 Thank you for coming to the last session of this mini course. Let me first share with you the files in the chat. And also, yes. So let's start. Here is where we were on Monday. So let's start with the gluing formula. The idea of the gluing formula is as follows. Roughly, the gluing formula states that we can glue two open curves along two opposite boundary components and form a bigger concatenated curve, like this. Then the count of the concatenated curve should be the product of the counts of the two initial curves by this formula. So this is more convincing evidence that our counts really reflect open curve counting. It is also an essential ingredient in the proof of the associativity of the mirror algebra. Now let's establish this gluing formula in three steps. Step one, given two spins, S1 and S2, in the essential skeleton of U, assume we have an infinite one-valent vertex, W1, and an internal infinite one-valent vertex, meaning that it's mapped to the essential skeleton instead of to the boundary. So assume we have such a vertex, W1 in S1, and respectively we have an internal infinite one-valent vertex, W2 in S2, such that the two vertices map to the same point in the essential skeleton, SKU. Next, we consider a spin delta with three infinite one-valent vertices, W1, W2, W, and mapping constantly to this point, where the previous W1 and W2 go. So we consider this spin, which maps constantly to the same point, and this spin has just one three-valent vertex and three infinite legs. So we can glue this constant spin delta to S1 and S2 along W1 and W2, and form a new spin S like this in the essential skeleton of U. Here, the infinite one-valent vertices, W1 and W2, they become nodes in the new spin after we glue. So in the new spin, this is an infinite edge containing a node, and this is another infinite edge containing a node. Now we have the following M. For any curve class gamma, we have the following equality, that is the count associated to the spin S and the curve class gamma by evaluating at this internal marked point W is equal to the sum over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to the spin S1, this spin S1, curve class gamma 1 times the count associated to the spin S2, curve class gamma 2. So for S1, we evaluate at the point W1 and for S2, we evaluate at the point W2. So this is the formula we have for the count of the glued spin. Since this part is constant, it doesn't contribute. The proof is not difficult. By passing to a big enough base field extension, this follows from a set of theoretical decomposition of the set of skeletal curves associated to S of the left-hand side to products of sets of skeletal curves associated to S1 and S2 respectively. So after big enough base field extension, it reduces just to a set of theoretical equality. One can just check by hand. So this is the first step of gluing. We glue two spines with this auxiliary spine at these infinite vertices. Now let's consider the second step of gluing. So in the second step, we are given two spines S1 and S2 in the essential skeleton of U, both transverse to walls. And assume we have a point P1 in gamma 1, the domain of S1, and P2 in gamma 2, the domain of S2, both in the interior of some edge. And we assume that they map to the same point in the essential skeleton, and also they do not meet wall. Since they map to the same point, we can glue, so we can glue S1 and S2 along the points P1 and P2 and obtain a new transverse spine S in the essential skeleton of U, like this. We just glue these two together at the point P1 and P2. Then we have a similar lemma saying that for any curve class gamma, we have the following equality, which says that the count associated to the glued span S and any curve class gamma is equal to the sum over all the compositions of gamma into gamma 1 plus gamma 2 of the count associated to the first span S1 and the curve class gamma 1 times the count associated to the second span S2 and the curve class gamma 2. Here is the proof for this equality. So for the proof, we add an infinite leg, W to S at P, because if you recall that in our definition of counts, we always need some internal mark point in order to evaluate. And we proved using skeletal curves, we proved a symmetry theorem seeing that the place where we evaluate eventually does not matter, but just first to construct the count, we always need some internal mark point. So let's add infinite leg W to S at P at the point we glue P and then we count this span by evaluating at this point W. And after adding this infinite leg, now we can deform by stretching this point P and make appear two small edges E1 and E2. So this is a small deformation of this by stretching the point P and now W is attached, the leg W is attached to the middle of this purple of the two new purple edges. And if we further stretch the two vertical edges E1 and E2 to make them of infinite length and contain a node, so we make each edge infinite length to contain a node, then we arrive at the gluing situation of step one. And note that in this stretching process, all spines are transverse, so their counts do not change by deformation invariance. Now we just conclude by the lemma we proved in step one we show and we conclude our proof for this equality. So this is the second step to establish the gluing formula. It uses deformation invariance for transverse spines as well as the previous lemma that we established for the counts of such a glued span with this auxiliary span delta in the middle. So finally, in step three for the gluing formula, we are given two spines, S1 and S2 in the essential skeleton of U, both transverse to wall, like this, S1 and S2. And assume we have finite one-valent vertices, V1 in the domain gamma1 of S1 and V2 in the domain gamma2 of S2, such that they map to the same point in the essential skeleton of U. And we assume that we have opposite derivatives, meaning that the derivative at V1 of H1 plus the derivative at V2 of H2 is zero. Since they have opposite derivatives and they map to the same point in the essential skeleton of U, we can glue S1 and S2 along V1 and V2 and obtain a new transverse span S, like this in the essential skeleton. And here is the final theorem, the final gluing formula. For any curve class gamma, we have the following equality, which says that the count associated to this glued span S and any curve class gamma is equal to the sum over all the compositions of gamma into gamma1 plus gamma2 of the count associated to the left part, S1, with curve class gamma1 times the count associated to S2 and the curve class gamma2. So this is the final gluing formula, and it is a generalization of the two-dimensional case in my previous paper, but here I'm presenting a more conceptual proof via deformation invariance. So the idea of the proof is the following. So we want to prove that when we glue these two together, we have this formula for the count. And let's make a small extension of S1, this is our S1, we make a small extension of our S1 at V1 to S1 hat by linearity. So we just extend linearly at this vertex V1, extend a little bit, this purple part is our extension. And similarly, we make a small extension of S2 at V2 by linearity to S2 hat. And by deformation invariance for this transverse truncated space, the count remains the same. The counts do not change when we make these small extensions as long as we do not meet walls. And now, after making the two small extensions, observe that if we glue S1 hat and S2 hat together by identifying V1 and V2, so let's glue S1 hat and S2 hat together at V1 and V2. We see that