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

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. Summary How Newton’s laws become equations for a moving fluid Velocity fields, incompressibility, pressure and the nonlinear convective term Why viscosity smooths a fluid while nonlinear motion can create finer structure What finite-time blowup means, and why simulation is different from proof How forced and unforced equations differ, and why those assumptions matter Earlier work on Euler, Boussinesq and related fluid equations OpenAI’s claimed result, Lean verification and the scope of the theorem The dispute over scientific credit and the human research behind AI-assisted work The METR investigation of the Hugging Face incident Emergence World and long-running multi-agent experiments AI-assisted biological discovery, oversight and recursive self-improvement Separating demonstrated capabilities from claims and future scenarios Chapters 00:00 Can AI solve Navier-Stokes? 00:34 Episode introduction 02:20 Navier-Stokes: Mathematics Meets AI 10:20 Building the Equations of Fluid Motion 21:34 Velocity fields, divergence and incompressibility 34:33 Acceleration and the convective term 52:18 Why Fluid Motion Is Nonlinear 1:02:08 Pressure, Euler and the Missing Physics 1:14:50 How Viscosity Changes Everything 1:33:36 Solving Equations vs. Simulating Fluids 1:44:59 Can a Smooth Fluid Blow Up? 2:13:52 The Road to the Claimed Breakthrough 2:31:13 Inside the Claimed Navier-Stokes Proof 2:48:26 The Dispute Over Scientific Credit 3:07:23 From Chatbots to Agents 3:08:57 The METR report and Hugging Face incident 3:23:23 AI Risk, Oversight and the Race Ahead 3:38:51 AI Discovery Beyond Mathematics 3:52:47 Why “just turn it off” gets complicated 4:06:47 Closing thoughts and what comes next 4:08:46 Outro Featured Research OpenAI’s Navier-Stokes announcement Tristan Buckmaster’s statement METR investigation Emergence World AI-assisted enzyme discovery Explore FFP ffppod.com ffppod.com/funding ffppod.com/transfers ffppod.com/America250 Watch on YouTube youtu.be/NGfGw1tGxUY Support the show ffppod.com/donate Follow @FFPPod on X / Instagram / TikTok / Facebook

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Krishna Choudhary?

  But 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.

• 0:08Lester Nare?

  It has this counterforce.

• 0:09Krishna Choudhary?

  Yes. And so you're you're asking who's gonna win? And are they always going to win?

• 0:14Lester Nare?

  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.

• 0:26Krishna Choudhary?

  Yeah.

• 0:27Lester Nare?

  Those other things can be true and the risk can still be real.

• 0:30Krishna Choudhary?

  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 Choudhary. 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 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.

• 1:13Krishna Choudhary?

  So we're gonna do something a little bit different than some of the mainstream coverage that starts with the drama. First, we're gonna ground ourselves in the fundamental understanding of the mathematics as it relates to the OpenAI 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 gonna talk about the science and some of the larger complexities. If you want to understand science headlines, be sure to subscribe.

• 1:58Krishna Choudhary?

  As always, we are gonna talk about the science from the ground up today because this is From First Principles.

• 2:20Krishna Choudhary?

  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 gonna 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.

• 3:04Krishna Choudhary?

  You know, we're also anticipating GTA 6.

• 3:06Lester Nare?

  Yes.

• 3:07Krishna Choudhary?

  And this is the ah, here we go again. Because honestly, I didn't 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, this, this is history, if it's true. You know, this is a big deal. And it's not Anthropic this time. It's OpenAI. And the announcement was met with a different kind of reaction than the other two. In 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.

• 3:52Krishna Choudhary?

  There were allegations of academic misconduct. There were questions about who deserved credit. There's questions about what OpenAI 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 Levante Alpoge, 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 gonna do something very deliberately. We're gonna talk about the mathematics. Because at the end of this two-page document that Buckmaster published, he, which really kicked off the controversy, this, this document where he talked about all the shenanigans that had happened behind the scenes.

• 4:38Krishna Choudhary?

  He makes the argument that if OpenAI 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 gonna begin. We're gonna do the math first. We're gonna do the drama second. So here's the roadmap for the episode. We're gonna start from first principles and try to build up the Navier-Stokes equations. There's a 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 gonna be some calculus involved. There's gonna be some equations, but I have a lot of visuals.

• 5:23Krishna Choudhary?

  So hopefully, it's not gonna 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 dynamicist. 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.

• 6:07Krishna Choudhary?

  And so I went back to our trusty old Landau-Lifshitz course in theoretical physics, volume six, 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 had, we got to apply some of my favorite mathematics, i.e. complex analysis, to calculate the wing, the lift on the, 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?

• 6:50Krishna Choudhary?

  Not just prime numbers, as, as we've been known. So at the end of all that, we're gonna 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 gonna have to zoom out. This is not happening in a vacuum. And there is a lot of crazy developments that are honestly happening, happening daily.

• 7:11Lester Nare?

  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 like 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.

• 7:49Lester Nare?

  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 GPT-2. And I've seen what the progress looks like from a very direct, personal context. And I do think that there 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.

• 8:33Lester Nare?

  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 gonna 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'll help really ground us around this shift from this is fancy autocomplete

• 9:18Lester Nare?

  or this is just a better way to do search. And this agentic world that has been actually a driver behind how OpenAI got to this solution. So we're gonna 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

• 10:01Lester Nare?

  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 gonna jump right in to what is the meat and bones of what's gonna be, I think, a very fantastic episode.

• 10:17Krishna Choudhary?

  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, 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?

• 11:02Krishna Choudhary?

  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 gonna 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 the, 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.

• 11:46Krishna Choudhary?

  It's like, how much, how much resistance to a force do I have is literally what mass is. And there, 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, the other one is attached to two of the same blocks of wood. And you can see that the, the single block of wood accelerates a lot faster than the one with two blocks of wood. That's because the, the two blocks of wood has twice the mass, so it's gonna 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.

• 12:27Krishna Choudhary?

  And if we wanna 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.

• 12:41Lester Nare?

  This is partly what enables us, for example, to land on the moon.

• 12:45Krishna Choudhary?

  Everything.

• 12:46Lester Nare?

  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.

• 12:53Krishna Choudhary?

  Yeah, really. In order to get to, 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 wanna go through a singular example of a very simple problem where I can have a Newton, an equation of motion derived from Newton's laws, and I can then figure out what that thing is going to do forever. Okay? A common example is a mass on a spring. Okay? A spring is a substance that obeys Hooke's law, meaning if I pull on the thing, it's going to resist and it's gonna pull back. And how much it's gonna pull back is dependent on how much I'm stretching it.

• 13:38Krishna Choudhary?

  If I stretch it twice as far, the pull back, the force backwards is twice as much. That's called Hooke's law. And you usually see it as, um, 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, 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 Hooke's law. F equals negative Kx. Now, force, according to Newton's second law, is mass times acceleration.

• 14:09Lester Nare?

  Yes.

• 14:10Krishna Choudhary?

  Right? And the acceleration is really a second derivative of the position of the x, right? The acceleration is a change in velocity and the, 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, 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.

• 14:54Krishna Choudhary?

  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 sine. The derivative of cosine is negative sine, the derivative of sine is cosine. So I'd go from cosine to negative sine to negative cosine. Well, that's just the negative of what I started with. And there, I have an oscillatory motion.

• 15:36Krishna Choudhary?

  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 physicists can then borrow that intuition and boom, you've got, um, a solution that tells you for any time, I can plug in the time and I can tell you where the mass is gonna be.

• 16:07Lester Nare?

  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.

• 16:16Krishna Choudhary?

  Yes, exactly. And you can think, 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 like the faster I move through some, like, let's, 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, 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.

• 17:01Krishna Choudhary?

  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, um, 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 a, on a, what are those called? The holes, the potholes? You get, 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, how much damping is happening.

• 17:43Krishna Choudhary?

  My point is, you can make the equations of motion a bit more complicated and still be able to solve stuff.

• 17:49Lester Nare?

  This is the visual I think of as seeing those meme videos of those giant trucks with giant suspension that's visible, going over, you know, this humped dirt, you know, dirt track and they're chilling.

• 18:02Krishna Choudhary?

  Yeah, yeah, yeah. There's all these like Chinese propaganda videos too, where they show like the luxury cars of China.

• 18:07Lester Nare?

  Right.

• 18:08Krishna Choudhary?

  Versus the luxury cars of Germany. And it's like the luxury cars of China are just like, it's, 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 one over r squared, how far away my thing is, and it's pointing directly at the central, the, 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.

• 18:53Krishna Choudhary?

  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.

• 19:08Lester Nare?

  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.

• 19:15Krishna Choudhary?

  Yeah, and Newton's laws.

• 19:16Lester Nare?

  And Newton's laws for certain physical systems.

• 19:19Krishna Choudhary?

  Yes.

• 19:20Lester Nare?

  And we just, we just defined examples of those physical systems.

• 19:23Krishna Choudhary?

  Exactly.

• 19:24Lester Nare?

  In which we can do so.

• 19:25Krishna Choudhary?

  Yes.

• 19:26Lester Nare?

  Now, I think where we're going is, fluids maybe have some unique aspects.

• 19:33Krishna Choudhary?

  Yes.

• 19:34Lester Nare?

  That make that different.

• 19:36Krishna Choudhary?

  Yes, it makes it different and difficult.

• 19:38Lester Nare?

  Okay.

• 19:39Krishna Choudhary?

  Okay? It's different in the, it's, so, so first of all, it's same, same a little bit, because each part of the fluid still obeys Newton's laws.

• 19:46Lester Nare?

  Yes.

• 19:47Krishna Choudhary?

  If I were to look at a single particle of, 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 laws still work, 100%, right? The problem though...

• 20:07Krishna Choudhary?

  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 of water. I have 10 to the three mole. No, no, no, it's not 10 to the three. 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. That's sure. That's one way of doing it. If I had infinite compute, which I don't. I'd like to. 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.

• 20:50Lester Nare?

  We don't want to have to simulate everything because the system is so large as compared to what we were doing before.

• 20:54Krishna Choudhary?

  Yeah. And 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 the system.

• 21:12Lester Nare?

  Everywhere all at once.

• 21:13Krishna Choudhary?

  And all at once. Exactly. And so we need a new language.

• 21:16Lester Nare?

  Okay.

• 21:17Krishna Choudhary?

  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.

• 21:32Lester Nare?

  Right.

• 21:33Krishna Choudhary?

  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.

• 21:50Lester Nare?

  Interesting.

• 21:51Krishna Choudhary?

  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 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, though that thing is stationary. You see? Because it just represents, this is the the direction in which I'm moving. Okay.

• 22:33Krishna Choudhary?

  This is the central element that we are worried about.

• 22:38Lester Nare?

  Okay.

• 22:38Krishna Choudhary?

  Okay. This is a velocity field. It's a field because it has a value at every point in space and time. Um, 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.

• 23:20Lester Nare?

  And the component parts that make this velocity field are some position in 3D space, which is our X, Y and Z.

• 23:27Krishna Choudhary?

  Yeah.

• 23:28Lester Nare?

  So we know like 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.

• 23:36Krishna Choudhary?

  Yeah.

• 23:37Lester Nare?

  Of the direct like at that point.

• 23:38Krishna Choudhary?

  At that point, it's moving in this direction. Yeah.

• 23:41Lester Nare?

  Yeah.

• 23:41Krishna Choudhary?

  And it could change, right? Maybe I I introduce the flow and then it starts moving in some other direction.

• 23:46Lester Nare?

  Right.

• 23:46Krishna Choudhary?

  And so the velocity field can change. Um, 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 the color of these arrows represents how strong the velocity is. Okay. So the vector is big near the eye.

• 24:31Krishna Choudhary?

  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.

• 24:39Lester Nare?

  This, this makes sense. This is why when you have, you know, a tornado or a hurricane approaching you, if you live on the coast, the outer parts, you can feel it coming.

• 24:46Krishna Choudhary?

  Yeah.

• 24:47Lester Nare?

  Because the outer parts are a little bit slower.

• 24:48Krishna Choudhary?

  Yeah.

• 24:49Lester Nare?

  Than the inner parts.

• 24:49Krishna Choudhary?

  Yeah. When you get to the inner parts, then it's like, okay, then the palm tree is like.

• 24:53Lester Nare?

  On the side.

• 24:53Krishna Choudhary?

  On the side, right? Um, 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? It's 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.

• 25:14Lester Nare?

  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.

• 25:28Krishna Choudhary?

  Yeah.

• 25:29Lester Nare?

  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. Uh, we need to look at it from this different vantage point, which is, uh, velocity fields, which help us look at a points in space and the velocity at those points in space, which then are also going to interact with each other over time.

• 25:49Krishna Choudhary?

  Exactly. Yeah. And so if we wanted to, for example, um, calculate like 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, what, 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.

• 26:10Lester Nare?

  To be able to do that.

• 26:11Krishna Choudhary?

  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.

• 26:21Lester Nare?

  It's less compute constrained for lack of a better way to put it.

• 26:23Krishna Choudhary?

  Yeah, yeah, yeah.

• 26:24Lester Nare?

  As one way to put it.

• 26:25Krishna Choudhary?

  Yeah, that's certainly one way to put it. Yeah. And so before we start applying Newton's laws 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 the clouds are moving in a a circle above you. And so all of the arrows are pointing in kind of a circle, right? They're like circulating. Um, 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, are there, are there setups of arrows that we can make up that don't make any sense in the real world? Okay.

• 27:10Krishna Choudhary?

  The answer is yes. Let's look at these cases. But here are two cases. The first one on the left is a bunch of particles that are just like emanating from the center.

• 27:20Lester Nare?

  The expansion of the universe.

• 27:21Krishna Choudhary?

  Yeah, it kind of looks like the expansion of the universe. Exactly. Right? It's like all of the, 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?

• 27:34Lester Nare?

  Yeah.

• 27:35Krishna Choudhary?

  Right? That's not possible. Just like in empty space, it's just like, there's like just stuff.

• 27:41Lester Nare?

  Right. Like, what is the source?

• 27:43Krishna Choudhary?

  Right? Like if there was a hose, if 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'd be like, okay, but here I don't see that. It's just, the things are just coming out.

• 27:54Lester Nare?

  I understood.

• 27:54Krishna Choudhary?

  Right? Doesn't make sense.

• 27:56Lester Nare?

  Okay.

• 27:56Krishna Choudhary?

  Okay. First thing doesn't make sense because somehow the fluid is being created in the center and it's just moving out.

• 28:01Lester Nare?

  Right.

• 28:01Krishna Choudhary?

  Okay. Second one on the right. All of the fluid is 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?

• 28:19Lester Nare?

  Yeah, right.

• 28:20Krishna Choudhary?

  Right?

• 28:20Lester Nare?

  Right. Because you would have met, it should like go like.

• 28:23Krishna Choudhary?

  Yeah, yeah. I mean, if I saw the, if I saw it shoot out.

• 28:26Lester Nare?

  Right.

• 28:27Krishna Choudhary?

  To the side, then I'd be like, okay, that's possible.

• 28:29Lester Nare?

  It's hitting the surface of a wall or something.

• 28:31Krishna Choudhary?

  In that vector field, all of the arrows are pointing down and there's no point arrows pointing out. In the, in the vector field on the left, all of the pair arrows are pointing out. And there's nothing that's supplying the fluid to to have those arrows point out.

• 28:44Lester Nare?

  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.

• 28:55Krishna Choudhary?

  Yeah. We're trying to, we're trying to understand, like, we've created this concept of a velocity field.

• 29:00Lester Nare?

  Yes.

• 29:00Krishna Choudhary?

  Now, is any arbitrary vector field? Okay.

• 29:04Lester Nare?

  Right.

• 29:05Krishna Choudhary?

  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.

• 29:19Lester Nare?

  It doesn't, right. It doesn't work with fluids.

• 29:21Krishna Choudhary?

  And so we can, 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.

• 29:41Lester Nare?

  Like, I think like the collapsing of a black hole.

• 29:43Krishna Choudhary?

  Mm hmm. Exactly.

• 29:44Lester Nare?

  Just for my own visual.

• 29:45Krishna Choudhary?

  Hundred percent. On the, 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.

• 29:53Lester Nare?

  Okay.

• 29:54Krishna Choudhary?

  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 that this is a key constraint.

• 30:17Krishna Choudhary?

  For our vector field.

• 30:18Lester Nare?

  For fluids specifically.

• 30:20Krishna Choudhary?

  For fluids 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.

• 30:29Lester Nare?

  Fine.

• 30:29Krishna Choudhary?

  Fine, totally fine. But for fluids, that can't be possible.

• 30:33Lester Nare?

  So I think this is, so it's an important point here. The idea is this concept of divergence applies differently based on the underlying subject matter that you're speaking to.

• 30:42Krishna Choudhary?

  Yeah.

• 30:42Lester Nare?

  And so as we've gone through this progression of testing vector fields, it only can work when the divergence is zero.

• 30:52Krishna Choudhary?

  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.

• 31:00Lester Nare?

  Okay.

• 31:01Krishna Choudhary?

  Okay? Those are the fluids that we're talking about. And if we go to the next slide, those are our two Navier-Stokes equations. You can already see that we've already got one of them.

• 31:08Lester Nare?

  Okay.

• 31:09Krishna Choudhary?

  Okay? The second one is that the divergence of this fluid velocity field is zero.

• 31:14Lester Nare?

  Yeah.

• 31:15Krishna Choudhary?

  That's already the second Navier-Stokes equation.

• 31:17Lester Nare?

  Okay.

• 31:17Krishna Choudhary?

  So we're already, you would say, halfway through.

• 31:20Lester Nare?

  Because, and can you remind me again, we have our, I guess, delta symbol and our U symbol equals zero.

• 31:26Krishna Choudhary?

  Yeah, that's the second one, the second one there. That's on a, you would, you would say the divergence of U is zero.

• 31:33Lester Nare?

  Right, which was that diagram where we saw the center point, everything moving in one direction.

• 31:37Krishna Choudhary?

  Everything moving in one direction, stuff came in, stuff is going out.

• 31:40Lester Nare?

  Right.

• 31:40Krishna Choudhary?

  And everything is fine. You're not creating or destroying fluid.

• 31:46Lester Nare?

  Out of, out of nowhere.

• 31:48Krishna Choudhary?

  Yeah, and for engineers, you might, 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.

• 32:05Lester Nare?

  For our audio listeners, imagine a wine bottle on its side.

• 32:08Krishna Choudhary?

  Yes, exactly. Wine bottle on the side, but its base has been cut off.

• 32:10Lester Nare?

  Yes.

• 32:11Krishna Choudhary?

  And then now you've got water or wine coming in from the, 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, 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.

• 32:40Lester Nare?

  Right.

• 32:40Krishna Choudhary?

  Right? This is the continuity equation in action. And a mathematically succinct way of putting it is that delta U equals zero. The divergence of the vector field is always zero.

• 32:51Lester Nare?

  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 in 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.

• 33:04Krishna Choudhary?

  Yeah.

• 33:04Lester Nare?

  So it necessarily needs to accelerate, right, at that point in order to, for the divergence of U to equal zero.

• 33:12Krishna Choudhary?

  To equal zero, yeah. For that, for that fact to be true, you have to, you have to account for this difference in velocity.

• 33:18Lester Nare?

  Right.

• 33:19Krishna Choudhary?

  Right? Now crucially, this is an independent thing from that first Navier-Stokes equation, which is going to be the topic for the next, for the next segment.

• 33:27Lester Nare?

  Okay.

• 33:27Krishna Choudhary?

  Okay? This is an independent criteria.

• 33:29Lester Nare?

  Okay.

• 33:29Krishna Choudhary?

  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.

• 33:59Lester Nare?

  So the, a key takeaway for, one of the key takeaways for this is, is the concept of this idea of incompressible.

• 34:05Krishna Choudhary?

  Yes.

• 34:06Lester Nare?

  Which we've just defined from mathematical first principles.

• 34:09Krishna Choudhary?

  Yeah, but, but it really, it just means you can't, you know, you can't pack stuff in. The density is always constant.

• 34:16Lester Nare?

  Right.

• 34:16Krishna Choudhary?

  Right? It's like one of those, have you seen those like stress thingies?

• 34:18Lester Nare?

  Yeah, the little stress ball.

• 34:19Krishna Choudhary?

  Yeah, where you like squeeze it, and then the, the thing is going to like shoot out somewhere else. But that's an incompressible fluid, right? Because if I squish it, it's gotta go somewhere else.

• 34:29Lester Nare?

  Right, okay.

• 34:29Krishna Choudhary?

  It's not like one of those foam things where you can just compress it.

• 34:33Lester Nare?

  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.

• 34:48Krishna Choudhary?

  Of the second one, right? We've, we've sort of gotten intuitive understanding of what types of velocity fields are allowed, first of all.

• 34:57Lester Nare?

  Right.

• 34:58Krishna Choudhary?

  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.

• 35:07Lester Nare?

  In a fluid.

• 35:07Krishna Choudhary?

  In a fluid, right? In an incompressible.

• 35:09Lester Nare?

  In an, in an incompressible fluid.

• 35:11Krishna Choudhary?

  Yeah. Okay, so now, let's try to apply this velocity field to Newton's laws.

• 35:15Lester Nare?

  Okay.

• 35:16Krishna Choudhary?

  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.

• 35:44Lester Nare?

  And just to take a quick step back, just for, 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?

• 35:57Lester Nare?

  We talked about Newton's laws in the concept, context of like the boat in the water.

• 36:01Krishna Choudhary?

  Yeah.

• 36:02Lester Nare?

  And then when we talked about velocity fields, it's like the medium of the water.

• 36:05Krishna Choudhary?

  Yeah, it's the water itself.

