By constructing a 'liquid computer' from the Navier-Stokes equations — a self-replicating fluid machine that transfers energy to smaller and smaller copies of itself — one could prove that finite-time blow-up is possible for the actual equations.

(00:00:00) The following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of math. (00:00:12) He won the Fields Medal and the Breakthrough Prize in Mathematics and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. (00:00:24) This was a huge honor for me for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. (00:00:35) It means the world. (00:00:36) And now a quick few second mention of each sponsor. (00:00:39) Check them out in the description or at lexfriedman.com slash sponsors. (00:00:44) It's the best way to support this podcast. (00:00:46) We got Notion for teamwork, Shopify for selling stuff online, NetSuite for your business, Element for electrolytes, and the AG1 for your health. (00:00:54) Choose Wes and my friends. (00:00:56) And now onto the full ad reads. (00:00:57) They're all here in one place. (00:00:59) I do try to make them interesting by talking about some random things I'm reading or thinking about. (00:01:05) But if you skip, please still check out the sponsors. (00:01:08) I enjoy their stuff. (00:01:08) Maybe you will too. (00:01:10) To get in touch with me for whatever reason, go to lextreatment.com slash contact. (00:01:14) All right, let's go. (00:01:16) This episode is brought to you by Notion, a note-taking and team collaboration tool. (00:01:21) I use Notion for everything, for personal notes, for planning this podcast, for collaborating with other folks, and for super boosting all of those things with AI because Notion does a great job of integrating AI into the whole thing. (00:01:35) You know what's fascinating is the mechanisms of human memory. (00:01:39) Before we had widely adopted technologies and tools, (00:01:45) for writing and recording stuff, certainly before the computer. (00:01:48) So you can look at medieval monks, for example, that would use the now well-studied memory techniques, like the memory palace, the spatial memory techniques, to memorize entire books. (00:02:00) That is certainly the effect of technology, started by Google search and moving to all the other things like Notion, that we're offloading more and more and more of the task of memorization to the computers, which I think is (00:02:13) probably a positive thing because it frees more of our brain to do deep reasoning, whether that's deep dive focused specialization or the generalist type of thinking, versus memorizing facts. (00:02:30) Although I do think that there's a kind of background model that's formed when you memorize a lot of things, and from there, from inspiration, (00:02:38) arises discovery. (00:02:40) So I don't know, it could be a great cost to offloading most of our memorization to the machines. (00:02:49) But it is the way of the world. (00:02:51) Try Notion AI for free when you go to notion.com slash Lex. (00:02:54) That's all lowercase notion.com slash Lex to try the power of Notion AI today. (00:02:59) This episode is also brought to you by Shopify, a platform designed for anyone to sell anywhere with a great looking online store. (00:03:06) Our future friends, (00:03:07) has a lot of robots in it. (00:03:09) Looking into that distant future, you have Amazon warehouses with millions of robots that move packages around. (00:03:17) You have Tesla bots everywhere in the factories and in the home and on the streets and the baristas. (00:03:23) All of that, that's our future. (00:03:25) Right now you have something like Shopify that connects a lot of humans in the digital space. (00:03:29) But more and more, there will be a (00:03:32) automated, digitized, AI-fueled connection between humans in the physical space. (00:03:39) Like a lot of futures, there's going to be negative things and there's going to be positive things. (00:03:44) And like a lot of possible futures, there's little we could do about stopping it. (00:03:49) All we can do is steer it in the direction that enables human flourishing. (00:03:55) Instead of hiding in fear or fear-mongering, be part of the group of people that are building (00:04:01) the best possible trajectory of human civilization. (00:04:06) Anyway, sign up for a $1 per month trial period at shopify.com slash lex. (00:04:10) That's all lowercase. (00:04:12) Go to shopify.com slash lex to take your business to the next level today. (00:04:17) This episode is also brought to you by NetSuite, an all-in-one cloud business management system. (00:04:24) There's a lot of messy components to running a business, and I must ask, and I must wonder, (00:04:30) at which point there's going to be an AI, AGI-like CFO of a company. (00:04:35) An AI agent that handles most, if not all, of the financial responsibilities or all of the things that NetSuite is doing, at which point will NetSuite increasingly leverage AI for those tasks. (00:04:50) I think probably it will integrate AI into its tooling, but I think there's a lot of edge cases that we need to (00:04:58) The human wisdom, the human intuition, grounded in years of experience in order to make the tricky decision around the edge cases. (00:05:07) I suspect that running a company is a lot more difficult than people realize, but there's a lot of sort of paperwork type stuff that could be automated, could be digitized, could be summarized, integrated, and used as a foundation for the said humans to make decisions. (00:05:24) Anyway, that's our future. (00:05:26) Download the CFO's Guide to AI and Machine Learning at netsuite.com/lex. (00:05:31) That's netsuite.com/lex. (00:05:34) This episode is also brought to you by Element, my daily zero-sugar and delicious electrolyte mix. (00:05:39) You know, I run along the river often and get to meet some really interesting people. (00:05:47) One of the people I met was preparing for his first ultra-marathon. (00:05:51) I believe he said it was 100 miles. (00:05:54) And that, of course, sparked in me the thought that I need for sure to do one myself. (00:06:00) Some time ago now, I was planning to do something with David Goggins, and I think that's still on the sort of to-do list between the two of us, to do some crazy physical feat. (00:06:11) Of course, the thing that is crazy for me is a daily activity for Goggins. (00:06:17) But nevertheless, I think it's important in the physical domain, the mental domain, and (00:06:23) all domains of life to challenge yourself. (00:06:26) And athletic endeavors is one of the most sort of crisp, clear, well-structured way of challenging yourself. (00:06:33) But there's all kinds of things. (00:06:35) Writing a book, to be honest, having kids and marriage and relationships and friendships, all of those, if you take it seriously, if you go all in and do it right, I think that's a serious challenge. (00:06:49) Because most of us are not prepared for it. (00:06:51) You can learn along the way. (00:06:52) And if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think that might as well be an ultra-marathon. (00:07:06) Anyway, get a sample pack for free with any purchase. (00:07:08) Try it at drinkelement.com/lex. (00:07:12) And finally, this episode is also brought to you by AG1, an all-in-one daily drink to support better health and peak performance. (00:07:21) I drink it every day. (00:07:23) I'm preparing for a conversation on drugs in the Third Reich. (00:07:30) And funny enough, it's a kind of way to analyze a Hitler biography. (00:07:35) It's to look at what he consumed throughout and Norman Oler does a great job of analyzing all of that and tells the story of Hitler and the Third Reich in a way that hasn't really been touched by historians before. (00:07:49) It's always nice to look at key moments in history through a perspective that's not often taken. (00:07:56) Anyway, I mention that because I think Hitler had a lot of stomach problems, and so that was the motivation for getting a doctor, the doctor that eventually would fill him up with all kinds of drugs. (00:08:09) But the doctor earned Hitler's trust by giving him probiotics, which is a kind of revolutionary thing at the time. (00:08:16) And so that really helped deal with whatever stomach issues that Hitler was having. (00:08:22) All of that is a reminder that war is waged by humans and humans are biological systems and biological systems require fuel and supplements and all of that kind of stuff. (00:08:32) And depending on what you put in your body will affect your performance in the short term and the long term. (00:08:37) With meth, that's true with Hitler to his last days in the bunker in Berlin. (00:08:44) all the cocktail of drugs that he was taking. (00:08:46) So I think I got myself somewhere deep and I'm not sure how to get out of this. (00:08:54) It deserves a multi-hour conversation versus a few seconds of mention. (00:08:58) But yeah, all of that was sparked by my thinking of AG1 and how much I love it. (00:09:05) I appreciate that you're listening to this and coming along for the wild journey that these ad reads are. (00:09:13) Anyway, AG1 will give you a one-month supply of fish oil when you sign up at drinkag1.com slash Lex. (00:09:21) This is the Lex Friedman Podcast. (00:09:23) To support it, please check out our sponsors in the description or at lexfriedman.com slash sponsors. (00:09:29) And now, dear friends, here's Terence. (00:09:32) Tao. (00:09:50) What was the first really difficult research-level math problem that you encountered? (00:09:55) One that gave you pause, maybe? (00:09:57) Well, I mean, in your undergraduate education, you learn about the really hard impossible problems, like the Riemann hypothesis, the Trin-Primes conjecture. (00:10:06) You can make problems arbitrarily difficult. (00:10:08) That's not really a problem. (00:10:09) In fact, there's even problems that we know to be unsolvable. (00:10:12) What's really interesting are the problems just on the boundary between what we can do easily and what are hopeless. (00:10:18) But what are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%? (00:10:28) I think as a PhD student, the Kakeya problem certainly caught