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Eva Miranda: Brand New Result Proving Penrose & Tao's Uncomputability in Physics!
July 7, 2025
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The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science, they analyze culture, they analyze finance, economics, business, international affairs across every region.
I'm particularly liking their new insider feature was just launched this month it gives you gives me a front row access to the economist internal editorial debates where senior editors argue through the news with world leaders and policy makers and twice weekly long format shows basically an extremely high quality podcast whether it's scientific innovation or shifting global politics the economist provides comprehensive coverage beyond headlines.
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, so happy to be here and looking forward to this new adventure and ready to disclose something new. Let's see if people like it. I'm very happy about the, 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. And it's a great pleasure to be here with all of you now. Okay.
I call this expect the unexpected. And what does it mean? Well, you know, we all know David Hilbert, the famous mathematician, right? Who said we, 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 very precise way this idea of precision of mathematics that is of course very important and formalization the idea of formalization of mathematics and 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 indeed.
There was David Hilbert, there was John von 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.
and i chose three different main characters today i chose alan turing i chose uh to explain as you said the chaos that the theory of chaos the classical theory of chaos and this is a nice story that can explain with butterfly and i want to explain also
How it affects our last month, we heard so much about all these asteroids that could fall on us. So 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 a little rubber duck and the rubber duck is going 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 the the tack for classical games. So interesting. Let's see.
How do I get to Alan Turing? I was talking about Alonso Chart 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 charge 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 Bletchley 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
Add algorid something that is on repetitive now imagine you entering a labyrinth and trying to getting 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 of the main okay and the question is the following is there a general recipe and algorid 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 like, 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? And or in a way, if we think of modern computers, OK,
Is there a super computer that can tell us if a given computer somewhere on the world will ever will stop which reach 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, 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 straight 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 no we
We can 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. OK, but the answer does exist.
Well, we don't know if they answered it. No, mathematically, 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 like the question is very strange. The existence of super computer 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 his thumb 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 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 old 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 computer began. So that's quite amazing and amazing everything Alan Turing did, cracking the Enigma code.
uh creating you know the the first theoretical model of computers and indeed his death was uh quite sad indeed uh you know the fact that he and uh and his team at blesley park break the enigma code was a secret for decades so when he when he when he died this was unknown he was not a hero at that moment
And however, all 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, the Turing Award. And this started in 1966. So to the Turing Award is the most important recognition in computer science. Summarizing a little bit, right, we go from this David Hilbert's
We will know, we must know, we will know in 1930 to this
You know so zones without without where we don't have internet where we don't have connection with our phones zones where we cannot know songs of darkness because logically there is no way to know under these i put the example of touring bad girls incompleteness theorems okay is another example and this goes back to nineteen thirty one okay.
So it did get us ideas at any sufficiently powerful mathematical system is incomplete. Okay, so It did more or less let's say we go from certainty to uncertainty We go from this idea that there is always and yes, no answer to we cannot know Okay, and now I want to explain this idea of on the side of things that are undecidable in a playful way So now I'm going to show you are not
And this is the art of a car and in this car, they explains a story of a, of a ship 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 and in this advertisement they tell us
Thousands of that's 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? 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 29 000 rubber ducks
But they were lost because of a storm. 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 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.
i appeared in scotland right so the truth is that just one of them appeared in scotland so there is some truth about the advertisement that of course based on true stories right so these 29 29 000 rubber ducks there is a very interesting story about them indeed you can buy i i was showing here before you can buy a book which is movie doc
okay which is the story tells you the story of the of the ducks and everything that has been that these ducks have uh helped to science because indeed uh there was a computer simulator to follow all the flotsam lost on the on the seas
which was developed
And, well, on one hand, thanks to the rubber duck, the friendly floaties, these rubber ducks, predictions about currents could be made, but not all the rubber ducks could be found. And 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 two percent because the message inside the bottle is you will get fifty fifty fifty dollars if you give us this note so maybe it's quite accurate that it's two percent so what does touring have to do with the rubber ducks and why i'm telling you this story of the rubber ducks because indeed there is this
The method 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 during proof that the whole thing problem It's a 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 that has been lost and very happy to be lost 15 years on the sea. He's in a cool vibe and
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. 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
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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 press on 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 is a solution of a differential equation corresponding to some physical problem okay 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 glass 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 of 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 he did the idea of, uh, 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. OK. And then they were they were finding that some data didn't make sense.
Okay, 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.
Okay and this goes very well with the idea of weather i mean we cannot predict the weather in any in any accurate way more than seven days well you can say ten days fifteen days not very very precisely and the problem of prediction of weather is related to these equations of lawrence 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 Lawrence 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. 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. And that's the idea of chaos. That's the idea of classical chaos. So now 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 okay. And we were very worried until some scientists came out with this map and then we were thinking are we 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 one point one point. In February it was getting high two point three point one on February 19. OK.
I remember on February 19 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 you know 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 right
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 while most likely is going to fall on water and this is what what happened okay crashed back on earth overnight on water everything went fine so indeed 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
And the idea of chaos is present indeed there is a whole theory called KM theory which is mathematical theory that controls that allows us to give some you know some probability that things are going to be alright somehow so going back to your question.
Unpredictable can be for two reasons. One because you don't have enough information about the initial conditions and then after a certain time because of chaos things can diverge completely or maybe you knock on a door and there is no answer.
uh 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
Now I'm going to provide some examples that a little bit go in this in this idea, right? We've seen that fluids like water or lava often rebel against as what's expected, right? We've seen this with tsunamis. We've seen this recently. I mean, nature is revealing, right? It's rebelling against against what's expecting. And 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 could 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 in his famous book.
And here we have Prismore 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.
uh, one of the most famous mathematicians, or I would say the most famous mathematicians of the world nowadays, uh, who is professor at UCLA. And he asked again this question and his motivation was a different one. His motivation was, can I use these, uh, 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. I mean, it's like 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. And one of them is the Navier-Stokes riddle. Navier-Stokes equations have been used forever
to model the movement of fluids. And 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 whether these equations have solutions or not,
and while 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 turnstile try to prove that these equations have blow up
Using this idea of associating a computer to the movement of fluids Okay, 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 are which he called the average Navier-Stokes because there were some integrals Okay, but the method
He couldn't, he tried to use exactly the same method for Navier-Stokes. Navier-Stokes, it's so hard and it didn't work, but he still, he raised the question, okay, I cannot find the 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 so you can have a billiard table with billiard balls and you can associate a Turing machine with
With the bouncing of the billiard balls. Okay. So then you think, can we do this with any physical system? 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 you almost got it you almost got it the last thing it's a bit more delicate so 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 use this idea 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 have, you are amplifying to the maximum the choices of initial conditions.
Okay and he wanted to put some conditions off. I will comment that later. Some additional conditions and 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 around that point. So this idea of concentration of energy, 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 i plan to explain this i need to explain yeah yeah no thanks a lot this is a very an excellent question and thank you for the question so exactly we need to go to from discrete to continuum and how are we going to do it i need to explain that it's like uh 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 a lot about your thesis. And this happened to Chris Moore, like he woke up one day and then, 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 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 counter set later. It's like it is the following. You take the interval. You split it in three and you drop the middle part and you continue. 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, 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.
whatever stays up there is the counter set but you 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 uh 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
I'm using that this animation too. So whatever stays up there is the counter set and the car. So 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 think the two dimensional counter set is like I multiply, take an axis, the X axis, which is the counter set and the Y axis is the counter set. OK, then the points that you can describe with the X and Y axis are going to be in what is called square counters. 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 that to compute with a Turing machine. So that 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.
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.
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And I don't want to get very technical about what a Turing machine is. But all of us think of Turing machine as a long, long tape, right? Long, long, long, long tape with zeros and ones. OK, 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 instructions tell you where to move left of 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 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 uh assignment
that changes the initial tape you want to change the zero by one okay and you want to say that this plus one 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 okay so this tells you that you need to that the current state is Q
prints on zero and now you change the zero by one and you move the tape you see to the left so the printer moves to the right this is the effect so let me show you again this this animation which is not exactly the same animation there is something i found on the internet you change zero one one 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. About what a mathematician would do with a touring with a bed this long tape is to try to put things in form in a mathematical form. A way to do it would be OK. 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 that are going to be an infinite collection of zeros and ones, and now, attention, it comes the miracle. I think of these 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 okay and then because of this ternary expand here where i have always two present observe that the the outcome of of the names here in which i plug zeros and ones okay i'm here putting my finger on the
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 one or a zero? So there's no blank. Well, the blanks you can put, 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 on one. 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 zero? Yeah, exactly. I would replace, for instance, a blank with a zero. So I just have some numbers and I think of these numbers as the coefficients of a ternary 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. The 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 Moore. 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 the 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 counters. 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. OK, so now it comes to and now I'm going to answer your question that was totally excellent to the point before like 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, it's moving on a continuous way. So you have to relate
you have
So you have in a way you have to associate to 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 okay 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 and 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
Right? And now you say, OK, I don't understand anything. Wait, you do, because you know that the holding problem is undecidable. OK, what did I say? I'm giving a definition that associates 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.