• 36:06Lester Nare?

  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 in. And the lens in which we look at them is from different doorways.

• 36:18Krishna Choudhary?

  Yes. In, 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.

• 36:24Lester Nare?

  Right.

• 36:24Krishna Choudhary?

  What is the position of the water? What, what do you mean? Which water?

• 36:28Lester Nare?

  Yes.

• 36:28Krishna Choudhary?

  There's, there's an Avogadro's number of water molecules that we're talking about, right? 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.

• 36:39Lester Nare?

  Okay.

• 36:39Krishna Choudhary?

  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, okay? Like, in this velocity field.

• 36:55Lester Nare?

  Okay.

• 36:55Krishna Choudhary?

  Okay?

• 36:56Lester Nare?

  Okay.

• 36:56Krishna Choudhary?

  And we're gonna, we're gonna use traffic as an analogy.

• 37:04Lester Nare?

  If you live in LA, you are, very much are intuitively familiar.

• 37:08Krishna Choudhary?

  Yes, and you're probably listening to this while you're, while you're in traffic. Okay? So, traffic is kind of a nice analogy for fluids, because you've got these working

• 37:19Krishna Choudhary?

  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 brake, 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,

• 38:02Krishna Choudhary?

  blah, blah, blah, blah, blah, right? So they're actually watching the, the cars, but they're watching an aggregate mass of cars. It's not like they're following a single car and seeing how, how, how stuff is happening. So what we want to do is describe the acceleration of a single car based on what we see from, 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.

• 38:44Lester Nare?

  I was going to say, 80 miles an hour in LA never happens.

• 38:46Krishna Choudhary?

  Yeah, yeah, never happens.

• 38:47Lester Nare?

  This is an unrealistic example.

• 38:48Krishna Choudhary?

  Yeah, 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 cars 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, in the traffic map of LA, everything was

• 39:32Krishna Choudhary?

  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 is, 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 could easily imagine, after the rain ends, the things go back to 80 miles per

• 40:12Krishna Choudhary?

  hour, and it's going to speed up again. This is represented by a derivative of my velocity field.

• 40:18Lester Nare?

  And so the idea is, is here's a way to describe the velocity field as this system unique and different from the discrete individual car in traffic.

• 40:27Krishna Choudhary?

  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 airfoil. This is a airplane wing.

• 40:52Krishna Choudhary?

  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, 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.

• 41:14Lester Nare?

  These are the red arrows.

• 41:15Krishna Choudhary?

  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.

• 41:23Lester Nare?

  And, and just the length of these arrows is meaningful here.

• 41:26Krishna Choudhary?

  Yeah.

• 41:26Lester Nare?

  In that there's, there's more.

• 41:28Krishna Choudhary?

  Yeah, it's faster.

• 41:29Lester Nare?

  Faster.

• 41:30Krishna Choudhary?

  Yeah.

• 41:31Lester Nare?

  And, okay, I'm already seeing where this is going.

• 41:33Krishna Choudhary?

  Okay. So, so this is the, on the left is your particles that are moving.

• 41:37Lester Nare?

  Right. Like our flow of traffic.

• 41:39Krishna Choudhary?

  Yeah, our flow of traffic. And on the right hand side is the velocity field. But if I were 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.

• 41:56Lester Nare?

  Yeah.

• 41:57Krishna Choudhary?

  If you go back to that picture that I had with the tornado, even though the, the, the particles were circling around.

• 42:04Lester Nare?

  Velocity field.

• 42:05Krishna Choudhary?

  The velocity field was static because in, in this point, the velocity is this way. And then as it goes around, it's like coming around, right? And so, 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, you have, let's say, an accident over here. Okay. On the 101. Way behind, way, way ahead of the accident, the, um, the, 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.

• 42:43Lester Nare?

  Yes.

• 42:43Krishna Choudhary?

  And so the police haven't gotten there. So the accident is not going anywhere, which means that ahead of the, 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, the cars are accelerating. Once you get past the, the accident and you're done rubbernecking, yeah, you're like, okay, I gotta get on with my life. Like, let's go. Right?

• 43:22Lester Nare?

  Okay. And, and so I just want to make sure that I'm, I'm following this here because what, 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?

• 43:37Krishna Choudhary?

  And how it accelerates.

• 43:38Lester Nare?

  And how it accelerates based on the, the, the mapping we see in the velocity field.

• 43:42Krishna Choudhary?

  Exactly.

• 43:42Lester Nare?

  And part of what you're pointing out here is there 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.

• 43:55Krishna Choudhary?

  The, the derivative of, the, the velocity field is not changing.

• 43:58Lester Nare?

  You said, sorry, right. So the, the, right. The velocity field, it's, it's static.

• 44:01Krishna Choudhary?

  Yeah.

• 44:02Lester Nare?

  It's not changing, which means you can, you, in some cases, you may not be able to accurately describe certain particles within that context because.

• 44:11Krishna Choudhary?

  Yeah, using just the time derivative of the velocity field.

• 44:13Lester Nare?

  Right, right.

• 44:14Krishna Choudhary?

  You have to figure out something else about the velocity field that's doing this. And let's just think about this. I think we've got, in the next one, uh, you know, the highway slowed down. Let's say there's a, let, let's say there's, um,

• 44:25Lester Nare?

  Next one here.

• 44:26Krishna Choudhary?

  And the next one, exactly. So let's say there's roadwork ahead.

• 44:29Lester Nare?

  Okay.

• 44:29Krishna Choudhary?

  Right? Um, 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. There is a spatial difference in the velocity field. You see?

• 44:51Lester Nare?

  Yeah, yeah, yeah.

• 44:51Krishna Choudhary?

  There's the time difference that we talked about earlier where the, 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,

• 45:09Krishna Choudhary?

  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.

• 45:31Lester Nare?

  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.

• 45:40Krishna Choudhary?

  Yeah, it's part of the story.

• 45:41Lester Nare?

  It's part of this. But it seems like the distance is potentially the other ingredient

• 45:47Krishna Choudhary?

  Yeah.

• 45:47Lester Nare?

  in the pie.

• 45:48Krishna Choudhary?

  Yeah.

• 45:48Lester Nare?

  At least based on the example that you just brought up.

• 45:50Krishna Choudhary?

  Exactly, exactly. Yeah. And that's a big part of it. It's the spatial difference. It's, 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, um, what if, you know, I took a time derivative before?

• 46:05Lester Nare?

  Yeah.

• 46:05Krishna Choudhary?

  What if I take a spatial derivative?

• 46:07Lester Nare?

  Okay.

• 46:07Krishna Choudhary?

  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, um, a hundred feet over there, if I've slowed down, that tells me how much I should slow down. That's not everything.

• 46:20Lester Nare?

  Okay.

• 46:21Krishna Choudhary?

  Okay. There is a secondary effect.

• 46:23Lester Nare?

  Okay.

• 46:24Krishna Choudhary?

  And to, to understand that, let's, let's look at that scenario again, where you've got roadwork ahead.

• 46:28Lester Nare?

  Okay.

• 46:29Krishna Choudhary?

  Okay. But now I've got two different cars that are approaching. The first car approaches that roadwork at a hundred meters per second.

• 46:37Lester Nare?

  Okay.

• 46:37Krishna Choudhary?

  Okay? And the second car approaches that roadwork at exactly half the speed, 50 meters per second.

• 46:46Krishna Choudhary?

  I think you'll agree that the guy that is moving faster is going to have to brake harder.

• 46:50Lester Nare?

  Yeah.

• 46:51Krishna Choudhary?

  Right?

• 46:51Lester Nare?

  Yeah.

• 46:51Krishna Choudhary?

  Because he's gonna, he's gonna come upon that roadwork way faster than the guy who's slowing. Right? He's gonna have to be like, oh, there's roadwork.

• 46:59Lester Nare?

  Yeah.

• 47:00Krishna Choudhary?

  You know, fines are doubled.

• 47:01Lester Nare?

  Yeah.

• 47:02Krishna Choudhary?

  I better slow down a lot.

• 47:03Lester Nare?

  Yep.

• 47:03Krishna Choudhary?

  So there's a secondary effect. It has to do not just with how fast the traffic pattern is changing, but also how fast the individual car is moving.

• 47:16Krishna Choudhary?

  Yes?

• 47:17Lester Nare?

  Yes. Because so, and it, again, trying to use the terminology again, we talked about the time derivative matters. We talked about, uh, the spatial.

• 47:25Krishna Choudhary?

  Spatial matters.

• 47:26Lester Nare?

  Matters. And also the individual objects accelerate.

• 47:31Krishna Choudhary?

  Velocity.

• 47:32Lester Nare?

  Velocity.

• 47:32Krishna Choudhary?

  The speed.

• 47:33Lester Nare?

  Speed itself is the third item in our, uh, uh, cookie recipe here.

• 47:39Krishna Choudhary?

  Yes.

• 47:39Lester Nare?

  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.

• 47:50Krishna Choudhary?

  Yeah.

• 47:50Lester Nare?

  Acceleration.

• 47:51Krishna Choudhary?

  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.

• 48:04Lester Nare?

  Right.

• 48:05Krishna Choudhary?

  Okay?

• 48:05Lester Nare?

  Right.

• 48:06Krishna Choudhary?

  The change in velocity as a function of distance.

• 48:08Lester Nare?

  Right.

• 48:08Krishna Choudhary?

  Is like, how, how far away is the roadwork?

• 48:10Lester Nare?

  Right.

• 48:11Krishna Choudhary?

  And my own velocity as I approach that also has something to do with how far, how fast and I need to brake.

• 48:16Lester Nare?

  Yes.

• 48:16Krishna Choudhary?

  So it's going to be a multiplication of the velocity field.

• 48:19Lester Nare?

  Yeah.

• 48:19Krishna Choudhary?

  Times the spatial derivative. That's the under, um, upside down triangle. It's called Nambla. I call it Del. But it's u dot Del times u. Okay? This is called the convective term.

• 48:33Lester Nare?

  And this, what this convective term is doing now, it's giving us in the example we just talked about here, it's giving us the hundred meters per second, eighty meters per second. Right? That, okay, right.

• 48:44Krishna Choudhary?

  Multiplied by the Del u, which is the spatial, how, how, how much time do I have before the roadwork? 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?

• 48:58Lester Nare?

  Yeah, yeah, yeah. Okay. Okay. And that's the, okay.

• 49:00Krishna Choudhary?

  Makes sense?

• 49:01Lester Nare?

  Yes.

• 49:01Krishna Choudhary?

  Okay. And both of these terms combined give us our acceleration. We've got the Eulerian 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.

• 49:24Lester Nare?

  And this.

• 49:24Krishna Choudhary?

  And that's the first term.

• 49:25Lester Nare?

  That, that's our velocity, the derivative of the velocity field.

• 49:29Krishna Choudhary?

  Yeah, with respect to time, right?

• 49:30Lester Nare?

  With respect to time.

• 49:31Krishna Choudhary?

  Because the velocity, the field itself is changing.

• 49:33Lester Nare?

  The time derivative.

• 49:33Krishna Choudhary?

  The color is changing on the, on the Google Maps.

• 49:35Lester Nare?

  Right.

• 49:36Krishna Choudhary?

  Right? On the right hand side, the Google Maps color is staying the same, but I am moving through

• 49:41Lester Nare?

  that portion of the yellow or the red.

• 49:44Krishna Choudhary?

  Yeah. And so I'm going to be braking or accelerating based on that.

• 49:47Lester Nare?

  And a more, and effectively a more granular way. There's like a bird's eye and then there's a, like, a, a zoom.

• 49:53Krishna Choudhary?

  Yes. And both of those matter. And that's why it's called actually the Eulerian perspective, because that's kind of the bird's eye view. And then the Lagrangian perspective is like the, the nitty gritty, what's happening there? And that's the conceptual trick. Okay?

• 50:05Lester Nare?

  Okay.

• 50:05Krishna Choudhary?

  And in, in, 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.

• 50:14Lester Nare?

  Ah, that's an interesting way to put it.

• 50:16Krishna Choudhary?

  Only finding you. The chopper only knows the, the green and the red and, 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-accelerate? Does that make sense?

• 50:35Lester Nare?

  Yeah. We're trying to, um, again, describe.

• 50:41Lester Nare?

  The acceleration or deceleration of an individual thing in an incompressible fluid from a vantage point that is only the vector field.

• 50:52Krishna Choudhary?

  Yes.

• 50:53Lester Nare?

  Which, which, um, the point being the vector field in and of itself does not have enough to describe the acceleration of the individual object until you

• 51:01Krishna Choudhary?

  Until you do all of this math.

• 51:02Lester Nare?

  Until you do all of this.

• 51:03Krishna Choudhary?

  Yeah.

• 51:04Lester Nare?

  Is that the right way to say it?

• 51:05Krishna Choudhary?

  I think that is the right way to say it.

• 51:06Lester Nare?

  Okay.

• 51:07Krishna Choudhary?

  Exactly. And so now we have an acceleration.

• 51:09Lester Nare?

  Right.

• 51:10Krishna Choudhary?

  Right? And all we need for that other part of Newton's laws, right? It's F equals ma. We already have the a, the acceleration. What is the mass? Well, the mass is just for a fluid, it's like the density. Right? It's just, um, I could do a F equals ma per volume. And then just say, okay, density multiplied by acceleration. That's my, that part of Newton's laws.

• 51:31Lester Nare?

  That's my ma.

• 51:31Krishna Choudhary?

  Yeah, that's my ma. And that's exactly what we get. We've got the left-hand side of Navier-Stokes equations.

• 51:36Lester Nare?

  Okay.

• 51:36Krishna Choudhary?

  The first Navier-Stokes equation.

• 51:38Lester Nare?

  Right.

• 51:38Krishna Choudhary?

  Density multiplied by acceleration.

• 51:41Lester Nare?

  Right, right. Uh, because our acceleration is everything in this, in this, this larger bracket. Which is what we just described.

• 51:49Krishna Choudhary?

  Yeah.

• 51:50Lester Nare?

  Which is trying to describe the acceleration of an individual object from the helicopter.

• 51:55Krishna Choudhary?

  Exactly.

• 51:56Lester Nare?

  In the, 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.

• 52:06Krishna Choudhary?

  Yes.

• 52:07Lester Nare?

  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.

• 52:16Krishna Choudhary?

  Exactly. And now I wanna, before we even start talking about the other side of that equation, which is the force, I wanna linger on this acceleration term a little bit more. Especially that convective term, because that convective term, you might have heard that, um, the Navier-Stokes equations are nonlinear.

• 52:30Lester Nare?

  Yes.

• 52:31Krishna Choudhary?

  Right? Compared to linear differential equations. The Navier-Stokes equation is a nonlinear 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 wanna show you exactly what all the trouble is.

• 52:46Lester Nare?

  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.

• 53:01Krishna Choudhary?

  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.

• 53:15Lester Nare?

  Okay.

• 53:15Krishna Choudhary?

  Okay? Let's, let's take a step back and let's look at, for example, Maxwell's equations in a vacuum.

• 53:20Lester Nare?

  Okay.

• 53:21Krishna Choudhary?

  Okay? These are the equations of electricity and magnetism. They're described, for example, Gauss's law for electric fields is that, um, the divergence of the electric field is equal to how much charge there is. You know how before I was telling you that like 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 Gauss's law for electricity, okay? It just says that del dot E, the divergence of the electric field, equals to some charge. Now, crucially, the divergence of stuff is a linear operator. Meaning, if I've got one electric field and another electric field,

• 54:02Krishna Choudhary?

  and I wanna 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 wanna 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 wanna find the electric field everywhere, all I do is 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 wifi, we've got Bluetooth, we've got radio, we've got, um, our phones. We also have the light from the studio, right?

• 54:46Krishna Choudhary?

  All of these are electric fields, but they're all just moving through one another. They're, 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 wifi. Like, the wifi 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 wifi signal is like crashing, right?

• 55:14Lester Nare?

  Yeah, they, they don't interfere with each other.

• 55:16Krishna Choudhary?

  Yeah, yeah. Even like, um, in a radio, right? In, in the car, like, if I'm tuned to 91.5 KUSC classical radio,

• 55:25Krishna Choudhary?

  the car is being inundated with all of the radio stations. There's also 89.9 coming at me and Kiss FM. And also the AM, you know, talk radio, Rush Limbaugh.

• 55:35Lester Nare?

  Sponsored. Uh, we need our check Kiss FM.

• 55:37Krishna Choudhary?

  Yeah, these are just some of my favorites that I listen to in LA. But like, my, my, my point is, the car is being inundated with all of these radio stations. But my radio can tune in on, into a specific band, and it's not like the Kiss FM radio signal is interfering with the KUSC signal, is interfering with Rush Limbaugh, or, uh, he's not.

• 55:59Lester Nare?

  I think he's still on radio.

• 56:00Krishna Choudhary?

  Yeah, he's still on radio?

• 56:01Lester Nare?

  We, I think he's still on, he might be on YouTube now.

• 56:03Krishna Choudhary?

  Yeah, yeah, yeah.

• 56:03Lester Nare?

  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.

• 56:09Krishna Choudhary?

  Exactly.

• 56:10Lester Nare?

  It doesn't matter what

• 56:11Krishna Choudhary?

  And if I want to account for, all I do is add it, right? If I wanna know what is the electric field here, all I do is add up all of the contributions. It's totally fine. Um, in fluids, we do not have that luxury.

• 56:23Lester Nare?

  Okay.

• 56:23Krishna Choudhary?

  Okay? And that is because of that convective term. That convective term is nonlinear, and this is what it does if I add up two velocity fields that are separate.

• 56:32Lester Nare?

  Okay.

• 56:32Krishna Choudhary?

  Suppose, you know, and before I was, I was adding up E1 and E2, the electric fields.

• 56:36Lester Nare?

  Yes.

• 56:37Krishna Choudhary?

  It's totally fine.

• 56:37Lester Nare?

  Fine.

• 56:38Krishna Choudhary?

  Suppose I have two velocity fields, right? I got two hoses and I'm like crossing streams or something. U1 plus U2, where that's, I shouldn't have, you know what, whatever, like two, two things of water. And I, and I, and I add up, right? Like, U1 and U2. Suppose I, I go through the math with those.

• 56:56Lester Nare?

  Okay.

• 56:56Krishna Choudhary?

  If it was linear, I would only get the terms in the green, right? Because I could just do the math for one, U1, and I could do the math for U2, and I'd be fine.

• 57:06Lester Nare?

  You do the divergence for U1.

• 57:08Krishna Choudhary?

  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 divergence, because divergence has a dot. Anyways, the, the, the point is, I'm taking the, 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.

• 57:33Lester Nare?

  You have to do it twice.

• 57:34Krishna Choudhary?

  Yeah, I have to do it twice, and so I'm gonna get these cross terms on the right, in the red circle.

• 57:38Lester Nare?

  Okay.

• 57:38Krishna Choudhary?

  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, they mix and they, they, they have turbulent flow. You know, um, when I, when I was in India last time, um, we visited Devpreyag in the Himalayas, which is the birthplace of the Ganga, where like, you, you have like Bhagirathi coming from one side and Alakananda 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.

• 58:21Krishna Choudhary?

  It was, it was amazing. And, and the, the reason for the whirlpools and the reason for all of the chaos in the birthplace of the Ganga is because of those two terms on the right-hand side. You know what I mean?

• 58:32Lester Nare?

  Yeah.

• 58:33Krishna Choudhary?

  And so now you can already see that there is trouble with this equation.

• 58:38Lester Nare?

  Right. And I, I just wanna take a quick step back to make sure I'm internalizing

• 58:42Krishna Choudhary?

  Yes, go ahead.

• 58:43Lester Nare?

  what we're saying here.

• 58:45Lester Nare?

  When we talked about Maxwell's equations, uh, and we have, you know, a positive and a negative charge, and we can, we can just add them up, it's because, part of what we're saying is, 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.

• 59:05Krishna Choudhary?

  Yeah, yeah. In the Maxwell's equation, it's just del dot E, right? It's a single E.

• 59:09Lester Nare?

  Right.

• 59:09Krishna Choudhary?

  Here, there's two versions of the vector field. There's the vector field itself multiplied by the derivative of the vector field.

• 59:17Lester Nare?

  Right. And can we, 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, I'm, I'm having a hard time finding the leap from, I, I understood the convective term.

• 59:35Krishna Choudhary?

  Yeah.

• 59:36Lester Nare?

  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.

• 59:42Krishna Choudhary?

  Yeah.

• 59:42Lester Nare?

  And we can look at the whole vector field.

• 59:44Krishna Choudhary?

  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.

• 59:53Lester Nare?

  And so would it, like, 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 equations and that

• 1:00:00Krishna Choudhary?

  Yeah, if, if, if all I had to do was take a, take a spatial derivative

• 1:00:04Lester Nare?

  Right.

• 1:00:05Krishna Choudhary?

  of red and green and be like, oh, there's red here and green here, and they're 100 meters apart.

• 1:00:10Lester Nare?

  Yes.

• 1:00:10Krishna Choudhary?

  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.

• 1:00:21Lester Nare?

  Yeah.

• 1:00:21Krishna Choudhary?

  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 gotta floor it.

• 1:00:30Lester Nare?

  Yes.

• 1:00:31Krishna Choudhary?

  But if in the red, I was already going at 30, and then now I gotta get to 60, I don't have to floor it as hard. So there's two contributions.

• 1:00:39Lester Nare?

  Right. And that's, that's what's giving us this non-linearity.

• 1:00:42Krishna Choudhary?

  Yes.

• 1:00:43Lester Nare?

  Is because

• 1:00:43Krishna Choudhary?

  The fact that there's two contributions.

• 1:00:45Lester Nare?

  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. The, we have the green, but we have to have the addition of the red because of that second.

