Theories of Everything with Curt Jaimungal - Brand New Result Proving Penrose & Tao's Uncomputability in Physics!

Episode Date: July 7, 2025

As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe Mathematician Eva Miranda returns with a groundbreaking new result: a real physical system (fluid motion) has been proven to be Turing-complete. This means some fluid paths are logically undecidable. In this mind-bending episode, she walks us through the implications for chaos theory, the Navier-Stokes equations, and uncomputability in nature, confirming long-held suspicions of thinkers like Roger Penrose and Terence Tao. Featuring rubber ducks, Alan Turing, and the limits of knowledge itself. Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Timestamps: 00:00 Introduction 01:10 Expect the Unexpected 02:52 Stories of Uncertainty 04:45 The Impact of Alan Turing 06:35 The Halting Problem Explained 09:29 Limits of Mathematical Knowledge 12:40 From Certainty to Uncertainty 16:19 The Rubber Duck Phenomenon 19:29 Unpredictability vs. Undecidability 20:18 Classical Chaos and the Butterfly Effect 27:12 Asteroids and Chaos Theory 34:32 The Navier-Stokes Riddle 41:18 The Cantor Set and Computation 46:18 Bridging Discrete and Continuous 49:21 Turing Completeness in Fluid Dynamics 1:02:39 The Quest for Navier-Stokes Solutions 1:06:53 The Role of Viscosity 1:12:09 Hybrid Computers and Fluid Dynamics 1:26:57 Unpredictability in Deterministic Systems 1:31:44 The Future of Computational Models Links Mentioned: •⁠ ⁠Eva’s First Appearance [TOE]: https://youtu.be/6XyMepn-AZo •⁠ ⁠Moby Duck [Book]: https://amzn.to/4ldoYsZ •⁠ ⁠Roger Penrose [TOE]: https://youtu.be/sGm505TFMbU •⁠ ⁠The Emperor’s New Mind [Book]: https://amzn.to/44jHpGK •⁠ ⁠Edward Frenkel [TOE]: https://youtu.be/RX1tZv_Nv4Y •⁠ ⁠Richard Borcherds [TOE]: https://youtu.be/U3pQWkE2KqM •⁠ ⁠Clay Mathematics Institute: https://www.claymath.org/ •⁠ ⁠Eva’s Papers: https://scholar.google.com/citations?user=werIoRQAAAAJ&hl=en •⁠ ⁠Topological Kleene Field Theories [Paper]: https://arxiv.org/pdf/2503.16100 •⁠ ⁠Ted Jacobson [TOE]: https://youtu.be/3mhctWlXyV8 •⁠ ⁠Stephen Wolfram [TOE]: https://youtu.be/0YRlQQw0d-4 SUPPORT: - Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Support me on Patreon: https://patreon.com/curtjaimungal - Support me on Crypto: https://commerce.coinbase.com/checkout/de803625-87d3-4300-ab6d-85d4258834a9 - Support me on PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 SOCIALS: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs #science Learn more about your ad choices. Visit megaphone.fm/adchoices

Professor Eva Miranda, I'm extremely excited to be here and be speaking with you again. The last time we spoke it went viral, so I'm super excited to have you on again because the audience just loves you. We're going to talk about hot topics, what you're presenting here. You're presenting for the first time in a manner that's introductory, so requires no background. The topics will include complexity, chaos theory, especially as contrasted with the standard chaos theory that the audience may already be

acquainted with, Navier-Stokes, of course, what it means to go beyond what's computational, and how all of this is connected to geometry, to physics, to the ideas of Penrose and Terry Tao. Welcome. Yes, thank you very much. I'm excited to be here again. I'm so happy to be here. Looking forward to this new adventure and ready to disclose something new.

Let's see if people like it. I'm very happy about all the followers, all the questions. I'm sorry I couldn't answer all the questions. I'll go and answer them little by little as I can. Great. It's a great pleasure to be here with all of you now. Okay. I call this expect the unexpected.

What does it mean? Well, you know, we all know David Hilbert, the famous mathematician, right? Who said, we must know, we will know. This was, let's say, his most famous sentence. And indeed, this was a little bit the idea of his idea that everything could be formalized mathematically in a very precise way. This idea of precision of mathematics that is of course very important and formalization, the idea of formalization of mathematics. What's very interesting is that Hilbert was in Göttingen.

I have been in Göttingen very recently and it was a pleasure to be there and to walk around the streets and to see the plaques where all these great mathematicians were living. There was David Hilbert, there was John Boyd Neumann, Robert Oppenheimer, who appeared in my last appearance in the theories of everything. And there was also Alonso Church, because Church and von Neumann went to visit Hilbert because they were excited about this formalization of mathematics. And of course, in their work, Hilbert's spaces were very, very important. But well, I put here a rubber duck.

We'll see what this rubber duck has to do with these big names. I want to tell you indeed today three different stories that have a common pattern. And the pattern is uncertainty. Uncertainty versus the certainty that David Hilbert was looking inside mathematics, that every question had to have an answer in a way.

I chose three different main characters today. I chose Alan Turing. I chose to explain, as you said, the chaos, the theory of chaos, the classical theory of chaos, and this is a nice story that can explain with butterflies. I want to explain also how it affects our last month. We heard so much about all these asteroids that could fall on us. I want to explain how this is connected. And finally, I want to connect this to Navid's talks to this unsolved problem in the list of unsolved problems in mathematics. And I will connect this to undecidable fluid paths. As you see here, I chose a little rubber duck and the rubber duck wants to be a little bit the

the tack, the new tack of logical chaos in the same way the butterfly is the tack for the The tack for classical chaos So interesting Let's see. How do I get to Alan Turing? I was talking about Alonzo Chart, right? And I was saying that Alonso Chart was visiting David Hilbert to learn about this formalization of mathematics. And he was there in the period 1927 and 28. And the funny thing is that Chart was indeed the PhD advisor of Alan Turing.

So well, Alan Turing is well known for cracking Enigma code. We've seen him in the films, the imitation game, and here we have the machine Enigma and the machine, the bomb, which was used to crack Enigma. So we could say that indeed Alan Turing and his team, it was all the team of Bledley Park, managed to solve one problem that looked impossible to solve. However, there is a problem that Turing found impossible. He was able to crack the enigma code with his team, but he found impossible the following problem, which is the halting problem. The halting problem looks a little bit strange.

I'm going to try to explain it in easy words. Imagine that you have a process that is happening everywhere on the world. Now we could say that this process was just switching on your computer, right? Putting some input and getting some output. But at this moment in which I'm talking there were computers that did not exist. So this could be an algorithm, something that is non-repetitive. Now imagine you entering a labyrinth and trying to get out of the labyrinth, right? So the input is the person entering a certain labyrinth,

okay, a certain maze, and the process is getting out the maze. Okay? And the question is the following. Is there a general recipe, an algorithm, that tells us whether an arbitrary given process with some specific input data will stop, which means will reach the holding state or continue to run forever? In the case of the labyrinth, is there a way to know if a certain person entering a labyrinth will be able to find out the way out in a certain amount of time or not. Or in a way, if we think of modern computers, is there a supercomputer that can tell us somewhere on the world will ever stop, which reached the holding estate or will continue to run forever.

So this is the question that had been interesting a lot of people working in logics at the beginning of the 20th century. And it was a problem that was not solved. It was solved in 1936 and it was solved by Alan Turing, who proved that the holding problem is indeed undecidable. Okay, so Turing was the one in 1936, was the one to prove that the holding problem is undecidable.

And you think undecidable, that's a very strange word. It means that it cannot be decided. So this is a question that doesn't have a yes-no answer. This is pretty surprising, right? So this contradicts a little bit this strict way of thinking of mathematics, as you know, a door where you can knock and there's going to be an answer, right? To be clear, for an undecidable problem, does the answer of yes or no exist, but we just don't have access to it?

We can't know that it's going to be yes or no in a finite time? We can't know. We can't know. It's like we have, you know, it's like we are with our mobile phone, right? And we are boarding an airplane and the airplane doesn't have, we don't have access to the connection, right? So that's exactly the situation. We cannot know, there is no logical way

to know the answer to this question. Okay, but the answer does exist. Well, we don't know if the answer is... No, indeed mathematically we cannot say that the answer exists. We say that the answer is not... it's impossible to know. So, let's say, look, the question is very strange. The existence of a supercomputer that tells you if a given computer on the world with some initial data will ever stop or not. Alan Turing proved that such a supercomputer couldn't exist, right?

