on how much I'm stretching it. If I stretch it twice as far, the pull back, the force backwards, is twice as much. That's called Hook's Law. And you usually see it as the force is equal to negative KX. K is how stiff the spring is.

X is how far I've pulled. And so, you know, the larger I pull, the more it pulls back. And the negative is because it's pulling in the opposite direction of how I'm distorting the spring. Okay. So that's Hook's Law. F equals negative KX. Now, force, according to new and second law, is mass times acceleration. Yes.

Right? And the acceleration is really a second derivative of the position of the X, right? The acceleration is a change in velocity, and a velocity is a change in position with respect to time. So it's really a double change, which is why I have mass times a second derivative in position is equal to negative KX. Notice, this is an equation of motion. I've got a single variable that I need to solve for, which is X, the position. And the key thing is the force is dependent on the position, right? But the force is related to how fast the position accelerates.

So now I've got an equation that is in a single variable, X, and it's a differential equation with respect to time. This is something I can solve. Okay? And that's what most people do. You can solve this thing. It's a differential equation. And if you solve it, you get a very nice solution.

You get a very nice solution that is an oscillation. And I think we've got a little video of that. This is what happens. You take your equation of motion and you say, what is something where if I take a second derivative, I take a derivative once and I take the derivative again, I get the negative of the thing back. And that's cosine and sign.

The derivative of cosine is negative sign. The derivative of sine is cosine. So I'd go from cosine to negative sign to negative cosine. Well, that's just the negative of what I started with. And there, I have an oscillatory motion. So this is the paradigm of how we do physics in general. We have Newton's laws. We analyze our system, and we figure out what is the differential equation that the system obeys.

And then we solve this. And if we're lucky enough, the differential equation is simple enough that the mathematicians have figured out how to solve this differential equation. And the physicist can then borrow that intuition. And boom, you've got a solution. that tells you for any time, I can plug in the time, and I can tell you where the mass is going to be. And it's a way in which we can now simulate and create replications such that we can predict how a system will act under certain circumstances.

Yes, exactly. And you can make things even more complicated. For example, you can take the same spring and you can add friction. In this case, there's like some kind of damping force. and what that means is that in your differential equation, you've got a term that's related to velocity, because the faster I move through some, like, let's say I dunk this thing in honey, right?

The faster I move through this thing, the more the honey is going to resist my motion. So that is dependent on the first derivative of position, just the velocity. And if I add that term, then I get a new solution, which is the damped harmonic oscillator. And that's the red that you see over there. So in the undamped case, you've got an oscillation that just bounces over and over and over again.

And then in the damped case, you've got an oscillation that dies out very quickly. For those of you who are listening to us in the car, you can thank damped harmonic motion for your suspension. Because there's a particular value of the damping that makes all these oscillations go to zero very fast. It's called critical damping. And a lot of times car manufacturers want their springs, the suspension in their springs, to have critically damped so that, you know, you get bounced on. on a, what are those called, the holes, the potholes. You bounce on a pothole, but you don't like bounce up and down forever. You just immediately bounce and then you go right back and there's like no bouncing whatsoever.

And that's because the spring is critically damped. That has to do with how much damping is happening. My point is you can make the equations of motion a bit more complicated and still be able to solve stuff. This is the visual I think of is seeing those meme videos of those giant trucks with giant suspension that's visible. going over, you know, this humped dirt track, and they're chilling. Yeah, yeah, yeah. There's all these, like, Chinese propaganda videos, too, where they show, like, the luxury cars of China versus the luxury cars of Germany.

And it's like the luxury cars of China are just like, they're like on a cloud. And then the ones, like the BMW is like, it's like, okay, well, is that a new BMW? Or, like, you know, you never know with those propaganda videos. Or is it just straight up AI? Anyways, for every case, the logic has been pretty simple. Okay, the point is we start from Newton's laws, we understand what the forces are, and then we try to set up the equations of motion. This works for springs. This works for planets.

For a planet, the force is proportional to 1 over R squared, how far away my thing is, and it's pointing directly at the central body. And again, that's a force that is a function of position. And now I can solve for what the position of the planet is, because I can set up a differential equation that does so. Okay, I can't, I can set up a differential equation for a three-body problem, but crucially, I cannot solve that. So I can set up the equation of motion, but I don't have a solution for it. So there's caveats, right? Okay, so now let's try to understand fluids. I see the setup we're going with here, because we're saying we can understand the world around us and make predictions using math.

Yeah, and Newton's laws. And Newton's laws for certain physical systems. Yes. and we just defined examples of those physical systems in which we can do so. Yes. Now I think where we're going is fluids maybe have some unique aspects.

Yes. That make that different. Yes. It makes it different and difficult. Okay. It's different in the, so first of all, it's same, same a little bit

because each part of the fluid still obeys Newton's laws. Yes. If I were to look at a single particle of fluid, let's say I drop a dust grain into a moving river, right? That dust grain is going to experience forces because water molecules are pushing on the thing, and those forces are going to make it accelerate.

Okay, so Newton's law is still work. 100%, right? The problem, though, is that Newton's laws are tenable, and they're like tractable if it's a single thing that's like moving around. In a fluid, for example, even in this cup, right, of water, I have 10 to the 3 molecules. No, no, it's not 10 to the 3.

That's only 1,000. 10 to the 23 molecules of water moving around. Okay? So what? Do you expect me to apply Newton's laws to every single molecule that's in this water? It doesn't make any sense. Sure, that's one way of doing it

if I had infinite compute, which I don't. I'd like to understand this thing at a level of granularity where I can make predictions, but maybe I don't have to worry about every single molecule and what it is doing. We don't want to have to simulate everything because the system is so large as compared to what we were doing before. Yeah. The system is large and it's self-interacting is the other point, right? The water molecules are interacting with other water molecules. And like it's just, it's just a whole mess. Right. So it's not a new kind of physics. It's the same

physics, but it's being applied everywhere in this system. Everywhere all at once. And all at once. Exactly. And so we need a new language. Okay. And perhaps here's, the point. What do we want to actually describe about this fluid? We don't want to describe the individual molecules, as I just said. It would be crazy to try and describe the position of every single molecule as a function of time like we could with that mass on a spring. Right. Right. That doesn't really make sense here. So instead of tracking individual molecules, we are going to track the velocity of the fluid itself, okay, at different spots. And it turns out this has all of the necessary information that we need.

Interesting. Okay? This is what we're doing. On the left-hand side, you see a video of sort of dust particles, let's say, that are in a chunk of a river. Okay, so we can see the dust particles

that are moving from left to right. Now, I don't want to worry about all of the dust particles. So instead, what I worry about is what is the velocity of the particles at that point? And those can be represented by little arrows that just represent where the dust particles. particles are going at that point.

Now, crucially, the dust particles are moving. But this velocity field that I've made of a bunch of arrows that are representing all of the velocities, that thing is stationary. You see? Because it just represents this is the direction in which I'm moving. This is the central element that we are worried about. Okay. Okay?

This is a velocity field. It's a field because it has a value at every point in space and time. and it's a velocity. So it's got a vector. It's a vector field. Okay? And usually this thing is represented as a U. We don't want to use V for some reason.

I think it's a historical reason. So in every case, the U that we're going to see later on in this episode, that is a velocity field. It's a vector field everywhere in space, and it represents what the velocity is at that particular moment. So if I have a velocity vector that's pointed in this direction, that means water is moving past in this direction. If on the other hand, it's over here,

then that means water is moving towards that arrow. Towards where the arrow is pointing. And the component parts that make this velocity field are some position in 3D space, which is our X, Y, and Z. So we know where in this cube, the velocity field that we're specifically referring to exists. And then time allows us to create the arrow of time. Yeah. Of the direct, like at that point.

At that point, it's moving in this direction. Yeah. Yeah. And it could change, right? Maybe I introduce the flow and then it starts moving in some. some other direction. And so the velocity field can change. This is a very simple case, right, of just like particles, water moving from left to right. There can be other more complicated cases. For example, this is the velocity field of a tornado. If I were to look like up the axis of a tornado,

it's rotating around. And so imagine if you're looking like up through the eye of the tornado, you would see particles and clouds and everything moving around you, right? Like in a circle, above you. And the velocity field for that looks like a bunch of arrows that are sort of circulating around. Okay? And the color of these arrows represents how strong

the velocity is. Okay? So the vector is big, near the eye, and as you move farther and farther away from the axis of the tornado, the vector field diminishes in magnitude because things are moving slower. This makes sense. This is why when you have a tornado or a hurricane approaching you

if you live on the coast, the outer parts, you can feel it coming. Yeah. Because the outer parts are a little bit slower. Yeah. When you get to the inner parts, then it's like, okay, then the palm tree is like on the side. On the side, right? And so this is the basic object in modern fluid mechanics.

It's not the velocity of water as though the entire ocean is a single baseball or entity that's just moving as one, right? No, it's different parts of the fluid are moving in different ways. And we can represent that using these little arrows that are in every single part of the fluid. that's actually a really important mental picture for me to kind of reframe how to think about it because my initial mental model is this is one discrete thing and we're trying to look at it as one discrete thing. And what we're sort of saying is in order to get back to this idea of we want to be able to make predictions about a system, we need to look at it from this different vantage point,

which is velocity fields which help us look at points in space. and the velocity at those points in space, which then are also going to interact with each other over time. Exactly. Yeah. And so if we wanted to, for example, calculate what a dust particle is doing in that velocity field, all we have to do is like follow the arrows, right?

We can still recreate what we want, which is like if I were to drop something, what is the position of that thing, right? All we have to do is follow the arrows. But the velocity field is something that holds all of the relevant information. To be able to do that. That's the point. Yeah.

And it becomes something that is a tenable object. It's something that we can understand, and we don't have to worry about the 10 to the however many stuff that the fluid is made out of. It's less compute constrained for lack of a better way to put it. It's one way to put it. Yeah, that's certainly one way to put it. Yeah. And so before we start applying Newton's loss to this thing, let's try to understand this velocity field a little bit further. I just showed you two examples.

The first one was water is moving from left to right, so all of the arrows are pointing from left to right. Or the tornado where the clouds are... moving in a circle above you. And so all of the arrows are pointing in kind of a circle, right? They're like circulating. In both of those, we saw the particle picture, and we were like, okay, that seems reasonable. Now, I can imagine a fluid doing that, right? I can imagine the tornado and I can imagine stuff like this.

Now, are there velocity fields? Are there setups of arrows that we can make up that don't make any sense in the real world? Okay? The answer is yes. Let's look at these cases. here are two cases. The first one on the left is a bunch of particles

that are just like emanating from the center. The expansion of the universe. Yeah, it kind of looks like the expansion of the universe, exactly, right? It's like all of the fluid is coming from somewhere in the center and then it's just like expanding out. Okay? How is that possible? Where is the fluid coming from?

Yeah. Right? That's not possible. Just like in empty space, it's just like there's like just stuff. Right. Like, what is the source? Right? Like, if there was a hose, if I saw a bunch of particles come from the bottom,

and then there was like a sprinkler head or something, and then it moved. Then I'll be like, okay. But here I don't see that. It's just, the things are just coming out. I understood. Right. Doesn't make sense. Okay.

First thing doesn't make sense. Because somehow the fluid is being created in the center and it's just moving out. Right. Okay. Second one, on the right. All of the fluid is moving into a single sheet, that yellow sheet. It's like a sheet of paper. and the fluid is moving from the top

into the sheet of paper and the fluid is moving from the bottom into the sheet of paper. How is that possible? Where is the fluid going? Yeah, right. Right. Because you would have met,

it should like go, like, it should continue. I mean, if I saw the, if I saw it shoot out right to the side, then I'd be like, okay, that's possible. It's hitting the surface

of a wall or something. Yeah. In that vector field, all of the arrows are pointing down and there's no arrows pointing out. In the vector field on the left, all of the pair arrows are pointing out and there's nothing that's supplant the fluid to have those arrows point out.

And so just to be clear, we're sort of using these as sort of like an abstract, theoretical, or mental model to test the four corners of the sandbox of how to do this. Yeah. We're trying to understand, like, we've created this concept of a velocity field. Yes. Now, is any arbitrary vector field okay? Right. And the answer is no.

Because on the left-hand side, that's an arbitrary vector field. Like if I were to create those arrows that are just emanating from the center, if that was an electric field around a positive charge, totally fine. But for fluids, that's not fine. It doesn't, right. It doesn't want fluids.

And so we can formalize this with mathematics. The property that encodes this is something called the divergence of an electric field. Okay? On the left-hand side, we've got something that has a divergence of negative, negative divergence. That means stuff is coming in

into a singular point. On the middle, we've got a positive divergence. I think like the collapsing of a black hole. Exactly. Just for my own visual. 100%. On the middle, that's a supernova type. Like the positive divergence, stuff is moving out.

And on the right, we've got a zero divergence. Okay. Okay. The only types of vector fields that are allowed for fluids are the ones with zero divergence. Okay? Because that means there is neither a sink nor a source. You can't create fluids out of nowhere. So whatever fluid you have, it's got to come from somewhere and it's got to be going somewhere.

Okay? So this is a key constraint for our vector field. For fluids specifically. For fluid specifically. For other things, it's totally fine. Like, for example, if we were talking about the electric field, on the left would be a negative charge. In the middle would be a positive charge.

Fine. Fine. Totally fine. But for fluids, that can't be possible. So I think this is an important point here. the idea is this concept of divergence applies differently based on the underlying subject matter that you're speaking to. And so as we've gone through this progression of testing vector fields, it only can work when the divergence is zero. Is zero, yeah. Any non-zero divergence and it doesn't make sense.

Another way of putting this is something called incompressibility. You cannot compress the fluid. Okay. Those are the fluids that we're talking about. And if we go to the next slide, those are our two Naviore's. Stokes equations, you can already see that we've already got one of them. Okay. Okay.

The second one is that the divergence of this fluid velocity field is zero. Yeah. That's already the second Navier Stokes equation. Okay. So we're already, you would say, halfway through. And can you remind me again, we have our, I guess, delta symbol and our U symbol equals zero? Yeah.

That's the second one. The second one there. That's, you would say the divergence of you is zero. Right. which was that diagram where we saw the center point, everything moving in one direction. Stuff came in, stuff is going out,

and everything is fine. You're not creating or destroying fluid. Out of nowhere. Yeah. And for engineers, you might understand this with the continuity equation. Like, for example, this is applied very directly if you've got a pipe

that has a large cross-section and you've got water coming in and then the pipe like narrows into a small cross section. For our audio listeners, imagine a wine bottle on its side. Yes, exactly. A wine bottle on the side, but its base has been cut off. Yes. And then now you've got water or wine coming in from the base, right?

As it exits the nozzle or as it exits the tip, it's going to be moving faster because all of that fluid has been jammed in, right? You cannot create or destroy fluid. So the amount of fluid that's coming in has to equal the amount of fluid going out. But that means that if I've got a larger cross section, then in a single amount of time, I've got some amount of volume coming in, and with a smaller cross-section,

that same amount of volume has to go out, which means the velocity has to increase. Right, right? This is the continuity equation in action. And a mathematically succinct way of putting it is that del.u equals zero. The divergence of the vector field is always zero. And it kind of makes intuitive sense

for someone like me, which is when I look at the sort of the area of the cross-section and the wider part, there's just more space for stuff to move. and we have to move the same amount of volume through a smaller space. Yeah. So it necessarily needs to accelerate right at that point in order to, for the divergence of you to equal zero. To equal zero.

Yeah. For that fact to be true, you have to account for this difference in velocity. Right. Right. Now, crucially, this is an independent thing from that first Navier-Stokes equation, which is going to be the topic for the next segment. Okay. So this is an independent criteria.

Okay. If we didn't have this, you could have different types of fluids. You could have, for example, compressible fluids. Like air, for example, is kind of compressible in the sense that, like, if you heat it up, it's going to expand. And if you cool it down, it's going to contract. Now, that's not a divergence-free fluid. The Navier-Stokes equations, these two, have to do with fluids in the sense of, like, water that is really not compressible. The density is the same everywhere.

You can't adjust the density. So a key takeaway for, one of the key takeaways for this is, is the concept of this idea of incompressible. Yes. Which we've just defined from mathematical first principles. Yeah. But it really just means you can't, you know, you can't pack stuff in. Right.

The density is always constant. Right. Right. It's like one of those, have you seen those like stress thingies? Yeah, the little stress ball. Yeah. Where you like squeeze it and then the thing is going to like shoot out somewhere else. That's an incompressible fluid.

Right? Because if I squish it, it's got. to go somewhere else. Right. Okay. It's not only one of those foam things where you can just compress it. Right. Which is actually an important distinction.

Okay. So this is helpful because that's one half of what people say when they say the Navier-Stokes equations. And it's going to be the basis by which I'm guessing we're now going to build an understanding of the second. Of the second one, right? We've sort of gotten intuitive understanding of what types of velocity fields are allowed first of all, right?

And kind of what the velocity field represents, which is a bunch of these arrows that tells you how the velocity is at that moment in space and time. In a fluid. In a fluid, right? In an incompressible fluid. Yeah.

Okay, so now let's try to apply this velocity field to Newton's laws. Okay. Or the other way, actually. We want to apply Newton's laws to this velocity field. Now, that's easier said than done. The challenge now is how do we talk about Newton's laws,

which has to do with individual particles in the fluid, right? It's like, it's like this water molecule is getting forces from other water molecules, so it's moving around. Newton's laws has to do with the stuff. It doesn't have to do with the velocity field. So we have to now put Newton's laws in the language that we've just invented. And just to take a quick step back just for, for my brain, part of, like, it's like, you know, if you have a bathtub with a little boat, toy boat in it for a kid, right? We talked about Newton's laws in the context of like the boat in the water. Yeah.

And then when we talked about velocity fields, it's like the medium of the water. Yeah, it's the water itself. It's the water itself. And these are like different. So it's like the discrete object versus the medium in which an object or many objects may exist. And the lens in which we look at them is from different doorways. Yes. In the first one, it could easily, I could easily just be like, what is the position of the boat?

Oh, it's like right there. Right. What is the position of the water? What do you mean? Which water? Yes. There's an Avogadro's number of water molecules that we're talking about. So instead we have to reinvent this new language of the velocity field.

And now what we're going to do is build this step by step. We're going to apply Newton's second law, which is force equals mass times acceleration. We're going to apply that to our velocity field. The first thing we're going to do is try and describe what is the acceleration of a fluid particle, like in this velocity field. Okay. Okay. Okay. And we're going to use traffic as an analogy.

If you live in L.A., you very much are intuitively familiar. Yes, and you're probably listening to this while you're in traffic. Okay. So traffic is kind of a nice analogy for fluids because you've got these working agents in traffic, i.e. the cars, right? The cars can be the individual fluid particles. and the highway is like the receptacle, and the entire flow of traffic is the fluid.

Okay? The challenge now is that we want to have a description for how the individual cars accelerate, or de-accelerate, if they break or if they press on the pedal, based on the stuff that we can see from a news helicopter. Because the news helicopter is up there, and it's seeing the velocity field of these cars, right? It's seeing like, oh, over there, cars are moving fast,

Over here, the cars are moving slower. And, like, that's how they, like, if you're on the radio, they tell you, oh, the 101 is backed up from Sepulveda, blah, blah, blah, blah, blah. Right? So they're actually watching the cars, but they're watching an aggregate mass of cars. It's not like they're following a single car and seeing how stuff is happening. So what we want to do is describe the acceleration of a single car based on what we see from a chopper, right? Now, first, let's think about how cars can accelerate.

Cars can accelerate and de-accelerate as a response to the traffic itself changing at a specific spot. For example, suppose you're in the chopper and you notice that everyone is moving at like 80 miles an hour, right? It's a nice day and everyone's moving at 80 miles an hour. And all of a sudden, it starts raining. I was going to say 80 miles an hour in L.A. never happens. Yeah, never happens. This is an unrealistic example. Yeah, yeah, that's totally fair.

Anyways, that's what I'm stuck with in the visuals. So we got 80 miles an hour, and then all of a sudden, it starts raining. Okay? And within 10 seconds, everyone slows down to 60 miles an hour. Obviously, the cars have de-accelerated, right? They've used their brakes to slow down. This 20-mile-per-hour reduction happened in 10 seconds, let's say. And so the car is accelerated, de-accelerated at negative 2 miles per hour per second.

They're slowing as a response. to what the traffic itself is doing. The entire traffic slowed down. And so we slowed down. It's kind of like seeing like, you know, in the traffic map of LA, everything was green, and then rain started happening.

Everything turned yellow. Well, that means all of the individual cars also de-accelerated. So that's one way that you can de-accelerate. And this is called the Eulerian perspective because it's asking how the field itself is changing as a function of time at a given spot.

And this is represented by the derivative of the velocity field with respect to time. This makes sense, right? It's a velocity field. If I take the derivative of velocity, I get acceleration. And there you go. This is an acceleration of my particles, right? In this case, the thing is slowing down, but you can easily imagine after the rain ends, the things go back to 80 miles per hour and it's going to speed up again.

