 The idea is to, I don't know, it's a way to find distinguished elements in a space, in some sense, okay? So we assume that in our set or a space, we have a notion of tangent space, in some sense. And at each point, at each structure, we have, I don't know, it's like a, well, it's a tangent vector, it's a direction of improvement. I don't know how to say this even in Spanish, but it's a preferred direction, we'll see in the examples, that in some sense, in this direction, the, the, the structure is improved in some sense, you can think in, I don't know, in a gradient of functional or something like that. Okay, so as usual, we also have an equivalence relation in the set, okay? And we denote the equivalence class of each element in this way. Okay, and so these, these three things are, are enough to define the concept of soliton, which is when the, when, when the preferred direction is tangent to the, to the equivalence class, okay? So it's, in some sense, is what, what we are, what we are saying is that the, I don't know, that the structure is, is nice enough, okay? In the sense that the, the direction, so what is, is supposed to, to improve is, is tangent to, to its equivalence class, okay? So it, it can not be improved in some sense, in, in, from this point of view. And well, usually the equivalence class is given by the action of a group, and if you is a Che Guevara, and that we will, we will assume that everything, always, then the structure is a soliton, if and only if the whole class is a, is a solid, is a, consists of solitons. So maybe this picture can, can clarify what I'm trying to say, recall that we are in a very heuristic way, heuristic situation. So imagine that this is the space of structure, you have a structure gamma here, and it's a equivalence class. This, these lines in color, Jill, I think he's called the color. Are the equivalence classes. And then at each structure, you have these prefer direction. Okay. So that's, that's it. color are the equivalence classes. And then at each structure, you have this preferred direction. So you have another equivalence class here. And so what we are saying that a soliton is this, is when the preferred direction is pointed to stay in the equivalence class. So it's not improving the structure. OK. Of course, if you assume enough differentiability in the setting, you get the flow, OK? An evolution equation, which is this. And so in this general setting, what we are saying is that a soliton is precisely the structure which evolves cell-similar, OK? That the solution stays in the equivalence class. So it's like you are not getting anything new with this flow. So of course, it's like, well, these flows are invented or considered to find usually the fixed points of the flow. And solitons are not fixed, OK? They are moving. So they can be a problem. They can stop you in the search for the fixed points. So they are a problem if you are trying to prove, for instance, the existence of fixed points. But on the other hand, if you are looking for what would be distinguished or what would be the best element, it can be very useful, OK? So let's start with some, I don't know, baby examples just to show the idea. We have considered the vector space of n by n matrices. Well, from some point of view, the nicest matrices are normal matrices, OK? Each matrix is semi-simple if and only if it's conjugate to a normal matrix. And normality is invariant by this action, OK? So let's take these as the equivalence class. Two matrices are considered equivalent if they are conjugate by an orthogonal transformation of the scale. So this would be, I mean, if you, the tangent space at the matrix of the GLN orbit of the conjugation class is the bracket between a and something. And then if you put that something as this, measuring how far is the matrix to be normal, this looks like a natural preferred direction. And well, what you get is that a is a soliton if and only if this q is a multiple of a. So this part is gone of the tangent space. And this is a very nice equation for a matrix. You, it's easy to see that then either a is normal and c is 0 or a is nilpotent, but not any nilpotent. An nilpotent matrix satisfying this equation. You have here like the Jordan blocks, which are the matrices. So these matrices are the solitons and any orthogonal conjugation of these. So instead of the expected, I don't know, Jordan blocks with 1111, these are actually the nicest matrices if you want after normal matrices, beyond normal matrices. And indeed, they are well in a conjugation class, there is in a nilpotent conjugation class, there is a unique always up to orthogonal conjugation and scaling. But it's unique and these solitons are minimal for this functional, measuring how far is a matrix from being normal. So are really the global minima on the whole conjugation class. For polynomials, take the vector space of homogeneous polynomials, I don't know, quadratic forms, binary, terminal forms. And in this case, the role of normal matrices of