 So, good morning everyone. My name is Jonas Fischer. I coordinate the visiting programs in the FAO here at OIST. And you might have noticed that we have call for applications open. So if you have any friends that might want to come to OIST like Sebastiano, or if you want to propose a program which is an entirely new part of the program, then let me know and we can talk about it. But today is, of course, Sebastiano Nucleoso-Golo's talk. He will talk about sub-Riemannian geometries everywhere, which I did not know, but let's see. Ivan just going to introduce him for a very short bit and then he will start his talk and we have Q&A after. So he is a post-doc in mathematics at the University of Huskala in Finland. Sorry that I didn't pronounce that right. He works on the interface between metric analysis and Lie group theory. He has worked in sub-Riemannian geometries and later expanded his interest to geometric measure theory on Lie groups equipped with non-Riemannian distances. He completed his PhD also at the same university. OK, so I'm looking forward to your talk. Please go ahead. Thank you. Thank you very much. So welcome, everybody, here. And thank you very much. So I have to thank very much OIST and all the program in the DVSP program who allow me to come here in Okinawa to have this great experience. And I really like this institute and I really like the fact that there is such a that I feel pushed to interact with different sector of science. And this is my talk wants to go in that direction. I want to just have like a loose, let's say a loose presentation for an overall audience trying to capture your curiosity. And because I think really sub-Riemannian geometry, other geometry is something that actually comes from control theory and it's really spreading in a lot of applications. And we will see some of them, some of theoretical and some are very practical. Yes, so I titled my talk, a sub-Riemannian geometry is everywhere with an extermination mark, which is maybe not very scientific. And but you will see that probably I'm right. So OK, I will show you some models. OK, so just the first model is a bit stupid. It's almost a joke, but it would be useful later to just understand the second model and what follows. So let's make a model, scientific model, of a point in space. It doesn't make much sense, but a point in space is a point in space. So we have three coordinates, x and y and z. And those coordinates, the three numbers determine completely where the point is in space. And then usually points move around. They like a lot to go from A to B. But these points goes also from A to B and usually points go from A to B in a straight line because that is the shortest line. But it's not really necessary, could go along any path from A to B. So we have a point given by three coordinates and it moves around quite freely. And there is a shortest path that is the straight line. So the next example is the monocycle. It's very typical for control theorem supplementary geometry. So I'm not able to, I tried once to go with a monocycle and I didn't go very far. So I will keep this imaginary monocycle with my heart. And my description of the monocycle is that it is somewhere on the plane. So it has two coordinates, x and y. And then it has a direction just given by an angle theta. For me, angles are actually elements of S1 of the circle. So this S1 is the circle in the plane. So the monocycle is given by two coordinates and a direction. And well, the monocycle cannot move completely because it cannot slide sideways. If you're pointing in this direction, you cannot go sideways, otherwise you fall. So you can only go back and forth or you can rotate. And these are the two allowed directions. I call it F or forward. And that is tangential. I mean, it's in the direction which we are pointing. So given theta is cross theta, the same theta. And then you can, well, I'm holding it with my hands. I can turn it. So then the second admissible direction is theta. And now the question is, OK, can I go from A to B? Now, A to B are not just points in the plane, but are points in the phase space, which are all points x, y, theta, which is like R3 before. So we have a point in a 3D space. And we wanted to go from A to B. But now it cannot go. Probably it cannot go in a straight line because there is one direction that is forbidden. There is one direction. And well, you can imagine that it can go everywhere. And we can do it with a picture. So if this is the monocycle seen from above. And I want to move it somehow in the forbidden direction. And how do I do? Well, I can turn by some angle theta. Then I can go forward. And then I can turn again, probably minus theta. And then I can go backwards. So the algorithm they say is to go to rotate for some angle, then go forward, then rotate back, and go forward, back. So go backwards. And a composition of these four movements is basically a commutator. So in mathematics, then this is what we do. We take this commutator, the Lie brackets of two vector fields, and which is, OK, now I'm talking a bit for mathematicians. But I'm deriving, basically, the cross theta that becomes a sine theta. And they have the cross, the minus sine, and cross theta. Sine theta becomes cross theta. And this is exactly the forbidden direction. This is the orthogonal direction to F. So Lie brackets is what allows us to get the forbidden direction. OK, this is very, this is a theorem, actually. And it's called the Chao theorem, or Chou