 Okay, let's get started because I stopped in the middle of things last time Let me just give a quick quick review of where we got to so we had a mirror pair u and v of log Calabi our varieties and We discussed the stromenge our Zaslo conjecture that says when you have a mirror pair The way to think about it geometrically is there should be a pair of dual Taurus vibrations a Lagrangian Taurus vibrations like this so over the same base U maps to be Via F V maps to be via G the fibers are The smooth fibers are a real n-tor I s1 to the n and they're dual in in the sort of obvious sense, so If I write a Taurus without choosing coordinates, I can write as H1 LZ Censor s1 and the dual Taurus just obtained by dualizing this will be in group H1 of L Okay, so that's the stromenge. I was as low conjecture Related to that is the homological mirror symmetry conjecture of can save it which says in this context if you just look at the Symplectic geometry of you there's a category you can associate to it called the Foucaille category and This category should be equivalent To the derived category of coherent sheaves on V So we're regarding now you just as a symplectic manifold and V is a complex manifold so what I explained last time is that This is compatible of the syz conjecture in the following way It's expected if you just take a point in V and you take the skyscraper sheet that corresponds to a fiber of the syz Fibration on the mirror the fiber Corresponding to the point P. So let's take so P lies in some fiber of V take the corresponding fiber of you and together with a local system so The objects of the Foucaille category a Lagrangian submanifolds together with the data of a of a local system on on the Lagrangian So that's just the same as a representation of the fundamental group of L into Into the group and the group of this local system is just the unitary group you want or the circle so this this element is nothing other than an element of the dual torus so it's just the same group as we had over here thinly disguised and Those should correspond under this identification. So that's actually corresponds the choice of the point P But the moral of all this is that somehow You know how does one construct V if you just given you and you want to construct the mirror and you try to do it Using the homological mirror symmetry conjecture what you say is there's some objects in the Foucaille category of you namely these Lagrangian Tori with with an arbitrary you on local system just take the modulite space of those objects That would correspond on the mirror symmetry just just the modulite space of skyscraper sheaves on V Which of course is V itself so that gives you a construction of the mirror at a conceptual level And so somehow this is point of view of motivate now how we try to describe this mirror variety V explicitly Okay, so first of all what I want to do is discuss the complex structure on V So at the moment I've said okay as a set told you what it is But you know it's basically a complex manifold So how do we construct the complex structure and this is going to be two versions? So there's the first version is the naive version And the second one will incorporate what are called instant on corrections So I'll say more about what that what that means And the first step But maybe I'll just say immediately that somehow this is where all the deep properties of mirror symmetry are sort of hidden So for instance if you are familiar with the fact that Gromov-Witton invariance on one side correspond to some periods on the others On the other side. This is really coming out of this second step, but how you modify the complex structure of V according to certain counts of Holomorphic curves, okay, but let's let's do the naive version first Okay, so let's draw the following picture. So this is a picture of the Boris vibration on you I want to describe local coordinates on this This set V the modular space of these Fibers of you together with you on local system. So let's pick a reference fiber over B zero and take another Point close by So let's draw the corresponding Fibers L zero and B. I'm going to describe some complex coordinates B Which I'm thinking of as this set of these various Fibers together with local systems. Okay, so what do I do? So just for any One cycle in the reference fiber. I'm going to lift it to a relative element. So I'm gonna choose a path from B zero To be here's my gamma in the fiber. I'm just going to extend it to a Two cycle like this So I'm working in some small neighborhood of B zero so I can do this In a way that sort of you know doesn't doesn't depend on Any choices at the level of of topology? And then I'm going to define The z z z to the exponent gamma that'll be my notation notation for the associated local coordinate If I evaluate it at a pair L row, I guess it's an L here What should it be? So it's a complex number in fact a non-zero complex number So it's the exponential the normalizing factor 2 pi I and then the integral of The Complexified Kayla form so this is what I discussed last time so omega