nice elements can be played by harmonic polynomials, which are the silos of the Laplace operator. And then we can see beyond harmonic polynomials, what would be a nice polynomial. So consider also this action. Harmonic polynomials are invariant by this action. And then, well, this is a natural preferred direction. The fixed points are precisely the harmonic polynomials. And so we may ask, what else is a soliton here? And well, it is well known that the space of harmonic is irreducible under the orthogonal action. And actually, these are the other irreducible components. And then we have here candidates for solitons. And indeed, if you take a polynomial here, which is not harmonic, but it's a multiple in this way, then it's actually an eigen polynomial of this operator. This is precisely the eigenvalue. And so it is a soliton. And well, it's very easy to see that these are all the solitons. This is very related with the spectrum of the Laplace-Beltrami operator of the sphere. But OK, only this. And if you consider the flow, it's very easy to see that if you start with the polynomial, with this projection non-zero, then you will converge to harmonic. But if this part is zero, then you will stop by one of these solitons in your way to a harmonic polynomial. OK, let's see a more geometric example. And of course, a very sophisticated example. Plane curves, the set of all plane curves, the equivalence we consider is if the traces are the same up to rotations, translation, and scaling all together. And so, well, let me show. So what's the tangent space of a curve? It's not that this, in this case, is not only one arrow. It's a field along the curve. That's a direction. And well, it's, of course, very natural to consider the curvature. The second derivative of the curve, assume that the curve is parametrized by arc length. And so the curvature, the second derivative, is measuring how curved is the curve at that point. OK, it's exactly that. And so it's a very natural, preferred direction. And as you see, we can already see in this picture that this, if you imagine the flow, the flow is kind of trying to make the curve convex. OK, to make the curve convex. And if you imagine already a convex curve, then the flow will try to shrink the curve to a point. We can already see that in this picture. OK, so this flow is very famous. It's called the curve shortening flow. And it may be considered as the gene of all geometric flows, I would say. And some of its nice properties are these. All these kind of curves are invariant by the flow. Also, it's precisely the negative gradient flow of the length. That's why the name, curve shortening. So it's the optimal direction to shorten a closed curve. This is in the closed case. And so what we were guessing are actually theorems. So Grayson proved that any simple curve first becomes convex. And then Gays and Hamilton, not then actually before in time, proved that once it's convex, then it converts to a round point. That means that it converts to a point. But asymptotically becoming a circle. OK, which are the solitons? A soliton from the dynamical point of view should be a curve which flows according to this equation without losing its shape. Just by rotations and translations and maybe expansion or shrinking in the scaling. So of course, straight lines are fixed points. Straight lines are precisely the fixed points of this flow. And circles, you can imagine that a circle will convert to a point and will always be a circle. But this is not a trivial question. Are there other solitons? I think it's not very hard to see that the only simple curves which are solitons are circles. But this is a very good question. And let me show you the first example. This example was found by Calabi. It's called the Green Ripper. And it's the unique translating soliton, the unique one. Look at the formula. The equation is very simple. But it's curvature. It's precisely the little wings you need at each point to move the whole thing without losing its shape. This is beautiful. We have also these solitons. These are shrinking. So the evolution is just the same flower all the time, just smaller and smaller. This is there is an infinite family depending on parameters on natural numbers, classified by average and longer. And the pictures are by Haldorsson. Thank you. And Haldorsson finished the classification of solitons in 2012. As you see, this is not from one century ago. This is very classic. And it's still a very on-going topic in this time. So I'm trying to show all the behaviors. But for each behavior, there are a list. But the complete classification is really a list. So they are very, very special curves. You have here three expanding solitons, just expand. And these rotate, rotate. And well, you can rotate and expand or rotate and shrink. This is tennis ball or yin and yang, as you prefer. OK, so let's start with a talk. It's OK. So let's go to a differentiable manifold. And well, OK, so we keep the setting in a very general, in this slide. Take