 Okay, so that data gives for us what's called a scattering diagram in the real vector space associated to N. That's the space where you would draw the fan of the toric variety. So what's a scattering diagram? Well, it's a set collection of walls. So there's the support of a wall. That's what I'm writing by this Gothic D. So that's just a real co-dimension one cone in the vector space Nr. And so we fix, just choose a sign here. So we fix an element of the dual which annihilates the cone. So it's determined up to a sign. Okay. So what's this F? So that's a function that's attached to the wall. To define that, first of all I need to have a vector. So there's a vector in the lattice N. It has this property. So if you apply the skew form on N corresponding to the holomorphic two-form on U, that's a multiple of N. This M up here. In particular, just notice since this is skew, in particular the V does lie in the hyperplane M, the linear span of this wall D. Okay. And so then my function is basically a polynomial in the corresponding character Z to the V on the dual torus. So here the dual torus is N dual, and we always write M for N dual. The characters of the dual torus are M dual in other words N. So this should be thought of as a character on the dual torus. So we take a polynomial in that variable and we add some formal parameters T1 up to TR to control convergence issues and we take the completion. So that's what this hat denotes with respect to these parameters TI. Okay. And so what you should think of is this is a function in the global functions on the mirror torus, the dual torus T dual, but we've had to work and introduce these parameters. So really it's that poor at torus cross with an affine space with parameters TI and then we're completing with respect to the TI. Okay. And so the direction of the wall is minus V. So that's sort of a part of the notation. Okay. And so associated to such a thing, if you give me a wall I can write down an automorphism of the dual torus and it's given by this formula which I wrote down last time. So if you take a character on the dual torus Z to the U, so user vector in N, then that's multiplied by a power of F given by this formula. So just pair U with this element M, which annihilates D. And so the sign is arranged in the following way. So this is correct if you're going from M positive to M negative. And just as a reality check, I should have written this down last time, but I didn't. Here's the sort of simplest possible example. So I'm in the plane and the wall is the X axis. I'm writing down the function 1 plus TX. So X and Y are characters of the dual torus. They're a dual basis. They're a basis of this ambient lattice. So they're a basis of N. This is a picture in N. And so this is a wall. If I put 1 plus TX on this wall so that the monomial X points in this direction, it means the direction of the wall is the opposite direction to the monomial. This sign change here. And if I choose to go from the upper half plane to the lower half plane, so my M would then be just given by the usual coordinates in the vertical direction. And so this formula will tell me that Y is multiplied by 1 plus TX to the power of 1. So that's the transformation. And you can see that's just the usual cluster transformation if you set T equals 1 for T equals 1 or T equals some finite value. And so let me just say what we sort of discussed last time. So this does two things. The first thing is it counts holomorphic disks in U ending on a Lagrangian torus fiber. And I should have said this last time as well. If you don't like to talk about holomorphic disks, then you can really avoid that in many cases. So equivalently maps, holomorphic maps from P1 to this compact variety X such that the inverse image of the boundary is a single point. So I shouldn't say equivalently, so that's sort of not a theorem, but at least in dimension two that's sort of certainly strongly suggested by a work of Gross, Pandary, Pandian, Siebert. Okay, so that's the first thing it does. It's some sort of enumerative data on U, some kind of open gromm of Whitton invariance. And the other thing it's doing, which we sort of saw here, is encodes the gluing of the torus charts in the mirror. B is a union of the dual torus. Okay. And I also wanted to mention, so this scattering diagram is a very complicated object, but there's a very nice paper which I already mentioned a few seconds ago, Gross, Pandary, Pandian, Siebert described it in the case of dimension two and related it to these gromm of Whitton invariance. So this is a very nice discussion of scattering diagrams in dimension two if you've never seen the concept before. Okay. So at this point I want to give some examples, but before I do that, are there any questions at this point? Yeah. Oh, ask just a positive integer. Yeah. So we'll see in a moment actually. More questions? So let's give some examples of scattering. Oh, actually, before I do that, let me also mention one thing we did last time. So this Konsevich Soyml and Gross Siebert Lemma, what it tells you is how do you get a scattering diagram? What you do, well, how do you get a useful