So we don't know the answer. We cannot know the answer. So this contradicts the saying of Hilbert that indeed I think it's written on Christomphe that we must know we will know, we must know we cannot know. That's the truth. It's an uncomfortable truth that you cannot know. So in a way it proves that mathematics also has its limits. So that's indeed a revolutionary idea because indeed in order to prove this statement Alan Turing invented what is the theoretical model of a computer because improving, you know, how did he prove this? He proved this by contradiction, assume that such an algorithm exists, okay?

And then you end up feeding the algorithm with the own machine you have created. The algorithm fills the algorithm and then you get a contradiction. And by doing so, I'm not going to do the proof here, okay? Because then I will lose all the audience here in this minute already. But in doing this proof already, he was giving the theoretical model of computers. And this is just in 1936. Can you imagine? And nowadays, everything we do with computer like this recording. So the beginning of computation is due to Alan Turing, even if he was a mathematician. So Turing machines, we could say, are the forerunners of today's computers. So indeed, with this question, the holding problem, will this process stop or not?

Will this question have a yes-no answer? The story of modern computers began. So that's quite amazing. And amazing everything Alan Turing did, cracking the Enigma code, creating the first theoretical model of computers. And indeed, his death was quite sad. The fact that he and his team at the Bledsley Park break the Enigma code was a secret for decades. So when he died, this was not known. He was not a hero at that moment. However, everything he has done for science, his legacy in computer science is so important that it's honored by the most important prize in computer science, which is the Turing Award. This started in 1966. To the Turing Award is the most important recognition in computer science.

Summarizing a little bit, right? We go from these David Hilvers, we will know, we must know, we will know in 1930 to these you know, zones without, without, where we don't have, where we don't have connection with our phones, zones where we cannot know, zones of darkness because logically there is no way to know. And this, I put the example of Turing, but Gödel's incompleteness theorems is another example and this goes back to 1931. So indeed Gödel's idea is that any sufficiently powerful mathematical system

is incomplete. So, indeed, more or less, let's say we go from certainty to uncertainty, we go from this idea that there is always a yes-no answer to we cannot know. And now I want to explain this idea of things that are undecidable in a playful way. So now I'm going to show you an ad, and this is the ad of a car. And in this car, they explain a story of a ship that lost all the cargo because of a storm. that lost all the cargo because of a storm and this cargo was formed in particular by 29,000 rubber ducks like the ones you see there.

And then the funny thing is that these rubber ducks appear in places that they were not expected. Ah, and in this advertisement, they tell us thousands of that appeared in the UK. And it says life is full of surprising endings. And this is an advertisement of a car that is no, you cannot longer buy, but of course you want to buy the motion, right? A life is full of unexpected endings and the unexpected endings, they are telling us a story that some rubber ducks, some rubber ducks like this rubber

duck here were lost because of a storm. And they tell us that 15,000 of those appeared or thousands of them appeared somewhere in the UK and there is some I mean this is based on a true story as everything that you see on the TV is based on a true story but it's not 100% true so what's true about the story that this advertisement is telling us is that in 92 there was a career called the Ever Laurel which was departing from Hong Kong and going to Tacoma and was carrying among the carriage 29,000 rubber ducks but they were lost because of a storm okay and then it was very strange the path that the rubber ducks followed after they were lost. Ten of them appeared in November.

This was January, so from January to November. Ten appear in Alaska. And then they have been appearing in several places. You can see here the map. And they have a funny name. They were called the friendly floaties because they have been appearing in places where they were not expected. And in the advertisement, they were telling us that thousands of them appeared in the British shores.

And this is what was expected. We can read it here in the news, thousands of rubber ducks to land on British shores after 15 year journey. And indeed, even the queen was waiting for them, but just one of them appeared and appeared in Scotland, right? So the truth is that just one of them appeared in Scotland. So there is some truth about the advertisement and of course based on true stories, right? So

these 29,000 rubber ducks, there is a very interesting story about them. Indeed, you can buy, I was showing here before, you can buy a book which is Moby Duck, okay? Which tells you the story. It's not Moby Dick, but it's Moby Duck. It tells you the story of the ducks. And everything that these ducks have helped to science, because indeed there was a computer simulator to follow all the flotsam lost on the seas, which was developed by an oceanographer called Jimmy Graham.

And indeed, Curtis S. Mayer, who was indeed studying ocean currents, but indeed just as an aficionado, indeed he used this computer simulator to try to track the rubber ducks and indeed the movement of the rubber ducks improved the computer simulator. But the truth is that this computer simulator couldn't find all these rubber ducks. So again, spec the unexpected. On one hand, thanks to the rubber duck, the friendly flotillae, these rubber ducks, predictions about currents could be made, but not all the rubber ducks could be found. Indeed, Eversmeyer and Ingraham have been working together and they did a lot of interesting experiments. For instance, they were throwing bottles with a

message inside the bottle and you know that they could see that only 2% of these bottles can be recovered. So how do you know that only 2%? Because the message inside the bottle is you will get $50 if you give us this note. So maybe it's quite accurate that it's 2%. So what does Turing have to do with the rubber ducks and why I'm telling you this story of the rubber ducks? Because indeed there is this fact that questions have no yes no answer, okay? With this idea that the rubber ducks didn't appear where they were expected. So what is the summary? Well, the method, this of course used by Ingraham and Ebes Meir, could not localize all these

lost rubber ducks. Only 2% of the messages of bottles are recovered. So maybe finding the rubber duck is also an undecidable problem. This looks like a very stupid question, right? Because probably these rubber ducks are in one of these, are lost in the middle of somewhere. But I want this to use the rubber ducks as a metaphor to explain this idea of undecidable. In a way, and the question is, how can I say that finding the rubber ducks is an undecidable problem? Well, remember that Turing proved that the halting problem, it's undecidable. Okay? So if maybe if I can associate a Turing

machine to the movement of the rubber ducks on the sea, okay, you see this is a rubber duck that has been lost and very happy to be lost 15 years on the sea, he's in a cool vibe, but you follow the rubber ducks, maybe following the rubber ducks is the same moving, the rubber duck moving on the water is the same as computing with a Turing machine. That's the question I want to answer. Now what would be the difference here between being undecidable and being unpredictable? That's where I'm going. Unpredictable, it's a word that can mean that you cannot predict something. And the question is not that you, it's not the fact that you cannot predict.

The question that you have to ask is why you cannot predict. And the answer to this why can be, there are two different answers. Like why you cannot predict because there is a logical barrier. This is undecidable. And why you cannot predict? Maybe I cannot predict because I don't have enough information. And this other way of being unpredictable is what is related to standard chaos, to the notion of classical chaos. Just a moment, don't go anywhere. Hey, I see you inching away. Don't be like the economy. Instead, read The Economist.

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So let's talk about classical chaos. The butterfly effect is a film. In that film, the main character wants to change something of his present. How many times we thought about this? If I could come back in time, right? Because if you are in a film, you can go back in time. So the main character tries to go back in time to change things in the past that could affect the future. And indeed, this is a very good way to explain what is indeed the butterfly effect in mathematics, which is the idea of chaos. Indeed, Edward Lawrence in 1972 gave a talk with the title, That's the Flap of a Butterfly Wing in Brazil,

Set off a Tornado in Texas. This looks like the title of a film but it's the title of a talk. And indeed there have been films, books about this idea of chaos and the idea of chaos is precisely the idea that if you can go back in time and change the initial conditions slightly, just a little tiny little bit, then in the future the change can be enormous. And that's the idea of chaos. And I'm going to put a simulation here. I want to thank Robert Rice for this simulation. Here we have this, this is a particle that is following a path and this path can

be the solution of a differential equation corresponding to some physical problem. And indeed I'm putting three different initial conditions that at the beginning, let me put this again, at the beginning they are very close. So here there are three balls, we don't see them. Right. They're visually indistinguishable. They are indistinguishable. But as the time goes on, these three balls, you see now we start to see it looks like a big ball. Okay. Now we see two balls. Well, with my glasses I would see better, but I took them out because people talk about the purple eyes. And this is a bit scary.