my eye. (00:10:31) And it just got solved, actually. (00:10:32) It's a problem I've worked on a lot in my early research. (00:10:35) Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in like 1918 or so. (00:10:43) So the puzzle is that you have a needle on the plane, or think like driving on a road, and you want to execute a U-turn, you want to turn the needle around, but you want to do it in as little space as possible. (00:11:00) So you want to use this little area in order to turn it around. (00:11:04) So, but the needle is infinitely maneuverable. (00:11:07) So, you can imagine just spinning it around it's it's a unit noodle, you can spin it around its center, and I think that gives you a disc of area, I think pi over 4. (00:11:18) 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 8. (00:11:24) So it's a little bit more efficient than a rotation. (00:11:28) And so for a while, people thought that was the most efficient way to turn things around. (00:11:32) But Bazakovich showed that, in fact, you could actually turn the needle around using as little area as you wanted. (00:11:38) So 0.001, there was some really fancy multi-back and forth U-turn thing that you could do. (00:11:46) that you could turn a needle around, and in so doing, it would pass through every intermediate direction. (00:11:50) Is this in the two-dimensional plane? (00:11:52) This is in the two-dimensional plane. (00:11:53) Yeah, so we understand everything in two dimensions. (00:11:56) So the next question is what happens in three dimensions? (00:11:58) So suppose like the Hubble Space Telescope is tube in space. (00:12:02) And you want to observe every single star in the universe. (00:12:05) So you want to rotate the telescope to reach every single direction. (00:12:08) And here's the unrealistic part. (00:12:09) Suppose that space is at a premium, which totally is not. (00:12:13) 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. (00:12:20) How small a volume do you need to do that? (00:12:23) And so you can modify Bezekovich's construction (00:12:26) And so if your telescope has zero thickness, then you can use as little volume as you need. (00:12:30) That's a simple modification of the two-dimensional construction. (00:12:33) 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? (00:12:45) So as delta gets smaller, as the needle gets thinner, the volume should go down, but how fast does it go down? (00:12:53) And (00:12:54) The conjecture was that it goes down very, very slowly, like logarithmically, roughly speaking. (00:12:59) And that was proved after a lot of work. (00:13:02) So this seems like a puzzle, why is it interesting? (00:13:05) So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics. (00:13:12) For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions. (00:13:19) But waves exhibit both particle and wave type behavior. (00:13:22) 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. (00:13:30) And so if you plot it into space and time, it occupies a region which looks like a tube. (00:13:36) 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. (00:13:44) Like you can imagine dropping a pebble into a pond and the rubble spread out. (00:13:48) 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. (00:14:02) And so it's possible to do that, and geometrically what's going on is that there's always sort of light rays. (00:14:08) So like if this wave represents light, for example, you can imagine this wave as the superposition of photons. (00:14:14) all traveling at the speed of light, they all travel on these light rays, and they're all focusing at this one point. (00:14:19) So you can have 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. (00:14:28) But potentially, if the pinjecture had a negative solution, so what that means is that there's a very efficient way to pack tubes pointing in different directions into a very, very narrow region of very narrow volume. (00:14:41) Then you would also be able to create waves that start out, 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, there'll be a lot of concentrations in space and time. (00:14:56) 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 non-linear. (00:15:08) And so in mathematical physics, we care a lot about whether certain equations in wave equations are stable or not, whether they can create these singularities. (00:15:16) There's a famous unsolved problem called the Navier-Stokes regularity problem. (00:15:19) So the Navier-Stokes equations, equations that govern the fluid flow or incompressible fluids like water. (00:15:25) The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point? (00:15:33) That's called a singularity. (00:15:34) We don't see that (00:15:37) in real life. (00:15:38) if you splash around water on the bathtub, it won't explode on you or have water leaving at the speed of light. (00:15:45) But potentially, it is possible. (00:15:49) 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. (00:16:03) But people have not yet been able to actually establish this. (00:16:06) The Clay Foundation has these seven millennium prize problems as $1,000,000 prize for solving one of these problems. (00:16:11) So this is one of them. (00:16:13) Of these seven, only one of them has been solved at the Point Gray conjecture at Perlman. (00:16:18) So the Kakei conjecture is not directly, directly related to the Navi-Stokes problem, but understanding it would help us understand (00:16:27) some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes problem better. (00:16:32) Can you speak to the Navier-Stokes? (00:16:34) So the existence and smoothness, like you said, millennial price problem. (00:16:38) You've made a lot of progress on this one. (00:16:40) In 2016, you published a paper, Finite Time Blow Up, for an averaged 3-dimensional Navier-Stokes equation. (00:16:47) Right. (00:16:48) So we're trying to figure out if this thing usually doesn't blow up. (00:16:52) Right. (00:16:53) But can we say for sure it never blows up? (00:16:56) Right. (00:16:56) Yeah. (00:16:57) So yeah, that is literally the $1,000,000 question. (00:16:59) Yeah. (00:16:59) So this is what distinguishes mathematicians from pretty much everybody else. (00:17:03) Like if something holds 99.99% of the time, that's good enough for most, you know, for most things. (00:17:11) But mathematicians are one of the few people who really care about whether like 100%, really 100% of all (00:17:18) situations are covered by, so most fluid, most of the time, water does not blow up, but could you design a very special initial state that does this? (00:17:29) And maybe we should say that this is a set of equations that govern. (00:17:33) in the field of fluid dynamics. (00:17:35) Trying to understand how fluid behaves and it's actually turns out to be a really complicated, you know, fluid is extremely complicated thing to try to model. (00:17:43) Yeah, so it has practical importance. (00:17:44) So this clay price problem concerns what's called the incompressible Navier-Stokes, which governs things like water. (00:17:49) There's something called the compressible Navier-Stokes, which governs things like air. (00:17:51) And that's particularly important for weather prediction. (00:17:54) Weather prediction, it does a lot of computational fluid dynamics. (00:17:56) A lot of it is actually just trying to solve the Navier-Stokes equations as best they can. (00:18:02) also gathering a lot of data so that they can get, they can initialize the equation. (00:18:06) There's a lot of moving parts. (00:18:07) So it's very important problem practically. (00:18:09) Why is it difficult to prove general things about the set of equations like it not blowing up? (00:18:17) Short answer is Maxwell's demon. (00:18:20) So Maxwell's demon is a concept in thermodynamics. (00:18:22) Like if you have a box with two gases, 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, right? (00:18:29) Then they will mix. (00:18:31) And (00:18:32) they should stay mixed. (00:18:33) There's no reason why they should unmix. (00:18:35) But in principle, because of all the collisions between them, there could be some sort of weird conspiracy like 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 then nitrogen goes to the other. (00:18:51) And you could have an extremely improbable configuration emerge, which we never see. (00:18:57) And statistically, it's extremely unlikely. (00:19:01) But (00:19:02) Mathematically, it's possible that this can happen, and we can't rule that out. (00:19:06) And this is a situation that shows up a lot in mathematics. (00:19:10) A basic example is the digits of pi, 3.14159, and so forth. (00:19:14) The digits look like they have no pattern, and we believe they have no pattern. (00:19:18) On the long term, you should see as many ones and twos and threes as fours and fives and sixes. (00:19:22) There should be no preference in the digits of pi to favor, let's say, 7 / 8. (00:19:28) But maybe there's some demon in the digits of pi that like every time you compute more and more digits, it sort of biases 1 digit to another. (00:19:36) And this is a conspiracy that should not happen. (00:19:39) There's no reason it should happen. (00:19:41) But there's no way to prove it with our current technology. (00:19:46) Okay, so getting back to Navier-Stokes, a fluid has a certain amount of energy. (00:19:49) And because the fluid is in motion, the energy gets transported around. (00:19:53) And water is also viscous. (00:19:55) 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. (00:20:03) And this is what happens when we actually experiment with water. (00:20:09) 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. (00:20:20) But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale. (00:20:26) And it will move faster and faster. (00:20:28) And at faster speeds, the effect of viscosity is relatively less. (00:20:31) And so it could happen that it creates some sort of what's called a self-similar blob scenario where, you know, the energy of the fluid starts off at some large scale and then it all sort of transfers its energy into a smaller (00:20:47) region of the fluid, which then at a much faster rate moves into an even smaller region and so forth. (00:20:55) And