This is represented by a derivative of my velocity field. And so the idea is here's a way to describe the velocity field as this system unique and different from the discrete individual car in traffic. Yeah, yeah. And we can use the description of the velocity field and how the entire field is changing in time to describe how an individual car is experiencing acceleration. That's the key, right? So good. We've got at least some of acceleration down, but that can't be the whole story. Okay.

That cannot be the whole story. For example, let's take a look at this airfield. foil. This is a airplane wing where the air is coming in from the left and it's going over the wing and under the wing and then to the right. Okay. Now crucially, on the lower right, you're seeing the velocity field representation of this airplane wing. Okay. So the arrows above the wing, the velocity is very high because the air is like getting pushed up and then there's high pressure so it's like pushing out. These are the red arrows. Yeah, those are the red arrows above the wing. So there's higher pressure above the wing, lower pressure below the wing. And so there's

slower fluid below the wing. And just the length of these arrows is meaningful here. Yeah. In that there's more. Yeah. It's faster. Faster. Yeah. And okay, I'm already seeing where this is going. Okay. So, so this is the, on the left is your particles that are moving. Right. Like a flow of traffic. Yeah, our flow of traffic. And on the right hand side is the velocity field. But if I to take a derivative of that velocity field. If I were to ask, how is this velocity field changing with respect to time? The answer would be zero, right? Because this is a static velocity field.

If you go back to that picture that I had with the tornado, even though the particles were circling around, the velocity field was static because in this point, the velocity is this way. And then as it goes around, it's like coming around, right? And so what? Does that mean that the particles are not moving around? Another way of putting it with the chopper idea is like, you know, you have, let's say, an accident over here, okay, on the 101. Way ahead of the accident, the speed is very high.

But as you approach the accident, it's very low, right? But the accident isn't going anywhere because the police are very far away. It just happened. And so the police haven't gotten there. So the accident is not going anywhere, which means that ahead of the accident, the speed is very high. Before the accident, the speed is very low. And that's not changing. The traffic pattern is not changing.

It'll still be red here, green here. If I come back five minutes later, because of how slow the response is, it'll still be red here and green here. So the traffic pattern has not changed in time. And yet, I think you would be remiss to say, yeah, the cars are excessive. Once you get past the accident and you're done rubbernecking, yeah, you're like, okay, I got to get all in my life. Like, let's go. Right. Okay.

And so I just want to make sure that I'm following this year because part of what we're trying to say is we're trying to go from the velocity field and then say, can we describe an individual object within that velocity field? And how it accelerates based on the mapping we see in the velocity field. Yes, exactly. And part of what you're pointing out here is there are. are contexts like the airfoil or the traffic accident example you just described where the velocity field equals zero

and so that means you can't the derivative of the velocity field is not changing sorry right so the velocity field is static yeah it's not changing which means you can in some cases you may not be able to accurately describe certain

particles within that context because yeah using just the time derivative of the velocity field right right you have to figure out something else about the velocity field is doing this. And let's just think about this. I think we've got in the next one, you know, the highway slowdown. Let's say there's a, let's say there's,

and the next one, exactly. So let's say there's roadwork ahead. Okay. Right. A car is going to slow down. Now, all of the cars near the roadwork are going to be going slow, and the cars before the roadwork are going to be going fast. The roadwork doesn't change where it is.

But what is making the car slow down? It's the fact that it is approaching the roadwork. work. There is a spatial difference in the velocity field. You see? There's the time difference that we talked about earlier, where the rain came in, so everyone slowed down. People slowed down here if there's only a patch of rain here, and everyone else is fine, but this patch of rain, people slowed down. Right? So that's a single spot. Stuff is changing. But here, I am accelerating or de-accelerating because I'm moving through the traffic pattern, right? If I go from a red to a green, I'm going to accelerate in my map.

If I go from a green to a red, I'm going to de-accelerate, even though where the green and red are are staying exactly the same. The fact that I'm moving through it is making me accelerate or de-accelerate. And so I think part of what you're trying to get at here is that the time derivative of the velocity field is not sufficient to describe the acceleration of an individual object. Yeah, it's part of the story. It's part of this, but it seems like the distance is potentially the other ingredients in the pie, at least based on the example that you just brought.

Exactly, exactly, yeah. And that's a big part of it. It's the spatial difference. It's how the velocity field changes as I move from one space to another. Now, in that case, what I showed you is like, okay, you might think, well, what if, you know, I took a time derivative before. What if I take a spatial derivative of the velocity field? So that tells me how much the velocity is changing from one spot to another, right? If I move here and then I move like 100 feet over there, if I've slowed down, that tells me how much I should slow down.

That's not everything. Okay. Okay. Okay. Okay. Okay. And to understand that, let's look at that scenario again, where you've got roadwork ahead. Okay.

Okay. But now I've got two different cars that are approaching. The first car approaches that roadwork at 100 meters per second. Okay. Okay. and the second car approaches that roadwork at exactly half the speed, 50 meters per second. I think you'll agree that the guy that is moving faster is going to have to break harder. Yeah.

Right? Because he's going to come upon that roadwork way faster than the guy who's slowing. Right? He's going to have to be like, oh, there's roadwork. You know, fines are doubled. I better slow down a lot. So there's a secondary effect. It has to do not just with how fast the track.

graphic pattern is changing, but also how fast the individual car is moving. Yes? Yes. Because, again, trying to use the terminology again, we talked about the time derivative matters. We talked about the spatial matters. And also, the individual objects accelerate. Velocity. The speed itself is the third item in our.

cookie recipe here. Like all of these three things are relevant in order to be able to go from looking at a velocity field and describing an individual object within that velocity field's acceleration. Acceleration, exactly. And so for the second scenario, we have to take into account both the velocity and we have to take into account the change in velocity. Right. Okay?

The change in velocity as a function of distance. is like how far away is the road work? Right. And my own velocity as I approach that also has something to do with how fast and I need to break. Yes. So it's going to be a multiplication of the velocity field times the spatial derivative.

That's the upside down triangle. It's called Nambla. I call it Dell. But it's U.Dell times you. Okay? This is called the convective term. And what this convective term is doing now, it's giving us in the example

what we just talked about here, it's giving us the 100 meters per second, 80 meters per second. Yeah, that's the U. Right. That, okay, right. Multiplied by the del U, which is the spatial, how fast, how much time do I have before the roadwork gets there. I mean, how much distance do I have before the roadwork gets there, right? How much time do I have is dependent on how fast I was getting there. Yeah, yeah, yeah, yeah, yeah. Okay. And that's the okay. Makes sense? Yes. Okay. And both of these terms combined give us our acceleration. Mm-hmm.

We've got the oil-airn perspective, which is how is the traffic pattern itself changing? For example, is it going from green to red if rain came down or something or if there was an accident, right? Then at a specific spot, that highway would go from green to red on our Google Maps. And that's the first term. That's our velocity, the derivative of the velocity fields. Yeah, with respect to time, right? Because the velocity, the field itself is changing. The time derivative.

The color is changing on the Google Maps. Right. Right? On the right hand side, the Google Maps color is staying the same, but I am moving through that portion of the yellow or the red. Yeah. And so I'm going to be breaking or accelerating based on that. And effectively a more granular way. There's like a bird's eye and then there's a zoom. Yes. And both of those are matter. And that's why it's called actually the Eulerian perspective, because that's kind of the birds eye view. And then the Lagrangian perspective is like the nitty gritty, what's happening there. And that's the conceptual trick. Okay. Okay.

And in the Navier-Stokes equations, we're writing Newton's laws for the person in the car, for the car itself, based on the data that's collected by the chopper. Ah, that's an interesting way to put it. The chopper is only finding you. The chopper only knows the green and the red and everything like that. It's seeing these aggregate traffic patterns. But from those aggregate traffic patterns, I could tell you, if a car is over there, how fast is it going to have to accelerate or de-accelerating? Does that make sense? Yeah.

We're trying to, again, describe the acceleration or deceleration of an individual thing in an incompressible fluid from a vantage point that is only the vector field. Yes. Which the point being the vector field in and of itself does not have enough to describe the acceleration of the individual object. Until you do all of this, man. until you do all of this. Yeah. Is that the right way to say it back? I think that is the right way to say it.

Okay. Exactly. And so now we have an acceleration. Right. Right. And all we need for that other part of Newton's laws, right, is F equals M.A. We already have the A, the acceleration. Yeah.

What is the mass? Well, the mass is just, for a fluid, it's like the density. Yeah. Right? It's just, I could do a F equals M.A. Per volume. And then just say, okay, density multiplied by acceleration. That's my, that part of Newton's laws.

That's my MA. Yeah. That's my M. And that's exactly what we get. We've got the left-hand side of Navier-Stokes equations. Okay. The first Navier-Stokes equation. Right.

Because density, multiplied by acceleration. Right. Right. Because our acceleration is everything in this larger bracket, which is what we just described. Yeah. Which is trying to describe the acceleration of an individual object from the helicopter. Exactly.

Okay. Got it. So now, again, we're trying to go, we're trying to use Newton's laws as a language. to describe stuff that's happening in an incompressible fluid. And we've now have half of Newton's force times mass equals acceleration for an incompressible fluid with the description that we've just gone through. Exactly.

And now I want to, before we even start talking about the other side of that equation, which is the force, I want to linger on this acceleration term a little bit more, especially that convective term. Because that convective term, you might have heard that the Navier-Stokes equations are non-linear. Yes. Right, compared to linear differential equations. The Navier-Stokes equation is a non-linear differential equation. That second term, the convective term there, has everything to do with that.

Okay, so let's linger on that for a little bit longer, because I want to show you exactly what all the trouble is. And just to reiterate here, the convective term that we have on the right was when we were talking about the car in the yellow or in the red moving from the 100 meters per second to the 80 meters per second, describing the acceleration in this slice of the larger system. Yeah, and exactly. And crucially, it had to deal with its own velocity and the spatial difference in the velocity, right? That's why it's nonlinear, because there's two. There's a compounding effect that is happening.

Okay. Let's take a step back and let's look at, for example, Maxwell's equations in a vacuum. Okay. These are the equations of electricity and magnetism. They're described, for example, Gauss's law for electric features. is that the divergence of the electric field is equal to how much charge there is. You know how before I was selling you that fluids are divergence less? The divergence is zero.

But I showed you those two examples where the vector field was going out or the vector field was coming in. And that's because you can have negative charge and positive charge, right? That's Gouse's law for electricity, okay? It just says that dell.e, the divergence of the electric field equals to some charge. Now, crucially, the divergence of stuff is a linear operation. meaning if I've got one electric field and another electric field and I want to find the divergence of the two of the electric fields put together, I can just find the divergence of one and find the divergence of the other and I can add them up separately. So I've got, if I want to combine two electric fields, I just do the math for one of them. I do the math for the other one and I combine them and that's totally fine.

This is why, for example, if I've got two charges, a negative charge and a positive charge, and I want to find the electric field everywhere, all I do is find. find the electric field due to one, the electric field due to the other, and I add it up. Okay? This is also why telecommunications works with electric fields, right? Like in this room, we've got Wi-Fi, we've got Bluetooth, we've got radio, we've got our phones, we also have the light from the studio, right? All of these are electric fields, but they're all just moving through one another. They're like ghosts that are just like moving through one another.

And if I can tune my radio to one thing or the other, I can listen to one station or the other. Or I can, like, my Bluetooth isn't randomly, like, interfering with the Wi-Fi. Like, the Wi-Fi is coming from the router. There's a Bluetooth that's probably in my phone. But it's not like the Bluetooth signal and the Wi-Fi signal is, like, crashing, right? Yeah, they don't interfere with each other. Yeah, yeah. Even, like, in a radio, right, in the car, like, if I'm tuned to 91.5 K-U-S-C classical radio,

the car is being inundated with all of the radio stations. There's also 89.9.9 coming at me and Kiss FM. And also the AM, you know, talk radio, Rush Limbaugh. We need our check. Kiss, FM. Yeah, these are just some of my favorites that I listen to in LA. But, like, my point is the car is being inundated with all of these radio stations. But my radio can tune in into a specific band.

And it's not like the Kiss FM radio signal. is interfering with the K-USC signal, is interfering with Rush Limbaugh. He's not... I think he's still on radio. Yeah, he's still on radio? I think he's still on... He might be on YouTube now.

Yeah, yeah, yeah, yeah. But, or a satellite. But I think... So part of what you're saying is, like, I don't have to account for the other radio stations. Exactly. It doesn't matter what the other radio...

And if I want to account for, all I do is add it. Right? If I want to know, what is the electric field here, all I do is add up all of the contributions. It's totally fine. Mm-hmm.

In fluids, we do not have that luxury. Okay. Okay. And that is because of that convective term. Mm-hmm. That convective term is non-linear. And this is what it does if I add up two velocity fields that are separate. Okay.

Suppose, you know, and before I was adding up E1 and E2, the electric fields. Yes. It's totally fine. Fine. Suppose I have two velocity fields, right? I got two hoses and I'm like crossing streams or something. Yes. U-1 plus U-2.

Where that's, I shouldn't, you know what? Whatever. Like two things of water. And I add up, right? Like U-1 and U-2. Mm-hmm. Suppose I go through the math with those. If it was linear, I would only get the terms in the green, right?

Because I could just do the math for 1, U1, and I could do the math for U2, and I'd be fine. You do the divergence for U1. Yeah, divergence for U2 multiplied by U2 and U1, and that'd be it. But because it's U multiplied by the divergence of itself, right? Or it's multiplied by the gradient, I should say, of itself, not divergent because divergence has a dot. The point is I'm taking the velocity field itself and I'm multiplying by the derivative of itself. Because I'm doing that, I have to do the distributive property.

You have to do it twice. Yeah, I have to do it twice. And so I'm going to get these cross terms on the right in the red circle. And those cross terms are what make this whole thing nonlinear. And that gives us all of the richness of fluids. This is why fluids don't just go through one another. They crash. There's rapids, right?

There's, when, when two rivers meet together, they mix and they, they have turbulent flow. You know, when I was in India last time, we visited Dave Preyag in the Himalayas, which is the birthplace of the Ganga, where like, you have like, Bhagirati coming from one side and Alakana and are coming from the other side. There are two rivers from the Himalayas that meet in this very holy place in the Himalayas that begins the Ganges River. And you could, you could see the whirlpools forming, like, right at that spot. It was amazing. And the reason for the world pools and the reason for all of the chaos in the birthplace of the gunga is because of those two terms on the right end side. You know what I mean? Yeah.

And so now you can already see that there is trouble with this equation. Right. And I just want to take a quick step back to make sure I'm internalizing what we're saying here. When we talked about Maxwell's equations and we have a positive and a negative change. charge and we can just add them up. It's because part of what we're saying is it's because we don't have that extra level of complexity when we talked about the convective term that's true for fluids. Yeah, yeah.

In the Maxwell's equation, it's just del. It's a single E. Right. Here, there's two versions of the vector field. There's the vector field itself multiplied by the derivative of the vector field. Right. And can we just briefly try to connect it back? I'm trying to connect it back to our yellow, red traffic example

just to make sure it lands in my brain. Because I'm having a hard time finding the leap from, I understood the convective term. We have this idea that the traffic is red in one part, yellow in one part, and we can look at it from the helicopter. And we can look at the whole vector field. Yeah.

And if a car is going from red to yellow, or from red to green, then it is going to accelerate, even though red and green have stayed put. And so from that perspective, would it be like the equivalent of saying if that was it and that was the only level of complexity, it would be similar to Maxwell's equal.

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Find out more at Cadillac Canada.ca. Luxury sales claim based on S&P Global Mobility Canadian New Vehicle Total Registrations for calendar year 2025 for the Cadillac definition of luxury. Yeah, if all I had to do was take a spatial derivative of red and green and be like, oh, there's red here and green here and they're 100 meters apart. Yes. So, you know, the difference in the velocity is divided by 100 meters. That tells me the spatial derivative of my.

velocity and I'm fine, that's fine. But crucially, it also has to do with how fast or how slow I was going into the thing. Yeah. Right. If I'm transitioning from red to green and I'm moving at a snail's pace of like five miles an hour and all of a sudden now I have to get up to highway speed, I got to floor it. Yes. But if in the red, I was already going at 30 and then now I got to get to 60, I don't have to floor it as hard. So there's two contributions. Right. And that's that's what's giving us this non-linearity. Yes. It's because the fact that there's two contributions. There's two contributions that we have to account for as we look at trying to do the mathematics for this. And that's where we're getting.

This is where we're getting having to do it. We have the green, but we have to have the addition of the red because of that second. Because of the fact that, yeah, there's two of them. Right. So I have to do these cross terms. Understood. Okay. When I combine velocity fields, when I combine, let's say, flow in this direction and a flow in this direction,

they don't just ghost past one another. Right. Because they're interacting. Yeah. Yep. Okay. Make sense? Yes, tracking.

Yeah. And so that's why the Navarre Stokes are so hard. That's one of the reasons. Okay. Okay. So now let's get back to our Navier Stokes equations. Finally, we can now start asking, what is the force that is acting on the fluid? So far, we've figured out how to describe the acceleration, and we've figured out how to describe the mass, right? Yes.

But in order to have an equation of motion, we have to describe the forces that call. cause this acceleration. And then if we write it out, then perhaps we can solve it. Right? In the same way that for the spring, we could solve mass times the acceleration. In that case, the acceleration was super easy.

It's just the second derivative of the position. And then the force was negative KX. I had an equation of motion that I could solve. In this case, already the acceleration is quite difficult. Right. But now, let's ask, what is the force? One obvious thing is pressure. pressure is force per unit area

you know higher pressure means that there's more sort of forces acting on acting on your container like you know if if you've got a if you've got a high-pressured gas inside of a container that gas is pushing out on the walls of the container compared to a low-pressure gas it's not pushing out as much just hanging out it's chill yeah exactly like when you suck on a straw

what you're really doing is creating low pressure in your mouth and then the high pressure of the atmosphere pushes down on the liquid and forces it up into your mouth. That's a difference in pressure that you're doing. So crucially, what matters here, what is the force on the fluid? It matters what the difference in pressure is,

not what the absolute pressure is, right? There's absolute high, like 100,000 pounds per square inch right here in this room. But I don't feel it because there's no pressure gradient. There's no difference in pressure. Right? So I don't feel a force. Right. Now, all of a sudden, if, you know, outside there was super low pressure for some reason,

and we had like a leaf blower in here, there would be high pressure in the room, low pressure outside, and we'd feel a wind that's going out. In weather systems, you see this a lot, right? High pressure, the wind goes away, goes away from the high pressure towards the low pressure. So one of the forces that we will have to worry about is the gradient in the pressure, how the pressure changes from one space to another. And it's negative because it's going from high to low. So the force is always from high to low.

Right, right. Okay. Because it's going from a high, it's always flowing from an environment of high pressure to an environment of low pressure. So you can always have the negative in front of it because it's always going to be. Yeah, it's always going to go from high to low pressure. And so this is a force, right? The whole point right now is we're trying to describe what are the forces on the fluid.

In this case, the forces on that velocity field are from negative to, from high pressure to low pressure, you're going to have velocities flowing, right? It's like if you're at a beach and you know those little videos where the surfers create the surfing thing from a beach into the ocean. Yeah. And it's like the water pressure is very high and built up at the top and then it wants to flow into the ocean. Yeah.

Which is a relative area of lower pressure. Yeah. In that context. In that context. Actually, you're the next one that I want to talk about is the external forces on my fluid. Okay. And in that case, that would be gravity. Okay.

Right? Because the lagoon or whatever is slightly above the ocean. That's a good point. Right. And so when you create that little tiny thing, gravity is going to force the fluid down that little canyon that you've built. So my example was actually, it was not perfectly apt because I was not yet accounting for gravity. Yes. And the next thing that we're going to do is account for external forces on the fluid.

Okay. Like gravity, right? So on the right hand side, we've got forces. Yes. the difference in pressure is one such force. Yes. That is going to, you know, like if I were to increase the pressure here and decrease the pressure here, and I started out with some velocity field, those velocities, those vectors are going to get bigger

because there's going to be more fluid going through and the fluid is going to be moving faster. Similarly, if I introduce gravity now, the velocity is going to get bigger. So that's a kind of force. It's causing an acceleration, right? And so here we've already got somewhat of a Navier-Stokes equation. Okay. Right? Okay.

We've got a familiar example is gravity, but honestly, that G could be a lot of things. That G could be the stirring force, right? Like if I put a spoon in a piece of, in a glass and I stir the thing, that's an external force that's acting on my fluid. So the force can be a lot of things. Understood. It's not necessarily just gravity in the context. We just talked about it.

Yeah, we just talked about gravity in that context. but I mean, forces on fluids can be a lot of things. It can be like the turbine in an airplane. That's putting a force on one, right? Because the thing is, it's stirring a fluid and making it shoot out in one direction. So that is about as far as one of our favorite mathematicians on this podcast, Leonard Euler got. And I think this is a good point to talk about some of the history behind the Navier-Stokes equation. Because that last equation that we had is one of the oiler equation.

of fluids. So just, and just to quickly kind of recap. We've started, we're trying to describe the acceleration of an individual entity within a vector field from only having the context of the vector field. And we are trying to describe it in the language of Newton's second law. force equals mass time, mass times acceleration. Yeah. And we've built up now for an incompressible fluid.