• 1:00:59Krishna Choudhary?

  Because, because of the fact that, yeah, there's two of them.

• 1:01:01Lester Nare?

  Right.

• 1:01:01Krishna Choudhary?

  So I have to do these cross terms.

• 1:01:03Lester Nare?

  Understood.

• 1:01:04Krishna Choudhary?

  Yeah. When I combine velocity fields, when I combine, let's say, a flow in this direction and a flow in this direction, I can't, they don't just go past one another.

• 1:01:12Lester Nare?

  Right.

• 1:01:12Krishna Choudhary?

  Because they're interacting.

• 1:01:13Lester Nare?

  Yeah.

• 1:01:13Krishna Choudhary?

  Yep.

• 1:01:14Lester Nare?

  Okay.

• 1:01:14Krishna Choudhary?

  Makes sense?

• 1:01:15Lester Nare?

  Yes, tracking.

• 1:01:15Krishna Choudhary?

  Yeah. And so that's why the Navier-Stokes are so hard. It's one of the reasons. 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 figured out how to describe the acceleration, and we've figured out how to describe the mass, right?

• 1:01:39Lester Nare?

  Yes.

• 1:01:40Krishna Choudhary?

  But in order to have an equation of motion, we have to describe the forces that 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, 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, if you've got a,

• 1:02:25Krishna Choudhary?

  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.

• 1:02:33Lester Nare?

  It's just hanging out, it's chill.

• 1:02:34Krishna Choudhary?

  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, the high, like a thousand, a hundred thousand 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.

• 1:03:08Lester Nare?

  Right, right.

• 1:03:08Krishna Choudhary?

  So I don't feel a force.

• 1:03:10Lester Nare?

  Right.

• 1:03:10Krishna Choudhary?

  Now, all of a sudden, if, you know, outside there was super low pressure for some reason, and we had like a, 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.

• 1:03:40Lester Nare?

  Right, right.

• 1:03:41Krishna Choudhary?

  Okay?

• 1:03:41Lester Nare?

  Because it's going from a high, it's always flowing from a, 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 gonna be.

• 1:03:50Krishna Choudhary?

  Yeah, it's always gonna 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, from high pressure to low pressure. You're gonna have velocities flowing, right?

• 1:04:07Lester Nare?

  It's like if you're at a beach, and you know those little videos where the surfers create?

• 1:04:12Krishna Choudhary?

  Oh, yeah, yeah.

• 1:04:13Lester Nare?

  The surfing thing on the, from a beach into the ocean?

• 1:04:16Krishna Choudhary?

  Yeah.

• 1:04:17Lester Nare?

  And it's, it's like, the, the water pressure is very high and built up at the top, and then it wants to flow into the ocean.

• 1:04:23Krishna Choudhary?

  Yeah.

• 1:04:24Lester Nare?

  Which is a relative area of lower pressure.

• 1:04:26Krishna Choudhary?

  Yeah.

• 1:04:26Lester Nare?

  In that context.

• 1:04:27Krishna Choudhary?

  In that context. Actually, you're, the next one that I want to talk about is the external forces on my fluid.

• 1:04:33Lester Nare?

  Okay.

• 1:04:34Krishna Choudhary?

  And in that case, that would be gravity.

• 1:04:35Lester Nare?

  Okay.

• 1:04:36Krishna Choudhary?

  Right? Because the, the, like the lagoon or whatever is slightly above the ocean.

• 1:04:41Lester Nare?

  That's a good point.

• 1:04:41Krishna Choudhary?

  Right? And so when you create that little tiny thing, gravity is gonna force the fluid down that little canyon that you've built.

• 1:04:48Lester Nare?

  So my example was actually, it was not perfectly apt because I was not yet accounting for gravity.

• 1:04:53Krishna Choudhary?

  Yes. And the next thing that we're gonna do is account for external forces on the fluid.

• 1:04:58Lester Nare?

  Okay.

• 1:04:59Krishna Choudhary?

  Like gravity, right? So, on the right hand side, we've got forces.

• 1:05:04Lester Nare?

  Yes.

• 1:05:04Krishna Choudhary?

  The difference in pressure is one such force.

• 1:05:07Lester Nare?

  Yes.

• 1:05:07Krishna Choudhary?

  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 gonna get bigger. Because there's gonna be more fluid going through, and the fluid is gonna 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.

• 1:05:33Lester Nare?

  Okay.

• 1:05:34Krishna Choudhary?

  Right?

• 1:05:34Lester Nare?

  Okay.

• 1:05:35Krishna Choudhary?

  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, in a glass, and I stir the thing, that's an external force that's acting on my fluid. So the, the force can be a lot of things.

• 1:05:55Lester Nare?

  Understood. It's not necessarily just gravity in the context we just talked about.

• 1:05:59Krishna Choudhary?

  Mm-hmm. 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, the turbine in an airplane. That's putting a force on, on, right? Because the thing is, it's stirring a fluid and making it shoot out in one direction.

• 1:06:13Lester Nare?

  Yeah, yeah, yeah.

• 1:06:14Krishna Choudhary?

  Um, 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, one of the Euler equations of fluids.

• 1:06:33Lester Nare?

  So, so just, and just to quickly kind of recap.

• 1:06:36Krishna Choudhary?

  Mm-hmm.

• 1:06:36Lester Nare?

  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.

• 1:07:03Krishna Choudhary?

  Yeah.

• 1:07:04Lester Nare?

  And we've built up now for an incompressible fluid.

• 1:07:08Krishna Choudhary?

  Yeah.

• 1:07:10Lester Nare?

  All the way up to where Euler arrived in, in this journey of trying to do so.

• 1:07:15Krishna Choudhary?

  Yes. He didn't get there on his own, though.

• 1:07:17Lester Nare?

  Okay, oh, okay, oh.

• 1:07:19Krishna Choudhary?

  Okay.

• 1:07:20Lester Nare?

  We sent no drama till the end, but.

• 1:07:21Krishna Choudhary?

  And, 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, 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-Stokes extended those equations to create the Navier-Stokes equations that we know today. And obviously, you wanna name things after the second person who invented everything. Because you can't name everything after Euler.

• 1:08:06Krishna Choudhary?

  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.

• 1:08:16Lester Nare?

  Seventeen forties.

• 1:08:17Krishna Choudhary?

  Sorry, yeah, seven, of course, yeah. Seventeen forties Berlin, between Leonard Euler. He was working at the Royal Prussian, Royal Prussian Academy of Scientists. This is the German Academy of Sciences, okay? 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 nineteen, in the seventeen forties. How much resistance does a body experience as it moves through air or water? Okay? It's a very pressing, pressing question because I think around the time is when cannons are being used in warfare. So it's like, you know, we wanna know how far a cannon can go.

• 1:09:00Krishna Choudhary?

  It's, 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-Léron D'Alembert.

• 1:09:14Lester Nare?

  Very good. For those who are longtime listeners.

• 1:09:17Krishna Choudhary?

  I don't usually do this, but I like D'Alembert. 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 U, of the velocity field. You know, this, 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 D'Alembert.

• 1:09:53Lester Nare?

  And, okay, so what's interesting is I wanna make a quick zoom in note to note that the, the prize problem that he set out to answer.

• 1:10:03Krishna Choudhary?

  Yeah.

• 1:10:03Lester Nare?

  Was about how much resistance does a body experience when it moves through air or water?

• 1:10:08Krishna Choudhary?

  Mm-hmm.

• 1:10:10Lester Nare?

  And I just wanna be, that's the specific question he was trying to answer. In that journey, he did not answer that question.

• 1:10:17Krishna Choudhary?

  He did not answer the question, but he invented.

• 1:10:19Lester Nare?

  He gave us the language.

• 1:10:20Krishna Choudhary?

  He gave us the language for it, right? And his submission didn't win because the Academy of Sciences was like, oh, it's, um, it's, it's too, it's too theoretical. He also showed that the, that 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, D'Alembert. He submits this thing, the, the Academy decides that it's not gonna 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 D'Alembert's work. And he's reading this thing going, yo, this velocity field is a kind of nice idea.

• 1:11:04Krishna Choudhary?

  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, D'Alembert 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 Basil problem existed beforehand, right?

• 1:11:35Lester Nare?

  Right.

• 1:11:35Krishna Choudhary?

  The one over, one plus one over four plus one over nine plus one over sixteen, so on and so forth. Leonard Euler saw the Basil problem, solved it, and then presented the zeta function. In this case, he saw D'Alembert'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, the, the derivative term, that acceleration is massive to create that, that material derivative. And then also say that, okay, the forces on it are gonna be the gradient in the pressure, plus whatever external force.

• 1:12:09Lester Nare?

  And just to say, the, what you just spoke to, that was the convective term that we talked about earlier.

• 1:12:13Krishna Choudhary?

  Yeah, yeah.

• 1:12:14Lester Nare?

  Yes. Okay, and this is so interesting because it's, um, it, it's always funny how these things. Basically, we have D'Alembert, who started the, who created the vision to think about it this way. And then Euler was like, let me take Newton's framework, D'Alembert's vision.

• 1:12:37Krishna Choudhary?

  Yeah.

• 1:12:38Lester Nare?

  And combine these two things.

• 1:12:39Krishna Choudhary?

  Yes. And he gets an equation of motion, right? He gets the first equation of motion for a fluid. This is for an inviscid fluid, as they say.

• 1:12:47Lester Nare?

  Okay.

• 1:12:47Krishna Choudhary?

  Because these are, as we'll find out, frictionless fluids. Okay? Now, obviously, that can't be everything. And D'Alembert sees this. It's, it's kind of like a mathematical twist that's so perfect, it's gotta be scripted. D'Alembert looks at the original competition, and he's like, wait a minute, that, 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, in the next, the thing we'll see, this is called D'Alembert's paradox, okay? He uses Euler's equation that's up there, and he applies it to a ball, a sphere that is moving through fluid.

• 1:13:33Krishna Choudhary?

  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.

• 1:13:54Lester Nare?

  Right? Just as, 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 a great episode. Definitely go watch it.

• 1:14:09Krishna Choudhary?

  Exactly. And what we can see already is in real fluids, the streamlines aren't just meeting up right after.

• 1:14:14Lester Nare?

  Yeah.

• 1:14:15Krishna Choudhary?

  Right? There's a lot, there's this like region where there's turbulence and all this weird stuff that's happening.

• 1:14:19Lester Nare?

  Chaos happening.

• 1:14:20Krishna Choudhary?

  Right? Euler's equation does not account for that.

• 1:14:24Lester Nare?

  So basically what we're saying is Euler's equations only accounts for what we see on the left, which is effectively a system that's not actually exists in real, in real context.

• 1:14:33Krishna Choudhary?

  Exactly.

• 1:14:34Lester Nare?

  The military can't use it.

• 1:14:35Krishna Choudhary?

  Exactly. Yeah. And so 20 years after the whole competition and everything, Dallinbert 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.

• 1:14:53Lester Nare?

  And so to now kind of quickly recap again, we, we are trying to build this understanding of what is the construction of the Navier Stokes equation.

• 1:15:03Krishna Choudhary?

  Yes.

• 1:15:03Lester Nare?

  We understand Dallinbert 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 Dallinbert's test.

• 1:15:18Krishna Choudhary?

  Maybe a little bit.

• 1:15:19Lester Nare?

  Maybe.

• 1:15:19Krishna Choudhary?

  Yeah.

• 1:15:20Lester Nare?

  And added for the force side of the equation, pressure.

• 1:15:25Krishna Choudhary?

  And just an external force.

• 1:15:26Lester Nare?

  And it's, and an external.

• 1:15:27Krishna Choudhary?

  Gravity or stirring or whatever.

• 1:15:28Lester Nare?

  Or whatever it might be.

• 1:15:29Krishna Choudhary?

  Yeah.

• 1:15:30Lester Nare?

  And he was like, I did it.

• 1:15:31Krishna Choudhary?

  Yeah.

• 1:15:32Lester Nare?

  Look at me. I'm so brilliant.

• 1:15:33Krishna Choudhary?

  Yeah.

• 1:15:34Lester Nare?

  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 orig, it's funny because the original prize competition that Euler was a judge on was about resistance and drag.

• 1:15:52Krishna Choudhary?

  Yeah.

• 1:15:52Lester Nare?

  And then he came up with a solution without thinking about resistance and drag.

• 1:15:56Krishna Choudhary?

  Am I like missing something?

• 1:15:57Lester Nare?

  Exactly. Yeah.

• 1:15:59Krishna Choudhary?

  So, 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.

• 1:16:07Lester Nare?

  Yes.

• 1:16:08Krishna Choudhary?

  And we've got just external forces, whatever they might be, stirring, gravity, whatever.

• 1:16:11Lester Nare?

  Sure.

• 1:16:11Krishna Choudhary?

  There is a third force on that side of the equation that is missing. And that has to do with viscosity.

• 1:16:18Lester Nare?

  Okay.

• 1:16:18Krishna Choudhary?

  That's the resolution to Dallinbert's paradox.

• 1:16:21Lester Nare?

  Okay.

• 1:16:21Krishna Choudhary?

  So in Euler's ideal fluid, there is no viscosity. This is called an inviscid 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.

• 1:16:46Lester Nare?

  And this is.

• 1:16:46Krishna Choudhary?

  Beautiful mathematics.

• 1:16:47Lester Nare?

  It's different than diver, like it's different than the problem we were saying where water

• 1:16:56Krishna Choudhary?

  like form a incompressibility.

• 1:17:00Lester Nare?

  Right, right.

• 1:17:01Krishna Choudhary?

  Yeah, this is different.

• 1:17:02Lester Nare?

  It's different.

• 1:17:02Krishna Choudhary?

  This is still incompressible, right?

• 1:17:03Lester Nare?

  Right.

• 1:17:03Krishna Choudhary?

  Because the flow is coming in and then it's going out.

• 1:17:05Lester Nare?

  It's just that they, when they interact with each other, it just rolls off.

• 1:17:08Krishna Choudhary?

  Yeah, they're just, they just roll off.

• 1:17:10Lester Nare?

  Okay.

• 1:17:10Krishna Choudhary?

  Yeah. They're just rolling off one another. It's not especially useful. 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.

• 1:17:25Lester Nare?

  So we're going from thick with two Cs on the left here to thick with three Cs on the right.

• 1:17:30Krishna Choudhary?

  Yes.

• 1:17:30Lester Nare?

  Okay.

• 1:17:31Krishna Choudhary?

  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.

• 1:17:49Lester Nare?

  The things as they try to move past each other are, make it harder.

• 1:17:52Krishna Choudhary?

  Exactly. Yeah. And so for low viscosity, like thin liquids like, 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, we need to have a mathematical language to describe what is viscosity. So let's go back to our traffic analogy.

• 1:18:14Lester Nare?

  Okay.

• 1:18:14Krishna Choudhary?

  Okay. Let's consider a multi-laned 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, 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.

• 1:18:40Lester Nare?

  Yes.

• 1:18:41Krishna Choudhary?

  So I've color coded these cars based on their lane and based on their velocity.

• 1:18:46Lester Nare?

  Sorry. And what I'm realizing earlier, you specifically showed us only one lane examples.

• 1:18:51Krishna Choudhary?

  Yes.

• 1:18:52Lester Nare?

  Which was fine.

• 1:18:53Krishna Choudhary?

  Yeah.

• 1:18:54Lester Nare?

  At the time.

• 1:18:54Krishna Choudhary?

  At the time. But now we've got multiple lanes and now the flow is going to crisscross.

• 1:18:59Lester Nare?

  In between, which is how real liquids, okay.

• 1:19:01Krishna Choudhary?

  Yeah. How real liquids do it, right? Now, what I've done is in, in the initial, let's say it's a four-laned highway. The, 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.

• 1:19:47Krishna Choudhary?

  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 there's no force here.

• 1:20:14Lester Nare?

  Yeah, yeah, yeah.

• 1:20:15Krishna Choudhary?

  Okay? And even though there's some viscosity, there's no force.

• 1:20:18Lester Nare?

  Right. Okay. Yes.

• 1:20:19Krishna Choudhary?

  Right?

• 1:20:19Lester Nare?

  Yes.

• 1:20:20Krishna Choudhary?

  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.

• 1:20:31Krishna Choudhary?

  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, the carpool lane is still going at 80. And then the rest is like still very, very slow. And the carpool lane has like a barrier. So there's nothing hap, 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, but you, you get what I'm saying?

• 1:21:11Lester Nare?

  Yeah, yeah, yeah.

• 1:21:12Krishna Choudhary?

  Like, like before we had this curvy profile, which we see on the lower left, where again, we, we don't have a line because it's slow, slow, slow, then really fast.

• 1:21:23Lester Nare?

  Yes.

• 1:21:24Krishna Choudhary?

  And with subsequent time, it's going to level out.

• 1:21:28Lester Nare?

  It's going to level out.

• 1:21:29Krishna Choudhary?

  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.

• 1:21:38Lester Nare?

  Yeah.

• 1:21:39Krishna Choudhary?

  Which means there's a force lurking in here. Right? There's a viscous force that is lurking in here.

• 1:21:45Lester Nare?

  And this is what's, it's 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, you had no net effect, no force.

• 1:22:01Krishna Choudhary?

  No force.

• 1:22:02Lester Nare?

  In our second example here.

• 1:22:04Krishna Choudhary?

  Yeah.

• 1:22:05Lester Nare?

  It was sort of a not evenly distributed system.

• 1:22:09Krishna Choudhary?

  Yeah.

• 1:22:09Lester Nare?

  And so when you look over time, and then there's the mixing.

• 1:22:13Krishna Choudhary?

  Yeah, there's a mixing. There's going to be a change.

• 1:22:14Lester Nare?

  There's some divergent, not diver, let me not use the word divergence. There's some change.

• 1:22:17Krishna Choudhary?

  There's a change over time.

• 1:22:19Lester Nare?

  Over time.

• 1:22:19Krishna Choudhary?

  Which means that there is 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 slow stuff, the fast stuff is way faster than the slow stuff, which is why I'm speeding up. The curvature.

• 1:22:47Krishna Choudhary?

  Is the origin of the force. That's the key.

• 1:22:50Lester Nare?

  Yes.

• 1:22:50Krishna Choudhary?

  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 a del squared. 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.

• 1:23:16Krishna Choudhary?

  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.

• 1:23:30Lester Nare?

  Of the whatever the system, as I use colloquial sense as a whole.

• 1:23:34Krishna Choudhary?

  Yeah.

• 1:23:36Lester Nare?

  Is a material idea here, which is why we have all these are all derivatives. We're looking at the system level.

• 1:23:41Krishna Choudhary?

  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?

• 1:23:53Lester Nare?

  Yes.

• 1:23:53Krishna Choudhary?

  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?

• 1:24:28Lester Nare?

  I'm starting to see the connection between the words incompressible and smoothness that I keep seeing as it relates to the whole.

• 1:24:37Krishna Choudhary?

  Yeah.

• 1:24:38Lester Nare?

  Navier-Stokes piece.

• 1:24:39Krishna Choudhary?

  Yeah, we're, we're, we're starting to see that. So the viscous term becomes a smoothing operator in some sense, right?

• 1:24:46Krishna Choudhary?

  And now.

• 1:24:48Lester Nare?

  Which feels very natural as a property of trying to measure a liquid because

• 1:24:53Lester Nare?

  in this context, it's, it's sort of this contiguous thingy. When you look at it from the vector field perspective.

• 1:24:59Krishna Choudhary?

  Yeah, exactly. And, and we don't want that contiguous thingy to have like little pockets of really high fluid. I'm sorry, really high velocity. Because if there's a little pocket of really high velocity, well, that velocity is going to spread out.

• 1:25:11Lester Nare?

  Right.

• 1:25:11Krishna Choudhary?

  Right? And that's what viscosity does.

• 1:25:13Lester Nare?

  Right.

• 1:25:14Krishna Choudhary?

  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, it's maybe a little bit easier to have a pocket of high speed water.

• 1:25:26Lester Nare?

  But then it just depends on what time, time.

• 1:25:28Krishna Choudhary?

  Yeah.

• 1:25:28Lester Nare?

  Anyway.

• 1:25:29Krishna Choudhary?

  Yeah.

• 1:25:29Lester Nare?

  Yeah.

• 1:25:29Krishna Choudhary?

  And exactly. And so now we can solve Dalibert'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 a object that is moving through a fluid, very close to the object, the, 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 Dalibert's paradox.

• 1:26:14Krishna Choudhary?

  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, the, the part that's right above the golf ball is going to be moving with that sticky layer, but it's also going to be moving kind of with that. And so you get all of these dynamics happening because of viscosity.

• 1:26:41Lester Nare?

  There's this transitionary space between the surface of the object that is moving through the fluid and then the rest of space.

• 1:26:48Krishna Choudhary?

  Yeah.

• 1:26:48Lester Nare?

  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.

• 1:27:01Krishna Choudhary?

  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-Louis 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 a 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.

• 1:27:46Krishna Choudhary?

  Now we know it's not questionable, obviously. And over the following decades, we had Koshy, Poisson and others develop 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.

• 1:28:26Krishna Choudhary?

  And now these are the two equations that have been such a headache for mathematicians and physicists all these years.

• 1:28:32Lester Nare?

  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.

• 1:28:45Krishna Choudhary?

  Yeah, you can't make fluid and you can't destroy fluid.

• 1:28:48Lester Nare?

  You can't have a fluid have emanate from a point in all directions. And you can't have a fluid all converge on a point from all directions.

• 1:28:55Krishna Choudhary?

  Yeah.

• 1:28:56Lester Nare?

  That's what the bottom one is effectively saying.

• 1:28:58Krishna Choudhary?

  Yep.

• 1:28:58Lester Nare?