each time it does this, it takes maybe half as long as the previous one. (00:21:00) And then you could actually converge to all the energy concentrating in one point in a finite amount of time. (00:21:09) And that's an always good finite time blow up. (00:21:13) So in practice, this doesn't happen. (00:21:16) So water is what's called turbulent. (00:21:18) 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. (00:21:27) It will transfer into maybe three or four. (00:21:28) And then those ones split up into maybe three or four small eddies of their own. (00:21:32) And so the energy gets dispersed to the point where the viscosity can then keep everything under control. (00:21:38) But if it can somehow concentrate all the energy (00:21:43) keep it all together and do it fast enough that the viscous effects don't have enough time to calm everything down, then this blowup can occur. (00:21:51) So there are papers who had claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just Navier-Stokes, but for many, many types of equations like this. (00:22:03) And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier-Stokes, which is the opposite of finite time blowup, that velocity stays smooth. (00:22:11) And it all failed. (00:22:12) There was always some sign error or some subtle mistake and it couldn't be salvaged. (00:22:18) So what I was interested in doing was trying to explain why we were not able to disprove finite time blow up. (00:22:26) I couldn't do it for the actual equations of fluids, which were too complicated. (00:22:29) But if I could average the equations of motion of the Navier-Sukes, so basically, if I could turn off certain types of ways in which water interacts and only keep the ones that I want. (00:22:40) 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 one and direct it only into this smaller eddy while still preserving the law of concentration of energy. (00:22:59) So you're trying to make a blow up. (00:23:00) Yeah, So I basically engineer. (00:23:03) blow up by changing the laws of physics, which is one thing that mathematicians are allowed to do. (00:23:07) We can change the equation. (00:23:08) How does that help you get closer to the proof of something? (00:23:10) Right. (00:23:11) So it provides what's called an obstruction in mathematics. (00:23:14) 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 non-linear, it makes it more regular and less likely to blow up. (00:23:27) But I found that by turning off a very well-designed (00:23:30) set of interactions, I could force all the energy to blow in finite time. (00:23:36) 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. (00:23:52) So it rules out certain approaches. (00:23:55) So (00:23:57) 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. (00:24:06) 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. (00:24:18) So having these counterexamples for nearby problems kind of rules out (00:24:24) It saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work. (00:24:31) How deeply connected is it to that specific problem of fluid dynamics or is it some more general intuition you build up about mathematics? (00:24:38) Right, yeah. (00:24:38) So the key phenomenon that my technique exploits is what's called supercriticality. (00:24:44) So in partial differential equations, often these equations are like a tug of war between different forces. (00:24:49) So in Navier-Stokes, there's the dissipation (00:24:52) force coming from viscosity, and it's very well understood, it's linear, it calms things down. (00:24:57) If viscosity was all there was, then nothing bad would ever happen. (00:25:02) But there's also transport, that energy in one location of space can get transported because the fluid is in motion to other locations. (00:25:11) And that's a non-linear effect, and that causes all the problems. (00:25:15) So there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term. (00:25:20) If the dissipation term dominates, if it's large, then basically you get regularity. (00:25:24) And if the transport term dominates, then we don't know what's going on. (00:25:29) It's a very non-linear situation. (00:25:30) It's unpredictable. (00:25:31) It's turbulent. (00:25:32) So sometimes these forces are unbalanced at small scales, but not imbalanced at large scales or vice versa. (00:25:39) So Navier-Stokes is what's called supercritical. (00:25:41) So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms. (00:25:47) So the viscosity terms are things that calm things down. (00:25:50) And so this is why the problem is hard. (00:25:54) In 2 dimensions, so the Soviet mathematician Lady Shinskaya, she in the 60s shows in two dimensions there was no blow up. (00:26:01) And in two dimensions, the Navier-Stokes equation is what's called critical. (00:26:04) The effect of transport and the effect of viscosity are about the same strength, even at very, very small scales. (00:26:09) And we have a lot of technology to handle critical and also subcritical equations and prove regularity. (00:26:15) But for supercritical equations, it was not clear what was going on. (00:26:18) And (00:26:20) 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 blob examples. (00:26:27) Once the non-linear effects dominate the linear effects at small scales, you can have all kinds of bad things happen. (00:26:32) 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. (00:26:41) I mean, that's a key qualitative feature that distinguishes (00:26:44) some equations were being sort of nice and predictable and planetary motion. (00:26:48) And I mean, there's certain equations that you can predict for millions of years or thousands at least. (00:26:54) 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. (00:27:01) Lots of really strange things are going on at very fine scales. (00:27:04) So whenever there is some huge source of non-linearity, (00:27:09) That can create a huge problem for predicting what's going to happen. (00:27:13) Yeah, and if non-linearity is somehow more and more featured and interesting at small scales. (00:27:19) I mean, there's many equations that are non-linear, but in many equations you can approximate things by the bulk. (00:27:24) So for example, planetary motion, you know, if you wanted to understand the orbit of the moon or Mars or something, you don't really need the microstructure of like the seismology of the moon or like exactly how the mass is distributed. (00:27:37) You just basically, you can always approximate these planets by point masters. (00:27:41) And just the aggregate behavior is important. (00:27:44) 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. (00:27:51) For supercritical equations, the finite scale confirmation is really important. (00:27:54) If we can just linger on the Navier-Stokes equations a little bit. (00:27:58) So you've suggested maybe you can describe it, that one of the ways to solve it (00:28:06) or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer, and then show that the halting problem from computation theory has consequences for fluid dynamics. (00:28:18) So show it in that way. (00:28:21) Can you describe this idea? (00:28:22) Right, yeah, so this came out of this work of constructing this average equation that blew up. (00:28:28) So one, (00:28:30) as part of how I had to do this. (00:28:32) So there's sort of this naive way to do it. (00:28:34) You just keep pushing. (00:28:37) Every time you get energy at one scale, you push it immediately to the next scale as fast as possible. (00:28:42) This is sort of the naive way to force blow up. (00:28:43) It turns out in five and high dimensions, this works. (00:28:47) But in three dimensions, there was this funny phenomenon that I discovered that if you keep, if you change the laws of physics, you just always keep trying to push. (00:28:58) the energy into smaller and smaller scales. (00:29:00) What happens is that the energy starts getting spread out into many scales at once. (00:29:05) 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. (00:29:16) You're trying to do everything at once. (00:29:18) And this spreads out the energy too much. (00:29:20) And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything. (00:29:27) So, it turns out this director doesn't actually work. (00:29:30) There was a separate paper by some of the authors that actually showed this in three dimensions. (00:29:36) So, what I needed was to program a delay. (00:29:40) So, kind of like air locks. (00:29:41) So, I needed an equation which would start with a fluid doing something at one scale. (00:29:47) It would push this energy into the next scale, but it would stay there until all the energy from the larger scale got transferred. (00:29:55) And only after you pushed all the energy in, then you sort of open the next gate and then you push that in as well. (00:30:01) 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. (00:30:08) And then it can resist the effects of viscosity because it's not dispersed. (00:30:13) So in order to make that happen, yeah, I had to construct a rather complicated non-linearity. (00:30:20) And it was basically like, (00:30:23) let's construct like an electronic circuit. (00:30:25) So I can thank my wife for this because she was trained as an electrical engineer. (00:30:29) And, she talked about... (00:30:33) he had to design circuits and so forth. (00:30:35) And if you want a circuit that does a certain thing, maybe have a light that flashes on and then turns off and then on and then off, you can build it from more primitive components, capacitors and resistors and so forth. (00:30:46) And you have to build a diagram. (00:30:47) And these diagrams, you can sort of follow up your eyeballs and say, oh, yeah, the current will brought up here and then it will stop and then it will do that. (00:30:56) So I knew how to build the analog of basic electronic components, like resistors and capacitors and so forth. (00:31:01) And (00:31:01) and I would stack them together in such a way that I would create something that would open one gate, and then there'll be a clock, and then once the clock hits the certain threshold, it would close it. (00:31:10) It would become a Rube Goldberg type machine, but just quite