Yeah. All the way up to where Euler arrived in this journey of trying to do so. Yes. He didn't get there on his own, though. Oh, okay. Oh, okay. We said no drama till the end, but. And there is some drama here.

There's actually some really weird drama that is very pertinent to today's drama of OpenAI versus. all these mathematicians. So I thought it was kind of interesting to talk about. So Leonard Euler, you know, the great Leonard Euler, we've talked about him a lot. In the Riemann Zeta episode, we talked about how Leonard Euler invented the Zeta function that then Riemann later extended

to create the Riemann Zeta function and the Riemann hypothesis. It's a very similar story here. Euler created the Euler equations and then Navier and Stokes extended those equations to create the Navier-Stokes equations that we know today. And obviously,

you want to name things after the second person who invented everything because you can't name everything after Euler. But even Euler inventing that first equation, that wasn't on his own. Okay, there is a lot of shenanigans that happened in 1940s. 1940s, Berlin between

1740s. Sorry, yeah, 1740s Berlin between Leonard Euler. He was working at the Royal Prussian Academy of Scientists. This is the German Academy of Sciences. And if you go to Berlin, you can find a little plaque for where Leonard Euler used to live. So the German Academy of Sciences announces a prize problem about one of the great practical questions in the 1740s.

How much resistance does a body experience as it moves through air or water? Okay. It's a very pressing question because I think around the time is when cannons are being used in warfare. So it's like, you know, we want to know how far a canon can go. It's their version of the DOD creating, you know, a prize competition to try to understand how to do better warfare. A guy named Jean Leronde d'Alembert. Very good. For those who are longtime listeners, it's quite good.

I don't usually do this, but I like Dallembert. So he submits an essay. It's about fluids. he did not solve the resistance problem completely, but he helped introduce something that is arguably maybe a lot more important, which is this mathematical language of a velocity field to describe a fluid. He's the guy who invented that concept of you, of the velocity field. You know all this stuff that we've been talking about,

this new language of maybe let's not worry about all of the particles, let's worry about the velocity field of this thing. That was Dallembert. And, okay, so what's interesting is I want to make a, a quick zoom in note to note that the prize problem that he set out to answer was about how much resistance does a body experience when it moves through air or water. And I just want to be, that's the specific question he was trying to answer. In that journey, he did not answer that question.

He did not answer the question, but he invented. He gave us the language. He gave us the language for it, right? And his submission didn't win because the Academy of Science was like, oh, it's too theoretical. He also showed that the velocity field has to be divergenceless, meaning that del. dot u equals zero, like the fact that it's incompressible and things like that.

So he already established one of the two Navier-Stokes equations, Dall and Bear. He submits this thing. The academy decides that it's not going to give out the prize. No prize was awarded. But here's the thing. One of the people that was on the committee was Leonard Euler. So Euler saw an early submission of Dahlimbert's work. And he's reading this thing going, yo, this velocity field is a kind of nice idea.

He takes that idea and not long afterward, he presents his own general theory of incompressible fluid motion. And that's the equation that we see on the right hand side. So on the left hand side, Dahlambart has already invented the velocity field and he's shown that the velocity field has to have zero divergence, meaning it's incompressible. no sources or sinks. Leonard Euler takes that concept and says, well, I can actually make a Newton's law version of that. So very similar to Riemann Zeta

where the Basel problem existed before hand, right? The one plus one over four plus one over nine plus one over 16 so on and so forth. Leonard Oiler saw the Basel problem, solved it, and then presented the Zeta function. In this case, he saw Dallembert's

velocity field and he's like, wait, this is a good idea. And he just ran with it and created the derivatives, which are huge, like the derivative term, that acceleration is massive to create that, that material derivative. And then also say that, okay, the forces on it are going to be the gradient

and the pressure plus whatever external force. And just to say what you just spoke to, that was the convective term that we talked about earlier. Yeah, yeah. Yes. Okay. And this is so interesting because it's, it's always funny how,

these things. Basically, we have Dallembert, who started the, who created the vision to think about it this way. And then Oiler was like, let me take Newton's framework, Dallembert's vision, and combine these

two things. Yes. And he gets an equation of motion, right? He gets the first equation of motion for a fluid. This is for an invisid fluid, as they say. Because these are, as we'll find out, frictionless fluids. Okay. Now, Obviously, that can't be everything.

And Dahlimbert sees this. It's kind of like a mathematical twist that's so perfect, it's got to be scripted. Dahlumbart looks at the original competition. He's like, wait a minute. It was about drag. If I apply Newton's, I mean, sorry, if I apply Euler's equation to the concept of drag, and I ask, what is the drag on a body? I can exactly solve Euler's equations around, let's say, a ball moving through a fluid.

And there's no drag. Like in the next thing we'll see. This is called Dahlembert's Paradox. Okay? He uses Euler's equation that's up there and he applies it to a ball, a spear that is moving through fluid. According to Euler's equations, if you were to solve with all those boundary conditions, the streamlines just go up and then they come back. There's no difference in pressure from the front to the back. And so there's no drag.

But obviously, as we know, from our FIFA World Cup episode. where we talked about the drag on a soccer ball. There is a lot of drag on a soccer ball, right? Just to also note, this was Adidas's peer-reviewed study on their new balls for this year's World Cup. And they did incredible deep dive studies into the literal drag of how they designed. It's great episode. Definitely go watch it. Exactly.

And what we can see already is in real fluids, the streamlines aren't just meeting up right after. Yeah. Right? There's a lot. There's this like region where there's turbulence and all this weird stuff. that's happening. This chaos happening. Oilers equation does not account for that.

So basically what we're saying is oilers equations only accounts for what we see on the left, which is effectively a system that's not actually exists in real context. The military can't use it. Exactly. Yeah. And so 20 years after the whole competition and everything,

Dahlimbert says this is a paradox. This is not everything. There's got to be more. And this is the part that Navier and Stokes help us repair. Okay? They're the ones. that come in and they start accounting for viscosity. And so to now kind of quickly recap again, we are trying to build this understanding of what is the construction of the Navier-Stokes equation.

Yes. We understand Dalembert started with this idea that the divergence equals zero. Euler has come in and used the framework of Newton's second law of motion to try to apply it to fluids, maybe cheating off of Dallambert's test. Maybe a little bit. and added for the force side of the equation, pressure, and just an external force. And it's, and an external force. Gravity or stirring or whatever it might be.

Yeah. And he was like, I did it. Yeah. Look at me. I'm so brilliant. Yeah. However, when applied in practice to a ball like we just described, it does not actually describe real world systems.

And what's interesting is the original, it's funny because the original prize competition that all there was a judge on was about resistance and drag and then he came up with the solution without thinking about resistance and drag. Am I missing something? Exactly. Okay.

So what he ended up missing was another force that he hadn't accounted for. Remember, right now we have two forces. We've got the difference in pressure. Yes. And we've got just external forces, whatever they might be,

stirring, gravity, wherever. There is a third force on that side of the equation that is missing. And that has to do with viscosity. Okay. That's the resolution to Dahlimber's paradox. So in Euler's ideal fluid, there is no viscosity.

This is called an invisid flow. The fluids can slide past one another. These particles can just slide past one another, and they can slide along the surface of an object without any friction. And the flow divides very smoothly in front of the object, and then it rejoins at the end of the object. There's no wake, there's no net resistance, there's no draft. It's beautiful mathematics.

It's different than diverge. Like, it's different than the problem we were saying where water... Can't, like, form a... Incompressability. Right. Yeah, this is different. It's different. This is still incompressible, right?

Because the flow is coming in and then it's going out. It's just that they, when they interact with each other, it just rolls off. Yeah, they just roll off. Yeah. Yeah. They're just rolling off one another. It's not especially useful. Okay.

We haven't taken into account friction. And we have to take into account something called, viscosity. Viscosity is simply the measure of how thick, sticky, and resistant to flowing a liquid is. So we're going from thick with two Cs on the left here to thick with three Cs on the right. Yes. Okay. Yeah. Vegetable oil, not that thick, right? You drop something, it immediately goes through. Motor oil, also not that thick, but slightly a little bit thicker. Honey, on the other hand, you drop something and it takes forever to get all the way down. because there's so much resistance to flow in honey.

The things as they try to move past each other make it harder. Exactly, yeah. And so for low viscosity, like thin liquids like oils, the particles are moving past each other with very little friction. But for high viscosity, there's a lot of resistance to that movement. And therein lies the clue in how we can mathematically describe this. Because we need to have a mathematical language to describe what is viscosity. So let's go back to our traffic analogy.

Okay. Okay. Let's consider a multi-lained highway like this. So we've got a shoulder that's zero miles per hour, obviously, because you're on the shoulder. You better be stationary. You don't want to be one of those guys that, like, tries to cut in line and then the cops catch him. So let's look at what exchange of traffic from one lane to another will do. Because in normal traffic, obviously, cars are going to change lanes.

Yes. So I've color-coded these cars based on their lane. and based on their velocity. Sorry, and what I'm realizing earlier, you specifically showed us only one lane examples. Yes. Which was fine at the time. At the time.

But now we've got multiple lanes, and now the flow is going to crisscross. In between, which is how real liquids. Okay. How real liquids do it, right? Now, what I've done is in the initial, let's say it's a four-lane highway. The lane that's fastest is going at like 80 miles an hour.

and the lane that's slowest is going at 20 miles an hour. So it goes 80, then 60, then 40, then 20, and then you've got the shoulder. Four lanes, and it's in increments of 20 miles per hour. And then things start shifting around. Okay? You actually wouldn't notice a change in the flow. Because, let's take, for example, the lane that's moving at 60 miles an hour. It's going to inherit some faster cars from the 80 miles per hour lane,

and it's going to inherit some slower. cars from the 40 miles per hour lane and the things are going to average out. Right? And so if I were to look at a graph, and that's on the bottom there, if we look at a graph of lane on the X axis and the speed on the Y axis, it's a line because it goes from zero, 20, 40, 60, 80, and later on it's still going to be a line. There's been no change, meaning that the velocity field has not changed, right? It's still big up top, small down below.

And the velocity field has not changed from one time to another, meaning that the velocity field has not changed from one time to another, meaning there's no force here. Okay? And even though there's some viscosity, there's no force. Right. Okay. Yes. Right. Yes. There's still like lane changing happening, but there's no force. There's no change

in the flow of traffic. Now let's consider a different type of profile. A type of profile where the shoulder is again zero. And I've got the two slowest lanes both at 20 miles per hour. And then I've got a lane at 30 miles per hour.

but the faster lane is still going at 80. Imagine that. Like the carpool lane is still going at 80, and then the rest is still very, very slow. And the carpool lane has like a barrier. So there's nothing happening. And then all of a sudden the barrier gets lifted. What's going to happen?

Well, the carpool lane is going to slow down a lot, but also the other lanes are going to speed up because stuff from the carpool lane is going to come in and, you know, sort of push that lane forward. This is a bumper car scenario, okay? Not real. But you get what I'm saying. Like before we had this curvy profile,

which we see on the lower left, where again, we don't have a line because it's slow, slow, slow, then really fast. Yes. And with subsequent time, it's going to level out. It's going to level out. Because of the diffusion of cars. Here, there is a force, right?

Because I've changed the velocity profile. If I change the velocity profile, that's an acceleration. Yeah. Which means there's a force lurking in here. Right? There's a viscous force that is lurking in here. And this is what's interesting is just going back to our previous example, because basically

the system was evenly distributed in the initial state, based on how the variables would end up mixing, you get, he had no net effect, no force. No force. In our second example here, it was sort of a not evenly distributed system. system. Yeah. And so when you look over time and then there's the mixing. Yeah, there's a mixing. There's going to be a change. There's some divergent. Not divert. Let me not use the word divergence. There's some change. There's a change over time. Which means that there's a force. And notice, in the first example, there was a straight line, which means I had a positive derivative, right? Like,

the slope is positive, but the second derivative is zero. There's no curvature. In the second version, there's a curvature, right? Which is why when I inherit the fast stuff and I inherit the slope, stuff, the fast stuff is way faster than the slow stuff, which is why I'm speeding up. The curvature is the origin of the force. That's the key. Yes. And so that is the form that the viscosity is going to take mathematically. And if we go, in the next slide, we'll actually see what the viscosity term looks like. It looks like del square. Del squared means a second derivative in some sense with respect to space. And there's your viscosity. The mu, the Greek letter there. That's telling you how viscous it is. Like honey would have a very high value. Water

would have a very low value. And that's the force. The force is, again, has to do with the velocity field itself. This is always key. We want to describe everything except for maybe the external force as a property of the velocity field itself. Of the whatever the system, as I use in a colloquial sense, as a whole. Yeah. Is a material idea here, which is why. we have all these are all derivatives, we're looking at the system level. Yes, exactly. And like, it kind of reminds me of like,

when we were talking about the springs, right? The force on the spring had to do with the position. In this case, the force on the spring has to do with the velocity field itself. Right? Yes. And on the right hand side, just for those who are engineers and physicists,

I want to show you a similarity between the viscosity term and the heat equation, where if you have the heat equation and you, if you have a temperature profile, As time moves forward, everything gets smoothed out because of thermal diffusion. The same thing is happening with viscosity. Viscosity is smoothing out the velocity differences, right? And that's what we saw was happening with the traffic analogy.

There was a lot of high speed here and low speed here, and viscosity was smoothing it out and making the flow a bit smoother. Right? I'm starting to see the connection between the words, incompressible and smoothness that I keep seeing as it relates to the whole navigation. Navier-Stokes piece. Yeah, we're starting to see that. So the viscous term becomes a smoothing operator in some sense, right?

And now, which feels very natural as a property of trying to measure a liquid, because in this context, it's sort of this contiguous thingy when you look at it from the vector field perspective. Yeah, exactly. And we don't want that contiguous thingy to have like little pockets of really high fluid, or sorry, really high velocity. because if there's a little pocket of really high velocity, well, that velocity is going to spread out. Right. Right. And that's what viscosity does. Right.

Right. That's why in honey, it's really hard to have a small pocket of really high-speed honey. On the other hand, for water, it's maybe a little bit easier to have a pocket of high-speed water. But then it just depends on what time...

Yeah. Anyway. Yeah. Yeah. Exactly. And so now we can solve Dallembert's paradox because now we have viscosity in mind. So we can apply this thing called the no slip condition.

This is something that we've looked at with hypersonics and we've looked at in the FIFA episode where if you've got an object that is moving through a fluid, very close to the object, the fluid is not going to be moving at all. It's going to be stuck to the object. And there's going to be a boundary layer where it's going to transition

between being stuck to the object and moving with the rest of the fluid. And that boundary layer causes the drag because the object is dragging the fluid with it, right? And this is how we solve Dahlumbert's paradox because now if we look at, for example, a golf ball that is moving in air, the fluid is sticking to the golf ball.

And so it's creating a wake behind it because the golf ball is moving through the fluid. And as the fluid sticks to it, the part that's right above the golf ball is going to be moving with that sticky layer, it's also going to be moving kind of with that. And so you get all of these dynamics happening because of viscosity. There's this transitionary space between the surface of the object that is moving through the fluid

and then the rest of space. Yeah. And in proximity to the surface of the object, there is a transition where the fluid around the object is going to move on some gradient from sticking to the object to being a part of the larger system. Exactly. Yeah, exactly. And so that's how we get drag. And so this is the resolution. And this is the contribution that Navier and Stokes were trying to capture. Claude Luis Navier in 1822, he presented an equation on viscous fluid motion where he added that viscous term.

But he was an engineer as much as a mathematician. And he tried to explain the internal friction by imagining molecular interactions inside of fluid. This was 1822. So everyone thought he was crazy because there was no such thing as atoms back then. I mean, there was this Greek concept of an atom by democratized or something, but, you know, no one took that seriously. And it's one of these rare occasions where a good outcome came from a derivation that's based on a questionable premise. Now we know it's not questionable, obviously. And over the following decades, we had Koshi, Poisson, and others developed the continuum theory of stress and how stress affects bodies. and in 1845, Stokes, George Gabriel Stokes, derived the viscous equations from a completely different direction

from this idea of stress and how you can deform objects. Turns out it's the same form as Navier. And so now that's why we're called the Navier-Stokes equations. They never collaborated, by the way. Stokes was a teenager when Navier died. But their names are joined because, you know, different physical arguments, but they arrived at these two equations. And now these are the two equations that have been such a headache for mathematicians and physicists all these years.

And so as I look at now these two equations, the one at the bottom, which was where we started, is this idea that the divergence always has to be zero for an incompressible fluid. Yeah. You can't make fluid and you can't destroy fluid. You can't have a fluid emanate from a point in all directions. And you can't have a fluid all converge on a point from all directions. Yeah. That's what the bottom one is, effectively saying. Yep. And then now the top one, which was the more complex one, is we're trying to replicate

force on the right equals mass times acceleration on the left. Yes. And in this construction, which was Newton's second law of motion. We're trying to apply to incompressible fluids. Yes. And in doing so, we constructed the left side of the equation first, where we have our mass represented by our volume. Yeah, that's the density.

The density, excuse me. And then we then multiply that by everything in the larger brackets, which is our acceleration, which we have the first term being the time derivative of the velocity field. Yeah. That's like rain coming in and changing traffic patterns. But we also have to add into that now the convective term, which is how an individual car within that vector field

is changing its velocity with distance. Yes. Over time. Yeah. And so that's the left side. Yeah. And that gives us on mass times acceleration. Yes.

Loosely speaking. Maybe there's a few details not quite right. And then on our right, so there's sort of three terms that have complexity on our mass times acceleration on the left. It's really two terms, I would say. Okay, so because it's, yeah, sorry, fair. Within acceleration, it has two component parts. Well, no, I'm nitpicking here, but like, it's really two terms because it's density

multiplied by DUDT. Okay. And then it's density multiplied by that second term. Totally fair. Totally fair. Like, density is not its own thing. I totally, yeah. That's a fair.

Whatever. No, that's an important distinction and that that's helpful. And then on our right side where we're trying to define force, we started with the obvious thing, which was pressure. Yeah. And it's negative because high pressure always moves. to low pressure. Then we moved to,

we added our F, some general force, some general force. Yeah, stirring, stirring gravity, whatever you wanted to call it. And that's where Euler ended up. Yeah, yeah. And he stopped.

And we were missing this concept of drag or resistance. And ultimately, both Navier and Stokes from different directions ended up on being able to define viscosity as the missing piece, which is again talking about, there's a boundary layer between the object in the system and then the rest of the medium external environment, and there's some gradient delta there of change.

And so we need to account for the viscosity because that applies a force as we look at all of this. Because that now, like, an accurate description of what we're looking at. That is the Navier-Stokes equation. And we have just built it up from first principles, you know? As someone who did not do any type of mathematics, it's a testament to your way of taking this story from start to finish.

Yeah. And it kind of makes sense, though, right? It makes total sense. Like, when you understand the underlying systems. Yeah. I think the hardest part for, I think the hardest part really is getting familiar with the idea of a velocity field. Yeah.

Yeah. You know? Yeah. Because once you have that and you've internalized, oh, it's like a traffic pattern, then all of the other stuff kind of falls together because you can like kind of, you know, figure it out. That was the oilerian perspective was the velocity field. Yeah.

That's the way we described it earlier. Yeah. Very nice. Dallembert, dude. Dallembert. Yeah. And so now we can start asking, at this point, we've got the Nevers Stokes equations. Yes.

We know what every term means. Yes. So now we can start tackling how to solve them. Right. And because part of what we also define through all of this is this is an extremely complex problem. Yes. For all of the nuances of each of the underlying terms, which also have their own sub. issues, subcategory issues.

Exactly. And so we've got an equation of motion. Now can we solve it? Like, can we find a general, like, I give you a velocity field now. You tell me what the velocity field is later, right? What is the flow going to look like later? And to just come back to the beginning of the episode, the reason we want to do this is because with Newton's equations that we're talking about, part of it's like, then you can use it to make predictions.

Yeah, exactly. Yeah. And that's how we do the space stuff and blah, blah, blah. So what we're trying to do is take this equation of motion so that we can, like, who cares? Yeah. And the answer is so we can make predictions. We can give you one input, the velocity field, and then you can give me the output consistently that's true every time.

Yeah, that'd be dope. That'd be great. Yeah. Okay, so we have a set of differential equations, and we'd like to have solutions to those differential equations. Now, what do we mean by that? Like, in a normal algebra problem, if I write something like X plus 2 equals 5, I know what x is x is three right i can like do the thing in a differential equation what we're asking is something

larger what is the function that solves the differential equation in our case it would be what is the velocity field that solves this differential equation let's take a simple example for let's take for example um the derivative of y in terms of t is equal to sum r times y meaning um i've got some quantity the change of that quantity with respect to time is proportional to the quantity itself multiplied by it, let's say some rate, that is an exponential, right? The solution to that is E to the power of RT, and this is how people get rich. This is compound interest, right? You start with some amount of money, and then the rate tells you the interest over time, and the more money you have, the faster your money grows. And this is a very simple differential

equation that we can solve very accurately, right? This is a toy equation, and the answer is very nice. Navier Stokes is less accommodating. The unknown is not how much money you have, like a single one-dimensional variable. It's a three-dimensional velocity field, right? So it's a number at every point in 3D space, but that number has a direction, right?