theorem. Depends on which mistake you want to do. And this condition, which is like, you can obtain all forbidden directions using Lie brackets. It's called the Lie Bracken Generative Condition, or Chao condition, Chao condition, or Hormander condition as well. Let me see where we are. Yeah, right. And this is the, so we have the phase, so we have the space of all configurations. In the space of all configurations, we have admissible directions. And then we have this condition that gives us controllability. So we can fully control the system, not going into straight lines, like for the point in the 3D space, but taking some more complicated algorithm, we can actually go from A to B, from any A to any B, in the phase space. And this is controllability. Another example, oh, yeah, this was for pictures. You can then work out what happens for a bicycle, just two wheels. And then here, I just try to write down the equations. So in this case, we have some controls. In this case, it's clear what are the controls. So I can go forward, or maybe also backward. And that is the S that is here, is the speed. And then I can control the angle. I can turn alpha. So when I am on the bike, I have control on S and alpha. And then the bike will satisfy, so the F and B are the front wheel and the back wheel. And they satisfy this equation. So when I cycle, I have control on alpha and S. And then I want, through this control, I want to move the bike, so the front and the back wheel, around in space. So, yeah, this is also controllable. Another typical example, which is even more complicated, but the idea is always the same, is with the car, that you know probably pretty well. So now a car is different from a motorcycle or for a bicycle, because it has four wheels. And you can turn only the two wheels in front. But the idea is still the same. You have two controls. You can turn the wheel, the steering wheel, or you can go back and forth. So you have two controls. But you're moving around in a 3D space, which is not this 3D space, but it's the three-dimensional space of all configurations. And so for instance, when you want to park a car, and you're in this situation, then you need to do a parallel parking. So in principle, you are not allowed to go down straight like that, but you know that with some complicated algorithm, you can actually go down. So with two controls, we are steering a system that has three degrees of freedom. And things get even more complicated if you are driving a truck, and then with maybe a trailer, or maybe with a second or third trailer. And then we go into really practical applications that are really softwares that help drivers to drive long trailers. And you can basically park it everywhere if you have enough space. So let's do a summary of this first part of the talk. We have said that we have, for all these models, we have a space of configurations. It's like all these configurations of a system. And this is usually a high-dimensional space. And by then, we have only a few rules. We have some rules for movement. So we have only a few controls. So the number of controls is smaller than the degree of freedom, degrees of freedom. And then what we check using Lee brackets, usually, and is the controllability of the system. So even though we have less controls, we can still steer the system from A to B. And then there are additional questions. Of course, once you know that you can, then you have to find a way to steer the system from A to B, which is not easy to implement. And usually, you want it to do with a software. So you need a theoretical tool to go through it. And then usually, you want to do it also optimally. So in the sense that you have probably a cost function that you want to minimize. It can be time. It can be energy or fuel consumption or distance. And because just an example, if you want to park your car parallel parking, you know that if you go a few centimeters forward, and you turn a little bit, and then you turn a little bit, you can go a few centimeters on the right. And then, of course, you can just repeat it this thousands times. And then you can move meters on the right. But that's probably not what you want to do when you are in a crowded city. You want to do it with a one smooth movement. And here is where sub-remanager comes. So yeah, I'm cheating a little bit, because what I'm talking about is actual control theory. But I am a geometry. And what I see in this control in this model is the sub-remanager geometry, which is the phase space with now a cost function. And when we go from A to B, we want to minimize the cost function. And this gives us a distance. So I have a geometrical space. And as a geometry, I want to study the geometry of this phase space and over with a cost function. And we will see towards the end, which will be probably the punchline of the whole talk, why geometry could be important. For now, the only geometrical object that we see is the optimality. So finding an optimal path, an optimal strategy, geometrically means finding a geodesic from A to B. Let's go on with some other examples. I like this one. This is very old example, just rolling a sphere on a plane. So imagine you have a sphere. Imagine like a planet. So you see all the continents or something like that. And then you roll the sphere on the floor. And you want to roll it without sliding and without spinning. And again, what is the phase space? In this case, the ball, the sphere in space is described by where it touches the