mathematically. (00:31:14) And this ended up working. (00:31:16) So what I realized is that if you could pull the same thing off for the actual equations, so if the equations of water supported computation, so (00:31:25) if you can imagine kind of a steampunk, but it's really water punk type of thing where, so modern computers are electronic, they're powered by electrons passing through very tiny wires and interacting with other electrons and so forth. (00:31:39) But instead of electrons, you can imagine these pulses of water moving at a certain velocity, and maybe it's, they're two different configurations corresponding to a bit being up or down. (00:31:50) probably that 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, that the output would depend in a very predictable way on the inputs. (00:32:04) And like you could chain these together and maybe create a Turing machine, and then you have computers which are made completely out of water. (00:32:13) And if you have computers, then maybe you can do robotics, you know, hydraulics and so forth. (00:32:19) And so you could create some machine, which is basically a fluid analog of what's called a von Neumann machine. (00:32:26) So von Neumann proposed, if you want to colonize Mars, the sheer cost of transporting people and machines to Mars is just ridiculous. (00:32:33) But if you could transport one machine to Mars, and this machine had the ability to mine the planet, create some more materials, smelt them, and build more copies of the same machine, then you could colonize the whole planet over time. (00:32:49) So if you could build a fluid machine, which, so it's a fluid robot. (00:32:56) And what it would do, it's purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state. (00:33:03) It wouldn't start just yet. (00:33:05) Once it's ready, the big robot configured water would transfer all its energy into the smaller configuration and then power down. (00:33:11) And then I clean itself up. (00:33:13) And then what's left is this newest state, which would then turn on and do the same thing, but smaller and faster. (00:33:19) And then the equation has a certain scaling symmetry. (00:33:21) Once you do that, it can just keep iterating. (00:33:23) So this in principle would create a blow up for the actual Navier-Stokes. (00:33:27) And this is what I managed to accomplish for this average Navier-Stokes. (00:33:30) So it provided this sort of roadmap to solve the problem. (00:33:33) Now, this is a pipe dream because there's so many things that are missing for this to actually be a reality. (00:33:41) So I can't create these basic logic gates. (00:33:45) I don't have these in these special configurations of water. (00:33:48) I mean, there's candidates that include vortex rings that might possibly work, but also, analog computing is really nasty compared to digital computing. (00:34:00) I mean, because there's always errors. (00:34:02) You have to do a lot of error correction along the way. (00:34:05) I don't know how to completely power down the big machine so that it doesn't interfere with the running of a smaller machine. (00:34:10) But everything in principle can happen, like it doesn't contradict any of the laws of physics. (00:34:16) So it's sort of evidence that this thing is possible. (00:34:20) There are other groups who are now pursuing ways to magnetic source flow up, which are nowhere near as ridiculously complicated as this. (00:34:29) 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. (00:34:40) There is a real leap of genius here to go from Navier-Stokes to this Turing machine. (00:34:46) So it goes from what the self-similar BLOB scenario that you're trying to get the smaller and smaller BLOB to now having a liquid Turing machine gets smaller and smaller and smaller and somehow seeing how that could be used to say something about a blow up. (00:35:07) I mean, that's a big leap. (00:35:08) So there's precedent. (00:35:09) 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. (00:35:19) But if the mathematical form is the same, you can draw a connection. (00:35:25) So there's a lot of previously on what is called cellular automata. (00:35:29) The most famous of which is Conway's Game of Life. (00:35:32) There's this infinite discrete grid. (00:35:34) and at any given time, the liquid is either occupied by a cell or it's empty. (00:35:37) And there's a very simple rule that tells you how these cells evolve. (00:35:40) So sometimes cells live and sometimes they die. (00:35:43) And this is, you know, when I was a student, it was a very popular screensaver to actually just have these. (00:35:49) these animations go on. (00:35:50) And they look very chaotic. (00:35:51) In fact, they look a little bit like turbulent flow sometimes. (00:35:54) But at some point, people discovered more and more interesting structures within this game of life. (00:35:58) So for example, they discovered this thing called a glider. (00:36:00) So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves at a certain direction. (00:36:06) And that's like this vortex rings. (00:36:09) Yeah, so this is an analogy. (00:36:10) The game of life is kind of like a discrete equation and (00:36:15) The fluid Navier-Stokes is a continuous equation, but mathematically they have some similar features. (00:36:21) And so over time, people discovered more and more interesting things that you could build within the Game of Life. (00:36:27) The Game of Life is a very simple system. (00:36:29) It only has like three or four rules to do it, but you can design all kinds of interesting configurations inside it. (00:36:34) There's something called a glider gun that does nothing to spit out gliders one at a time. (00:36:40) And then (00:36:41) After a lot of effort, people managed to create AND gates and OR gates for gliders. (00:36:48) Like 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. (00:36:58) If maybe if both of the streams have gliders, then there will be an output stream. (00:37:05) But if only one of them does, then nothing comes out. (00:37:08) So they could build something like that. (00:37:10) And once you could build these basic gates, then just from software engineering, you can build almost anything. (00:37:19) You can build a Turing machine. (00:37:20) I mean, it's like an enormous steampunk type things. (00:37:23) They look ridiculous. (00:37:25) But then people also generated self-replicating objects in the game of life. (00:37:29) A massive machine, a polynomial machine, which over a huge period of time and all these little glider guns inside doing these very steampunk calculations, it would create another version of itself. (00:37:39) which could replicate. (00:37:40) It's so incredible. (00:37:42) A lot of this was like community crowdsourced by like amateur mathematicians, actually. (00:37:46) So I knew about that work. (00:37:48) And so that is part of what inspired me to propose the same thing with Navier Stokes. (00:37:54) That was just a much, as I said, analog is much worse than digital. (00:37:57) Like it's going to be, you can't just directly take the constructions in the game of life and plunk them in. (00:38:03) But again, it just, it shows it's possible. (00:38:06) You know, there's a kind of (00:38:07) emergence that happens with these cellular automata, local rules, maybe it's similar to fluids, I don't know, but local rules operating at scale can create these incredibly complex dynamic structures. (00:38:25) Do you think any of that is amenable to mathematical analysis? (00:38:30) Do we have the tools to say something profound about that? (00:38:34) The thing is, you can get this emergent very complicated structures, but only with very carefully prepared initial conditions. (00:38:40) So these glider guns and gates and sort of machines, if you just plunk on randomly some cells and you're looking at that, you will not see any of these. (00:38:49) And that's the analogous situation with Navier-Stokes again, you know, that with typical initial conditions, you will not have any of this weird computation going on. (00:38:59) But (00:39:00) Basically, through engineering, by specially designing things in a very special way, you can make clever constructions. (00:39:07) I wonder if it's possible to prove the sort of the negative of like, basically prove that only through engineering can you ever create something interesting. (00:39:16) 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. (00:39:25) They look like random, like the digits of pi. (00:39:27) Well, we believe is a good example. (00:39:30) But there's a very small number of things that have patterns. (00:39:33) But now you can prove something as a pattern by just constructing, like if something has a simple pattern and you have a proof that it does something like repeat itself every so often, you can do that. (00:39:44) And you can prove that, for example, you can prove that most sequences of digits have no pattern. (00:39:49) So like if you just pick digits randomly, there's something called the low large numbers. (00:39:52) It tells you you're going to get as many ones as twos in the long run. (00:39:57) But (00:39:59) We have a lot few. (00:40:00) If you are tools to, 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? (00:40:07) Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured. (00:40:16) So some functions are what's called additive, like if you have a function of natural numbers, the natural numbers. (00:40:20) So maybe, you know, two maps to four, three maps to six, and so forth. (00:40:25) Some functions are what's called additive, which means that if you add (00:40:28) If you add 2 inputs together, the output gets added as well. (00:40:31) For example, I'm multiplying by a constant. (00:40:33) If you multiply a number by 10, if you multiply A + B by 10, that's the same as multiplying A by 10 and B by 10 and then adding them together. (00:40:41) So some functions are additive. (00:40:44) Some functions are kind of additive, but not completely additive. (00:40:47) 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. (00:40:54) So 10 by square root 2 is like 14 point something, so 10 up to 14. (00:40:59) 20 up to 28. (00:41:01) So in that case, additivity is true