 Okay, so welcome to the second half of the sugar seminar with Daping Wen telling us about the cluster structures for Legendian links, please take it away. Yeah, thanks. So now we moved on to the more technical part of the talk. So first cluster structures, so it's a bit hard to give precise definition of cluster structures in the seminar talk, normally require more than half of the seminar to do it. So I'm going to follow the tradition of not giving the precise definition but giving you the rough idea of what the cluster structure is. So a cluster variety, roughly speaking, is an f5 variety, be an atlas of torus charts. So this is very analogous to how we think about smooth manifolds. So a manifold is a topological space, be an atlas of charts, they are difumofic to Rn. A cluster variety is an f5 variety, be an atlas of charts, and each of them is isomorphic to a torus, to an adverbic torus, torus charts isomorphic, c star to the n for some n. And moreover, we require that each chart is equipped with a quiver and a collection of local coordinates. So local coordinates are also known as cluster variables. You may have heard of these names, but cluster variables are local coordinates. And local coordinates in the sense that this c star to the n is regarded as a spec of this c join x1 plus minus dot dot xn plus minus. So the corner ring of an adverbic torus is a Laurent polynomial ring, and these generators are the local coordinates. These are the cluster variables. Moreover, we also require that the gruing map of these torus charts are governed by a commutorial process called mutation. So under mutation require both data of a chart to change. So we have one chart, we have one chart here and another chart here. So the quiver of one chart has to undergo a quiver mutation to get to the quiver of the next chart. And the cluster variables of one chart undergo cluster mutation to get to coordinates of the next chart. So the commutorial process called mutation. So in some senses, I mean, in a way, it sounds very restrictive. It's hard to imagine that this kind of definition is ever going to be interesting. But it turns out that this actually exists in a very wide range of mathematics. Many spaces are known to be cluster, Quasmanian, black varieties, Barstamersen varieties, and so on. And if you algebraic geometry, you're probably going to point out my mistake. These are not affine varieties. They are projective varieties, what you're talking about. Well, I should mention that in order to get a cluster variety, you have to toss out something. You always have to delete some divisor in order to get a cluster variety. So the real cluster variety is going to be Quasmanian, throw away a divisor, black variety, throw away some divisors, and Barstamersen varieties throw away some divisors. So the remaining part is an affine variety. And let me also mention that cluster structures actually is very closely related to Toric geometry as well. You should really regard a cluster variety as a deformed version of the torus inside a Toric variety. So in a Toric variety, there's only one torus in the middle. And the Toric variety is often described by some sort of fan structure. And each chamber in the fan is going to tell you some coin that's on the torus. But the good news is that when you go to the next chamber, it's still going to be the same torus, but just rotate it a little bit. So the corner change is a monomial change. In the cluster variety, when you go to the next chamber, the torus is actually not just rotated a little bit, but goes a bit sideways so that the change of coordinates is no longer monomial. It's going to be a binomial change coordinate. And that's what gives rise to this kind of torus charts overlapping together. So there's a whole parallel between cluster structures and Toric variety. And whenever you see a statement in cluster variety, cluster structures, you probably want to think about what's the counterpart counter statement in the Toric setting. That usually will give you a very good idea of what the type cluster structure is talking about. Okay, so that's a very general overview of what a cluster structure is. Let me state the main theorem. So the main theorem of today's talk is that for many parsed-grade betas, the flag modular space m beta is a cluster variety and moreover, some exact Lagrangian feelings of this minus one closure naturally induces induced cluster charts in this flag modular space. And if two exact Lagrangian feelings are Hamiltonian isotopic, then they must induce the same cluster chart. So in this sense, the cluster structure is giving you a very effective way of comparing of distinguishing exact Lagrangian feelings. If you can show that the two feelings give you two different cluster charts, then they cannot be Hamiltonian isotopic. So here the language is not very precise. Let me also say what I mean by for many betas. In my joint work with Honghao Gao and Lin Hui Shen, we look at rainbow closures in this case. So all rainbow closures, this statement is true for all rainbow closures. So I mentioned during the break rainbow closure, the closure looks like this, and there are all instances of minus one closures. On the other hand, in this later joint work with Magica South, we look at something more general than rainbow closures. So we look at some minus one closures. Some minus one closures are related to what we call grid payback graphs. Closures related to grid payback graphs. But later on, there's a joint work by Kassel, Gorski, Lin Hui Shen, Ian Lei, Lin Hui Shen, and then Jose Simantau. They proved it for all minus one closures of the form gamma w naught, where the delta zoo product of gamma is also w naught. So they further, they prove a more general result than what I had with my collaborators. So this is the most general up to date. So now we know that all minus one closures of the form gamma w naught with gamma itself, having a delta