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Terence Tao

lines on the record: 252
first: 2025-06-15
most recent: 2025-06-15
sittings or episodes: 1
sources: podcast

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“There's a lot less sort of speculation about. Suppose I did this, what would happen? Planning and modeling, speculative fiction maybe is one other place. But that's about it, actually. Most of the things we do in life is conclusion-driven, including physics and science. I mean, they want to know where is this asteroid going to go? What is the weather going to be tomorrow? My fax also has this other direction of going from the axioms.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Mathematics is concerned with the models. Science collects the observations, and it proposes the models that might explain these observations. What mathematics does we stay within the model and we ask what are the consequences of that model? What observations, what predictions would the model make of future observations or past observations, does it fit observed data? So there's definitely a symbiosis. I guess mathematics is unusual among other disciplines that we start from hypotheses like the axioms of a model and ask what conclusions come up from that model. In almost any other discipline, you start with the conclusions, I want to do this, I want to build a bridge, I want to make money, I want to do this. And then you find the path to get there.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So I think science in general is interaction between three things. There's the real world. There's what we observe of the real world, our observations, and then our mental models as to how we think the world works. So we can't directly access reality. All we have are the observations which are incomplete and they have errors. There are many, many cases where we would want to know, for example, what is the weather like tomorrow. We don't yet have the observation and we'd like a prediction. And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the methodical models.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“The downside is that the Fantast proves are just much, much messier. So the infinite ones I found first usually, like decades earlier, and then later on people fantasize them.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“More recent years, people have started taking results that are true in infinite limits and what's called finetizing them. So you know that something's true eventually, but you don't know when. Now give me a rate. If I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for Hamlet to come out? And that's a more quantitative question. And this is something that you can attack by purely finite methods and you can use your finite intuition. And in this case, it turns out to be exponential in the length of the text that you're trying to generate. And so this is why you never see the monkeys create Hamlet. You can maybe see them create a four-letter word, but nothing that big. And so I personally find once you finetize an infinite statement, it does become much more intuitive. And it's no longer so weird.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, so there's a lot of pitfalls. So we spend a lot of time in undergraduate math classes teaching analysis and analysis is often about how to take limits and whether so for example A plus B is always B plus A. So when you have a finite number of terms and you add them, you can swap them and there's no problem. But when you have an infinite number of terms, these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity. You have to introduce these epsilon's and deltas. And there's a certain type of way of reasoning that helps you avoid mistakes.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Well, you can think of infinity as just an abstraction of a finite number of which you do not have a bound for. So nothing in real life is truly infinite. But you can... You can ask yourself questions like what if I had as much money as I wanted, or what if I could go as fast as I wanted? And a way in which mathematicians formalize that is mathematics has found a formalism to idealize instead of something being extremely large or extremely small to actually be exactly infinite or zero. And often the mathematics becomes a lot cleaner when you do that. I mean, in physics, we joke about assuming spherical cows. reward problems have got all kinds of real world effects, but you can idealize, send certain things to infinity, send some things to zero. And the mathematics becomes a lot simpler to work with it.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“The popular version of the monkey theorem is that if you have an infinite number of monkeys in a room with each with a typewriter, they type out text randomly. Almost surely one of them is going to generate the entire school of Hamlet or any other finite string of text. It will just take some time, quite a lot of time, actually. But if you have an infinite number, then it happens. Basically, the theme is that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge. It may take a long time, but it will eventually happen. In particular, ethnic progressions of any length will eventually happen, but you need an extremely long random sequence for this to happen.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorists, but occasionally they give colorful names.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“But Zamaniism also applies to random sets. If I take the set of all numbers and I flip a coin for each number and I only keep the numbers for which I got a heads for coins, I just randomly take out half the numbers I keep one half. So that's a set that has no patterns at all. But just from random fluctuations, you will still get a lot of ethnic progressions in that set.