OK, then I say that this association is good if the whole thing is equivalent to the trajectory entering and opens. But I know that the whole thing 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? 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 yes it's going to do this content yeah yeah that's correct but what we are doing is to extend this idea so that it does this as smoothly i totally agree with you
Okay, so then when we scroll down now to the next slide, the slide after the next one, 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.
uh 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 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 i i really want to show you how we did it because we answered the question of terian style i mean we answered his question i indeed i sent him an email dear terry we know how to do that okay
Sorry then it's like this idea like it's quite wild because you have to jump from the screen to 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 would be helpful to the audience if you don't mind okay so here i just coated up something using clod 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 going to act and then you see it just jumping around. This is exactly what I, this is fantastic. This is 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. Yeah. But if you represent that, that's very good when you have that. No, this is amazing. This is amazing. This is great. Oh my God. This is fantastic.
uh indeed when you do this the mapping that richard more that uh that chris more that chris more is saying is um the mapping that assigns the point to the point okay and you have you have been playing with it and you have been doing these kind of jumps and 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 this is these two square countersets are points that you are doing inside a square. You agree with me. These live inside the square. OK, so this mapping and this is going to make your mind blow like this mapping is this continuous when you think like when you see this job,
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, yeah, I find this amazing. So you did that two very good questions. The final one is that you were asking me
If this was canonical and the answer is no, you have many, many ways to produce this. Okay. And 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 turns down 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 data. I cannot do it with the sprint data.
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. So I imagine that's going to get associated with the symplectic. That's that's that's great. That's great. Exactly. That's 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 and 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.
And this was answered by Gregory Paramon, who you have here, who answered correctly this question, but refused to get $1 million. Okay. 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, you've talked, I think, maybe about the Riemann hypothesis in one of your form. No.
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 to become known as 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, OK, all these problems are pending. So if you can solve these problems,
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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 dot org, KurtJaimengel dot org. So Navier-Stokes equation. Let's talk about Navier-Stokes equations.
These are the equations that model the motion of fluids, which are 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,
Okay.
the case in which you have viscosity 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 stout was asking if he was asking can i find some initial conditions of navier stokes that are touring complete oiler 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 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 effort, but have some natural constraints that are given with by some equations, but correspond to physical systems. OK, so we want that to determine if all initial conditions give rise to a smooth solutions that evolve smoothly. OK.
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. 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, in a way, if you think about this, the fact that these equations, if these equations, if we are able to prove that these blow up or these explosions exist,
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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. 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 this 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. Now why can't someone say well look even from your old podcast yeah
We talked about quantum mechanics, so it's clear that Navier-Stokes isn't fundamental. Quantum mechanics is more fundamental. There may be something underneath that, but what if someone just retorts at that? Yeah, I don't know. Don't make me choose. I think both are fundamental problems.
You know for if you ask this question for somebody who has been working in average talks or 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. This is a but I think this is a I mean, this is an important problem This is one of the seven problem seven problems in the list for the Millennium Chosen by the Clive Foundation. So this is important
I agree that 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. Yes. OK, let's go on. So as I said, this problem is 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 all the ladies and skyer already in 1958 She proved that the solutions were always a 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 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
Okay. You can try to use artificial intelligence and that's a big, a big, big, uh, approach to this problem. Okay. But let me talk, talk, uh, to about one approach to this problem, which is the approach that Terence Tao had in 2019 in which he thought, okay, I, 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 turnstiles, right, working hard. And here I have the Matrioshkas, which is like a copy of, you know, of a doll. You open the doll, you get another doll, a doll, another doll, another 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. Okay. 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 flows, to the Euler equations?
So the equations, the earlier equations would be the case of Navier-Stokes when the viscosity is zero. Okay. And that's, 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
Thinking ocean in a way this is what we are trying to find a thinking ocean right in a way i'm thinking computing ocean. So. Let's go to this idea now i want to show you how i answered how we answered into the question of turnstile 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 equations? 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 the hydrodynamics is capable of performing computations.
so he asked whether you know we can use fluids to compute yes and now we have this story of the 29 000 rubber ducks lost in the ocean okay uh in 92 okay 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 three, which can simulate any Turing machine. And that's the way, 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
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 these answers, I think it's going to clarify your question about how to go from discrete to continuous. OK. And. 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, which was, OK, more had worked on the squares.
Squares are on dimension two. 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 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 Moore. So this is indeed very well known in mathematics and it's called the Poincare section. So we think of Moore's transformation as a Poincare section of a vector field, well, of a velocity field.
I see. You see?
And that's how you go. Then you say, OK, but the mapping of more, if you are taking the velocity of a fluid, then the intersection is not necessarily. Is it on the counter or is not OK? What we do is to extend this initial mapping on the square counter set to a mapping on the disk. To go from the screen to continuous. So you're creating a terrible not. I'm well, yes.
Yes, indeed. And then you see the rubber duck. So each 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 duck. 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, OK. By the way, what I just said, so I was analogizing it to a knot, but 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. OK. Indeed, we are thinking more of this going back to this to this. So you see the connection to go from dimension two to dimension three. That's the idea of what we do. Yeah.
Okay. And then, uh, in order to perform a certain computation, if you think of a Turing machine, maybe you have to go infinite number of times. It's like, if you think of this method as a method to compute, it's not, it looks, it looks very strange, but it's, it would be a computational method at the end of the day. You can think that you are representing, it's a, it's a representation of the Turing machine. So this movement of the, of the, of the particle.
Okay, 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. Okay, 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 and 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. And 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 will be a particular type of a 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 symplectic geometry indeed it's the odd equivalent of symplectic geometry and the way that it's usually represented again thank you very much uh grist for giving me robert grice for giving me this this picture this picture represents the what is called the contact the contact yeah the contact the 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 a contact form alpha. So here, the important thing is that
There is 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 solution, 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 was showing equations, but maybe we don't need to show equation. It's a concept. So this is what I called a mirror. OK, this was proved indeed by Robert Greist and John Endier long time ago in 2000. And the first time I learned about this correspondence, it was in the in one in a mini course that
Daniel Peralta Salah was teaching and I was attending that mini course and 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 we started to, this is how we started to collaborate with this idea. This is a beautiful, beautiful theorem that tells you that there is a correspondent. It is a magic mirror that I expressed.
here in a 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. Okay.
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 And 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 this puzzle of the character sets. And then what we do is to extend this from dimension two to dimension three, first to extend also these from discrete to continuous. OK. And then look for this vector field that that that solves the right equation. OK. And then so we solve this problem geometrically. OK. You think this contact geometry, which is a which is the other dimensional version of simplistic 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, is a solution of the Euler equation, is a Beltrami field. Okay. And this Beltrami field is too incomplete. Why? Because the initial construction of Chris Moore was too incomplete and the extension that we are doing with this is also too incomplete. 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 theory? And then what are the consequences of this theory? Well, the theory we prove is that there exists solutions to the other equations that can simulate any Turing machine. Okay, but Turing had proved in 36 that the holding problem is undecidable. Therefore, if we put everything together,
because we know that 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 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 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, right? And now we apply this now assume that the machine that we have got this fluid computer
Corresponds to the movement of this rubber ducks that this is not the case, but let's assume that it does Okay This is not the case because in our case we change the metric in a in a very small place So this is not 100% physical but this would explain why this rubber ducks did not show up in in the UK as Everybody was waiting. Okay indeed
These rubber ducks, I have used them as an excuse, you know, 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 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, not logical chaos. No, this is not the same. 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. OK, there is a list of undecidable problems. And I'm very happy to know that somebody added who are problems in the list. So that's nice. OK, so we are 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 any Turing complete dynamical system will inherit such undecidability. Yeah. Okay, then what you've shown is that there are certain 3D oiler flows that are Turing complete fluid computers. Exactly, exactly, exactly. So some solution that I said solutions
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.
Interested another question. Did you get one million dollars, right? You say oh, why do you need one million dollars? You proved a beautiful theorem Okay, still the question makes sense and the answer is the short answer is no so this this construction we did was not good enough to give us the blow up of novice talks and and the short answer no, the long answer is read my paper and
but let's say we were able to pluck indeed 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. Okay. But okay, we were answering the question of Terence Tao, but well, 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 like 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. OK, so now let's see. It's time to think like what's outside the Valtrami box. Right, so that's me.
That's me working with my colleagues. Probably one of them is Daniel Peralta Salas and the other one is probably Ángel González Prieto who is our new member of the group. Sorry, which one is you? The one by the door? I have two copies of myself because I believe in this quantum
Okay, so there is a copy of myself inside Having a coffee with glasses and comfortably looking at these straight line a goes to be 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 cool This is difficult this system I see actually so instead of spin up spin down you have hair up hair down exactly Okay
Exactly. So let's say, okay, what's outside of our tramy box? Because we did with, uh, with, uh, Daniel, we have been working also on solutions on proving undecidable, uh, indecidability in free 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 that while pending,
What was pending for a long time is like, well, what about Navier-Stokes, right? Well, Navier-Stokes, the idea of 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, Plum Plum. 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, okay. Yeah, so it did look we have proved So what we got proofs on that and 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 a space in the space right thinking of this rather So no matter how potent your computational problem is, right? In other words, if you want the problem is not computable Okay
So, well, 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 Burnais, Daniel Grasse and Emmanuel Henry. Excuse me because I don't pronounce Henry well.