scattering diagram? So you start with an initial scattering diagram. There's this assumption the wall should be hyperplanes and that gives you a unique, a larger scattering diagram. So this D containing D in such that all the walls in this new diagram are outgoing walls in a way I made precisely last time such that the new walls are outgoing and if you take the corresponding automorphism of the torus for any loop, for any loop, you have got some complicated scattering diagram. I take a loop composing the scattering automorphisms, I get some automorphism, the dual torus. I want that to be the identity. It's saying that when I do this gluing, it's consistent. But you get a unique diagram with that property. Okay, so basically what's going to happen in our context is we're going to be given an initial diagram corresponding to the toric model of our cluster variety. So for us, the N is given by the data for the toric model. Let me just explain. So first of all, what is it? Let's write it down very simply. Just take all the hyperplanes MI per 1 plus TI Z to the VI. So I'll tell you in a moment what these M's and V's are. So for each M and V, what do I do? So I remember how does the toric model look? I start with X bar. This is a toric boundary divisor. Let's call it C as usual, a component of the toric boundary. That's the component of the toric boundary D bar corresponding to the vector V in N. Just as usual, if you have an array in N, that corresponds to a boundary divisor of your toric variety. Okay, then remember we took inside that divisor a co-dimension to subset Z and that was given by an equation, given by a character. So if I use this notation Z to the M for the character associated to an element M, this is just the locus where that character is equal to some scalar lambda and I do a blow up. So what the picture is, at least this is some drawing dimension 2, but in general the picture just looks the same except there's a product with a trivial direction. So I'll get the strict transform of this divisor given by the character and I'll get some exceptional divisor. So what's happening here is if I draw a more faithful picture, let's assume we're in dimension 2. So this is a CP1 and here I've got a copy of C and these are my disks which are corresponding to the, this is the disk corresponding to this wall. M perp and 1 plus T to the Z. So you just write down a bunch of walls which correspond to these disks which are generated by the construction of the toric model. And if you draw this in N, so the picture in N is that, you know, so if I draw this picture it's sort of a bit more boring. You've got the hyperplane in M perp corresponding to the zero locus of this character, Z to the M or rather, you know, the zero locus, the, the, this translate of the hyper torus Z to the M equals 1. Tropically that corresponds to just the hyperplane M perp and you've got this direction V, remember, so that's inside, that's pointing towards the divisor. You think about it tropical, it's pointing towards the divisor in the, that this disk comes from. And so the direction of the wall which is minus V is the, is the direction in which the disk, the disk's area increases. So somehow it's the opposite of this direction. If you go in this direction and you change the Lagrangian you want this disk to end, the area of the disk is increasing. That's, that's the meaning of the direction of the wall. Okay. So that's, that's what we're going to plug into this Conceivich-Slovenman Lemma. I'm going to get some, some crazy scattering diagrams out of it. So somehow the initial data however is completely trivial, but the scattering diagram produces some rather amazing data. So let's see the examples. So the first example is going to correspond to just taking this cluster variety. So just take P2, this will be my Toric Model X bar D bar, and I blow up two points. So that's a very simple cluster variety. So what does the scattering diagram look like for that guy? So let me first draw the initial diagram. Looks like this. I've got two walls to begin with and these are the directions, these arrows. This is 1 plus T1 X inverse, 1 plus T2 Y inverse, where again I've chosen X and Y to be a basis of my, of the lattice in this vector space. And so a short computation shows that if you add one ray in the direction 1, 1, and with attached function 1 plus T1, T2 X inverse Y inverse, then that's got the consistency property. So this is the, this is the scattering diagram associated to that initial scattering diagram. Maybe let me just say, so what's the picture, enumerative picture? So again you think about these incoming rays as corresponding to two disks like this, and what you're doing is roughly speaking gluing on a cylinder like that and perturbing so that this disk becomes holomorphic. You get some disk that looks like this. This is the new disk associated to this wall here. And if you don't like holomorphic disks, you can do this in algebraic geometry as well. Okay, so I'm, I'm blowing up this, these two points. Let's look at the line that goes through those two points. So I'll draw it as a kind of amoeba, just the line. I guess you