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Explaining this sort of partial pattern that you have. And so if you have these who inverse theorems, it creates this sort of dichotomy that either Objects that you study are either have no structure at all, or they are somehow related to something that is structured. And in either way, in either case, you can make progress. A good example of this is that there's this old theorem in mathematics called Semeretti's theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers. The pattern is Avemic progression, things like 3, 5, and 7 or 10, 15, and 20. Amereti, Andreas Amiri proved that any set of numbers that are sufficiently big, what's called positive density, has alpha progressions in it of any lengthy wish. So, for example, the odd numbers have set at density 1 half, and they contain arithmetic progressions of any length. So in that case, it's obvious because the odd numbers are really, really structured. I can just take 11, 13, 15, 17. I can easily find arithmetic progressions in that set.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Some companies are kind of additive, but not completely additive. So, for example, if I take a number n, I multiply by the square root of 2, and I take the integer part of that. So 10 by square root of 2 is like 14 point something, so 10 up to 14, 20 up to 28. So in that case, additivity is true then. So 10 plus 10 is 20 and 14 plus 20 is 28. But because of this rounding, sometimes there's round of errors and sometimes when you add a plus b, this function doesn't quite give you the sum of the two individual outputs, but the sum plus minus 1. So it's almost additive, but not quite additive. So there's a lot of useful results in mathematics, and I've worked a lot in developing things like this to the effect that if a function exhibits some structure like this, then it's basically, there's a reason for why it's true. And the reason is because there's some other nearby function, which is actually completely structured, which is”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Inverse theorems that give tests for when something is very structured. So some functions, what's called additive, like if you have a function that must be natural numbers, the natural numbers. So maybe two maps to four, three maps to six and so forth. Some functions are what's called additive, which means that if you add two inputs together, the output gets added as well. For example, multiplying by a constant. If you multiply a number by 10, if you multiply 8 plus b by 10, that's the same as multiplying a by 10 and b by 10 and adding them together. So some function additive.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness. That most objects that you can generate in mathematics are random. They look like random, like the digits of pi. Well, we believe is a good example. But there's a very small number of things that have patterns. But now you can prove something as a pattern by just constructing if something has a simple patent and you have a proof that it does something like repeat itself every so often, you can do that. And you can prove that, for example, you can prove that most sequences of digits have no pattern. So like if you just pick digits randomly, there's something called low large numbers. It tells you you're going to get as many ones as twos in the long run. But we have a lot fewer tools, if I give you a specific pattern, like the digits of pi, how can I show that this doesn't have some weird pattern to it? Some other work that I spend a lot of time on is to prove what are called structure theorems or in”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Thing is, you can get this emergent very complicated structures, but only with very carefully prepared initial conditions. So these glider guns and gates and software machines, if you just plonk out randomly some cells and you're looking at that, you will not see any of these. That's the analogous situation of Navier Stokes again that with typical initial conditions, you will not have any of this weird computation going on. basically through engineering by specially designing things in a very special way, you can pick clever constructions.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Of this was like community crowdsourced by amateur mathematicians actually. So I knew about that work. And so that is part of what inspired me to propose the same thing with Navier Stokes. As I said, analog is much worse than digital. It's going to be you can't just direct Plump them in. But again, it shows it's possible.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So maybe if both of the streams have gliders, then there will be an output stream. But if only one of them does, then nothing comes out. So, they could build something like that. And once you could build these basic gates, then just from software engineering, you can build almost anything. You can build a Turing machine. I mean, it's an enormous steampunk type thing. They look ridiculous. But then people also generated self-replicating objects in the game of life. A massive machine, a polynomial machine, which over a huge period of time and always look like gladder guns inside doing these very steampunk calculations, it would create another version of itself which could replicate.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Which evolves and it just moves at a certain direction. And that's like this vortex rings. So this is an analogy. The game of life is kind of like a discrete equation and the Navy Sok is a continuous equation. But mathematically, they have some similar features. And so over time, people discovered more and more interesting things that you could build within the game of life. The game of life is a very simple system. It only has like three or four rules to do it. But you can design all kinds of interesting configurations inside it. There's something called a glider gun that does nothing to spit out gliders one at a time. And then after a lot of effort, people managed to create AND gates and all gates for gliders. There's this massive, ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce extreme gliders coming out.