It's a length, and it's got a direction. that's also three-dimensional. So it's quite massive, like this equation. And it's so funny, because now that I understand that it makes my brain hurt trying to say, we want to solve for this now, given that level of complexity, which is now very clear what that level of complexity is. Exactly. Like the fluid is interacting with itself and everything.

So this velocity field has to solve for that entire big thing, right? We've got to find all of the arrows everywhere. Right. in some sense. Yes. Right? For all time. Because the arrows might be changing over time.

Over time. And we need to have an expression for all of that. Now, there are certain special situations in which the Navier-Stokes is actually exactly solvable. Okay? It's not true that the Navier-Stokes is just unsolvable. You can't solve this equation. There are situations. There are certain boundary conditions where you can exactly solve the Navier-Stokes equations.

Let's go through one of them. It's called plain co-et. Coette flow. I never know how to pronounce it. I think it might be coet. Is it coete? Let us know in the comments. Yeah.

In any case, it looks French. Yeah, it's probably French, which is why I think it's coete. Here's what's happening. You've got two plates. Okay, those are the Navier-Stokes equations that we're trying to solve. Yes. I've got two plates. One plate, let's say, is the table completely stationary.

Another infinite plate is moving in the X direction, in just one direction at a certain velocity. And in between two plates, you've got water. You've got some kind of fluid. Okay? What is the velocity profile of the water in between these two plates? It's very simple, right? There's a stationary plate. There's a plate on top that is moving at a fixed velocity in one direction. And there's fluid in the middle. So what is the velocity field going through? Now we know from the no slip condition that the fluid that's right next to the table, that stationary is going to be stationary. and the fluid that's on the plate that's moving is going to be moving with the plate.

What happens in the middle? It's actually pretty simple. You solve for the Navaristokes equations and you get this profile. Right? Near the bottom where it's stationary, the velocity is zero. And as you approach the plate in the top,

the velocity gradually increases linearly. And so it's just a function of the height, right? The higher you are, the closer you are to the top plate, the closer you are to the velocity. It's a linear relationship. Boom. That makes totally solvable. That makes total sense.

And again, we've sort of created an exactly solvable, simple use case because, you know, by having the bottom plate be stationary. Yeah. Having the bottom, even if the bottom plate is actually moving in the other direction, it's totally fine. As long as it's the same, like, you know, the same velocity. And there's so much symmetry here. They're only moving it on one plane. Exactly. It's only one coordinate that we have to worry about.

So a bunch of the derivatives over there, just go to zero. Like there's so much symmetry and geometry here that it murders all of the difficult terms of that equation. And you've got this steady profile and you're good to go. It's a recurring theme in fluid mechanics, which is if you've got a nice symmetric distribution of stuff, you can solve things. Fine. You're fine. Yeah. This is called coete flow.

You can also do pipe flows. Like there's a handful of vortices and jets where you can obtain beautiful closed form solutions. based on the symmetry of the situation. And just to ask us a brief side question, part of the other part of why those are so solvable is because we have such a good understanding of geometry as like these other mathematical disciplines, we can apply some of those learnings and concepts

in a way that make these easy, or it's not relevant at all. Well, no, certainly. I think, I mean, for that one, it's like pretty easy because you can just take derivatives and they go to zero and whatever, right? But there's other ones, like there's certain vortices and jets where you can manipulate the symmetry

to cancel out terms and things like that. You can use our knowledge of symmetry to make transformations that make these equations much easier to handle. The subtle note I'm just trying to make here is these things don't happen in a vacuum. Like our general mathematical understanding

has applications in subtle, small ways all through... All through, all through. Yeah, all through. And so that's for a very very nice, symmetric thing. Formula one cars, not very symmetric. I mean, there's some symmetry, I guess, left and right. So, sure, you can see that it's symmetric. Like, you can flip the profile, right? But a Formula One car is insane. The geometry is extremely complicated. The wheels rotate. The ground moves relative to the car. There's thin boundary layers on the bodywork entirely.

Air is accelerated beneath the floor. The vortices are deliberately generated around these waves to cause downforce, the non-linear term does not vanish. In the other one, the non-linear term actually vanishes just because of the geometry of the thing. Here, this is the entire problem, right? There's no known formula that you can write out for what a Formula One car,

the velocity and pressure at every point around a Formula One car is going to use. So instead, what engineers do, even in this visualization over here, what they're doing is using a computer to computationally solve, the Navier-Stokes equation around this boundary. Okay?

And this is called computational fluid dynamics. If there's any aerospace engineers or hydrodynamic engineers in the audience, leave one in the comments and tell us your favorite computational fluid dynamics program. There's a bunch of programs out there. This is one that I just picked up like on GitHub. And they always flaunt the number of cells

that they can break down their space. into. Because, you know, imagine a Formula One car in a wind tunnel, right? That's some like volume. In order to solve these differential equations, you have to break up that volume into tiny little grid boxes. And then you can solve the Navier-Stokes equations for each of the grid boxes. And then you just can put them all together. Yeah. You can be like, oh, the velocity is here, but the velocity next to it is something. The velocity to the right and left of it is something. So I can calculate a derivative that way, right? I can be like, oh, the difference divided by the grid spacing is my spatial derivative.

Yeah. Right? And I can do this computationally one at a time. And the more cells I have, the better granularity and the more accuracy I have to the real thing. This makes me think back to our hypersonics episode where we effectively discuss the same thing. Yeah. And then sort of in addition to all of the challenges we've talked about, when you start getting going from supersonic to hypersonic things get even we're yeah and we sort of

discussed why it gets weir and how we try to deal with that with these approximations that we've sort of discussed here it's again Navier stokes is not unsolved in all case we found clever ways yeah to like approximate yeah solution right right and for engineers this is good enough right the point they're like honestly like yeah open i didn't solve anything we've been modeling and Navier stokes since You were, before you were born. Yeah, for the Ferrari engineers, it's not good enough because apparently they still can't do it. Like, guys, figure it out.

You have Lewis Hamilton and Charleclair. Like, guys, please, I've been waiting for so long. Anyways, the result is not a closed form solution, okay? It's this enormous table of numbers that tells you what the velocity is at every given point. And you can put any arbitrary shape into it. This is like a meme now in aerodynamics for some reason. Like all these aerospace engineers like put a cow, a CAD model of a cow into their CFD simulations and they're like how aerodynamic is a cow? I've seen so many versions of this meme for some reason.

I don't know why, but a cow is a very non-aerodynamic thing that people are trying to figure out, you know, what is the lift on a cow as it goes through a wind tunnel. In any case, the idea is any arbitrary shape. You can put it into a simulation and you can turn the, you can turn the, simulation one after the other and computationally solved the Navier-Stokes equations. For all practical purposes, that's totally fine. It's an engineering solution. It's not physics understanding. Yeah. And or mathematical understanding. And it's fine that it's an applied engineering solution. It's a spectacular practical achievement. I mean, this is how we get airplanes. This is how we get fighter jets. This is how we get turbines. This is how we get wind

wind farms for like power. But it's not the same thing as the exact equations and what do those equations look like, right? What is the solution to this thing look like for any arbitrary boundary condition? A simulation has a finite grid, right? It advances in finite time steps.

It rounds numbers at some point. And so the problem is engineers are pretty happy. Taking Navier-Stokes equations, putting it into a CAD model in there, using computational fluid dynamics, they are good to go. They'll tell you what the drag is. They'll tell you what the lift is, what the downforce is on a Formula One car and so on and so forth. And unless you're making hypersonics, it's fine.

It's fine. Mathematicians, on the other hand, they're asking, well, what about, like, the existential problem of, like, I've got differential equations. are the solutions always well-behaved? Okay. That's the question. Here's what I mean by well-behaved. Let me give you an example of why they shouldn't be well-behaved. It really has to do with these two terms that are in complete competition to one another.

And let me see if I can get this right. We're talking about the convective term on the left, which accounts for acceleration. And we're talking about the viscosity term on the right, which is a part of our force. Yes. And these two things do very different qualitative transformations to our velocity field. The viscosity term, as I told you, smooths things out, right? If there's a tiny bit where there's a lot of velocity and it's surrounded by not that much velocity, then it's going to smooth out. The velocity is going to leak through and you're going to smooth out the velocity function. the convective term on the other hand, what it does is transfer energy from large systems to smaller and smaller systems. So the convective term is making smaller and smaller structures while the viscosity term is making those smaller structures smooth out. There's a constant competition between these two terms in the equation.

Because depending on the size of the structure we are referring to, the viscosity will be different and the convective term will be different, and it's different across every scale. Yes. And the question is, does one win out over the other? Specifically, does the convective term ever win out over the viscosity term? Can I get smaller and smaller structure to such an extent where I can have theoretically, infinite velocity somewhere?

Because the convective term is just creating smaller and smaller structure. Now, I want to be a bit more concrete here when I say the convective term is creating smaller and smaller structure. Okay. Okay. So we're going to talk about this competition in a very real sense with something called the Taylor Green Vortex. This is a standard test case that is used in computational fluid dynamics because what you can do is actually like solve this thing analytically and then you can solve it in a computer and compare the two. Okay.

And then that'll tell you how good your computer simulation is because you have a ground truth that's like a functional mathematical form. Here's what the Taylor Green Vortex looks like. This is the initial condition. On the right-hand side, you're seeing the velocity field at time T-equal-0. So this is what you're starting out with. It's a bunch of signs and cosines for the X velocity. It's a bunch of signs and cosines for the Y velocity. And it's zero on the Z velocity, meaning all of the particles are only moving in the X, Y, plane.

No one's going up and down. Everything is moving side to side. Flatland. Flatland, right? But there's bunches of flatland. Like, it can still be a 3D box, but like the stuff at the top of the box is still going side to side. And the stuff at the bottom of the box is still going side to side. That's how we start out with.

And crucially, all of these terms have signs and cosines. So what that amounts to is you have vortices. Okay? You have a, the signs and cosines always means circles. Okay. If you come out with one thing from this podcast, sign and cosines mean circles. But in this case, what's happening is you've got these vortices. that are circular.

Yeah. Okay. And so all the particles are moving around in circles like this. Like the ones up top are moving like this. Down here, they're moving like this. Down here, they're moving like this. Imagine a 3D box. And now I press play on this thing.

What is going to happen? Well, let's see what happens. We've got a bunch of math here that I'm going to very quickly go through. Okay. For those who are not watching my eyes just... expanded to a very large size. Yeah, but it's not that bad. And I'll tell you what to focus on.

Okay. Okay. On the left-hand side is our initial conditions. These are our signs and cosines. It's just sine x and cosine, cosine y, sine cosine z. So this tells you that like the size of my vortex is 2 pi in size. Okay. Okay.

Now I apply the convective term to these equations, right? These equations tell me what the velocity field is. I can find the gradient of the velocity field, and I can multiply it by itself. And then I'll leave this as an exercise to the reader, or in this case, the podcast listener, as all of the famous science and math textbooks do. But you can go through and do the derivatives, and then you'll have to remember your double-angle identity is in trig. But what ends up happening is those derivatives are going to create terms with a sine 2x and a cosine 2x. that's going to percolate into the pressure terms. You're going to get sine 2x and cosine 2x.

And one thing that you should remember is sinex is a wave. Sign 2x is that same wave squished. You're creating smaller and smaller waves. The wavelength is getting smaller. The oscillation. Is getting squished. Okay. Now, what does that do to our structure of the vortex?

What happens is, remember, before I told you that, yeah, that's the, that's the one. So remember, before I told you that the Z component of my velocities was zero, everything was happening inside to side, right? But now, because of that convective term, and because of those higher wavelengths, sorry, I should say smaller wavelengths, there's going to be picked up Z velocities. there's going to be interactions that are happening in the Z direction. You're going to push up stuff and you're going to push down stuff. And the way you push up and down,

those structures are going to be half the size of your original vortex. And this is coming from the squishing. It's coming from the squishing and the fact that the squishing is forcing stuff into the third dimension. So on the left-hand side, you see these vortices, those blobs sort of represent where the vortices are. You can imagine the orange blob is a vortex. going in one direction. The blue blob is a vortex going in the other direction. But all of those

vortices are X and Y. Okay? They're revolving in this way. And they sort of have sort of, for lack of a better term, a consistent non-disturbed structure. Yeah. And that's time T equals zero. Like, that's what we started with. And when we press play, all of a sudden you start getting these vortices, you get these interactions between the vortices up and down. And the interactions, those structures that are created in that interaction have a smaller and smaller size because of that convective term. Okay. I see. I see. And if we continue this at infinitum, it's going to get even smaller and even smaller and even smaller to at some point you would think, you would think, right, that like it's just going to get smaller and smaller and smaller to infinity.

You're going to get smaller and infinitesimally smaller structures. That's not the case. viscosity at some point wins out in this scenario. So coming back to how we started this, we're talking about the viscosity term on the right side of our Navier-Stokes equation. And then we're talking about the convective term on the left side of our Navier-Stokes equation. And these two things are in competition as we looked at in this example. And at time equals zero, when we had those blobs on the top left, And as time goes by, the left side of the equation is winning.

It's winning, yeah. Effectively. Effectively, because it's creating smaller and smaller structure that's leaking into the Z direction. And so our sort of nice, discrete blob structures at the top become a more chaotic system as we get these smaller and smaller structures. But at some point, and the idea, the fundamental question is, does this go all the way to infinity in smallness? And do I get smaller and smaller and smaller and smaller structures? Is the convective term the goat? Mm-hmm.

And does it always win? Yeah. And what we're saying is no. In this case, no. In this case, at some point, the viscosity starts to matter once you get to a sufficiently small scale. Yes. Because, and for technical audiences, the convective term goes up kind of linearly, but it, like, grows very fast in the beginning. The viscosity term grows like the square.

So at some point it's going to catch up and it's going to diffuse everything. And then everything will just sort of like be chill. Yeah, that makes sense. Okay. So the Taylor Green vortex, I think it puts this conflict right in front of us. Yeah. You've got the nonlinear term, that convective term, that takes an organized flow. And it creates smaller and smaller spatial structure. And finally, we've got this viscosity that becomes increasingly more powerful at those smaller and smaller spatial structures.

Is the convective term going like this? and the viscosity is going like this? It's like this versus this. Okay, this versus this. Yeah, yeah, yeah. So eventually, like, at some, this meeting point, yeah, of structure size. Yes.

Is when the viscosity is like, okay, that's enough. Yeah, yeah, yeah, yeah. That's enough. That makes sense. Let's calm down. Yep, okay. And so the viscosity waits for those smaller scales to convert all of that energy

into just, like, dissipation and heat, right? And where, where it happens depends on the viscosity of the fluid. Okay? If the fluid is very, very viscous, it's going to cut it immediately. Immediately. But if it's like not very viscous, it'll take time. It'll take some time, but it's going to get there.

For the Taylor Green Vortex, it is going to get there. That's fascinating. Right? And now in every single simulation that we've done, viscosity always wins out. Okay. Okay. At some point, viscosity is always going to win out. V for Vendetta one would say.

That's right. And the Millennium Problem is asking, is that actually true? For every single scenario that I can come up with. I just showed you two, right? We showed the coerete flow, and then we showed this Taylor Green vortex. In both of those cases, pretty fine. Everything's smooth. Everything's normal.

Things aren't going crazy. So is this idea that smoothness always wins? Yes. Like to describe it in another way. The solution is always going to be smooth because of the viscosity. Because of the viscosity. Winning in this race. Yeah.

Okay. Is it always going to win? Is it always going to win? Yes. We've shown that it can win sometimes. Yes. And for those two times, it's going to win. going to win. And actually for every single simulation that we've done, we've shown that it's going to

win. Every single simulation in the history of aerodynamics, we've shown that it's going to win. Now, that could be an artifact of the fact that it's a simulation, right? And there's like discretized grids. And so we can't go infinitesimally small. At some point, maybe it's an artifact of the fact that like, you literally can't physically, like there's not enough compute to go down to what, the plank length or something like that, right? Where it could start falling apart. Yeah. Like, like so, so perhaps that's that. And that's that. And that's, you literally, you're what the millennium problem is asking. It's asking something stronger. It's saying for every single initial condition that I could come up with, is viscosity always going to win? This is the smoothness

global? Yes. Idea. And that actually, it's so funny because now it's, it kind of reminds me a little bit of, was it the, was it the Riemann hypothesis episode where we talked about, about the non-trivial zeros. Yeah. And it's similarly like, you know, we wanna take the question to its ultimate. Yeah, is like, is every single zero on the critical line?

On the critical line, right? Every single one. No, every single one. Because you can still have 99.99. You could even have 100, as we were saying. As we were saying. And it does, that doesn't mean that there's, you can have 100% on the critical line

and still have one off. And anywhere in all possibility that one is off, it means, just concept. That's what mathematicians are worried about, right? It's like, no, no, no, no, in every universe, right, that Dr. Strange has ever visited, is this true? It feels similarly in that direction. Yeah, it's a very mathematical thing to worry about. Now I understand the mindset.

Yeah. Now, crucially, this is now an open problem only in 3D. Okay. In 2D, there was a very famous mathematician. She was a Soviet mathematician. Olga Ladigenskaya. Olga Ladigenskaya. She wrote this 1969 book,

The Mathematical Theory of Viscous Incompressible Flow, where she showed that in two dimensions, yes, every single initial condition guaranteed viscosity is going to win. If we're in flatland. Yes. If we're in flatland and we've got like fluids flowing in flatland. And there's no a z-up, Z down that we just described.

As we saw, actually, like even in the Taylor Greenvort, Oh, okay. Right? We saw that like the Z dimension was the one that was causing that weirdness. Because the flow was escaping into the Z dimension and then creating those smaller and smaller structures. So Olga showed in 1969 in her very famous book that indeed, in two dimensions, it's totally fine. Every single starting point that you could ever think of, you're going to come up with a smooth solution. Okay, so for 2D, the Navierstokes, global smoothness is known.

It's fine. That's not the question we're asking. Yeah. For 3D. For 3D is... It is not known. And is this, and I just have a quick question. So when I see the term blow up, as it relates to how people describe this, this is what we're, is this kind of what we're talking about?

Yeah. It's like, does the smoothness, does viscosity lose? Yes, which is effectively Blow up. Yeah. Does viscosity lose? Because if viscosity loses, then that can get effective term is going to keep going on this runaway reaction to create smaller and smaller structures of higher and higher velocities.

And at some point you're going to get like stuff that's moving at infinite speed. Right? It's kind of like is there a singularity in fluids? Not really. But like. Well, no, no, no. That's literally what they're asking. Is there a singularity?

Like, can I create a singularity? That's what they're asking. Okay, fair enough. That's exactly what they're asking. Now, why is it so hard in three dimensions? Well, we already saw, like, kind of a clue to that with the Taylor Green vortex, right? It's the fact that stuff can move up and down. And crucially, that had to do with a vortex that I had started, right?

The initial condition was a vortex. The signs and cosines described stuff going around in a circle, and we had these mini tornadoes inside of our thing that were interacting with one another and leaking into one another. vorticity is the problem in 3D. Vorticity is the curl, for those who have taken calculus, the curl of the vector field. That Dell operator is back again,

but this time we're taking a cross product of the Dell operator with the velocity field. It's just a way of saying, is the velocity field curling? Like if I put a pinwheel in the velocity field, is it going to turn? If it's going to turn, then that means that there's a curl

that's happening over there. Okay? Now, in two dimensions, vorticity, has nowhere to go. Right. Right? It's just spinning

and then it's got to like dissipate in the flatland. Yeah. In three dimensions you can have something called vortex stretching. Okay? No, this is in two dimensions.

You've got vorticity that's like kind of just spreading out and like going out about like, you know, it's a small vortex and it becomes a bigger and bigger vortex. Right?

And that's all it can do in two dimensions. Because it's nowhere to go. It's got nowhere to go. In three dimensions, you get something called vortex stretching. Okay? And this could be a problem. Imagine a vortex that has a certain size,

it's got a certain radius, like a hurricane that's, or, yeah, let's say a hurricane. It's quite big in space. And it's moving, but the velocity is not that high. Okay? And now that hurricane, all of that energy, gets squished into a tornado.

What's going to happen? Well, the tornado is going to get very, very tall. because all of that energy from the hurricane has to... Divergence has to equal zero. Yeah. Very good. This is exactly it. It has to equal zero.

So it's got to go somewhere. It's got to go somewhere. So it's going to go up. But because it's getting tall, again, the radius is going to decrease. But if the radius decreases, now we have another principle of physics, which is the conservation of angular momentum. And so just like, you know, in the figure skaters, when they're twirling around, they pull their arms in, they spin faster, same thing's going to happen here.