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. So. And that's very interesting, OK, because OK, 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, so this is
You can put here as flashy, you can put flashy neons there. Yes, this is this isn't it. Yeah, 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. OK.
So 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 cosimplectic geometry. And this is something that should be online soon, I hope very soon. But we know how to do it. And the idea again comes from geometry. So we have been able to produce a Turing complete
I have more things to explain you
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 El País, 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 from this solution to incomplete Navier-Stokes flow. This is something we have done with Ángel González Prieto from Universidad Complutense de Madrid and with Daniel Peraltas 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, assume 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 assume like 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 Ángel González Prieto doing pictures like this one, which are the picture 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, 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 Cosimple geometry is Navier-Stokes. 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 in 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 more 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 the downhill 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.
What's next? What's in our agenda? Well, of course we want to
What the blow-up of Navier Stokes was never in my agenda but now it's in the agenda somehow because this is a bit related. We are working, we are thinking around this with Daniel Peralta Salas in this picture and Ángel González-Pieto. We are here, all of us are smiling, this is good.
Yes. 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 yours here. Yeah. Yes. We have to arrange a visit of yours here. I look forward to it. So this is Terence Tao who visited us last September and we organized a conference and Terence was giving, yeah, he 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 these 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.
lost in the completely lost and now if i go back to the asteroid that's another question is 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 or not, 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. 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? 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, right? So you want to reconcile or you want to relate, you want to see if these two are related or not, right? We have understood many things about the relation looking at entropy, 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 girls, 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. Yeah. Thank you. Thank you, Eva. How about one quick question? Yeah. Do these fluid computers violate the church Turing thesis?
Do these Turing computers? No, no, no, no, that's a very, that's a super good. 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 of, uh, if this idea that if you have any system, all of these systems should be equivalent.
In terms 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, right? When we talk about quantum supremacy, we're talking about efficiency of computation.
So that would be another question. So 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. That you have several ways to like this church during 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, three years. Okay. Yes. That's what quantum computer does. So now what we want to do is to do this even quicker than a quantum computer using maybe this idea of, 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 my our students is going this week
With the 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 students, I'm going to tell his name, did a very nice simulation of these of these constructions for particular types. Now we want to do the to make this on a great scale and to be able to see
that this version, this topological game 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.
Hi there, Kurt here. If you'd like more content from Theories of Everything and the very best listening experience, then be sure to check out my sub stack at kurtjymungle.org. Some of the top perks are that every week you get brand new episodes ahead of time
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Well, while I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. And it's the perfect way to support me directly. KurtJaymungle.org or search KurtJaymungle substack on Google. Oh, and I've received several messages, emails and comments from professors and researchers saying that they recommend theories of everything to their students.
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You should know this podcast is on iTunes, it's on Spotify, it's on all the audio platforms. All you have to do is type in theories of everything and you'll find it. I know my last name is complicated, so maybe you don't want to type in Jymungle, but you can type in theories of everything and you'll find it.
Personally, I gain from rewatching lectures and podcasts. I also read in the comment that toll listeners also gain from replaying. So how about instead you relisten on one of those platforms like iTunes, Spotify, Google podcasts, whatever podcast catcher you use. I'm there with you. Thank you for listening.
▶ View Full JSON Data (Word-Level Timestamps)
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"text": " 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."
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"text": " 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."
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"text": " Welcome. Yes. Thank you very much. I'm excited to be here again. I'm so, so happy to be here and looking forward to this new adventure and ready to disclose something new. Let's see if people like it. I'm very happy about the, 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. And it's a great pleasure to be here with all of you now. Okay."
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"text": " I call this expect the unexpected. And what does it mean? Well, you know, we all know David Hilbert, the famous mathematician, right? Who said we, 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"
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"text": " his idea that everything could be formalized mathematically in a very very precise way this idea of precision of mathematics that is of course very important and formalization the idea of formalization of mathematics and what's very interesting is that Hilbert was in Göttingen I have been in Göttingen very recently"
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"text": " 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 indeed."
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"text": " There was David Hilbert, there was John von 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."
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"text": " 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."
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"text": " and i chose three different main characters today i chose alan turing i chose uh to explain as you said the chaos that the theory of chaos the classical theory of chaos and this is a nice story that can explain with butterfly and i want to explain also"
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"text": " How it affects our last month, we heard so much about all these asteroids that could fall on us. So 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."
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"text": " And I will connect this to undecidable fluid paths. As you see here, I chose a little a little rubber duck and the rubber duck is going 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 the the tack for classical games. So interesting. Let's see."
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"text": " How do I get to Alan Turing? I was talking about Alonso Chart 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."
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"text": " And the funny thing is that charge 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 Bletchley Park."
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"text": " 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."
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"text": " 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"
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"text": " Add algorid something that is on repetitive now imagine you entering a labyrinth and trying to getting 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 of the main okay and the question is the following is there a general recipe and algorid that tells us whether"
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"text": " 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 like, 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? And or in a way, if we think of modern computers, OK,"
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"text": " Is there a super computer that can tell us if a given computer somewhere on the world will ever will stop which reach 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."
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"text": " 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, 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."
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"text": " This is pretty surprising right so this contradicts a little bit this straight 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 no we"
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"text": " We can 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. OK, but the answer does exist."
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"text": " Well, we don't know if they answered it. No, mathematically, 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 like the question is very strange. The existence of super computer that tells you if a given computer on the world with some initial data will ever stop or not."
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"text": " 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 his thumb 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"
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"text": " That's 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,"
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"text": " You end up feeding the algorithm with the old 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."
},
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"text": " 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."
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"text": " 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 computer began. So that's quite amazing and amazing everything Alan Turing did, cracking the Enigma code."
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"text": " uh creating you know the the first theoretical model of computers and indeed his death was uh quite sad indeed uh you know the fact that he and uh and his team at blesley park break the enigma code was a secret for decades so when he when he when he died this was unknown he was not a hero at that moment"
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"text": " And however, all 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, the Turing Award. And this started in 1966. So to the Turing Award is the most important recognition in computer science. Summarizing a little bit, right, we go from this David Hilbert's"
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"text": " You know so zones without without where we don't have internet where we don't have connection with our phones zones where we cannot know songs of darkness because logically there is no way to know under these i put the example of touring bad girls incompleteness theorems okay is another example and this goes back to nineteen thirty one okay."
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"text": " So it did get us ideas at any sufficiently powerful mathematical system is incomplete. Okay, so It did more or less let's say we go from certainty to uncertainty We go from this idea that there is always and yes, no answer to we cannot know Okay, and now I want to explain this idea of on the side of things that are undecidable in a playful way So now I'm going to show you are not"
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"text": " And this is the art of a car and in this car, they explains a story of a, of a ship 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,"
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"text": " Thousands of that's 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? 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"
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"text": " 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."
},
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"text": " 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 29 000 rubber ducks"
},
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"text": " But they were lost because of a storm. 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 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."
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"text": " 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."
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"start_time": 1013.507,
"text": " i appeared in scotland right so the truth is that just one of them appeared in scotland so there is some truth about the advertisement that of course based on true stories right so these 29 29 000 rubber ducks there is a very interesting story about them indeed you can buy i i was showing here before you can buy a book which is movie doc"
},
{
"end_time": 1060.435,
"index": 42,
"start_time": 1039.104,
"text": " okay which is the story tells you the story of the of the ducks and everything that has been that these ducks have uh helped to science because indeed uh there was a computer simulator to follow all the flotsam lost on the on the seas"
},
{
"end_time": 1090.93,
"index": 43,
"start_time": 1060.93,
"text": " which was developed"
},
{
"end_time": 1120.452,
"index": 44,
"start_time": 1090.93,
"text": " And, well, on one hand, thanks to the rubber duck, the friendly floaties, these rubber ducks, predictions about currents could be made, but not all the rubber ducks could be found. And 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."
},
{
"end_time": 1146.254,
"index": 45,
"start_time": 1121.561,
"text": " so how do you know that only two percent because the message inside the bottle is you will get fifty fifty fifty dollars if you give us this note so maybe it's quite accurate that it's two percent so what does touring have to do with the rubber ducks and why i'm telling you this story of the rubber ducks because indeed there is this"
},
{
"end_time": 1169.019,
"index": 46,
"start_time": 1146.715,
"text": " The method could not localize all these lost rubber ducks. Only 2% of the messages of bottles are recovered."
},
{
"end_time": 1191.664,
"index": 47,
"start_time": 1169.582,
"text": " 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,"
},
{
"end_time": 1221.305,
"index": 48,
"start_time": 1191.988,
"text": " How can I say that finding the rubber ducks is an undecidable problem? Well, remember that during proof that the whole thing problem It's a 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 that has been lost and very happy to be lost 15 years on the sea. He's in a cool vibe and"
},
{
"end_time": 1248.131,
"index": 49,
"start_time": 1221.647,
"text": " 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"
},
{
"end_time": 1275.418,
"index": 50,
"start_time": 1248.643,
"text": " 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. Why you cannot predict maybe i cannot predict because i don't have enough information."