plane, so the x, y coordinates, and how it is oriented in space, which is a rotation, an element of SO3. And so space of configuration is x, y, and a rotation, which is an element of SO3. And so without slipping or spinning, these are our, so let's go in order. So the space of configuration has dimension 5. So R2 has dimension 2 and SO3 has dimension 3. And SO3 is the Lie algebra. So it's generated by these three infinitesimal rotations, which are the infinitesimal. Ix is a matrix, but it's the rotation around the x-axis. And Iy is the rotation around the y-axis. And Iz is the rotation around the z-axis. And these three, these are the infinitesimal version, the rotations generate all the rotations. And so SO3 dimension 3, the phase space dimension 5, and we have now two controls. Because what we can do with the ball, we can just roll it either along x or along y, I mean on the plane, so in two dimensions. And the two conditions, the two allowed directions are these two, x and y. Now I talk maybe more to mathematicians here. What it means is that I can move along x, dx, part of a factor. But when I move along x, I have to rotate. So I imagine here that I plug in my ball from the right. So r is a rotation. So this is the initial ball I plug in from the right. Then I apply r, so it gets the ball where I am. It's like this. And then if I move along x, the ball we start rotating along the y-axis. Yes, it's correct. So then it starts rotating along the y-axis. And it's kind of quantitative. Because after time 1, here it did a quarter of a rotation. So it had, if the radius of the ball is r, then it went pi r divided by 2 meters along the x-axis. And similarly for the other one. So we have two expressions, matrices, et cetera. And then we can do the lead brackets. And one starts computing, blah, blah, blah. These are matrices, so somehow one should be able to do it. I call it mathematical yoga. Just go down. And you check that at the end, you get 1, 2, 3, 4, 5 vectors that are linearly independent. So this total space is five dimensional, so we get all directions. So this system is controllable. Even the sphere here and the sphere there, in any orientation, I can go from here to there just by rolling the sphere somehow. And let's see, even a more complicated example is when I want to roll, forget about the plane, I want to roll a ball a sphere over another sphere. So the rule is always the same without slipping and twisting or whatever. And I drew this plane just to suggest you if you want to get the equations here, you can go back to the previous example, because you have two balls that both balls are rolling on a plane, and their rolling is linked to each other. Anyway, we focus on the two balls, so then we have that the phase space in this case is s o 3 times s o 3, so all the rotations of one and all the rotation of the other. Now the dimension is 6. And we still have only two directions and two controls. And again, we can do this mathematical yoga and compute all these brackets. And at this point, there is a twist in the story, because if you look at it, if r and s are equal, it should be a minus and a plus anyway. If r and s are equal, then I don't see it, because I'm too close to it. But two vectors that become the same as the previous ones. So these two are equal, and if r and s are equal, they become proportional to the first one. And also computer determinant if you want. So what happens is that if r and s are equal, these leap brackets and then iteratively also all the other leap brackets are not able to span six dimensions. So the system is not controllable. So if you have two spheres that are of the same radii, then you cannot control it. So you cannot go from A to B in this case. But if r and s are different, then suddenly you can control it. And there is also another example where you put a sphere inside another sphere. So a smaller sphere that is rolling inside the other sphere. So it's an interesting yoga exercise. All right, so there are then more examples that you can find in literature. And for instance, I know in quantum mechanics, I didn't find exactly an example that I met a few years ago. But OK, I wouldn't have time anyway. But OK, in quantum mechanics, you can imagine that you have probably a quantum system. And you have control over certain quantities. And you want to steer the quantum system from A to B. And this is in particular something that they try to do, or at least theoretically in quantum computing, where you have these qubits that have no idea what they are, but you want to control them. And also in terms of dynamics, that's actually a very old theory. And that's one of the first, maybe the first examples in control theory that was made already by Kara Theodori and at the beginning of 20th century, I think. And from that theory comes, for instance, a very important name in subrimane geometry, which is Karno, which is the Karno from thermodynamics. And in subrimane geometry, we have Karno groups, which are actually the infinitesimal models of all subrimane geometries. And now I want to present you another example, which is now of a different type, even though then we end up with the same general framework. And it's a folding cut. So this is a very old, it's actually a movie. And it's like the first movie where a cut is appearing. This is Wikipedia says. And so they let a cut fall. And you know that a cut falls always on its legs. And that's quite amazing. I never tried it because I'm very scared of it. And