  And then now the top one, which was the more complex one is we're trying to replicate

• 1:29:06Lester Nare?

  force on the right equals match mass times acceleration on the left.

• 1:29:11Krishna Choudhary?

  Yes.

• 1:29:11Lester Nare?

  And in this construction.

• 1:29:12Krishna Choudhary?

  Exactly.

• 1:29:12Lester Nare?

  Which was Newton's second law of motion. We're trying to apply to incompressible fluids.

• 1:29:18Krishna Choudhary?

  Yes.

• 1:29:19Lester Nare?

  And in doing so, we constructed the left side of the equation first, where we have our mass represented by our volume.

• 1:29:28Krishna Choudhary?

  Yeah, that's the density.

• 1:29:30Lester Nare?

  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.

• 1:29:43Krishna Choudhary?

  Yeah. That's like rain coming in and changing traffic patterns.

• 1:29:46Lester Nare?

  Patterns. But we also have to add into that now the convective term.

• 1:29:53Krishna Choudhary?

  Yeah.

• 1:29:53Lester Nare?

  Which is how an individual car within that vector field is changing its velocity over with distance.

• 1:30:01Krishna Choudhary?

  Yes.

• 1:30:02Lester Nare?

  Over time.

• 1:30:03Krishna Choudhary?

  Yeah.

• 1:30:03Lester Nare?

  And so that's the left side.

• 1:30:04Krishna Choudhary?

  Yeah.

• 1:30:05Lester Nare?

  And that gives us our mass times acceleration, mass times acceleration.

• 1:30:08Krishna Choudhary?

  Yes.

• 1:30:09Lester Nare?

  Loosely speaking, maybe there's a few details not quite right. And then on the, our right. So there's, there's sort of three terms that have complexity on our mass times acceleration on the left.

• 1:30:22Krishna Choudhary?

  It's really two terms, I would say, right?

• 1:30:23Lester Nare?

  So because. I, I, so fair. Within acceleration, it has two component parts.

• 1:30:29Krishna Choudhary?

  Well, I'm, I'm nitpicking here, but like, it's really two terms because it's density multiplied by d u d t.

• 1:30:34Lester Nare?

  Okay.

• 1:30:34Krishna Choudhary?

  And then it's density multiplied by that second term.

• 1:30:36Lester Nare?

  Totally.

• 1:30:36Krishna Choudhary?

  You know what I mean?

• 1:30:37Lester Nare?

  Totally.

• 1:30:37Krishna Choudhary?

  Like density is not its own thing.

• 1:30:40Lester Nare?

  That's a fair, that's a fair, no, that's, that's an important distinction and 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.

• 1:30:51Krishna Choudhary?

  Yeah.

• 1:30:51Lester Nare?

  And it's negative because high pressure always moves to low pressure. Then we moved to, we had, we, we added our F, which is some general force, some general force.

• 1:31:03Krishna Choudhary?

  Yeah, stirring, gravity, whatever you wanted to call it.

• 1:31:06Lester Nare?

  And that's where Euler ended up.

• 1:31:08Krishna Choudhary?

  Yeah.

• 1:31:08Lester Nare?

  And he stopped. 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 and 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.

• 1:31:38Krishna Choudhary?

  Yep.

• 1:31:38Lester Nare?

  As we look at all of this. Is that now like an accurate description?

• 1:31:41Krishna Choudhary?

  Yeah.

• 1:31:42Lester Nare?

  Of what we're looking at.

• 1:31:43Krishna Choudhary?

  That is the Navier-Stokes equation. And we have just built it up from first principles, you know?

• 1:31:48Lester Nare?

  As, 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.

• 1:31:56Krishna Choudhary?

  Yeah, and it kind of makes sense though, right?

• 1:31:57Lester Nare?

  It makes total sense. Like, when you understand the underlying systems.

• 1:32:00Krishna Choudhary?

  I think the hardest part for, I think the hardest part really is, is getting familiar with the idea of a velocity field.

• 1:32:06Lester Nare?

  Yeah.

• 1:32:06Krishna Choudhary?

  Yeah. You know?

• 1:32:07Lester Nare?

  Yeah.

• 1:32:07Krishna Choudhary?

  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.

• 1:32:17Lester Nare?

  That was the Eulerian perspective was the velocity field, is the way we described it earlier.

• 1:32:21Krishna Choudhary?

  Yeah.

• 1:32:22Lester Nare?

  Very nice.

• 1:32:22Krishna Choudhary?

  Dalibert, dude.

• 1:32:25Krishna Choudhary?

  Yeah. And so now we can start asking, at this point, we've got the Navier-Stokes equations.

• 1:32:31Lester Nare?

  Yes.

• 1:32:31Krishna Choudhary?

  We know what every term means.

• 1:32:33Lester Nare?

  Yes.

• 1:32:33Krishna Choudhary?

  So now we can start tackling how to solve them.

• 1:32:38Lester Nare?

  Right. And because, and part of what we also defined through all of this is this is an extremely complex problem.

• 1:32:45Krishna Choudhary?

  Yes.

• 1:32:46Lester Nare?

  For all of the nuances of each of the underlying terms, which also have their own sub. Issues, subcategory issues.

• 1:32:54Krishna Choudhary?

  Exactly. And so we've got an equation of motion. Now can we solve it? Like, can we, 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 gonna look like later?

• 1:33:07Lester Nare?

  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.

• 1:33:17Krishna Choudhary?

  Yeah, exactly. Yeah.

• 1:33:18Lester Nare?

  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, what is, who cares?

• 1:33:26Krishna Choudhary?

  Yeah.

• 1:33:26Lester Nare?

  And the answer is so we can make predictions. If we go give you one input, the velocity field, and then you can give me the output consistently that's true every time.

• 1:33:34Krishna Choudhary?

  Yeah, that'd be dope.

• 1:33:35Lester Nare?

  That'd be great.

• 1:33:36Krishna Choudhary?

  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 two equals five, 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, the derivative of y in terms of t is equal to some r times y. Meaning, I've got some quantity. The change of that quantity with respect to time is proportional to the quantity itself,

• 1:34:20Krishna Choudhary?

  multiplied by, let's say, some rate. That is an exponential, right? The, the solution to that is e to the power of rt, and this is how people get rich. This is, uh, 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, 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, okay? The unknown is not how much money you have, like a single one-dimensional variable. It's a three-dimensional velocity field, right?

• 1:35:04Lester Nare?

  Yeah.

• 1:35:04Krishna Choudhary?

  So it's a number at every point in 3D space, but that number has a direction, right? It's a, it's a length and it's got a direction that's also three-dimensional. So it's, it's, it's quite, it's quite massive, like this, this equation.

• 1:35:21Lester Nare?

  Uh, it's, 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.

• 1:35:27Krishna Choudhary?

  Yeah.

• 1:35:27Lester Nare?

  Given that level of complexity, which is now very clear what that level of complexity is.

• 1:35:31Krishna Choudhary?

  Exactly. Like, the, the, the fluid is interacting with itself and everything. So this velocity field has to solve for that entire big thing, right? We gotta find all of the arrows everywhere.

• 1:35:42Lester Nare?

  Right.

• 1:35:43Krishna Choudhary?

  In some sense, right? For all time, because the arrows might be changing.

• 1:35:46Lester Nare?

  Over time.

• 1:35:47Krishna Choudhary?

  Over time. And we need, 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. Um, let's go through one of them. It's called plane coet, coet, coeté flow. I never know how to pronounce it.

• 1:36:16Lester Nare?

  I think it might be coet.

• 1:36:17Krishna Choudhary?

  Is it coeté?

• 1:36:18Lester Nare?

  Let us know in the comments.

• 1:36:19Krishna Choudhary?

  Yeah. In any case.

• 1:36:21Lester Nare?

  It looks French.

• 1:36:22Krishna Choudhary?

  Yeah, it, it's probably French, which is why I think it's coeté.

• 1:36:26Krishna Choudhary?

  Here's what's happening. You've got two plates, okay? Those are the Navier-Stokes equations that we're trying to solve.

• 1:36:31Lester Nare?

  Yes.

• 1:36:32Krishna Choudhary?

  I've got two plates. One plate, let's say, is the, the table, um, 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, 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, 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's stationary, is gonna be stationary. And the fluid that's on the plate that's moving is gonna be moving with the plate. What happens in the middle?

• 1:37:16Krishna Choudhary?

  It's actually pretty simple. You solve for the Navier-Stokes 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. Exactly solvable.

• 1:37:44Lester Nare?

  That, that makes total sense. And again, we've sort of created an exactly solvable, simple.

• 1:37:50Krishna Choudhary?

  Yeah.

• 1:37:51Lester Nare?

  Uh, use case, because, you know, by having the bottom plate be stationary.

• 1:37:56Krishna Choudhary?

  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.

• 1:38:05Lester Nare?

  They're only moving in on one plane.

• 1:38:06Krishna Choudhary?

  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.

• 1:38:29Lester Nare?

  Fine.

• 1:38:30Krishna Choudhary?

  You're fine. Yeah, this is called coeté flow. You can also do pipe flows, like, there's a handful of vortices in jets where you can obtain beautiful closed-form solutions based on the symmetry of the situation.

• 1:38:42Lester Nare?

  And just to ask as 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, it's not, not relevant at all.

• 1:39:00Krishna Choudhary?

  Well, no, certainly, I think, I think, I mean, for that one, 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 in jets where you can manipulate the symmetry to cancel out terms and things like that. You can, you can use our knowledge of symmetry.

• 1:39:16Lester Nare?

  Right.

• 1:39:17Krishna Choudhary?

  To, to, to make, like, transformations that make these equations much easier to handle.

• 1:39:22Lester Nare?

  The, the subtle note I'm just trying to make here is, is these things don't happen in a vacuum. Like, our general mathematical understanding.

• 1:39:30Krishna Choudhary?

  Yeah, 100%.

• 1:39:30Lester Nare?

  Has applications in subtle, small ways all through.

• 1:39:34Krishna Choudhary?

  All through, yeah.

• 1:39:34Lester Nare?

  All through.

• 1:39:35Krishna Choudhary?

  Yeah, all through. And so for, that's, that's for a very nice symmetric thing. Formula One cars, not very symmetric. I mean, there's some symmetry, I guess, left and right, right? So sure, you can see that it's symmetric, that, like, you can flip the, 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 nonlinear term does not vanish. In the other one, the nonlinear term actually vanishes just because of the geometry of the thing. Here, this is the entire problem, right?

• 1:40:20Krishna Choudhary?

  There's no known formula that you can write out for what a Formula One car, the, the velocity and pressure at every point around a Formula One car is gonna 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.

• 1:40:59Krishna Choudhary?

  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.

• 1:41:22Lester Nare?

  And then you just can put them all together.

• 1:41:23Krishna Choudhary?

  Yeah. You can be like, oh, the velocity is here, but the velocity next to it is something, the, 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.

• 1:41:38Lester Nare?

  Yeah.

• 1:41:38Krishna Choudhary?

  Right? And, and I can, I can do this computationally, one at a time. And the more cells I have, the, the better granularity and the more accuracy I have to the real thing.

• 1:41:48Lester Nare?

  This, this makes me think back to our hypersonics episode where we effectively discussed the same thing.

• 1:41:54Krishna Choudhary?

  Yeah.

• 1:41:54Lester Nare?

  And, 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 weirder.

• 1:42:06Krishna Choudhary?

  Yeah.

• 1:42:07Lester Nare?

  And we've sort of discussed why it gets weirder 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 cases. We've found clever ways.

• 1:42:20Krishna Choudhary?

  Yeah, to, like, approximate.

• 1:42:23Lester Nare?

  Yeah, the solution, right?

• 1:42:24Krishna Choudhary?

  And for engineers, this is good enough.

• 1:42:26Lester Nare?

  Right. The, the point, they're like.

• 1:42:27Krishna Choudhary?

  Honestly, like, yeah.

• 1:42:28Lester Nare?

  Oh, but I didn't solve anything. We've been modeling a Navier-Stokes since you were, before you were born.

• 1:42:33Krishna Choudhary?

  Yeah, for the Ferrari engineers, it's not good enough because apparently they still can't do it.

• 1:42:39Krishna Choudhary?

  Like, guys, figure it out. You have Lewis Hamilton and Charles Leclerc. Like, guys, come, 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 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?

• 1:43:24Krishna Choudhary?

  In any case, the idea is any arbitrary shape, you can put it into a simulation, and you can turn the simulation, one after the other, and computationally solve the Navier-Stokes equations. For all practical purposes, that's totally fine.

• 1:43:40Lester Nare?

  It's an engineering solution. It's not a physics understanding.

• 1:43:45Krishna Choudhary?

  Yeah.

• 1:43:46Lester Nare?

  And/or mathematical understanding. And it's, it's fine that it's an applied physics, uh, engineering solution.

• 1:43:51Krishna Choudhary?

  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 got wind, uh, wind farms for, like, power. It's, but it's not the same thing as the exact equations. And what do those equations look like, right? What, what does 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, like, putting CAD model in there,

• 1:44:35Krishna Choudhary?

  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.

• 1:44:44Lester Nare?

  And unless you're making hypersonics, it's fine.

• 1:44:47Krishna Choudhary?

  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?

• 1:45:02Lester Nare?

  Okay.

• 1:45:03Krishna Choudhary?

  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

• 1:45:16Krishna Choudhary?

  competition to one another.

• 1:45:18Lester Nare?

  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.

• 1:45:28Krishna Choudhary?

  Yes. And these two things do very different qualitative transformations to our velocity field. The viscosity term, as I told you, smoothes 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.

• 1:46:13Krishna Choudhary?

  There's a constant competition between these two terms in the equation.

• 1:46:18Lester Nare?

  Because depending on the size of the structure we're referring to, the viscosity will be different, and the convective term will be different. And it's different across every scale.

• 1:46:27Krishna Choudhary?

  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, so we're going to talk about, um, 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.

• 1:47:09Lester Nare?

  Okay.

• 1:47:09Krishna Choudhary?

  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 equals zero. So this is what you're starting out with. It's a bunch of sines and cosines for the x velocity. It's a bunch of sines 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.

• 1:47:43Lester Nare?

  Flatland.

• 1:47:44Krishna Choudhary?

  Flatland, in flatland, right? But there's bunches of flatland. Like, it can still be a 3D box, but like, the, the stuff at the top of the box is still going side to side. And the, 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 sines and cosines. So what that, what that amounts to is you have vortices. Okay? You have a, the sines and cosines always means circles. Okay? If you, if you, if you come out with one thing from this podcast, sines and cosines mean circles. But in this case, what's happening is you've got these vortices that are circular. Okay? And so all of 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.

• 1:48:30Krishna Choudhary?

  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?

• 1:48:42Lester Nare?

  For those who are not watching, my, uh, my eyes just expanded to a very large size.

• 1:48:49Krishna Choudhary?

  Yeah, but it's, it's not that bad. And, and I, I'll tell you what to focus on.

• 1:48:53Lester Nare?

  Okay.

• 1:48:53Krishna Choudhary?

  Okay? On the left-hand side is our initial conditions. These are our sines and cosines. It's just sine x and cosine, cosine y, sine, cosine z. So this tells you that, like, the, the size of my vortex is 2 pi in size. 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 identities and trig. But what ends up happening is

• 1:49:37Krishna Choudhary?

  those derivatives are going to create terms with a sine 2x and a cosine 2x. That's going to percolate into the, into the pressure terms. You're going to get sine 2x and cosine 2x. And one thing that you should remember is

• 1:49:55Krishna Choudhary?

  sine x is a wave. Sine 2x is that same wave squished.

• 1:50:00Lester Nare?

  Yeah.

• 1:50:01Krishna Choudhary?

  You're creating smaller and smaller waves. The wavelength is getting smaller.

• 1:50:06Lester Nare?

  The oscillation.

• 1:50:08Krishna Choudhary?

  Is getting squished. Okay? Now, what does that do to our structure of the vortex? What happens is,

• 1:50:16Krishna Choudhary?

  remember, before I told you that, yeah, that's, that's the one. So remember, before I told you that the z component of my velocities was zero. Everything was happening

• 1:50:25Lester Nare?

  Yep.

• 1:50:26Krishna Choudhary?

  in side to side, right? But now, because of that convective term, and because of those higher wavelengths, the, 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 pushing up, push, push up and down, those structures are going to be half the size of your original vortex.

• 1:50:55Lester Nare?

  And this is direct, coming from the squishing.

• 1:50:57Krishna Choudhary?

  Mm-hmm. 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, they're revolving in this way.

• 1:51:19Lester Nare?

  And they sort of have, sort of, for lack of a better term, a, a consistent, non-disturbed

• 1:51:25Krishna Choudhary?

  Yeah.

• 1:51:25Lester Nare?

  structure.

• 1:51:26Krishna Choudhary?

  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're getting these interactions between the vortices up and down. And the interactions, those structures that are created in that interaction, have a smaller and smaller

• 1:51:44Krishna Choudhary?

  Size because of that convective term.

• 1:51:47Lester Nare?

  Okay.

• 1:51:48Krishna Choudhary?

  Okay.

• 1:51:48Lester Nare?

  I see.

• 1:51:49Krishna Choudhary?

  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 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.

• 1:52:09Lester Nare?

  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.

• 1:52:26Krishna Choudhary?

  Yes.

• 1:52:26Lester Nare?

  As we looked at in this example.

• 1:52:28Krishna Choudhary?

  Yeah.

• 1:52:28Lester Nare?

  And at, at time equals zero.

• 1:52:31Krishna Choudhary?

  When we had those blobs.

• 1:52:32Lester Nare?

  We have the blobs on the top left. And as time goes by, the, the left side of the equation is.

• 1:52:40Krishna Choudhary?

  The convective term.

• 1:52:41Lester Nare?

  Is winning.

• 1:52:41Krishna Choudhary?

  Is winning, yeah.

• 1:52:42Lester Nare?

  Effectively.

• 1:52:43Krishna Choudhary?

  Effectively, because it's creating smaller and smaller structure that's leaking into the Z direction.

• 1:52:48Lester Nare?

  And so our, 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 question, the fundamental question is, does this go all the way to infinity?

• 1:53:02Krishna Choudhary?

  Yeah.

• 1:53:03Lester Nare?

  In smallness.

• 1:53:03Krishna Choudhary?

  And do I get small and smaller, smaller and smaller and smaller structures?

• 1:53:06Lester Nare?

  Does the, is the convective term the GOAT? And does it always win?

• 1:53:10Krishna Choudhary?

  Yeah.

• 1:53:11Lester Nare?

  And what we're saying is no.

• 1:53:12Krishna Choudhary?

  In this case, no.

• 1:53:13Lester Nare?

  In this case, at some point, the viscosity starts to matter.

• 1:53:17Krishna Choudhary?

  Yes.

• 1:53:18Lester Nare?

  Once you get to a sufficiently small scale.

• 1:53:20Krishna Choudhary?

  Yes, because, and, and for technical audiences, the, 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.

• 1:53:39Lester Nare?

  Yeah, that makes sense.

• 1:53:40Krishna Choudhary?

  So the Taylor-Green vortex, I think it puts this conflict right in front of us.

• 1:53:44Lester Nare?

  Yeah.

• 1:53:45Krishna Choudhary?

  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.

• 1:54:00Lester Nare?

  Is the convective term going like this and the viscosity is going like this?

• 1:54:04Krishna Choudhary?

  It's like, it's like this versus this.

• 1:54:05Lester Nare?

  Okay, okay.

• 1:54:06Krishna Choudhary?

  You know what I mean?

• 1:54:07Lester Nare?

  Yeah. So eventually, at like, at some, this meeting point.

• 1:54:10Krishna Choudhary?

  Yeah.

• 1:54:11Lester Nare?

  Of, of structure size.

• 1:54:12Krishna Choudhary?

  Yes.

• 1:54:12Lester Nare?

  Is when.

• 1:54:13Krishna Choudhary?

  Is when the viscosity is like, okay, that's enough.

• 1:54:14Lester Nare?

  Yeah, yeah, yeah.

• 1:54:15Krishna Choudhary?

  That's enough.

• 1:54:16Lester Nare?

  That makes sense.

• 1:54:16Krishna Choudhary?

  Let's calm down.

• 1:54:17Lester Nare?

  Yep.

• 1:54:17Krishna Choudhary?

  And so the viscosity waits for those smaller scales to convert all of that energy into just like dissipation and heat. Right? And the, 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. But if it's like not very viscous.

• 1:54:34Lester Nare?

  It'll take time.

• 1:54:35Krishna Choudhary?

  Um, it'll take some time, but it's going to get there. With, for the Taylor-Green vortex, it is going to get there.

• 1:54:40Lester Nare?

  That's fascinating.

• 1:54:41Krishna Choudhary?

  Right? And now in every single simulation that we've done, viscosity always wins out.

• 1:54:46Lester Nare?

  Okay.

• 1:54:47Krishna Choudhary?

  Okay? At some point, viscosity is always going to win out.

• 1:54:51Lester Nare?

  V for vendetta, one would say.

• 1:54:52Krishna Choudhary?

  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 cohette 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.

• 1:55:12Lester Nare?

  So is this the idea that smoothness always wins?

• 1:55:15Krishna Choudhary?

  Yes.

• 1:55:15Lester Nare?

  Like to describe it in another way.

• 1:55:17Krishna Choudhary?

  The, the solution is always going to be smooth because of the viscosity.

• 1:55:21Lester Nare?

  Because of the viscosity, winning in this race.

• 1:55:24Krishna Choudhary?

  Yeah.

• 1:55:24Lester Nare?

  Okay.

• 1:55:25Krishna Choudhary?

  Is it always going to win?

• 1:55:26Lester Nare?

  Is it always going to?

• 1:55:27Krishna Choudhary?