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So this precedent, I mean, so the thing about mathematics is that it's really good at spotting connections between what you think of what you might think of as completely different problems. But if the mathematical form is the same, you can draw a connection. So there's a lot of work previously on what I call cellular automata. The most famous of which is Conway's Game of Life. There's this infinite discrete grid. And at any given time, the grid is either occupied by a cell or it's empty. And there's a very simple rule that tells you how these cells evolve. So sometimes cells live and sometimes they die. And there's when I was a student, it was a very popular screensaver through just have these animations go on. And they look very chaotic. In fact, they look a little bit like turbulent flow sometimes. But at some point, people discovered more and more interesting structures within this game of life. So, for example, they discovered this thing called a glider. So a glider is a very tiny configuration of like four or five cells.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Is really nasty. It compared to digital computing. I mean, because there's always errors. You have to do a lot of error correction along the way. I don't know how to completely power down the big machine so that it doesn't interview the running of the smaller machine. But everything in principle can happen. It doesn't contradict any of the laws of physics. So it's sort of evidence that this thing is possible. There are other groups who are now pursuing ways to magneto explore up which are nowhere near as ridiculously complicated as this. They actually are pursuing much closer to the direct self-similar model, which can, it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Version of itself in some sort of cold state, it wouldn't start just yet. Once it's ready, the big robot convection of water would transform all its energy into the smaller configuration and then power down. And then I clean myself up. And then what's left is this newer state, which would then turn on and do the same thing, but smaller and faster. And then the equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this in principle would create a blow up for the actual Navier Stokes. And this is what I managed to accomplish for this average Navier Stokes. So it provided this sort of roadmap to solve the problem. Now, this is a pipe dream because there are so many things that are missing for this to actually be a reality. So I can't create these basic logic gates. I don't have these special configurations of water. I mean, there's candidates that include vortex rings that might possibly work. But also analog computers.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Create a Turing machine, and then you have computers which are made completely out of water. And if you have computers, then maybe you can do robotics. Hydraulics and so forth. And so you could create some machine, which is basically a fluid analog, what's called a vonomian machine. So Von Norman proposed, if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous. But if you could transport one machine to Mars and this machine had the ability to mine the planet, create some raw materials, smelt them, and build more copies of the same machine, then you could colonize the whole planet over time. So if you could build a fluid machine, which is a fluid robot. And what it would do, its purpose in life, it's programmed so that it would create a small”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Goldberg type machine, but describe mathematically. And this ended up working. So, what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation, so like if you can imagine kind of a steampunk versus really water punk type of thing where modern computers are electronic, they're powered by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of water moving at a certain velocity. And maybe they're two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, they would come out with some new configuration, which would be something like an AND gate or OR gate. The output would depend in a very predictable way on the inputs. And you could chain these together and maybe.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So, in order to make that happen I had to construct a rather complicated nonlinearity. And it was basically like it was constructing like an electronic circuit. So I actually thanked my wife for this because she was trained as an electrical engineer. And she talked about she had to design circuits and so forth. And if you want a circuit that does a certain thing, like maybe have a light that flashes on and then turns off and then on and off, you can build it from more primitive components, capacitors and resistors and so forth. And you have to build a diagram. And these diagrams, you can sort of follow up with your eyeballs and say, oh yeah, the current will build up here and then it will stop and then it will do that. So I knew how to build the analog of basic electronic components like resistors and capacitors and so forth. And I would stack them together in such a way that I would create something that would open one gate and then there would be a clock and then once the clock hits the threshold, it would close it. It's kind of a rubber.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And this spreads out the energy too much. And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything. So it turns out this direct abortion doesn't actually work. There's a separate paper by some other authors that actually showed this in three dimensions. So what I needed was to program a delay. So kind of like airlocks. So I needed an equation which would start with a fluid doing something at one scale, it would push its energy into the next scale, but it would stay there until all the energy from the larger scale got transferred. And only after you pushed all the energy in, then you sort of opened the next gate and then you pushed that in as well. So by doing that, it kind of, the energy inches forward scale by scale in such a way that it's always localized at one scale at a time. And then it can resist the effects of viscosity because it's not dispersed.