 Right, so welcome everyone to the Schubert seminar today. We're happy to have the pink bank from UC Davis Telling us about cluster structure for Legendian links Yes, thanks for the Yeah, thanks for the introduction and thanks for the invitation of me to speak at this seminar Yes, it's there. We'll talk about cluster structures for the Legendian links and the main references are two papers that they I Wrote jointly with the whole car new mission and other one with the Rojeka cells So here's the plan of today's talk. I was told that the first half should be Accessible to graduate students. So I'm going to give some backgrounds on Legendian links and their exact Lagrangian feelings and also Define the main object of study for today's talk the black modular space and then I'm gonna move on to the third and the fourth part, which are a bit technical So there will be lots of details omitted. But if you're interested, I'm more than happy to discuss afterward or Over zoom at some other time Yeah, and the main goal is to talk about this filling cluster correspondence Okay, so without further ado me begin and Also at any point if I know my handwriting is not perfect If there's anything that's not clear or you you have a question, please feel free to interrupt and ask the question I'm more than happy to elaborate Okay, so Legendian links and exactly I wanted feelings. So here's the definition a Legendian link is a link in our three With coordinates x y z satisfying the equation that y is d z over dx Okay, given this condition you can project the link on to the x z plane and The core the white corner can be recovered by computing the slope So for example, if you see a crossing on the x z plane projection You can compute the slope and you know that this one has a positive slope And this one has a negative slope and if you pull out your right hand, you know that the y axis is pointing into the paper So the positive slope should be behind the paper and the negative slope should be above the paper So you're going to see a crossing like this Okay, so any crossing on the x z projection. It's always going to be like this in reality and Also because why is the slope so when you project on to the x z plane You cannot have this kind of vertical tangent, you know that if you have a vertical tangent the slope is infinite That's not allowed. So typically you're going to see this kind of cusp returns so you have this one and then on the right going to have something like this and In a 3d picture, it's more like this with the horizontal tangent here, but then this tangent is actually along the y axis So when you project, you're not going to see anything. You're going to see this kind of cusp here, but it's not really a cusp It's still smooth. The length is still smooth Okay, so this is some typical features of the tangent links, right? So therefore when I draw its projection on the x z plane, I do not really need to specify the crossings and All the returns are going to be Don't look like this kind of cusps Okay, so now let me introduce you a big family of the tangent links called the minus one closures of a positive braid so what a positive braid is so a braid is a braiding of strengths and Parts it means that all crossings of the same form. So for example, he's a positive braid Just gonna saw some random example So he's a braiding of three strengths and you see that at all crossings. It's always this kind of positive crossing So it's called positive braid and The minus one closure is the following. So I'm gonna take a copy of this and then I'm gonna close it up like this So remember I have to do it with cusps It's gonna look something like this So it's got a minus one closure of break of beta This is gonna be Lambda sub beta to the on the on both on the two sides you're gonna see Cusps that's Going along So the lowest strands gonna have the lowest strand here The next one is going to correspond the next one and so on so you're gonna close it like like this There'll be a another closure that's sometimes going to mention in the talk. It's called a rainbow closure And the difference between the rainbow closure And the minus one closure is the following so the rainbow closure of a braid We also put it here. Oh Good news is that I can string this a bit So rainbow closure look something more like this I'm gonna When I close it up I close it Like this they call rainbow closure Okay it's I'm gonna mention it at some point, but just keep in mind that all rainbow closures are minus one closures as we will see Using by the master moves Oh Both closures minus one closures So my one closures are more general Any questions so far? All right, so next Since the ginger links has this additional tangential condition, right the right the mice and moves you have to also modify We know that two links are the same because it'll be the same if you can Transform one into the other using right the mice the moves so for the ginger links They're also three right the mice the moves and they can't look kind of similar to the ordinary right the mice the moves So the first right the mice the moves looks like the following so you can create this kind of Twist above or below Which direction it's up to you oops This is a right the mice of one And for