don't normally draw the line through PQ in that way, but okay. Now I blow up those two points, P and Q, and you see this disk, well actually what you see is you see a P1 in the interior. That's just the strict transform of L. And the disk is obtained by just truncating. So that sort of explains what's happening in this picture from enumerative point of view. Okay, and so if you see, see that example, you might think, oh, well this is going to be easy, you know, it's very simple. But very quickly you realize that scattering diagrams can be much, much worse. So let's see another example. So just sort of slightly change what I did. So example two, just blow up two points on each boundary divisor. Okay, and so in terms of the scattering diagram, what's happening? Now I've just changed this, so I've got two factors on each rays. This will be 1 plus T1 X inverse times 1 plus T2 X inverse, and similarly on the Y axis, so 1 plus. And so for simplicity let's just set all the T's equal. I just want to try to give you the flavor of this diagram. So what does this diagram look like? Well first of all, there are infinitely many rays like this. We do still have that ray of slope one, but we also have rays approaching that from both sides. So infinitely many rays approaching this given ray, so explicitly what are the rays? Take these, these guys. So in the discrete part of the diagram, the functions are simple as before, and the symmetric ones on the other side. And you've got this one ray that's kind of more exciting. So this has attached function, which is a power series. And it's just, if you want to expand, it's a power series with positive coefficients, just using the binomial theorem. So this, this function attached to this ray is really, it's a power series, it's not an algebraic function. Well, I mean it's an algebraic function, but it's not a, it's not a polynomial. So maybe I just mention, you know, at least remember one thing I did say last time was that although this diagram isn't finite in general, there is finiteness if you reduce mod t to the n for some n. So you know, so mod t to the l will be finite. Meaning, you know, for, for all but finitely many walls, the function will be congruent to one, and the automorphism will be trivial. That's a general property of how these diagrams are defined. Okay. And so one more example, actually two more, but one will be very quick. So example three. Okay. So now let's go, go one more. So let's just take m blowups on each, each component and let's say, you know, m, m at least three. So what happens is if you look at this diagram, so you have the same picture as before in, in for m equals two, that you have some discrete series approaching this wall. I won't give the exact formulas, but you can. Somehow it's very easy to understand the discrete part of the diagram. Okay. But there's now, these arrays now are irrational. So this is sort of chi, rational, quadratic irrational. That's what it is. But inside this irrational cone, what Mark, uh, Gross likes to call the badlands, we, uh, we have almost no control. So in the cone, in this cone, what we expect is that every rational array appears and, um, you know, the functions are a power series. Somehow that's, that's a part of the diagram that's very difficult to analyze explicitly. Okay. And let me give one, one final example. I won't say much about it in detail, but again, it looks like a pretty innocuous example from some points of view. So take now P two and I do two blowups on each line. So this is, um, a cubic surface of a triangle of lines. So that's R X D. And then in this case here, so what's the initial scattering diagram? It looks like this. So I just choose the usual, uh, let's say I choose the fan for P two like this. Um, okay. So then my, my rays look like this, my initial walls look like this, the opposite direction. Um, this is one plus. Oh, I'm sorry. So yes. So this is one plus T X inverse, one plus T Y inverse and one plus T X inverse Y inverse. Oh, maybe X Y. So that's your initial diagram. And so you, you scatter, you get the scattering diagram D and here every rational ray appears. So this is even worse than when we just saw somehow, you know, there's no good region of the scattering diagram after you scatter every rational ray appears with some power of T. So that's just a, that's right. That's right. Yeah. It's part of the definition. Yeah. Any questions at this point? Um, so I remember I'm blowing up two points on each of the three devices earlier. I was only blowing up on the, on two of the devices. Yeah. Okay. Okay. So now I want to talk about, okay. So let's, what we're trying to do is use this diagram to produce these global functions on cluster varieties. Uh, so first of all, I need to describe a mirror construction for cluster varieties due to fucking Gunsharov. This is work of fucking Gunsharov. And I want to immediately give a warning. Uh, this is really only valid in special cases. So not always valid. And we expect it's okay if our variety U is affine. I'm going to start with a cluster variety U. If this is not an affine variety, this, um, description needs, um, modification. Okay. So