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, so this came out of this work of constructing this average equation that blew up. So as part of how I had to do this, so there's all this naive way to do it. You just keep pushing every time you energy at one scale, you push it immediately to the next scale as fast as possible. This is sort of the naive way to force blow up. It turns out in five and high dimensions, this works. But in three dimensions, there was this funny phenomenon that I discovered that if you change laws of physics, you just always keep trying to push the energy into smaller and smaller scales. What happens is that the NG starts getting spread out into many scales at once. So you have energy at one scale, you're pushing it into the next scale, and then as soon as it enters that scale, you also push it to the next scale, but there's still some energy left over from the previous scale. You're trying to do everything at once.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, and if nonlinearity is somehow more and more featured and interesting at small scales, I mean, there's many equations that are nonlinear, but in many equations you can approximate things by the bulk. So for example, planetary motion, if you wanted to understand the orbit of the moon or Mars or something, you don't really need the microstructure of the seismology of the moon or exactly how the mass is distributed. You can almost approximate these planets by point masses. And just the aggregate behavior is important. But if you want to model a fluid, like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this. For supercritical equations, the finest confirmation is really important.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“I did a lot of work, and then there's been a lot of follow up showing that for many other types of supercritical equations, you can create all kinds of blow-up examples. Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. So this is sort of one of the main insights of this line of work is that supercriticality versus criticality and subcriticality, this makes a big difference. I mean, that's a key qualitative feature that distinguishes some equations from being sort of nice and predictable and like planetary motion. I mean, there's certain equations that you can predict for millions of years or thousands at least. Again, it's not really a problem, but there's a reason why we can't predict the weather past two weeks into the future, because it's a supercritical equation. Lots of really strange things are going on at very fine scales.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Sometimes these forces are in balance at small scales, but not in balance at large scales or vice versa. So NavisOx is what's called supercritical. So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. So the viscosity terms are things that calm things down. And so this is why the problem is hard. In two dimensions, so the Soviet mathematician Ladishin Skaya, she in the 60s, shows in two dimensions there was no blow-up. And in two dimensions, the Navy-Socus equations is what's called critical, the effect of transport and the effect of viscosity about the same strength, even at very, very small scales. And we have a lot of technology to handle critical and also subcritical equations and proof regularity. But for supercritical equations, it was not clear what was going on.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, yeah. So the key phenomenon that my technique exploits is what's called supercriticality. So in partial differential equations, often these equations are like a tug of war between different forces. So in Nabia Stokes, there's the dissipation force coming from viscosity, and it's very well understood. It's linear. It calms things down. If viscosity was all there was, then nothing bad would ever happen. But there's also transport, that energy from in one location of space can get transported because of fluid is in motion to other locations. And that's a nonlinear effect. And that causes all the problems. So there are these two competing terms in the Navy-Soke's equation, the dissipation term and the transport term. If the dissipation term dominates, if it's large, then basically you get regularity. And if the transport term dominates, then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem. But it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counter examples for nearby problems, kind of rules out, it saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. So it provides what's called an obstruction in mathematics. So, what I did was that basically if I turned off the certain parts of the equation, which usually when you turn off certain interactions make it less nonlinear, it makes it more regular and less likely to blower. But I found that by turning off a very well-designed set of interactions, I could force all the energy to blow in finite time. So what that means is that if you wanted to prove global regularity for Navier Stokes for the actual equation, you must use some feature of the true equation, which my artificial equation does not satisfy. So it rules out certain approaches. So the thing about math is it's not just about finding, taking a technique that is going to work and applying it. But you need to not take the techniques that don't work.