So now we have the three balls going away and away and you see the trajectories of these three balls, they are very far away. So think of these trajectories of these three balls as the life of the main character of this butterfly effect. You want to change something in the past, it's enough to change maybe slightly, slightly one tiny decision that you took and in the future, the difference is big. And that's the idea of chaos. And indeed, the idea of chaos, chaos theory was discovered in a way by chance. And it was discovered, well, it's attributed to Edward Lawrence, but in the group

of Edward Lawrence, there was Ellen Fetter and Margaret Hamilton. We know Margaret Hamilton because she did all the software engineering that took us to the moon, right, in the Apollo missions. But before that, before that, she was in the team of computations of Edward Lawrence, and they were making some studies in meteorology. And then they were finding that some data didn't make sense. And it's not that they didn't make sense, is that if we change some initial conditions slightly, long term, the changes are big. This goes very well with the idea of weather.

We cannot predict the weather in an accurate way, more than seven days, well you can say 10 days, 15 days, not very precisely. The problem of prediction of weather is related to these equations of Lorentz, indeed. So when you ask what is the weather going to be, that's a difficult question, right? So let's keep in mind this definition that Edward Lorentz gave of chaos. When the present determines the future, but the approximate present, so changing slightly the initial conditions, does not approximately determine the future. So if you change the conditions just a little bit, the outcome can be very different.

So that's the idea of unpredictable in the sense that we don't have enough capacity to measure the initial conditions of the particle. That's the idea of chaos. That's the idea of classical chaos. Let me talk about something real that we were living some months ago. We were living this in December last year, we were told, or maybe in January, around December or January, we were told that there was going to be an asteroid falling on us, which was the why for our asteroid.

We were all very worried until some scientists came out with this map, and then we were thinking, oh, we are away from this danger zone, right? Indeed, there was a moment when, well, we've seen films, Armageddon is based on this idea that an asteroid falls on us and it's not very pleasant. And we've seen here we have a simulation of the danger of this asteroid to fall on us. And as you can see, this is evolving. In January it was 1.5. In February it was getting high, 3.1 on February 19th. I remember on February 19th I was indeed attending a program on TV and they were asking me about

that. People were scared. So the probability was very high and then it went down. So now nobody is talking about these asteroids. On the other hand, the odds of dying of an asteroid against other causes, well, here we have a summary. I mean, the odds of dying of an asteroid impact is very low. It's easier to fall from a certain height or to get shocked by eating or having a bicycle accident. So the question is, can we relax? Now we know that this asteroid is not going to fall on us. Immediately, this was happening in May.

We heard about a satellite that was to crash on Earth. And then everybody was very worried. If you were reading Wales online, it said it could hit Wales or London or Catalonia or, or, or. Everybody was very nervous. I remember I had that week I had to give a talk and I was thinking, where is it going to fall? Well, most likely it's going to fall on water. And this is what happened. It crashed back on Earth overnight on water. Everything went fine.

How is the idea of chaos connected to the asteroids? It's closely connected because all the measures that we do of a certain satellite or asteroid depend on initial conditions. The idea of chaos is present. Indeed, there is a whole theory called KAM theory, which is mathematical theory that controls that allows us to give some probability that things are going to be all right somehow. Going back to your question, unpredictable can be, the answer is well something can be unpredictable for two reasons. One because you don't have enough information about the initial conditions and then after a certain time this because of chaos things can diverge completely or

Maybe you knock a door and there is no answer. Okay It's because we cannot provide an answer because there is a logical barrier to a yes or no answer And this puts us very nervous, right? Because we want to control the world, right? We human beings want to control the world and we as scientists want to understand everything, but there are frontiers that you just cannot cross. And that's exactly the idea of undecidable events. Now I'm going to provide some examples that a little bit go in this idea. We've seen that fluids like water or lava often rebel against what's expected. We've seen this with tsunamis. We've seen this recently. Nature is revealing. It's rebelling against what's expected. The question is, well, can we use this kind of power of nature to compute? Indeed, that's a question that looks very wild, but it's a question that we could say

it was already formulated by Roger Penrose. Roger Penrose was asking what are the limits of computation? Can physical systems compute? This is already present in his famous book, On the Emperors, New Mind. Here we have Chris Moore in the middle who really precisely asked this question. He asked, are fluids complicated enough to perform computations? And he asked this in a very particular context. And this was in the 90s. And then very recently, here we have Terence Tao. Terence Tao, one of the most

famous mathematicians, or I would say the most famous mathematicians of the world nowadays, who is professor at UCLA, and he asked again this question, and his motivation was a different one. His motivation was, can I use this computational power to answer one of the unknown questions. Because we mathematicians sometimes we cannot answer because we don't know how to do the things. That's true. There are problems that we cannot solve and we don't know if they cannot be solved or not. It's not a question of undecidable. It's that we don't have still the mathematical power to solve them. One of them is the Navier-Stokes riddle. Navier-Stokes equations have been used forever to model the movement of fluids. They are used all the time. We use them. We use them. Engineers use them.

But we mathematicians, attention, we don't know if these equations have solution. What? How is this possible? We know that these equations have solution short-term. These equations are more complicated than differential equations. They are partial differential equations. And we know they have solutions short-term but not long-term. So these are equations we are actually using to model the movement of fluids and we don't know if they have solutions long-term. So there is an open problem and I'll discuss about this problem in some minutes. I will give more details.

There is an open problem if whether these equations have solutions or not. The community was somehow divided but now it's more or less clear that these solutions are these equations are may have may have some kind of disruptive behavior that we called blow up. Okay. And then here we have Terence Tao try to prove that these equations have blow up using this idea of associating a computer to the movement of fluids. So he explicitly asked this question in 2019. Because himself he was able to find some kind of this blow up phenomena, but not for Navier-Stokes but for other equations which he called the average Navier-Stokes because there were some integrals.

But the method he tried to use exactly the same method for Navier-Stokes and Navier-Stokes it's so hard and it didn't work but he still raised the question, I cannot find a blow up but if I could associate maybe some Turing some computer or some Turing machine to the movement of fluids maybe I would be able to use that to produce this kind of blow up and that's then the question became like people started to discuss these like This question starts to be relevant also concerning these other problems. Just a moment, I want to see if I have the question correct in my head from Terence Tao.

So you can have a billiard table with billiard balls and you can associate a Turing machine with the bouncing of the billiard balls. Okay, so then you think can we do this with any physical system? Exactly. And you think, well, in the fluid system, can we associate some initial conditions and then evolving forward with the Turing machine? And then the fact that Navier-Stokes sometimes produces blowups, can we make an analogy between that and undecidable problems? Yeah, well, you almost got it. You almost got it. The last thing, it's a bit more delicate. He was asking, can we associate

Turing machines, as you said, to this physical system, as you are describing, perfect, to fluids? But his goal was a bit more, I didn't explain it. The way he wanted to produce this blow up is much more complicated because he wanted to use this kind of computer as initial conditions of some Navier-Stokes equation. So he wanted to plug some initial conditions that were super powerful computationally in such a way that you see that you can, the idea is that you are amplifying to the maximum the choices of initial conditions. He wanted to put some conditions off, I will comment that later, some additional conditions that in a way physically you could see that if you think of the idea of blow-up, which mathematically means that some equations stop being smooth, physically means that the

energy of the fluid increases to a way in which it explodes. This could be a little bit. The energy is concentrated a lot, it's concentrated around that point. So this idea of concentration of energy, it's like, it's very compatible with this idea of associating a Turing machine. A small point of confusion. So in the Navier-Stokes equation, you assume smoothness or continuity, but Turing machines are discrete so do you have to put some bounds on whatever your initial conditions are or some constraints? Yes, I mean yeah I mean yeah yeah indeed you are totally right we are going ahead I plan to explain this I need to explain yeah yeah no thanks a lot this is a very excellent question and thank you for the question. So

exactly, we need to go from discrete to continuum. How are we going to do it? I need to explain that. It's like, this is a very good question indeed. So this is exactly, I mean, you are guessing my mind. This is fantastic. So this is exactly what I'm going to explain now. I'm going to explain indeed what Moore did in his thesis. You know what, like the dream of a PhD student, right, is to wake up one day and discover people care about, a lot about your thesis. And this happened to Chris Moore, like he woke up one day and then you could see, I mean, people were talking about the results in his thesis, mathematicians discover a more complex form of chaos.