then. (00:41:04) So 10 plus 10 is 20 and 14 plus 14 is 28. (00:41:06) But because of this rounding, sometimes there's round-off 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 or minus 1. (00:41:17) So it's almost additive, but not quite additive. (00:41:21) So there's a lot of useful results in mathematics, and I've worked a lot on 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 explaining this sort of partial pattern that you have. (00:41:44) And so if you have these sort of inverse theorems, it creates this sort of dichotomy that either (00:41:49) the objects that you study are either have no structure at all, or they are somehow related to something that is structured. (00:41:56) And in either way, in either case, you can make progress. (00:42:01) A good example of this is that there's this old theorem in mathematics called Zemeredi's theorem, proven in the 1970s. (00:42:07) It concerns trying to find a certain type of pattern in a set of numbers, the patterns have progression. (00:42:11) Things like 3, 5, and 7, or 10, 15, and 20. (00:42:15) And Zemeredi, Andrei Zemeredi proved that (00:42:18) any set of numbers that are sufficiently big, what's called positive density, has arithmetic progressions in it of any length you wish. (00:42:28) So for example, the odd numbers have a set of density one-half, and they contain arithmetic progressions of any length. (00:42:34) So in that case, it's obvious because the odd numbers are really, really structured. (00:42:38) I can just take 11, 13, 15, 17. (00:42:41) I can easily find arithmetic progressions in that set. (00:42:46) But Zermanism also applies to random sets. (00:42:48) 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, because I just flip coins, I just randomly take out half the numbers, I keep one half. (00:43:01) So that's a set that has no patterns at all. (00:43:04) But just from random fluctuations, you will still get a lot of arithmetic progressions in that set. (00:43:10) Can you prove that? (00:43:13) There's arithmetic progressions of arbitrary length within a random. (00:43:17) Yes. (00:43:18) Have you heard of the infinite monkey theorem? (00:43:20) Usually mathematicians give boring names to theorems, but occasionally they give colorful names. (00:43:24) Yes. (00:43:24) The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out text randomly, almost surely one of them is going to generate the entire square of Hamlet or any other finite swing of text. (00:43:38) It will just take some time, quite a lot of time actually. (00:43:41) But if you have an infinite number, then it happens. (00:43:44) So basically the thing is that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge. (00:43:54) It may take a long time, but it will eventually happen. (00:43:57) In particular, arithmetic progressions of any length will eventually happen. (00:44:00) Okay, but you need an extremely long random sequence for this to happen. (00:44:04) I suppose that's intuitive. (00:44:06) It's just infinity. (00:44:08) Yeah, infinity absorbs a lot of sins. (00:44:11) Yeah. (00:44:11) How are we humans supposed to deal with infinity? (00:44:15) Well, you can think of infinity as an abstraction of a finite number for which you do not have a bountiful. (00:44:23) That, you know, I mean, so nothing in real life is truly infinite. (00:44:27) But, you know, you can, you know, you can ask these old questions like, what if I had as much money as I wanted? (00:44:33) Or what if I could go as fast as I wanted? (00:44:36) 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. (00:44:47) And often the mathematics becomes a lot cleaner when you do that. (00:44:50) I mean, in physics, we joke about assuming spherical cows. (00:44:55) You know, like reward problems have got all kinds of reward effects, but you can idealize, send some things to infinity, send some things to 0. (00:45:03) And the mathematics becomes a lot simpler. (00:45:05) to work with it. (00:45:06) I wonder how often using infinity forces us to deviate from the physics of reality. (00:45:16) Yeah, so there's a lot of pitfalls. (00:45:19) So, you know, we spend a lot of time in undergraduate math classes teaching analysis. (00:45:23) And analysis is often about how to take limits and whether you, you know, so for example, A + B is always B + A. (00:45:31) So when you have a finite number of terms, you add them, you can swap them and there's no problem. (00:45:35) But when you have an infinite number of terms, there are these sort of show games you can play where you can have a series which converges to 1 value, but you rearrange it and it suddenly converges to another value. (00:45:44) And so you can make mistakes. (00:45:46) You have to know what you're doing when you allow infinity. (00:45:49) You have to introduce these epsilons and deltas and there's a certain type of way of reasoning that helps you avoid mistakes. (00:45:58) In more recent years, people have started taking results that are (00:46:02) true in infinite limits and what's called finitizing them. (00:46:07) So you know that something's true eventually, but you don't know when, now give me a rate. (00:46:11) Okay, so it's just that 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 how long to come out? (00:46:20) And that's a more quantitative question. (00:46:22) And this is something that you can attack by purely finite methods, and you can use your finite intuition. (00:46:30) And in this case, it turns out to be exponential in the length of the text that you're trying to generate. (00:46:34) And so this is why you never see the monkeys create Hamlet. (00:46:39) You can maybe see them create a four-letter word, but nothing that big. (00:46:42) And so I personally find once you finitize an infinite statement, it does become much more intuitive and it's no longer so weird. (00:46:51) So even if you're working with infinity, it's good to finitize so that you can... (00:46:56) have some intuition. (00:46:57) Yeah, the downside is that the finitized groups are just much, much messier. (00:47:00) And so the infinite ones are found first, usually, like decades earlier, and then later on, people finitize them. (00:47:07) So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding, of seeing the world? (00:47:17) Maybe we can throw an engineering in there. (00:47:19) You mentioned your wife is an engineer, give it new perspective on circuits. (00:47:23) So there's a different way of looking at the world, given that you've done mathematical physics, so you've worn all the hats. (00:47:30) Right. (00:47:30) So I think science in general is an interaction between three things. (00:47:34) There's the real world. (00:47:37) There's what we observe of the real world, our observations, and then our mental models as to how we think the world works. (00:47:45) So (00:47:46) we can't directly access reality. (00:47:49) All we have are the observations, which are incomplete, and they have errors. (00:47:54) And there are many, many cases where we want to know, for example, what is the weather like tomorrow? (00:48:00) And we don't yet have the observation and we'd like to make a prediction. (00:48:03) And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. (00:48:09) Those are the mathematical models. (00:48:11) Mathematics is concerned with the models. (00:48:13) Science collects the observations. (00:48:16) and it proposes the models that might explain these observations. (00:48:20) What mathematics does is we stay within the model and ask what are the consequences of that model? (00:48:25) What observations, what predictions would the model make of future observations or past observations? (00:48:32) Does it fit observed data? (00:48:35) So there's definitely a symbiosis. (00:48:38) It's (00:48:40) I guess mathematics is unusual among other disciplines is that we start from hypotheses, like the axioms of a model, and ask what conclusions come up from that model. (00:48:50) In almost any other discipline, you start with the conclusions, you know, I want to do this, I want to build a bridge, you know, I want to make money, I want to do this, okay? (00:48:58) And then you find the path to get there. (00:49:04) There's a lot less sort of speculation about, suppose I did this, what would happen? (00:49:09) planning and modeling. (00:49:12) Speculative fiction maybe is one other place, but that's about it, actually. (00:49:17) Most of the things we do in life is conclusions driven, including physics and science. (00:49:20) I mean, they want to know, where is this asteroid going to go, or what is the weather going to be tomorrow? (00:49:27) But also has this other direction of going from the axioms. (00:49:32) What do you think? (00:49:33) There is this tension in physics between theory and experiment. (00:49:37) What do you think is a more powerful way of discovering truly novel ideas about reality? (00:49:42) Well, you need both, top down and bottom up. (00:49:45) Yeah, it's a real interaction between all these things. (00:49:47) So over time, the observations and the theory and the modeling should both get closer to reality. (00:49:53) But initially, and I mean, this is always the case, they're always far apart to begin with. (00:50:01) But you need one to figure out where to push the other. (00:50:04) So if your model is predicting anomalies that are not picked up by experiment, that tells experimenters where to look, to find more data, to refine the models. (00:50:17) So it goes back and forth. (00:50:21) Within Mathematica itself, there's also a theory and experimental component. (00:50:24) It's just that until very recently, theory has dominated (00:50:28) almost completely, like 99% of mathematics is theoretical mathematics. (00:50:32) And there's a very tiny amount of experimental mathematics. (00:50:34) I mean, people do do it, you know, like if they want to study prime numbers or whatever, they can just generate large data sets. (00:50:42) So once we had computers, we began to do it a little bit. (00:50:46) Although even before, well, like Gauss, for example, he discovered, he conjectured the most basic theorem in number theory to call the prime number theorem, which predicts how many primes, up to a million, up to a