also my cut was very scared, so we didn't do it. But it seems that you can do it and you can find many videos. But what is going on? If you look at it naively, you say, OK, he's smart. But let's be a bit more intelligent and think that the cut is starting without angular momentum, derangular momentum. And during the fall, it's free-falling. And at the end, it made a rotation, a full rotation. So it means that it looks like he got some angular momentum. And we know that at the end, it has to, in physics, angular momentum is always conserved. So where does it come from? And well, it cannot come from air because the drag of air is too small to get so much. And the cut is not a bird. And the cut knows it because it understood it when he was on the top of the tree that he was not a bird. And it was too late. And so it's not a bird, so it cannot fly. And it cannot get this angular momentum from the air. So where does it get from? Get it from. And it cannot break the rules of physics. So actually, it's not getting any angular momentum. And what he's doing is something very smart. So what he's doing is changing his shape. You can see it. And I cannot draw a cut, so I drew these three cuts, which is a model of the cut. So you have the center of the body and the two sides of the ends of the body, more or less. And what the cut can do is it can, while flying, change the shape in this way. So first it bends like this. And then at that point, it can basically rotate the two opposite sides. And so this picture is very schematic. But it makes there is something that becomes very clear that if this part rotates in one direction, the body is all connected, the other part has to rotate it in the other direction. And so the total angular momentum is 0. So it's not breaking the angular momentum because the two parts are rotating, canceling each other. And you cannot imagine how you could do it. If you were floating in space, you can, using your muscles, your core muscles, you can rotate in this way. This is what the cut is doing. And well, and then magically, at this point, if it goes back, it's facing down. There is force facing up and here is facing down. And a way to describe it is the following. So for each, so this is the space of configurations which are the cut in space, which are some shape in space and somewhere with some orientation. But for every configuration of the cut in space, the cut has a shape. The shape is independent from the space. It's just like the abstract shape. And you see that. And at the beginning, it has this shape. And at the end, it has actually the same shape. And between is changing shape. So what the cut can do is moving around in this space of shapes can change this shape. And then what we have here in this bigger space of configurations is a lift of the curve from here to there. And basically, this represents the controls and this represents the phase space. But it's still always the same thing. But now we see it as a lift. So a curve in the shape space lifts to a curve in the space of configurations. And what the cut is doing is exploiting, I think this is called like this, it's exploiting the non-triviality of the holonomic group. So a loop here, so it starts and ends at the same shape, lifts to something that is not a loop. Yeah, maybe because I don't know. It's like if you are on the big circle and then above the circle, we have this spiral. And then if I start here, and so I'm here on the spiral, and then if I make one round around the circle on the spiral above, what happens is that I go up, so I don't end up in the same space. This is exactly the same situation. Very good. Some extra examples that follow more or less the same philosophy is, OK, I call them space robots. Imagine like a robot floating in space, so it can probably move around like a cat, but maybe more freely. And another example are microspheres, which means leaving things or robots that are very small and float in some liquid water. Like very small means like bacteria or something. And what happens is that at that, it's like the opposite of the cut in the air. At that scale, drug forces are very strong, and you don't have inertia. So a body floating so small in water can like, if it wants to go to constant speed, has to exert a constant force. And as soon as it stops making any effort, it will stop moving because the drug is very strong. It's like the opposite of the cut. And this is what happens is that many of these microorganisms can move around, change in their shape. And then I try to make a picture to explain how that can happen. So if you go from a round ball, and then you squeeze yourself into like a saddle, like that, bike saddle, then what happens is that what was the, can I use this one? The, no. And I have another one here. So what I draw here are the water that was here. So here there was some water. And now when this thing changes its shape, this water has to flow here. And if some water flows there, then the body has a force on the opposite direction, so it moves. That's how it moves. So then now we want to, a control is like, we want to understand how to go really from A to B, changing the shape to go there. And there is this book by Richard Montgomery, a tour of sub-reminion geometries, their geodesics and applications, where you can find most of these examples. And it's a very good book. And it's also quite thin and contains a lot of stuff. Another, another very interesting example is Celestial Mechanics. And here, very good