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, yeah. So I basically engineer by changing laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“The viscosity, and you can keep everything under control for not just the navier stokes, but for many, many types of equations like this. And so in the past, there have been many attempts to try to obtain what's called global regularity for Nabiostokes, which is the opposite of finite time blower, that a velocity stays smooth. And it all failed. There was always some sin error or some subtle mistake, and it couldn't be salvaged. So what I was interested in doing was trying to explain why we were not able to disprove Palatine Blower. I couldn't do it for the actual equations of fluids, which were too complicated. But if I could average the equations of motion of the Navy So basically, if I could turn off certain types of waves in which water interacts and only keep the ones that I want. So in particular, if there's a fluid and it could transfer its energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Converge to all the energy concentrating one point in a finite amount of time. And that's now he's got Fanny Blow-Up. So in practice, this doesn't happen. So water is what's called turbulent. So it is true that if you have a big eddy of water, it will tend to break up into smaller eddies, but it won't transfer all the energy from one big eddy into one smaller eddy. It will transfer into maybe three or four. And then those ones split up into maybe three or four small eddies of their own. And so the energy gets dispersed to the point where the viscosity can then keep a thing under control. But if it can somehow concentrate all the energy, keep it all together, and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blow up can occur. So there were papers who had claimed that, oh, you just need to take into account conservation energy and just carefully use”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Actually, experiment with water. You splash around, there's some turbulence and waves and so forth, but eventually it settles down and the lower the amplitude, the smaller the velocity, the more calm it gets. But potentially there is also a demon that keeps pushing the energy of the fluid into a smaller and smaller scale and it will move faster and faster. And at faster speeds, the effect of viscosity is relatively less. And so it could happen that it creates some sort of self-similar blow up scenario where the energy of the fluid starts off at some large scale and then it all sort of transfers energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth. And each time it does this, it takes maybe half as long as the previous one. And then you could actually”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“3.14159 and so forth. The digits look like they have no pattern, and we believe they have no patent. On the long term, you should see as many ones and twos and threes as fours and fives and sixes. There should be no preference in the digital pi to favor, let's say, seven over eight. But maybe there is some demon in the digits of pi that every time you compute more and more digits, it biases one digit to another. This is a conspiracy that should not happen. There's no reason it should happen. There's no way to prove it with our current technology. Okay, so getting back to Navier Stokes, a fluid has a certain amount of energy. And because the fluid is in motion, the energy gets transported around and water is also viscous. So if the energy is spread out over many different locations, the natural viscosity of the fluid will just damp out the energy and it will go to zero. And this is what happens when we...”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Short answer is Maxwell's demon. So Maxwell's demon is a concept in thermodynamics. Like if you have a box or two gases in oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen on the other side, but there's no barrier between them. Then they will mix and they should stay mixed. There's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that maybe there's a microscopic demon called Maxwell's demon that will, every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and the nitrogen goes to the other. And you could have an extremely improbable configuration emerge, which we never see, and statistically it's extremely unlikely, but mathematically it's possible that this can happen and we can't rule it out. And this is a situation that shows up a lot in mathematics, a basic example is the digits of pi.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, so it has practical importance. So, this high price problem concerns what's called the incompressible Napo-Stokes, which governs things like water. There's something called the compressible Nabio Stokes, which governs things like air. And that's particularly important for weather prediction. Weather prediction, it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Nabi-Stokes equations as best they can. Also gathering a lot of data so that they can initialize the equation. There's a lot of moving parts. So it's very important problem practically.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, yeah. So, yeah, that is literally the million dollar question. Yeah, so this is what distinguishes mathematicians from pretty much everybody else. Something holds 99.99% of the time, that's good enough for most things. But mathematicians are one of the few people who really care about whether 100%, really 100% of all situations are covered by most fluid, most of the time water does not blow up, but could you design a very special initial state that does this?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And in fact, in recent years, the consensus has drifted towards the belief that in fact for certain very special initial configurations of, say, water, that singularities can form. But people have not yet been able to actually establish this, the Clay Foundation has these seven Millennium Fries problems, has a million-dollar prize for solving one of these problems. This is one of them. Of these seven, only one of them has been solved at the Poincar ⁇ conjecture. So the Kaya conjecture is not directly, directly related to the Navy-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navy Sorks problem better.