trillion. (00:50:56) It's not an obvious question. (00:50:58) And basically what he did was that he computed, I mean, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic, to compute the 1st 100,000 frames or something and made tables and made a prediction. (00:51:15) That was an early example of experimental mathematics. (00:51:18) But until very recently, it was not, I mean, theoretical mathematics was just much more successful. (00:51:24) I mean, because (00:51:25) Doing complicated mathematical computations was just not feasible until very recently. (00:51:31) And even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically. (00:51:37) There's something called the combinatorial explosion. (00:51:39) If you want to study, for example, Zermedys theorem, you want to study all possible subsets of the numbers 1 to 1,000. (00:51:45) There's only 1,000 numbers. (00:51:46) How bad could it be? (00:51:46) It turns out the number of different subsets of 1 to 1,000 is 2 to the power of 1,000. (00:51:51) which is way bigger than any computer can currently can enumerate, any computer ever, or ever can enumerate. (00:51:59) So you have to be, there are certain math problems that very quickly become just intractable to attack by direct brute force computation. (00:52:08) Chess is another famous example. (00:52:10) The number of chess positions we can't get a computer to fully explore. (00:52:16) But now we have AI. (00:52:18) We have tools to explore this space, not with 100% guarantees of success, but with experiment. (00:52:25) So like we can empirically solve chess now. (00:52:29) For example, we have very, very good AIs that can, they don't explore every single position in the game tree, but they have found some very good approximation. (00:52:38) And people are using, actually, these chess engines to make, to do experimental chess. (00:52:44) That they're revisiting old chess theories about, oh, you know, when you do this type of opening, this is a good type of move, this is not. (00:52:50) And they can use these chess engines to actually refine, in some cases, overturn conventional wisdom about chess. (00:52:58) And I do hope that mathematics will have a larger experimental component in the future, perhaps powered by AI. (00:53:05) We'll, of course, talk about that. (00:53:06) But in the case of chess, (00:53:08) And there's a similar thing in mathematics. (00:53:10) I don't believe it's providing a kind of formal explanation of the different positions. (00:53:17) It's just saying which position is better or not that you can intuit as a human being. (00:53:21) And then from that, we humans can construct a theory of the matter. (00:53:27) You've mentioned the Plato's cave allegory. (00:53:30) So (00:53:31) in case people don't know, it's where people are observing shadows of reality, not reality itself, and they believe what they're observing to be reality. (00:53:42) Is that in some sense what mathematicians and maybe all humans are doing is looking at shadows of reality? (00:53:51) Is it possible for us to truly access (00:53:54) reality? (00:53:55) Well, there are these three ontological things. (00:53:58) There's actual reality, there's observations, and our models. (00:54:04) And technically, they are distinct, and I think they will always be distinct. (00:54:08) But they can get closer over time. (00:54:14) So, and the process of getting closer often means that you have to discard your initial intuitions. (00:54:21) So, (00:54:23) Astronomy provides great examples, like an initial model of the world is that it's flat because it looks flat, and that it's and it's big, and the rest of the universe, the skies is not, like the sun, for example, looks really tiny. (00:54:38) And so you start off with a model which is actually really far from reality, but it fits kind of the observations that you have. (00:54:45) so things look good, but over time, as you make more and more observations, bring it closer to reality, the model gets dragged along with it. (00:54:52) And so over time, we had to realize that the Earth was round, that it spins, it goes around the solar system, the solar system goes on the galaxy, and so on and so forth. (00:55:00) And the universe seems expanding. (00:55:03) Expansions are still expanding, accelerating. (00:55:05) And in fact, very recently in this year, even the expression of the universe itself is evidence that is non-constant. (00:55:12) And the explanation behind why that is... (00:55:16) It's catching up. (00:55:18) It's catching up. (00:55:18) I mean, it's still, you know, the dark matter, dark energy, this kind of thing. (00:55:22) We have a model that sort of explains, that fits the data really well. (00:55:25) It just has a few parameters that you have to specify. (00:55:30) But so people say that's fudge factors, with enough fudge factors, you can explain anything. (00:55:35) But the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set. (00:55:43) So if you have a model with 10 parameters that explains 10 observations, that is a completely useless model. (00:55:48) It's what's called overfitted. (00:55:50) But like if you have a model with, you know, (00:55:52) Two parameters and it explains a trillion observations, which is basically, so the dark matter model, I think it has like 14 parameters and it explains petabytes of data that the astronomers have. (00:56:06) You can think of a theory, like one way to think about a physical methodical theory is it's a compression of the universe. (00:56:14) and data compression. (00:56:16) So, you have these petabytes of observations, you'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters, and if it can fit to reasonable accuracy, you know, almost all of your observations. (00:56:29) I mean, the more compression that you make, the better your theory. (00:56:32) In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all. (00:56:38) It's the unreasonable effectiveness of mathematics. (00:56:40) Yeah, Einstein had a quote like that, the most incomprehensible thing about the universe is that it is comprehensible. (00:56:45) Right, and not just comprehensible. (00:56:47) You can do an equation like E equals MC squared. (00:56:49) There is actually some mathematical possible explanation for that. (00:56:54) So there's this phenomenon in mathematics called universality. (00:56:57) So many complex systems at the macro scale are coming out of lots of tiny new interactions at the macro scale. (00:57:02) And normally because of the common form of explosion, you would think that the macro scale equations must be infinitely, exponentially more complicated than the macro scale ones. (00:57:12) And they are, if you want to solve them completely exactly. (00:57:15) Like if you want to model (00:57:18) all the atoms in a box of air. (00:57:20) That's like Avogadro's number is humongous. (00:57:23) There's a huge number of particles. (00:57:24) If you actually have to track each one, it'll be ridiculous. (00:57:27) But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the microscale, or only depend on a very small number of parameters. (00:57:35) So if you want to model a gas of, you know, (00:57:39) frontilian particles in a box. (00:57:41) You just need to know its temperature and pressure and volume and a few parameters, like 506, and it models almost everything you need to know about these 10 to 23 or whatever particles. (00:57:53) So we have (00:57:56) We don't understand universality anywhere new as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. (00:58:06) Most basic one is the central limit theorem. (00:58:09) That explains why the bell curve shows up everywhere in nature. (00:58:11) But so many things are distributed by what's called a Gaussian distribution, famous bell curve. (00:58:16) There's now even a meme with this curve. (00:58:18) And even the meme applies broadly. (00:58:20) The universality to the meme. (00:58:22) Yes, you can go meta if you like, but there are many, many (00:58:26) processes, for example, you can take lots and lots of independent random variables and average them together in various ways. (00:58:33) You can take a simple average or more complicated average, and we can prove in various cases that these bell curves, these Gaussians emerge. (00:58:40) And it is a satisfying explanation. (00:58:44) Sometimes they don't. (00:58:45) So if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from the bell curve show up. (00:58:51) And this is also important to know whether this element fails. (00:58:55) So universality is not a 100% reliable thing to rely on. (00:59:01) The global financial crisis was a famous example of this. (00:59:05) People thought that mortgage defaults had this sort of Gaussian type behavior that if you ask if a population of, you know, 100,000 Americans with mortgages, ask what proportion of them would default on their mortgages. (00:59:21) If everything was decorrelated, it could be an asset bill curve and you can manage risk with options and derivatives and so forth. (00:59:27) And it is a very beautiful theory. (00:59:30) But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non-Gaussian behavior. (00:59:39) And this wasn't fully accounted for in 2008. (00:59:44) Now I think there's some more awareness that this is a systemic risk is actually a much bigger issue. (00:59:49) And (00:59:50) Just because the model is pretty and nice, it may not match reality. (00:59:55) So the mathematics of working out what models do is really important, but also the science of validating when the models fit reality and when they don't. (01:00:06) I mean, you need both. (01:00:09) But mathematics can help because it can, for example, these central limit theorems, it tells you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other, (01:00:19) then you have these Gaussian behaviors, things are fine. (01:00:22) It tells you where to look for weaknesses in the model. (01:00:25) So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copulas or whatever to model default risk, if you're mathematically trained, you would say, okay, but what are the systemic correlation between all your inputs? (01:00:41) And so then you can ask the economists, you know, how much of a risk is