is a symplectic form and B is the so-called B field on On our variety you so if you want to ignore the B field feel free could set B equals zero and we're just going to integrate that over this cycle gamma and Multiply so the final term is how does the local system contribute so it's row times the boundary of gamma This is an element of C star and so so that's a definition associated to one element of h1 and if I choose a basis, so if Gamma one up to gamma n is a basis of h1 then we get local coordinates Z1 up to Zn on V as a complex manifold and The local model in these coordinates is just you take the torus with n and coordinate Z1 up to Zn and the map to be It's just given by um the So let's see so have I done this I'm going to Take the following coordinates on the base, so I'm just going to take the absolute value and then do the following Normalization so these are my coordinates on the base and so just unpacking what I've done there in other words. This is Sorry, I'm just inverting the transformation there. This is the integral over gamma i of omega so these are the coordinates on the base, so we have Distinguished coordinates on the base given just by the areas of these cylinders with respect to The symplectic form omega Okay, and so why did I choose this particular Normalization of the coordinates on the base the point is that these are what are so-called integral affine coordinates remark Is that the yj? What's called integral affine coordinates on on the base bit? Or what does that mean? So it just means if I have a No, I cover my manifold by patches of this sort if I Change from one chart to another the transition map lies in the integral affine group glnz semi-direct product with Rn, but what I mean is you know, what was the Ambiguity in choosing these coordinates. Well, first of all, I had to pick a basis of H1 of the torus, so that's just an element of glnz and I also have to pick my base point So if I translated the base point that would change these areas by a constant That's the translation here Okay, so these are special distinguished coordinates on the base associated to Symplectic form on the selected manifold and the structure of this Lagrangian torus vibration And so in the theory of interval systems. These are called action angle coordinates. Okay, so that's the naive complex structure And as I said that doesn't quite work it needs to be corrected So what's the problem? So this does not extend So let me go back to the same notation. I had yesterday. So what I really have is You know a Lagrangian torus vibration over an open subset of the Fibers and there'll be some subset of singular fibers where this construction Does not apply. So what I'm really good at doing is getting Like structure on this open set So the problem is that this complex structure Does not extend to a complex structure on Maybe I should be more precise about it. I haven't told you what V is yet But you know, this is some complex manifold and there's no way to extend it to any sort of reasonable complex analytic space So, you know, if you try to Do that, you'll just find there aren't enough functions to make this a proper space. So because Why is that because of the monotron around the singular fibers? So the point being if I go around a singular fiber these sort of coordinates I've written down will undergo some Non-trivial transformation and it means that, you know, you won't be able to extend that to a local function near near the singularity Okay, so what's the solution and again, this was in the original paper of Strominger-Johannes Aslo And it was motivated by considerations in string theory So what you should do is modify the gluing of these local charts in the following way. So using counts Ending on the fibers of the SYZ fibrations. So the picture is, you know, my SYZ fiver. I've got some holomorphic disc whose boundary lies on that SYZ fiber and a so typical picture is How does this disc arise? So somehow It's a good picture. If you have a singular fiber like this a pinched torus Such that this cycle is the vanishing cycle for this degeneration Then at least topologically you can see there is a disc like this and in certain situations That disc can be made to be can be come holomorphic for appropriate choice of particular disc So let's see an example So I want to go back to the example we discussed yesterday So if you remember yesterday, I had a construction of the following type. So I started with Basically started with C2 and I just blew up a single point On the boundary, let's say 1-0 for instance, and I'm thinking about my log club. Yeah, I was being the complement of the boundary So, you know, so my U, this is my Xd U is X-2 So D being the the strict transform of the boundary And so what I said last time is that if you think about this from the point of view of the SYZ fibrations You get a picture like this for the torus vibration So, SYZ fibrations It's a bit difficult to draw, but let me try So first of all the base Like this, so like a quadrant and there's a A singularity So the monodromy of the cycles in