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“At some point, that's called a singularity. We don't see that in real life. If you splash around water on the bathtub, we won't explode on you or have water leaving at the speed of light. But potentially it is possible.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And you could create what's called a blow up where these waves, their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear. And so in mathematical physics, we care a lot about whether certain equations and wave equations are stable or not, whether they can create these singularities. There's a famous unsolved problem called the Navier-Stokes regularity problem. So the Navier Stokes equations equations that govern the fluid flow of incompressible fluids like water. The question asks if you start with a smooth velocity field of water, can it ever concentrate?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And so it's possible to do that. And geometrically, what's going on is that there's always light rays. So if this wave represents light, for example, you can imagine this wave as a superposition of photons all traveling at the speed of light. They all travel on these light rays, and they're all focusing at this one point. So you get a very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and it separates. But potentially if the conjuncture had a negative solution, so what that meant is that there's a very efficient way to pack tubes pointing in different directions, a very, very narrow region, very narrow volume, then you would also be able to create waves that there'll be some arrangement of waves that start out very, very dispersed, but they would concentrate not just at a single point, but there'll be a large concentrations in space and time.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“But ways exhibit both particle and wavety behavior. So you can have what's called a wave packet, which is like a very localized wave that is localized in space and moving a certain direction in time. And so if you plot in both space and time, it occupies a region which looks like a tube. And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time. Like you can imagine dropping a pebble into a pond and will spread out. But then if you time reverse that scenario and the equations of wave motion are time reversible, you can imagine ripples that are converging to a single point and then a big splash occurs, maybe even a singularity.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And so you can modify it because the Copish's construction. And so if your telescope has zero thickness, then you can use as little volume as you need. That's a simple modification of the two-dimensional construction. But the question is that if your telescope is not zero thickness, but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta? So as delta gets smaller, as you need or gets thinner, the volume should go down. But how fast does it go down? And the conjecture was that it goes down very, very slowly, like logarithmically, roughly speaking. And that was proved after a lot of work. So this seems like a puzzle. Why is it interesting? So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, common forex. For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“You want to observe every single star in the universe. So you want to rotate the telescope to reach every single direction. And his unrealistic part, suppose that space is at a premium, which totally is not. You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky. How small a volume do you need to do that?”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So you can imagine just spinning it around as the unit noodle. You can spin it around its center. And I think that gives you a disc of area, I think pi over 4. Or you can do a three-point U-turn, which is what we teach people in their driving schools to do. And that actually takes area pi over eight. So it's a little bit more efficient than a rotation. And so for a while, people thought that was the most efficient way to turn things around. But Bazakovic showed that, in fact, you could actually turn the needle around using as little area as you wanted. So 0.001, there was some really fancy multi back and forth U-turn thing that you could do that you could turn a needle around. And in so doing, it would pass through every intermediate direction. Is this in the two-dimensional plane? This is in the two-dimensional plane. So we understand everything in two dimensions. So the next question is what happens in three dimensions? So suppose the Hubble Space Telescope is tube in space.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“A needle on the plane. Think like driving on a road or something. And you wanted to execute a U-turn. You want to turn the needle around. But you want to do it in as little space as possible. So you want to use as little area in order to turn it around. But the needle is infinitely maneuverable.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“What was the first really difficult research level math problem that you encountered? One that gave you pause, maybe? Well, I mean, in your undergraduate education, you learn about the really hard impossible problems, like the women hypothesis between Prime's conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just on the boundary between what we can do relatively easily and what are hopeless, but what are problems where existing techniques can do 90% of the job and then you just need that remaining 10%. I think as a PhD student, the care problem certainly caught my eye. It just got solved, actually. It's a problem I've worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Suji Kakea in like 1918 or so. So the puzzle is that you have”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source