that? (01:00:46) And then you can go look for that. (01:00:48) So (01:00:49) There's always this synergy between science and mathematics. (01:00:52) A little bit on the topic of universality. (01:00:56) You're known and celebrated for working across an incredible breadth of mathematics, reminiscent of Hilbert a century ago. (01:01:03) In fact, the great Fields Medal-winning mathematician Tim Gowers has said that you are (01:01:10) the closest thing we get to Hilbert. (01:01:14) He's a colleague of yours. (01:01:15) Oh yeah, good friend. (01:01:16) But anyway, so you are known for this ability to go both deep and broad in mathematics. (01:01:22) So you're the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics? (01:01:30) Is there a kind of deep underlying structure to all of mathematics? (01:01:36) There's certainly a lot of connecting threads and (01:01:39) a lot of the progress of mathematics can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections. (01:01:50) An ancient example is geometry and number theory. (01:01:54) So in the times of ancient Greeks, these were considered different subjects. (01:01:58) I mean, mathematicians worked on both. (01:02:00) You know, Euclid worked both on geometry, most famously, but also on numbers. (01:02:06) But they were not really considered related. (01:02:10) I mean, a little bit like, you could say that this length was five times this length because you could take 5 copies of this length and so forth. (01:02:16) But it wasn't until Descartes who really realized that to develop analytic geometry, that you can parameterize the plane, a geometric object, by two real numbers. (01:02:26) Every point can be, and so geometric problems can be turned into problems about numbers. (01:02:33) And today, this feels almost (01:02:37) trivial. (01:02:37) Like, there's no content to this. (01:02:39) Like, of course, a plane is XX and Y, because that's what we teach and it's internalized. (01:02:46) But it was an important development that these two fields were unified. (01:02:52) And this process has just gone on throughout mathematics over and over again. (01:02:56) Algebra and geometry were separated and now we have a spirit algebraic geometry that connects them and over and over again. (01:03:01) And that's certainly the type of mathematics that I enjoy the most. (01:03:05) So I think there's sort of different styles to being a mathematician. (01:03:08) I think hedgehogs and fox, a fox knows many things a little bit, but a hedgehog knows one thing very, very well. (01:03:14) And in mathematics, there's definitely both hedgehogs and foxes. (01:03:17) And then there's people who are kind of, who can play both roles. (01:03:22) And I think ideal collaboration between mathematicians involves a very, you need some diversity, like a fox working with many hedgehogs or vice versa. (01:03:32) So, but I identify mostly as a fox, certainly. (01:03:36) I like arbitrage somehow, like learning how one field works, learning the tricks of that wheel, and then going to another field, which people don't think it is related, but I can adapt the tricks. (01:03:49) So see the connections between the fields. (01:03:52) Yeah. (01:03:52) So there are other mathematicians who are far deeper than I am. (01:03:55) They're really hedgehogs. (01:03:57) They know everything about one field, and they're much faster. (01:04:01) and more effective in that field, but I can give them these extra tools. (01:04:05) I mean, you said that you can be both the hedgehog and the fox, depending on the context, depending on the collaboration. (01:04:11) So what, can you, if it's at all possible, speak to the difference between those two ways of thinking about a problem? (01:04:18) Say you're encountering a new problem, you know, searching for the connections versus like very singular focus. (01:04:26) I'm much more comfortable with the (01:04:30) The Fox paradigm, yeah. (01:04:31) So, I like looking for analogies, narratives. (01:04:37) I spend a lot of time, if there's a result, I see it in one field, and I like the result, it's a cool result, but I don't like the proof. (01:04:44) It uses types of mathematics that I'm not super familiar with. (01:04:48) I often try to reprove it myself using the tools that I favor. (01:04:53) Often my proof is worse, but by the exercise of doing so, (01:04:58) I can say, now I can see what the other proof was trying to do. (01:05:02) And from that, I can get some understanding of the tools that are used in that field. (01:05:07) So it's very exploratory, very doing crazy things in crazy fields and like reinventing the wheel a lot. (01:05:14) Whereas the hedgehog style is, I think, much more scholarly. (01:05:18) You know, you're very knowledge-based. (01:05:20) You stay up to speed on like all the developments in this field. (01:05:23) You know all the history. (01:05:25) You have a very good understanding of exactly the strengths and weaknesses of each particular technique. (01:05:32) Yeah, I think you'd rely a lot more on sort of calculation than sort of trying to find narratives. (01:05:39) So yeah, I mean, I could do that too, but there are other people who are extremely good at that. (01:05:44) Let's step back and maybe look at a bit of a romanticized version of mathematics. (01:05:53) So (01:05:54) I think you've said that early on in your life, math was more like a puzzle-solving activity when you were young. (01:06:03) When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it? (01:06:14) That's a good question. (01:06:16) When I came to graduate school in Princeton, so John Conway was there at the time. (01:06:20) He passed away a few years ago. (01:06:22) But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. (01:06:28) So Conway just had this amazing way of thinking about all kinds of things in a way that you would normally think of. (01:06:33) So he thought of proofs themselves as occupying some sort of space. (01:06:38) So if you want to prove (01:06:41) something, let's say that there's infinitely many primes. (01:06:42) Okay, you have all different proofs, but you could rank them in different axes, like some proofs are elegant, some proofs are long, some proofs are elementary and so forth. (01:06:51) And so this is cloud, so the space of all proofs itself has some sort of shape. (01:06:57) And so he was interested in extreme points of this shape, like out of all these proofs, what's one of those, the shortest at the extent of everything else, or the most elementary, or whatever? (01:07:09) And so he gave some examples of well-known theorems, and then he would give what he thought was the extreme proof in these different aspects. (01:07:19) I just found that really eye-opening, that, you know, it's not just getting a proof for what was interesting, but (01:07:27) But once you have that proof, trying to optimize it in various ways, that proofing itself had some craftsmanship to it. (01:07:40) It's something for my writing style that, like when you do your math assignments and as an undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works, okay? (01:07:50) And then hand it in, as long as it gets a tick mark, you move on. (01:07:55) But if you want your (01:07:56) your results to actually be influential and be read by people. (01:08:00) It can't just be correct. (01:08:01) It should also be a pleasure to read, you know, motivated, be adaptable to generalize to other things. (01:08:10) It's the same in many other disciplines, like coding. (01:08:12) There's a lot of analogies between math and coding. (01:08:15) I like analogies if you haven't noticed. (01:08:18) But, you can code something spaghetti code that works for a certain task and it's quick and dirty and it works. (01:08:25) But there's lots of good principles for writing code well so that other people can use it, build upon it, and so then has fewer bugs and whatever. (01:08:34) And there's similar things with mathematics. (01:08:37) So. (01:08:37) Yeah, first of all, there's so many beautiful things there, and Kamal is one of the great minds. (01:08:44) in mathematics ever and computer science. (01:08:47) Just even considering the space of proofs and saying, okay, what does this space look like? (01:08:53) And what are the extremes? (01:08:56) Like you mentioned, coding is an analogy is interesting because there's also this activity called the code golf, which I also find beautiful and fun. (01:09:06) where people use different programming languages to try to write the shortest possible program that accomplishes a particular task. (01:09:12) And I believe there's even competitions on this. (01:09:15) And it's also a nice way to stress test not just the... (01:09:23) sort of the programs, or in this case, the proofs, but also the different languages. (01:09:27) Maybe that's a different notation or whatever to use to accomplish a different task. (01:09:31) Yeah, you learn a lot. (01:09:32) I mean, it may seem like a frivolous exercise, but it can generate all these insights, which if you didn't have this artificial objective to pursue, you might not see. (01:09:43) What to use the most beautiful or elegant equation in mathematics? (01:09:48) I mean, one of the things that people often look to in beauty (01:09:52) is the simplicity. (01:09:53) So if you look at E equals MC squared. (01:09:56) So when a few concepts come together, that's why the Euler identity is often considered the most beautiful equation in mathematics. (01:10:05) Do you find beauty in that one, in the Euler identity? (01:10:08) Yeah, well, as I said, I mean, what I find most appealing is connections between different things that you did. (01:10:14) So if you e to the pi i equals -1. (01:10:17) So, people use all the fundamental constants. (01:10:20) OK, that's I mean, that's cute, but to me, so the exponential function was to measure exponential growth, compound interest or decay, anything which is continuously growing, continuously decreasing growth and decay or dilation or contraction is modelled by the exponential function. (01:10:39) Whereas pi comes around from circles and rotation. (01:10:42) If you want to rotate a needle, for example, 100 degrees, you need to rotate by pi radians. (01:10:47) And I, complex