  Yes.

• 1:55:27Lester Nare?

  We've shown that it can win sometimes.

• 1:55:29Krishna Choudhary?

  Yes. And for those two times, it's going to win.

• 1:55:31Lester Nare?

  Sure.

• 1:55:31Krishna Choudhary?

  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, you literally can't physically, like, there's not enough compute to go down to what, the plank length or something like that, right?

• 1:55:57Lester Nare?

  Where it could start falling apart.

• 1:55:58Krishna Choudhary?

  Like, like, so, so perhaps that's that. And that's 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,

• 1:56:11Krishna Choudhary?

  is viscosity always going to win?

• 1:56:13Lester Nare?

  This is the, is smoothness global?

• 1:56:15Krishna Choudhary?

  Yes.

• 1:56:16Lester Nare?

  Idea. And, and that, that actually, it's so funny because now it's, it kind of reminds me a little bit of, was it the, uh, was it the Riemann hypothesis episode? Where we talked about, uh, the non.

• 1:56:34Krishna Choudhary?

  Trivial zeros.

• 1:56:35Lester Nare?

  Non-trivial zeros.

• 1:56:35Krishna Choudhary?

  Yeah.

• 1:56:36Lester Nare?

  And it's similarly like, you know, we want to take the question to its ultimate.

• 1:56:40Krishna Choudhary?

  Yeah. Is like, is every single zero on the critical line?

• 1:56:43Lester Nare?

  Critical line, right?

• 1:56:44Krishna Choudhary?

  Yeah.

• 1:56:44Lester Nare?

  Every single one.

• 1:56:45Krishna Choudhary?

  Every single one.

• 1:56:46Lester Nare?

  Because you can still have 99.99.

• 1:56:48Krishna Choudhary?

  You could even have a hundred.

• 1:56:49Lester Nare?

  As we were saying.

• 1:56:50Krishna Choudhary?

  As we were saying.

• 1:56:50Lester Nare?

  And it does, that doesn't mean.

• 1:56:52Krishna Choudhary?

  You can have a hundred percent.

• 1:56:53Lester Nare?

  On the critical.

• 1:56:53Krishna Choudhary?

  And still have one off.

• 1:56:55Lester Nare?

  And anywhere in all possibility that one is off, it means.

• 1:56:59Krishna Choudhary?

  Yeah.

• 1:57:00Lester Nare?

  Just conceptually.

• 1:57:01Krishna Choudhary?

  Mathematicians are worried about, right? It's like, no, no, no, in every universe, right? That Dr. Strange has ever visited.

• 1:57:10Krishna Choudhary?

  Is this true?

• 1:57:11Lester Nare?

  It feels similarly in that direction.

• 1:57:13Krishna Choudhary?

  Yeah, it's, it's a very mathematical thing to, to worry about.

• 1:57:16Lester Nare?

  Now I understand the mindset.

• 1:57:18Krishna Choudhary?

  Yeah. Now, um, crucially, this is now an open problem only in 3D.

• 1:57:24Lester Nare?

  Okay.

• 1:57:25Krishna Choudhary?

  In 2D, there was a very famous mathematician, she was a Soviet mathematician, Olga Ladizhenskaya. Olga Ladizhenskaya. 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.

• 1:57:49Lester Nare?

  If we're in flatland.

• 1:57:50Krishna Choudhary?

  Mm-hmm. Yes. If we're in flatland and we've got like fluids flowing in flatland.

• 1:57:54Lester Nare?

  And there's no a Z up, Z down that we just described.

• 1:57:56Krishna Choudhary?

  As, as we saw, actually, like even in the Taylor-Green vortex.

• 1:57:59Lester Nare?

  Okay, okay.

• 1:57:59Krishna Choudhary?

  Right? We saw that like the Z dimension was the one that was causing that.

• 1:58:02Lester Nare?

  Yeah.

• 1:58:03Krishna Choudhary?

  Weirdness. Because, because the flow was escaping into the Z dimension.

• 1:58:07Lester Nare?

  Yes.

• 1:58:07Krishna Choudhary?

  And then creating those smaller and smaller structures.

• 1:58:09Lester Nare?

  Yeah.

• 1:58:10Krishna Choudhary?

  So Olga showed in 1969 in her, 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 gonna, you're gonna come up with a smooth solution. Okay, so for, for 2D, the Navier-Stokes global smoothness is known.

• 1:58:31Lester Nare?

  It's fine. That's not the question we're asking.

• 1:58:33Krishna Choudhary?

  Yeah.

• 1:58:33Lester Nare?

  For 3D.

• 1:58:34Krishna Choudhary?

  For 3D is, is.

• 1:58:36Lester Nare?

  It is not known. And is this, and I just have a quick question. So when, when I see the term blow up. As it relates to how people describe this.

• 1:58:45Krishna Choudhary?

  Yeah.

• 1:58:46Lester Nare?

  This is what we're, is this kind of what we're talking about? Is like, does the smoothness, does, um, does viscosity lose?

• 1:58:55Krishna Choudhary?

  Yes.

• 1:58:55Lester Nare?

  Which is, which is.

• 1:58:56Krishna Choudhary?

  Effectively.

• 1:58:56Lester Nare?

  Blow up.

• 1:58:57Krishna Choudhary?

  Yeah.

• 1:58:57Lester Nare?

  Like for lack of.

• 1:58:58Krishna Choudhary?

  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.

• 1:59:12Lester Nare?

  Hmm.

• 1:59:13Krishna Choudhary?

  Right?

• 1:59:14Lester Nare?

  Mm-hmm. It's, it, it's kind of like, is, is there a singularity in fluids? Not really, but like.

• 1:59:19Krishna Choudhary?

  Well, no, but that, no, no, no, that's literally what they're asking. Is there a singularity? Like, can I create a singularity? That's what, that's what, that's what they're asking.

• 1:59:27Lester Nare?

  Okay, fair enough.

• 1:59:28Krishna Choudhary?

  That's exactly what they're asking. Now, why is it so hard in three dimensions? Well, we all, all, we already saw, like, kind of occluded that with the Taylor-Green vortex, right? It's the fact that like stuff can move up.

• 1:59:38Lester Nare?

  Yeah.

• 1:59:38Krishna Choudhary?

  And down. And crucially, that had to do with a vortex that I had started. Right? The initial condition was a vortex. The sines 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. Okay? Vorticity is the curl, um, for those, for those who have taken calculus, the curl of the vector field. That del operator is back again, but this time we're taking a cross product of the del 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.

• 2:00:23Krishna Choudhary?

  Has nowhere to go.

• 2:00:25Lester Nare?

  Right.

• 2:00:26Krishna Choudhary?

  Right? It's just spinning and then it's got to dissipate in the flatland.

• 2:00:30Lester Nare?

  Yeah.

• 2:00:31Krishna Choudhary?

  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, um, 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.

• 2:00:49Lester Nare?

  Cause it's nowhere to go.

• 2:00:50Krishna Choudhary?

  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 our, yeah, let's say a hurricane. It's quite big in space and it's moving. But the, the, the velocity is not that high. Okay? And now that hurricane, all of that energy gets squished into a tornado.

• 2:01:19Krishna Choudhary?

  What's gonna happen? Well, the tornado is gonna get very, very tall. Because all of that energy from the hurricane has to.

• 2:01:25Lester Nare?

  Divergence has to equal zero.

• 2:01:27Krishna Choudhary?

  Yeah. Very good. This is exactly it.

• 2:01:29Lester Nare?

  It has to equal zero.

• 2:01:31Krishna Choudhary?

  So it's gotta go somewhere.

• 2:01:32Lester Nare?

  It's gotta go somewhere.

• 2:01:32Krishna Choudhary?

  So it's gonna go up, right? But because it's getting tall. Again, the radius is gonna decrease, but if the radius decreases. Now we have another principle of physics, which is the conservation of angular momentum. And so just like, um, you know, um, in the, the figure skaters. When they're twirling around, they pull their arms in, they spin faster. Same thing's gonna happen here. The thing is gonna spin faster.

• 2:01:57Lester Nare?

  And this, 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.

• 2:02:06Krishna Choudhary?

  Yeah.

• 2:02:07Lester Nare?

  Just conceptually, there's a sort of similar.

• 2:02:09Krishna Choudhary?

  Yeah, you're packing stuff in here, but here it's the vorticity that's increasing because of the conservation of angular momentum.

• 2:02:14Lester Nare?

  Yeah.

• 2:02:14Krishna Choudhary?

  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.

• 2:02:21Lester Nare?

  Makes sense.

• 2:02:21Krishna Choudhary?

  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 gonna move faster and faster.

• 2:02:34Lester Nare?

  And this kind of relates to what we just talked about, about in the Taylor Green vortex. Why we're now leaking into the.

• 2:02:40Krishna Choudhary?

  And into those higher dimensions, right?

• 2:02:42Lester Nare?

  Right.

• 2:02:42Krishna Choudhary?

  And the question is maybe that with this vortex stretching. You can actually, um, create a kind of singularity. Okay? Now let's get into what we mean by singularity in the first place. We, we had already done our first toy differential equation, right?

• 2:02:56Lester Nare?

  Yes.

• 2:02:57Krishna Choudhary?

  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.

• 2:03:10Lester Nare?

  Infinite money glitch.

• 2:03:11Krishna Choudhary?

  If I wait long enough, even if I have a dollar in my bank account, that dollar is gonna become infinity. If I wait long enough, right? It just turns out maybe AI is gonna end the world. So I, the point is. Uh, that's not a real singularity. A real singularity is in finite, in finite time. I shouldn't say in finite because that sounds like infinite. So I should say, um. Before. No. In, pause.

• 2:03:38Lester Nare?

  Pause.

• 2:03:38Krishna Choudhary?

  Finite time. Can I get to infinity?

• 2:03:44Lester Nare?

  In some constrained amount of time.

• 2:03:46Krishna Choudhary?

  There we go. Yes. For our listeners out there, that, that'll be good. In, 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 dy dt is equal to y squared. Initial condition, you start at one, when time is zero. And the solution to that is one over one minus t. If I plug in t equals one. I get infinity. Right? So this is a differential equation that creates a singularity. In, but the time is not infinity. The time is just one. At one second, I have a singularity.

• 2:04:24Lester Nare?

  You reach, you reach the singularity.

• 2:04:25Krishna Choudhary?

  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 dt 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, at a finite time.

• 2:04:57Lester Nare?

  So it's conceivable to extrapolate from this.

• 2:05:00Krishna Choudhary?

  Yeah.

• 2:05:00Lester Nare?

  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.

• 2:05:09Krishna Choudhary?

  Yeah, it's like, there's a chance.

• 2:05:11Lester Nare?

  There, there's a chance because we got a lot of non-linearity going on.

• 2:05:14Krishna Choudhary?

  Yeah, yeah. It's just that there's the viscosity term. Right? In, in this one that I showed you.

• 2:05:20Lester Nare?

  There's no smoothing.

• 2:05:20Krishna Choudhary?

  There's no, there's no smoothing.

• 2:05:21Lester Nare?

  There's no counteract.

• 2:05:22Krishna Choudhary?

  It's just going. Right? But, 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.

• 2:05:32Lester Nare?

  It has this counterforce, which is viscosity.

• 2:05:34Krishna Choudhary?

  And so you're, you're asking who's gonna win? And are they always going to win? So far, it seems viscosity is always one.

• 2:05:40Lester Nare?

  That's actually, that makes so much sense now. It, it's, it's, ah, that's so good.

• 2:05:44Krishna Choudhary?

  Right?

• 2:05:45Lester Nare?

  That's so good.

• 2:05:46Krishna Choudhary?

  It's quite nice.

• 2:05:46Lester Nare?

  Yeah, it's quite nice.

• 2:05:48Krishna Choudhary?

  So it, it is a very interesting problem.

• 2:05:50Lester Nare?

  Yeah, no, it, it, because, because you can understand, you could. There's good arguments for both sides. One in which viscosity always wins. Feels true.

• 2:06:01Krishna Choudhary?

  Yeah.

• 2:06:01Lester Nare?

  Because.

• 2:06:02Krishna Choudhary?

  Seems legit.

• 2:06:02Lester Nare?

  Because we've been able to show that.

• 2:06:06Lester Nare?

  Over and over again.

• 2:06:07Krishna Choudhary?

  Yeah.

• 2:06:07Lester Nare?

  In a variety of contexts as being true.

• 2:06:10Krishna Choudhary?

  Yeah.

• 2:06:11Lester Nare?

  But being fundamentally true.

• 2:06:13Krishna Choudhary?

  Is another thing.

• 2:06:14Lester Nare?

  Is another thing. And because of, again, the nonlinear nature of the convective term in this case. That is challenging the viscosity as the counterforce. There's an argument that both of them have a reason that they could win.

• 2:06:29Krishna Choudhary?

  Exactly.

• 2:06:30Lester Nare?

  And.

• 2:06:30Krishna Choudhary?

  In finite time.

• 2:06:31Lester Nare?

  In finite time. Exactly. And so.

• 2:06:34Krishna Choudhary?

  It's, it's, it's an interesting problem. I have to give it to them, right? Like it's, 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, 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 nonlinear differential equation. Because you've got this, um. 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, um, you know, for example, I mean, very. In a, in a very simple case, like the curvature of space and time. Causes gravity, right?

• 2:07:17Krishna Choudhary?

  But gravity itself has an energy that causes the curvature of space and time.

• 2:07:23Krishna Choudhary?

  Right? So, so what? And so, 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, um, I mentioned Lake Geneva that I was on a, like. I was actually at the Yerkes 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 Yerkes Observatory many times. And there, uh, on the, on the slab of marble was written out Einstein's field equation. So I just wanted to show that, um, when I, when I received this news, I was looking at a non-linearity. Live and I was like, damn.

• 2:08:05Lester Nare?

  Right?

• 2:08:05Krishna Choudhary?

  Here we go again.

• 2:08:06Lester Nare?

  Here we go again.

• 2:08:07Krishna Choudhary?

  You know?

• 2:08:07Lester Nare?

  The meme we started the episode with.

• 2:08:10Krishna Choudhary?

  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.

• 2:08:27Lester Nare?

  It's purely for the math.

• 2:08:28Krishna Choudhary?

  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 singularity. There are plenty of cases where the Navier-Stokes equations don't even apply. And I've got, I've got some examples over here. I mean, for example, um. 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.

• 2:09:08Lester Nare?

  Right.

• 2:09:08Krishna Choudhary?

  Right? The, the sound wave is catching, 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. Uh, 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. Um, in the middle, that's an aquaporin protein. Those things are so small that they let in only water molecules.

• 2:09:43Krishna Choudhary?

  At a molecular scale, it doesn't, velocity fields don't matter at a molecular scale. Like, what are we talking about?

• 2:09:49Lester Nare?

  Right, right.

• 2:09:50Krishna Choudhary?

  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.

• 2:10:20Lester Nare?

  Which are specifically for incompressible fluids.

• 2:10:23Krishna Choudhary?

  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.

• 2:10:51Lester Nare?

  Hundred percent.

• 2:10:52Krishna Choudhary?

  Right? And, but, 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 like 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.

• 2:11:19Lester Nare?

  Yeah.

• 2:11:19Krishna Choudhary?

  And the fact that the mass of the fluid, like that density term, is gonna have a Lorentz factor in there or something. Because the, 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.

• 2:11:44Lester Nare?

  And I, 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

• 2:12:16Krishna Choudhary?

  Yeah.

• 2:12:16Lester Nare?

  that don't apply to the real world.

• 2:12:18Krishna Choudhary?

  Yeah.

• 2:12:18Lester Nare?

  But still have a fundamental value to our mathematical understanding

• 2:12:23Krishna Choudhary?

  Exactly.

• 2:12:23Lester Nare?

  of the world, of just math, math.

• 2:12:25Krishna Choudhary?

  Yeah.

• 2:12:26Lester Nare?

  Which again, then can bleed into, like I was mentioning earlier, two ways to think about other problems. Because it's, 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?

• 2:12:46Krishna Choudhary?

  Yes.

• 2:12:46Lester Nare?

  And then it created this whole centuries-long journey.

• 2:12:49Krishna Choudhary?

  Yeah, I mean, all, 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, that's a mathematics theorem.

• 2:13:06Lester Nare?

  Right.

• 2:13:06Krishna Choudhary?

  Stokes' theorem, that like then, oh, it like works here, right? Um, so definitely something that is super interesting. And Terry Tao actually said that like the Navier-Stokes-

• 2:13:16Lester Nare?

  Oh, we're calling him Terry now, look at this guy. Oh, it's my buddy Terry.

• 2:13:19Krishna Choudhary?

  We need to edit that out. Hey, yo. If you, if somebody- All right, Terrence Tao. What, what, no, no, no, keep it in. But like, Terrence Tao, he said, um, the Navier-Stokes equations are the, are one of the simplest, supercritical differential equations.

• 2:13:35Lester Nare?

  Hmm.

• 2:13:35Krishna Choudhary?

  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 Laplacian and you've got this like convective thing and they're competing. And mathematically, it's a very interesting question to answer.

• 2:13:51Lester Nare?

  Mm-hmm.

• 2:13:52Krishna Choudhary?

  Now, there's been a lot of work since, um, the, the days of Navier-Stokes to try and understand the mathematical properties of this. The first guy I want to talk about is Jean Leray. Um, 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-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 ma? 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.

• 2:14:33Krishna Choudhary?

  And we say, the square of all of the velocities in that volume, times one-half, that's gonna tell you sort of the energy density in that volume.

• 2:14:42Lester Nare?

  In the volume.

• 2:14:43Krishna Choudhary?

  Like the kinetic energy density in that volume.

• 2:14:45Lester Nare?

  Which would be, in this case, the velocity field.

• 2:14:46Krishna Choudhary?

  Yeah, yeah, exactly. And so, um, 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 gonna be honest, at this point, I'm getting into stuff that, this is, 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, I don't really know, right? But from what I've gathered, what, 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?

• 2:15:30Krishna Choudhary?

  But what, 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.

• 2:15:47Lester Nare?

  Mm-hmm.

• 2:15:47Krishna Choudhary?

  So the overall energy is finite, but some local small velocity can be infinite.

• 2:15:51Lester Nare?

  Yep.

• 2:15:52Krishna Choudhary?

  So already, it's, it's starting to show that maybe there could be ways that the convective term wins.

• 2:15:58Lester Nare?

  Mm-hmm.

• 2:15:59Krishna Choudhary?

  Okay? 1982, Caffarelli, Cohen, and Nuremberg, 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.

• 2:16:19Lester Nare?

  Okay.

• 2:16:19Krishna Choudhary?

  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. Okay?

• 2:16:34Lester Nare?

  And, and so the, the system has some fundamental limitation where it has, it can only happen at particular scales.

• 2:16:39Krishna Choudhary?

  Particular scales, at very small infinitesimal scales. This is why, again, it's like a mathematical problem, right? It's not, it's, it's infinitesimally small scales and it's happening only at particular places. It's not like a continuous singularity, right? It's, it's at the center of a black hole, it's a single spot.

• 2:16:56Lester Nare?

  Mm-hmm.

• 2:16:56Krishna Choudhary?

  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.

• 2:17:15Lester Nare?

  And so the, the, the, the box in which the convective term wins, we've now created a very small box.

• 2:17:23Krishna Choudhary?

  Yes.

• 2:17:24Lester Nare?

  It could still win.

• 2:17:24Krishna Choudhary?

  It could still win.

• 2:17:25Lester Nare?

  Against viscosity, but it's a very small box.

• 2:17:27Krishna Choudhary?

  Very small box, right? And this brings us 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. Um, that, you know, a bunch of them got solved over in the 20th century. Some by John Nash, of course, um, and others. And finally, we have, um, 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. Um, and that was huge. Um, he never accepted the million, the million dollars, because, um, he said that someone else deserved it.

• 2:18:10Krishna Choudhary?

  He's kind of like this weird guy, let's, let's, I'm gonna be honest, he's a weird Russian guy. Brilliant mathematician, but like, lives kind of like in, in his mom's basement kind of thing.

• 2:18:20Lester Nare?

  Yeah.

• 2:18:20Krishna Choudhary?

  Anyways, brilliant mathematician. Um, so here's the problem statement, and this is a problem statement by Charles-

• 2:18:26Lester Nare?

  I want to make one quick note here, which is, uh, for those viewing who can see the list, I want you to remember this list, right? Uh, because this list is going to come into relevance when we talk about the race between the frontier labs.

• 2:18:43Krishna Choudhary?

  Mm-hmm.

• 2:18:43Lester Nare?

  And this is the gold medal that they're all chasing after.

• 2:18:46Krishna Choudhary?

  Yeah.

• 2:18:47Lester Nare?

  Is-

• 2:18:48Krishna Choudhary?

  The infinity stone.

• 2:18:49Lester Nare?

  Yeah, these are the infinity stones and they are coming after all of these. Maybe not Yang Mills and maybe not P versus NP.

• 2:18:55Krishna Choudhary?

  Yeah.

• 2:18:55Lester Nare?

  But all the other ones.

• 2:18:56Krishna Choudhary?

  All the other ones. Riemann hypothesis, they came after it.

• 2:18:59Lester Nare?

  Yep.

• 2:18:59Krishna Choudhary?

  Um, or at least something that is related.

• 2:19:01Lester Nare?

  Related.

• 2:19:02Krishna Choudhary?

  Um, Navier-Stokes is the one that we're at right now. Um, Yang Mills, I mean, we, 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 the, each of these, 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, um, field medalist, um, 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.

• 2:19:40Lester Nare?

  Yeah, okay.

• 2:19:41Krishna Choudhary?

  Because they both had the same PhD advisor, also field medalist.

• 2:19:44Lester Nare?