what is it? Start with my U and it's going to be mirror to V, which is Y, Y minus E the same, the same way. So use a union of Tori. V will be a union of the dual Tori. And, you know, um, let's just recall the notation. So here T is n, 10 to C star and dual E T, dual is the dual M, 10 to C star. Uh, what's the other data that comes into a cluster variety? So there's a holomorphic two form, sigma bar in wedge to M C. This is the holomorphic two form or strictly speaking. It's restriction to T and on the mirror side. So remember, I assume this guy was non-degenerate so I can just take the corresponding form by which I mean, you know, apply sigma, um, after identifying the two lattices using, using sigma. So we're assuming non-degenerate. Okay. And so, so what happened? So we had these gluing mutations on this side. So this is tells me how to glue the charts together. It's given by an element M of M and V of N. M is in M, V is in N and they're related in the usual way. So they must be related like this for some scale and new, but it's got this very simple formula. And so, um, again if we write it in this invariant fashion, it's just that mu upper star of Z to the alpha is Z to the alpha times one plus Z to the M times the pairing of alpha with V for alpha an element of M and of words, a character of T. So this is just the usual cluster mutation in, uh, without choosing coordinates. Okay. But now it's sort of easy to see. I'm supposed to write down some dual, uh, gluing on the other side. Anybody got any ideas? Maybe this is a bit fast, but look, this is determined by an element of N and an element of N, these dual lattices. All that happens on the other side is those two lattices are reversed. So you can just switch to say this is going to be given by mu of N, M, sorry, V, M. And the only thing, uh, I want to say is this actually is convenient to put a sign in here. But if you just switch V and M, and I'll explain later the sort of sign convention that makes it a minus there. Okay. So that's, that's, uh, the Fock-Gontrav mirror of a cluster variety. So now I want to talk about a rather surprising fact that if you think about it the right way, if you look at the scattering diagram of a cluster variety, you do have a discrete part of the diagram, uh, just as we did in this example, example three, there's some nice part of the diagram, which is discrete. So cluster varieties, uh, some work after passing to related auxiliary variety. Um, the scattering diagram D has a discrete region. Of course that's a bit vague, but what I'm trying to say is you can somehow reduce to this kind of case where you have a nice part of the diagram. Okay. So working towards that, let me, um, explain how that works. So let's assume, um, the following condition. If I look at the log two forms on my variety XD, that's just one dimensional. So it's generated by our, our given form Sigma. So in the compact case, this will be what we called an irreducible holomorphic symplectic variety or hypercaler manifold. And it's sort of understood by the so-called Bogomolov decomposition theorem that this is the essential case. And so that should be true in the non-compact case as well. In fact, it should be substantially easier. Okay. So we'll assume that, but then a sort of short calculation of the way that you constructed the cluster variety means that this forces a Sigma to be rational up to scale. So this is a form with, um, with constant coefficients on the tourists. We can assume that that's actually rational. And so scaling, so we can assume this is just in wedge two M. So I don't know if you remember a long time ago, Don complained that the condition looked rather strange. So now I'm, I'm sort of going to this, um, more special case where that, that form is integral. It's not, it's not a complex value form. So in this case, remember we had this condition between the M's and the V's, this condition here. So we're, we're new is some multiple, but now this is a rational number. And we can assume replacing M with minus M, but it's positive. So let me just say what's this M goes to minus M. Remember how did M enter? It was, you know, we had a, sorry. Oh, we may assume. I'm sorry. You obviously weren't educated in Cambridge. They're fond of their will logs and you know, all that kind of stuff. Um, yeah, sorry, we may assume. And so, yeah, the point is how did M enter? It was giving, you know, we're just writing down, um, uh, divisor, uh, you know, using a character, but of course I can just in, you know, this is a, this is a non zero scale. I can just, you know, take the inverse character so I can assume this positivity. This will be important, um, in a moment. So if you make this assumption and you do a short calculation using the scattering diagram theory, so now scattering calculation implies that for all walls, remember you've got this rational polyhedral cone inside a wall defined by some, um, inside a hyperplane, defined by some element of M, we have, again, choosing the sign of M appropriately. This is just a positive combination of the initial walls, of the initial M's. So if these guys are linearly