S as the set of geometric structure of some time. I don't know, Riemannian matrix, all moscalar matrix, spin-7 structures, some class of structures. This set is usually a subset of a vector space of tensors, and sometimes it's open. For instance, Riemannian matrix is an open set of the vector space of two symmetric tensors, for example. And well, a preferred direction will be some, I don't know, curvature tensor with respect to some connection, the gradient of a natural functional or a Laplacian or, OK, it's again a preferred direction or direction of improvement. The equivalence will be given by defiomorphism, but sometimes by a subgroup of, for instance, if you are considered Hermitian matrix, you have to take a holomorphic defiomorphism, right? OK, and so, well, again, these three, the set, the equivalence, and the direction gives you the concept of soliton, OK? And of course, we can assume that this Q is natural enough to satisfy this defiomorphism, OK, at least for these Hs. OK, this is a soliton in differential geometry, and OK, the first example is, of course, Ritchie solitons, which the name of soliton was used for the first time by Hamilton was introduced by Richard Hamilton. And so in this case, you have the space of Riemannian matrix, as I said, the symmetric two sensors. Here, you have all the defiomorphism and scaling. And OK, this is a preferred direction to go toward the Ritchie tensor. In principle, it's not degrading of any functional or anything, OK? So it's just natural. But well, in this case, the flow is the very famous Ritchie flow, we all know, introduced by Hamilton and OK, used by Perlman as a primary tool to prove all these very, very important results. OK, so let's go to other types of structures. But before, since we know how to compute this tangent space, then we get this formula for a soliton, OK? This is the lead derivative of the tensor with respect to this field x, because this is a typical element in this tangent space. And well, here, we also have a flow, at least an equation. But OK, this talk is more about solitons than flows. For instance, the first problem is, I don't know, maybe your preferred direction is not defined in flow, that you cannot prove existence of solutions. But so sometimes, you will get solutions uniting the short-time existence, but sometimes you don't. An example of this is the Ritchie flow on pseudo-Riemannian geometry that you can study solitons. There are many, many results and papers about that, but you don't have a flow. So you can use this soliton concept without the flow. Of course, usually most of the times, you have the famous flow associated. But OK, so with respect to the evolution structure, it's a soliton if and only if it's a self-similar solution. This is the concept, that the solution starting at the soliton is self-similar. So if you expect it to improve gamma with this evolution, you're not. You're not. You're getting all equivalent germane. OK, well, you have these notions of expanding steady and shrinking, depending on the sign of this C here, this constant C. And you get this kind of solutions for the solitons with these very poetic names. OK, so let's start with Chen Ritchie solitons. Here, you have a fixed complex manifold. So not only M is fixed, it's fixed as a complex manifold. The complex structure J is fixed. And you consider all Hermitian metrics on M. And then, well, instead of all symmetric, you have allomorphic symmetric two sensors. As I say, biolomorphic diffeomorphis for the equivalence. And well, a very natural preferred direction is the Chen Ritchie tensor. This is there is a connection called the Chen connection, which is the only Hermitian connection. Hermitian means that the metric and the J are parallel. And this is the only Hermitian connection such that the torsion is anti-J invariant. So the 1, 1 part is 0. So this is a soliton. This is the equation for a soliton. We also have these pluriclossed solitons. In this case, we take the Hermitian metrics such that this condition holds. These are called also SKT metrics, a strong Kailer retortion. And in this case, you may choose as a preferred direction the Bismuth connection, the Bismuth Ritchie tensor. This is another Hermitian connection in almost Hermitian geometry, but this is characterized by having a total skew symmetric torsion. OK. If you fix a symplectic manifold and take the set of all compatible metrics, and of course, take all everything is symplectic here, take symplectomorphism for your equivalence, well, Lea and Wang define this preferred direction, this flow, which is to take the anti-J invariant part of the Ritchie tensor. It's very natural. But I think this is another case that you have traveled to get a general short time existence theorem for solutions. But you can, of course, classify and study solitons, remember, without the flow. So this is very natural in symplectic geometry. And in this case, you don't fix a complex or a symplectic structure. You just fix M and consider the set of