H1L around this point given by this matrix And so, you know, so the picture, so this is the base and here's U So over this point, there's a singular fiber and if I sort of look at this path here That's Roughly speaking, so over some path of this sort, I see the exceptional divisor This this line here corresponds to my boundary, one of my boundary divisor, then you have a boundary divisor I guess I don't care about This is um the x-axis That's that's the rough picture, but he's my exceptional curve. Okay, and so now one can see There are two disks in this picture Which end on the fibers of the SYZ vibration so maybe Oh, sorry. So this is what I did last time So if you remember we had just the moment map image of C2 just the positive quadrant in R2 We removed A small triangle And then we glued via this map to form this space B. Does this sound familiar? So I want to emphasize that, you know, this is a um this this should not be thought of as embedded in the blackboard. It's some kind of You know, so this this is a planar picture and now I've done some funny gluing So it's really, you know, think of it more as an abstract space as opposed to Okay, so what I want to sort of explain is there are two disks in this picture. So let's sort of see so where are they You take this line. So this is the line. I guess x equals 1 And you take it strict transform So now let's sort of um fatten this up to to a more reasonable picture the sort that a complex geometry might draw So here's here's so this is um Right. What does this look like? It looks like a Uh a disc So cp1 of one point removed. So here's a disc Coming out of the singularity of the singular fiber here and similarly Um, you know, here's my cp1 And you know, this is the other disc. So coming out that way So there are two disks um emanating from the singular point one going this way one going the other way And so when I say two discs, uh, what I mean is Um two families of discs. So depending on which fiber I want it to end on, you know, I stop at a different point So this is the singular fiber. This is a general fiber Over this um Over a wall in the base So what we have is this kind of picture there's a wall um Real co-dimension one over which there are do exist holomorphic discs Ending on the fiber So there exist Discs exactly exactly one Ending on the fiber again, this is a picture of b again So there's a singularity here and the discs are emanating in the two directions from the singularity Okay, so now I want to try to show how do we correct this problem with the local coordinates So now I'm going to introduce a branch cut So that on the complement of this cut Uh in in the in the variety you I can define these functions Uh z one and z two globally away from the cut So define coordinates Z one z two Um away over the complement of the cut and so um First of all, let me just say what's the monodromy? So remember these are associated to um The homology of um the torus These correspond to a basis gamma one gamma two of l So what's the monodromy as I go around? This point so you have to be a little bit careful to go back to um So what we did last time so this has monodromy this gives you the monodromy on the integral tangent vectors So this is the dual space these correspond to coordinates. So this is cotangent vectors So that's dual To the tangent space Um at a point So I'll write tz for an integral tangent space. So this was the monodromy On the tangent space. So the monodromy here. So in the opposite Direction will just be given by the transpose So if we take our z one z two to correspond to the two basis vectors or the dual basis of the basis there This is the monodromy. So it's saying z one Goes to z one And z two goes to z one inverse c two. Oh, I'm sorry. I forgot to do the transpose I should have written it out. So this is the transpose Okay, now we're good. So z one goes to z one z two inverse And z two goes to z two. Sorry. I I I should say so this is of course, um the monodromy written Additively and here we're writing it multiplicatively. So the map gamma goes to z gamma is a homomorphism. So this is just writing H one of l multiplicatively. Okay, so that's that's the transformation across here And so what do we attach to the wall? So so as I said on this wall, we're going to associate an automorphism to crossing this wall So what is it? It's just the standard cluster transformation that we've seen already So on this side, it's given by the following formula. So z one is fixed And z two goes to z two times one plus z one inverse And above the wall Above the singular point. So on the other connected component of the wall I'm running out of space. Maybe I'll erase this matrix if you don't mind So z one z two similar formula, but now it's one plus z one Let me try to explain. So where did these functions here? So we think of those two functions as attached to the wall one plus z one and one plus z one inverse Where did those functions come from? They're basically just recording The class of this disc Um, you know, so this is some holomorphic disc You can look at the