independent, what we get is, um, you know, the sum region which is completely outside the scattering diagram or C plus and there's also C minus of course and the scattering diagram wherever it, whatever it is is in this other region. A chamber in the scattering diagram. So it tells you that, you know, we're in this nice situation where there's some part of the diagram which is nice. Okay, but it's better than that because now we can look at another tourist chart. Oh, sorry before I do that, I actually have to say, made this big assumption. So for instance, this guy wasn't true. In the last case, I drew the last example of the cubic surface where it was everywhere dense, the diagram. So obviously that, that did not satisfy this condition that these three vectors were linearly independent. So, um, how can we reduce to this case? This is a two universal constructions. Okay, so I start with, um, so the point of these will be that we can assume these are linearly independent. So what? Okay, so what you do, I'm going to make a mild assumption just to make this discussion a little bit easier. So let's assume for simplicity. When I look at this, um, map, z to the r goes to m, just given by the vectors mi, this is subjective. And of course, if you're a geometry, you'd like to know what on earth does that mean. And it's actually completely equivalent to requiring that the mirror, the point gunter of mirror is, um, has h1 equals zero. So it's simply, actually it'll, it'll be simply connected. Okay, but now associated to this map, there are two maps of Tori. So on the one hand, we just tensor this map with c star. I'm getting a suggestion, uh, from c star to the r onto t dual. Uh, alternatively if I dualize this map, so the kernel, the co-kernel is torsion, um, but dualize this map and tensor by c star. That's an injective map by, by this assumption. And so I get an injection of t into c star to the r. Yeah, so this map's just given by tensor by c star. This map's given by homin to c star. Okay, but now associated to those things, you can just do those, uh, do that construction for each torus and you'll glue to get embeddings of u in some script u. So remember, u is just a union of Tori. Similarly for v, union of t dual. Um, and for v, yes, I'm going to get a cover. Let me call it v tilde maybe, which on each torus chart is just given by this, this map. So what are these maps? Okay, so here, uh, what's happening is this is the universal deformation u. So remember, you, when you define the cluster variety, you blew up some, uh, translates of hyper Tori and the boundary devices, but of course you can move the position of the hyper Torus. You can vary this parameter lambda. And this is exactly what this family is doing. It's just saying, this is the total space of the definition of u. You get by varying those lambdas in all possible ways. And here, this is, uh, also universal construction. It's what's called the universal torso. So what's the best way to say that, uh, basically what you do is just take the Picard group of v, actually by assumption, it's a torsion free, a Boolean group, take a basis of line bundles and you just take the, uh, C star bundle or C star, I guess, what is it, to the R minus N bundle given by those, uh, that basis of the Picard group. So it's sort of, uh, the universal torus bundle over v in some sense. Yeah. And so some, if you know this sort of theory, it's sort of related to this cocks ring construction. Um, same kind of thing. Okay. But the point is, why did I do this? I can now pass to these other two, um, cluster variety. So these are again cluster and u is mirrored to v tilde. So I can sort of reduce to this case. Yeah. And the point being, uh, you know, so what I'm trying to do at the moment is construct global functions on v. Um, but what I'm actually going to do is construct global functions on v tilde and then, you know, the, the functions I want on the, you'll be some, some invariant functions under this torus action. So everything will be an equivalent for the torus. I can just take the invariant functions, get my canonical basis on the, and so of course, in this case, you know, the MI now, just a basis of the corresponding, uh, M tilde. Oh, good point. Thank you. Yeah. I meant to say that. So no, notice, thank you very much. Uh, you know, I was always assuming this non degeneracy. Now we've lost non degeneracy. So of course what you do is the form is just pulled back and Sigma tilde is just a pullback or maybe this is Sigma tilde dual. So now, you know, this is no longer non degenerate. And here, you know, I can have my Sigma, uh, Sigma on you. It's not even a form anymore. It's a relative form, you know, relative to form. So, you know, it's well defined on each fiber, but it's not, it's not, doesn't come from a global two form on you. And you know, on each fiber, it'll still be holomorphic symplectic, but you know, you sort of moved out of this, uh, so we've moved away from this assumption where Sigma is non degenerate. But, you know, actually, um, that doesn't cause any trouble with this scattering diagram. You can still, you can still apply our method. So, so I should