},
{
"end_time": 1289.428,
"index": 51,
"start_time": 1275.93,
"text": " 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"
},
{
"end_time": 1313.148,
"index": 52,
"start_time": 1289.906,
"text": " Don't be like the economy, instead read the economist. I thought all the economist was was something that CEOs read to stay up to date on world trends. And that's true, but that's not only true. What I found more than useful for myself personally is their coverage of math, physics, philosophy, and AI, especially how something is perceived by other countries and how it may impact markets."
},
{
"end_time": 1337.176,
"index": 53,
"start_time": 1313.148,
"text": " For instance the economist had an interview with some of the people behind deep seek the week deep seek was launched no one else had that another example is the economist has this fantastic article on the recent dark energy data which surpasses even scientific americans coverage in my opinion they also have the charts of everything like the chart version of this channel it's something which is a pleasure to scroll through and learn from."
},
{
"end_time": 1355.026,
"index": 54,
"start_time": 1337.176,
"text": " Links to all of these will be in the description of course. Now the economist commitment to rigorous journalism means that you get a clear picture of the world's most significant developments. I am personally interested in the more scientific ones like this one on extending life via mitochondrial transplants which creates actually a new field of medicine."
},
{
"end_time": 1377.108,
"index": 55,
"start_time": 1355.026,
"text": " Something that would make Michael Levin proud. The Economist also covers culture, finance and economics, business, international affairs, Britain, Europe, the Middle East, Africa, China, Asia, the Americas, and of course the USA. Whether it's the latest in scientific innovation or the shifting landscape of global politics, The Economist provides comprehensive coverage"
},
{
"end_time": 1404.138,
"index": 56,
"start_time": 1377.108,
"text": " and it goes far beyond just headlines. Look, if you're passionate about expanding your knowledge and gaining a new understanding, a deeper one of the forces that shape our world, then I highly recommend subscribing to The Economist. I subscribe to them and it's an investment into my, into your intellectual growth. It's one that you won't regret. As a listener of this podcast, you'll get a special 20% off discount. Now you can enjoy The Economist and all it has to offer."
},
{
"end_time": 1432.159,
"index": 57,
"start_time": 1404.138,
"text": " Ford BlueCruise hands-free highway driving takes the work out of being behind the wheel allowing you to relax and reconnect"
},
{
"end_time": 1458.933,
"index": 58,
"start_time": 1432.807,
"text": " So, let's talk about classical chaos. The butterfly effect is a film. In that film,"
},
{
"end_time": 1487.381,
"index": 59,
"start_time": 1459.258,
"text": " The main character wants to change something of his press on 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."
},
{
"end_time": 1516.459,
"index": 60,
"start_time": 1487.671,
"text": " 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,"
},
{
"end_time": 1542.517,
"index": 61,
"start_time": 1516.954,
"text": " 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"
},
{
"end_time": 1569.07,
"index": 62,
"start_time": 1542.824,
"text": " and this path can be the solution of is a solution of a differential equation corresponding to some physical problem okay 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"
},
{
"end_time": 1587.995,
"index": 63,
"start_time": 1569.07,
"text": " You see now we start to see it looks like a big ball okay now we see two balls well with my glass 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."
},
{
"end_time": 1614.326,
"index": 64,
"start_time": 1588.439,
"text": " Of these three balls, they are very far away. So think of these trajectories of these three balls of 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 he did the idea of, uh, of chaos, chaos theory."
},
{
"end_time": 1620.691,
"index": 65,
"start_time": 1614.753,
"text": " was discovered in a way by chance and it was discovered"
},
{
"end_time": 1651.101,
"index": 66,
"start_time": 1621.152,
"text": " 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. OK. And then they were they were finding that some data didn't make sense."
},
{
"end_time": 1661.527,
"index": 67,
"start_time": 1651.374,
"text": " Okay, 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."
},
{
"end_time": 1684.718,
"index": 68,
"start_time": 1661.954,
"text": " Okay and this goes very well with the idea of weather i mean we cannot predict the weather in any in any accurate way more than seven days well you can say ten days fifteen days not very very precisely and the problem of prediction of weather is related to these equations of lawrence indeed so when you ask what is the weather going to be."
},
{
"end_time": 1713.2,
"index": 69,
"start_time": 1685.299,
"text": " That's a difficult question, right? So let's keep in mind this definition that Edward Lawrence 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."
},
{
"end_time": 1739.957,
"index": 70,
"start_time": 1713.541,
"text": " So that's the idea. 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. And that's the idea of chaos. That's the idea of classical chaos. So now let me talk about something real that we were living some months ago. We were living this in December last year."
},
{
"end_time": 1762.466,
"index": 71,
"start_time": 1741.015,
"text": " 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 okay. And we were very worried until some scientists came out with this map and then we were thinking are we away from this danger zone right."
},
{
"end_time": 1791.971,
"index": 72,
"start_time": 1762.944,
"text": " 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 one point one point. In February it was getting high two point three point one on February 19. OK."
},
{
"end_time": 1807.108,
"index": 73,
"start_time": 1792.415,
"text": " I remember on February 19 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."
},
{
"end_time": 1825.964,
"index": 74,
"start_time": 1807.261,
"text": " on the other hand you know 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 right"
},
{
"end_time": 1851.578,
"index": 75,
"start_time": 1826.459,
"text": " 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"
},
{
"end_time": 1882.022,
"index": 76,
"start_time": 1852.108,
"text": " I remember I had that week I had to give a talk and I was thinking where is it going to fall while most likely is going to fall on water and this is what what happened okay crashed back on earth overnight on water everything went fine so indeed 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"
},
{
"end_time": 1901.237,
"index": 77,
"start_time": 1882.398,
"text": " And the idea of chaos is present indeed there is a whole theory called KM theory which is mathematical theory that controls that allows us to give some you know some probability that things are going to be alright somehow so going back to your question."
},
{
"end_time": 1925.811,
"index": 78,
"start_time": 1901.903,
"text": " Unpredictable can be for two reasons. One because you don't have enough information about the initial conditions and then after a certain time because of chaos things can diverge completely or maybe you knock on a door and there is no answer."
},
{
"end_time": 1953.524,
"index": 79,
"start_time": 1925.913,
"text": " uh 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"
},
{
"end_time": 1980.299,
"index": 80,
"start_time": 1954.036,
"text": " Now I'm going to provide some examples that a little bit go in this in this idea, right? We've seen that fluids like water or lava often rebel against as what's expected, right? We've seen this with tsunamis. We've seen this recently. I mean, nature is revealing, right? It's rebelling against against what's expecting. And the question is,"
},
{
"end_time": 2009.872,
"index": 81,
"start_time": 1980.998,
"text": " 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 could 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 in his famous book."
},
{
"end_time": 2035.247,
"index": 82,
"start_time": 2010.589,
"text": " And here we have Prismore 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."
},
{
"end_time": 2062.619,
"index": 83,
"start_time": 2035.691,
"text": " uh, one of the most famous mathematicians, or I would say the most famous mathematicians of the world nowadays, uh, who is professor at UCLA. And he asked again this question and his motivation was a different one. His motivation was, can I use these, uh, computational power to answer one of the unknown questions"
},
{
"end_time": 2091.834,
"index": 84,
"start_time": 2063.166,
"text": " Because we mathematicians sometimes we cannot answer because we don't know how to do the things. That's true. I mean, it's like 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. And one of them is the Navier-Stokes riddle. Navier-Stokes equations have been used forever"
},
{
"end_time": 2122.585,
"index": 85,
"start_time": 2092.585,
"text": " to model the movement of fluids. And 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."
},
{
"end_time": 2149.548,
"index": 86,
"start_time": 2122.995,
"text": " 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 whether these equations have solutions or not,"
},
{
"end_time": 2176.766,
"index": 87,
"start_time": 2149.753,
"text": " and while 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 turnstile try to prove that these equations have blow up"
},
{
"end_time": 2206.425,
"index": 88,
"start_time": 2177.108,
"text": " Using this idea of associating a computer to the movement of fluids Okay, 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 are which he called the average Navier-Stokes because there were some integrals Okay, but the method"
},
{
"end_time": 2232.329,
"index": 89,
"start_time": 2206.783,
"text": " He couldn't, he tried to use exactly the same method for Navier-Stokes. Navier-Stokes, it's so hard and it didn't work, but he still, he raised the question, okay, I cannot find the 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."
},
{
"end_time": 2252.978,
"index": 90,
"start_time": 2233.268,
"text": " 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 so you can have a billiard table with billiard balls and you can associate a Turing machine with"
},
{
"end_time": 2274.309,
"index": 91,
"start_time": 2253.677,
"text": " With the bouncing of the billiard balls. Okay. So then you think, can we do this with any physical system? 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?"
},
{
"end_time": 2300.947,
"index": 92,
"start_time": 2274.957,
"text": " yeah well you you almost got it you almost got it the last thing it's a bit more delicate so 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 use this idea to produce this blow up is much more"
},
{
"end_time": 2329.343,
"index": 93,
"start_time": 2301.527,
"text": " 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 have, you are amplifying to the maximum the choices of initial conditions."
},
{
"end_time": 2348.268,
"index": 94,
"start_time": 2329.497,
"text": " Okay and he wanted to put some conditions off. I will comment that later. Some additional conditions and 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."