once we do that, this gluing is actually just equal to the glued span S to the gluing of S with some small straight span L. So this gluing S1 hat, S2 hat at V1 and V2, it's equal to the gluing of S, this red S together with this purple edge L at the point V. And both sides of this formula are gluing of two spans. So both sides are gluing at the gluing situation of step two. So we can apply step two to both sides of the above equality. And we obtain this, which says that the sum over all the compositions of gamma into gamma 1 and gamma 2 of the count associated to S1 hat and gamma 1 times the count associated to this small extension S2 hat and gamma 2 is equal to the sum over all the compositions of gamma into beta 1 plus beta 2 of the count associated to S, red S and curve class beta 1 times the count associated to the L, this purple, small straight span L and curve class beta 2. So now we have to let's compute explicitly the contribution of this part. So we can explicitly compute that the count associated to L and the beta 2 is equal to 1 if beta 2 is 0 and it's equal to 0 otherwise. And if we substitute, if we substitute this explicit computation into the equality above, we obtain the gluing formula in R0. Oh, sorry. Do you understand you have this purple interval L maps to a point? Is essentially the only contribution you have, yeah? L doesn't map to a point. L maps to a small interval. Ah, small interval. I see, yeah. Yeah, so it's really, yeah, it's not clear what does correspond to, in simply typology because you don't consider p1s but kind of like annular, yeah. It's, yeah, it's a very small annular map into a place without walls, without any walls. I see. Yeah, so that place is like just a torus. So we have, it's like we count annulus just in an algebraic torus. Yes. So the count is really the simplest. Yeah. Either one, if curve class is 0 or 0 if it has that, if we put some non-trivial curve class. Because it just maps to a place without walls so there is nothing interesting happening. Yeah, so via these three steps, we obtain this gluing formula 0. And let me remark that similar idea can be applied to show that our accounts are independent on the choice of the torus when we impose the toric tail condition. Assume we have two torus embeddings, tm in u and tm prime in u, just two different embeddings leading to two different toric tail conditions, t and t prime. So recall that in the definition of our naive accounts of skeletal curves, in order to obtain a finite dimension or modular space, we need to impose some extra condition and some extra regularity condition on the boundary via analytic continuation. And that condition called the toric tail condition was formulated according to the choice of some torus embedding. Now we want to show that it's independent of the torus embedding using the same idea in the proof of the gluing formula. So assume we have two torus embeddings, tm in u and tm prime in u leading to two different toric tail conditions, quality and quality prime. And now we consider a span S like this in the essential skeleton of u with a finite one valent vertex v. Here the span S is the whole graph, including the purple part. This is our span S. It has a finite one valent vertex v. We want to show that the count of this span S does not depend on which tail condition we impose at this end v. For this, let's pick w, some point w very close to v and let L denote the restriction of the span S to this small interval w v. So this purple interval is just a restriction of S to a small neighborhood of v. And then we pick any point x in the middle of w and v and we consider the gluing of S with L. So it's similar to the gluing we considered a moment ago. So here when we consider gluing of S and L, we get like this part gets doubled. It's like they're double. Now, by step two above, we obtain the following equality. So here you see that we are gluing two spans at some interior point of edges, which is the situation of step two. And so we can apply the formula in step two. And now, if we apply the formula in step two, we obtain the following. So the left hand side. Yeah, so when we apply we think like this. For the left hand side is the sum over all the compositions of gamma into beta plus delta of the count associated to S using the tail condition, the first tail condition T everywhere times the count associated to L using the tail condition T prime at V and T at the other end of L. So this left hand side, so we can think of L as for the left hand side, we put the toric tail condition T everywhere on S. But for L, we put the toric tail condition T at W but tail condition T prime at V. And then we apply the step two above to this gluing. Now for the right hand side, we just think that for the gluing we switch this part of L with this part of S. Which means that it's the sum over all the compositions of gamma into beta plus delta of the count associated to our span S where we use the tail condition T prime at V and T in all other places. Now at times the count associated to this small interval L where we apply a tail condition T at both ends. So it's really so the left hand side as I said the difference between the left hand side and the right hand side, it's how we think of the gluing. It's like we are switching that half of L with that last piece of S. So one piece has the toric tail condition T attached and the other piece has toric tail condition T prime attached. And we switch them in this gluing we obtain and when we apply step two, we obtain this formula, then this equality. So now it remains to compute explicitly the contributions of the small this L piece. So we can compute as before that the count associated to L using the tail condition T and the curve class delta is equal to one if delta is zero and if it's equal to zero for all other delta. So the same holds when we count the small interval L using tail condition T prime at V and T at W the same equality the same count works also for the other one. Now we substitute this explicit computation into the equality above we obtain the following theorem for tail condition with varying torus. So we have proved by substituting this explicit computation into the equality we have proved that the count associated to our spine S using tail condition T everywhere. And for some fixed class gamma is equal to the count associated to the spine S equal to the count associated to the spine S using tail condition T prime at V and tail condition T everywhere else. So two counts are equal. So here we are just switching tail condition at one vertex. Then, of course, one can switch at all other vertices and we can even apply different torus tail conditions at all finite vertices. This, in other words, the theorem shows that the count of skeletal curves is independent is independent on the choice of the torus TM inside you. So just a small remark concerning the computation for the count associated to this small interval where we put different torus tail conditions at both ends. So the explicit computation for this count using two different tail conditions is a bit more subtle than for the count where we put the same tail conditions. We need to use a result concerning the gluing of no Archimedean poly annual light in my previous paper. If we put the two different tail conditions, because if we put the same tail condition then the count is easy. It's like something. Yes. Maybe you mean that on one part, half of interval area put condition T prime and other T, but your notation don't say this, it's kind of like T prime everywhere. Yeah, but actually it