say, so earlier somebody asked, you know, um, why do you assume it's non degenerate? And I just feel that the mirror symmetry is more, um, transparent if you phrase it that way. But, you know, in the cluster literature, they, they never assume this form was non degenerate. And so somehow they were always working, tend to work with these, these varieties up here. So in terms of, fuck, gone to our notation. Um, this script U is what they call X and V tilde is what they call X. You know, so these were the, the varieties were sort of classically studied, well, classically, by fucking Goncharov and, for me, Zelovinsky, um, initially in the sort of development of cluster theory. Okay. Oh, it's, uh, you can say explicitly. I mean, it's just you took, it's just the quotient, right? So C star to the R, modulo T. Um, so, you know, what this means is you choose your landers in here, which give you the positions of the hypertorei in the boundary. But then you quotient by T, that's the automorphism group of the toric variety. So, you know, this is, this is like the modulite space. All right. So now we can go back to, um, this chamber decomposition and talk about the cluster complex. So at the moment we have one cluster chamber, this C plus, but we can get more by doing mutations. So the support of this diagram will be invariant under mutation. Um, so remember what we had is, you know, if I have a mutation of two tori, then I get an associated tropical tropicalization, which is a piecewise linear map, um, between the corresponding, um, real vector spaces associated to N, yeah, some kind of shear. Um, and what I'm saying is that, you know, so I could have built this correspond maybe should call this T prime or something. So these are two copies of, of the same tourist T corresponding to different toric models. I get this piecewise linear transformation. Now, if I took those toric models, I would build two different scattering diagrams D and D prime. And what I'm claiming is if I apply the tropicalization map, I mean, there's some change in the attached functions, which is fairly minor, but at least if I just look at the support, the two things are the same. And you know, heuristically that comes from this enumerative interpretation that they're both counting the same things. They're counting these holomorphic disks or, or if you like P ones meeting the boundary in a single point. So somehow this really comes from the enumerative interpretation, at least herestically. Of course, the way you prove it is you actually just check, check it's true. Um, okay, but now, uh, so, so now when you do this, you know, you see that what happens is you, you sort of produce a whole collection of chambers, one for each cluster tourists in the scattering diagram. So let me just draw the picture. So I have my C plus and here's some plus. Uh, let's apply this tropicalization of you. What happens is, well, one of the hyperplanes, you know, the new, the new C plus here is, um, this is C plus prom is adjacent to one of the hyperplanes. And it's just, of course, the hyperplane corresponding to the, the, um, the choice of, um, divisor, which you, um, mutate it. So what you get is you get the cluster complex delta inside D. This is a simplificial fan and the cones correspond to the cluster tour. And so, you know, again, so going back to the example we drew earlier, this is the picture you should have in your mind. There's some nice part of the diagram where everything sort of trivial, you know, the attached functions are sort of really obvious. Um, and those are just corresponding to these cluster charts. And then there's some part of the diagram where we have absolutely no idea what's going on. But there is this nice part, that's the cluster complex. Okay. But now we can build our, our, um, mirror manifold using the scattering diagram. Just say, take V to be the union of the T-dual over the, um, chambers in the cluster complex. Um, and we're just going to use the scattering functions. So, you know, if I want to glue two guys, you know, the scattering diagram tells me how to do that. And notice, you know, so these, these, as I said, these functions, when you set T equals one, they're just the usual cluster transformations. So there's no problem because these functions aren't power series. You know, in this region, the functions are, are, are the simple functions we've seen already. So, you know, here I guess I'm setting T i equals one, or if I want, um, a general cluster variety, just to, T i is some, some constant corresponding to the choice of the positions that I blow up. And so, this is just the same construction as, as before, in a different interpretation, but just the usual construction of a cluster variety obtained by gluing Torai together. Okay, so what have we gained? So, there's this whole part of the diagram that's kind of totally nuts, um, but we haven't used it yet. Now, now, now we're going to use it. So, this is the notion of broken lines. And again, let me say, so what I talked about yesterday, this is some version of the symplectic, um, heuristic we discussed. I've got a fiber of the S Y Z vibration. I've got a