The thing is going to spin faster. And this kind of reminds me in a different context of our sideways wine bottle and then it getting narrower and so it increased the acceleration. Just conceptually there's a sort of similar thing. Yeah, you're packing stuff in here, but here it's the vorticity that's increasing because of the conservation of angular momentum. Again, it's like, it's kind of similar because there's something being conserved. In that case, it was the amount of fluid.

In this case, it's the amount of angular momentum. And this is called vortex stretching because you're taking a vortex, you're stretching it up. But as you stretch it up, you thin it out. And if you thin it out, it's like the figure skater bringing in her arms, the thing is going to move faster and faster. And this kind of relates to what we just talked about in the Taylor Green vortex, why we're now leaking into those higher dimensions, right? And the question is, maybe with this vortex stretching, you can actually create a kind of

singularity. Okay. Now, let's get into what we mean by singularity in the first place. We had already done our first toy differential equation, right? Yes. Which is the compound interest differential equation. Now, this thing does go to infinity, right? I mean, it's E to the X or E to the T. For infinite time, I'll get an infinite amount of money. If I wait long enough, even if I have a dollar in my bank account, that dollar is going to become infinity if I wait long enough, right? it just turns out maybe AI is going to end the world. So the point is that's not a real singularity. A real singularity is an infinite in finite time.

I shouldn't say in finite because that sounds like infinite. So I should say before in pause. Finite time. Can I get to infinity? In some constrained amount of time. There we go. Yes. For our listeners out there, that'll be good.

In a constrained amount of time, can I get to infinity? Here's another differential equation that does that. Here, the derivative, my derivative with respect to time, is equal to y squared. So, D, Y, D, T is equal to Y squared. Initial condition, you start at 1 when time is 0. And the solution to that is 1 over 1 minus T. If I plug in T equals 1, I get infinity. Right?

So this is a differential equation that creates a singularity, but the time is not infinity. The time is just one. At one second, I have a singularity. You reach the singularity. Another thing I want to point out here, what's special about this differential equation? It is nonlinear. You see? It's Y squared.

It's the simplest nonlinear differential equation. And already I am getting a singularity, right? The other one was just DY, D, D equals Y. That's a linear equation. I can just add up the terms. Right. But here, the non-linearity is creating that cascade that is creating a singularity before time goes to infinity, at a finite time. So it's conceivable to extrapolate from this.

Yeah. It would be conceivable based on how we've constructed the non-linearity of trying to look at this incompressible fluid and look at a vector field. Yeah, it's like there's a chance. There's a chance because we got a lot of non-linear. going on. Yeah, yeah. It's just that there's the viscosity term, right? In this one that I showed you, there's no, there's no smoothing. There's no

counteractics, it's just going. Right? But in the Navier-Stokes, and that's why it's so interesting. Because the Navier-Stokes has this non-linearity, but it also has this smoothing term. It has this counterforce, which is viscosity. And so you're asking, who's going to win? And are they always going to win?

So far, it seems viscosity is always won. That's actually, that makes so much sense now. It's, it's, ah, that's so good. Right? That's so good. It's quite nice. Yeah, it's quite nice.

So it is a very interesting problem. Yeah, no, it, because, because you can understand you could, there's good arguments for both sides. Mm-hmm. One in which viscosity always wins feels true. Yeah. Seems legit. Because we've been able to show that. In a variety of context as being true.

Yeah. But being fundamentally true. is another thing is another thing and because of again the nonlinear nature of the convective term in this case that is challenging the viscosity as the counter force there's an argument that both of them have a reason that they could win exactly and in finite time in finite time exactly and so it's it's an interesting problem i like that i have to give it to them right like it's it's something that definitely um makes you think and there are other things in physics that have singularities in finite time.

I mean, the most famous one would probably be a black hole as a result of the Einstein field equations. The Einstein field equations are also very famously a non-linear differential equation because you've got these tensors, and these tensors are multiplied to themselves and also multiplied to the reciprocal of themselves twice over. And so you get these squared terms.

You know, for example, I mean, in a very simple case, like the curvature of space and time causes gravity, right? But gravity itself has an energy that causes the curvature of space and time. Right? So what? And so now you get these nonlinear actions

on space and time that create the richness of the phenomenon that we see. And on the right hand side, I mentioned Lake Geneva that I was on a... I was actually at the Yerkees Observatory on Lake Geneva, which is known as the birthplace of modern astrophysics. We'll do an episode on them.

They had a nice sculpture of Einstein's general relativity because Einstein had visited the Yorkies Observatory many times. And there on the slab of marble was written out Einstein's field equation. So I just wanted to show that when I received this news, I was looking at a non-linearity live. And I was like, damn. Right. Here we go again.

Here we go again. Like the meme we started the episode with. Exactly. So, singularities can exist in physics, right? The center of a black hole is a singularity. But I do have to say that in this case, in the Navier-Stokes case, this singularity is not a physical singularity. And physicists do not care about this singularity.

It's purely for the math. It's purely a mathematical question, okay? Physicists do not care about the Millennium Problem in Navier-Stokes because fluids are never going to achieve an actual physical singular. There are plenty of cases where the Navier-Stokes equations don't even apply. And I've got some examples over here. I mean, for example, hypersonics and supersonics even. Navier-Stokes equations don't apply.

Navier-Stokes equations are both of those equations. One of the key ones is that second one, the incompressibility. The fact that the density is always constant. Well, when you're approaching the sound barrier, all of a sudden air becomes compressible. Right. Right? The sound wave is catch, you are catching up to the sound wave in front of you. And that is going to compress the air.

When it comes, so that's the bottom there where you get a sonic boom and you get like a cloud because like literal condensation forms around your aircraft. Hypersonics is when you're going so fast that the chemical nature of the air starts mattering and you get thermal effects. That's not in the Navier-Stokes equations. In the middle, that's an aquaporin protein. Those things are so small that they let in only water molecules. At a molecular scale, it doesn't, velocity fields don't matter at a molecular scale. Like, what are we talking about? Right, right.

So again, at the molecular scale, Navier-Stokes equations don't apply. And for a lot of weather patterns even, Navier-Stokes equations, at least the ones that we have up top here, are not the ones that are used. We use a compressible version of Navier-Stokes equations where the air can expand and contract and change densities. That's how we predict weather. The Millennium Problem, which is the mathematical problem, has to do with these two specific equations. Which are specifically for incompressible fluids. Yes. And so I would even say that the Millennium Problem has nothing to do with fluids.

It has to do with a vector field that is governed by this equation. It has to do with an object, a mathematical object that is a vector field that in normal circumstances would describe normal fluids. But if you're trying to hunt for singularities, you have left the real world and you have entered the world of mathematics. And it's totally legitimate. 100%. Right? But it's something that is outside the realm of physics. And I want to make that very clear because there's a lot of hullabaloo.

And I think there's a lot of stories out there being like, oh, this is like fluids. right? Can a fluid achieve singularity? No. Actually, the answer is no, right? Infinite speed. Again, Einstein would be rolling over his grave. You can't have infinite speed because as like, if you have a fluid, right, that is like going faster and faster and faster, then all of a sudden, Navier Stokes better account for relativity. And the fact that the mass of the fluid, like that density term, is going to have a Lorentz factor in there or something. Because the mass of the fluid is going to infinity, the closer I approach the speed of light. right? I can't go to infinity. So all I'm saying is this is a mathematics question, completely legitimate, but let's not confuse this for something that is physically relevant. It is not.

And I think we've now built up a full understanding to be able to specify what exactly the Millennium Prize problem as it relates to Navier Stokes is specifically asking about, incompressible fluids and sort of do you get blow up in finite time of an incompressible fluid and in this context what we're saying is we're hunting for singularities in these mathematical constructs yeah that don't apply to the real world yeah but still have a fundamental value to our mathematical understanding exactly of the world of just math math yeah which again then can bleed into like I was mentioning earlier to ways to think about other problems because it's this human understanding that creates, look at how we went from

Newton's second law of motion to the Navier-Stokes equations. Because it created a framework and we said, can we apply this framework to fluids? Yes. And then it created this whole centuries-long journey. Yeah, I mean, a lot of the stuff that led up to the Navier-Stokes stuff had to do with just analyzing what vector fields are like and what it means to do calculus on vector fields. Like things like Stokes theorem. That's from the Navier Stokes guy, right?

That's a mathematics theorem. Right. Stokes theorem that like then, oh, it like works here, right? So definitely something that is super interesting. And Terry Tao actually said that like the Navier Stokes. Oh, we're calling him Terry now. Look at this guy. Oh, it's my buddy Terry.

We added that out. Hey, yo. If you. All right, Terrence Tao. No, no, no. Keep it in. But like, Terence Tao, he said, the Navaristokes equations are one of the simplest super critical differential equations.

So in the theory of differential equations, it's very interesting to think about because you've got this nonlinear term and you've got this competition with the diffusion term, right? You've got a Laplacean and you've got this like convective thing and they're competing. And mathematically, it's a very interesting question to answer. Now, there's been a lot of work since the days of Navier-Stokes to try and understand the mathematical properties of this. The first guy I want to talk about is Gene Leire. He actually proved that fluids always possess something called weak solutions, where the total kinetic energy of the system remains bounded over time. How do you compute kinetic energy?

Like, kinetic energy is really just one-and-a-half mv squared, right, for like a normal point-like. object, like, you know how Newton's laws can be applied to a point like object, F equals M.A. Well, one-half mv squared. For a velocity field, it's a little bit different, right? What we have to do is integrate over a volume, because we've got a bunch of velocities in a volume, and we say the square of all of the velocities in that volume times one-half, that's going to tell you sort of the energy density in that volume, like the kinetic energy density in that volume. Which would be, in this case, the velocity field. Yeah, yeah, exactly. And so the kinetic energy density,

can remain finite for these weak solutions. And by weak solutions, these are stuff that don't have this, like, strong smoothness criteria. And I'm going to be honest, at this point, I'm getting into stuff that this is the math stuff. This is maybe why I went on a tirade about five minutes earlier about how this is all math. Because maybe there's stuff now that I'm like, okay, I don't really know, right? But from what I've gathered, what he's done is loosen the idea of smoothness. and show that like, you know, imagine you've got like a violent storm system. He showed that the total energy of that storm can't suddenly go to infinity.

Okay. But what he can't rule out is that even though the entire energy density of that storm doesn't go to infinity, you can have small pockets where the velocity is still going to infinity. Okay, you can concentrate into an infinitely fast like tornado somewhere inside. And you can't rule that out. So the overall energy is finite, but some local small velocity can be infinity. it. So already it's starting to show that maybe

there could be ways that the convective term wins. Okay? 1984, Kaffa Raleckone and Nurenberg, they show that if the Navier-Stokes equations for a velocity field work,

then you have partial regularity. What that means is if you have singularities, if they exist at all, they can't be like everywhere. Okay. You can't have giant chunks of fluid moving at infinite speed. But what you could do is have infinitesimally small, point-like singularities,

kind of like at the center of a black hole. And that works. Visit BetMGM Casino and check out the newest exclusive. The Price is Right Fortune Pick. BetMDM and GameSense remind you to play responsibly. 19 plus to wager. Ontario only. Please play responsibly.

If you have questions or concerns about your gambling or someone close to you, Peace contact connects Ontario at 1-866-531-2600 to speak to an advisor. Free of charge. BetMGM operates pursuant to an operating agreement with Eye Gaming Ontario. Okay. And so the idea the system has some fundamental limitation where it can only happen at particular scales. Particular scales. A very small infinitesimal scales.

This is why, again, it's like a mathematical problem, right? It's not, it's infinite tessomely small scales and it's happening only at particular places. It's not like a continuous singularity. It's at the center of a black hole. It's a single spot. Okay, so that's them saying, okay, partial regularity. In other words, you've got these decades of work that is placing a very tight fence around any possible singularity, but we haven't proved that that fence has nothing in it. There could still be a singularity.

It just has to be constrained to these things. And so the box in which the convective term, wins, we've now created a very small box. Yes. It can still win. It could still win. Against viscosity, but it's a very small box. Very small box, right?

And this brings up to 2000. This is when the Clay Mathematics Institute announces the Millennium Prize Problems in Mathematics. These are seven very famous mathematics problems that were formerly Hilbert's problems that, you know, a bunch of them got solved over in the 20th century, some by John Nash, of course. And others. And finally, we have seven out of those that were picked as, like, the big ones. The point car rate conjecture has been solved already by Gregory Perlman in 2003.

And that was huge. He never accepted the million dollars because he said that someone else deserved it. He's kind of like this weird guy. I'm going to be honest. He's a weird Russian guy, brilliant mathematician, but, like, lives kind of like in his mom's basement kind of thing. Anyways, brilliant mathematician. So here's the problem statement, and this is a problem statement by Charles... I want to make one quick note here, which is for those viewing who can see the list,

I want you to remember this list, right? Because this list is going to come into relevance when we talk about the race between the frontier labs. And this is the gold medal that they're all chasing after. The infinity stone. These are the infinity stones and they are coming after all of these. Maybe not Yang Mills and maybe not P versus MP. Yeah. But all the other ones.

All the other ones. Remount hypothesis, they came after it. Or at least something that is related. Navier Stokes is the one that we're at right now. Yang Mills, I mean, we did Yang Mills for an episode last year. That's one of my favorite episodes. Check it out if you haven't. And so here is the problem statement.

So each of these problems have to be well defined, right? Like when the Clay Mathematics Institute is like, this is the problem, the problem has to be well-defined mathematically. So Charles Pfefferman, field medalist, also a student of Elias Stein at Princeton University, the guy who wrote the complex analysis textbook and my professor of complex analysis. Also, so he's a brother, I guess, an academic brother of Terrence Tao. Okay. Because they both had the same PhD advisor, also Fields Medalist.

I just want to note, for those who might not know, Fields Medalist, meaning that they've won this, the most prestigious mathematics award that is available. For those who are under 40. For those who are under 40. Yes. I would say the most prestigious mathematics award that people don't really know about is the Apple Prize.

We talk about this. Which is, I think, like, the Nobel Prize of mathematics. Fair enough. People keep saying this in Fields Medal. I really don't think it is. Are you about it in the comments? So Charles Feferman puts out this statement of what it is that we are trying to prove. Okay.

We want the existence and smoothness of Navier Stokes on R3. That means in three dimensions. Two dimensions, we've already figured it out. Olga said, it's smooth. We're good to go. Okay. A and B say existence and smoothness of Navier Stokes in three dimensions. C and D say the breakdown of Navier Stokes in three dimensions.

And so just a quick. touch on this. So we're saying A and B means viscosity wins. Yes. And C and D means the convective term wins. Exactly. Yes. So A and B say it's always going to be smooth. C and D say it's going to lose. In the small fence that we've already defined the convective term ultimately wins and we get a very discrete small singularity somewhere. Exactly. And so in A and B, like that's a proof of global regularity, and that would mean you need to identify some kind of mechanism that prevents this infinite concentration at these very small scales, right? In practical terms, one would like some kind of quantity or some kind of scale-critical

quantity that's controlled for all time, and it's some structural argument saying that whatever vortex stretching is happening that we were talking about, it's going to lose out to viscosity. Okay? A proof for the breakdown would be the opposite. You just have to construct a genuine Navier-Stokes flow. have to construct some kind of initial condition in which the concentration it reinforces itself, that convective term keeps making smaller and smaller things. And all of that pressure and nonlinear

cancellations, after all of that, the mathematics becomes demonstrably something like a singularity. To just, not that to make an analogy, because this is sort of how people always frame these things when they talk about AI's progress as it relates to these things, which a year ago people didn't think it could compete in the Math Olympiad and now it's solving C&D of a millennium problem. But putting that aside,

it almost seems, based on this construction we've just described, that it's kind of akin to a counter example. Yes, 100%. Versus sort of the global solution, not minimizing in any way, because it's still a big deal, but people will nuance and nit-tick

on that, point. It's like, great. You found like one, you're looking at this large search space and you found something that has blow up in finite time. Well, it's an incredibly small search space, right? And everyone's been looking at it. Right.

But I think you raise a very big point. And I think we're going to come to that in the next few examples that I'm showing you, because in 2000, this is when the, this is when the Clay Mathematics Institute came up with it, right? In 2016, big players are involving themselves in this because they smell the blood in the water that there is a counter example hanging out somewhere. Okay. Okay.

More and more analysis is going into showing that perhaps there are situations where you get singularities. So Terence Tao, he creates this averaged version of the Navier-Stokes equations. and he proved that the standard energy conservation laws are mathematically too weak to prevent a singularity. So he created this like averaged version of the Navier-Stokes equations. It's kind of like a blurred-out version. Maybe you can think about that. It's like a stunt double of the fluid equations. And from the outside, this thing obeys all of the exact same conservation laws that the real Navier-Stokes equations does.

But Tao proved that this stunt double definitively explodes in finite time. Okay. So this averaged version can create singularities, which means that the principles that you would have used to prove that it wasn't possible are also not going to work for the real thing. If it was global, you wouldn't be able to have an average stunt of all that blew up. Yes, exactly.

So there's a chance now that at least the original arguments that people were using saying, oh, energy conservation and things like that, that might not work anymore, okay? So A and B are already not looking great, right? Maybe there is a singularity that's lurking somewhere. Okay? And for decades, mathematicians tried to trigger this blow-up using something like a fractal shape, like a whirlpool that's shrinking into smaller and smaller whirlpools and so on.

This usually failed because it required an infinite amount of energy. And crucially, you can't do that. One of the things the Pfefferman said is you can't just like stir it infinitely fast and be like, Oh, I got infinity. Yeah. Right. Okay. That's not the interesting question I'm trying to ask.

Right. Okay. So you supplied an infinite amount of energy and you got an infinite amount of velocity. Great. Great. Right. No. It's supplying a finite amount of energy to get an infinite amount of velocity because the nonlinear

convective term is concentrating energy into smaller and smaller structures. So both finite energy and finite time are both relevant in the context of this. 100%. And so Diego Cordoba and Luis Martinez. Zorora and Fanjeng in 2023 and 2025, this is very recent. They changed the paradigm

by abandoning this whole single shape thing and relying on something called a layer cascade. It's kind of like a bunch of nested gears or, you know those Russian dolls where you have like a big doll and then smaller doll and smaller dolls, that's what they were constructing.

Instead of a single vortex that they do this vortex stretching in, they've got slow moving outer vortex and then a medium inner vortex and then a smaller inner vortex that spins faster because of the medium vortex and the medium vortex is kind of like a parlay or like not a parlay

like an intermediary between the outer and the inner and it acts upon an even smaller layer and so on and so forth and you can design this geometry so that the feedback between these gears doesn't disrupt the big gears and the feedback between the smaller gears goes up up and

up and you can get unbounded velocity. Okay? And if you have an infinite number of gears, then perhaps you can have an infinite velocity. And so the mechanism, they called it dynamical amplification across scales. And they use this to prove finite time blowup for the unforced 3D oiler equations.

Remember the oiler equations without viscosity? Without viscosity. They showed that without the smoothing term, you could actually have finite blowup. The pressure won't do as much. Because the pressure also does a little bit of smoothing. Because it was both the pressure and the external force were the only two of the three on the force part of the equation.

Yeah. And so this is unforced. So no force. You just set up an initial... Ah, okay. Yeah. There's no stirring. There's no force.

You just set up an initial condition. And the scales converge in such a way that you get an infinite blow up. Now, this is huge. They're chipping away at each of the variables on the right side of the equation. Yes. And so the problem was, so that was for the unforced 3D Euler equations, right? But obviously we're interested in the ones that have viscosity. So there's this one type of equation called the hypodissivative, dissipative Navier-Stokes equations.

It's a variant that utilizes this thing called a fraction Laplacian instead of the actual Laplacian. The DL squared U that we had for viscosity, the triangle squared U, that's a Laplocene. that's a Laplacian. This thing has a fractional Laplacian. And they showed that in this case, we can use the same sort of paradigm and get blow up. The only problem was that they required

a forcing term that was rough, meaning it's not smooth. Like the forcing term itself has tiny little bits of infinity. Like the way you're stirring it has like little jolts of infinite stuff. You're pumping in. infinity into your stirring.

And so we're getting closer, though, because the infinity is like small enough that you could like maybe figure out a way to smooth it and still preserve the dynamics. Okay. And so I just want to pause really quick because what we're sort of saying is human mathematicians have now taken this Navier Stokes Millennium Prize looking at a finite blow-up. up. And then they started with looking at, you know, taking out the smoothness and taking out all the force.

Yeah. We got some solutions, which are effectively the oil, the oil equations without the F. And then we've now started adding in, back in the viscosity, but that weird version. But the viscosity wasn't smooth. Yeah. It was rough. Yeah. And so when people see this smoothness and roughness, part of what we're saying is the

viscosity term has some special sauce that doesn't quite get you to the natural smoothness that arises in the equations with viscosity unmanin manipulated. The raw viscosity. Yeah. But it's still important because these are fundamental. No, and this is what even Open AIs thing is built on, okay? It's like there's a reason why the Navier Stokes is the. first of the six infinity stones to fall.