says that it's one end is T, the other end is T prime. So I hope it's understandable. Okay. Otherwise the notation is a bit complicated. Yeah. Yeah, so I'm saying that if at both ends we put the same tail condition, then the count is easy to do because yeah, yeah, yeah. As Maxime remarked, we just map to a torus, so then it's just counting something in the toric variety. But if we put different tails at the two ends, then the counting is a bit more complicated because it maps to some gluing of two different torus. But one can show that actually one can decompose the automorphism of some annulus into two parts where one part can be extended to an automorphism of some disk times polyannulus. And another part can be extended to automorphism of another opposite disk times the annulus. So after applying this automorphism, we see that we are back to the previous situation, just counting something in P1 times some polyannulus. It's a computation of automorphisms of this automorphisms of some affinoid algebras. Yeah, so this is the idea for the proof of the gluing formula and also for the proof of changing tail condition. And in fact, this proof of changing tail condition also works if we eventually use more general tail conditions without using any embedded torus. So it's easier to carry out this proof than one can imagine using deformation invariance for it feels a bit more complicated. So now let's turn to the next section, structure constants and associativity of mirror algebra. We want to apply our gluing formula and all the other techniques we have developed so far to study structure constants and the associativity of the mirror algebra. So let us recall the setting from the first lecture where we have log kala b yaw variety u containing some torus tm and we have some SNC compactification of u in y. We have monoid ring r over this effective curve classes assembling all possible curve classes together and the mirror algebra a it is it has basis the integer points in the essential skeleton as an arm module. So this was our setup in the first lecture. And again recall that given some integer points in the skeleton, we write the product in the mirror algebra a as this product of the theta functions, theta p1 to theta pn in the mirror algebra a as this sum. So first we sum over all integer points in the essential skeleton of the basis vector theta q and then we sum over all curve classes of all effective curve classes of the basis element z to the gamma. It's just a notation for the basis elements. And we denote the coefficient by chi p1 to pn q gamma and this guy is called a structure constants of our mirror algebra a and let's also recall how it was defined in the first lecture. The structure constant chi was defined as follows. We had the so first we figure out what was the class delta of the added toric tail and we denote the total class beta to be gamma plus delta beta is supposed to be the class of the. The closed p1 after the toric tail extension. We also had the z equal to the opposite of q in the essential skeleton and we had top of PZ putting P p1 to pn and z together and we use this top of PZ to specify intersection numbers with the boundary. Then we considered the modular space. This HPZ beta with marked points labeled as P1 to PN ZS. So this is the modular space of maps of maps from P1 with marked points P1 to PN ZS. And we say that we specify the intersection of the P1. So it's the modular space of maps of P1 into Y. And we specify the intersection numbers at these marked points with the boundary D using the top of PZ. And we had a natural map phi. From this modular space to. Here, taking domain. Taking domain and evaluation at the last marked point S. And we also had a special point to kill the other in the identification of the target. Which was given as a pair of mu and q. We give mu, we just specify what was mu in the first factor corresponds to some divisory evaluation and q is just the given q. So the given q is the integer point in the essential skeleton. In particular, it's a point in the identification of you. So we had this special point q tilde. And then finally, we had a sub space F in the fiber of the map phi. Over q tilde, which is a finite analytic space. And if we take its lens. The lens was by definition the structure constant. In the first lecture, first I give a heuristic picture of what we count for the structure constant about counting discs with some conditions on the derivative of the disk at the boundary and and and after that, I give a precise construction of the structure constant in using algebraic using no Archimedean geometry. So this was just a recall of what we did. And the definition is quite straight forward. But let's remark the following. Due to the choice of this specific point q tilde inside the target. So we used this q tilde to take fiber and then take subspace. Due to the choice of this specific point, the curves in F responsible for structure constants, although highly generic in the algebraic sense are in fact very special. In other words, non transverse from the tropical viewpoint, because this special point. It's a generic point in the algebraic world, but it's a very non generic point in the tropical world. So this results. Very generic curves in the algebraic sense but very special curves from the tropical in the tropical picture. And this was convenient for giving a quick definition of structure constants, but it's impractical for proving any properties about them. For example, associativity fan and this all these properties they are out of reach from this quick definition. We must deform the curves in F into more transverse positions by to perturbing this special point q tilde. Because when we want to prove properties. For example, if we want to apply the gluing formula or deformation variance. We, we usually need the assumption that the spine. The spins are transverse, but if we do the counter using this specific point q tilde. We will not get transverse spines, so we must perturb, perturb these curves. By varying the point q tilde. Do we do it like this proposition, we label the marked points of metric trees in. This modular space, so this is a modular space of tropical curves, rational tropical curves with n plus one and plus two legs. Here, abstract tropical curves are the same as metric trees. And we label the n plus two points, marked points as P1 to PN ZS. And let Vm in the modular space. Be the subset consisting of metric trees, whose Z lag and S lag are incident to a single three valent verdicts. Here is the picture, we have such a metric tree, it has a lot of legs. By leg we mean infinite one valent verdicts. Or, yeah, so, or more precisely, we mean this at the edges containing infinite one valent vertices. So here, we consider the subset where the Z lag and the S lag, they are incident to a single three valent verdicts. So the Z lag and the S lag, they meet first. And before the tree branches to other legs. Yeah, and we observe that this Vm is a neighborhood of this special choice of the modulus mu. And next, we want to figure out a neighborhood of this special point q. So let's consider a polyhedral subdivision, sigma of the essential skeleton, given by the set of walls in the essential skeleton. Here, we can assume the set of walls to be finite polyhedral by bounding the degree of twigs by the fixed curve class beta. So in general, wall, the set of walls, they are infinitely many walls and it can be dense in the essential skeleton. But if we bound some degree, we get a finite set of walls. So we consider the polyhedral subdivision