This is a fantastic title, right? What did Chris Moore did in his thesis? Well, he did something that is very, very nice. He associated a Turing machine to another idea which is quite wild, which is the Cantor set. The idea of the Cantor set is a mathematical concept which can be explained very quickly. I have a presentation of the Cantor set later. It's like this is the following. You take the interval, you split it in three, and you drop the middle part, and you continue

this process. I have this, wait, where do I have this? Here I'm going to put it here. What's the counter set? So you divide your interval in three, you drop the middle, and what is left, you do the same. You divide in three, you drop the middle, you divide in three, you drop the middle, and what is left, you do the same. You divide in three, you drop the middle, you divide in three, you drop the middle. Whatever stays up there is the counter set.

But this is a process that does not stop. You divide in three, and you drop the middle. Oh, I very much like this animation. Yeah, this is also a Greist animation. Same credit? Yeah, same credit. It's amazing, it's on YouTube, and he is a Greist animation. Same credit? Yeah, same credit. It's amazing. It's on YouTube and he is fantastic with the animation.

So I'm using this animation too. So whatever stays up there is the counter set and the counter set has some very interesting properties. I will not make here talk a lot about the counter set. I just need the definition. And indeed, what he realized more is that, okay, instead of taking the counter set, I just take the counter set and multiply and take, you know, in terms of taking the one dimensional counter set, I take the two dimensional counter set. It's like I multiply, I take an axis, the x-axis, which is

the counter set, and the y-axis is the counter set. Then the points that you can describe with the x and y-axis are going to be in what is called square counter set. So what he discovered is that it's the same, the following two facts are the same, it's the same to disorganize this square counter set in a particular way than to compute with a Turing machine. So this disorganization is a kind of puzzle. I don't want to be very technical, but imagine that you are playing puzzles, but instead of doing a puzzle with normal pieces, you draw this puzzle on the square counter set. Why would you do that? Because you are Chris Moore. Then you go on the newspaper, you do your thesis, and that's what you do. So he discovered that it was the same to do a kind of puzzle with this square counter set than to compute with a Turing machine.

Moreover, that this play game of doing the puzzles with a square counter set could simulate any Turing machine. Not a given one, but any. So it was like a kind of universal thing. This is what he did. And you say, oh, this looks like magic. I mean, how could he do that? I'm going to tell you and you're going to realize that the idea is quite intuitive. I need to think about Turing machines, and I don't want to get very technical about what a Turing machine is. But all of us think of Turing machines as a long, long tape, right? Long, long, long, long tape with zeros and ones. Okay? I want you to think of a Turing machine as a printer.

We have here some states. Here you see this image. We have some states. And we are printing the states. The printer is called the Turing machine, on a long infinite tape full of zeros and ones. And how the printer works tells me how I have to change maybe these zeros by one and move the tape to the left or the right. So let's say this printer, which is the Turing machine, comes with some instructions.

And the instructions tell you where to move left or right. Of course, here I have a definition, but I don't want to bore you with a definition. I don't know why I put a definition. What I just did is the user's guide. There is a user's guide, which is the definition we mathematicians say. I shouldn't put this. But let me show an example here. Here I have, let's say the user guide is this kind of assignment that changes the initial tape. You want to change the 0 by 1, okay? And you want to say that this plus 1 tells you that the tape has to move to the left.

You say, but this is not very intuitive because you are moving to the left and you put a plus one. You move to the left, the tape moves to the left, but the printer moves to the right. So you put a plus one. So this tells you that the current state, it's Q, prints on 0, and now you change this 0 by 1, and you move the tape, you see, to the left, so the printer moves to the right. This is the effect. Let me show you again this animation, which is not exactly the same animation.

There is something I found on the internet. You change 0 and 1, and then the printer moves to the right, so you have this plus one. So, that's the most basic form of computation. And now, I want to, in a way, this idea of Chris Moore looks very strange, but it's not. Because if you think about what a mathematician would do with a bit of this long tape is to try to put things in a mathematical form. A way to do it would be, okay, now I have a lot of zeros and ones, so I'm going to do two packs.

I take the zeros and ones on the left, I am starting at a certain point, and I pack all the ones on the left. There are going to be an infinite collection of zeros and ones. Now, attention, it comes the miracle. I think of these numbers as coefficients of a series, numbers as coefficients of a series, of a ternary expand. This is a way to export to a number. We are used to expressing it on base 10, but we could express it on base 3. And then because of this ternary expand here, where I have always two present, observe that the outcome of the names here in which I plug zeros and ones, okay, I'm here putting my finger on the screen, I shouldn't do this. Just a quick question. So most often when Turing machines are first taught,

the tape has blanks, and then you can write a one or a zero, but are you saying that initially it's just going to all be written with a 1 or a 0? So there's no blank? Well, the blanks, you can put the blanks, yeah. But mathematically, it doesn't affect. You can simplify and forget about the blanks. It's true that from a computer scientist point of view, we put the blanks, but you can forget and put the zeros and ones. It's going to be equivalent what you do with them. Well, the reason is, let's say i3 was a blank, well then what would I put inside the equation

for x, for the i, if it was blank? Would it be 0? Yeah, but here I put a, yeah, exactly, I would replace for instance a blank with a 0. So I just have some numbers, and I think of these numbers as the coefficients of eternity expand. So what does it mean? At the end of the day, it means that this gives me a number. With this particular choice, you see that this gives me the number 2 over 3 plus 2 over 9. That if you look at my former description of the Cantor set, this lies on the Cantor set because of these two. Okay? And why I do the same. So this is x is the number where I have replaced

the numbers, the long series of numbers on the left, these zeros are one as coefficients, and I do the same for the right and I put them as coefficients. Okay? Interesting. So this gives me what? This gives me two numbers. This second number is two over nine. And this is an example. So these two numbers are on the counter set, right?

So if I represent them, they are going to be on the famous square counter set. So this idea blows my mind in a way. So this is the idea of Chris Moove. Go to a Turing machine with your phone, take a picture, okay? And when you take this picture, take the numbers of zeros and ones, this produces a point on the square counter set. So what I have explained now is that just a picture of a Turing machine working on a certain moment is the same as a point on the square counter set. So when this Turing machine is going to start moving, this point, this red point that you

see here, is going to be jumping, and this is something a simulation I should do at some point, is going to be jumping on this square counter set. And this idea of jumping of the square counter set is what we mathematicians call mapping. So indeed, that's the key point in Morse construction. A universal Turing machine can be associated to transformations of this square counter set. And this transformation is a mapping. And indeed, in order to have this good property of universality, you need that this mapping satisfies some properties that I'm not going to spell out. But for instance, it needs to preserve the area. That's one of the conditions. Now I'm going to answer your question that was totally excellent to the point before.

You are thinking of a Turing machine as a discrete object, and you are thinking of nature as a continuous object. Your movement of a particle on the water is moving on a continuous way, so you have to relate, you have to find a way to associate, to define, to give a proper definition of what is to associate a Turing machine to the movement of a particle, for instance, on water. So here in this picture, on the left you have the Turing machine working, and on the right you have the trajectories, for instance, of the rubber duck. You have to associate, you need to mark some points on the trajectory and moving from one point to the other in the trajectory of this rubber duck should be the same as computing on the Turing machine.

Of course, you are completing. You go from discrete to continuous. So, you need to know if this assignment is correct in a way. And the way to define this is, OK, this assignment is good, and this is a key important point of my talk today. This assignment is good if you do the following thing. You mark an area, for instance, a neighborhood. Now imagine that this green line here wants to represent in the map a neighborhood of the British islands. Okay?