all almost scalar structures. OK, so everything will be moving here. So these are the almost Hermitian structures that the omega is closed. So of course, you are always in a symplectic manifold, but omega is changing. So it's not fixed, the symplectic manifold. And well, the vector space of tensors is this one. This condition will make, I mean, you need this to stay in the set of almost scalar structures. And in this case, you can take all the diffeomorphies because nothing is fixed. And this is a very natural flow defined by Street-Santiane. You also take the Chern connection and the Chern Ritchie form instead of the tensor, which is closed. And then this is pointing, I mean, you will keep having a symplectic strato. So omega will be closed in time. But then according to this tangent space, you are forced to put as the one-one part of this coordinate. And then as the anti-j-invariant part, you can put this very natural. And so this is the symplectic carbate flow. And well, the equation for a soliton is actually two equations. And well, the fact that everything is flowing is really challenging. And of course, the classification of complex surfaces is the main problem, I don't know, motivating all this, the study of all these flows. And well, let's go to some exceptional olonomy. I'm just giving some examples. There are many, many more. G2 Laplacian solitons. So in this case, you are in dimension seven. You take the set of G2 structures, which is a definite three-form in the sense that it defines a determinism metric and the volume form. The set of G2 structure is open in the space of three forms. And well, this is a very natural direction where delta is the Ho Chi Laplacian. And well, this is a soliton. This flow was introduced by Robert Bryant to starting with a closed G2 structure. The idea is to flow in order to find the fixed points, which are the parallel G2 structures. And then you get that the metric has a lot of it containing in G2. So this flow was introduced to try to find a lot of G2 metrics. OK, let's start with some, with lead groups. So we come back to the general case, not a specific example, but now our manifold is a lead group. We assume simply connected. And then, well, everything works mostly for homogeneous space is actually, but the presentation is much more technical. So let's stay in a lead group. But remember that for homogeneous space, this most of what follows works. So accordingly, we consider left invariant geometric structures, invariant by the action of the group on itself by left translation. And we can consider any of the above. In this case, since the tensors are left invariant, then G will be a finite dimensional vector space on the lead algebra. And well, also, accordingly, the equivalence will be defined using these, if you want, algebraic defiomorphism, which are the automorphies of the lead group. So you consider that two, I don't know, metrics are equivalent if they are equivalent via an automorphism of the group, not only a defiomorphism, to keep everything in the context. And the preferred direction can be any of the above. And so the solitons are called semi-algebraic. And you get this equation. So it's very easy. Why? Because the tangent space at gamma to this orbit is precisely this lead derivative. But now the field is not any field. It's a very algebraic field, if you want. Are these field defined by derivations? They are generalizations of linear fields on our end. They are not left invariant because of the identity, they vanish. But they are, OK, these fields, defined by these automorphies, attached to a derivation. And actually, the lead derivative of the tensor is equal to this very algebraic. This is the action of glg on the set of tensors. So everything is very unwell. Algebraic solitons are, if you, in addition, have this additional property for the derivation. But as you can see, this is full of algebra. All in this steel color is algebra everywhere. And well, the equation is very nice, because on the left, you have the geometry. Any geometry as above. And on the right, this term is very algebraic. It depends on the derivation of a lead algebra. And well, the most important is the quality part. OK, so in the Ricci flow or Ricci soliton case, Jablonski proved that every semi-algebraic is algebraic. And actually, every homogeneous Ricci soliton is isometric to an algebraic. So we will be talking only about algebraic Ricci solitons. They are called nil solitons on nilpotently groups. And well, there are many results on this. You have uningness, so it's a way to provide an nilpotently group, which is impossible to have an isometric on an nilpotently group. But you may have one of these solitons and it's unique. So it's a way to provide an nilpotently group with a canonical metric if you want. And they maximize this functional, this Ricci pinching functional, measuring how far is the metric from being Einstein. And so that's a very nice property of solitons, which is independent of the definition. And well, we