induced by the set of walls. And we let Vq be the open star of the point q in sigma. In other words, the union of open cells in sigma whose closure contains q. We have q and we have a polyhedral decomposition, we just take a cone around the point q. q might lie inside a wall, but it doesn't matter. We take the star of q in sigma, then q becomes an interior point in this open star. And we said that by construction, q tilde, which was given as a pair, mu q, vm is a neighborhood of mu, vq is a neighborhood of q. So vm times vq is a neighborhood of q tilde inside the product of the tropical modular space and the essential skeleton. And we already remarked when we talk about essential skeletons that using Tamkin's naturalization theory, one can show that product of skeleton is homomorphic to skeleton of product. And this lies in the identification of the product. Yeah, so using this construction, we figure out two natural neighborhood of sorry, we figure out a natural neighborhood of the first factor of q tilde and a natural neighborhood of the second factor of q tilde. So we figure out a neighborhood of q tilde. Recall that our goal is to perturb the point q tilde. So we will be perturbing q tilde inside this neighborhood vm times vq. And now let this be the pre-image of vm times vq by the map phi analytic from this modular space by taking domain and evaluation at the last mark point s. Before we took pre-image of a single point q tilde. Now we take pre-image of this neighborhood of q tilde. And now we take curly f to be the subset in the pre-image satisfying the toric tail condition. Then the proposition says that phi analytic is finite at all on the neighborhood of this subset and whose degree gives the structure constant. So this is how we perturb the special point q tilde into general position by allowing q tilde to vary inside this neighborhood vm times vq. And we prove that if we allow q tilde to vary and when we impose toric tail condition, then this map phi is still is good. It's finite at all on the neighborhood of this f. So the degree is well defined. And it gives the structure constant. Since it's finite at all, so we get a well defined degree. And the degree at the fiber over the point q tilde, it was our quick definition of the structure constant. So here, the structure constant is reinterpreted as some degree of finite at all map. For its proof, we use the toric tail proposition for almost the transverse of spines, because here it feels like deformation invariance that we are moving the spines for the structure constants inside this conical neighborhood. But they do not state, it's not always a transverse. Sometimes it becomes non transverse, especially at the point q tilde, for example, but we had developed this proposition in the last lecture, not just for transverse spines. Especially adapted to the situation here where we can go across walls. So after this perturbation into generic position, now we can prove the following theorem. The multiplication rule given by the structure constants here is commutative and associative. Here is the sketch of proof. Commutativity is obvious, because the definition of the structure constant chi is a symmetric with respect to the PIs. And associativity means that the product C type P1, C type P2 to C type PN does not change if we add arbitrary parentheses. So let us now sketch the proof of the following equality. Where the left hand side means we first take product of C type P1 and C type P2, and then we take product with C type P3. While the right hand side means we take the product C type P1, C type P2, C type P3 together using the multiplication rule. So we want to prove this equality. And we just rewrite the products using the multiplication rule. We substitute the multiplication rule into the equality. And we see that the equality becomes equivalent to the following equality. For every integer point q in the essential skeleton and every curve class gamma. Now observe that the right hand side of the equality star is given by counts of skeletal curves associated to the spines of this shape. Here we have three infinite legs with derivatives P1, P2, P3 respectively. And also we have one finite leg with inward derivative q. So three infinite legs with outward derivative P1, P2, P3 and one finite leg with inward derivative q. And by the above proposition about perturbing q tilde, we can deform the modulus of the domain here by stretching this point. So we deform this point into a small path L. And then we further stretch this path L very, very long like this. So if we stretch the path L very long, we see that the point u near the top of the path L will map sufficiently close to the ray 0R inside the essential skeleton of u. And we needed to be sufficiently close to the ray because we wanted to lie inside the cone VR in the essential skeleton of u as in the above proposition for the structure constant chi P1, P2, R eta. Recall that in our quick definition of structure constants, we say that the marked point just go directly to the point q tilde and that was two non-transverse, two rigid. Now we allow q tilde to move a little bit around, but we still we always needed to be sufficiently close to the ray 0Q. So it should not move outside across some walls around this ray. Otherwise, it doesn't give the correct structure constant. So here we stretch this path L very long so that finally some point u near the top of the path will be sufficiently close to the ray 0R. Then that will be good enough for defining the structure constant chi P1, P2, R eta. So we can cut at the cross u, apply the gluing formula and obtain the left hand side of the equality star. So similarly, given two spans responsible for the product in the left hand of the equality star, we can glue them to form a highly stretched span. Like this, that is responsible for the right hand side, and this completes the proof of associativity. So before the break, let me quickly sketch another important property of our structure constants, which is the convexity property. So here is the theorem of the convexity property. Let F be a Cartier divisor on Y and consider an analytic disk in general position responsible for the structure constant chi P1 to Pn q gamma. Then the following hold. First, since F is a Cartier divisor, we can take its tropicalization and obtain a real valued function fTrop on the essential skeleton. Then first, we have that the sum of fTrop at overall Pi minus fTrop at q is equal to the intersection number between f and gamma minus the degree of f and litified restricted to the punctured disk, meaning the disk minus all the marked points. And the second, if f is an F and minus f restricted to you is effective, then we have fTrop of q is less than or equal to the sum of fTrop at every Pi. And furthermore, assume f is ample and minus f restricted to you is effective, then the above equality inequality is an equality if and only if f maps the punctured disk into the torus. And the proof uses some detailed computation using semi stable models of curves. The convexity theorem implies the following finiteness result. Finiteness result one, given P1 to Pn in the essential skeleton of you some integer points in the essential skeleton of you, then there are at most finitely many pairs q gamma where q is an integer point in the essential skeleton of you and the gamma is a curve class, such that the structure constant is non zero. Let me give a quick proof how the finiteness follows from the convexity. So, since you is a fine, we can find the regular functions x1 to XL on you, such that the