},
{
"end_time": 2378.37,
"index": 95,
"start_time": 2348.66,
"text": " 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 around that point. So this idea of concentration of energy, 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"
},
{
"end_time": 2386.015,
"index": 96,
"start_time": 2379.053,
"text": " But Turing machines are discrete, so do you have to put some bounds on whatever your initial conditions are or some constraints?"
},
{
"end_time": 2415.043,
"index": 97,
"start_time": 2386.288,
"text": " yes i mean yeah i mean yeah yeah indeed you are totally right we are going ahead i i plan to explain this i need to explain yeah yeah no thanks a lot this is a very an excellent question and thank you for the question so exactly we need to go to from discrete to continuum and how are we going to do it i need to explain that it's like uh it's like this is a very good question indeed so this is exactly i mean you are guessing my mind this is fantastic"
},
{
"end_time": 2421.049,
"index": 98,
"start_time": 2415.708,
"text": " So this is exactly what I'm going to explain now. I'm going to explain indeed what Moore did in his thesis."
},
{
"end_time": 2446.834,
"index": 99,
"start_time": 2421.442,
"text": " You know what, like the dream of a PhD student, right, is to wake up one day and discover people care a lot about your thesis. And this happened to Chris Moore, like he woke up one day and then, 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 fantastic title, right?"
},
{
"end_time": 2475.862,
"index": 100,
"start_time": 2448.234,
"text": " 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."
},
{
"end_time": 2505.333,
"index": 101,
"start_time": 2476.766,
"text": " I have a presentation of the counter set later. It's like it is the following. You take the interval. You split it in three and you drop the middle part and you continue. 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, 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."
},
{
"end_time": 2526.954,
"index": 102,
"start_time": 2505.845,
"text": " whatever stays up there is the counter set but you 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 uh 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"
},
{
"end_time": 2552.142,
"index": 103,
"start_time": 2527.517,
"text": " I'm using that this animation too. So whatever stays up there is the counter set and the car. So 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"
},
{
"end_time": 2579.019,
"index": 104,
"start_time": 2552.654,
"text": " You know, in terms of taking the one dimensional counter set, I think the two dimensional counter set is like I multiply, take an axis, the X axis, which is the counter set and the Y axis is the counter set. OK, then the points that you can describe with the X and Y axis are going to be in what is called square counters. So what he discovered is that it's the same"
},
{
"end_time": 2605.623,
"index": 105,
"start_time": 2579.65,
"text": " The following two facts are the same. It's the same to disorganize this square counter set in a particular way that to compute with a Turing machine. So that 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."
},
{
"end_time": 2633.951,
"index": 106,
"start_time": 2606.22,
"text": " 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."
},
{
"end_time": 2654.787,
"index": 107,
"start_time": 2634.343,
"text": " Running a business comes with a lot of what ifs."
},
{
"end_time": 2684.804,
"index": 108,
"start_time": 2655.128,
"text": " But luckily, there's a simple answer to them. Shopify. It's the commerce platform behind millions of businesses, including Thrive Cosmetics and Momofuku, and it'll help you with everything you need. From website design and marketing to boosting sales and expanding operations, Shopify can get the job done and make your dream a reality. Turn those what-ifs into... Sign up for your $1 per month trial at Shopify.com slash special offer. I need to think about Turing machines."
},
{
"end_time": 2713.046,
"index": 109,
"start_time": 2685.077,
"text": " And I don't want to get very technical about what a Turing machine is. But all of us think of Turing machine as a long, long tape, right? Long, long, long, long tape with zeros and ones. OK, 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."
},
{
"end_time": 2729.701,
"index": 110,
"start_time": 2713.456,
"text": " 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"
},
{
"end_time": 2759.394,
"index": 111,
"start_time": 2730.435,
"text": " comes with some instructions and instructions tell you where to move left of 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 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 uh assignment"
},
{
"end_time": 2789.735,
"index": 112,
"start_time": 2759.906,
"text": " that changes the initial tape you want to change the zero by one okay and you want to say that this plus one 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 okay so this tells you that you need to that the current state is Q"
},
{
"end_time": 2819.48,
"index": 113,
"start_time": 2790.333,
"text": " prints on zero and now you change the zero by one and you move the tape you see to the left so the printer moves to the right this is the effect so let me show you again this this animation which is not exactly the same animation there is something i found on the internet you change zero one one and then the printer moves to the right so you have this plus one so that's the most basic form of computation"
},
{
"end_time": 2847.398,
"index": 114,
"start_time": 2820.094,
"text": " 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. About what a mathematician would do with a touring with a bed this long tape is to try to put things in form in a mathematical form. A way to do it would be OK. Now I have a lot of zeros and ones, so I'm going to do two packs."
},
{
"end_time": 2868.933,
"index": 115,
"start_time": 2848.2,
"text": " 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 that are going to be an infinite collection of zeros and ones, and now, attention, it comes the miracle. I think of these numbers as coefficients of a series."
},
{
"end_time": 2893.848,
"index": 116,
"start_time": 2869.138,
"text": " 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 okay and then because of this ternary expand here where i have always two present observe that the the outcome of of the names here in which i plug zeros and ones okay i'm here putting my finger on the"
},
{
"end_time": 2922.312,
"index": 117,
"start_time": 2894.224,
"text": " 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 one or a zero? So there's no blank. Well, the blanks you can put, 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."
},
{
"end_time": 2950.879,
"index": 118,
"start_time": 2922.619,
"text": " but you can forget and put the zeros on one. 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 zero? Yeah, exactly. I would replace, for instance, a blank with a zero. So I just have some numbers and I think of these numbers as the coefficients of a ternary expand."
},
{
"end_time": 2977.756,
"index": 119,
"start_time": 2951.476,
"text": " 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"
},
{
"end_time": 2998.097,
"index": 120,
"start_time": 2978.319,
"text": " 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. The second number is two over nine"
},
{
"end_time": 3020.913,
"index": 121,
"start_time": 2998.575,
"text": " 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 Moore. Go to a Turing machine with your phone, take a picture. Okay."
},
{
"end_time": 3033.916,
"index": 122,
"start_time": 3021.425,
"text": " And when you take this picture, take the numbers of zeros and ones, this produces a point on the square counter set. So the what I have explained now is that."
},
{
"end_time": 3062.517,
"index": 123,
"start_time": 3034.48,
"text": " just a picture of a Turing machine working on a certain moment is the same as a point on the square counters. 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."
},
{
"end_time": 3090.418,
"index": 124,
"start_time": 3062.944,
"text": " 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."
},
{
"end_time": 3119.394,
"index": 125,
"start_time": 3091.459,
"text": " That's one of the conditions. OK, so now it comes to and now I'm going to answer your question that was totally excellent to the point before like 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, it's moving on a continuous way. So you have to relate"
},
{
"end_time": 3149.735,
"index": 126,
"start_time": 3120.111,
"text": " you have"
},
{
"end_time": 3161.22,
"index": 127,
"start_time": 3150.162,
"text": " So you have in a way you have to associate to you need to mark some points on the trajectory and moving."
},
{
"end_time": 3185.486,
"index": 128,
"start_time": 3161.51,
"text": " 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."
},
{
"end_time": 3215.418,
"index": 129,
"start_time": 3185.879,
"text": " and the way to define this is okay 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 and 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"
},
{
"end_time": 3241.425,
"index": 130,
"start_time": 3217.329,
"text": " 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"
},
{
"end_time": 3264.428,
"index": 131,
"start_time": 3241.766,
"text": " 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"
},
{
"end_time": 3294.974,
"index": 132,
"start_time": 3265.009,
"text": " of the"
},
{
"end_time": 3323.097,
"index": 133,
"start_time": 3295.794,
"text": " Right? And now you say, OK, I don't understand anything. Wait, you do, because you know that the holding problem is undecidable. OK, what did I say? I'm giving a definition that associates 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."
},
{
"end_time": 3353.046,
"index": 134,
"start_time": 3323.507,
"text": " OK, then I say that this association is good if the whole thing is equivalent to the trajectory entering and opens. But I know that the whole thing 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? So take a look at this red dot here."
},
{
"end_time": 3380.196,
"index": 135,
"start_time": 3353.404,
"text": " 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 yes it's going to do this content yeah yeah that's correct but what we are doing is to extend this idea so that it does this as smoothly i totally agree with you"
},
{
"end_time": 3402.398,
"index": 136,
"start_time": 3381.084,
"text": " Okay, so then when we scroll down now to the next slide, the slide after the next one, 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"
},
{
"end_time": 3423.148,
"index": 137,
"start_time": 3402.91,
"text": " 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."
},
{
"end_time": 3451.544,
"index": 138,
"start_time": 3423.404,
"text": " uh 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 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 i i really want to show you how we did it because we answered the question of terian style i mean we answered his question i indeed i sent him an email dear terry we know how to do that okay"
},
{
"end_time": 3474.087,
"index": 139,
"start_time": 3451.783,
"text": " Sorry then it's like this idea like it's quite wild because you have to jump from the screen to 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."
},
{
"end_time": 3500.964,
"index": 140,
"start_time": 3474.377,
"text": " 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 would be helpful to the audience if you don't mind okay so here i just coated up something using clod 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"
},
{
"end_time": 3527.534,
"index": 141,
"start_time": 3501.305,
"text": " Okay, so let's press play and the Turing machine is going to act and then you see it just jumping around. This is exactly what I, this is fantastic. This is 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. Yeah. But if you represent that, that's very good when you have that. No, this is amazing. This is amazing. This is great. Oh my God. This is fantastic."