point in the base, you know, corresponding to the fiber. I've chosen a boundary divisor, C in my boundary. And what I'm trying to do is count holomorphic disks that end on this L. And they're going to intersect, um, the divisor D here with some contact order. This is a holomorphic disk. Now, these, and we get some global functions, uh, theta, I think I called it theta C comma N. This was in H zero. So, global function on the mirror. And this, I should have said, this is, of course, remember, this is a picture of the, of the original variety. So, now we want to sort of translate that into, into our picture. This is the notion of a broken line. So, we need some data. So, first, we're just going to take any non-zero vector in our lattice, N. Okay. And in terms of the data over there, maybe I just say, so, of course, if I have any non-zero vector, I can write it as an integer N times a primitive vector. So, that's what these, uh, this data over there will be. So, N will be the contact order of the disks with the boundary divisor. And the boundary divisor is given by, well, it's the one corresponding to this, this primitive vector. You have your troic variety, um, after blowing up, you can assume that this is part of the fan, this ray, and there'll be a boundary divisor corresponding to that vector. Okay. And we're going to choose a point P in Nr. So, that's going to be a general point. And again, that's the same thing as over there. It corresponds to the point, choice of the point in the base SYZ, the SYZ base. At least, you know, the tropical version of that. Okay. So, just give a definition. So, a broken line for V with N point P. So, it's a piecewise linear object. So, it's a map from the negative, uh, real axis, real, real interval, to Nr. So, continuous and piecewise linear. Together with, for each linearity domain, an attached monomial. Um, so, where does this monomial live? So, I'll write C sub L times Z to the VL. And C L is in our coefficient ring. So, C, but we've joined these formal parameters, T1 up to TR. And V is just an element of N. This is a character on the dual torus. Okay. So, that's satisfying the following properties. So, such that, so first of all, the initial monomial is just given by V. So, that's, that's how this thing starts. And so, for every, um, linearity domain L, and any point in L, and look at the derivative of, um, this, uh, you know, the, the direction of the path. This is given by the negative of the vector V. So, that's the usual thing about the direction being the opposite of this vector V. The wall crossing. So, now what happens if I try to cross a wall? Let's say I'm going in this direction. This is L and L prime. And let's say that the wall crossing automorphism here is theta. Then, what do you do? So, it's the obvious thing. You start with CL, Z to the L, Z to the VL. You apply theta and just pick out one monomial term. So, in here, this is just C, C L prime, Z to the VL prime, just a monomial. So, to expand this, um, thing in a power series, um, take, take, take a monomial. So, maybe I just draw this previous thing. So, remember this is, this is VL, and then it changes. And so, let's just draw, uh, oh, sorry, finally, one more question, one more thing. It's got n point p, of course. So, gamma zero is p. Okay, what's the picture? So, you know, it's just some piecewise linear picture. It's very simple. Here's my scattering diagram. I'm going to draw some part of it. Here's my gamma. It comes in like this, hits some walls, bends, ends at p. And, you know, so here, this is the direction V. That's how, that's how the thing starts. And so, it starts with, you know, the attachment monomial here is Z to the V, but it becomes more complicated as you apply these wall constant transformations. And, you know, so again, this is something we've sort of seen before, if you, if you sort of, um, believe this symplectic picture, what's happening is you, you start off with some holomorphic disc. So, maybe draw it like this. So, at infinity, it hits this boundary device at C. It comes in, it hits this wall associated to this wall with some disc attached to the wall. You glue on that disc, that changes its direction, and then you glue on another disc when you hit the next wall, changes its direction again, and then it ends on the SYZ fiber. So, that's what it's supposed to be doing. Of course, we don't, we don't prove that that's correct, but that's, that's the heuristic. So, let's write M gamma for the final monomial. Okay, and now just following what the symplectic guys did, we just define this local expansion of the theta function as the sum of these monomial terms over each gamma. So, it's the local expression for this global function, and then what you can show, so this is work, uh, is that these actually give a global function. So, you get theta v in global functions on this, um, plus the variety, uh, v, the mirror, I'm sorry, so these are two v, this is a small v, and this is a big v. So, what I mean to say is, you know, this is the, uh, for p in a chamber, in a chamber of the cluster complex, this gives the restriction of theta v to, uh, the corresponding torus, the