have many classification results in low dimension by Will and Fernandez Kulma. And as you see, a lot of people, I must be forgetting someone, I'm very sorry, have worked on these nil solitons. And we still have, after 20 years, unfortunately, this painful question. We don't know. We don't know general abstractions, just that the nilpotently algebra must be n graded. But it's incredible. As usual, on nilpotently algebras, you get to a point that no answers or no nice answers, at least. OK. Well, solitons are on solvably groups, more general. And we also have uningness. So you can equip a solvably group without an Einstein metric, without admitting an Einstein metric with these solitons, especially unimoderal solvably groups, for instance. And they are also maxima for this rich pinching functional. This functional is equal to the dimension if and only if the metric is Einstein, right? This is because she shows us. But well, at least in these cases, we already know that for some reason we don't understand. But solitons are nice from this point of view. And also, for instance, nice from this other point of view, very different, which is symmetries. All these people have proved very, very nice results on maximal symmetry, that the solv solitons, among all left invariant metrics in the given lig group, they have maximal symmetry. So they are, in a sense, the nicest ones. And well, the only examples we know are solv solitons. But we also have some structural results on homogeneous spaces. And this famous Alevsesky conjecture, one of the main problems in the area, in the field, is in the homogeneous geometry, is that any Einstein homogeneous is actually a metric on a solvably group, is very relating with these structural results on solitons, on algebraic solitons. Beyond Riemannian geometry, we have examples on examples of algebraic solitons, in the Chen-Ritchie case, in the Simpleton-Kerbatov flow case. As you see, we found shrinking examples, because in Riemannian geometry, all solitons, which are not Einstein, are expanding. But in these geometries, you may have shrinking solitons. In the Plutic-Lose flow, for instance, Arroyo and La Fuente found semi-algebraic solitons, which are not algebraic. So this is also different from the Riemannian case. And for the G2 Laplacian flow, Niccolini found also semi-algebras, which are not algebraic, and also a curve, a continuous family of solitons on a given league group. So uniqueness doesn't hold anymore, at least in this case of Laplacian. We also have this shrinking, and it's really interesting that as an application of these algebraic solitons, these shrinking solitons are the only known solutions in Chen, in general, for the G2 Laplacian flow, which explodes in finite time. So this can be viewed as an application of algebraic solitons through the flow. So by now, these are the only known solution with the finite time singularity. And well, also Fino Rafferro worked on these Laplacian solitons. And for instance, to relate with some other property, you have these, they are called extremely richy pinch. They were introduced by Brian, close G2 structure. So as you see, are the close G2 structure close to behind time, in a sense. So they are very special. And we prove with Niccolini that they are automatically steady Laplacian soliton. We don't know why, but this is the other way around. From a very special geometric pinching structure, you get a soliton. OK, the time is so, well, underlying all these point of view of algebraic solitons is this approach that we call the moving bracket approach, which is an approach which allows us to apply GIT, geometric invariant theory results as moment map, the stratification, stable points, stability, I don't know, null cons. And the idea is, this is the last page, to say something about this, which is very related to algebraic solitons. The idea is that in the Lie group, you have a Lie group and a geometric structure, which is just a tensor on the Lie algebra. And then in the simply connected case, what you have is just a Lie bracket, a vector space, if you want, fixed, a Lie bracket and a tensor. And then you may say, well, why don't we fix gamma on the vector space and vary the Lie brackets? Instead of varying the structures, let's vary the Lie algebras. And then consider the variety. This is an algebraic subset because the Jacobi condition is just polynomials. And identify each point in the variety, each mu, with this geometric structure. So you take the simply connected Lie group with the algebra mu and always the same gamma as a tensor on the Lie algebra. And then this isomorphism is the key part of the approach. H is giving you an equivalence between these two geometric structures. So as you see here, you're moving, you're obtaining all the structure of the same time, all the metrics, all the G2 structures, all most Hermitian structures, if you want. And on the left, you are in the GLG orbit of mu. So you are getting all Lie brackets, which are isomorphic to me. And then in