},
{
"end_time": 3557.227,
"index": 142,
"start_time": 3528.422,
"text": " uh indeed when you do this the mapping that richard more that uh that chris more that chris more is saying is um the mapping that assigns the point to the point okay and you have you have been playing with it and you have been doing these kind of jumps and because this is discrete the funny thing is that you can extend and that's the point these maps this is a map"
},
{
"end_time": 3584.957,
"index": 143,
"start_time": 3557.688,
"text": " And that's exactly the precise way to say it. That's a map between two square countersets. I can think that this is these two square countersets are points that you are doing inside a square. You agree with me. These live inside the square. OK, so this mapping and this is going to make your mind blow like this mapping is this continuous when you think like when you see this job,"
},
{
"end_time": 3609.343,
"index": 144,
"start_time": 3585.606,
"text": " 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, yeah, I find this amazing. So you did that two very good questions. The final one is that you were asking me"
},
{
"end_time": 3630.538,
"index": 145,
"start_time": 3610.555,
"text": " If this was canonical and the answer is no, you have many, many ways to produce this. Okay. And 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 turns down is using"
},
{
"end_time": 3656.527,
"index": 146,
"start_time": 3631.271,
"text": " 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 data. I cannot do it with the sprint data."
},
{
"end_time": 3685.384,
"index": 147,
"start_time": 3656.783,
"text": " 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. So I imagine that's going to get associated with the symplectic. That's that's that's great. That's great. Exactly. That's 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."
},
{
"end_time": 3715.23,
"index": 148,
"start_time": 3686.101,
"text": " 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 and 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."
},
{
"end_time": 3741.51,
"index": 149,
"start_time": 3715.623,
"text": " And this was answered by Gregory Paramon, who you have here, who answered correctly this question, but refused to get $1 million. Okay. 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, you've talked, I think, maybe about the Riemann hypothesis in one of your form. No."
},
{
"end_time": 3760.026,
"index": 150,
"start_time": 3741.971,
"text": " Yes, exactly. I saw the one of Frankel. It's amazing. And so you've heard part of it."
},
{
"end_time": 3789.309,
"index": 151,
"start_time": 3760.333,
"text": " So what happens is that what is the next big problem, next riddle that is going to to become known as 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, OK, all these problems are pending. So if you can solve these problems,"
},
{
"end_time": 3818.712,
"index": 152,
"start_time": 3789.974,
"text": " 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."
},
{
"end_time": 3844.138,
"index": 153,
"start_time": 3818.712,
"text": " 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 dot org, KurtJaimengel dot org. So Navier-Stokes equation. Let's talk about Navier-Stokes equations."
},
{
"end_time": 3870.247,
"index": 154,
"start_time": 3844.292,
"text": " These are the equations that model the motion of fluids, which are 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,"
},
{
"end_time": 3893.114,
"index": 155,
"start_time": 3870.64,
"text": " Okay."
},
{
"end_time": 3923.285,
"index": 156,
"start_time": 3893.473,
"text": " the case in which you have viscosity 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 stout was asking if he was asking can i find some initial conditions of navier stokes that are touring complete oiler flows"
},
{
"end_time": 3951.544,
"index": 157,
"start_time": 3923.763,
"text": " 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,"
},
{
"end_time": 3973.353,
"index": 158,
"start_time": 3951.988,
"text": " 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 the formulation of the questions, you can get the formal formulation if you go to the website of the Clay Foundation."
},
{
"end_time": 3992.705,
"index": 159,
"start_time": 3973.729,
"text": " 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"
},
{
"end_time": 4020.179,
"index": 160,
"start_time": 3993.302,
"text": " Give rise to smooth solutions. Well, these initial conditions, if you go to the formulation of effort, but have some natural constraints that are given with by some equations, but correspond to physical systems. OK, so we want that to determine if all initial conditions give rise to a smooth solutions that evolve smoothly. OK."
},
{
"end_time": 4043.2,
"index": 161,
"start_time": 4021.203,
"text": " 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. We are three dimensional."
},
{
"end_time": 4065.998,
"index": 162,
"start_time": 4043.66,
"text": " 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, in a way, if you think about this, the fact that these equations, if these equations, if we are able to prove that these blow up or these explosions exist,"
},
{
"end_time": 4096.869,
"index": 163,
"start_time": 4067.142,
"text": " Hola, Miami! When's the last time you've been in Burlington? We've updated, organized and added fresh fashion. See for yourself Friday, November 14th to Sunday, November 16th at our Big Deal event. You can enter for a chance to win free wawa gas for a year, plus more surprises in your Burlington. Miami, that means so many ways and days to save. Burlington. Deals. Brands. Wow! No purchase necessary. Visit bigdealevent.com for more details. Then, what this does tell me?"
},
{
"end_time": 4125.486,
"index": 164,
"start_time": 4097.295,
"text": " 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. Okay. Tsunamis. I mean, you can get them some minutes before some time before not a lot of time before, but you have some"
},
{
"end_time": 4148.063,
"index": 165,
"start_time": 4126.032,
"text": " information so then this is telling us this 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. Now why can't someone say well look even from your old podcast yeah"
},
{
"end_time": 4168.131,
"index": 166,
"start_time": 4148.439,
"text": " We talked about quantum mechanics, so it's clear that Navier-Stokes isn't fundamental. Quantum mechanics is more fundamental. There may be something underneath that, but what if someone just retorts at that? Yeah, I don't know. Don't make me choose. I think both are fundamental problems."
},
{
"end_time": 4193.763,
"index": 167,
"start_time": 4168.677,
"text": " You know for if you ask this question for somebody who has been working in average talks or 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. This is a but I think this is a I mean, this is an important problem This is one of the seven problem seven problems in the list for the Millennium Chosen by the Clive Foundation. So this is important"
},
{
"end_time": 4223.899,
"index": 168,
"start_time": 4194.531,
"text": " I agree that 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. Yes. OK, let's go on. So as I said, this problem is 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"
},
{
"end_time": 4254.275,
"index": 169,
"start_time": 4225.009,
"text": " Was solved by all the ladies and skyer already in 1958 She proved that the solutions were always a 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 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"
},
{
"end_time": 4284.241,
"index": 170,
"start_time": 4254.701,
"text": " Okay. You can try to use artificial intelligence and that's a big, a big, big, uh, approach to this problem. Okay. But let me talk, talk, uh, to about one approach to this problem, which is the approach that Terence Tao had in 2019 in which he thought, okay, I, this is an approach to find a control example. This means to find a construction where there is blow up. Okay."
},
{
"end_time": 4312.432,
"index": 171,
"start_time": 4284.991,
"text": " And here I have turnstiles, right, working hard. And here I have the Matrioshkas, which is like a copy of, you know, of a doll. You open the doll, you get another doll, a doll, another doll, another 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."
},
{
"end_time": 4329.633,
"index": 172,
"start_time": 4312.91,
"text": " 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."
},
{
"end_time": 4358.012,
"index": 173,
"start_time": 4330.247,
"text": " And then in his counter example, he was working by hand. Okay. 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 flows, to the Euler equations?"
},
{
"end_time": 4383.541,
"index": 174,
"start_time": 4359.701,
"text": " So the equations, the earlier equations would be the case of Navier-Stokes when the viscosity is zero. Okay. And that's, 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"
},
{
"end_time": 4406.732,
"index": 175,
"start_time": 4384.07,
"text": " Thinking ocean in a way this is what we are trying to find a thinking ocean right in a way i'm thinking computing ocean. So. Let's go to this idea now i want to show you how i answered how we answered into the question of turnstile with a yes."
},
{
"end_time": 4431.63,
"index": 176,
"start_time": 4407.056,
"text": " 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 equations? 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."
},
{
"end_time": 4461.869,
"index": 177,
"start_time": 4432.483,
"text": " 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 the hydrodynamics is capable of performing computations."
},
{
"end_time": 4491.63,
"index": 178,
"start_time": 4462.176,
"text": " so he asked whether you know we can use fluids to compute yes and now we have this story of the 29 000 rubber ducks lost in the ocean okay uh in 92 okay 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"
},
{
"end_time": 4519.667,
"index": 179,
"start_time": 4492.056,
"text": " and Francisco Presas, we proved that there exist solutions of the Euler flow in dimension three, which can simulate any Turing machine. And that's the way, 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"
},
{
"end_time": 4541.766,
"index": 180,
"start_time": 4520.538,
"text": " 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."
},
{
"end_time": 4572.142,
"index": 181,
"start_time": 4542.466,
"text": " So now these answers, I think it's going to clarify your question about how to go from discrete to continuous. OK. And. 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, which was, OK, more had worked on the squares."
},
{
"end_time": 4600.026,
"index": 182,
"start_time": 4572.363,
"text": " Squares are on dimension two. 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 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"
},
{
"end_time": 4626.613,
"index": 183,
"start_time": 4600.538,
"text": " When it comes back, each time it comes back, it corresponds exactly to the mapping of Chris Moore. So this is indeed very well known in mathematics and it's called the Poincare section. So we think of Moore's transformation as a Poincare section of a vector field, well, of a velocity field."