zoo product of w naught, all have this phenomenon. And how does this, how would this help us solve the infinite many feeling conjecture? Well, and we also mentioned that cluster structures are much better, we have much better control than the exact Lagodian feelings. And the reason is that at the beginning of cluster theory, for me the Rinske already gave a finite type classification of cluster varieties. So cluster varieties with finally many cluster charts are called finite type, and like anything finite type in algebra, they are classified by thinking diagrams. So other than a few, two infinite family basically and some sporadic ones, or everything else is infinite type. All right, any question about this main theorem? So part three is a bit, be quick if there's any question about cluster theory in general. Happy to answer. If not going to move on to the feeling cluster correspondence. So maybe I'll just ask a quick question, just a really basic example. If you just had Beto as say one, two, then you just get a point in the modulite space. What do you get as a modulite space in this case? Well, it's going to be very stacky. If it's just one, two, I guess you're talking about three strand braid. Sorry, and I'm thinking exactly. Yeah. So it's going to be very stacky. It's a point quotient by some group. I see. Yes. So in general, the interesting cluster case is for longer braids. Yes, but this is also interesting in some other way. Right. Yeah. All right, thanks. Yeah. All right. So now let's talk about the feeling cluster correspondence. Okay. So let's look at this positive braid, one, two, one, one, one, two, one. It's basically W naught third time, third power. So what if I want to write it, write down the chain of flags, it's going to satisfy this relative position. So it's going to be one, two, one, one, two, one, and so on. But instead of writing it this way, let me actually use some colored edges to separate the flags. Oops. I'm going to use blue for one and red for two. So I can do one, two, one, one, two, one, and then one, two, one. And have twos here. Okay. So this will carry the same, the same data instead of having a dash with a labeling. I'm going to use edges to separate the flags. But then this looks like this is asking us to, to draw the edges further into the disc, the fur in the disc. So this is, this leads us to what's called a Gengen N-weave. So a Gengen N-weave is a planar graph with edges colored by one, two, all the way to N minus one, to N correspond to N minus one here. I should think of this as the cluster generators. And special kinds of vertices. So there are three kinds of vertices that we allow. There's a monochromatic trivalent vertex. So it sounds like a trivalent vertex in the same color edges. And then there's a hexa-valent vertex with adjacent colors. So for example, if red means two and blue means one, right? So I can have a hexa-valent vertex like this with adjacent colors. Or you could have a tetravalent vertex, tetra means four with far away colors. For example, I can use this as one and this as three. So far away means that the late, the, the cluster generators are not next to each other. It's more than one apart. So allow these three kinds of vertices. And let me point out that the last two kinds of vertices, it's a resemblance of the Bray relation. So this is kind of like S1, S2, S1. And it goes S2, S1, S2. So you see one to one on this side and two and two on this side. And this is resembling S1, S3 equals S3, S1. But these two are a resemblance of, of the Bray relations. And then the first one has something to do with the, the Damazou product. So take S1 squared equals S1. This is the Damazou product. So we allow these vertices. So let's do a practice and, and try to fill in this picture here. I could do a hexavalent change here. And the hexavalent change here. Then I can join this. And I can do another hexavalent change here. And join this. And then I have a trivalent vertex in the middle. So for example, this will be a Lagrangian and we've a three. So you see that the, all the, all the vertices are the, the types that we mentioned. Any question about this example? All right. So now, when we can go how we define the flag marginalized space for passive Bray, for a weave, W, I can also define a flag marginalized space, MW2B. So it's going to be, a flag satisfying relative position imposed by W. And then you quotient again by the GIL action. What do I mean by flag satisfy the relative position imposed by W? Well, if you look at this picture up here, you see that there are still some gaps in between. So for example, F9, F10, F11. So you need to fill in those regions. And you have to fill it in the way such that the flag satisfy the relative position imposed by the edges. So for example, F0 and F9 is separated by a red edge. That means that the two has to be relative position two apart from each other. And F9, F11 has to be of relative position one away from each other. So you have to, the flags have to satisfy this relative position conditions. And then you again quotient out by the GIL action. So this is the flag marginalized space of a weave. Any question? Okay. So now, not all weaves are useful to us. So the ones that are useful are called free weaves. So let me give you a practical definition of what a free weave is. This is not the actual definition given by Kassel and Zasso, but it works equivalently the same. Okay. So weave is free if the flag marginalized space is an algebraic torus, and the interior flags are uniquely determined by the boundary flags. So when W is free weave, then we can do a restriction map. We can take the marginalized space for the weave and then I'm going to forget everything inside and just look at the boundary that gives you an open embedding of an algebraic torus. Where beta is the boundary braid of W. So when you have a weave, you can go around the boundary of the weave and read off to the part of