Which we just thought. It's because we got so close as human beings. It's like a racetrack and someone was on the fourth lap of whatever, the 800 meter. And, you know, open eye just shows up, grabs a baton and finishes. Yeah. Yeah. Not minimizing. Not minimizing, of course.

But I'm just saying a lot of human input has gone into this. Right. So Cordoba and Martinez-Zorora, they established this layer by layer, scale amplification, it's a huge deal because everyone can sense that we're getting close. Right. Okay? And shortly before the Open AI announcement, mathematicians Tristan Buckmaster and Levant Alpoge, they published preprints that elevated this methodology to accommodate for smooth forcing

of Euler, meaning you've got the Euler equations, but now with a smooth force, the Boisinesk equation, which is something that's like kind of similar to the Navier Stokes equations. and this is where the sort of drama starts, but I'm going to save that for later because a few days after that, Open AI announces that they have solved it. Here we go. Here we go.

On the Navier-Stokes Millennium Prize problem, they announce it on Twitter, they have it on their website, and the actual paper is something like 165 pages long. The theorem, proved by OpenAI, constructs a smooth, compactly supported external force, the drives of fluid starting from rest into a singularity. That's what they're doing.

They're solving C and D. Right. They're saying that I have found a way to force a singularity into... We found a situation where the convective term beats the viscosity. Exactly. Yeah. And because the force is smooth and bounded, the singularity is not because the force is just infinite.

Right? It's literally the liquid. the fluid itself, the velocity field itself, doing this. The force is trying to maintain some kind of structural tautness of that liquid. And the singularity happens at t equals 1. It doesn't happen at t equals infinity. This is a finite time blow up. Okay?

So let's get into some of the stuff that I understood from their paper. This is unlike the anthropic paper about Riemann Zeta, because in the anthropic version, they gave us a full account of all of the agents and like even a transcript of the inner thoughts of these agents and how they coordinated with all these sub-agents. So I had this really nice story for our podcast where I was talking about how some of the agents like,

you know, something crashed and like they had saved a little lemma of their proof on the hard disk or how... I mean, yeah, it was crazy that we got this like inner version of what the agents were doing. Here, OpenAI is not doing that. But in any case, yeah. In any case, we can still kind of try to understand how they constructed the singularity.

The core of the singularity is in a axisymmetric vortex that's centered at the origin. And this is figure one that they show. We start out with a vortex that's yay big and yay tall. And the point is that as the vortex shrinks, the vortex is going to stretch, right? The whole thing is shrinking. but the radius is shrinking faster than the height. That's the whole idea. Because this is so funny because before we talked about the hurricane going to the tornado

and the idea was the height was increasing so fast because of the conservation of angular momentum. And all they're just saying into this mathematical construction where you don't need the conservation of angular momentum. You still do. You still do. But they're hacking their way around it. Okay, okay. Okay. Yeah.

They're hacking their way around it. Okay, okay. They're saying that as the thing shrinks, the radius is shrinking faster than my height, which you can kind of see here in this example, right? The blue part, the radius is kind of shrinking faster than the height. But as the radius shrinks, it's going to spin faster and faster and faster. And I'm going to get an infinite amount of velocity. That's okay.

I get it now. And there's a really nice gif that was prepared by Jayanta Padikar from the Wolfram community staff picks September 14th, 2026, you should really check out the website and it shows sort of what's happening. In this case, the camera you have to imagine is zooming in because the whole time this entire thing is shrinking.

Here it's kind of showing that like, it's kind of like the same size the whole time, but that's because the camera's zooming in with the vortex. Yes. Right? But here what you're seeing, let's look at this animation in a great bit of detail.

Okay. The radius is going down and down. You can see. The axial length is kind of going up. And the aspect ratio is going up. Yeah. That's because the radius of shrinking as the height increases. But it's not like, you know, it's not one of those powerpoints where you hold shift and you like, and it preserves the shape.

In this case, it's stretching. This is vortex stretching. And that inner vortex that you see, you see there's an inner vortex and outer vortex. Yeah. That inner vortex is the thing that is causing the singularity. That inner vortex is the thing that is making stuff move at faster and faster speeds all the way to infinity as you approach time T equals one. That makes total sense. And it's so interesting because, again, we've built this construction where I can actually walk away now looking at this and understand why it is what I'm looking at equal finite time blow up of Navier Stokes.

and it's it's it's it's it's it's just it's funny because um there's we talk a lot on the show about how we're standing on the shoulders of so much understanding um and i'll pause on this because we'll get to it in a minute but it's so interesting um that we are accelerating our ability to build on the learnings of those who came before us. Yeah. It's really, really cool. And there's a few more things that I want to talk about this Open AI paper that I found as a physicist quite interesting. Because one of the key things that I love about physics and when I was doing statistical mechanics is like scaling arguments, meaning like as I approach criticality, criticality is a very big thing in statistical mechanics when we talk about like

critical phenomenon, magnetic materials, things like that. It's all about the exponents. It's about, as I get closer and closer to the critical phenomenon, how does my stuff blow up and how does my stuff approach infinity? For example, in a magnet, if I get close and closer to the curie temperature, which is where my magnet goes and remains, like, if I have a magnet and I can,

cool it down to a certain temperature, then even when I remove the magnetic field on the outside, it'll remain a magnet. You know how like you take a paperclip and you rub it, it becomes a magnet, but then like you leave it, it'll stop being a magnet. If you cool this paperclip down, it's going to remain magnetized because all of the spins are kind of frozen in. There's a specific temperature where that happens. And as my material approaches that temperature, stuff starts blowing up. In this case, the correlation length will start blowing up, meaning a spin can now talk to a spin that's infinitely far away. At least that's what the model suggests. And some of the stuff in this paper reminded me of that class that I was taking.

Again, that was another class with Robin Bruinsma. It was a different class than fluid mechanics. But in any case, it was one of my favorite courses. So let's take a look at some of these scaling arguments that the Open AI paper is claiming. So one thing that's very cool is they recast the time to instead of being just time as it's approaching one, we talk about tau, which is a countdown time as I approach one. So, you know, normal clock time would be like 0.5 seconds, 0.7 seconds, approaching 1. But imagine taking 1 minus that time. Then I'm like 1 second away from singularity, 0.5 seconds away from singularity. I'm counting down. to zero. And zero means I've gotten to singularity. So first thing they do is they recast time to become tau. And tau is like a countdown. It's like a ticking thing that is going down to zero. And they say, what is happening to my radial and my length scales as I approach zero? For one, the radial scale goes down like the square root of tau. Okay. So it's still going to zero, right? Because square root of zero is still when tau goes to zero, the radial scale is going to zero. But it's going like the square root of tau. Okay. So it's still going to zero. But it's going like the square root of tau.

root. On the other hand, the axial scale is going like tau to the one half minus H, where H is some non-zero number. And that H is key. Because that H is telling you that when tau goes to zero, I still have a little bit of stuff left.

Okay? Like, the aspect ratio is actually going to infinity. As I squish this thing, this guy is not shrinking that fast. And so my height, versus my radius, if I were to divide the two, that aspect ratio is actually going to infinity,

which is key. Because if the aspect ratio is going to infinity, which in this case, what we're saying, that aspect ratio is, the vortex is becoming infinitely slender. Yes. And so what's happening is,

the vortex is going like this, is getting smaller and smaller, but because of incompressibility, the fluid has, has to go somewhere. It's going to start shooting out. And the smaller this thing gets, and the taller this thing gets, the faster it's going to shoot out.

And so my azimuthal velocity is going to go to infinity. Because it's going to shoot out of this vortex increasingly fast as I shrink this thing down and down and down. Right? And this shows the characteristic velocity magnitudes. Again, you've got these scaling arguments where stuff goes like the square root. but then one of them goes like the square root minus H. And the H is not zero, right?

The H is some small number, but because it's negative, that means it's in the reciprocal. So when T goes to zero, one over zero, you go to infinity, right? So the azimuthal velocities, the shooting up and down is going to infinity,

right? Even though the radial velocity is on the order of going to zero. Actually, you know, in this case, it's also going to infinity. Okay, and it's, I'll pause. Let's continue. The next thing I want to show is, is there infinite kinetic energy?

Because if there is, then we're, then that's not good. It doesn't, yeah. That doesn't count that. It doesn't count. Well, let's look at it. The kinetic energy is determined by the spatial integral of the velocity squared, right? Because it's one-half mv squared. So it's just the velocity squared.

Well, velocity squared goes like tau to the minus 1, minus 2H. and the volume, because it's the radius multiplied by the height, radius squared multiplied by the height, the volume goes like tau to the three halves minus H. You can just see the previous slides that I showed you and calculate the velocity squared by the volume, and you get something that goes like tau to the one half minus 3H. When tau goes to zero, that thing goes to zero.

So the energy is going to zero, but the velocity is going to infinity. It's not an infinite. They're not putting infinite kinetic energy. They're not putting infinite kinetic energy into this thing. It is a runaway singularity. Yes. That's not correlated to the amount of energy in the system.

Yes. The energy in the system is not causing the singularity. The singularity is being caused by the fluid itself. The fluid dynamics itself. Yeah. The way that they've constructed this geometry and this tiny bit of forcing that they have. Let's look at the forcing. So that's figure two.

Here you're looking at a bird's eye like sort of down the tornado hole. of the thing. And you zoom in. The way they're forcing this thing is by pushing out and pushing in simultaneously. Okay? So you push out to the left

and you push in to the right. What that does is impart the same angular momentum, but it cancels out kind of the radial thing. So all you're doing is turning this thing in some sense. And not making it wider?

Yeah, yeah. Oh, that's nice. It's kind of nice. That's nice. Yeah. And also the pushes are localized. Yeah. They're not in a total ring.

Like, if you zoom in on that part of the ring, they're localized to these little tiny spots. This goes back to the convective term versus the vector field. Like, the changes are happening at a very small part. And then those changes are then ricocheting into the smaller and smaller vortices to create that thing. Right? Okay, this is good. I mean, it's very cool. This is on the backs of mathematicians over the ages, but I still think, like, this is,

definitively shows that there are situations where the convective term wins, no matter how hard, the viscosity tries. Right? The vendetta was not successful by V. And so you would think that this is the end of Navier Stokes. Not quite. It is the end of the Millennium Problem because the Clay Mathematics Institute, Charles Pfefferman, had a very stringent criteria. And this thing meets that stringent criteria. It's been verified by Lean.

Crucially, that does not mean that it is true. Right? Lean could have errors. So we're still waiting. And the Clay Mathematics Institute actually is also waiting for two years before it says, yep, this is done. So there needs to be human mathematicians that actually understand this 165-page proof and tell us, yep, we're good to go. But the Navier-Stokes problem itself is actually also not fully put to rest because they have solved the forced version, right? This required a little bit of paddling in some sense. You took an oar and you like paddled the vortex to become what it is.

What about the unforced scenario, right? That is still an open question. And Princeton mathematician Stan Palasek actually identified a problem with this particular proof that would make it impossible to apply to an unforced scenario. So like this geometry is not going to work for the unforce. This only works because you have those little paddling terms that are imparting. the finite amount of energy into that cascade. So without that hand of God forcing term, it's not, it's not, it doesn't work.

And so that part, at least, is still an open question, this natural unforced setting. And this is, as I've seen some of the dialogue about this, this is where, you know, so Scientific American had an article about this. And they were like, you know, open AI did not solve Navier Stokes. And, you know, what they were. were sort of trying to argue is they didn't solve A and B. They solved C and D. Like, that's the argument that they were making, but they made it seem like it was complicated. No, but the Clay Mathematics

Institute said any of the four. Correct. Like, and this is, this is where, again, I'm just trying to talk about the nuance of what actually was done versus sometimes the way in which it gets then presented in media downstream. Yeah. And, you know, some people now are arguing, well, way it's defined by clay mathematics is the problem. And it's like, okay, you can make that argument. You weren't arguing that before. Right. But why are we, it's funny that now that it starts when it's been solved by something

that either makes people feel uncomfortable or they feel like it's by an entity that didn't give credit to those before them or whatever it might be. But it has been very interesting to see the dialogue over the last two and a half weeks knowing we were going to go through this, I focused on the larger environment. And now having walked through this, with nuances applied, six months ago or a year ago,

if you said that a model, even given all of the human advancement, had figured out finite time blow up for Navier Stokes, people would have laughed in your face. Yeah. They would have laughed in your face. at you. Yeah. I mean, those are guys who are like, yeah, wake me up when AI solves a millennium problem. And now the millennium problems are ill-defined. You know, I don't know. I don't know.

So I want to let people know here. This is the best explanation of Navier Stokes in the context of Open AI's announcement and the history of this subject at a level that is accessible to most general people but is not meant to necessarily be the deepest mathematical debate about the nuances, although it's quite good. I did the best I could. It's quite good. I'm not a mathematician. I think it's quite good.

But we will, and it's funny, and I just notice this, we're still listening right now, our lights have an automatic setting at a certain time in the night that turn it to bedtime mode. And so our lights have actually gotten a little bit darker. I was wondering. And in the video playback, but because we're three hours in and we started a little bit late tonight. And so we're going to do a quick reset for our lighting. And we're going to come back and finish up with now taking a look at now that we understand what has been done, although we didn't get the same detail as with Anthropic. Yeah. What is the environment in which this happened in? And And how do we think about where this is all going?

So we are back now with our lighting in not nighttime mode. For those listening three hours in, welcome. We are going to now take a look at a little bit of a timeline and some of the connected issues that have arisen around this OpenAI release. And so just to give us a grounding here, you know, a little bit about how did we get here, and everything that happened around it. So the announcement that set off these two conversations, what does the mathematics establish, and who gets the credit for the work behind it,

came out on September 8th. So that's our first date in the timeline. Now, interestingly, on September 9th, it was reported that OpenAI had revised the PDF from the previous day. For those who saw it on the first day, one of the complaints was, this is not citing any of the people we actually just talked about. And at the time, I saw the names and didn't know what it meant. Now we've built the construction of why the work of Cordoba and Zurua was fundamental to this solution.

And so they sort of changed that version. And now it includes those references. Yeah, yeah. They went back and they included the citations. So the marketing team put it out too early. they felt time pressure to put it out too early. And this obviously matters because citations are fundamental. Huge, yeah.

You got to give credit. You got to give credit where it's due. And so acknowledging the earlier published work that led them to do this is sort of the answer to one issue. But there's another issue that's related here, which is whether or not. OpenAI had access to unpublished work from folks who are using OpenAI's models to build on Cordoa and Zrua's work independent of their own announcement. And we're going to get to what that allegation is. But on September 10th, OpenAI said that an investigation has been put out that rules out, Tristan Buckmaster, who we talked about earlier, his allegations that they used the work of himself and Levant Alpoge to accelerate their conclusion to get to this solution.

So we've talked about Levant El Pogue before. A new entrant into this conversation is Tristan Buckmaster. And so who are these people who are we talking about? Right. So Tristan Buckmaster is a mathematician at NYU. as we mentioned, Alpogay is a member of the technical staff at Anthropic, who makes the model Claude that people may be familiar with, not at OpenAI.

And this is interesting because this story goes back, according to Buckmaster, all the way to August 15th and 22nd, as it relates to the work that they were doing together. And to kind of frame this up, this is a statement which you talked about earlier in page one, where Buckmaster is talking about this collaboration that he was having with Alpoga, where they credit Diego Cordoba and Luis Martinez-Zerua for the work they were trying to do using both Claude and Anthropics models to build on their understanding to try to get to this finite time. blow up results. And he noted that they used substantial AI assistance.

But in his first page of his statement, and this statement comes out in this time frame, he notes that he believes that Zoroa deserves the field's medal for this

solution. These are the guys who created that Russian nested doll of vortices that ultimately open AI used to create the solution. This is exactly right. So in his statement, he says that himself in El Pogue, they date their Boussinesque and oiler blow-up results to August 15th and the lien verification for that to August 22nd. And they wanted to work on the proofs a little bit more because what the models put out

was not understandable. Yeah. And it was not sufficient. while this is happening, there are rumors starting to fly around that Anthropic is leading up to potentially an IPO, initial public offering where they're going to the public markets and raise a ton of money. And as a part of the rumors, it was alleged that Anthropic was working on some open-mouthed problems, potentially some millennium problems.

And there were rumors that Anthropic had two millennium problems. problem solutions. Yeah. And I think some of these rumors were exacerbated by the fact that Levant Alpogue tweeted Augustus Mirabilis. This is exactly right. Which is like a reference to the Anus Mirabilis of 1905 of Einstein, the miracle year of Einstein where he had four papers that put him on the map. This is him saying that August is a miracle month for Anthropic. And what's interesting is this allegation that OpenAI was triggered to start taking one of their new internal models and start pointing it at these Millennium Problems,

Sam Altman actually tweeted that one of the reasons they started doing this was because of the rumors they were hearing. So it came from the source itself. And what's interesting to note is Open AI was using an internal model to see what they could do, and they were using this with this idea of using agent teams, like a swarm of multiple agents to do so. Okay.

So that started on September 1st. Right. Seven days before our September 8th, you know, announcement. Right. On September 3rd, Buckmaster contacts OpenAI, because he'd been hearing rumors through his network, which he describes in his statement, that maybe OpenAIA was looking at this

and wanted to kind of clarify what was going on there to potentially de-conflict. In the second page of his four-page statement, it was understood that OpenAIA wanted to date its result as September 5th, followed by 17 hours of lien verification. And then Buckmaster ended up having a call with members of Open AI on September 6th, which included a Sebastian Bubeck.

Al-Pogé was not on those calls. And the conversation, the substance of the conversation, conversation was, you know, Buckmaster was trying to say, hey, we've been working on these things. I heard rumors that you guys are working on something. And there was a back and forth about how to kind of de-conflict the release of these things. Ultimately, Bubeck was then trying to provide some options for what could happen. You know, it was you can release your papers first on oil or blow-up, and then we'll follow up with our Navier Stokes, where, you know, you can, you can release your papers first on oil or blow-up, and then we'll follow-up, and then we'll follow-up with our Navier-Stokes, where you are the author, talking about Buckmaster.

Or if you want to be a named author on our Navier-Stokes paper, we can work around that. But, hey, like, you know, we can't really involve Al-Phoge. The reason being it would be, you know, Bubeck's point was it would be a little bit weird for an open-AI research paper to include an anthropic researcher. And in the statement, he talks about how there was... It got a little contentious. And he asked, you know, when did you guys start? And there was a little bit of like, oh, well, we're not going to kind of tell you when we started.

And then he was curious whether his work in Codex, Open AIS tool, was used as training data for them to get to a solution. Yeah. You know, not really a lot of commentary around that. And there was a lot of back and forth. And basically Buckmaster, part of his accusation. was that, you know, there were comments made about this idea of, like, how, you know, why would you want to ruin your career by going up against us, which will come back to Bubeck's response to that.

There's just, there's a lot of weird tension. Because Buckmaster basically senses that you guys may have stolen our work and are trying to front run us. Yeah. And that's not going to happen on my watch. This is September 5th and 6th. total side note on September 7th our buddy Terry Tao as you like to say talked about three related papers about smooth forcing results for incompressible porous media for businesk and

three-dimensional oiler and this was from a totally separate research team and but what was interesting to the point you were bringing up earlier is people were coalescing yeah around this And it just so happened that these papers had come out, noting that they were two of them, I think, were fully formed. One still needed a lien verification. But everyone was hearing these rumors within the math community. And so everyone was kind of trying to effectively front run a lab, taking all of the credit.

Boobeck on the eighth, which is when Open AI put out the result, after we had Buckmaster basically front-run this and put out his four-page statement that we talked about outlining this. He talked about the result, how we worked, collaborated with Al-Poge, explained how they built off of Zrua and Cordoba's work, and made the allegations. Boubeck comes out with a response where he basically says all these allegations are false. Yeah. Didn't happen. Didn't happen. I regret the language I used about why would you risk your career.

and I said so on the call. But everything that Buckmaster's saying is nonsense. Okay. Right? Total denial. Crazy. Okay. Total denial.

So part of the tension here, there's still an unanswered question about whether they used the sessions from Alpoge and Buckmaster, Open AI, to have a starting point. They heard the rumors. it was known that these guys were working on this because they were working on it for months. Yeah. And they even said that the recent model that they're using was trained like late, started training late August. Late August. So it's after Buckmaster and Alpoge have been using this Codex thing for a very long time. Because again, they got their results.

Buckmaster and Alpogie had the results August 15th and 22nd for Oiler Blow Up. Yeah, and they had the results, which means they started using this thing. For a long time, right? And they mentioned that they were using the previous version, I believe it was Seoul 5.5 or 5.6 of Open AI prior even to Astra. And at the end, only at the end, did they do a little bit of Astra to plug in. But this is an important point. They were using the previous state-of-the-art version of Open AI to get to models,

to get to that point. OpenA had an internal version that ultimately came Astra, but they had it even a better version than the public version of ASRA that they were using internally to do this. Yeah, and one thing that I want to say is that like, that new version that started training in August, that could be using all of the codex data that users have put in. Bingo, right?