We are in a neighborhood of the United Kingdom with this green mark here on the map. So we say that a vector field would be the velocity of the particle. The velocity of the particle is said, we say that this velocity is compatible with associating a Turing machine. When we say that this association is good, we say it's Turing complete if it can simulate any Turing machine. And in order to test this, because this definition looks very, very suspicious and very hard to test, we say the holding of any Turing machine with a certain input is equivalent to a certain trajectory of the vector field. So it's equivalent to the trajectory of the rubber duck. Okay. To enter a certain open set.

This open set here is a neighborhood of the British islands of the United Kingdom. So we say, well, this association is good. If it's the same to enter in this, uh, United Kingdom, then this is equivalent to the Turing machine holding. Right? And now you say, okay, I don't understand anything. Wait, you do a Turing machine to the movement of the rubber duck, or a trajectory on water, or a trajectory. Here I say water, but in general, let's say the velocity of a particle. Then I say that this association is good if the holding is equivalent to the trajectory entering and opens. But I know that the holding problem is undecidable. So what's the conclusion?

This tells me that it's undecidable to know if the rubber duck will enter or not United Kingdom. You see? Did I convince you a little bit? Kingdom. You see? Did I convince you a little bit? So take a look at this red dot here. That represents the state of the tape. Yes. Okay, great. The Turing machine acts. Then this red dot is going to jump around to some other point. It's going to immediately. It's going to do so discontinuously. Like it could go all the way up to the left, to the top left.

Is that correct or no? Yes, it's going to do discontinuously. Yeah, yeah, that's correct. But what we are doing is to extend this idea so that it does this smoothly. I totally agree with you. Okay, so then when we scroll down now to the next slide, the slide after the next one, see this continuous motion with the, yes. So where are you getting this mapping?

Is this supposed to correspond to the cantor set or what? Yeah, no, that mapping is what we call the encoding. Okay, so that mapping you have to come with it, you have to associate. Okay, you have to find a way to associate the movement of the Turing machine to the velocity. And that's a canonical association? Like there's just one up to some isomorphism or? No, no, no, no, no. It's not canonical in our way. And I see your point. How do you jump from discrete to continuous?

Okay. Like this is a dog and you will see how I do it in the case of the... I want to show you today. I came here in full force. I mean, I have all my, I have all my rubber ducks with me. So I came, you know, I really want to show you how we did it because we answered the question of Terian's style. I mean, we answered his question.

I indeed, I sent him an email. Dear Terry, we know how to do that. Okay. So then it's like, I, this idea, like it's quite wild because you have to jump from this creature continuous and there is, and it's not, there is no recipe that tells you how to do it. There is a recipe that tells you once you have done it, if your recipe is good enough. Okay. This is something. So this is what I call Touring Complete. So actually, actually, I would appreciate if right now you unshared because there is something I want to show you, which I think can be helpful to the audience, if you don't mind. Okay, so here,

I just coded up something using Claude. This is fantastic. And this is the Cantor Set and of course we can make the Cantor Set more dense. You have to tell me how you do it. I will, I will. Okay, so down here this is a tape. Okay, so let's press play and the Turing machine is gonna act and then you see it just jumping around. This is exactly what I, this is fantastic. Exactly, and what I was confused about, which you are going to get to, is that these jumps are as discontinuous as one can be. Yeah. Yeah.

Yeah. Yeah. But if you represent that that's very good when you have done all this is amazing. This is amazing. This is great. Oh my God. This is fantastic. Indeed, when you do this, the mapping that Richard Moore, that Chris Moore, that Chris Moore is saying is the mapping that assigns the point to the point, okay?

And you have been playing with it and you have been doing these kind of jumps. Because this is discrete, the funny thing is that you can extend and that's the point. These maps, this is a map and that's exactly the precise way to say it. That's a map between two square countersets. I can think that these two square countersets are points that you are doing inside a square. You agree with me. This live inside the square. Okay.

So this mapping, and this is going to make your mind blow like this mapping is discontinuous when you think like, when you see this jump, you think that this is discontinuous, but you can extend that to a smooth mapping between the square itself. Interesting. Okay. This is great that you did this program. It's amazing. This is amazing. Great. I hope it's useful to people. Yeah, I find this amazing. So you did two very good questions. The first one is that you were asking me if this was canonical and the answer is no. You have many, many ways to produce this. The way we think of

this is we extend this mapping from the discrete to a continuous set because the way we answer the question of Terence Tao is using this kind of geometry I was also talking the other day, symplectic geometry, right? We were using that for quantization. And we can also use symplectic geometry to answer that question to Terence Tao. And in order to make this work, I really need continuous that. I cannot, I cannot do it with this screen. That. Yeah.

So I don't see that connection right now. And I know you're going to get to it. But the only hint that I've seen so far is that you said something was area or volume preserving. Yes. I imagine that's going to get associated with the symplectic. That's, that's, that's great. That's great. Exactly. Good. Fantastic. Yeah. But now let's talk

about money. Let's relax a little bit. We have been working hard. So let's see if we get some money out of this. And let's talk up million dollars for a correct answer. That's what the Clay Foundation wanted to give mathematicians if they were able to solve one of the seven problems on this list. For each problem they would give one million dollars. This problem was announced in 2000 and it's 2025. We mathematicians work hard, but we were only able to answer one of the questions, which is the Poincare conjecture.

This was answered by Gregory Perraman, who you have here, who answered correctly this question but refused to get $1 million. This is a good definition of a mathematician. When you get it, then you don't get the money. So these are problems that are still pending in the literature. And while you've talked, I think, maybe about the Riemann hypothesis in one of your form. No, no, no, you didn't talk about this, but A tad with Edward Frankel.

Exactly. Exactly. And also Richard Borchards. I'll put the links on screen to those. Yes, exactly. I saw the one of Frankel. It's amazing. And so you've heard part of it. So what happens is that what is the next big problem, next riddle that is going to become known, that will have a solution. People say, oh, it's the Riemann hypothesis, blah, blah, blah, blah.

Some people say it's the Navier-Stokes equation. So I will talk about the Navier-Stokes equations. But just for the audience to know that, okay, all these problems are pending. So if you can solve these problems, you are welcome to do that. It's like, it's a very interesting world to be working on. Hi everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access

to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here, hit subscribe, and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L.org. Kurtjaimungal.org. So Navier-Stokes equations. Let's talk about Navier-Stokes equations. These are the equations that model the motion of fluids, which are incomp incompressible so that you cannot compress, and viscous,

so you have some viscosity. And you have here the equations, today I put some equations, not a lot, these are the equations here of Navier-Stokes, and you see that there is some force, which is the external force, and you see the viscosity here, that you have this mu. Mu is the viscosity. So if we are dealing with water, the mu is zero and these equations have a different name. They are called Euler equations. And if you prove the blow up of Euler equations, that's very good for you, but you don't get this $1 million, okay? Because it's not in this list of problems, okay? So let's say that the case in which you have viscosity is zero, so for instance, that you

don't have any viscosity, then these equations are, let's say, easier to deal with, though they are very complicated, and this would be a different problem, but I will go back to Euler equations because indeed That's the question that Terence Tao was asking If he was asking can I find some initial conditions of Navier-Stokes that are Turing complete Euler flows He was asking can you provide a Turing complete Euler flow? So well these are the equations that we have been using for many things, because the movement of incompressible viscous fluids is important, right? Meteorology, movement of water, the water that you get at home, like, this is governed

by these equations. And we mathematicians know what the equations are but we don't know if this has a long time solution. This again proves that this is one of the questions that is pending. Let's say that the formulation of the questions you can get the formal formulation if you go to the website of the Clay Foundation, and then Fefferman, in the case of the Navier-Stokes, was the one who gave the very precise definition of what it means to prove or disprove these questions. So, the question is the regularity of Navier-Stokes equations. And the problem is to determine whether all initial conditions give rise to smooth solutions. Well, these initial conditions, if you go to the formulation of FFERMAN, have some natural constraints that are given by some equations,

but correspond to physical systems. So we want that to determine if all initial conditions give rise to smooth solutions that evolve smoothly, or whether solutions may degenerate and blow up after a certain time. And as I said, this explosion corresponds to what we mathematicians call the appearance of singularities. And physically, this would correspond to regions of space, because this equation we are moving in three dimensions here, okay? We are three-dimensional. And

this would correspond to regions of space where the energy of the fluid becomes concentrated to the point of becoming infinite. This is the physical idea. So in a way, if you think about this, the fact that these equations, if we are able to prove that these blow up or these explosions exist, then what this does tell me? This tells me that these equations are not good enough to express the movement of fluids. Okay. Why? Because we're all here. We are all here. Exactly. We are because we are all here. Exactly. Because if this was true, then if I do this thing, I would have a tsunami maybe, right? And okay, tsunamis, I mean, you can get them some minutes before, some time before, not

a lot of time before, but you have some information. So then this is telling us, if one confirms that the blow up exists for these equations, this is going to tell us that these equations are not good to describe the physical proof. Now, why can't someone say, well, look, even from your old podcast, we talked about quantum mechanics. So it's clear that Navier-Stokes isn't fundamental. Quantum mechanics is more fundamental than maybe something underneath that. But what if someone just retorts at that?