},
{
"end_time": 4655.145,
"index": 184,
"start_time": 4627.176,
"text": " I see. You see?"
},
{
"end_time": 4683.916,
"index": 185,
"start_time": 4655.93,
"text": " And that's how you go. Then you say, OK, but the mapping of more, if you are taking the velocity of a fluid, then the intersection is not necessarily. Is it on the counter or is not OK? What we do is to extend this initial mapping on the square counter set to a mapping on the disk. To go from the screen to continuous. So you're creating a terrible not. I'm well, yes."
},
{
"end_time": 4710.213,
"index": 186,
"start_time": 4684.411,
"text": " Yes, indeed. And then you see the rubber duck. So each 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 duck. 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."
},
{
"end_time": 4738.012,
"index": 187,
"start_time": 4711.22,
"text": " Yes, OK. By the way, what I just said, so I was analogizing it to a knot, but 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. OK. Indeed, we are thinking more of this going back to this to this. So you see the connection to go from dimension two to dimension three. That's the idea of what we do. Yeah."
},
{
"end_time": 4768.166,
"index": 188,
"start_time": 4738.695,
"text": " Okay. And then, uh, in order to perform a certain computation, if you think of a Turing machine, maybe you have to go infinite number of times. It's like, if you think of this method as a method to compute, it's not, it looks, it looks very strange, but it's, it would be a computational method at the end of the day. You can think that you are representing, it's a, it's a representation of the Turing machine. So this movement of the, of the, of the particle."
},
{
"end_time": 4797.739,
"index": 189,
"start_time": 4768.729,
"text": " Okay, 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. Okay, 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."
},
{
"end_time": 4825.708,
"index": 190,
"start_time": 4798.097,
"text": " 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."
},
{
"end_time": 4850.111,
"index": 191,
"start_time": 4826.186,
"text": " and here we change the way to measure and given any metric we have some associated Euler equations and this modification of the metric is important because there is a correspondence between red vector fields and solutions to the Euler equations"
},
{
"end_time": 4879.735,
"index": 192,
"start_time": 4850.503,
"text": " which are Beltrami fields. And 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."
},
{
"end_time": 4905.247,
"index": 193,
"start_time": 4880.265,
"text": " And then Beltrami fields will be a particular type of a 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."
},
{
"end_time": 4929.889,
"index": 194,
"start_time": 4905.572,
"text": " that you can associate to symplectic geometry indeed it's the odd equivalent of symplectic geometry and the way that it's usually represented again thank you very much uh grist for giving me robert grice for giving me this this picture this picture represents the what is called the contact the contact yeah the contact the structure is"
},
{
"end_time": 4958.882,
"index": 195,
"start_time": 4930.179,
"text": " 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 a contact form alpha. So here, the important thing is that"
},
{
"end_time": 4987.363,
"index": 196,
"start_time": 4959.206,
"text": " There is 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 solution, a stationary solution of the Euler equations to a red vector field in contact geometry. This is"
},
{
"end_time": 5014.77,
"index": 197,
"start_time": 4988.012,
"text": " Fantastic. This is a fantastic idea. And indeed, I was showing equations, but maybe we don't need to show equation. It's a concept. So this is what I called a mirror. OK, this was proved indeed by Robert Greist and John Endier long time ago in 2000. And the first time I learned about this correspondence, it was in the in one in a mini course that"
},
{
"end_time": 5038.746,
"index": 198,
"start_time": 5015.367,
"text": " Daniel Peralta Salah was teaching and I was attending that mini course and 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 we started to, this is how we started to collaborate with this idea. This is a beautiful, beautiful theorem that tells you that there is a correspondent. It is a magic mirror that I expressed."
},
{
"end_time": 5062.193,
"index": 199,
"start_time": 5039.121,
"text": " here in a 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. Okay."
},
{
"end_time": 5093.302,
"index": 200,
"start_time": 5063.558,
"text": " 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 And 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"
},
{
"end_time": 5122.466,
"index": 201,
"start_time": 5093.524,
"text": " and this is the mapping. This is the famous mapping in this puzzle of the character sets. And then what we do is to extend this from dimension two to dimension three, first to extend also these from discrete to continuous. OK. And then look for this vector field that that that solves the right equation. OK. And then so we solve this problem geometrically. OK. You think this contact geometry, which is a which is the other dimensional version of simplistic geometry."
},
{
"end_time": 5149.855,
"index": 202,
"start_time": 5123.046,
"text": " 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, is a solution of the Euler equation, is a Beltrami field. Okay. And this Beltrami field is too incomplete. Why? Because the initial construction of Chris Moore was too incomplete and the extension that we are doing with this is also too incomplete. So the proof, it's very easy to understand. I have"
},
{
"end_time": 5178.456,
"index": 203,
"start_time": 5150.435,
"text": " 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 theory? And then what are the consequences of this theory? Well, the theory we prove is that there exists solutions to the other equations that can simulate any Turing machine. Okay, but Turing had proved in 36 that the holding problem is undecidable. Therefore, if we put everything together,"
},
{
"end_time": 5207.756,
"index": 204,
"start_time": 5179.753,
"text": " because we know that 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 important corollary. There exists undecidable fluid paths and this was something that nobody was expecting because you were not expecting"
},
{
"end_time": 5234.838,
"index": 205,
"start_time": 5208.353,
"text": " to have out of some equations that are written and you say you have to have a solution 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, right? And now we apply this now assume that the machine that we have got this fluid computer"
},
{
"end_time": 5260.555,
"index": 206,
"start_time": 5235.094,
"text": " Corresponds to the movement of this rubber ducks that this is not the case, but let's assume that it does Okay This is not the case because in our case we change the metric in a in a very small place So this is not 100% physical but this would explain why this rubber ducks did not show up in in the UK as Everybody was waiting. Okay indeed"
},
{
"end_time": 5283.422,
"index": 207,
"start_time": 5261.476,
"text": " These rubber ducks, I have used them as an excuse, you know, 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 also construction and that construction therefore is 100% physical."
},
{
"end_time": 5302.176,
"index": 208,
"start_time": 5283.422,
"text": " It seems like this is revolutionary. So is this the first example of standard chaos? Sorry, not standard chaos, not logical chaos. No, this is not the same. No, this is not the first example. So let's say you have examples of undecidable physical systems, for instance, the spectral gap."
},
{
"end_time": 5330.52,
"index": 209,
"start_time": 5302.483,
"text": " This was proved to be undecidable before in 2000 or something. And if you go to Wikipedia, you look for undecidable. OK, there is a list of undecidable problems. And I'm very happy to know that somebody added who are problems in the list. So that's nice. OK, so we are 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."
},
{
"end_time": 5359.838,
"index": 210,
"start_time": 5331.357,
"text": " 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 any Turing complete dynamical system will inherit such undecidability. Yeah. Okay, then what you've shown is that there are certain 3D oiler flows that are Turing complete fluid computers. Exactly, exactly, exactly. So some solution that I said solutions"
},
{
"end_time": 5388.729,
"index": 211,
"start_time": 5360.111,
"text": " 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."
},
{
"end_time": 5413.319,
"index": 212,
"start_time": 5389.411,
"text": " Interested another question. Did you get one million dollars, right? You say oh, why do you need one million dollars? You proved a beautiful theorem Okay, still the question makes sense and the answer is the short answer is no so this this construction we did was not good enough to give us the blow up of novice talks and and the short answer no, the long answer is read my paper and"
},
{
"end_time": 5436.937,
"index": 213,
"start_time": 5413.814,
"text": " but let's say we were able to pluck indeed 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"
},
{
"end_time": 5463.763,
"index": 214,
"start_time": 5437.295,
"text": " 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. Okay. But okay, we were answering the question of Terence Tao, but well, this answer does not lead to the blow-up."
},
{
"end_time": 5491.476,
"index": 215,
"start_time": 5464.172,
"text": " This does not mean that other answers can lead to the blow up. Because you have like 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. OK, so now let's see. It's time to think like what's outside the Valtrami box. Right, so that's me."
},
{
"end_time": 5513.148,
"index": 216,
"start_time": 5492.602,
"text": " That's me working with my colleagues. Probably one of them is Daniel Peralta Salas and the other one is probably Ángel González Prieto who is our new member of the group. Sorry, which one is you? The one by the door? I have two copies of myself because I believe in this quantum"
},
{
"end_time": 5542.773,
"index": 217,
"start_time": 5513.677,
"text": " Okay, so there is a copy of myself inside Having a coffee with glasses and comfortably looking at these straight line a goes to be 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 cool This is difficult this system I see actually so instead of spin up spin down you have hair up hair down exactly Okay"
},
{
"end_time": 5569.616,
"index": 218,
"start_time": 5543.063,
"text": " Exactly. So let's say, okay, what's outside of our tramy box? Because we did with, uh, with, uh, Daniel, we have been working also on solutions on proving undecidable, uh, indecidability in free 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 that while pending,"
},
{
"end_time": 5600.282,
"index": 219,
"start_time": 5570.623,
"text": " What was pending for a long time is like, well, what about Navier-Stokes, right? Well, Navier-Stokes, the idea of 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, Plum Plum. 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."