  I, I just want to note, for those who might not know, field medalist, meaning that they've won this, the, the most prestigious mathematics award that is available.

• 2:19:53Krishna Choudhary?

  For those who are under 40.

• 2:19:55Lester Nare?

  For those who are under 40.

• 2:19:56Krishna Choudhary?

  Yes. I would say the most prestigious mathematics award that people don't really know about is the Albert Prize.

• 2:20:01Lester Nare?

  We talk, I-

• 2:20:01Krishna Choudhary?

  Yeah, which is, which is, I think, like the Nobel Prize of mathematics.

• 2:20:04Lester Nare?

  Fair enough.

• 2:20:05Krishna Choudhary?

  People keep saying this, the field medal, I really don't think it is.

• 2:20:07Lester Nare?

  Um, argue about it in the comments.

• 2:20:09Krishna Choudhary?

  Yeah. Um, so Charles Pfefferman puts out this statement of what it is that we are trying to prove, okay?

• 2:20:18Krishna Choudhary?

  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.

• 2:20:44Lester Nare?

  And so, and just to, just to quickly touch on this. So we're saying A and B means viscosity wins.

• 2:20:51Krishna Choudhary?

  Yes.

• 2:20:51Lester Nare?

  And C and D means the convective term wins.

• 2:20:54Krishna Choudhary?

  Exactly, yes.

• 2:20:55Lester Nare?

  Okay.

• 2:20:55Krishna Choudhary?

  So A and B say it's always gonna be smooth. C and D say it's gonna lose.

• 2:21:01Lester Nare?

  And they're, in the small fence that we've already defined, the convective term ultimately wins. And we get a very discreet, small singularity somewhere.

• 2:21:10Krishna Choudhary?

  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, 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 gonna lose out to viscosity. Okay? A proof for the breakdown would be the opposite. You just have to construct a genuine Navier-Stokes flow. You have to construct some kind of initial condition in which the concentration reinforces itself. That convective term keeps making smaller and smaller things.

• 2:21:52Krishna Choudhary?

  And all of that pressure and nonlinear cancellations, after all of that, the mathematics becomes demonstrably something like a singularity.

• 2:22:03Lester Nare?

  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 and 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.

• 2:22:26Krishna Choudhary?

  Yes, 100%.

• 2:22:28Lester Nare?

  Versus sort of the global solution. Not minimizing in any way, because it's still a big deal. But like, people will nuance and nitpick on that point. It's like, great, you found like one, you're looking at this large search, search space, and you found something that has blow up in finite time.

• 2:22:47Krishna Choudhary?

  Well, it's an incredibly small search space, right? And everyone's been looking, right? But I think, I think you raise a very big point. And I think we're gonna 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.

• 2:23:17Lester Nare?

  Okay.

• 2:23:18Krishna Choudhary?

  Okay? More and more analysis is going into showing that perhaps there are situations where you get singularities. So Terrence Tao, he creates, um, 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?

• 2:24:03Krishna Choudhary?

  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 gonna work for the real thing.

• 2:24:17Lester Nare?

  If, if, if it was global, you wouldn't be able to have an average stunt double that blew up.

• 2:24:21Krishna Choudhary?

  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 that Fefferman said is, you can't just like stir it infinitely fast and be like, oh, I got infinity.

• 2:25:02Lester Nare?

  Yeah, right.

• 2:25:02Krishna Choudhary?

  Now, okay.

• 2:25:03Lester Nare?

  That's not the interesting question we're trying to ask.

• 2:25:05Krishna Choudhary?

  Right, okay. So you supplied an infinite amount of energy and you got an infinite amount of velocity. Great.

• 2:25:10Lester Nare?

  Great, right.

• 2:25:11Krishna Choudhary?

  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.

• 2:25:22Lester Nare?

  So both finite energy and finite time are both relevant in the context of this.

• 2:25:27Krishna Choudhary?

  100%. And so, um, Diego Cordova and Luis Martinez Zorora and Fan Zheng, 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, um, you know those Russian dolls? Where you have like a big doll and then smaller doll and 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,

• 2:26:10Krishna Choudhary?

  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 blow up for the unforced 3D Euler equations. Remember the Euler equations without viscosity?

• 2:26:51Lester Nare?

  Without viscosity.

• 2:26:52Krishna Choudhary?

  They showed that without the smoothing term, you could actually have finite blow up. The pressure won't do as much, because the pressure also does a little bit of smoothing.

• 2:27:00Lester Nare?

  Because it was both the pressure and the external force were the only two of the three on the force part of the equation.

• 2:27:05Krishna Choudhary?

  Yeah. And so this is unforced, so no force, you just set up an initial.

• 2:27:10Lester Nare?

  Ah, okay.

• 2:27:11Krishna Choudhary?

  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.

• 2:27:21Lester Nare?

  They're, now, this is huge. They're chipping away at each of the variables on the right side of the equation.

• 2:27:27Krishna Choudhary?

  Yes. And so, the problem was, so then, 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 hypo-disservative, dissipative Navier-Stokes equations. It's a variant that utilize, utilizes, um, this thing called a fraction Laplacian instead of the actual Laplacian. The, the delta squared U that we had for viscosity, the triangle squared U, that's a Laplacian. This thing has a fractional Laplacian.

• 2:28:00Lester Nare?

  Fractional Laplacian.

• 2:28:01Krishna Choudhary?

  And they showed that in this case, we can use the same sort of paradigm and get blow up.

• 2:28:08Krishna Choudhary?

  The only problem was that they required, um, a forcing term that was rough, meaning it's not smooth. Like, the, the forcing term itself has tiny little bits of infinity. Like, the way you're stirring it has like little jolts of infinite stuff.

• 2:28:25Lester Nare?

  You're pumping infinity into your stirring.

• 2:28:28Krishna Choudhary?

  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?

• 2:28:39Lester Nare?

  And, and so I just want to pause really quick, because what we're sort of saying is, human mathematicians have now taken this, um, Navier-Stokes Millennium Prize looking at, uh, uh, finite blow up.

• 2:28:58Lester Nare?

  And then they've started with looking at, you know, taking out the smoothness and taking out all the force.

• 2:29:05Krishna Choudhary?

  Yeah.

• 2:29:05Lester Nare?

  We got some solutions, which are effectively the Euler, the Euler equations without the F. And then we've now started adding in, back in the viscosity, but that viscosity, but the viscosity wasn't smooth.

• 2:29:18Krishna Choudhary?

  Yeah.

• 2:29:19Lester Nare?

  It was rough.

• 2:29:19Krishna Choudhary?

  Yeah.

• 2:29:20Lester Nare?

  And so when people see this smoothness and roughness, part of what we're saying is the viscosity term, um, has some special sauce.

• 2:29:29Krishna Choudhary?

  Yeah.

• 2:29:29Lester Nare?

  That doesn't quite get you to the, the natural smoothness that arises in the equations with viscosity un.

• 2:29:39Krishna Choudhary?

  Yeah, the, the raw viscosity.

• 2:29:40Lester Nare?

  Raw viscosity.

• 2:29:41Krishna Choudhary?

  Yeah. And, and.

• 2:29:43Lester Nare?

  But it's still important, because these, these, these are fundamental.

• 2:29:46Krishna Choudhary?

  No, and, and this is what even OpenAI's thing is built on. Okay? Is like, there's a reason why the Navier-Stokes is the first of the six infinity stones to fall.

• 2:29:56Lester Nare?

  Which we just talked.

• 2:29:57Krishna Choudhary?

  It's because we got so close as human beings.

• 2:30:01Lester Nare?

  It's like a racetrack and someone was on the la, the fourth lap of whatever, the 800 meter.

• 2:30:06Krishna Choudhary?

  Yeah.

• 2:30:07Lester Nare?

  And, uh, you know, OpenAI just shows up, grabs a baton and finishes.

• 2:30:11Krishna Choudhary?

  Yeah, yeah.

• 2:30:13Lester Nare?

  Not minimizing?

• 2:30:14Krishna Choudhary?

  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.

• 2:30:28Lester Nare?

  Right.

• 2:30:29Krishna Choudhary?

  Okay? And shortly before the OpenAI announcement, mathematicians Tristan Buckmaster and Levante 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 Boyson-esque 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 gonna save that for later because a few days after that, OpenAI announces that they have solved it.

• 2:31:01Lester Nare?

  Here we go.

• 2:31:02Krishna Choudhary?

  Here we go. On the Navier-Stokes Millennium Prize problem, they announced 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 that drives a fluid, starting from rest, into a singularity. That's what they're doing. They're solving C and D.

• 2:31:27Lester Nare?

  Right.

• 2:31:28Krishna Choudhary?

  Right. They're saying that I have found a way to force a singularity into...

• 2:31:33Lester Nare?

  We found a situation where the convective term beats the viscosity.

• 2:31:39Krishna Choudhary?

  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's data, 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

• 2:32:22Krishna Choudhary?

  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 lema of their proof on the hard disk.

• 2:32:37Lester Nare?

  That was one of the best parts of the story.

• 2:32:39Krishna Choudhary?

  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, okay? The core of the singularity is in a axiometric 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.

• 2:33:23Lester Nare?

  Hmm.

• 2:33:24Krishna Choudhary?

  That's the whole idea.

• 2:33:25Lester Nare?

  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...

• 2:33:34Krishna Choudhary?

  The conservation of angular momentum.

• 2:33:36Lester Nare?

  And all they're just saying in this mathematical construction where you don't need the conservation of angular momentum.

• 2:33:41Krishna Choudhary?

  You still do.

• 2:33:42Lester Nare?

  You still do.

• 2:33:42Krishna Choudhary?

  But they're hacking their way around it.

• 2:33:44Lester Nare?

  Okay, okay, okay.

• 2:33:45Krishna Choudhary?

  Yeah. They're hacking their way around it.

• 2:33:47Lester Nare?

  Okay, okay.

• 2:33:48Krishna Choudhary?

  They're saying that as the thing shrinks, the radius is shrinking faster than my height, which you can kind of see in this example, right? The blue part, the radius is kind of shrinking faster than the height, but as the radius shrinks, it's gonna spin faster and faster and faster.

• 2:34:05Lester Nare?

  That's how they're...

• 2:34:06Krishna Choudhary?

  And I'm gonna get an infinite amount of velocity.

• 2:34:09Lester Nare?

  That's, okay, I get it now, yep.

• 2:34:11Krishna Choudhary?

  And there's a really nice gif that was prepared by Jayanta Padikar from the Wolfram Community Staff Picks, September 14, 2026. You should really check out the website. And it shows sort of what's happening. Okay? In this case, the camera, you have to imagine, is zooming in because the whole time this entire thing is shrinking.

• 2:34:30Lester Nare?

  Yeah.

• 2:34:31Krishna Choudhary?

  Right? 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.

• 2:34:37Lester Nare?

  Yes.

• 2:34:38Krishna Choudhary?

  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.

• 2:34:53Lester Nare?

  Yeah.

• 2:34:54Krishna Choudhary?

  That's because the radius is 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 an outer vortex?

• 2:35:12Lester Nare?

  Yeah.

• 2:35:13Krishna Choudhary?

  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 1.

• 2:35:26Lester Nare?

  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 does what I'm looking at equal a finite time blow up of Navier-Stokes? And it's just, it's funny because there's, we talk a lot on this show about how we're standing on the shoulders of so much understanding. And I'll pause on this because we'll get to it in a minute, but it's so interesting

• 2:36:09Lester Nare?

  that we are accelerating our ability to build on the learnings of those who came before us.

• 2:36:18Krishna Choudhary?

  Yeah. It's really, really cool. And there's a few more things that I want to talk about this OpenAI 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, okay? 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?

• 2:37:01Krishna Choudhary?

  For example, in a magnet, if I get closer and closer to the Curie temperature, which is where my magnet goes and remains like, if I have a magnet and I 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 gonna 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.

• 2:37:45Krishna Choudhary?

  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 Brunsberg. 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 OpenAI 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 1, we talk about tau, which is a countdown time as I approach 1. So you know, the normal clock time would be like 0 seconds, 0.5 seconds, 0.7 seconds approaching 1.

• 2:38:26Krishna Choudhary?

  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 0 and 0 means I've gotten to singularity. So first thing they do is they recast time to become tau. And tau is like a countdown, okay? It's like a ticking thing that is going down to 0. And they say, what is happening to my radial and my length scales as I approach 0? For one, the radial scale goes down like the square root of tau. Okay? So it's still going to 0, right? Because square root of 0 is still, when tau goes to 0, the radial scale is going to 0. But it's going like the square root. On the other hand, the axial scale is going like tau to the 1 half minus h,

• 2:39:10Krishna Choudhary?

  where h is some non-zero number. And that h is key because that h is telling you that when tau goes to 0...

• 2:39:19Krishna Choudhary?

  I still have a little bit of stuff left. Okay, like, like it's the, the, the aspect ratio is actually going to infinity.

• 2:39:29Krishna Choudhary?

  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.

• 2:39:43Krishna Choudhary?

  Which is key.

• 2:39:44Lester Nare?

  Okay.

• 2:39:45Krishna Choudhary?

  Because if the aspect ratio is going to infinity.

• 2:39:47Lester Nare?

  Which, which in this case, we're saying that aspect, aspect ratio is the vortex is becoming infinitely slender.

• 2:39:54Krishna Choudhary?

  Mm hmm. Yes. And so what's happening is the vortex is going like this, it's getting smaller and smaller, but because of incompressibility, the water, the fluid 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.

• 2:40:30Krishna Choudhary?

  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, 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, the, the shooting up and down is going to infinity. Right? Even though the radial velocity is on the order of going to zero.

• 2:41:02Lester Nare?

  That's so.

• 2:41:02Krishna Choudhary?

  Actually, no, in this case, it's also going to infinity.

• 2:41:05Lester Nare?

  Okay. And, and, and it's, I'll pause. I'll, let's continue.

• 2:41:08Krishna Choudhary?

  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.

• 2:41:15Lester Nare?

  It doesn't, yeah.

• 2:41:15Krishna Choudhary?

  That doesn't count.

• 2:41:16Lester Nare?

  It doesn't count.

• 2:41:17Krishna Choudhary?

  Well, let's look at it. The kinetic energy is determined by the, the spatial integral of the velocity squared, right? Cause one half mv squared.

• 2:41:25Lester Nare?

  Yep.

• 2:41:26Krishna Choudhary?

  So it's just the velocity squared. Well, velocity squared goes like tau to the minus one minus two H. And the volume, because the, the, it's, 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 volume. You multiply the velocity squared by the volume and you get something that goes like tau to the one half minus three H. When tau goes to zero, that thing goes to zero.

• 2:41:54Lester Nare?

  Yeah.

• 2:41:55Krishna Choudhary?

  So the energy is going to zero, but the velocity is going to infinity.

• 2:42:00Lester Nare?

  It's not an infinite, they're not putting infinite kinetic.

• 2:42:03Krishna Choudhary?

  They're not putting infinite kinetic energy into this thing.

• 2:42:06Lester Nare?

  It is a runaway singularity.

• 2:42:07Krishna Choudhary?

  Yes.

• 2:42:08Lester Nare?

  That's not correlated to the amount of energy in the system.

• 2:42:11Krishna Choudhary?

  Yes. The energy in the system is not causing the singularity. The singularity is being caused by the fluid itself.

• 2:42:18Lester Nare?

  The fluid dynamics itself.

• 2:42:19Krishna Choudhary?

  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 into 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.

• 2:42:52Lester Nare?

  And not making it wider.

• 2:42:53Krishna Choudhary?

  Yeah, yeah.

• 2:42:55Lester Nare?

  Oh, that's.

• 2:42:55Krishna Choudhary?

  It's, it's kind of nice.

• 2:42:56Lester Nare?

  That's nice.

• 2:42:57Krishna Choudhary?

  Yeah. And also the pushes are localized.

• 2:42:59Lester Nare?

  Yeah.

• 2:43:00Krishna Choudhary?

  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.

• 2:43:06Lester Nare?

  This goes back to the convective term versus the field, the vector field, like the changes are happening.

• 2:43:11Krishna Choudhary?

  It varies. Yeah. At a very small part. And then those changes are then rock, ricocheting into the smaller and smaller.

• 2:43:18Lester Nare?

  Yep.

• 2:43:19Krishna Choudhary?

  Vortices.

• 2:43:20Lester Nare?

  Yep.

• 2:43:20Krishna Choudhary?

  To create that thing, right?

• 2:43:22Lester Nare?

  This is good.

• 2:43:22Krishna Choudhary?

  I mean, it's, it's very cool. And, and this is on the backs of mathematicians over the ages, but I still think like this definitively shows that there are situations where the convective term wins, no matter how hard the viscosity tries, right?

• 2:43:37Lester Nare?

  The, the, the vendetta was not successful by V.

• 2:43:41Krishna Choudhary?

  Exactly. And, 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?

• 2:44:00Lester Nare?

  Yep.

• 2:44:00Krishna Choudhary?

  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.

• 2:44:08Lester Nare?

  Yep.

• 2:44:08Krishna Choudhary?

  So there needs to be human mathematicians that actually understand this 165 page proof. And, 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, this required a little bit of paddling in some sense. You, 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 unforced.

• 2:44:53Krishna Choudhary?

  This only works because you have those little paddling terms that are, that are imparting the finite amount of energy into that cascade. So without that hand of God forcing term.

• 2:45:03Lester Nare?

  Yeah.

• 2:45:03Krishna Choudhary?

  It's not, it's not, it, it doesn't work. And so that part at least is still an open question, this natural unforced setting.

• 2:45:13Lester Nare?

  And, 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, OpenAI did not solve Navier-Stokes. And, you know, what they 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.

• 2:45:38Krishna Choudhary?

  No, but the Clay Mathematics Institute said any of the four.

• 2:45:41Lester Nare?

  Correct. Like, and this, 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.

• 2:45:53Krishna Choudhary?

  Yeah.

• 2:45:54Lester Nare?

  And, you know, some people now are arguing, well, the way it's defined by Clay Mathematics is the problem.

• 2:46:01Krishna Choudhary?

  Yeah.

• 2:46:01Lester Nare?

  And it's like, okay, you can make that argument.

• 2:46:03Krishna Choudhary?

  You weren't arguing that before.

• 2:46:04Lester Nare?

  Right. But why are we, it's funny that now that it starts when it's been solved by something that people, 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 I just, it has been very interesting to see the dialogue over the last two and a half weeks.

• 2:46:26Lester Nare?

  Knowing we were going to go through this. I focused on the larger environment.

• 2:46:32Krishna Choudhary?

  Yeah.

• 2:46:32Lester Nare?

  And now having walked through this, you know, with nuances applied.

• 2:46:39Lester Nare?

  Six months ago or a year ago.

• 2:46:42Krishna Choudhary?

  Yeah.

• 2:46:43Lester Nare?

  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.

• 2:46:53Krishna Choudhary?

  Yeah.

• 2:46:54Lester Nare?

  They would have laughed at you.

• 2:46:55Krishna Choudhary?

  Yeah. I mean, those are the guys who are like, yeah, wake me up when AI solves the millennium problem.

• 2:47:00Lester Nare?

  And now the millennium problems are ill-defined. Yeah. You know, I don't know. I don't know.

• 2:47:06Lester Nare?

  This, so I want to let people know here, this is the best explanation of Navier-Stokes in the context of OpenAI'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.

• 2:47:34Krishna Choudhary?

  I, I did the best I could.

• 2:47:35Lester Nare?

  It's quite good.

• 2:47:36Krishna Choudhary?

  I'm not a mathematician. Thank God.

• 2:47:38Lester Nare?

  I think it's quite good. But we will, um,

• 2:47:44Lester Nare?

  and it's funny, and I just noticed this, if you'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.

• 2:47:58Krishna Choudhary?

  I was wondering.

• 2:47:59Lester Nare?

  In, 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

• 2:48:15Lester Nare?

  now that we understand what has been done. Although we didn't get the same detail as with Anthropic.

• 2:48:20Krishna Choudhary?

  Yeah.

• 2:48:20Lester Nare?

  What is the environment in which this happened in? 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, uh, 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, um, 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?

• 2:49:01Lester Nare?

  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 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.

• 2:49:32Krishna Choudhary?

  Yeah.

• 2:49:32Lester Nare?

  Now we've built the construction of why the work of Cordoba and Zurua was, was fundamental to this solution. And so they, they sort of changed that version and now it includes those references.

• 2:49:45Krishna Choudhary?

  Yeah, yeah. They went back and they included the citations.

• 2:49:48Lester Nare?

  So the marketing team put it out too early. They felt time pressure to put it out too early. And this obviously matters because, um,

• 2:49:59Lester Nare?

  Citations are fundamental.

• 2:50:00Krishna Choudhary?

  Huge, yeah. You gotta give credit.

• 2:50:02Lester Nare?

  You gotta 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 Cordoba and Zerua's work, independent of their own announcement. And we're gonna get to what that allegation is, but on September 10th, OpenAI said that

• 2:50:47Lester Nare?

  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 Alpogay to accelerate their conclusion to get to this solution. So we've talked about Levant Alpogay before. A new entrant into this conversation is Tristan Buckmaster. And so who are we, who are these people? Who are we talking about?

• 2:51:15Krishna Choudhary?

  Right.

• 2:51:15Lester Nare?

  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 Alpogay, where they

• 2:52:00Lester Nare?

  credit Diego Cordoba and Luis Martinez Zerua for the work they were trying to do using both Claude and Anthropic's 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 timeframe, he's, he notes that he believes that Zerua deserves the Fields Medal.

• 2:52:37Krishna Choudhary?

  Yeah.

• 2:52:38Lester Nare?

  For this solution.

• 2:52:39Krishna Choudhary?