that orbit in the isomorphist class of mu, you can identify with the space of all geometric structure on a fixed Lie group. But the variety can be identified with the space of all Lie groups with all geometric structures of a given type. And then the nice thing is that, well, the usual convergence of brackets with the topology in this vector space sometimes gives you very, very nice notions of topology of convergence for the geometric structures. For instance, pointed or Chigagromov or smooth converges up to the theomorphist. And a very interesting thing is that you can with a sequence of Lie brackets, you can go the limit of a sequence of Lie brackets can be at the boundary of the orbit. So you may get a new lambda, which is not isomorphic to mu. So this allows you to have convergence of geometric structures on one Lie group to the geometric structure on a different Lie group, which can be even non-homomorphic. So that's very interesting. And the relation with algebraic solitons is that if you translate your flow to brackets, it's called the bracket flow, then you can study the solution for this flow now. And the fixed points of this flow are precisely algebraic solitons. So this is something which makes algebraic solitons very, very special. And well, I think I will stop here. Thank you. OK, thank you very much for a lovely talk. OK, that's my video. So now I hope we will have some questions and some discussion. So because we are almost 70 participants, maybe we can, maybe if you have a question, you can just write in the chat that you have a question. And I can just call each one of you in order. So let me just know. OK, so Claudio has a question. So you can start. I'm just trying to see this. OK, yes, thanks a lot, Claudio. I have a couple of questions. So the first one is, in the first part, you motivated this interest for solitons, like thinking to a flow. And then in the example, in these algebraic examples, you mentioned this property about maximizing, for example, this functional s squared over itchy squared. I mean, and then you said as a comment, I mean, this means that they are as far as possible from being Einstein, in some sense. So would that mean that you will not get these solitons by flowing? I mean, they are kind of self-similar solutions to the itchy flow, for example, but as soon as you move, they are kind of repulsive for the flow? I don't know if I understood the question. If you use lights, you mentioned in some example about the algebraic solitons, they are global maxima. So they are as close as possible to be Einstein, as close as possible. OK, so maximizing them, they are close. Because this function is always least or equal to the dimension n and equality holds if and only if is Einstein. So you are as close as possible. OK. But the itchy flow is not the gradient flow of this function. That's the problem. It's very difficult to see the itchy flow as you can. I mean, Perelman did as a gradient flow. But you have to consider the homo-thety classes. But yeah, you may have a lot of these natural functionals. But in some cases, for instance, this function is not monotone. It's not always increasing along the itchy flow in the homogeneous case. Sorry. And I have another quick question. Are there also nonexistence results? Yes, yes. I mean, is there some general conjecture about when they should exist and when they shouldn't? Well, that depends a lot on the setting. But in the itchy flow case, we have this tricky question that, yeah, we know that many importantly algebra do not admit, but it's very hard to get nonexistence results already in this case. But in the other context, yeah, we also have some league groups that we can prove that there are no algebraic solitons, but sometimes we don't know if there is a general soliton. Of course, algebraic solitons are solitons. I mean, they are solitons. They are just special. But sometimes, maybe a league group can admit a very exotic soliton, which is not algebraic. But yeah. Thank you. OK, so we have several other questions here. So the next question was, I think I will let Imogen Kan ask this question first, and then Fernando, and then Ludwig Tablasti. So Imogen Kan, if you have your question. Hi. My question is that imagining a soliton as a physical object, I have difficulty understanding what does semi-algebraic soliton, how is semi-algebraic soliton different from the normal soliton? Well, I can show you the definition. I mean, I can answer that from a physical point of view, of course. But the difference is that if you see, I mean, here you can see the difference that the general soliton, you can put any field here. But for the semi-algebraic, you can put only these fields, which are defined by derivations. And from the point of view of flowing, maybe it's more clear that the solitons are, we say that are those extracted flowing in this self-similar way. So for instance, rich solitons is when you flow with F of t, a one-parameter family of defiomorphism. A semi-algebraic soliton is