set of points in the essential skeleton where the norm of Xi at B is bounded by some real number C for all I this set is a is bounded for any real number C. Now, if the structure constant is non zero by the convexity statement two, we apply the convexity statement to the Cartier divisor given by these regular functions. We obtain this equality, the norm of Xi of the function Xi at q is less than or equal to a sum of is less than or equal to sum over all j of x the norm of Xi at PJ. And this shows that given P1 to Pn, there are at most finitely many q such that the structure constant is non zero for some gamma. So this bounds q. And the next let's bound the gamma. We want to bound both q and the gamma. And the assumption that you is a fine implies that there is an ample divisor F on why such that minus F restricted to you is effective. And now we apply the convexity theorem one, the statement one, this statement. And we obtain this the following equal the following equality and the inequality. Which says that the intersection number between F and the gamma is equal to the sum of F drop at every PI minus F drop at the queue, plus the degree of the analytic F restricted to the punctured disk. And since minus F restricted to you is effective. And this degree is non positive. So we see that this is less than equal to this. And this is fixed. The right hand side. This bonds gamma by the ampolness of F. So this is how we deduce the finite this result from the convexity property. And the finite this result. Is the important because it implies that the two sums in the multiplication rule. Here they are finite sums. So the multiplication rule gives an R algebra structure on the free R module. Instead of just some formal algebra structure. And in fact, we have the following stronger finite and it's result to which says that the mirror algebra is a finitely generated R algebra. And for its proof, we need to result to the equivalent boundary torus action on the mirror algebra. So here, due to time constraints, I will admit this boundary torus action and the finite generation result in this lecture. And after the break, I will explain the application towards cluster algebra. And also, I will explain the wall crossing how to get scattering diagrams using these kinds of analytic curves. So let's make five minutes break. Okay, thank you. So here's the plan for the last part of this lecture. First, I'll explain how to construct a scattering diagram via infinitesimal analytic cylinders. And the second, I'll prove the property of theta function consistency for the scattering diagram. Third, we will set all curve classes to zero so that we no longer care about the compactification. And the fourth, I'll explain the class. We will apply the above to the case of cluster algebra where I need to introduce two new notions, sea twigs and the sea walls, especially for the cluster case. And finally, I'll explain the comparison with the work of gross hacking keel condensevich with the work of gross hacking keel condensevich for cluster algebras. So let's start with first scattering diagram via infinitesimal analytic cylinders. Both in the, yes. Both in the original suggestions by condensevich-sopoma and in the gross seabird program, the construction of mirror variety relies on the combinatorial algorithmic construction of scattering diagram, also known as wall crossing structure. Our construction of the mirror algebra by counting non-archimedean analytic disks, as in the previous lectures, completely bypasses any use of scattering diagram. Nevertheless, our geometric approach also allows us to give a direct construction of the scattering diagram by counting infinitesimal analytic cylinders without the step-by-step condensevich-sopoma algorithm. And this has three implications. First, it gives a geometric interpretation of the combinatorial scattering diagram. Second, conversely, we obtain a combinatorial way for computing the non-archimedean curve counts. And third, it paves the way for the comparison with the work of gross hacking keel condensevich on cluster algebras. Let me also remark that there are recent works of Argus gross seabird give another geometric interpretation of scattering diagram based on the theory of punctured log curves developed by Abramovich-Chen gross and seabird. Now let us sketch our construction of the scattering diagram via infinitesimal analytic cylinders. Recall, we have our log-calabiou containing some torus tm and is contained in some SNC compactification y. And we denote by n the du of m. So, definition, given a hyperplane n-purp in mr, m tensor with r, and any generic point x in the hyperplane n-purp, generic means that it's not contained in any other n-purp. It's only contained in this one hyperplane. And say we are given two vectors v and w in m minus n-purp and the curve class alpha. Let vxvw be the infinitesimal spine here, bending at x with incoming direction w and outgoing direction v. And this gives rise to the associated count of analytic curves n v alpha. So, we consider this infinitesimal spine bending once at a generic point x in the hyperplane n-purp with specified incoming and outgoing direction. And we consider the associated count of analytic curves. Then, using all these counts for any x in some hyper, for any generic point x in some hyperplane n-purp, we define the following wall crossing transformation. Psi xn acts on the basis vector z to the v as follows. For all v in m, which pairs with n positively, we define the value of psi xn at z to the v to be the sum over all possible vectors of every vector w in m which pairs with n positively. And over all curve class of the basis vector z alpha, zw, with coefficient, the count we just defined. So, in other words, we just sum over all possible incoming direction w and all possible curve class. And if for all v in m, which pairs with n trivially, which lies in the hyperplane, we just define it to be z to the v. It doesn't. The wall crossing transformation does nothing for this v. And unlike in the multiplication rule, this sum is not a finite sum. But this converges in the natural eddyc topology, which we describe here. Since this cone of curves may not be polyhedral, we fix a strictly convex toric monoid q containing this effective curve classes. And let our hat be the completion of the monoid ring over q direct sum m with respect to the maximum monomial ideal i. In other words, the ideal generated by monomials z qzm with q nonzero and arbitrary m. So, we use this completion to express the convergence lemma convergence lemma, the formal sum in the wall crossing transformation lies in our head. So the reason is that when we give a bound on curve classes, it implies bound on the combinatorial types of the two weeks of all analytical curves contributing to these counts. And this in turn gives a bound on the incoming directions w. So now, by linearity over this monoid ring, we can extend the wall crossing transformation transformation to a map from the monoid ring over all curve classes plus direct sum with the subset of the lattice m which pairs with m nonactively. And the map goes to our head. Here we defined what it does on the basis vectors and we extended by linearity over the monoid ring. Theorem wall crossing homomorphism theorem. The map psi xn is a ring homomorphism. So it means that if we pick arbitrary m1 m2 in m that pairs with n nonactively, we want to show that we have a psi applied to z to the power m1 plus m2 is equal to psi of z to the m1 times psi to the z of m2. It's what means for the map to be a ring homomorphism. So