Yeah, I don't know. Don't make me choose. I think like Navier- I mean, don't make me choose. I think both are fundamental problems. And, you know, for, if you ask this question for somebody who has been working in Navier-Stokes all their life, which is not me,

because I came to this problem by chance, and I will explain you how. But then, of course, this is a big problem. But I think this is an important problem. This is one of the seven problems in the list for the millennium chosen by the Clive Foundation. So this is important. I agree that many other problems are important, but we have one life, so we need to choose some problems to solve. We cannot solve them all. All right.

Yes. Okay. Let's go on. So as I said, this problem, it's the movement, which we need to think that it's three-dimensional. This is very important. Here dimensions are important. We are moving in three dimensions, X, Y, and Z. And why I say that? Because in dimension two, the same problem was solved by Olga Ladies and Skye already in 1958. She proved that the solutions were always smooth in dimension

two. However, the three-dimensional problem is still open and people are getting nervous about it. I mean, and now we've read on the newspaper very recently, but this is something that has been going on for the last couple of years. People are trying to use machine learning indeed to prove the existence of blow-up. You can try to use artificial intelligence. That's a big, big approach to this problem. But let me talk about one approach to this problem, which is the approach that Terence

Tao had in 2019, in which he thought, okay, this is an approach to find a control example. This means to find a construction where there is blow-up. Okay? And here I have Terence Tao's working hard. And here I have the Matrioshkas, which is like a copy of a doll. You open the doll, you get another doll, a doll, and a doll, and a doll. And I have Matrioshkas here at home. So indeed I have this, I bought this because I like this idea. So this is the idea of recursion, which is a very mathematical idea. And this idea of recursion is inside the idea of the Turing machines, indeed, in the idea of self-replicating machines. So in a way, he had solved Tenenstahl. He found a counter example, a blow-up situation in this average Navier-Stokes equation. And then in his counter

example, he was working by hand, but he thought, okay, maybe if I could associate indeed a Turing machine to the initial conditions of Navier-Stokes, I would be able to get this flow. And more precisely, he was asking, can I find a Turing complete solution to the Euler equations. So the equations, the Euler equations would be the case of Navier-Stokes when the viscosity is zero. Okay? And that's his question. So in a way here, I have this third picture is like a picture of, you know, it's a picture that there is a film called Solaris, which comes from a book of course, in which there is this old idea of a thinking ocean. In a way, this is what we are trying to find, a thinking ocean, right? In a way, while thinking, a

computing ocean. So let's go to this idea. Now I want to show you how I answered, how we answered, to the question of Terence Tao with a yes. Not the blow up, but the question. So the question of the blow up had his approach is, can I find a solution to the Euler equation, so the Navier-Stokes equations with zero viscosity that are Turing-complete, which is this condition, can I associate things in a good way? And we answer this question with a yes. Though then the second question is can you use this construction, Eva, that you did to find a counter example or to find a blow-up to the Navier-Stokes equations? And you'll see the answer to the second one. You can guess that it's no, otherwise you would know.

But let me do a small summary of where we are. In 91, Moore asked if hydrodynamics is capital for performing computations. So he asked whether we can use fluids to compute. Yes. And now we have this story of 29,000 rubber ducks lost in the ocean in 92. And in 2007, just a rubber duck showing up in Scotland. So, in December 2020, we proved with my collaborator, Daniel Peralta-Salas, my former student Robert Cardona and Francisco Presas, we proved that there exist solutions of the Euler flow in dimension 3 which can simulate

any Turing machine. And that's the statement of the theorem. But then we had an interview with somebody who was working for El País and he said, you should call this fluid computer. And I thought, oh, that's great. That's why we need some media here. So people in the media call this the fluid computer and I think it's a great idea.

So it's a computer in a way that works with fluids, but where is the computer? Do I have the computer in my house? No, why? Could I construct this computer? No, not yet. And we'll get to that. So now, this answers, I think it's going to clarify Kurt, your question about how to go from discrete to continuous. And our idea was very simple. Usually simple ideas don't work. And in this time, I think this is the only time in which the first idea that we had is the idea that worked for the proof.

More had worked on squares. Squares are on dimension 2. They are on a plane. They are not three dimensional, right? So in a way, our idea is let's go from the construction on the plane that Chris Moore did to a construction in dimension three. And our idea is we are looking for the velocity field of the velocity of a particle, okay, such that when it comes back, each time it comes back, it corresponds exactly to the mapping of Chris Morph. Okay?

So this is indeed very well known in mathematics and it called a pumpkin a section so we think of most transformation as a pumpkin a section of a of a vector field. What are the velocity field. Which is the velocity of a particle moving in three dimensions so the velocity of the rubber duck moving in three dimensions okay and in such a way way that each time this rubber duck goes back through this, I fix a plane, and each time the rubber duck goes back to the plane, it hits in a different point, and this point where it hits is exactly the mapping of Moore. I see. You see? And that's how you go, then you say, okay, but the mapping of Moore, if you are taking the velocity of a fluid, then the intersection is not necessarily, is it on the counter

set or is not? Okay, what we do is to extend this initial mapping on the square counter set to a mapping on the disk to go from discrete to continuous. So you're creating a terrible knot. Well, yes. Yes, indeed. And then you see the rubber duck. So it's time I fix here my hand is what is called the Poincare section.

So it's a perfect plane, two dimensional. And this is the trajectory of the fluid, the rubber that each time it goes through here, it hits. Each time it hits, I think of each time it hits as the mapping of Chris Moore. Yes, okay. By the way, what I just said, so I was analogizing it to a knot, but knots are false because you can deform a knot and still call it the a knot, but knots are false because you can deform a knot and still call it the same knot, but the exact points here on the plane actually matter. Yes, yes.

Okay. Indeed, we are thinking more of this going back to this. So you see the connection to go from dimension two to dimension three. That's the idea of what we do. Yeah. Yeah. Okay? And then in order to perform a certain computation, if you think of a Turing machine, maybe you have to go infinite number of times.

If you think of this method as a method to compute, it looks very strange, but it would be a computational method at the end of the day. You can think that you are representing, it's a representation of the Turing machine through this movement of the particle. So indeed, how we did it is like, okay, any velocity field was not good enough because we need, and you remember this, we need this idea of preserving the area. So we needed a particular type. And we needed what we called a rep vector field. And this is related to Euler equations, and well here I got very technical, I shouldn't be showing equations. Okay but if you think of classical Euler equations that sometimes people study very early in undergraduate degrees are these equations so in a way we are working with classical Euler equations

but with a twist because we can change the metric. The metric of the Euler equations, we don't see a metric, but when we don't see a metric, is that the metric that we use is the Euclidean metric, the standard one. And here we change the way to measure, and given any metric, we have some associated Euler equations. This modification of the metric is important because there is a correspondence between red vector fields and solutions to the Euler equations which are Beltrami fields. This is very technical so I don't want to get into this. But particular solutions of these equations are called Beltrami fields, and these are very, very particular because if you are a mathematician and you want to put yourself in the easiest possible case, then you would ask the vector field, the velocity field x, not to depend on time.

When this happens, the solution is called stationary because it doesn't move and then Beltrami fields would be a particular type of stationary solutions of the Euler flow and for them for this vector field then there is a correspondence between Beltrami fields and red vector fields. So there is a way to associate red vector fields is a vector field of a certain geometry that you can associate to simplistic geometry. Indeed, it's the odd equivalent of simplistic geometry and the way that it's usually represented, again, thank you very much, Robert Grice, for giving me this picture. This picture represents what is called the contact structure.