the braid. And then in the marginalized space, the flag marginalized space of the weave, it's going to open inject, open embed into the flag marginalized space of the boundary part of the braid. And this embedding is an embedding of a torus because if it's free, it is an algebraic torus. So you get a torus inside the flag marginalized space and beta. So we say in the main theorem that this is a cluster of variety. So in order to be a cluster of variety, there should be torus charts and these are the candidates of the torus charts. But on the other hand, this is called a filling cluster correspondence. Now we get the cluster. What about the fillings? How do you get exactly 100 fillings from weaves? Well, it turns out that each weave actually describes an immersed surface in X1, X2, Z. So how? Well, you should think of a weave drawn on the disk as telling you a generic end to end coverling of the disk where the edges are the singular locus. So if you see an edge here, that means that the two sheets, the bottom sheet and the second to the bottom sheet cross each other versus if you see a red edge that tells you that the second and the third sheet cross each other along the edge. So along an edge, the sheets cross each other. So that's how you see an edge. What about vertices? Well, for trivalent vertex, there are actually two sheets crossing each other around the trivalent vertex. And one way to think about it is by looking at the graph of the imaginary part of Z to the 3 halves. So you can use a graphing calculator and graph this multivariate function. Generically, you're going to see two points in the graph. But then these graphs actually cross each other along three lines. And that's what this surface is like. So this is trivalent vertex. What about hexavalent vertex? So the hexavalent vertex, you would look at the corner hyperplanes in R3. And if you put place yourself at the passive octant and look towards the origin, generically, you're going to see three points. But then the sheets cross each other along the corner axis. In particular, the top two sheets cross along the passive axis. And the bottom two sheets cross along the negative axis. So if you stand in the passive octant and look towards the origin, you're going to see a local picture of these three sheets crossing each other. That's a hexavalent vertex. And what about a forward and vertex? Well, forward and vertex is not really playing that much. It's like you have two sheets crossing in one direction and other two sheets crossing in a different direction. And they are far apart. So they don't even talk to each other. So that's what a forward and vertex is like. So in this way, a weave is going to give you a merced surface in R3, where the z direction is the height. So the z is the height direction. OK. And there's a reason I use x1, x2, z, rather than x, y, z. And that's because I'm going to do what I did with the Legendre link. So I'm going to get a Legendre surface by taking a cotangent lift. So I'm going to set yi to be the partial derivatives of z with respect to xi. And this way, we're going to get a Legendre surface in R5. And the projection of the Legendre surface onto R4 gives an exact Lagrangian surface. So this R4 is going to be simplistic. And the image, so by forgetting the z coordinate, the image in x1, x2, y1, y2, it's going to be an exact Lagrangian surface if w is free. And moreover, this exact Lagrangian surface is actually a filling of the minus 1 closure of its boundary braid. So it fills the braid. And using this connection to exact Lagrangian fillings, you can actually view the algebraic torus. That's the flag Lagrangian space of the weave as the Lagrangian space of rank 1 local systems on the exact Lagrangian surface. So as we know, rank 1 local systems on the surface is going to be home from h1 of the surface to C star, which is also h1 of the surface with C star coefficient, which is the torus. All right. Now I have, oh, good. I'm doing good with time. Any question? All right. So now we can outline how to prove this infinitely many filling conjecture with cluster theory. So there are two steps. One, we need to find a Lagrangian isotope. So this isotope can be constructed using the right of mice to moves. And you want to prove that this isotope induces a cluster automorphism on the flag modernized space. So a cluster automorphism is an automorphism on the variety, such that it moves one chart onto another chart. So any automorphism should preserve the structure that you're interested in. In this case, the cluster automorphism should preserve the cluster structure. So you should preserve the atlas of cluster charts, but then they preview the charts. It moves one chart to the other. And then you have to argue that induced automorphism is of infinite order. So it means that if you start with one chart and then you move to another chart and move to another chart, you're never going to come to another chart ever again. So now with this in place, what we can do is we can start with the link lambda beta, take any feeling that corresponds to a cluster. And now the Lagrangian isotope can be viewed as a cylinder acting on the link, because the isotope just moves the link around. So it is a cylinder. And then you can attach this cylinder to this feeling. But since this feeling corresponds to a cluster already, and this is the cluster automorphism, so this new feeling is going to give you another cluster that's different from the first one. And if R is infinite order, then you can get new feeling every time. And that's how you'll be able to construct infinitely many feelings by using cluster theory. So I think this is all I want to say. One advertisement, I'm giving a course in the cluster algebra summer school at the University of Connecticut. So if you're interested, you can search cluster algebra summer school, 2024 online, and you should apply. Thanks. Thank you very much. Have a very nice talk.