right the mice the two You can pass a cuffs through another strength with a bigger or small much bigger much smaller slope So for example, this strength is it has a very negative slope, right? So I can pass this cups through So this is the same as Like this Okay, you always pass cuffs through the strength. That's more negative or very positive And right the mice of three is the same as before just Just like that. Oh not this one. It's the same as before Like this, so this is right the mice of three So any the ginger links that can be related by this by the mice and moves are the ginger isotopic to each other And we consider them the same link any question okay, so Now we talk about we have talked about links which are one-dimensional objects So now let's move one dimension up and talk about the feelings exactly like engine feelings So let lambda be a legengine link Right, so the engine link remember it's in R3. So I'm going to place R3 as the boundary as T goes to infinite infinity in R4 So yeah, the symplatic R4 x y z t and I view R3 as the asymptotic boundary at T go as T goes to infinity and Suppose I have a link there and Exact Lagrangian filling is an exact Lagrangian surface in this empletic R4 That asymptotically tends to Lambda as T goes to infinity, right? So for example, it may look something like this This is an exact Lagrangian filling of an idea of an exact Lagrangian filling of the link I Should also explain what exact means so exact means that so first we have a symplatic form which is the Differential of a Louisville form one form and exact means that the Louisville form is exact on On on the surface So that's what what exact Lagrangian means. So Lagrangian means that it's half-dimensional Right, so R4 half-dimensions up is two-dimensional surfaces and And the symplatic form vanishes along that surface That's what Lagrangian means an exact means that the Louisville form is exact okay and a Theorem of shunt train proved that All-orientable exact Lagrangian feelings of the same link of same Lagrangian link at the same genus So this is quite typical quite special for the genus links. So for for topological links The the feeling the genus of the feeling can always increase you can always Handle attachment to increase the genus of feelings But that's not true for Exact Lagrangian feelings or some of symplatic sort of the engine links So all all links have the same genus So topologically you cannot really differentiate them. So how do we different? How do we differentiate them? So we differentiate them up to what's called Hamiltonian isotope and the reason is that this is exact Lagrangian feelings actually Defines objects in the rock focaya category of the of the R4 Okay, and this is some something that people care a lot in mirror symmetry, but I'm not going to dive into this in today's talk But to exactly keep in mind that to exact line and feelings Define the same object if they are Hamiltonian isotope isotopic So what this means is that there's a Hamiltonian function Such that it's Hamiltonian flow moves One to the other that's what What Hamiltonian isotopic means so it is an isotope generated by a Hamiltonian flow, okay All right So so so far There are very little is proved about classification of Exact Lagrangian feelings of for the engine links The only the engine link with a complete classification is the max TB are not so it's not that looks like this So that's the only The only link that we have a full classification of exact Lagrangian feelings And the classification kind of boring it says there's only one Exact Lagrangian feeling and it's what you think it and that's what you think it should be. It's a bowl It's it's a disc that's attached to this and not it's proved by the Ashford and Potorovic Okay All right, so so this is quite hard question There's also related Conjecture called the infinity many feeling conjecture, right? So it's more about the lower bound of number of feelings for the engine link But the question is does there exist an engine link with infinity many non-Hamiltonian isotopic exact Lagrangian feelings So this question remains open For a long time until the year 2020 where when everybody was locked down locked locked in in door by cove it and Then three groups Kind of overlapped, but we use slightly different methods Prove this result in the same year. So in January Rocheka cells and Honghao Gao they proved it For the tourist links case in July Rocheka cells and Eric Zaslow proved it for three stranded break closures And later in September Honghao Garlin, Huixian and I proved it for rainbow closures or part of breaks and all three approaches used cluster theory in some way and and this shows the power of this connection between cluster algebra and and the and the study of Lagrangian links and so Today, this is going to be the result that i'm going to talk about and also I should mention that one and this last result actually covered the first two cases as well in in this paper, we gave a Sort of a classification. We divide positive breaks into two families infinite type and finite type and we prove that for anything in the infinite type They all have infinite many feelings All right before we move on to The second part are there any questions? The break