boundary of the disc that has a class in h one And under our normalizations that corresponds to the coordinate z one Similarly here, you know this disc has the opposite Boundary as an oriented You know disc and that corresponds to z one inverse Okay, and so you can now check just see That now the gluing is consistent Meaning if I go around a loop this Composition is the identity So that's the um simplest example of this fixing the problem with monodromy using holomorphic discs And in general, you know, this is kind of an amazing Thing that sort of came out of string theory Um, I should say I'll give you know later in a moment I'll give a sort of more mathematical reason. Why should this work? So why does this work if you just correct using holomorphic discs? But let me just say now in general What happens? So holomorphic discs In you ending on the fibers So these lie over Real co-dimension one walls in the base Possibly thickened So it's not In some limit they they become real co-dimension one But uh, they might be sort of slightly thicker that might be some kind of amoeba like structure transverse to the wall And what's the we attached to function so here alpha is the class of the disc Um, so let me just say so now I'm sort of writing a slightly more general notation So let's write it down. So what's z z to the alpha? It's really the same definition as before But now I have a global homology class to integrate over Times the holonomic. I'm sorry. What did I call it row row of delta? So in terms of what I did before this is just a constant Times z to the delta alpha The boundary of alpha So, you know the original coordinates were in terms of h1l now I've got an actual class in a relative class Is that max to h1l just by taking the boundary? But uh, here I'm getting a A function which takes account of the the the area of this of this disc And now this function is some kind of generating function Counting multiple covers of the disc Discs associated with the wall multiple covers Okay, and um What's the gluing automorphism? So it can be written in these coordinates z gamma The following way They just take the product with this generating function raised to a power So you take the boundary of gamma And then there's a class Let me call it c So this um, maybe c c sub c sub c So what's c? So this is a a class in h n minus 1 of the fiber And it's just the um swept out by the boundary of the discs And so if we're in dimension two, this would just be Delta r for itself the boundary Okay, so this looks like a crazy thing to do So let me try to explain why this is going to work Why? Why does this work? reasonable question So the point is that with the corrected gluing we can write down global functions global holomorphic functions on v In the following way Of course now i'm really in the non-compact case that we're most interested in Otherwise there won't be any global holomorphic functions at all But yeah, so we're really thinking about the log log collabial case now So what do I do? So let's take some compactification Let's let c be some A component of the boundary in some uh log collabial compactification What usually v is inside um So Yeah, I think it's y ye right Oh, I'm sorry. I'm sorry. I'm sorry. I'm on the wrong side Is x set minus d So Just the usual picture. I've got my u and I compactify it to some projective variety x with a normal crossing boundary That's infinity And just take some component of the boundary And I'll take a positive integer as well Then what's my function? So incidentally we call these a theta functions because um You know the analogous construction for an abelian variety produces the usual Theta functions from classical geometry So again, it's the same a similar formula to before So if I have A point in v so it's given by a Lagrangian and a local system I take a sum of the following sort So it will be a sum over classes in the relative The relative homology group h2xl of a count of disks n beta times this um Local function z to the beta defined in the same way as over here. So this is just the The definition of z to the beta And what's n beta so where? n beta Is the number of holomorphic disks I'll use this blackboard bold d for the the usual complex disk of radius one Close disk in the plane So mapping to the pair xl So a holomorphic disk in x now the compact guy with boundary on the Lagrangian With the following properties so it's in class beta It has a relative homology class That's this beta here Such that so several things So first of all it meets c in a single point Multiple st so contact order m The local intersection number Is this positive integer m Secondly It's disjoint from the rest of the boundary It doesn't intersect the boundary anywhere else and finally One more condition which fixes the expected dimension to zero Is that it passes for a general point of l So remember the boundary lies on l I wanted to pass through a particular Point of that Lagrangian so then that's expected dimension zero And so the point is that if you try to do these counts Um, you know use the naive gluing of v