The contact structure is, in dimension three, we can think a contact structure is just a collection of planes in dimension three, but these planes don't glue in a very nice way. In a way, there is no surface such that these planes are tangent to the surface. So, the way mathematicians have a way to explain this very geometrical idea using forms, and this is contact form alpha. So here, the important thing is that the idea of preservation of area implies the preservation of volume form that is important for this geometry. Here I'm getting very technical and this is too technical for our talk today,

but there is a correspondent, there is a magic mirror that associates a stationary solution of the Euler equations to a red vector field in contact geometry. This is fantastic. This is a fantastic idea and indeed I what I called a mirror. This was proved indeed by Robert Greist and John Endyer a long time ago in 2000. And the first time I learned about this correspondence, it was in a mini course that Daniel Peralta-Solo was teaching and I was attending that mini course. And Daniel Peralta and solo was teaching and I was attending that mini course. Daniel Peralta and I met for a long time, but we were not collaborating. And then I told him, but this is fantastic. We have to work on this together. So this is how we started to collaborate with this idea.

This is a beautiful, beautiful theorem that tells you that there is a correspondent. This is a magic mirror that I expressed here in this Disney way between particular solution to the other equations, which are these Beltrami fields, which are therefore particular also solutions of a particular case of Navier-Stokes, let's say, and solutions of a geometric problem, okay? It's a problem, no, it's problem, which is the red vector field. This is fantastic because if you are good in geometry and you are not good in fluid dynamics, then what you can do is try to apply your knowledge in geometry to solve the problem. So that's what we did. What we did and our proof, our construction follows the following idea.

Here we have the mapping of Chris Moore. Okay. That has this point that you represented so well with this program. I'm amazed with Claude. And this is the mapping. This is the famous mapping in the, this puzzle of the character sets. And then what we do is to extend this from dimension 2 to dimension 3. First to extend also this from discrete to continuous.

And then look for this vector field that solves the right equation. So we solve this problem geometrically using this contact geometry, which is the odd dimensional version of simpletting geometry, and then we translate this using this mirror. So if you have a solution of the red vector field, then this solution is an Euler. It's a solution of the Euler equation, it's a Beltrami field. And this Beltrami field is Turing complete. Why? Because the initial construction of Chris Moore was Turing complete and the extension that we are doing with this is also Turing complete. So the proof, it's very easy to understand. I have came here with full force and

I gave this proof to the audience. Okay. So this is the way that we prove. So what is the theorem? And then what are the consequences of this theorem? Well the theorem we prove is that there exist solutions to the Euler equations that can't simulate any Turing machine. But Turing had proved in 36 that the holding problem is undecidable. Therefore, if we put everything together, because we know that this simulation, that this association with the Turing machine is equivalent that the trajectory enters an open set even only if the Turing machine holds, what is the condition? The condition is that there, and that's the important corollary,

there exists undecidable fluid paths. And this was something that nobody was expecting because you were not expecting to have out of some equations that are written, and you say you have to have a solution, there exists undecidable fluid paths. So there is no algorithm that can decide There exists on the side of our fluid path, so there is no algorithm that can decide whether a trajectory will enter an open set or not. So now we go back to this rubber duck. And now we apply this, now assume that the machine that we have, this fluid computer, corresponds to the movement of these rubber ducks. That this is not the case.

But let's assume that it does. This is not the case because in our case we changed the metric in a very small place. This is not 100% physical. But this would explain why this rubber dux did not show up in the UK as everybody was waiting. Indeed, these rubber ducks, I have used them as an excuse as a metaphor to explain this theorem. But the truth is that later on with Robert Cardona and Daniel Peralta-Salas, we provided a construction which

was totally Euclidean. And there the proof is completely different. So we have an Euclidean, and there the proof is completely different. So we have an Euclidean also construction, and that construction therefore is 100% physical. It seems like this is revolutionary. So is this the first example of standard chaos? Sorry, not standard chaos, logical chaos? No, this is not the first example.

So let's say you have examples of undecidable physical systems. For instance, the spectral gap. This was proved to be undecidable before in 2000 or something. And if you go to Wikipedia, you look for undecidable. Okay. There is a list of undecidable problems. And I'm very happy to know that somebody added who our problems in the list. So that's nice.

Okay. So we are not the first, but this was, let's say, the first that corresponds to the movement of fluids. That's the first time that was observed in fluid dynamics. Okay. So let me see if I got this straight so far. So firstly, we know that universal Turing machines have undecidable halting behavior. Okay. Yeah. We know that. We know that universal Turing machines have undecidable halting behavior. Okay, we know that. We know that any Turing complete dynamical system will inherit such undecidability. Okay, then what you've shown is that there are certain 3D Euler flows that are Turing complete fluid computers. Exactly. Exactly. Exactly. So some solutions, there exist solutions of these, exactly, solutions of these earlier equations, these earlier flows that you described very well, okay, that are too incomplete,

therefore there exist undecidable fluid pumps. So the determinism of nature or of physics hides algorithmically undecidable, hence fundamentally unpredictable outcomes. Absolutely. That's exactly the perfect, perfect summary. Fantastic. That's exactly what we did. Interesting. So now the question, did you get $1 million? Right? You say, oh, why do you need $1 million? You proved a beautiful theorem. Okay, still the question makes sense. And the answer is, the short answer is no. So this construction we did was not good enough to give us the blow up of Navier-Stokes.

And the short answer no, the long answer is read my paper. But let's say we were able to plug these initial solutions in these equations here, Navier-Stokes, and then indeed we were able to follow the path, and then what happens is that we have this exponential decay. So this exponential decay shows that you have a total control of the smoothness. Also, because you have this exponential decay, then this also is not good enough to find a Turing complete solution of Navier-Stokes equations. So it's a no-no. But okay, we were answering the question of Terence Tao, but this answer does not lead to the blow-up.

This does not mean that other answers can lead to the blow-up. Because you have the set of initial conditions, right? It's a matter of finding on this set of initial conditions some initial conditions that may give this blow up. So now let's think. It's time to think. What's outside the Valtrami box? So that's me. That's probably, I mean, that's me working with my colleagues. Probably one of them is Daniel Peralta Salas and the other one is probably Angel Gonzalez Prieto who is our new member of the group.

Sorry, which one's you? The one by the door? I have two copies of myself, because I believe in this quantum thing. There is a copy of myself inside having a coffee with glasses and comfortably looking at these straight line A goes to B, right? Things are beautiful. Yes. Things are easy. And there is another one, me also, this other quantum version of me going outside the door and saying, oh, this is difficult. This is the issue. So instead of spin up, spin down, you have hair up, hair down. Exactly. So let's say, okay, what's outside of our Tramibox? Because we did with Daniel Peralta-Salas,

we have been working also on solutions, on proving undecidable, undecidability in fluid dynamics, not necessarily of a stationary solution. This is something we have been working on. We have a number of results. But what is now, let's say, pending, what was pending for a long time is like, well, what about Navir's talks? Right? Well, Navir's talks, the idea of Ter Terence Tao was let's try to look for initial conditions,

and then let's see if we get blow up. These initial conditions maybe are Euler, plume, plume. Then we could also try to think, well, Navier-Stokes are like a perturbation of Euler equations, and the perturbation that you are doing depends on the viscosity of the fluid. Well, yeah. So indeed, look, we have proved, what we have proved so far is that some systems which are of Euler type, okay, it's not possible to decide if certain particle will reach certain regions in the space, thinking of this rabble. So no matter how potent your computational problem is, in other words, if you want, the problem is not computable. So this construction only works in the case in which we don't have viscosity.

But then let's try to think if we can use the viscosity as a perturbation. Well, there is a result, and indeed we were thinking for a long time in this idea of perturbation, but there is a result by computer scientists, Olivier Bournès, Daniel Grasse and Emmanuel Henry. So, excusez-moi parce que je prononce pas bien Henry. So this theorem shows that it's not possible to construct a Turing complete system with finite energy, which is Robespierre perturbation. So in other words, if you add viscosity to the system, you can completely distract the computational power. And that's very interesting.