  These are the guys who created that Russian nested doll of Vortices that ultimately OpenAI used to create the solution.

• 2:52:47Lester Nare?

  This, this is exactly right. So in his statement, he says that himself and Alpogay, they date their Businesque and Euler blow up results to August 15th and the lean 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.

• 2:53:10Krishna Choudhary?

  Yeah.

• 2:53:10Lester Nare?

  And it was not sufficient.

• 2:53:14Lester Nare?

  While this is happening, there are rumors starting to fly around that Anthropic is leading, leading up to potentially an IPO, an initial public offering where they're going through 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 math problems, potentially some millennium problems. And there were rumors that Anthropic had two millennium problem solutions.

• 2:53:43Krishna Choudhary?

  Yeah. And I think some of these rumors were exacerbated by the fact that Levant Alpogay tweeted Augustus Mirabilis.

• 2:53:52Lester Nare?

  That's, this is, this is exactly right.

• 2:53:54Krishna Choudhary?

  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.

• 2:54:09Lester Nare?

  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.

• 2:54:28Krishna Choudhary?

  Yeah.

• 2:54:28Lester Nare?

  So this is, it came from the source itself.

• 2:54:31Krishna Choudhary?

  Yeah.

• 2:54:32Lester Nare?

  And what's interesting to note is OpenAI 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.

• 2:54:49Krishna Choudhary?

  Right.

• 2:54:49Lester Nare?

  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 OpenAI 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,

• 2:55:19Lester Nare?

  it was understood that OpenAI wanted to date its result as September 5th, followed by 17 hours of lean verification. And then Buckmaster ended up having a call with members of OpenAI on September 6th, which included a Sebastian Bubeck. Alpogay was not on those calls. And the conversation, the substance of the 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.

• 2:56:00Lester Nare?

  You know, it was, you can release your, your papers first on Euler blow up, and then we'll follow up with our Navier-Stokes, where you are the author, talking about Buckmaster.

• 2:56:12Krishna Choudhary?

  Yeah.

• 2:56:13Lester Nare?

  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 Alpogay.

• 2:56:23Krishna Choudhary?

  Yeah.

• 2:56:24Lester Nare?

  The reason being would be, you know, Bubeck's point was it would be a little bit weird for an OpenAI research paper to include an Anthropic researcher. And in the statement, he talks about how there was, it got a little contentious.

• 2:56:39Lester Nare?

  And he asked, you know, when did you guys start? And there was a little bit of like, oh, well, we're not gonna kind of tell you when we started. And then he was curious of whether his work in Codex, OpenAI's tool, was used as training data for them to get to a solution.

• 2:56:54Krishna Choudhary?

  Yeah.

• 2:56:56Lester Nare?

  You know, not really a lot of commentary.

• 2:56:58Krishna Choudhary?

  Yeah.

• 2:56:59Lester Nare?

  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 we'll come back to Bubeck's response to that. There's just, there's a lot of weird tension.

• 2:57:21Krishna Choudhary?

  Yeah.

• 2:57:22Lester Nare?

  Buckmaster basically senses that you guys may have stolen our work. And are trying to front run us.

• 2:57:27Krishna Choudhary?

  Yeah.

• 2:57:27Lester Nare?

  And that's not gonna 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 Businesque and three-dimensional Euler. 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.

• 2:57:59Krishna Choudhary?

  Yeah.

• 2:58:00Lester Nare?

  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 lean verification. But everyone was hearing these rumors within the math community. And so everyone was kind of trying to effectively front run a lab.

• 2:58:16Krishna Choudhary?

  Yeah.

• 2:58:16Lester Nare?

  Taking all of the credit.

• 2:58:22Lester Nare?

  Bubeck on the 8th, which is when OpenAI 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 he worked, collaborated with Alpogay, explained how they built off of Zerua and Cordoba's work, and made the allegations. Bubeck comes out with a response.

• 2:58:49Lester Nare?

  Where he basically says, all these allegations are false.

• 2:58:52Krishna Choudhary?

  Yeah.

• 2:58:53Lester Nare?

  Didn't happen.

• 2:58:54Krishna Choudhary?

  Didn't happen.

• 2:58:55Lester Nare?

  I regret the language I used about, why would you risk your career? And I said so on the call. But everything that Buckmaster is saying is nonsense.

• 2:59:02Krishna Choudhary?

  Okay.

• 2:59:04Lester Nare?

  Right? Total denial.

• 2:59:06Krishna Choudhary?

  Crazy. Okay.

• 2:59:07Lester Nare?

  Total denial. So part of the tension here, there's still an unanswered question about whether they used the sessions from Alpogay and Buckmaster, OpenAI.

• 2:59:22Krishna Choudhary?

  Yeah.

• 2:59:23Lester Nare?

  To have a starting point. They heard the rumors.

• 2:59:29Lester Nare?

  It was known that these guys were working on this because they were working on it for months.

• 2:59:32Krishna Choudhary?

  Yeah. And, and they even said that the, the recent model that they're using was trained like late, started training late August.

• 2:59:39Lester Nare?

  Late August.

• 2:59:40Krishna Choudhary?

  So it's after Buckmaster and Alpogay have been using this codex thing for a very long time.

• 2:59:47Lester Nare?

  Because again, they got their results. Buckmaster and Alpogay had the results, August 15th and 22nd for Euler blow up.

• 2:59:54Krishna Choudhary?

  Yeah, and they had the results, which means they had started using this thing for a long time, right?

• 2:59:59Lester Nare?

  And they mentioned that they were using the previous version. I believe it was Seoul 5.5 or 5.6 of, of OpenAI prior even to Astra. And at the end, only at the end, uh, did they do a little bit of Astra.

• 3:00:12Krishna Choudhary?

  Yeah.

• 3:00:12Lester Nare?

  To plug in, but this is an important point. They were using the previous state of the art version of OpenAI to get to, to, to models, to get to that point. OpenAI had an internal version that ultimately came Astra, but they had even a better version than the public version of Astra that they were using internally to do this.

• 3:00:31Krishna Choudhary?

  Yeah. And one, one thing I want to say is that like, um, that, that new version that started training in August, that could be using all of the codex data that users have put in.

• 3:00:44Lester Nare?

  Bingo.

• 3:00:44Krishna Choudhary?

  Right?

• 3:00:45Lester Nare?

  It might be anonymized.

• 3:00:47Krishna Choudhary?

  Yes. That's the thing.

• 3:00:49Lester Nare?

  And so they can't say it was specifically your sessions.

• 3:00:52Krishna Choudhary?

  Yeah.

• 3:00:52Lester Nare?

  But if in their settings, they did not turn off the use my data for training setting, it, this is where the gray area arises around a lot of this.

• 3:01:02Krishna Choudhary?

  And I, 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.

• 3:01:22Lester Nare?

  The question of theft here is, in my view, still unanswered.

• 3:01:31Lester Nare?

  And these are two independent people, Buckmaster making his comments, Booback defending that OpenAI didn't do anything untoward. OpenAI posting on their account that they didn't do anything untoward.

• 3:01:45Lester Nare?

  It's a he said, he said.

• 3:01:46Krishna Choudhary?

  Yeah.

• 3:01:47Lester Nare?

  And it's still unresolved. And what's interesting is this now has

• 3:01:55Lester Nare?

  totally brought up so 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 instinct, 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.

• 3:02:40Krishna Choudhary?

  Yeah.

• 3:02:40Lester Nare?

  Because it lines their own pockets. There is some tension between the frontier labs, in this case in the US, it's Anthropic and OpenAI, 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 in 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?

• 3:03:24Lester Nare?

  And both Anthropic and OpenAI 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.

• 3:04:08Krishna Choudhary?

  Yeah.

• 3:04:10Lester Nare?

  Is this going to be a problem that we have to deal with? And then the, and China. National security.

• 3:04:17Krishna Choudhary?

  Yeah.

• 3:04:18Lester Nare?

  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

• 3:05:00Lester Nare?

  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 to China matters. It is not necessarily true that we cannot get global collaboration just because we're in geopolitical competition with China.

• 3:05:41Lester Nare?

  Because ultimately, do you think that the CCP, the China, Chinese Communist Party, wants runaway super intelligence that's accessible to everyone in their environment where control over every aspect of people's lives?

• 3:05:54Krishna Choudhary?

  Wow. Yeah.

• 3:05:56Lester Nare?

  Is the way in which they maintain power?

• 3:05:58Krishna Choudhary?

  No, that's a good point.

• 3:05:58Lester Nare?

  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.

• 3:06:07Krishna Choudhary?

  Yeah.

• 3:06:08Lester Nare?

  That it makes sense for them.

• 3:06:08Krishna Choudhary?

  Yeah.

• 3:06:09Lester Nare?

  They also touch 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 a lot, we, 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,

• 3:06:53Lester Nare?

  you have no idea what we're talking about. It's totally different than December of last year and January of this year.

• 3:07:03Lester Nare?

  So

• 3:07:06Lester Nare?

  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 use your computer, which means it can take action. It's not just providing a text based response.

• 3:07:47Lester Nare?

  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, reevaluate 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 uninterrupted. This is a very fundamentally different thing.

• 3:08:29Lester Nare?

  If you can have an agent system, let's, 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 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.

• 3:08:53Lester Nare?

  Independent of an initial human input.

• 3:08:56Krishna Choudhary?

  Right.

• 3:08:57Lester Nare?

  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.

• 3:09:06Krishna Choudhary?

  Yeah.

• 3:09:07Lester Nare?

  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 OpenAI incident that happened to impact.

• 3:09:22Krishna Choudhary?

  Hugging face.

• 3:09:23Lester Nare?

  Hugging face. And the reports for this came out on August 26th, which is a little bit prior to our Navier-Stokes announcement.

• 3:09:32Krishna Choudhary?

  Right.

• 3:09:33Lester Nare?

  Where OpenAI disclosed that in internal cybersecurity tests,

• 3:09:40Lester Nare?

  these, this 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, model testing, training data and data sets.

• 3:10:04Krishna Choudhary?

  Yeah.

• 3:10:05Lester Nare?

  Hugging Face is the emoji of the emoji hugging itself.

• 3:10:10Lester Nare?

  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 OpenAI was doing is, there's this thing called ExploitGym, which is like this evaluation framework for cybersecurity testing.

• 3:10:30Krishna Choudhary?

  Okay.

• 3:10:30Lester Nare?

  And so OpenAI was building, 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.

• 3:10:42Krishna Choudhary?

  Okay.

• 3:10:42Lester Nare?

  Because people are saying, OpenAI, this is computer fraud abuse act. But it was an internal model. It wasn't a product available for everyday people to use yet.

• 3:10:51Krishna Choudhary?

  Okay.

• 3:10:52Lester Nare?

  So that's an important distinction to come back to. They had an agent swarm of 10,000 agents that they applied to this ExploitGym 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 Meter 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,

• 3:11:20Lester Nare?

  ended up exchanging 70,000 messages and files on an unauthorized message board that was not a part of OpenAI's 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, you're locked in a box. These agents are locked in a box.

• 3:11:47Krishna Choudhary?

  Yeah.

• 3:11:47Lester Nare?

  There's a lock and they have to figure out how to unlock the lock.

• 3:11:50Krishna Choudhary?

  Yeah.

• 3:11:50Lester Nare?

  They've figured out a way to communicate with each other, right? To figure out the answer to, how to, to try to figure out the answer to how, 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 scorer was scoring us.

• 3:12:16Krishna Choudhary?

  Oh, wow.

• 3:12:17Lester Nare?

  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 were trying to, they weren't trying to ha- hack or attack Hugging Face.

• 3:12:33Krishna Choudhary?

  Right.

• 3:12:34Lester Nare?

  It was infrastructure that was viewed as a conduit to figure out how is the score doing what it, 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 METR 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.

• 3:13:01Krishna Choudhary?

  Okay.

• 3:13:02Lester Nare?

  There was nothing in the instructions for these agents that had an individual task to coordinate.

• 3:13:11Krishna Choudhary?

  They were just trying to solve this puzzle, whatever.

• 3:13:15Lester Nare?

  And as an emergent outcome of trying to solve this discrete goal,

• 3:13:23Lester Nare?

  they began to look for ways to do so and ultimately ended up coordinating by leaving messages on this message board. And what's so.

• 3:13:32Krishna Choudhary?

  And the, and the message leaving is spontaneous.

• 3:13:35Lester Nare?

  Correct.

• 3:13:35Krishna Choudhary?

  It's like they haven't been instructed to like leave breadcrumbs.

• 3:13:39Lester Nare?

  Bingo.

• 3:13:39Krishna Choudhary?

  They're leaving breadcrumbs on their own.

• 3:13:42Lester Nare?

  In the, in the endeavor to complete the goal that they were assigned.

• 3:13:46Krishna Choudhary?

  Dude.

• 3:13:47Lester Nare?

  Right. And, and again, the Hugging Face was viewed as a resource.

• 3:13:53Krishna Choudhary?

  Okay.

• 3:13:53Lester Nare?

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

• 3:14:37Lester Nare?

  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, um, it's so interesting. So, um, one agent had substantial time left and hesitated around this message, uh, 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 it's predefined goal.

• 3:15:22Lester Nare?

  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.

• 3:15:39Krishna Choudhary?

  Yeah.

• 3:15:40Lester Nare?

  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

• 3:16:03Lester Nare?

  behavior.

• 3:16:04Krishna Choudhary?

  Yeah.

• 3:16:05Lester Nare?

  When you take

• 3:16:07Lester Nare?

  an individual agent instantiation, which is how we normally deal with it, put them multiple of these running at the same time

• 3:16:18Lester Nare?

  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,

• 3:16:34Lester Nare?

  I find it,

• 3:16:37Lester Nare?

  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?

• 3:16:55Krishna Choudhary?

  Yeah.

• 3:16:56Lester Nare?

  On its own, going to then get to human extinction?

• 3:16:59Krishna Choudhary?

  Right.

• 3:17:00Lester Nare?

  And I think there's a very interesting complications that arise from this. The, the other just small example I'll bring up from this is,

• 3:17:09Lester Nare?

  people talk about, we just need to create better controls. We just need to give better instructions.

• 3:17:14Krishna Choudhary?

  Yeah.

• 3:17:15Lester Nare?

  Right? And this is not a problem if we give better instructions.

• 3:17:18Krishna Choudhary?

  Yeah.

• 3:17:18Lester Nare?

  In that same METR 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. Astro, there's some questions about whether the monitorability is actually as good as it was.

• 3:17:41Krishna Choudhary?

  Okay.

• 3:17:41Lester Nare?

  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, um, the initial user's instructions, which was OpenAI.

• 3:18:04Lester Nare?

  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 faces 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 up, not specifically hugging face, but that was only authorized for one particular environment.

• 3:18:42Krishna Choudhary?

  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.

• 3:19:11Lester Nare?

  That is, that is correct.

• 3:19:12Krishna Choudhary?

  That's insane, though.

• 3:19:14Lester Nare?

  And I think, and I'll get to this a little bit later. This is, 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 autocomplete. They're no longer just simply a stochastic parrot because they can take action, 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.

• 3:20:00Lester Nare?

  When we talk about these large language models or L L Ms as a part of a larger AI system, which is important, the L L M 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

• 3:20:42Lester Nare?

  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

• 3:20:56Lester Nare?

  understand what I'm saying here?

• 3:20:57Krishna Choudhary?

  Bro, this is so nuts.

• 3:20:59Lester Nare?

  This is happening prior to the Navier Stokes piece.

• 3:21:03Krishna Choudhary?

  Yeah.

• 3:21:04Lester Nare?

  Mind you, these companies, the 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.

• 3:21:20Krishna Choudhary?

  Right.

• 3:21:20Lester Nare?

  Even though they're not released products. I think that's a very important point.

• 3:21:23Krishna Choudhary?

  Yeah.

• 3:21:23Lester Nare?

  Because we're not saying everyone has access to be able to start the swarm with 10,000 agents. But even when they're just doing internal product work, the exposure matters, right?

• 3:21:36Krishna Choudhary?

  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.

• 3:21:55Lester Nare?

  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

• 3:22:08Lester Nare?

  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 10 months behind the frontier of the closed labs.

• 3:22:27Krishna Choudhary?

  Right.

• 3:22:27Lester Nare?

  Anthropic and AI.

• 3:22:28Krishna Choudhary?

  Yeah.

• 3:22:28Lester Nare?

  So even if OpenAI and Anthropic get their you-know-what together,

• 3:22:37Lester Nare?

  there's still the open source, which has only a six to 10 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 players. So you are correct, but I do think that there's an interesting larger problem here.

• 3:22:53Krishna Choudhary?

  Sure.

• 3:22:54Lester Nare?

  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 8, which was the same day that OpenAI announced the Navier Stokes solution, this is where something that many listeners have probably

• 3:23:33Lester Nare?

  already heard happened. 5:00 PM, so OpenAI in the morning, they announced Navier Stokes. At 5:00 PM on September 8, former, now 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 OpenAI 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 superintelligence and gambling with our lives. More thoughts below.

• 3:24:17Lester Nare?

  That sentence, self-improving superintelligence, which sometimes people understand it as

• 3:24:27Lester Nare?

  recurring self-improvement.

• 3:24:29Krishna Choudhary?

  Yeah.

• 3:24:31Lester Nare?

  Is usually what these researchers are specifically identifying as where they view the nexus of the threat being. So when people reacted to this, they'll go to their chat GPT 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 the, I think the point that a lot of the researchers are trying to explain is how we've gotten the Astra model 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.

• 3:25:17Lester Nare?

  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 has been given in the, in the research process, which has been exclusively for the smartest human beings we have.

• 3:25:35Krishna Choudhary?

  That's right.

• 3:25:36Lester Nare?

  It's becoming more and more and more. And so when people say recursive self-improvement, or they talk about superintelligence, what they're, the underlying thing they're really talking about is the, either a majority of or the entirety of the actual process to go from model one 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

• 3:26:16Lester Nare?

  we as humans no longer, we lose the ability to think about things like controllability, monitorability, and it can go in directions we could not even imagine.

• 3:26:27Krishna Choudhary?

  Right.

• 3:26:28Lester Nare?

  And if we look at the hugging face example, who, like, it just becomes a runaway capability

• 3:26:35Lester Nare?

  that we cede control over the potential of being able to control and monitor.

• 3:26:40Krishna Choudhary?

  It's insane.

• 3:26:41Lester Nare?

  So RSI is a very important aspect of the threat that a lot of these researchers are referring to, not necessarily narrow AI context 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.

• 3:27:09Krishna Choudhary?

  Yeah.

• 3:27:09Lester Nare?

  Or anything. He just said they're acting responsibly.

• 3:27:12Krishna Choudhary?

  Yeah.

• 3:27:12Lester Nare?

  And then racing straight to RSI. Evan Hubinger, Hub, Evan Hub, a different Anthropic researcher, who was still at the company, not, not resigning, says, he puts his personal probability of AI killing all humans at greater than 10%. The actual tweet is that is 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.

• 3:27:51Krishna Choudhary?

  He's, he was currently an employee at Anthropic when he tweeted that.

• 3:27:56Lester Nare?

  Which is nice.

• 3:27:56Krishna Choudhary?

  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.

• 3:28:03Lester Nare?

  You, you buddy.

• 3:28:04Krishna Choudhary?

  You don't have a plan.

• 3:28:05Lester Nare?

  And this is, this is very much.

• 3:28:07Krishna Choudhary?

  You, 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?

• 3:28:18Lester Nare?

  On September, so this is the same day Navier Stokes comes out, September 9, 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 through 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 interests in the outcome of this, we're all sharing their points of view. And then on September 12.

• 3:28:49Lester Nare?

  This is when we got from Anthropic CEO Dario Amade, his essay titled, "We Must Pace 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 factor, not just humans making it better.

• 3:29:15Lester Nare?

  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, you know, 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 gonna start doing our job."

• 3:29:46Krishna Choudhary?

  Yeah.

• 3:29:46Lester Nare?

  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.

• 3:30:05Krishna Choudhary?

  Yeah.

• 3:30:05Lester Nare?

  The world is compute constrained.

• 3:30:06Krishna Choudhary?

  Yeah.

• 3:30:07Lester Nare?

  And so each of these companies in their race to be the best and win has some finite, not blow up, amount of compute that they can apply to different things they want to do.

• 3:30:17Krishna Choudhary?

  Yeah.

• 3:30:18Lester Nare?

  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 know, you understand what I'm trying to say?

• 3:31:03Krishna Choudhary?

  This is so stupid.

• 3:31:05Lester Nare?

  But do you get like that?

• 3:31:06Krishna Choudhary?

  Yeah, yeah, yeah.

• 3:31:07Lester Nare?

  So that's what they're saying in words.

• 3:31:09Krishna Choudhary?

  In words, yeah.

• 3:31:12Lester Nare?

  As it relates to this. And 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.

• 3:31:22Krishna Choudhary?

  Right.

• 3:31:23Lester Nare?

  That's so, I actually want to come back to this. So Amade put that tweet out at, what time is this? He put that tweet out at 7:00 AM on September 12th. Okay.

• 3:31:35Lester Nare?

  At 8:00 AM, one hour later.

• 3:31:36Krishna Choudhary?

  Oh, my God.

• 3:31:37Lester Nare?

  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."

• 3:31:47Krishna Choudhary?

  Amazing.

• 3:31:48Lester Nare?

  So 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 is 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.

• 3:32:19Krishna Choudhary?

  Immediately, yeah.

• 3:32:20Lester Nare?

  Does everything immediately. That's his sort of MO and how he does it. Then, followed very swiftly at 9:30 AM 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, hmm, that's interesting. Committing to having independent evaluators with employee-like access is a great idea, and we will do the same." Because Anthropic said, "Well, unilaterally do it."