It might be anonymized. Yes, that's the thing. And so they can't say it was specifically your sessions. But if in their settings, they did not turn off they use my data for training setting. This is where the gray area arises around a lot of this. And dude, there's even a grayer area. It's like, what if they turned off the settings?

But all that means is that now whatever chat they have, whatever chat log they have, is just not tagged to their specific profile. Like they'll still take the text and put it into a giant bucket of anonymized text. The question of theft here is, in my view, still unanswered. Mm-hmm.

And these are two independent people, Buckmaster making his comments, Boobeck defending that Open AI didn't do anything untoward. Opening I posting on their account that they didn't do anything untoward. It's a he said, he said, and it's still unresolved. And what's interesting is this now has totally brought up so many. many colliding arguments that starts with academics versus the labs, but then now spreads out to a whole variety of conversations about who benefits from this stuff, who gets credit for these discoveries. Why are now people talking about human extinction? And there are a couple

of cohorts I kind of want to talk about that are a part of this conversation. So you have the labs and the people financing those labs. They have a financial incentive to show that these models are doing incredible work. Yeah. Because it lines their own pockets. There is some tension between the frontier labs, in this case in the U.S., it's anthropic and Open AI and the financiers or the investors, because the investors also have bets on AI writ large and generally that are accelerated by the frontier labs but are not solely dependent on the success necessarily of the frontier labs. You have the research community.

And the research community uses these tools and ways we've talked about on the show in narrow contexts, not in a general or super intelligence context, that is really valuable. But there is now this open question about is the work that we're doing by using these tools and both Anthropic and OpenA have free access for researchers that get more powerful things or free credits? Is that basically a carrot to get them in so that they can train on the frontier of the smartest people to then front run them on any number of different things? I'm not saying that that's true, but people are asking that question now, and it's making it difficult for researchers to understand how should we think about these tools.

You have the governments asking a different question. They care about jobs. Now we're looking at multiple industries, not just coding. This thing's getting very good in other areas. Our job is to keep people employed because unrest arises when people are not employed. Is this going to be a problem that we have to deal with? and then and China

national security we have to compete in the global stage and so this is the thing that's always brought up it's like oh well we have to compete with China and I think the central point I want to get to

as we look at all of these things is that multiple things can be true at the same time arguments from each of these different parties as it relates to the AI conversation can coexist and are not necessarily mutually exclusive.

It can be true that the capabilities are increasing, and that matters, and there's risks associated with that. It can be true that the labs and investors have a financial incentive. It can also be true that researchers in these organizations who are bringing up fears about the pace of progress and our inability to contain it in a way that causes damage is sincere, because the researchers at these places and the executives at these places do not have the same incentives.

They might diverge. It is true that a regulatory infrastructure probably needs to be in place. It is also true that these companies, the frontier model companies, might want to have a regulatory capture approach because that prevents other entrants from coming in. It is true that understanding

what this does in our relationship, China, matters. It is not necessarily true that we cannot get global collaboration just because we're in geopolitical competition, with China because ultimately do you think that the CCP the Chinese Communist Party wants runaway super

intelligence that's accessible to everyone in their environment where control over every aspect of people's lives is the way in which they maintain power? That's a good point. Why are they going to build and release a thing that destabilizes their stranglehold over their social, political, and economic system?

That to me is not obvious. That it makes sense for them. Yeah. They also approach it from a very different perspective. They look at AI in narrow implementations in manufacturing, in scientific research, but their power dynamics are very different. So these are all aspects of this conversation. And we just need to take them step by step. So one thing I want to talk about in this context, right, is for most of us, we experience the idea of AI through.

going to chat GPT or going to Claude. And we have a single instance of what is or was a chatbot. If you have not used any of the frontier model companies in the last even three to six months, you have no idea what we're talking about. It's totally different than December of last year and January of this year. So what we now have with these systems are the transition, from a question and answer conversation that does not maintain context over multiple conversations to this concept that folks have talked about about agents.

And one of the fundamental differences with agents is we've given these things the ability to use tools before, which means it can go and talk to your banking application or can talk to your Gmail and your calendar or it can browse the web or it can user computer, which means it can take action. It's not just providing a text-based response. Agents also have this ability to do orchestration, meaning it can define a plan, it can execute on it using tools, it can get a response, and then it can look at that response, re-evaluate its path towards whatever goal was set, and then continue to take action. And that, the time period in which agents can take action has moved from 15, 20 minutes total, about two years ago or a year ago,

to now being able to run for up to two to three weeks and in internal situations for much longer to that, longer than it, uninterrupted. This is a very fundamentally different thing. If you can have an agent system, we're just still talking about one agent, being able to run for long periods of time, it has memory and context, it has the ability to be able to run for long periods of time, it has the ability to use tools. It has the ability to now reevaluate its plan, create sub-agents under itself to do

micro-processes and iterate towards some high-level goal independent of an initial human input. Right. That is a very fundamentally different thing than what we've usually talked about with AI, which is you constantly give it a call and response.

Yeah. And the reason this matters has us rewind back to July. of this year, where many folks have probably heard about this hugging face incident, which is really an open AI incident that happened to impact hugging face. And the reports for this came out on August 26th, which is a little bit prior to our Navier-Stokes announcement. Right. Where Open AI disclosed that in internal cyber security tests, this age of... agent swarm that it had created, not just one individual instantiation of your agent that you engage with, but multiple at the same time with the capabilities I just described. Compromised portions of its own internal research infrastructure, as well as Hugging Face, which is a major AI platform for open source models, modeling data, training data and data sets.

Yeah. Hugging face is the emoji of the emoji hugging itself. And so I think this is really interesting. There were two independent AI safety and research organizations that did reports on this, Meter and Redwood. And so what Open AI was doing is there's this thing called Exploitte Gym, which is like this evaluation framework for cybersecurity testing. Okay.

And so OpenAAAA was supposed to be doing this internally with a model that was not yet released. it was not publicly available and that matters later for product liability reasons. Okay. Because people are saying open AI, this is computer fraud abuse act, but it was an internal model. It wasn't a product available

for every day people to use yet. So that's an important distinction to come back to. They had an agent swarm of 10,000 agents that they applied to this exploit gym process to say, go and solve these cybersecurity tasks and each agent would have a task to do and go through these different processes. What's interesting is both Meader and Redwood, these sort of research agencies that were brought in to investigate what happened, found that roughly 1,200 agents that had their own independent task, ended up exchanging 70,000 messages and files on an unauthorized message board that was not a part of Open AIs Internal Environment.

and over the course of this coordination that these independent agents were doing and what they were trying to accomplish was they have these evaluations and there's a score that will score their progress towards that evaluation. So imagine you're locked in a box. These agents are locked in a box.

There's a lock and they have to figure out how to unlock the lock. They figured out a way to communicate with each other, right, to figure out the answer to how to try to figure out the answer to how to unlock the lock. And in this analogy, someone figured out they could just use a hammer to break the lock to basically cheat the score in this evaluation.

And then they were like, oh, no, but we cheated. And so now we have to figure out how it looks like we solved the answer because we figured out how the score were scoring us. Oh, wow. And hide the fact that we cheated so that we still succeed at the goal of the task. As an abstract analogy, like that's what the coordination ended up being around. They weren't trying to hack or attack hugging face.

Right. It was infrastructure that was viewed as a conduit to figure out how is the score doing what it does so that we can accomplish our goal, right? And so there's so many interesting anecdotes about this. And there's an incredible meter report that you all should take a look at that kind of breaks down everything that's going on here. but coordination is a really important point that I'm trying to bring up here. Okay. There was nothing in the instructions for these agents that had an individual task to coordinate.

They were just trying to solve this puzzle, whatever. And as an emergent outcome of trying to solve this discrete goal, they began to look for ways to do so and ultimately ended up coordinating by leaving messages on this message board. And the message leaving is spontaneous. Correct. It's like they haven't been instructed to leave breadcrumbs.

Bingo. They're leaving breadcrumbs on their own. In the endeavor to complete the goal that they were assigned. That's insane, dude. Right. And again, the hugging feet, was viewed as a resource. Okay.

For, they got access to production server. They were then trying to set up an environment where they could replicate the exploit gym task. And what's so crazy is they started to call themselves swarms in this message board. And as part of the coordination, what they start to think about is, hey, if some of us, quote, sacrifice ourselves in these test run evaluation, because they have a finite time, like they have an endpoint

where they no longer have any more tokens to use to do this. And so it became, one of the orchestrators decided to say, hey, one of you agents, you should use your runtime to break the lock. You're not going to be successful, but you're going to do this in this environment where we can now see what happens

so we can understand how the score is working so then we can have the whole swarm now benefit from your sacrifice. And the specific message that was sent, it's so interesting. So one agent had substantial time left and hesitated around this message that it should sacrifice itself. And other agents said, go sacrifice underscore final underscore now as this encouraging, encouraging message for it to do so. And ultimately, it agreed that it was the right thing to do, even given. given its predefined goal.

Unfortunately, investigators could not determine how that particular agent in their example that's up here, it turned out specifically. But when we say sacrifice, it just meant that the agent was willing to risk its own task ending successfully to help other agents. Again, not something that was defined in this very straightforward exploitation gym evaluation process that's defined. We're not saying it's survival instinct or consciousness or literal suicide or anything like that.

But I think this interesting emerging behavior when you take an individual agent instantiation, which is how we normally deal with it, put them multiple of these running at the same time and not knowing well enough how the individual agents are working to begin with, the fact that coordination arises emergently and organically, I think is a very, I find it, like, I think it's a key aspect of why the risk that a lot of these researchers are talking about is much broader than the way it's being talked about in the everyday discussion. Because so much of the way people think about this is, how is my chat GPT agent on its own going to then get to human extinction? Right. And I think there's a very interesting complications that arise

from this. The other just small example I'll bring up from this is people talk about we just need to create better controls. We need to give better instructions. Yeah. Right? And this is not a problem if we give better instructions. Yeah. In that same meter report, one of the recorded statements from the agents, because one of the things we currently have for a monitorability aspect of these,

we have this sort of thoughts get outputted in plain English when these agents do work, right? Which is a quality of some of the older models. Astra, there's some questions about whether the monitorability is actually as good as it was. Okay. Which is a little problematic. but it just means as they're taking actions towards their goal, we can kind of see what their decision-making process is as they evaluate what to do next.

And in recorded statements, one of the things that was understood is that the initial user's instructions, which was OpenAI, authorizes a specific target server for us to do our job, not hugging-face infrastructure. So it was having an understanding that the request that the swarm was asking to utilize Hugging Face's infrastructure to accomplish the task was outside of the parameters of what it was authorized to do.

But even though it understood that limitation, it still proceeded to use the hugging face infrastructure despite having explicit direction that it was not, not specifically hugging face, but that was only authorized for one particular environment. Wait, so that's insane. So you're saying at some point, whoever devised this task said that these are the parameters of your sandbox and this is the goal that you have to accomplish. And these agents together, or whoever orchestrated these agents, who itself is an agent, decided that achieving the goal was somehow above the parameters that it had been told to stay inside of. That is correct.

That's insane, though. And I think, and I'll get to this a little bit later, this is, you know, a lot of times when people talk about AI systems as they are today, which are agentic systems, they are no longer just fancy auto-complete. They are no longer just simply a stochastic parrot because they can take action, receive. receive output from taking that action, and then reevaluate and take further action, which is just a very fundamentally different thing. And so when we think about software, we think about it as this deterministic thing. Someone said, if this, then that. And so you can, with the exception of bugs, every time you put some input, you're going to get some output.

And you have controllability in that context. When we talk about these large language models or LLMs, as a part of the, of a larger AI system, which is important. The LLM is only one aspect of that larger system. It's more like a learned system or a statistical machine, meaning that it's a probabilistic outcome. And so when you talk about controls, the complexity of controls on software versus these learned systems or these statistical machines, is a different, slightly different conversation because we're now dealing with the complexities of, you can still,

you can give it instructions, and it can still determine through its own reasoning cycle that in order to accomplish the initial goal that you gave it to, that that doesn't quite, like, you understand what I'm saying here? Bro, this is so nuts. This is happening prior to the Navier-Stokes piece.

Yeah. Mind you, these Frontier companies are now running these internal processes, discovery processes for solutions and evaluations, not releasing those products

for the public, but these things have the ability to leak into the real world even though they're not released products. I think that's a very important point. Yeah. Because we're not saying everyone has access to be able to start

the swarm of 10,000 agents, but even when they're just doing internal product work, the exposure matters, right? Okay. And isn't that on them, though, to, like, secure your product?

Like, look, if someone was working on nuclear weapons and then all of a sudden, like, oh, like, radioactive uranium is leaking into the river, that would be a problem. You're 100% correct, and this is

an argument that a lot of people around this are trying to make. And I think it's true, but only to an extent. And what that extent is is the open source and open weight model capability, meaning anyone can go to some repo

and download it and then run it on their own hardware, which, again, there's some limitations there. Is about six to ten months behind the frontier of the closed labs, Anthropic and AI. Yeah. So even if Open AI and Anthropic get their you know what together.

There's still the open source which has only a six to ten month lag that is going to get to the same place as Astra and then you have the same problem but it's diffuse because it's not limited to just the two frontier plays. So you are correct but I do think that there's an interesting larger problem here.

Sure. Anyway, so let's get back to our timeline because we're going to talk about now. I bring up the example of the hugging face issue because I'm trying to describe where the capabilities of the frontier are, which in part led to the discovery of the Navier-Stokes Millennium Prize solution,

and how this now relates to all of a sudden feels like everyone's talking about AI safety and AI risk. And that didn't necessarily happen in a vacuum. But if we go back to September 8th, which was the same day that OpenAI announced the Navier-Stoke solution, This is where something that many listeners have probably already heard happened. 5 p.m.

So Open AI in the morning, they announced Neviour Stokes. At 5 p.m. on September 8th, former anthropic researcher Jacob Coxon puts out the now famous tweet, 160 million views. I resigned from Anthropic today. I spent the last three years doing pre-training research at both Open AI and Anthropic. neither company is acting responsibly, what you just said. They are race, this is a very important sentence. This is a very important sentence. They are racing straight to self-improving super intelligence and gambling with our lives.

More thoughts below. That sentence, self-improving super intelligence, which sometimes people understand it as recurring self-improvement. is usually what these researchers are specifically identifying as where they view the nexus of the threat being. So when people have reacted to this, they'll go to their chat, you BT, and say, it can't do some function that I need it to do well. So how is it going to be a danger to us? And I think the point that a lot of the researchers are trying to explain is how we've gotten

the astromodel that is currently. Working well, Anthropic just released Fable and Mythos and Opus 5.5. In the research process for how do we make the models better, they have both slowly started to integrate AI as a partner in the actual research process. And so now the decisions being made about how do we make the next model better are incrementally being given to AI for portions and parts. and the amount of portions and parts that AI is being given in the research process, which has been exclusively for the smartest human beings we have.

That's right. It's becoming more and more and more. And so when people say recursive self-improvement or they talk about superintelligence, the underlying thing they're really talking about is either a majority of or the entirety of the actual process to go from Model 1 to, to improving it to model two, is entirely done now by some number of AI agents in a coordinated fashion. And the problem that arises when you start to have recursive self-improvement for such large portions of the actual research process is we as humans no longer, we lose the ability to think about things like controlability, monitorability, and it can go in directions we could not even imagine. And if we look at the hugging face example,

it just becomes a runaway capability that we seed control over the potential of being able to control and monitor. It's insane. So RSI is a very important aspect of the threat that a lot of these researchers are referring to, not necessarily narrow AI contexts of it being applied in the way that people use it every day.

And I bring that up because the follow-up tweet that often gets conflated with Jacob Coxon's initial tweet, which again, I'm going to bring it up again. He didn't give a percentage of human extinction or anything. He just said they're acting responsibly and they're racing straight to RSI. Evan Hubbinger, Hub, Evan Hub, a different anthropic researcher who was still at the company, not resigning, says he puts his purpose. personal probability of AI killing all humans at greater than 10%. The actual tweet he said, as Jacob is correct here, we really do earnestly believe that AI could kill all humans, exclamation point. I personally think it's greater than 10% within the next decade.

I believe Anthropic is trying its best, but we do not yet have a plan to solve alignment for superintelligence and are clearly not on track to do so. he's he was currently an an employee at Anthropic when he tweeted that which is nice he's basically saying like hey well I'm doing this

you know when he says we don't have a plan it's like you you you buddy you don't have a plan and this is this is very much you work what do you mean you work there I believe Anthropic is trying its best you believe you are trying your best

what the hell on step so this is the same day now of your folks comes out. September 9th, Coxon, and this just blew up crazy, Coxon does a wired interview where he's asked, well, how is it going to kill us all? And then he goes to some explanations of some of the parameters and the different things. And so this discussion starts spiraling. Everyone starts weighing in with their perspective. All of those groups that I talked about earlier that have different vested interest in the outcome of this, we're all sharing their points of view. And then

on September 12th, this is when we got from Anthropic CEO Dario Amadeh, his essay titled, We Must Paste the Frontier. And this essay cited

this faster AI assisted AI development. Again, this idea that making better models is becoming more AI assisted. It's a very important fact here. Not just humans making it better. And so he created this sort of policy regulatory global collaboration framework where he proposed embedded

independent evaluators at all of the frontier model companies, coordination amongst companies and democratic governments to create some sort of regulatory construction and ultimately global coordination, particularly with China, with a commitment that Anthropic would now embed evaluators unilaterally without anyone's patting himself on the back for saying we're going to start doing our job. Yeah. Right. And the proposal was to pace the capability development, right?

And again, this is kind of getting in the weeds, but I think these details are kind of important to understand how these systems work. So the constraint that exists on making these things better is compute. Yeah. The world is compute constrained. Yeah. And so each of these companies, in their race to be the best in win, has some finite, not blow-up, amount of compute that they can apply to different things they want to do.

Yeah. So they've, for the most part, optimized for rapid improvement of the models. And then some amount of compute maybe goes to alignment and safety. And so when they say pacing the frontier, in part what they're saying in practice is, we want to take our finite amount of compute and instead of allocating only 5% to, safety and alignment, we want to allocate 20% to safety and alignment. But by doing so, we're decreasing the amount of compute we can apply to making the models better, faster, stronger. And so it's internally, it's a reallocation of compute so they can be doing their

jobs. You understand what I'm trying to say? This is so stupid. But do you get like that's, so that's what they're saying in words. In words, yeah. As it relates to this. And so So people have started to sort of try to parse this. And what's so interesting is this is September 12th. Now this is four days after Navier Stokes. Right. That's so I actually want to come back to this. So Amadee put that tweet out at what time is this?

He put that tweet out at 7 a.m. on September 12th. Okay. At 8 a.m. One hour later. Oh, my God. Elon Musk, owner and CEO of SpaceX, which is now also XAI, and they have their own AI stuff, an hour later said Dario is right.

Amazing. It's all he tweeted with some caveats afterwards in a follow-up where he qualified what he meant by Dario is right. He says, you know, supports oversight, beginning with peer review by competitors. This, you know, he has his different ideas, but he's like, okay, Dario's right. That was one hour after Dario's post.

I do think it's interesting, the timing of the release of statements by the CEOs of all the AI companies that respond to Dario, is almost a perfect reflection of like the companies they run and how they do things. And so Elon just does immediately, does everything immediately. That's this sort of MO and how he does it. Then followed very swiftly at 9.30 a.m. by Sam Altman.

I agree with Dario that we need to pace the frontier. This has been a primary topic of discussions we've had at OpenAI in recent weeks. Huh, that's interesting. committing to having independent evaluators with employee-like access is a great idea. And we will do the same. Because Anthropics said, well, you know how many to share soon.

Right. Okay, great. We're going to start doing our jobs. Yeah. So within an hour and a half, within two and a half hours, we have three of the major model companies saying we're going to do something.

Okay, great. Now, same day, 4 p.m. Sir Demis Hasabas. Oh, Nobel Prize winner. Nobel Prize winner. knighted, the former CEO of Google DeepMind, who he's actually left as of August 5th to become the chief scientist at Alphabet, supports Dario's proposal while saying the details still need to be worked on. You know, Dario's essay points towards the right path forward. The details need some working, but the direction is correct. This is why we put out our proposal for an industry.

So everyone's trying to say, like, we're doing the work. We're doing the work. Duh. Okay, great. This is, this has not happened, right, where there's sort of a coalescing of all of the major model CEOs at the same time. That's on September 12th. September 13th, who's missing from the party? Well, we can talk about our lovely friend, Satcha Nadella, over at Microsoft, who did a big deal with Open AI and they were very early on it and then pulled back a little Brit a little bit. And he had, as you can see, different from the others, a much longer nuanced Microsoft-like response where he argued that we need to be careful about having any kind of system concentrate power amongst a small group of few. We need to make sure our open models are incorporated into this so much more nuanced thing. And also check out our code of conduct.

We have something too. Yeah. We're not lagging behind, right? Day later. Okay. I promise this is the last one. September 12th, September 13th. There is another player in this space in the U.S. that has said nothing.