let's try to prove this. We let's fix any vector e in m and any curve class alpha and let's prove the equality of the coefficients of the alpha z inside the formula above. Note that it's obvious the equality is obvious when both m1 and m2 lies in n-perf because if they lie in n-perf, then the wall crossing transformation does nothing on them. So we may assume that one of them pairs with n positively. So assume that the pairing between n and m1 is positive and the pairing between n and m2 is non-strictly positive. And we consider the count of analytic pair of pants associated to this spice with three just various infinitesimal spine with three legs with three ends near x with directions m1 m2 minus e at the three ends. So claim suppose x is contained in the wall sigma, then the count of analytic pair of pants is independent of which side of sigma the three valent vertex of this our spine maps to. Here the left picture shows this spine mapping to the essential skeleton and this blue line is a wall. And the left picture shows the situation where the three valent vertex maps to the left is three valent vertex maps to the left and the right picture shows the situation where the three valent vertex maps to the right. So the claim says that the count is independent of which side the three valent vertex go. Let's further observe that in the left picture when we specify the directions m1 m2 and e, this the shape of the spine is unique, which we denote by SL, because there is only one possible band and that band is determined by the three directions. However, in the right picture, the shape of the spine is not unique because we have two bands and this band when we deform from left to right, this band is decomposed into two bands and all possible ways of decomposition are allowed. So, for the right hand picture, we are actually summing over many, we are summing over many different shapes of spines, which we denote by SRI. Sorry, Tony. Yeah, the source of stability is that when m2 is parallel to the wall, which is, yes, yes, this is the trickiest, this is the trickiest possibility, but I allow m2 to be parallel to the wall here. What is your doubt? Where is your doubt? No doubt, no, no. It's allowed. Yeah, yeah. And it's exactly the possibility where m2 is parallel to the wall, that possibility implies that like the preservation of volume elements. We see. So, yeah, so we keep in mind that we also have the possibility m2 parallel to the wall. Yeah, so as Maxim remarked that here in the proof, we use the proposition for tauic, for tauic conditions in families in the last lecture. And the trickiest case is when one of m1 or m2 lies in m-perp. One of them is parallel to the wall, where we need to use the deformation invariance for almost transverse spines. So now let's go back to the proof of the theorem. This is proof for the claim about deformation invariance when we move from left to right. And now let's back to the proof of the wall crossing homomorphism theorem. Note that for any span L disjoint from walls, the count nL gamma is 1. If gamma is 0 and is 0, otherwise, because if it's disjoint from walls, we are essentially in the tauic situation and everything can be computed explicitly. Keeping this in mind. Now we consider first the left picture. If we cut the span SL at the cross by the gluing formula, then by the gluing formula, the count associated to SL, this red span SL, and any curve class alpha gives the coefficient of Z alpha ZE in this wall crossing transformation. Because when we cut at this cross, the left part is disjoint from wall, so it doesn't contribute. While the right part has a band, and the right part is exactly the infinitesimal cylinders that we use in the definition of the wall crossing formula. So therefore, when we count analytic curves associated to SL, we get the coefficient in this wall crossing transformation. And the next, if we cut, next let's consider the right picture, where the three valent vertex maps to the right. If we cut the span SRI at the two crosses, but then by the gluing formula, so we cut at these two places, now the span is broken into three pieces. Note that the right piece is disjoint from any walls, so the right piece doesn't contribute. But the two left pieces, they both contribute, and they both have a band, and we just add their contributions together. So by the gluing formula, we see that the count associated to this span SRI is equal to the sum over all the compositions of curve class alpha into alpha 1 plus alpha 2 of the count associated to this upper left span times the count associated to the lower left part. And we remarked that when the three valent vertex maps to the right of the wall, we have different shapes parametrized by I. And the next, we sum over all possible shapes SRI, and we see that the sum of counts associated to SRI gives exactly the coefficient of Z alpha ZE inside the product of the two wall crossing transformations. So we conclude the proof of wall crossing homomorphism by the claim. Yeah, so this shows that wall crossing transformation is a ring homomorphism, and we remarked that the immediate consequence of the ring homomorphism is the following. Given any X generic, any generic X in some hyperplane n perp and any vector V in m, which was pairing with n equals one. In particular, this means that n is necessarily primitive vector. We just rewrite psi xn at ZV to be ZV times some function. This is just a rewriting with some function in our head. Then the fact that wall crossing is a ring homomorphism implies that the function fx and V does not depend on V. And geometrically, this means that the count of infinitesimal analytic cylinders depends only on the amount of band independent of incoming or outgoing direction. So for the count of this, we can change incoming or outgoing direction as long as we have the same amount of band, we always have the same count. In particular, this implies that the wall crossing transformation preserves the standard volume form on tori. And moreover, we have the equality between the function fx and V and fx minus and V. In other words, it's independent of the orientation of wall. The function is independent of orientation of wall. So we can denote fx, so we can denote fx and V just by fx since it's independent of n and also of V. For any choice of n primitive whose prep contains x and any choice of V with the pairing between n and V equals 1. And we call fx the wall crossing function attached to x. So here is the conclusion. Now we can write the wall crossing transformation applied to zV simply as zV times fx to the power the pairing between n and V for any vector V in m that pairs with n non-actively. And this is, it resembles like to the more classical wall crossing formula. And using this formula, we see that the wall crossing transformation per se xn extends to an automorphism of the fraction field of our hat. Let me remark that this may not give an automorphism of our hat since the wall crossing function need not to be invertible in our hat because of curve classes. It might not start with one, it might just start with a curve class and that curve class doesn't is not invertible. It doesn't matter. First, this is just how it is if we take into account curve classes and it will become invertible when we set all curve classes to zero. Now, definition. Now we have all the wall crossing transformations, wall crossing functions, we can define what is our scattering diagram. So let D be the set of pairs xfx where x is any generic point in m-perp, in the hyperplane n-perp for any non-zero n and fx is the