when here F of t are automorphism of the lig group. They are defiomorphism, very special for the lig group. Since you are in a lig group, it's very natural to consider that, to ask for automorphism instead of defiomorphism. And that's a difference. But the Jablonsky's results tells you that in the rich flow case, it's more or less the same. OK. Thanks. And there's another general question, if you can answer that regarding the Poincaré conjecture, and its solution, which was done by Gregory Perlman. So I just want to ask if you have any idea whether solitons were used in that proof or are related to Poincaré conjecture or not? Yes, yes. One of the things that Perlman proved is the non-existence of solitons in dimension three. Right, right. Thank you. OK. So I think the next question was Fernando Rodriguez Villegas. Yes, hi. Thank you for the talk. No, I have a very naive question from an outsider. I always thought that solitons were these things that this guy riding a horse in Edinburgh saw in a canal and that they followed it for whatever long it was. Is this related to this? I get a sense that it is, but you didn't mention it. And so I was wondering. No, yeah, that's the problem. In PDE, the word soliton is much older. And it refers to this solution, quarterly debris equations that does water waves flowing without losing the shape. But yeah, in the differential geometry, Hamilton started to use this word in a more general sense for self-similar solutions. That would be the correct name in PDE. But is it the case that the PDE soliton satisfy your action? Yeah, yeah, it's the idea is these water waves that flow without losing its shape. Thank you. OK, so it's the same idea as in the curve shortening flow. That the soliton is to flow with the same shape. Yeah, it's related. But yeah, I think PDE people doesn't like the use of soliton in this. But it's already 30 years and spread. OK, so the next question is from Ludwik Dabrowski. Hi, thanks for the word of examples. I have also kind of a naive question. So when you started with examples on the curves on the plane, you spoke about rigid motions and scalings. And when you went to manifolds, you also had you substituted the rigid motions by default morphes. But scaling remain numbers. Later, they may depend on flow, on time, on parameter T. But they are numbers. Why they are not functions on the microphone? Is there some reason for that? Oh, no, no, because yeah, you mean like to consider conformal metrics. Yeah, that would be like a too big equivalence relation but for some purposes. But in the curves, we also consider just scaling by a number. So my question was there must be some reason. There's two groups of transactions. Group would be too big, yeah, that's what you said. Yeah, I think so, yeah. But yes, yes. Thanks. OK, so Yalong, Yalong, you had your hand raised. But do you want to answer? OK, you still have a question? Yeah, OK, OK. Thanks. I recall that you mentioned that for the trend-rich flow, you constructed the shrinking type soliton, right? For trend-rich solitons? Yeah. I remember you stressed that you constructed shrinking soliton, right? Yes. Also, I think you mentioned like for the new manifold, like also there are a lot of solitons. So I remember that actually you proved that there's no shrinking solitons on new manifold, right? But for rich? Yeah, for rich solitons. For rich solitons, yeah. Yeah, yeah, this is very different, yeah. So can you explain what's the reason why this kind of difference happens? Oh, right now? Maybe. Yes, on new manifold, you mean? Yeah, on new manifold. Yeah, because for trend-rich, there is something even more strange that any new manifold is a trend-rich soliton. So you have to go to solvably groups to make sense, because anything is a soliton on new manifolds for trend-rich. It's not a very strong condition. OK. So, but yeah, for rich, you prove very easily in one line that the constant C must be positive, so you get always expanding. OK. I don't know the reason to say that why in this order you may get shrinking is because it's very different. But right now, I can just give you a better explanation. OK. OK, thank you. You're welcome. OK, so, were there any other questions? Someone else who wants to ask something? OK. So it doesn't seem so. So, yeah, give you a second. OK, so if there are no more questions, then I would like to thank you again for this excellent talk. And I will just remind everybody again that this is a seminar that we will try to continue every week from now on for some time to come. And there will very soon be a kind of schedule that we'll put on the web page. And the next seminar is in one week's time, the same time, on Thursday at 2 o'clock, Central European Time. And I hope to see all of you here. Then it will be Elizabeth Gasparin, who will be the speaker. And I will see you all there. Thank you very much. Bye. Bye. Thank you, Jorge. Thank you, Sacarias. Thank you, Claudia. Thank you. See you next week. See you next week. See you. Bye.