dual which is associated to that chamber, um, and I should say also, so I've also got to, uh, say, assuming convergence. So, at this point, if I want to get my v back, I have to set the ti to some finite values. Okay, so that, but that's the construction, and now let me state our theorem. This is, um, with gross, uh, keel and concevich, so if you want to see on the archive, it's this paper. So, this works, so h0 v o v, these are global functions, and they give a basis, uh, if, so under one of the following conditions, so either, so the first condition is that this cluster complex, if you look at it in n, it's not contained in a half space, and the second condition, um, if you just work in the dimension 2 case. So what we expect is this, this, expect this is sort of optimal. Could you have five more minutes? I just wanted to give some examples. What does the chair think? I think, sorry, I should say to the audience, you've got to leave, that's fine, I don't get it. I'm just saying that it's horrible to give a state of theorem and not give a single example, um, okay, I'm sorry, I, anyway, um, okay, so just a couple of examples. So, example zero, this is what's called finite type, um, so these are, this was done by Fomin and Zelovinsky in 03, so, so called finite type plus the varieties, these correspond to, uh, root systems or Dinkin diagrams, including the non-simply laced. So here, the cluster complex is the whole of D, so there's no, there's no bad region at all, and so the theta basis is just equal to, uh, the basis that these guys knew, the cluster variables. And more generally, if I take a chamber in the cluster complex, and I take a point in that chamber, and I expand, you know, I take a, a ray in the chamber and expand, so you know, to his, I'm in this situation, so I have a, a, a point v that lies in a cluster chamber, I take a point in the chamber and expand, there's just a single broken line, and so what it shows you is that you're a theta function in that chamber, you know, so theta v restricted to this corresponding torus, is just a monomial, that's the cluster monomial. So there's some part of our basis that is just something that was already known, um, so-called, um, sorry, cluster monomials, I should say. So it's a monomial in the cluster variables with positive, positive exponents. So, so yeah, so in, in the finite type case, that gives you the whole basis, in general it gives you some subset. So example one, so this was worked out quite recently by Grosson and some, some other guys, a lot of people actually. So let's go back to the examples we started talking about at the beginning. So I have a bunch of points m1 and m2 on the two, um, coordinate lines in, in a2, and I blow up, that gives me my x d. So remember the scattering diagram was insane, you know, some really nasty part here we have no absolutely nothing about, um, but nonetheless, you can prove in this case that the theta basis is, is something that was already known, it's what's called the greedy basis, which was described by Lee, Lee and Zelovinsky. And so the reference is, so it's Mark Grosson and a bunch of other people, um, uh, 1508. So that's, that's another example where everything is known. Unfortunately all the examples where we know what it is, it's something that people already knew about, but, but presumably there are lots of cases where that's not true. Um, okay so the final example, let's go back to the cubic surface, remember this was the, this was the scattering diagram where like every rational ray appears, what, what on earth are you going to do? Well the point is that again, uh, remember when we, when, if, in order to see these chambers you had to pass to this other associated variety, if you do pass that, that of a variety there is some nice region, this, uh, cluster region where you can do, where you can work and that gives you the result for the cubic. So let me just say what it is, this is my XD, very beautiful thing and basically this, in this case it's due to fucking Gontra originally. So if you just look at the interior in this case, it's a fact, this is a modularized space of local systems. On the sphere with four punctures for the group SL2C and this depends on parameters, the parameters are given by fixing the traces, traces of the monodromy around the punctures. So cubic surfaces depend on four parameters, those are the four parameters in this, in this case, in this interpretation. So yeah, so you can read about that if you're interested in a nice paper of camtap and lauray, that's this archive, and so in this case, what's the feature basis? It turns out that it's just the traces, the monodromy, around simple loops, simple closed curves in S2-4 points, possibly with a multiplicity, meaning you know, go around the loop several times. Okay, so I should have said, yeah so this is sort of basically what happened is fucking Gontra I've knew this already, this is fucking Gontra of, this rather long paper, which is actually very readable, and what we did is just checked, oh sorry, not me, this is Sean Keele and Andy Niteski checked that this is the same thing. Okay, thank you very much.