numbers, represents the swapping between your imaginary axis of a 90 degree rotation, so a change in direction. (01:10:53) So the exponential function represents growth and decay in the direction that you already are. (01:11:00) When you stick an I in the exponential, now it's instead of motion in the same direction as your current position, it's the motion as a right angles to your current position, so rotation. (01:11:11) And then so e to the pi equals minus 1 tells you that if you rotate for time pi, you end up at the other direction. (01:11:17) So it unifies geometry through dilation and exponential growth, dynamics, through this act of complexification, rotation by eye. (01:11:25) So it connects together all these tools, mathematics, dynamics, geometry and complex and complex and the complex numbers, they're all (01:11:33) considered almost, they're all next-door neighbors in mathematics because of his identity. (01:11:37) Do you think the thing you mentioned is cute, the collision of notations from these disparate fields, it's just a frivolous side effect? (01:11:47) Or do you think there is legitimate value in when the notation, all our old friends come together at night? (01:11:54) Right. (01:11:54) Well, it's confirmation that you have the right concepts. (01:11:58) So when you first study anything, (01:12:01) you have to measure things and give them names. (01:12:04) And initially, sometimes, because your model is, again, too far off from reality, you give the wrong things the best names. (01:12:11) And you only find out later what's really important. (01:12:14) Physicists can do this sometimes. (01:12:16) I mean, but it turns out okay. (01:12:18) So actually, with physics, so equals mc squared, okay? (01:12:20) So one of the big things was the E, right? (01:12:23) So when Aristotle first came up with his laws of motion and then Galileo and Newton and so forth, (01:12:31) they saw the things they could measure. (01:12:32) They could measure mass and acceleration and force and so forth. (01:12:35) And so Newtonian mechanics, for example, was the famous Newton's second law of motion. (01:12:40) So those were the primary objects. (01:12:41) So they gave them the central billing in the theory. (01:12:44) It was only later, after people started analyzing these equations, that there always seemed to be these quantities that were conserved. (01:12:51) So in particular, momentum and energy. (01:12:55) And it's not obvious that things happen in energy. (01:12:58) It's not something you can directly measure the same way you can measure mass and velocity, so both. (01:13:02) But over time, people realized that this was actually a really fundamental concept. (01:13:05) Hamilton, eventually in 19th century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object. (01:13:15) Once you know how to measure the Hamiltonian of any system, you can describe completely the dynamics, like what happens to all the states. (01:13:22) It really was a central actor, which was not obvious initially. (01:13:28) And this helped, actually, this change of perspective really helped when quantum mechanics came along. (01:13:34) Because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt their Newtonian thinking because everything was a particle and so forth to quantum mechanics, because everything because it was a wave, it just looked really, really weird. (01:13:50) Like you ask, what is the quantum version of F equals MA? (01:13:53) And it's really, really hard to give an answer to that. (01:13:57) But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics, that there's also an object called a Hamiltonian. (01:14:09) It's a different type of object. (01:14:10) It's what's called an operator rather than a function. (01:14:14) But again, once you specify it, you specify the entire dynamics. (01:14:17) So there's an uncle showing this equation that tells you exactly how quantum systems evolve once you have a Hamiltonian. (01:14:23) So side by side, they look completely different objects. (01:14:26) One involves particles, one involves waves, and so forth. (01:14:30) But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. (01:14:36) So for example, in classical mechanics, there's this thing called Noether's theorem. (01:14:40) Every time there's a symmetry in a physical system, there is a conservation law. (01:14:44) So the laws of physics are translation invariant. (01:14:46) Like if I move 10 steps to the left, I experience the same laws of physics as I was here. (01:14:50) and that corresponds to conservation of momentum. (01:14:54) If I turn around by some angle, again, I experience the same laws of physics. (01:14:57) This corresponds to conservation of angular momentum. (01:15:00) If I wait for 10 minutes, I still have the same laws of physics. (01:15:04) So this time transition invariance, this corresponds to the law of conservation of energy. (01:15:08) So there's this fundamental connection between symmetry and conservation. (01:15:12) And that's also true in quantum mechanics, even though the equations are completely different, but because they're both coming from the Hamiltonian, the Hamiltonian controls everything. (01:15:20) Every time the Hamiltonian has a symmetry, the equations will have a conservation law. (01:15:25) So it's once you have the right language, it actually makes things a lot cleaner. (01:15:33) One of the problems why we can't unify quantum mechanics and general relativity yet, we haven't figured out what the fundamental objects are. (01:15:38) Like, for example, we have to give up the notion of space and time being these almost Euclidean type spaces. (01:15:43) And it has to be, you know, and you know, we kind of know that at very tiny scales, there's going to be quantum fluctuations, there's a space-time foam. (01:15:52) And trying to use Cartesian cords XYZ is going to be, it's just, it's a non-starter. (01:15:58) But we don't know how to, what to replace it with. (01:16:02) We don't actually have the mathematical concepts. (01:16:06) The analog of the Hamiltonian that sort of organized everything. (01:16:09) Does your gut say that there is a theory of everything, so this is even possible to unify, to find this language that unifies general relativity and quantum mechanics? (01:16:19) I believe so. (01:16:20) I mean, the history of physics has been that of unification, much like mathematics over the years. (01:16:25) You know, electricity and magnetism were separate. (01:16:27) theories and then Maxwell unified them. (01:16:29) Newton unified the motions of heavens for the motions of objects on the earth and so forth. (01:16:34) So it should happen. (01:16:36) It's just that the, again, to go back to this model of the observations and theory, part of our problem is that physics is a victim of its own success. (01:16:44) That our two big theories of physics, general relativity and quantum mechanics, are so good now. (01:16:51) So together, they cover 99.9% of sort of all the observations we can make. (01:16:56) And you have to either go to extremely insane particle accelerations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories to the point where you can actually figure out how to combine them together. (01:17:12) But I have faith that we've been doing this for centuries. (01:17:16) We've made progress before. (01:17:17) There's no reason why we should stop. (01:17:18) Do you think you will be a mathematician that develops a theory of everything? (01:17:24) What often happens is that when the physicists need something about mathematics, there's often some precursor that the mathematicians worked out earlier. (01:17:35) So when Einstein started realizing that space was curved, he went to some mathematician and asked, is there some theory of curved space that the mathematicians already came up with that could be useful? (01:17:45) And he said, oh yeah, I think Riemann came up with something. (01:17:49) And so, Riemann had developed Riemannian geometry, which is precisely, a theory of spaces that are curved in various general ways, which turned out to be almost exactly what was needed by Einstein's theory. (01:18:01) This is going back to Dubing this unreasonable effectiveness on mathematics. (01:18:04) I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems. (01:18:12) Ultimately, they're just sort of both ways of organizing data in useful ways. (01:18:17) It just feels like you might need to go some weird land that's very hard to intuit. (01:18:23) Like, you have like string theory. (01:18:25) Yeah, that was a leading candidate for many decades. (01:18:28) I think it's slowly falling out of fashion because it's not matching experiment. (01:18:33) So one of the big challenges, of course, like you said, is experiment is very tough. (01:18:38) Yes. (01:18:38) Because of how effective both theories are. (01:18:42) But the other is like, just, you know, you're talking about (01:18:47) you're not just deviating from space-time. (01:18:49) You're going into like some crazy number of dimensions. (01:18:52) You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started at, like you mentioned. (01:18:59) Yeah, And now we're just, it's very hard to use our limited ape descendants of cognition to intuit what that reality really is like. (01:19:10) This is why analogies are so important, you know. (01:19:12) I mean, so yeah, the round earth is not intuitive. (01:19:15) because we're stuck on it. (01:19:17) But, round objects in general, we have pretty good intuition a little bit. (01:19:21) And we have intuition about light works and so forth. (01:19:23) And it's actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth can be really easily explained by round earth and round moon, you know, and models. (01:19:36) And you can just take, you know, a basketball and a golf ball and a light source and actually do these things yourself. (01:19:43) So the intuition is there. (01:19:46) But you have to transfer it. (01:19:47) That is a big leap intellectually for us to go from flat to round earth because our life is mostly lived in flat land. (01:19:55) Yeah. (01:19:55) To load that information and we're all like take it for granted. (01:19:58) We take so many things for granted because...