Because this tells you that you cannot find a Turing complete solution to Navier-Stokes if by perturbation. However, now I can announce, and this is our new result, I'm announcing here. You can put here as flashy, you can put flashy neons there. Yes. This is for one of your shorts. This is completely a new thing. This is totally revolutionary. Now I can say we know how to construct a Turing complete Navier-Stokes solution to Navier-Stokes. We can construct a Turing complete Navier-Stokes flow using, again, the power of geometry, but this time it's not contact geometry, it's co-symplectic geometry.

This is something that should be online soon, I hope very soon. But we know how to do it. The idea, again again comes from geometry. So we have been able to produce a Turing complete Navier-Stokes flow. So this is completely new. What we don't know, well, the one that we have produced, we know it doesn't give the blow up. But it's an interesting path also for the blow up, for trying to find the blow up, but it's an interesting path also for the blow up, so for trying to find the blow up.

And now, I don't know, we have been talking for a long time, but I have more things to explain you. Please. Okay. Now, it comes like a new idea. Now I learned from the journalists that you have to find the right names for your mathematical objects. Remember when I was explaining to this guy of Al-Pais, I was explaining, no look, what I did is to find a Turing complete solution to the Euler flow. No, what you did

is a fluid computer. So I said, okay, I buy it. So now we have done something also new, different, different from this solution to incomplete Navier-Stokes flow. This is something we have done with Ángel González-Pietro from Universidad Complutense de Madrid and with Daniel Peralta Salas. And it's totally revolutionary, I would say. And it's what I call a hybrid computer. And the idea is that the initial idea was to create a hybrid of this fluid computer with a quantum computer,

but what we have done in the end is much more powerful than this. But just as a first idea,ume that I call the former construction this idea of going around this idea of this machine that I explained, the Turing complete Euler flow. Now imagine that I call this a way to compute and I call this a fluid, right? In the same way we have the qubit. I call the fluid the basic unit of computation with a fluid computer. And now I assume I can assemble these pieces using the rules of Feynman of quantum computing, right? So one day I find myself in my office with my colleague Angel González Prieto doing pictures like this one which are the pictures used in topological quantum field theory.

So the idea is that use the tools of topological quantum field theory to improve your computational method. And how do you think of this? Well, you think of our method, the one I described before this pancare, as putting things on a cylinder, indeed. Right? But cylinders are not enough. This is just a piece of your puzzle. And also you could put in this category this new construction that I do with Cosimplect, this is also a potential piece that you could plug here and try to plug them with some algebraic rules, like the same one that Feynman gave for his lecture of computation. What do you get? Well, we get a new model of computation that we call topological Kinfield

theory. And that's our result that we posted on the archive. This is very recent. This is from the month of March. We posted on the archive. You can find it. So now it's public. I gave some talks about this and also did Angel and Daniel and well now I just like this is a bit too technical to explain what we do but what I can tell you is that this is much more powerful than this construction of the fluid computer okay And because in a way you are putting the techniques of quantum with fluid, you are improving both. However, this is something we were not... So what's next? What's in our agenda? Well, of course we want to... Well, the, the blow up of Navier Stokes was never in my agenda, but now it's in in the agenda somehow because this is a bit related.

Uh, we are working, we are thinking around this with Daniel Peralta Saras in this picture and I can go for the Pieto. We are here. All of us are smiling. This is good. Yes. And this, this picture is taken in Barcelona. By the way, Kurt, I'm still waiting for you. Whenever you come here, please let me know. I'll let

you know. We have to arrange a visit of your here. Yeah. You have to arrange a visit of yours. Yeah. I look forward to it. So this is Terence Tao who visited us last September. Uh, and we organized a conference and Terence was giving, uh, I, I, yeah Terence gave a couple of wonderful talks. We spent some time, we didn't have a lot of time because there were a lot of talks, but some time discussing about these other ideas. I remember we explained to Terence this idea of finding a Turing complete solution to the Navier-Stokes and also this new idea which we finally wrote up. So what's next?

What's next is to try to find these ideas, to use these ideas to find really the blow up of Navier-Stokes and indeed while doing this new computational models, the natural question is, are these computational models better than quantum computing? So, will the hybrid computer be the quantum supremacy? And this is something we are now working on. And these are the rubber ducks completely lost. And now, if I go back to the asteroid, that's another question. Do we have this idea of undecidable?

I was discussing the idea of the asteroid as another idea of unpredictable events. But now I want to fish it back and think, look, we thought of the asteroids as related to the idea of classical chaos. Can we put on the same problem, can we find on the same physical problem, classical chaos and logical chaos? In other words, do you have undecidable problems in celestial mechanics? What does it mean? Well, think of the following. You have the asteroid and now you put the rubber duck on your asteroid, right? So knowing whether this asteroid will fall on the earth of naught, if this rubber duck, so the idea I put here,

the rubber duck to give this idea of undecidability, can we prove that some trajectories, some problems in celestial mechanics are undecidable too? Because for instance, the Bernoulli shift has been used on many problems, in restricted problems on the three-body problem. Okay. And now the question is, well, it's known that many of these problems display classical chaos. Do they also display this idea of logical chaos? Okay.

So that's the kind of problems in which we are thinking. So in a way, this would be the equivalent of putting the butterfly on the rubber duck. So you want to reconcile or you want to relate, you want to see if these two are related or not. We have understood many things about the relation looking at entropy. Thinking entropy is the big word, we use it for physics, we use it for mathematics and it means that we measure disorder, I mean we measure order, right? So using entropy we have been able to understand somehow the relation between these two chaos but it's not so clear. We have all these complexities to be unveiled somehow.

So, yeah, with this I'm done. That was beautiful. That was wonderful. Thank you so much. Thank you. So thanks a lot. I don't know if you want to ask some more questions on... I say the audience, I'm sure you all have questions. Please write it in the comments because Eva will be making another appearance and I'll ask those questions on your behalf.

Thank you, thank you Eva. How about one quick question? Do these fluid computers violate the church touring thesis? Do these touring computers violate? No, no, no, no. That's a super good question indeed. The Church-Turing thesis is the thesis, and that's the nice thing about computer science that you don't call something a theorem.

You say, it's a thesis. I love it. Okay. It's this idea that if you have any system, all of these systems should be equivalent in computational terms. Doing a computation with a Turing machine should be equivalent with doing a computation with another system. However, one thing is doing the computation and the other thing is efficiency. When we talk about quantum supremacy, we're talking about efficiency of computations. That would be another question. These new constructions, this is a very good

question by the way. Thanks for posing it. That's something I should have maybe mentioned, but I didn't want to make this longer. You have have several ways to, like this Church-Turing thesis is like you have all these several ways of computing. They are all equivalent in a way, whatever, but this is theoretically. Now what we want is to put this computer, and I want in one second to get my computation, that maybe with a classical computer it takes three years. That's what quantum computers are. So now what we want to do is to do this even quicker than a quantum computer, using maybe this idea of the hybrid computer, of putting pieces of the computer that are fluid and some pieces of the computer to accelerate the computation.

This is something we are working on, on proving that this, we did this topological clean field theory construction and indeed we have some people working on the simulations and looking forward because we are doing some simulations, this is quite abstract what we did. So we now have some simulations and indeed I have one of our students is going this week with Stephen Wolfram in one of his schools and Stephen Wolfram is looking forward to having indeed a simulation of this. This is great to have the interest of Wolfram in this. So our student, Isaac Ramos, I'm going to tell his name, did a very nice simulation of these constructions for particular types. Now we want to make this on a great scale and to be able to see that this version,

this topological clean field theory allows to produce systems that are better in the sense that quicker than quantum computers. For that we will need to accelerate. So we need to work a little bit harder than this model that is there on the archive. We have an idea of how to do it and we need to introduce some kind of accelerators of the computations and we know how to do it. We are working on this. More soon. Okay, more soon. Speak to you soon. Take care. Okay. Thanks a lot. Thanks a lot for everything, Kurt. are that every week you get brand new episodes ahead of time. You also get bonus written content exclusively for our members. That's C-U-R-T-J-A-I-M-U-N-G-A-L dot org. You can also just search my name and the word

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