researchers, Emmanuel Trela, the book source of the European space industry. And what he studied, in particular, is how to move from, so you have a space captain in space. And when it's not in its happen, it's just traveling along an orbit. And then you want to change orbit, which is where you want to go from orbit A to orbit B, and you want to do it maybe in an optimal way. Or actually, you want to be able to do it, which is not very easy. And then you can also do more complicated stuff like traveling from one planet to another one, trying to minimize the fuel consumption. And then there are very nice theories that shows you, you can almost go around for free in space, if you have time. Last example of complicated. This is the V1 visual cortex and imagery construction. Well, let's start with this example. So here you see two pictures, one on the left and one on the right. So this on the left is the Canis-Canitsa triangle. And if you look at it carefully, what you should see, so the triangle is the one that is inside here, that is covering these three disks. But in fact, if you look carefully, there is no triangle there. And the right-hand side picture should help you to understand that. So there's nothing happening here. They just rotate it. But between these disks, you don't see anything. It's just pure white. But between these disks, probably, you should see some line here. Sounds crazy. But it is actually like that. I mean, they did experiments. People do see usually a line there. Somehow a very faint line. Like the white is changing. Like it's darker outside and lighter inside them, I guess. That is how I see it. So you are not crazy. We are not crazy. There is something there. And let's keep it there a moment. And let's do the following exercise. So we want to, so I'll show you how to reconstruct an image that has a hole. So let's start with an image. It's just a gray-scale image. And with the gray that is changing so that there has always a non-zero gradient. Now, if you fix a point there, you look at the direction. Well, you can look at the direction where the color changes the most, the gradient. Or you can also look at what I drew there, or I tried to draw that, is the direction where the color remains constant. And basically, this is a boundary. In that picture, of course, inside the black, it's just black. But in the interface between the black and the white, there is a direction of a change. And we take that tangential direction, which is this one. And now we lift it in. So for each point of the image, we take an S1 again. So it's like an interval with two endpoints identified. So it's a circle. And it gives us this direction. So every point in the image, we have a direction there. So if we do this for every point, we get a surface in a three-dimensional space, which is, again, let's see if I do it, R3, R2 times S1. So here, this is more or less the scheme. If you have a start with a picture, it's actually the same. And you see the lines where the color remains constant. And now we lift it into this R2 times S1. Then we get a surface there that describes the picture. And now, there was a hole. We didn't know what to do there. But when you see a surface, it's easier to think what you can put there. Just feel the hole of the surface. But now this new surface should actually describe some gray-scale coloring. It is actually a quite drastic procedure because, OK, I didn't show it here. You can imagine, for instance, if you see from above a highway with some bridges, and then the highway crosses another highway. If you see it from above, what you see is that the lines of the roads, on the certain point, some road gets interrupted. And the bridges of the highway are actually like the same type of lift, in a certain sense, into the three-dimensional space. So then when you see it from above, you don't understand what is going on. So it is like a cross. But then when you see in three-dimension, you see that there is one road that goes up and one road that goes below. And you can complete the road below. Just go straight. That's very useful because it's used to build up algorithms that are used to study the veins blood circulation in the retina. That is linked to some illnesses. And what you want to do is have a computer program that you give it a picture of the retina, which is something you should imagine, like a reddish with these veins that cuts each other. And it should be able to understand if they're to distinguish different veins when they cross. It is more or less, but you get it. And OK, how to fill this? There are many ways to fill this hole. But we apply this principle. And it's like a propagation of the information. It means the following. Suppose that we are here, this blue dot. And there is no violet. So the violet is in the hole. But we have this black line. So we know that up to now, this was the boundary. This was the line of a constant color. And what can happen next? So one thing is that it can go straight, continue straight line, or maybe it can change direction. But it would not change direction completely all of a sudden. It should change direction maybe more smoothly, like this, as I drew that. And OK, so if I am one pixel, and then there is another person that makes another pixel. And here between us, there is a hole. And I see I have a line here, a straight line. They have also a straight line there. And then I say, look, here I have a straight line. So probably we go in this direction, like this. And the