is after the second part, but you're free to ask questions now This is very good. All right So now, uh, let's talk about some geometry, uh flag modular spaces So what is a flag? A flag in cn is a nested chain of vector subspaces So zero being the smallest vector subspace In some f1 in some f2 and so on all the way to fn being the full space such that each Under k step the dimension of the vector subspace is k And the reason it's called flag is because if you think of the flag in, uh The deciding point of flag is a point contained inside a line and then contained inside a plane So you see a flag right there. That's why flags are called flags Of course, if you keep increasing in dimension, then then no longer looks like a Flag that we know but it's still called flag Okay So, uh, given two flags, you can talk about their relative position um, the relative position between two flags are encoded by, uh, permutations Um, but the simplest kind of relative position is when the two flags only differ at a single step So here, um, I'm going to give two examples. So these are going to be flags in c3 Okay, so a flag in c3 is going to be, uh, a point which is the origin contained in a line contained in a plane and then contained in c3 And we write that two flags are of relative position one That means that the only subspace that they differ is the one dimensional subspace So the two flag looks like this. So you have so they share the same plane And then maybe one flag has this, uh One dimension line and this one has this one dimensional line. So these two flags Will be of relative position one So so one of the flag is the point contained in the line contained the plane And then the other flag is the point contained the line contained the plane So you see that the two flags share the same plane and the same point but have different lines Versus If you have two flags The relative position two That means that they share the same line But they have different planes one flag may have a plane It's like this another flag may have a plane that's like this so that's And then of course, there's the already there's the point in the origin So the one flag is going to be a point containing the line containing the plane And then the other flag is the point containing the line containing the plane And you see that the two flags share the same point the same line, but have different planes So these are the right the simplest kind of ready positions So and they have this they have the same three-dimensional space? Yes, yes, there are flags in c3. Yes. Yeah, good. The three-dimensional space. Okay. Yes But of course if you are in higher cn, then you could have more choices You can position one two all the way up to n minus one Thanks for the question all right So now I can define the the main focus of today's talk the flag more try space So let beta I want I2 so these are This tells you where the crossing this right now for example Great that we had before Maybe this one So if I want to write this braid in in this form that I'll write it so write it as 212112 Because it's crossing at a second level first level second level first first second Um Let me think I probably want to call it the other way around but Sorry, let me let me let me let me write it like one two one two two one. Maybe I prefer this way So so I'm going to call call this level one and this level two So this is level one two one two two one So I can we call the braid like a positive braid like this This is a positive braid of with n strands Then the flag more try space m of beta is defined to be The space right where you have a bunch of flags f zero That's I1 from f1 That's I2 from f2 and so on all the way to n minus one so l minus one used l i sub l minus one and then the last i sub l i'm going to go back to f zero So this is a cyclic chain. So I start with f zero and then I end at f zero again And all these are flags in in cn that these are flags in cn satisfying this ready position condition and I'm going to caution out by The diagonal gln action So gln acts on cn. So it moves flags around. So there's a simultaneous action on all flags Then we're cautioned out by this action It's called a flag more dry space and You point out that the flag more dry space is actually a attention link invariant of lambda beta of this minus one closure of beta this So the reason is because it is a modularized space of simple objects in the kashuar asha pura category of constructible sheaves with microlocal support along lambda beta so I'm not going to dive into the dimension of this category Just treat it as a black box, but let me Let me emphasize that this is a the legend link invariant. So it is quite meaningful object to consider Any question About the definition of flag more dry space. Can you remind us what was lambda beta? Yeah, lambda beta is you take the break beta and then you do the minus one closure. So you look something like this And draw it up here. Yes, no problem But this is a lambda beta All right. So this is the part when I designed the break is What the break is? Yeah, uh, we should take a five minutes break or something Yeah, uh, thank you. Yeah, let's take uh, six minutes break. It's actually until five o'clock. So it's four. Yeah, all right now sounds good