associated wall crossing function. We call this set of pairs xfx, we call it the scattering diagram associated to u with respect to the compactification y and also with respect to the torus tm. So scattering diagram depends on the torus. And we remarked that by the eddy convergence of fx of the wall crossing function, if we mod ik, if we mod out some power of i, where i was the maximum monomial ideal, then we obtain a finite scattering diagram dk. In other words, finite means that we will only have finitely many polyhedral walls in dk. Here in D, we have infinitely many walls, so it no longer makes sense to say the shape of each wall. They are so small, it's better just to give the wall crossing functions attached to each generic point. But once we mod out some power of the maximum monomial ideal, we get finite scattering diagrams, dk, with finitely many polyhedral walls, and we call dk the case or the approximation of d. And one important property for scattering diagram is consistency, consistency property. So let's try to establish consistency. First, let us establish a variant of consistency property, which we call theta function consistency. We introduce a new definition. Choose any generic point x in M R and two vectors M E in M. And let S P X M E be the set of spines in M R with domain minus infinity to zero, like this, such that minus infinity maps to the boundary, with derivative minus M and zero maps to X with derivative minus E. So we consider the set of all such redspines, starting from infinity, ending at X with with directions M and E at the two ends. And this is related to the notion of broken line in GHKK. We define the local theta function, theta X M to be the formal sum over all vector E and all such redspines S and all curve class of the vector Z alpha, the basis vector Z alpha Z E. And with the coefficient, the count associated to the redspine S and alpha. And again, we have attic convergence. This lives in our head. And we have the following theta function consistency theorem. The scattering diagram D is theta function consistent in the following sense. Given any K, we consider the case order approximation. Given any K, we consider the case order approximation of our scattering diagram, D K. And we take a polyhedral wall, sigma with attached wall crossing function f sigma inside D K. And we choose a vector n such that the wall is contained in the hyperplane and perp. And we consider two points A and B, two general points near another general point X in the wall on two sides of the wall. So we pick a general point X in the wall and two general points on two sides of the wall, A and B near X such that A pairs with N positively and B pairs with N negatively. Then we have the following. If we apply the wall crossing transformation psi to the local theta function at A with infinite direction M, we obtain the local theta function at B with the same infinite direction. And conversely, if we apply the wall crossing transformation psi with respect to the opposite orientation of the wall to the local theta function theta BM, then we get back the local theta function theta AM. This is what we call the theta function consistency property. And we use the gluing formula and the deformation invariance for the proof. So our next step is to forget all curve classes. We want to get closer to the classical wall crossing structure or classical scattering diagram without always thinking about curve classes in our compactification. So let's set all curve classes to zero. As I said, our wall crossing transformation and the scattering diagram depends on the compactification U inside Y via the usage of curve classes in Y. So this is more refined information, but we can remove this dependence by setting all curve classes to zero. In order to have attic convergence, if we set all curve classes to zero, we need to impose a condition on the band of infinitesimal analytic cylinders. Without extra condition, we will lose attic convergence if we forget the curve classes. The assumption is the following. For any non-zero count associated to some infinitesimal spine at X with the incoming direction outgoing directions V and W. If the count is non-zero, then the band W minus V lies in the strictly convex monoid P inside M. We need this assumption in order to have attic convergence when we forget curve classes. And here is the new attic convergence we consider thanks to the monoid M. We consider J, the maximum monoidal ideal in the monoid ring over P, the monoid P. And let L not hat be the genetic completion of ZP. And finally, let L hat be L not hat tensor with ZM over ZP. In other words, we are taking completion of ZM in the direction of the strictly convex monoid P. We allow infinite sums in the P direction and only finite sums in all other directions. So now we will be forgetting curve classes. For a position, under the quotient from the monoid ring associated to Q direct sum P to the monoid ring associated to P, when we ignore all curve classes, the wall crossing functions fx in R, they map to this quotient wall crossing functions fx bar in L hat, meaning that when we ignore curve classes, they have attic convergence in L hat. And the wall crossing transformations Psi xn, which used to be an automorphism of the fraction field of R hat, they become automorphisms of L hat. So we no longer need to take fraction field of L hat because when we forget curve classes, the wall crossing transformations will automatically become invertible. And furthermore, the local theta functions in R hat, they map to the quotient theta functions in L hat. So we have this genetic convergence for all these things. Before, and moreover, we get the invertibility of the wall crossing transformation. So now we denote du to be the set of pairs x fx bar where x is any generic point in any hyperplane and perp, and the fx bar is the associated wall crossing function ignoring all curve classes. And we call this the scattering diagram associated to u with respect to the tors tm. Now we have the following consistency result which says that the scattering diagram du is consistent in the sense of conserved sobal man. In other words, for any general loop inside the MR, the composition of wall crossing automorphisms after ignoring curve classes along the loop is just the identity. So for the proof, we observe that the subring of L hat generated by all local theta functions is genetically dense. And then the theorem follows from the theta function consistency. So that's the general constructions of the scattering diagram using counts of infinitesimal analytic cylinders. And now let us apply our constructions above to the case of cluster algebras. Now here is the cluster data. We have lattice m with an integer-valued skew symmetric form. And we have s prime, a basis of m, and s inside s prime, a subset of s prime. This gives a seed for a skew symmetric cluster algebra of geometric type where s corresponds to unfrozen variables and s prime minus s corresponds to frozen variables. And the seed gives rise to curly A, a fork contour of A type cluster variety, which is the gluing of tori via cluster mutations. And we let A up be the algebra of global functions on A called the upper cluster algebra. Let us assume that the skew symmetric form is unimodular. We assume this for, mainly partially for convenience, this holds in the principal coefficient case. Because this holds in the principal coefficient case, and we will deduce more general cases from the principal coefficient case. And we also assume that the spec of the upper cluster algebra, which we