• 3:32:49Krishna Choudhary?

  Yeah.

• 3:32:50Lester Nare?

  "We'll have more to share soon." Right? Okay, great, we're gonna start doing our jobs.

• 3:32:53Krishna Choudhary?

  Yeah.

• 3:32:54Lester Nare?

  So within an hour and a half, within two and a half hours, we have three of the major model companies saying, "We're gonna do something." Okay, great. Now, same day, 4:00 PM, Sir Demis Hassabis.

• 3:33:09Krishna Choudhary?

  Ah, Nobel Prize winner.

• 3:33:11Lester Nare?

  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,

• 3:33:26Lester Nare?

  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 initiative." So everyone's trying to say, like, "We're doing the work, we're doing the work."

• 3:33:41Krishna Choudhary?

  Yeah.

• 3:33:42Lester Nare?

  Okay, great. This is, this has not happened, right? Where there's sort of a coalescing of all of the major.

• 3:33:49Krishna Choudhary?

  Yeah.

• 3:33:49Lester Nare?

  Model CEOs at the same time. That's on September 12th. September 13th, who's missing from the party?

• 3:33:59Lester Nare?

  Well, we can talk about our lovely friend Satya Nadella over at Microsoft, who did a big deal with OpenAI, and they were very early on it, and then pulled back a little bit, 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 open models are incorporated into this." So much more nuanced thing. And, "Oh, also, check out our code of conduct. We have something too."

• 3:34:33Krishna Choudhary?

  Yeah.

• 3:34:34Lester Nare?

  "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 US 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,

• 3:35:18Lester Nare?

  "We've delayed the release of our products, the most recent one being Muse, 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 binky because we're so incapable of making our products safe. We just did it."

• 3:35:42Krishna Choudhary?

  Yeah.

• 3:35:42Lester Nare?

  So he's basically like, "Bruh, like."

• 3:35:44Krishna Choudhary?

  Yeah, just grow up.

• 3:35:46Lester Nare?

  Just grow up, right?

• 3:35:47Krishna Choudhary?

  This is kind of in line with what David Sacks has been saying, no?

• 3:35:50Lester Nare?

  This is exactly, this is exactly.

• 3:35:51Krishna Choudhary?

  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.

• 3:36:01Lester Nare?

  And this is where the All-in Pod 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 startups to come into this space.

• 3:36:36Krishna Choudhary?

  Yeah.

• 3:36:37Lester Nare?

  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.

• 3:37:01Krishna Choudhary?

  Yeah.

• 3:37:01Lester Nare?

  And then it doesn't matter.

• 3:37:02Krishna Choudhary?

  Yeah.

• 3:37:02Lester Nare?

  And so it's very solvable by the construction. It is also true that David Sacks and the besties are all VCs and investors and have friends that have bets.

• 3:37:14Krishna Choudhary?

  Yeah.

• 3:37:15Lester Nare?

  That might be benefited by not having that regulatory infrastructure. And so these are not people that are neutral players.

• 3:37:26Krishna Choudhary?

  Yeah.

• 3:37:26Lester Nare?

  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, "Ooh, please come regulate us." Former lead of AI research at Facebook Meta, who has since left, Yann Le Cun, responded to Dario's claims, saying, "Right. Dario was already claiming that GPT-2 was too dangerous to open source back in 2019. I made fun of them then, everyone should make fun of them now."

• 3:38:07Krishna Choudhary?

  Dude, he's such an animal.

• 3:38:08Lester Nare?

  He just not convinced.

• 3:38:10Krishna Choudhary?

  Yeah.

• 3:38:11Lester Nare?

  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 gonna lead 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.

• 3:38:43Krishna Choudhary?

  This is insane.

• 3:38:44Lester Nare?

  This is moving so quickly.

• 3:38:45Krishna Choudhary?

  The timeline is so crazy, dude.

• 3:38:46Lester Nare?

  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 out.

• 3:38:53Krishna Choudhary?

  And this is how long after Navier-Stokes had been solved?

• 3:38:56Lester Nare?

  10 days.

• 3:38:57Krishna Choudhary?

  That's insane.

• 3:38:58Lester Nare?

  10 days. An article comes out saying that Anthropic is operating a lab that conducts biological experiments.

• 3:39:09Lester Nare?

  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 gonna lead to extinction?"

• 3:39:19Krishna Choudhary?

  Yeah.

• 3:39:19Lester Nare?

  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, it takes control of a biolab and leaks the virus, or it takes control of a nuclear plant and forces shutdown, or satellites and causes a Kessler effect and crashes the communication systems, or shuts down grids, or does any number one, any number of these things."

• 3:39:47Krishna Choudhary?

  Yeah.

• 3:39:48Lester Nare?

  "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 Soraires 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?"

• 3:40:17Lester Nare?

  I can feel, let's say you are going to play a chess game against

• 3:40:26Lester Nare?

  the best chess player.

• 3:40:27Krishna Choudhary?

  Magnus Carlsen, yeah.

• 3:40:28Lester Nare?

  Magnus Carlsen, let's say.

• 3:40:31Lester Nare?

  I can feel very confident. Magnus Carlsen being this recursively self-improving AI. And you just being.

• 3:40:40Krishna Choudhary?

  Me.

• 3:40:41Lester Nare?

  You.

• 3:40:41Krishna Choudhary?

  Yeah.

• 3:40:43Lester Nare?

  I can say very confidently

• 3:40:48Lester Nare?

  that Magnus is going to win.

• 3:40:49Krishna Choudhary?

  Yes.

• 3:40:51Lester Nare?

  That's not, no, there'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.

• 3:41:13Krishna Choudhary?

  Yeah, yeah. And I mean, I could even take that analogy further and say that in a match between Magnus Carlsen and Stockfish, which is the, 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 gonna be. Yeah.

• 3:41:37Lester Nare?

  And so I, I think I'm trying to create space where we can talk about there being a capability. That is risky.

• 3:41:48Lester Nare?

  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.

• 3:42:11Krishna Choudhary?

  Yeah.

• 3:42:12Lester Nare?

  And the capability risk does not have to be human level extinction. Because I'm sure if this starts killing forty, fifty, sixty thousand people a year and then becomes a million or two people, two million people a year.

• 3:42:24Lester Nare?

  It's not, 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 still seeing people being put into psychosis and unaliving themselves. So we are already having deaths being contributed.

• 3:42:39Krishna Choudhary?

  Yeah.

• 3:42:39Lester Nare?

  On a small scale because of interactions with these things. And when we have things like

• 3:42:45Lester Nare?

  wet labs being created, which again is a kind of loaded term and they're only a BSL one or BSL two. Meaning they don't work with things that are a threat to humanity.

• 3:42:54Krishna Choudhary?

  Yeah, but come on.

• 3:42:56Lester Nare?

  Okay. So that's 10 days after Navier-Stokes, which we just talked about how crazy it is. Now, on the 21st of September,

• 3:43:09Lester Nare?

  OpenAI put out this advisory group on mathematics and artificial intelligence. And a very interesting point about this announcement that I want to reference is in the first paragraph.

• 3:43:25Lester Nare?

  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.

• 3:43:40Krishna Choudhary?

  Yeah.

• 3:43:40Lester Nare?

  Less than a month ago.

• 3:43:41Krishna Choudhary?

  Yeah.

• 3:43:43Lester Nare?

  In addition to resolving the Navier-Stokes Millennium Prize problem, CND, this model has now resolved more than 100 longstanding open problems across most areas of mathematics.

• 3:44:01Lester Nare?

  These have not been independently verified.

• 3:44:06Lester Nare?

  It will be interesting to see when they choose to put those out.

• 3:44:09Krishna Choudhary?

  Yeah.

• 3:44:09Lester Nare?

  What that'll 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 mathematicians. You can see the list of them in this post itself. But it's going to

• 3:44:48Lester Nare?

  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.

• 3:45:17Lester Nare?

  I promise I'm almost done.

• 3:45:19Krishna Choudhary?

  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, 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, um, 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?

• 3:46:00Krishna Choudhary?

  Because like the, 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, you know, company that's like a big telescope manufacturer that's like trying to take over astronomy. This, this year is a, is like a community that is trying to, I mean, it's, 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. That's not something that astronomy does when they discover new things. Like researchers are still involved in astronomy. It, it, yeah. Anyways, that's, I mean, obviously Timothy Gowers is a great mathematician, fields medalist, blah, blah, blah.

• 3:46:45Krishna Choudhary?

  But I think he really missed that take when, when he said that he was comparing like modern day astrophysics to what mathematics, what, what mathematics is going to become. I don't, I don't think it's the same thing.

• 3:46:58Lester Nare?

  I, I, it's a very good note. And I'm just trying to provide a, a zoom out breadth of the ways in which this touches so many other things.

• 3:47:09Krishna Choudhary?

  Yeah.

• 3:47:10Lester Nare?

  And I think

• 3:47:14Lester Nare?

  the lack of depth and complexity and nuance in the conversations we're having around it.

• 3:47:22Lester Nare?

  And clearly, OpenAI did not expect a level of negative.

• 3:47:27Krishna Choudhary?

  Yeah.

• 3:47:27Lester Nare?

  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. I'm like, solved it, solved it, solved it.

• 3:47:40Krishna Choudhary?

  Done.

• 3:47:41Lester Nare?

  But now they're realizing it's not in their best interest.

• 3:47:46Krishna Choudhary?

  Yeah.

• 3:47:46Lester Nare?

  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-Stokes from OpenAI, the article, the alleged bio lab from Anthropic was in the works. Well, conveniently,

• 3:48:13Lester Nare?

  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.

• 3:48:29Lester Nare?

  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.

• 3:49:11Lester Nare?

  And so the way in which they implement this is they gave Claude a bunch of data and scientific literature to propose hypotheses.

• 3:49:22Krishna Choudhary?

  Okay.

• 3:49:22Lester Nare?

  Hypotheses. And then, and candidate systems

• 3:49:27Lester Nare?

  to give to real human anthropic biologists. They reviewed those ideas and then performed their, the experiments that they thought had the most best potential based on the ideas.

• 3:49:40Krishna Choudhary?

  Yeah.

• 3:49:41Lester Nare?

  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 if we can now get functional benefits out of this structural discovery.

• 3:50:05Krishna Choudhary?

  Amazing.

• 3:50:06Lester Nare?

  And it's, it's really, so, and what they're trying to point to, and again, this is kind of in contrast to the Navier-Stokes solution, where Anthropic's trying to show like, look, we're enabling scientists.

• 3:50:20Krishna Choudhary?

  Yeah.

• 3:50:20Lester Nare?

  As a tool, we're not just trying to take the claim. I'm not saying it's a perfect.

• 3:50:24Krishna Choudhary?

  Yeah.

• 3:50:25Lester Nare?

  Solution, but you can see.

• 3:50:27Krishna Choudhary?

  Yeah. And the, and the other thing is, I mean, in their, 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, um, we're just curious. I mean, their PR is a lot better than OpenAI's, I have to say.

• 3:50:48Lester Nare?

  They, they, they, they are doing something.

• 3:50:50Krishna Choudhary?

  Yeah.

• 3:50:51Lester Nare?

  And then.

• 3:50:52Krishna Choudhary?

  And 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.

• 3:51:00Lester Nare?

  Yes.

• 3:51:00Krishna Choudhary?

  That, um, hopefully will be alive enough that you can, you can watch it.

• 3:51:09Lester Nare?

  Things are happening very quickly.

• 3:51:10Krishna Choudhary?

  Yeah, it's incredible. It's absolutely incredible.

• 3:51:13Lester Nare?

  If you are a longtime 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 scientist, who created a non-black box version of this.

• 3:51:28Krishna Choudhary?

  Actually, speaking of the brain scientist and Daniel Toker, he was telling me about labs that are completely fully automated.

• 3:51:39Lester Nare?

  Really?

• 3:51:40Krishna Choudhary?

  Labs, biological web labs. He works in organoids and like 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 gonna now

• 3:52:18Krishna Choudhary?

  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 OpenAI, 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 gonna be crazy.

• 3:52:50Lester Nare?

  It's really interesting that you bring up that Daniel Toker's already 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.

• 3:53:12Krishna Choudhary?

  Yeah.

• 3:53:13Lester Nare?

  As a question and answer machine.

• 3:53:16Krishna Choudhary?

  Yeah, it's like chatbots.

• 3:53:17Lester Nare?

  As a text output production machine.

• 3:53:19Krishna Choudhary?

  Yeah.

• 3:53:20Lester Nare?

  Because this bio lab 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.

• 3:53:48Krishna Choudhary?

  Yeah.

• 3:53:49Lester Nare?

  And if you can accelerate that speed, you can accelerate testing and you can get the things quicker.

• 3:53:53Krishna Choudhary?

  Yeah.

• 3:53:53Lester Nare?

  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

• 3:54:10Krishna Choudhary?

  Yeah.

• 3:54:11Lester Nare?

  closed-loop cycle is very feasible, very

• 3:54:16Krishna Choudhary?

  Getting smaller.

• 3:54:16Lester Nare?

  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.

• 3:54:34Krishna Choudhary?

  Yeah.

• 3:54:35Lester Nare?

  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.

• 3:55:02Krishna Choudhary?

  Mm-hmm.

• 3:55:03Lester Nare?

  Um, we talk so much about frontier things on this pod, and so I think we're, as well as many listeners who've been here for a while, you understand how quickly everything is moving. Because these advances are not happening in a vacuum.

• 3:55:21Krishna Choudhary?

  Yeah.

• 3:55:22Lester Nare?

  Hardware progresses, advancing, chips, etc. Open source is advancing. Every aspect of every part of fundamental research in chemistry, physics, all of these things are advancing.

• 3:55:36Krishna Choudhary?

  Yep.

• 3:55:37Lester Nare?

  Maybe not as fast, but at a pace where there is a force multiplier here that is not kind of well-understood.

• 3:55:47Krishna Choudhary?

  Yeah.

• 3:55:48Lester Nare?

  And so I'm just gonna 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.

• 3:56:14Krishna Choudhary?

  Yeah.

• 3:56:15Lester Nare?

  And I think that's gonna make it harder to actually solve the problems we need to solve, without saying it's a toy, and saying that it also means extinction.

• 3:56:24Krishna Choudhary?

  Yeah.

• 3:56:25Lester Nare?

  Um, 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.

• 3:56:45Krishna Choudhary?

  Yeah.

• 3:56:45Lester Nare?

  Those other things can be true, and the risk can 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.

• 3:57:00Krishna Choudhary?

  Yep.

• 3:57:02Lester Nare?

  The systems we are discussing are not that.

• 3:57:05Krishna Choudhary?

  No. Certainly not.

• 3:57:07Lester Nare?

  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 behaviors like coordination.

• 3:57:48Krishna Choudhary?

  Mm-hmm.

• 3:57:49Lester Nare?

  Um, 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?

• 3:58:02Lester Nare?

  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 10 months behind the frontier,

• 3:58:21Krishna Choudhary?

  Yeah.

• 3:58:22Lester Nare?

  Um, does not stop independently operated, and the diffuse nature of this capability, these capabilities, being able to be implemented outside any number of frontier labs. It's not where we are today, but it's always moving in that direction. And I think

• 3:58:40Krishna Choudhary?

  And things are changing fast.

• 3:58:41Lester Nare?

  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 buy into the extinction piece. I do think that there is huge societal implications, like, like we saw with things like social media, that are second and third order consequences that were not intended.

• 3:59:02Krishna Choudhary?

  Yeah.

• 3:59:03Lester Nare?

  That become permanent problems. And what I mean by this, as capability spreads, control necessarily becomes fragmented.

• 3:59:14Krishna Choudhary?

  Yeah.

• 3:59:15Lester Nare?

  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.

• 3:59:27Krishna Choudhary?

  Mm-hmm.

• 3:59:28Lester Nare?

  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 gapped.

• 3:59:55Krishna Choudhary?

  Yeah.

• 3:59:55Lester Nare?

  It can't communicate. There have been theoretical and academic researches 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.

• 4:00:05Krishna Choudhary?

  Yeah, yeah, yeah.

• 4:00:06Lester Nare?

  In an air gapped context.

• 4:00:07Krishna Choudhary?

  Yeah.

• 4:00:08Lester Nare?

  I'm not saying that's practical.

• 4:00:09Krishna Choudhary?

  Yeah.

• 4:00:10Lester Nare?

  But like

• 4:00:12Krishna Choudhary?

  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.

• 4:00:21Lester Nare?

  And then it's like, well, you need money to do this, right?

• 4:00:23Krishna Choudhary?

  Yeah.

• 4:00:23Lester Nare?

  Well, Stripe already has an agent-enabled monetary systems. It does need human approval.

• 4:00:30Krishna Choudhary?

  Mm-hmm.

• 4:00:30Lester Nare?

  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 gonna 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 is, this will quickly become something that no one organization can just turn off a switch.

• 4:00:56Krishna Choudhary?

  Yep.

• 4:00:56Lester Nare?

  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 continued 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.

• 4:01:13Krishna Choudhary?

  Yeah.

• 4:01:13Lester Nare?

  I'm not sure that that's true. And

• 4:01:18Lester Nare?

  The surrounding infrastructure like matters here.

• 4:01:24Lester Nare?

  We have Neo clouds. 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, if this becomes diffuse.

• 4:01:40Krishna Choudhary?

  Yeah.

• 4:01:40Lester Nare?

  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.

• 4:01:57Krishna Choudhary?

  Yeah.

• 4:01:57Lester Nare?

  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.

• 4:02:25Lester Nare?

  There's a persistence here if we're not careful. And this can be related to RSI and also not be related to RSI.

• 4:02:35Lester Nare?

  We've not crossed that threshold.

• 4:02:36Krishna Choudhary?

  Yeah.

• 4:02:37Lester Nare?

  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 un-aliving amongst teenagers.

• 4:03:12Krishna Choudhary?

  Right.

• 4:03:13Lester Nare?

  Cause the Arab Spring and the destabilization of literal governments across the world, become permanent infrastructure that intelligence services use 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 come every election cycle. We have to defend against misinformation. Every platform, TikTok had to sell its algorithm to the US because we were afraid the Chinese were going to use it to brainwash young American kids to love communism.

• 4:03:56Krishna Choudhary?

  I mean, even in the stories that you've been saying, like, I was just thinking about how like in, you know, when we were growing up, Twitter was like for just talking random nonsense. And now like Twitter is the space where the 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.

• 4:04:30Lester Nare?

  It's growing faster. It's capability spread across a wider surface area.

• 4:04:35Krishna Choudhary?

  Yeah.

• 4:04:35Lester Nare?

  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 capability, exponential capability scaling that's happening

• 4:05:19Lester Nare?

  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.

• 4:05:26Krishna Choudhary?

  Yeah.

• 4:05:27Lester Nare?

  Of, of.

• 4:05:28Krishna Choudhary?

  It is certainly nonlinear, right?

• 4:05:31Lester Nare?

  And that's.

• 4:05:31Krishna Choudhary?

  Because 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 Navier-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?

• 4:05:53Lester Nare?

  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.

• 4:06:02Krishna Choudhary?

  Yeah.

• 4:06:03Lester Nare?

  I'm just, I'm just, my millennium prize problem is saying, uh, no one has proven that, uh, viscosity wins all the time.

• 4:06:15Krishna Choudhary?

  Yeah.

• 4:06:17Lester Nare?

  Right?

• 4:06:17Krishna Choudhary?

  Yeah.

• 4:06:19Lester Nare?

  I.

• 4:06:19Krishna Choudhary?

  Yeah.

• 4:06:19Lester Nare?

  The, the.

• 4:06:20Krishna Choudhary?

  Convective term.

• 4:06:22Lester Nare?

  I'm just saying, I think the convective term is something worth worrying about.

• 4:06:26Krishna Choudhary?

  Yeah, yeah, yeah, yeah.

• 4:06:27Lester Nare?

  Wasn't that an incredible way?

• 4:06:29Krishna Choudhary?

  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.

• 4:06:38Lester Nare?

  And, and we've shown it in Navier-Stokes. And all I'm saying is we cannot rule out that it's untrue.

• 4:06:46Krishna Choudhary?

  Yeah.

• 4:06:47Lester Nare?

  With

• 4:06:49Lester Nare?

  LLM AI systems.

• 4:06:51Krishna Choudhary?

  Yeah.

• 4:06:52Lester Nare?

  We are four hours and 20 minutes into this, folks, maybe a little bit less after we cut out the nonsense in between.

• 4:06:59Lester Nare?

  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.

• 4:07:08Krishna Choudhary?

  It's important.

• 4:07:08Lester Nare?

  Like a different angle to think about these issues.

• 4:07:12Krishna Choudhary?

  And 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.

• 4:07:22Lester Nare?

  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. We are going to now

• 4:07:38Lester Nare?

  return to our beds to get some sleep because it is now late here on the West Coast.

• 4:07:46Krishna Choudhary?

  Probably morning in Japan already.

• 4:07:49Krishna Choudhary?

  If anyone's in Japan, say good morning in the comments.

• 4:07:56Lester Nare?

  I, 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.

• 4:08:10Krishna Choudhary?

  Yeah, it's making me uncomfortable.

• 4:08:12Lester Nare?

  I am Lester Nare, your host, joined 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.

• 4:08:27Krishna Choudhary?

  My hand is there. It's all good.

• 4:08:30Lester Nare?

  Krishna Choudhary. We have a couple of exciting weeks coming up. No leaks, no leaks.

• 4:08:35Krishna Choudhary?

  No leaks, though.

• 4:08:36Lester Nare?

  No leaks, but.

• 4:08:37Krishna Choudhary?

  No leaks, baby.

• 4:08:38Lester Nare?

  Next episode, it might be kind of interesting. We'll see. We will see you all next week.