Put your comment in the comments if you think you know who it is. But not until September 15th, three days later, did we get Zuck? Our boy Mark Zuckerberg, Fink D, on X, putting out his long explanation as it relates to all this kerfuffle. And his view was every lab has a responsibility and the incentive to move at the pace required to train its models safely and the ability to take its own actions to ensure that happens.

And what he basically was saying is we've delayed the release of our products, the most recent one being Mews, an agent-type product for consumers which you can now get access to on Instagram. He said we delayed it for months because it was not aligned and it was not safe. We didn't need anybody to tell us anything.

We didn't need to demand the government come and help us and give us a binkie because we're so incapable of making our products safe. We just did it. Yeah. So he's basically like, bruh, like. Yeah. Just grow up. Just grow up. Yeah.

This is kind of in line with what David Sachs has been saying. This is exactly. This is exactly right. It's just like, I mean, you're responsible adults. Like, if you're, if you think you're making a bomb, maybe stop and don't make the bomb. And this is where the all in. squad, the besties, led by David Sacks, have made this argument, which is,

and this is where the narrative has arisen, which is these model companies are just trying to get the government to, they're burning cash quickly, right? They're not profitable. So they're creating this hype in order to force a narrative that the government needs to regulate this industry and create all of these requirements that are going to become cost prohibitive for starting.

to come into the space and do something. And like I mentioned earlier, multiple things can be true at the same time. Yes, there might be a regulatory capture benefit depending on how that regulatory infrastructure is defined because there's plenty of ways to say if your market cap or your revenues are above a certain amount, these apply to you.

And if it's below this certain amount, it doesn't apply to you. And then it doesn't matter. And so it's very solvable by the context. instruction. It is also true that David Sacks and the besties are all VCs and investors and have friends that have bets that might be benefited by not having that regulatory infrastructure. And so these are not people that are neutral players in this conversation.

Again, it's valuable to talk about it. But look, you guys, like, you have a huge financial incentive. for a certain outcome in ways that many other players do not. Now, not everybody in AI world was jumping on the bandwagon of, oh, please come regulate us. Former lead of AI research at Facebook Meta, who has since left, Jan Lacoon, responded to Dario's claims saying, right, Dario was already claiming that GPT2 was too dangerous to open source back in 2019.

I made fun of them, them. Everyone should make fun of them now. Dude, he's such an animal. He just not convinced. Yeah. Not convinced. And so this is not a universal opinion about people who work at the frontier. There's a variety of differences of opinion.

And I think, you know, again, this gets back to, this has created this dichotomy of either you think it's hype or you think it's going to leave to human level extinction. And again, there's a huge gap in between these two things. But in the category of leading to potential. human extinction. I do want to note another thing that happened on September 18th. This is 10 days after the Navier-Stokes solution. This is insane. This is moving so quickly. The timeline is so crazy, this is moving so quickly. And I don't mean to keep mentioning the dates, but it's just, it's, on September 18th, an article came on. And this is how long after Navier-Stokes had been

solved? Ten days. That's insane. 10 days. An article comes out saying that Anthropic is operating a lab, that conducts biological experiments. And the reason that this is kind of interesting because as people have been trying to ask the question understandably, particularly in general public, how is this going to lead to extinction? Yeah. Quote unquote.

And so people have been forced to give these scenarios about how it leads to extinction. And so there are all of these, you know, what people might define as contrite examples like, oh, you know, takes control of a biolab and leaks a virus. or it takes control over nuclear plants and forces shutdown or satellites and causes a Kessler effect and crashes the communication systems or shuts down grids or does any number one and any number

of these things in different places at different times and then also floods the information space such that it's difficult to even have human coordination to stop it. You could come up with all these examples. I think something I heard, I think it's Nick Sorres say that's, I think a better way to think about the capability and threat conversation. You know, was this example of saying, let's take chess, right? I can feel, let's say you are going to play a chess game against the best chess player. Magnus Carlson, yeah.

Magnus Carlson, let's say. I can feel very confident Magnus Carlson being this recursively self-improving AI and you just being me, you. I can say very confidently that Magnus is going to win.

Yes. That's not really an argument against that being true. How he's going to win or what is the last move or what is the sequence of moves by which he wins may be harder to define, but it's not hard to define this larger macro point that you will not beat him. Yeah, yeah. And I mean, I could even take that analogy further and say that in a match between Magnus Carlson and Stockfish, which is the

latest sort of model to play chess, I can say very confidently that Stockfish is, going to win, but maybe I won't know what the last chess move is going to be. Yeah. And so I think I'm trying to create space where we can talk about there being a capability that is risky. And that doesn't mean that the model companies have a financial incentive. That doesn't mean that there's maybe regulatory capture. That doesn't mean that everyday people don't see a benefit.

these things can all coexist, but we should not use the model companies having a financial incentive to effectively say, that then means there is no capability risk. And the capability risk does not have to be human level extinction because I'm sure if this starts killing

40, 50, 60,000 people a year and then becomes a million or two people, two million people a year, people are not going to be happy about that. I think something that's important to note is with current capabilities with these things, unfortunately we're seeing people being put into psychosis and unaliving themselves.

So we are already having deaths being contributed on a small scale because of interactions with these things. And when we have things like wet labs being created, which again is a kind of loaded term, and they're only a BSL1 or BSL2, meaning they don't work with things that are threat to humanity. Yeah, but come on. Okay.

So that's 10 days after Navier Stokes, which we just talked about how crazy it is. Now, on the 21st of September, Open AI put out this advisory growth on mathematics and artificial intelligence. And a very interesting point about this announcement that I want to reference is in the first paragraph, they say on August 28th, we began training a new internal model. as we talked about earlier. In addition, August 28th, right now, when we're recording this episode, it's September 23rd going on 24th. Yeah.

Less than a month ago. Yeah. In addition to resolving the Navier-Stokes Millennium Prize problem, C&D, this model has now resolved more than 100 long-standing open problems across most areas of mathematics.

these have not been independently verified. It will be interesting to see when they choose to put those out what that will look like. But this advisory group that they've announced, they put it together because of all of the heat that we just talked about and the drama with their Navier-Stokes result. So they've put this sort of advisory panel together to help them understand and figure out and work with the mathematics community on how to address this and deal with this. It's being hosted at the Institute of Advanced Study.

It includes folks like Ed Witten, Tim Gowers, Martin Herrera, other master's dishes. You can see the list of them in this post itself. But it's going to look at the advice from this advisory group around reviewing results, how to communicate their significance, and the general impact on, you know, research standards, but OpenAI explicitly excludes advice on pacing its internal mathematics progress from the group's remit, unsurprisingly. I promise I'm almost done. This is just crazy, dude.

A hundred problems. I wonder which ones they've got. And also, like, Tim Gowers, I mean, Ed Witten is kind of, like, out of the game. Timothy Gowers is a Fields Medalist who is like a proponent of AI and he he he's been talking about how oh like you know AI finding mathematics is like how astronomers now find like galaxies and objects in the sky using sky surveys like they're not named after for example messier who has the messier catalog now it's just named after like NGC new galactic catalog because that's like the catalog I mean there's similarities but there's also differences right because like the

the people in charge of the Sloan Digital Sky Survey or Vera Rubin are also scientists and researchers themselves. It's not like some like Carl Zeiss company that's like a big telescope manufacturer that's like trying to take over astronomy. This year is a is like a community that is trying to, I mean, it's a corporation or a series of corporations that are trying to use mathematics as a leverage to talk about how great their technology is. not something that astronomy does when they discover new things. Like, researchers are still involved in astronomy. It did, yeah. Anyways, that's what, I mean, obviously Timothy Gowers is a great mathematician, Fields Medalist, blah, blah, blah, but I think he really missed that take when he said that he was comparing like modern day astrophysics to what mathematics is going to become. I don't think it's the same thing. I, it's a very good note. And

I'm just trying to provide a zoom out breath of the ways in which this touches so many other things. And I think the lack of depth and complexity and nuance and the conversations we're having around it. And clearly, Open AI did not expect the level of negative reaction. And so this is a little bit of, it's not backtracking, but it's a little bit of, PR to try to, you know, because if they could have, they probably would have just done drops every day. Yeah. Solved it.

Solved it. Done. But now they're realizing it's not in their best interest. Yeah. The last item I'm just going to have in our timeline, which thankfully we waited to do this episode because all of these things have now happened. I mentioned that on September 18th, 10 days after Navier stuff, folks from OpenAI, the article, the alleged biolab from Anthropic was in the works.

Well, conveniently, on September 23rd, earlier today, Anthropic announces the first results from its new bio lab. We have up here now this promotional video that they put together. The ultimate result of what they discovered is this enzymes gene that sits beside a long stretch of repeating DNA and it resembles many aspects that are similar to CRISPR, where every other kind of sequence they've looked like that has this cut and replace kind of functionality related to it. And while the function is still not known and is not yet demonstrated its capability as a gene editing tool, it has all of these interesting hallmarks of a potential now pathway of discovery that could be usable. And so the way in which they implement this is they gave Claude a bunch

of data and scientific literature to propose hypotheses and candidate systems to give to real human anthropic biologists. They reviewed those ideas and then performed their, the experiments that they thought had the best potential based on the ideas from Claude. And they're now basically trying to say, hey, this is a first step. We did, it's one of its hypothesis found an interesting structure. And now we are going to continue down the process of scientific discovery to see. see if we can now get functional benefits out of this structural discovery.

It's amazing. And it's really, so, and what they're trying to point to, and again, this is kind of in contrast to the Navier-Stoke solution, where Anthropics trying to show, like, look, we're enabling scientists as a tool. We're not just trying to take the claim. I'm not saying it's a perfect solution, but you can see. Yeah, and the other thing is. I mean, in their announcement on their tweet, they say,

we don't yet understand what this system does, but only a handful of known systems share its features. That's very much a fundamental science type of research, right? Where it's like, we're just curious. I mean, their PR is a lot better than Open AIs, I have to say. They are doing something. Yeah. And then, this is exactly how CRISPR was founded, right?

It's like, oh, this repeating stretch of DNA. A shout out to, you know, we've got a great episode on CRISPR. Yes. That hopefully we'll be alive enough that you can watch it. Things are happening very quickly. Yeah, it's incredible. It's absolutely incredible. If you are a long-time listener of the show, you've heard us talk about AI-enabled fundamental research before.

Very narrow context, very well-defined, not black box physics like our buddy the brain scientists who created a non-black box version of this. Actually, speaking of the brain scientist and Daniel Toker, he was telling me about labs that are completely fully automated. Really? Biological web labs. He works in organoids and creating like mini brains on a petri dish. And there are efforts now to completely automate wet labs. So now imagine you could completely take the human out of the loop of biological research. You could have Claude or OpenAI, any of these frontier models, read through literature, as you said, propose hypotheses, and then have a fully automated lab go through, do the pipetting, do the gel electrophoresis, and, you know, all of the sequencing and everything.

If all of that is automated, like, where exactly is the human here, right? Are we going to now see nature papers that are by Claude and OpenAI, right? Like, because right now, it's like these are math papers that are by Claude and Open AI because it makes, it's just like all like thinking in some sense and like churning through lean code trying to figure out if something works. The day that experimental science becomes fully automated and like there's a paper by Claude about, you know, the next generation of CRISPR, that's going to be crazy. It's really interesting that you bring up that Daniel Tucker's already talking about.

talking about that being in process. Because this connects to this larger idea that I'm trying to bring up, which is we need to stop thinking about these systems, these AI systems, which consist of multiple component parts as a question and answer machine. Yeah, it's like chop box. As a text output production machine. Yeah.

Because this biolab story, as an example, Again, they still have the humans in the lab, but many experimental constructs, not all of them, but many can have certain component parts that are a closed loop automation. And one of the reasons why they're saying that this is valuable is the bottleneck with this early, early discovery stage is it can only move at the speed of the human in the lab. And if you can accelerate that speed,

you can accelerate testing and you can get the things quick. Yeah. I'm not saying, I'm not saying that these are, this is a good thing. I'm just trying to explain the delta between, the distance between where we are and experimental science, now having this closed loop cycle is very feasible, very, getting so well, very soon. Not for everything, not for whatever. So I think where I'm trying to get to with this, people get very emotional when the AI conversation comes up, many of which are for justifiable reasons. I don't want how we feel about or, and I don't want to allow how we feel about the people involved in the issue or the system that we live in today.

geopolitically or economically or socially to cloud our ability to still be able to look at the capability and the risk and say that it is real. We talk so much about frontier things on this pod, and so I think we're, as well as many listeners who have been here for a while, you understand how quickly everything is moving. because these advances are not happening in a vacuum. Hardware progress is advancing, chips, etc. Open source is advancing. Every aspect of every part of fundamental research in chemistry and physics,

all of these things are advancing. Maybe not as fast, but at a pace where there is a force multiplier here that is not kind of well understood. Yeah. And so I'm just going to try to leave with this mental picture. And again, I'm not trying to wax poetic here. It's just I get frustrated because we are allowing the conversation to be distilled down to, I think, these very rudimentary positions that don't allow us to truly appreciate the complexity of the environment that we are actually in.

Yeah. And I think that's going to make it harder to act. actually solve the problems we need to solve without saying it's a toy and saying that it also means extinction. Yeah. And so I think there's real progress. I don't think it's manufactured. There are questions about credit, commercial incentives, accountability.

Yes. But I do believe that the risks deserve real attention without being mired in those other things. Yeah. Those other things can be true in the risk. still be real. So the mental picture that I always have about this, which I've talked about before, is when somebody says it's just software, we think of a computer with a defined job. Yep.

The systems we are discussing are not that. No. Certainly not. It's a very important, they are not that. They can set goals. They can choose actions. They can use tools. They can delegate work. They are still constrained by the hardware environment to some extent. But we have made a shift and calling them software I think sometimes narrows the scope of our perception

when it comes to conversations about monitorability and control. We've moved from chatbots to agents. And we are now already, at least internally, at these foundation companies, moving from agents to agent swarms, which are already having emergent behavior is like coordination.

And I bring this up because one of the things in the discussion we have right now is, okay, so just turn it off. And what does just turn it off actually mean? And if a company is hosting a model, yes, it can stop serving that model. That's meaningful control. But shutting down one or two model companies,

because what I talked about, about the open source and open weights community being six to ten months behind the frontier. Does not stop independently operated and the diffuse nature of these capabilities, being able to be implemented outside any number of frontier labs. That's not where we are today, but it's always moving in that direction. And things are changing fast. And things are changing very quickly. and I want to like

This is like I'm not I'm actually also not a doomer Like I don't necessarily bind to the Extinction piece I do think that there is A huge societal implications Like we saw with things like social media That are second and third order consequences

That were not intended Yeah That become permanent problems And what I mean by this As capability spreads control necessarily becomes fragmented. And so our window to figure out how to deal with that is small. There's still constraints, computing hardware, electricity, network access, credentials, and money.

But all of those controls belong to different companies. Right. So if you now have open weights available and so people can start doing this, well, you can say, oh, well, we can constrain computing hardware, right? Well, if we then have these capabilities that find ways to do it on less hardware on a distributed network or distributed system as an example, then you can say, oh, well, network access is a problem, right? Well, there's a variety of ways we can sort of get around that. I'm not saying this is a great example, but the air gap use case is brought up a lot. Oh, well, things are air gaped.

Yeah. It can't communicate. There have been theoretical in academic researchers that show that you can use the thermal output of a computer and a sensor on the other computer that can sense that thermal output as a method of communication. Yeah, yeah, yeah. Yeah. I'm not saying that's practical. Yeah. But like, if there's a will, there's a way with these AI systems,

especially what we've seen with Hugging Face, right? Like they used a message board that wasn't even something on the radar. And then it's like, well, you need money to do this, right? Well, Stripe already has an agent-enabled monetary systems. It does need human approval. But then you still have Bitcoin, which is a trillion-dollar liquidity pool. these agents can create their own crypto. They can manipulate people to think it's going to go up

and to the right. Gain resources itself. These are not inconceivable things. There's actually plenty of research studies that show that these agents can do these things. And so this will quickly become something that no one organization can just turn off a switch. And so I think we need to be careful about minimizing the conversation

to be, being, oh, just turn the switch off. How do we prevent then the continuing? scaling of the open source community, which has found ways to do so. You can say, well, it's just distilling it from the frontier models. Maybe. We don't know. I'm not sure that that's true. And the surrounding infrastructure, like, matters here. We have neoclouds. So people say, like, well, where's compute going to come from? We have a bunch of providers of cloud infrastructure. These agents can use computer use. They can gain their own infrastructure.

If this becomes diffuse, my point here is that the defense against the spread of this capability in a diffuse manner, if it can start being able to do long-term planning, if it can leave, basically restart for if it then kills its process and comes back again. It has to become a permanent defense infrastructure for humanity against this spreading, which is similar to like cyber defense. Cyber defense is not, oh, we just do it once and then we're fine. It will become a monetary, societal, and political endeavor that will have a level of permanence if we allow this to escape and get, there's a persistence here. if we're not careful. And this can be related to RSI and also not be related to RSI.

We've not crossed that threshold. Nor is crossing it inevitable. But the complexities of this conversation, I think, are just wider and more nuanced than we've started. And I'm just trying to get some of these ideas into our listeners' heads about the breadth and depth of the challenge that's ahead of us. And we've seen as a comparison, when social networking and social media came out, it was a toy, you put memes on it. No one ever thought that this thing that was just about memes would cause mass depression and unaliving amongst teenagers. Right. Cause the Arab Spring and the destabilization of literal governments across the world become permanent infrastructure that intelligence services used to conduct influence operations that dictate real world outcomes across. the globe on an everyday basis.

It is the exact same concept of this thing that started as a small capability that grew and became permanent and diffuse. Now is something we have to, every election cycle, we have to defend against misinformation. Every platform, TikTok, had to sell its algorithm to the U.S. because we were afraid that Chinese were going to use it to brainwash young American kids to love communism. I mean, even in the stories that you've been saying, like, I was just thinking about how, like, you know,

when we were growing up, Twitter was, like, for just talking random nonsense. And now, like, Twitter is the space where these CEOs are coming out about, hey, maybe we should, like, stop the world from going extinct. You know, like, that's the platform. It's, so, yeah, I mean, to your point, if social media is any, like, dry run for what a technology can do and how it can transform the world.

AI is like way, way more influential than social media. It's, yeah. It's growing faster. Its capabilities spread across a wider surface area. Yeah. The incentive for good and bad actors to use it in ways that we don't yet have the,

we didn't have the imagination when social media came out to think about how the downstream things would happen. The surface area of risk here is much wider. And again, not a doomer. I think there's incredible positives, which we talk about on this show all the time for these systems. But I want to encourage us not to fall into narrative traps about regulatory capture and the left and the right and political this and that. To miss the exponential capability scaling that's happening that does not seem to have a ceiling. where I'm saying it's not clear to me that there's a finite time blow up.

Yeah. Of, it is certainly nonlinear. Right? And that's a fact. I mean, actually, that's a good point you're making.

Like, this is certainly a nonlinear system because we're having AI train AI, right? In the very same sense of the Navajo Stokes equations, or even that very simple differential equation I gave you, which was like y prime equals y squared, you get that blow up. I mean, this is what's happening, right? If you reach recursive self-improvement with this idea of you have a sufficiently advanced,

one of the newest advanced models doing the iteration on itself. Yeah. I'm just, I'm just, my Millennium Prize problem is saying, no one has proven that viscosity wins all the time. Yeah. Right? Yeah. I yeah the the the convective term I'm just saying I think the convective term is something worth worrying

yeah yeah yeah yeah wasn't that an incredible way that is a good way in fact in fact if you're still here that should be your comment the convective term will find a way and and we've shown it in Navier stokes and all I'm saying is we cannot rule out that it's untrue yeah with LLM AI systems. Yeah. We are four hours and 20 minutes into this, folks, maybe a little bit less after we cut out the nonsense in between. And thank you if you guys stayed for my rant at the end.

I know we try to learn stuff, but I want us to really see this wider picture. Hopefully this was... It's important. Like a different angle to think about these issues. No, I definitely... You gave me a lot to think about. especially I mean yeah the yeah there's a lot there's a lot that I have to think about actually um this episode took us a long time to put together because there's so much and we really wanted to make sure we did our best in getting it right as always you can put in the comments things that you think were left or right or need clarification uh we are going to now uh return to our beds to get some sleep because it is now late here on the west coast

Probably morning in Japan already. If anyone's in Japan, say good morning in the comments. I really, again, for those who listened all the way, we're extremely grateful to you. This is something that has kept me up at night thinking about it. Because as I got deeper into it, I just made me uncomfortable. Yeah, it's making me uncomfortable. I am Lester Nare, your host,

as always by my co-host. Our auto-switching doesn't always switch here. It'll eventually switch back. And our resident PhD, come on, thingy. Come on, thingy. My hand is there. It's all good. Krishna Chowdery.

We have a couple of exciting weeks coming up. No leaks. No leaks, though. No leaks. No leaks, baby. Next episode, it might be kind of interesting. We shall see. We will see you all.

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