other one says, oh, I go in this direction. And then we meet. That was good. So then we can say that between us, there is a straight line. Or maybe there is a line that bends a little bit like that. And that's how we want to do. So we propagate the information on the hole along these curves. And if we look now up in the lift, what it means is that now we are in the point x, y there, and with some direction. And from there, we can go either with the same direction. Now, yeah, now the height is the direction. So we go there, we go to another direction, et cetera. And so these are the two directions of propagation of the information. And should ring you a bell, because I already wrote these two vector fields at the beginning of the seminar. And for the monocycle, this is exactly the same geometry of the monocycle. But remember there, we were interested basically in geodesics, so it's minimizing curves. Now we are interested in surfaces, in minimizing surfaces. And we feel the hole. I mean, they feel the hole because they didn't do it, but there are some papers out there. And they do it using, for instance, the heat flow. That's the propagation of the information. And that's where now geometry, some reminding geometry comes into the game, because we really want to understand the spaces. And for instance, we want to study the heat equation, whatever it means, using these two vector fields and the intrinsic geometry of them on the space. No, I'm sorry. There's something wrong. What come next? This is an example of an image reconstruction. So above, OK, it's a picture. And then they just made a mess with it, taking away a lot of data. And then there's just a computer program that fills in the holes. And yeah, there are even more striking examples that we didn't find it yesterday. And I think this is really quite amazing, because it's just a computer program. An algorithm is like a heat flow, and then you get that. And not only, so this is just an image reconstruction, but the inspiration of that model comes really more or less from the study of the brain, so the V1 cortex, which is like a part of the brain here in the back. And it's one of the first process of this, one of the first moments in the image process of what we see. There is the V1, there is the V2, V3, I don't know. So then the image processing in our brain is very complicated. And this is one of the first things that happens. And it's really tries, so the brain tries to find contours and complete them. And this is a picture there. So the idea is that above every point, let's say a pixel of the retina, you should have a column of neurons. And on the column of neurons, one of the neurons gets excited when, so every neuron in the column has a preferred direction. And when the contour has that direction at that point, that neuron gets excited. And then when it gets excited, it tries to excite other neurons around, all with the two directions of before. And that's how in the brain, then you get this flow of information inside the brain. But then actually in the brain, these columns are not really all columns, but they are flat down. So that's why this picture. So different columns correspond to different directions. And this is a map. And I think this was made really experimentally. Like you take a person and you show a picture with a line, and you see some parts that are more active than others. And changing the direction, you see this part changing. So you can make a map which neurons are excited by what direction, by which direction. And so I don't have it here anymore. But if we go back to this one, yeah, this is what is happening. This is at least one explanation. So here the brain is recognizing this border, this contour. And then there is this automatic system that tries to fill it. And tries to fill it here very sharp. And then the information loses strength. So then here is not so sharp. But then here is also sharp. And they meet in the between. There are even more examples, more complicated, where you really see that the brain finds lines in a very messy picture. Just because it does this process. So you see, I find it amazing because the line is not there, but you see it, you really see it. It's a real perception. It's not, they call it illusion, but it's a perception. So I think this is the end of it. Thank you very much. Questions or remarks, please? Maybe they want you to use a microphone. Where do you ask, why do you have this word sub in front of this subring, Manny? And what is sub about this? Ah, yeah. OK, the sub is about the fact that the number of controls is less the dimension of the configuration space. So if you have as many controls as the dimension of the configuration space, which means the very first example I gave of the point in space, then that is like re-management. But now if you have less controls, then you have sub-re-management. Mathematically, it means that we have like a re-man tensor, like a metric tensor, but it's not defined on the old tangent bundle, but only on a sub-bundle. That is a very important thing I should have said. Thank you very much. Other questions or comments? Yeah. Yes, yes, yes, yes. So in Riemannian geometry, you have a geodesic equation, which gives you like a geodesic flow on the tangent bundle. Now, that doesn't work in the sub-bremagnum setting, but then you have a Hamiltonian approach to it. So you get an Hamiltonian. And the difference, so the Riemannian Hamiltonian is not much different from the scalar product