},
{
"end_time": 5629.906,
"index": 220,
"start_time": 5601.561,
"text": " Well, okay. Yeah, so it did look we have proved So what we got proofs on that and 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 a space in the space right thinking of this rather So no matter how potent your computational problem is, right? In other words, if you want the problem is not computable Okay"
},
{
"end_time": 5658.166,
"index": 221,
"start_time": 5630.23,
"text": " So, well, 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 Burnais, Daniel Grasse and Emmanuel Henry. Excuse me because I don't pronounce Henry well."
},
{
"end_time": 5687.705,
"index": 222,
"start_time": 5658.49,
"text": " 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. So. And that's very interesting, OK, because OK, this tells you that you cannot find"
},
{
"end_time": 5708.387,
"index": 223,
"start_time": 5688.268,
"text": " 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, so this is"
},
{
"end_time": 5732.892,
"index": 224,
"start_time": 5708.848,
"text": " You can put here as flashy, you can put flashy neons there. Yes, this is this isn't it. Yeah, 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. OK."
},
{
"end_time": 5762.773,
"index": 225,
"start_time": 5733.234,
"text": " So 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 cosimplectic geometry. And this is something that should be online soon, I hope very soon. But we know how to do it. And the idea again comes from geometry. So we have been able to produce a Turing complete"
},
{
"end_time": 5791.34,
"index": 226,
"start_time": 5763.131,
"text": " I have more things to explain you"
},
{
"end_time": 5816.067,
"index": 227,
"start_time": 5791.783,
"text": " 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 El País, 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."
},
{
"end_time": 5840.623,
"index": 228,
"start_time": 5816.544,
"text": " So now we have done something also new, different from this solution to incomplete Navier-Stokes flow. This is something we have done with Ángel González Prieto from Universidad Complutense de Madrid and with Daniel Peraltas Salas and it's totally revolutionary I would say and it's what I call a hybrid computer. And the idea is that"
},
{
"end_time": 5869.411,
"index": 229,
"start_time": 5841.152,
"text": " 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, assume that I call the former construction this idea of going around this idea of this machine that I explained, the Turing complete Euler flow."
},
{
"end_time": 5893.575,
"index": 230,
"start_time": 5870.213,
"text": " 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 assume like i can assemble these pieces using the rules of Feynman of quantum computing right"
},
{
"end_time": 5914.104,
"index": 231,
"start_time": 5894.104,
"text": " So one day I find myself in my office with my colleague Ángel González Prieto doing pictures like this one, which are the picture used in topological quantum field theory. So the idea is that use the tools of topological quantum field theory to improve your computational method."
},
{
"end_time": 5929.684,
"index": 232,
"start_time": 5915.077,
"text": " 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, right? But cylinders are not enough. This is just a piece of your puzzle."
},
{
"end_time": 5958.797,
"index": 233,
"start_time": 5930.418,
"text": " and also you could put in this category this new construction that I do with Cosimple geometry is Navier-Stokes. 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 in his lecture of computation. What do you get? Well, we get a new model of computation"
},
{
"end_time": 5989.616,
"index": 234,
"start_time": 5959.821,
"text": " 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 more 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 the downhill 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."
},
{
"end_time": 6015.555,
"index": 235,
"start_time": 5991.442,
"text": " What's next? What's in our agenda? Well, of course we want to"
},
{
"end_time": 6035.691,
"index": 236,
"start_time": 6016.254,
"text": " What the blow-up of Navier Stokes was never in my agenda but now it's in the agenda somehow because this is a bit related. We are working, we are thinking around this with Daniel Peralta Salas in this picture and Ángel González-Pieto. We are here, all of us are smiling, this is good."
},
{
"end_time": 6061.783,
"index": 237,
"start_time": 6035.879,
"text": " Yes. 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 yours here. Yeah. Yes. We have to arrange a visit of yours here. I look forward to it. So this is Terence Tao who visited us last September and we organized a conference and Terence was giving, yeah, he gave a couple of wonderful talks."
},
{
"end_time": 6086.015,
"index": 238,
"start_time": 6062.142,
"text": " 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?"
},
{
"end_time": 6116.067,
"index": 239,
"start_time": 6086.63,
"text": " 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 these 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."
},
{
"end_time": 6140.981,
"index": 240,
"start_time": 6117.278,
"text": " lost in the completely lost and now if i go back to the asteroid that's another question is 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"
},
{
"end_time": 6163.933,
"index": 241,
"start_time": 6142.329,
"text": " 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,"
},
{
"end_time": 6192.381,
"index": 242,
"start_time": 6164.36,
"text": " 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 or not, 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?"
},
{
"end_time": 6218.985,
"index": 243,
"start_time": 6192.995,
"text": " Because, for instance, the Bernoulli shift has been used on many problems, in restricted problems, on the three-body problem. 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? So that's the kind of problems in which we are thinking. So in a way,"
},
{
"end_time": 6245.282,
"index": 244,
"start_time": 6219.462,
"text": " This would be the equivalent of putting the butterfly on the rubber duck, right? So you want to reconcile or you want to relate, you want to see if these two are related or not, right? We have understood many things about the relation looking at entropy, entropy, thinking entropy is the big word. We use it for physics, we use it for mathematics and it means that we measure"
},
{
"end_time": 6274.343,
"index": 245,
"start_time": 6245.674,
"text": " Disorder I mean we measure order, right? So using entropy we have been able to understand somehow the relation Between these two girls, 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"
},
{
"end_time": 6292.722,
"index": 246,
"start_time": 6274.787,
"text": " 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. Yeah. Thank you. Thank you, Eva. How about one quick question? Yeah. Do these fluid computers violate the church Turing thesis?"
},
{
"end_time": 6321.886,
"index": 247,
"start_time": 6294.531,
"text": " Do these Turing computers? No, no, no, no, that's a very, that's a super good. 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 of, uh, if this idea that if you have any system, all of these systems should be equivalent."
},
{
"end_time": 6340.725,
"index": 248,
"start_time": 6322.688,
"text": " In terms 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, right? When we talk about quantum supremacy, we're talking about efficiency of computation."
},
{
"end_time": 6370.998,
"index": 249,
"start_time": 6341.101,
"text": " So that would be another question. So 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. That you have several ways to like this church during thesis is like you have all these several ways of computing. They are all equivalent in a way."
},
{
"end_time": 6399.718,
"index": 250,
"start_time": 6371.527,
"text": " 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, three years. Okay. Yes. That's what quantum computer does. So now what we want to do is to do this even quicker than a quantum computer using maybe this idea of, of the hybrid computer of putting"
},
{
"end_time": 6426.954,
"index": 251,
"start_time": 6400.299,
"text": " 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"
},
{
"end_time": 6432.432,
"index": 252,
"start_time": 6427.585,
"text": " and indeed i have one of my our students is going this week"
},
{
"end_time": 6461.186,
"index": 253,
"start_time": 6432.824,
"text": " With the 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 students, I'm going to tell his name, did a very nice simulation of these of these constructions for particular types. Now we want to do the to make this on a great scale and to be able to see"
},
{
"end_time": 6491.715,
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"text": " that this version, this topological game 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."
},
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"text": " Hi there, Kurt here. If you'd like more content from Theories of Everything and the very best listening experience, then be sure to check out my sub stack at kurtjymungle.org. Some of the top perks are that every week you get brand new episodes ahead of time"
},
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"text": " 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 sub stack on Google. Since I started that sub stack, it somehow already became number two in the science category. Now, sub stack for those who are unfamiliar is like a newsletter, one that's beautifully formatted. There's zero spam."
},
{
"end_time": 6574.053,
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"start_time": 6546.869,
"text": " This is the best place to follow the content of this channel that isn't anywhere else. It's not on YouTube. It's not on Patreon. It's exclusive to the Substack. It's free. There are ways for you to support me on Substack if you want and you'll get special bonuses if you do. Several people ask me like, hey, Kurt, you've spoken to so many people in the field of theoretical physics, of philosophy, of consciousness. What are your thoughts, man?"
},
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"end_time": 6603.387,
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"text": " Well, while I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. And it's the perfect way to support me directly. KurtJaymungle.org or search KurtJaymungle substack on Google. Oh, and I've received several messages, emails and comments from professors and researchers saying that they recommend theories of everything to their students."
},
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"text": " That's fantastic. If you're a professor or a lecturer or what have you, and there's a particular standout episode that students can benefit from, or your friends, please do share. And of course, a huge thank you to our advertising sponsor, The Economist. Visit economist.com slash totoe to get a massive discount on their annual subscription."
},
{
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"text": " I subscribe to The Economist, and you'll love it as well. Toe is actually the only podcast that they currently partner with, so it's a huge honor for me. And for you, you're getting an exclusive discount. That's economist.com slash toe. And finally,"
},
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"text": " You should know this podcast is on iTunes, it's on Spotify, it's on all the audio platforms. All you have to do is type in theories of everything and you'll find it. I know my last name is complicated, so maybe you don't want to type in Jymungle, but you can type in theories of everything and you'll find it."
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"text": " Personally, I gain from rewatching lectures and podcasts. I also read in the comment that toll listeners also gain from replaying. So how about instead you relisten on one of those platforms like iTunes, Spotify, Google podcasts, whatever podcast catcher you use. I'm there with you. Thank you for listening."
}
]
}No transcript available.