denote by u, is smooth. So for example, double brouhaha cells in semi-simple complex league groups satisfy this smoothness assumption. I think it's possible to extend our work also to cover the non-smooth case without too much extra effort. But I haven't checked all the details. Then u is a log-calabiol containing a torus tm, so we can apply our theory to u and obtain a mirror algebra as well as a canonical scattering diagram by counting non-alchymidian analytic curves. And here is the question, how shall we compare with the constructions in the paper of Gross-Hacking-Kiehl-Konzer-Sevitz? The idea is the following. In GHKK, the mirror algebra is built from the scattering diagram. And the scattering diagram is built by specifying the initial walls and using the Konzer-Sevitz-Sobermann algorithm. Therefore, for comparison with GHKK, by the uniqueness property of the Konzer-Sevitz-Sobermann algorithm, it suffices to compare the set of incoming walls. However, we have defined walls simply as images of tweaks of tropical curves, and we do not have a notion of incoming or outgoing walls in this generality. Therefore, here in the cluster case, we need to introduce a more restrictive notion of tweaks and walls, which will allow us to distinguish incoming versus outgoing walls and to better control the monomials in the scattering functions. We will call them C-tweaks and C-walls, where C stands for cluster. So a C-tweak is a tweak such that each infinite leg maps to a hyperplane E-perp with derivative sum multiple of E for sum E in this basis. Corresponding to unfrozen variables. It's easy to see that for all stable maps in the modular space M-smooth that we are interested in, the tweaks of the tropical curve associated to such stable maps, they are all C-tweaks in the cluster case. And next, we define a C-wall to be a pair, sigma n, where n lies in p, the sub-monoid of M generated by s, and sigma in the hyperplane m-perp is a closed convex rational polyhedral cone. A C-wall is called incoming if n lies in sigma and is called outgoing otherwise. So, C-wall has a little bit more refined information of this direction vector n, and now we can construct a collection of C-walls by induction. So, we start with w0, the collection of C-walls of form E-perp n, where E is any basis vector corresponding to unfrozen variable, and n is any positive multiple of E. We call these the initial C-walls. And then by induction, we define, assume we already have w0 until wt, and we define wt plus 1 as follows. For all pairs, for two C-walls, all pairs of C-walls in wt, such that either the pairing of the two direction vectors is nonzero, or the direction vectors are parallel. We define the sum of the C-walls of the two C-walls like this, where the support of the wall is sigma1 intersects sigma2 minus all positive multiples of n1 plus n2, and the direction of the C-wall is just n1 plus m2. We check that it is a C-wall, and we add all such sums to wt. And finally, we let w be the union of all wt. So here, it's important that we do not add walls when the two direction vectors as pairing zero, but they are nonparallel. Because this corresponds to a non-transverse situation. So two consequences of the construction are the following. First, the incoming C-walls of w are exactly the initial C-walls. And second, for any generic C-tweak and any edge, there is always some C-wall in our collection w, such that the image of the edge lies in the support of the C-wall, and the derivative is equal to the direction of the C-wall. And we remarked that for the scattering diagram associated to u, the second consequence implies that the wall crossing function always has this form, where the exponent is just multiples of this n. Whose perp contains x. So a priori, the exponents can be quite arbitrary, but from the notions of C-tweaks and C-walls, we see that we have a strong restriction on the shape of the scattering functions. Now we are ready to deduce the comparison with GHKK. So let dGHKK be the scattering diagram in GHKK. It is produced by the condensate disturbance algorithm from the set of initial walls, e-perp with scattering function 1 plus z to the e for all basis vector e corresponding to the unfrozen variables. Therefore, by the uniqueness property of the condensate disturbance algorithm and by the identification of incoming C-walls in our setting just above, it remains to show that our C-walls have the same wall crossing functions. And in other words, it's enough to show the following claim that for each e in S and for any point x in e-perp generic, the attached scattering function for getting curve classes is just 1 plus z to the e. And that can be done by first figuring out what do the tweaks look like and then by an explicit computation. So from this we deduce immediately the comparison theorem for the scattering diagrams between our scattering diagram and the GHKK scattering diagram. And then we have the comparison theorem in the both A cluster case and also the X cluster case. So let us recall that in the first lecture I mentioned the five consequences of the comparison theorem. And roughly the comparison theorem gives geometric interpretations of and also more conceptual understandings of many constructions in GHKK and also proves some of their conjectures. Since I don't have much time for in the first lecture I mentioned another application of our theory to the study of for modular spaces of Calabiol pairs. So probably I can explain it in some other occasions. Thank you very much for your attention. Thank you. Thank you very much for your lectures and maybe some people want to ask some questions. Please just unmute yourself and ask questions directly. For me the last part was very familiar so I don't really have questions. It's too easy. Maybe the general questions in the situation we don't have nice assumptions about affinus. When we don't have affinus the mirror algebra is not an algebra. Yeah. It's just a formal. But still yeah it's still algebra of some form of posterior syndrome. Yes. So probably you want it to be some affinoid algebra. Yeah without affine. But in any case we need some positivity assumption. Yeah. Because as we have seen in the proof of the deformation invariance we want to prevent bubbles moving from the interior to the boundary. So we want to count analytic disks in the log Calabiol variety U. We don't want to count analytic disks that has something to do with the boundary. Yeah. And if we don't have any affinus or other sort of positivity assumption then we cannot count analytic disks in U. Because it can have bubbles or it can deform and touch the boundary. So I guess in order to be able to count analytic disks we need at least that the log Calabiol to be proper over some affine. Yeah. You need some kind of similarity to the boundary. Yeah. It's actually very proper over affine. The proper affine it's sufficient. Yeah. But the proper over affine is equivalent to the condition that we have some positive we have some positive combination of boundary components which is enough. Yeah. But they feel good conditions. If they both conditions they are equivalent and I think it's. Yeah. Also such kind of variety appears in JT series usually JT you produce. I think that's a satisfactory setting. Yeah. To assume proper over affine. It also projective. Projective over affine. Yeah. And it also covers just the projective case over a point. Yeah. Okay. So there's no more questions then we can virtually thank Tony for the nice series of lectures and. Thank you Maxine. Thank you. And thank you everyone for your attentions.