itself, the quadratic form, but the sub-bremagnum Hamiltonian has now a single asset at 0, basically in the non-admissible direction of your book. And that's the singularity. So you should be careful. So you can get most of geodesics using this Hamiltonian approach, but not everyone, not every geodesic. And that's like a big open problem in sub-bremagnum geometry, because there are these geodesics that are called singular, or strictly normal geodesics that do not come from the Hamiltonian theory. And I mean, yeah, because they don't come from that. So we don't even know if they are smooth. First, thanks for the talk. When you said that, for example, that you find that these equations in a real system, and then that you want to understand the geometry, what do you mean by understanding the geometry? I'm not a mathematician, so. Yeah, OK, yeah. So the geometry of the space is really everything and nothing. So here in this talk, maybe I try to show two things. And two cases. So you can look at geodesics, so minimizing curves. And when you look at geodesic, then it means I want to know, given two points, how many geodesics there are, if they are smooth, if they satisfy some PDEs, some ODE that I can solve. And when they meet, and what happens when they meet. And yeah, and then using the geodesic and this minimizing course, you get a distance between two points. So probably you cannot compute explicitly the distance, but you want to have an estimate of the distance from above and from below, more or less asymptotically. And so this is more or less for a path geometry. But then we have also this surface geometry, or then we want to understand what is the area, for instance, of our surface, and what does it mean to minimize that area. Or more in PDEs, we can have the heat flow, which should. I mean, if you think of it like now, maybe the path geometry is quite clear by now. But then you can do a Brownian motion, a Brownian motion using only the allowed directions. And then you should get at the end a heat kernel in this way. But in this heat kernel should have also PDE with a laplacian, or something that works like a laplacian. And so then you want to know if there exists a heat kernel, what is its regularity, what its estimates, the decay, or things like that. And you can study also the wave equation and just the laplacian equation and so on. Yeah, this is the geometry. And usually, this is a trend of the last 40 years now. Every year, that's longer. And try to boil down everything to the distance, because what mathematicians understood is that most of the analysis that you do, like mathematical analysis, like derivatives, integrals, PDEs, is something that what you really need is just the distance between points, not for everything, but for most qualitative properties. You only need to know the distance. So once tries really to make a very abstract theory using only the distance. How could I somehow get a hint that if I'm studying something, if I'm studying something, a system, whatever it is, how could I get a hint that, oh, there might be a sub-Riemannian geometry here. I should call you and ask you for help. Yeah, yeah, yeah, yeah. Just call me and ask me. Because maybe it's not a sub-Riemannian geometry, but I could still help. Or maybe not, and then I tell you one thing. Probably one thing is that if you have a space that is bigger than the number of controls, probably these are the questions. There's a question from the Zoom chat. So amateur question, from falling cat's space of shapes, is it possible to simulate human bipedal walking? Yeah, probably this is what they also do, right? So walking can be seen as a control problem in the same way. So now think of me like floating in the space. I can change my shape as I want. And then I make a strategy, and then I put myself on the ground, and I see what happens when I do this strategy and see how far I go. And then I would like to go very fast. And so then I try to optimize my movement. I'm changing just my shape. I move in my shape space, but then my movement in the shape space is translating into a movement into the space. Yeah, there's exactly the same thing as the cat folding. And the whole onomy work, right? Are there questions? I have one more from Zoom. Has any study been done when the face space is fractal? Yeah, so what does it mean fractal? So usually for me, fractal means that I use the word fractal to say that the house of dimension, which is like the metric dimension. The dimension of the metric space is different from the topological dimension. Topological dimension is just the very basic dimension of where you are. So all the dimensions that I stated here are topological. The dimension of this SO3, SO3 is 6. But now if we take these two directions, these two controls, and the subrimane geometry implied by that, on that space, SO3 cross SO3, now we get a metric space. And the house of dimension there is 10, I think. It's 10. So actually, that subrimane manifold, that subrimane space is a factor for me. Then probably the question was not really this about this. It was more like, well, then if the face space, instead of SO3 cross SO3, you have a funk curve or something like that. And no, I don't know. I know nothing about it. I tried to give the answer to the question that I knew, for which I knew the answer. Thank you. So any other questions? OK, if not, then I think we thank the speakers. Thank you very much, Sebastian. And give them a round of applause. Thank you very much.