so you don't make any of these corrections This function will jump Um as you cross a wall and the gluing correction is um defined in such a way that that that it corrects for that jump So that these become global holomorphic functions on v Um, so let's give an example So back to our our running example So again, I'll sort of draw the base And here's our wall remember There are these two discs emanating from the wall Um, and so if I try to let's let's make this our boundary divisor c I try to do this computation on one side of the wall over here I'll just get a single disc So I'll sort of draw it like this sort of It's just um Think about it, you know, it's just one of the lines y equals constant That's your disc If you're on the other side of the wall, you have that you have that disc again, you know the same thing Another disc y equals constant, but there's another one as well looks like this So I'll sort of draw it like an amoeba. That's what it looked like under this projection Some kind of amoeba like this There's another disc of this sort And again, let me try to sort of say intuitively what you should think take the original disc we had There's a disc coming out along the wall Sort of glue those discs together Together with a cylinder So here there'll be some kind of balancing condition Is that the sum of the boundaries zero h1l And so you could imagine you you glue together some topological surface like that But then the assertion is okay near to that topological surface. There's an actual holomorphic disc And since we're in such a simple example, you can actually write down this disc explicitly So let's go back to the toric variety. So remember that this was uh, you know originally we had the Toric variety we're just blowing up a point I guess I'm sort of slightly abusing notation here I was originally talking about the base, but there's also of course the the variety living over it So we blow we blew up a point Let's construct what would this disc look like down here So it would just be a disc like this There's a disc intersecting the boundary in two points and it's ending. So this is our fiber here ending on a Lagrangian torus fiber l And the way that we constructed this Lagrangian vibration I didn't give you all the details last time But away from this wall the Lagrangian vibration is just the same away from some Compact neighborhood of the of the wall So in fact, this will just be the usual usual torus fiber um for a toric variety just you know the absolute value of the the Coordination is constant And so then uh, you know my disc Be the following disc. So remember this is the point. What did I call it one zero? um So what's the disc one way to write it down like this? Yeah, so that's a a disc Which goes through this point one zero And if you look at the boundary of the disc, this is D mapping to C2 Um, you'll check that the way I've cooked things up. So the boundary of the disc maps exactly to this some Lagrangian torus fiber And so really the only thing that's going into this which probably most people know Is this a thing from complex analysis 101? In the so-called blashka factor So that's what we're saying is this map gives a map from the unit disc to itself Um, which sends alpha to zero In fact, you only if you only know this one fact you can basically completely classify holomorphic discs in toric varieties ending on a on a on a Lagrangian torus fiber So that was done in a paper Of cho and oh Let me just give the reference So, you know, I'm just giving this reference if in case you want to see more but I guarantee you if you don't see it immediately you can do this example on your own and see that this is this is the this is a Such a disc Okay So I want to switch gears now and so before I do are there any questions at this point So maybe I just quickly review what we said so the point is that You know a mathematician can understand why this gluing is the right thing because what it does It makes sure that the complex manifold you're constructing has lots of global functions Right, so somehow if I want to extend a complex manifold, you know, I have have some say structure on a On a complement of some small co-dimension sets in a complex manifold And I want to know if it extends to a complex variety over that puncture Then you know one way to do that is just ensure that you have enough functions to define the thing You know so you can take sort of spec of the ring of global functions essentially just to extend it And so this construction sort of explains from a mathematical point of view why this instanton correction will solve this problem That the complex structure doesn't extend over over the discriminant locus Okay, but now I want to Explain so, you know our work is basically taking this picture and trying to make it a rigorous mathematical Proof of this conjecture for cluster algebras. So construction of global functions on cluster algebras So the next step is to translate this picture into algebraic geometry or really more accurately tropical geometry And use it to prove To construct canonical basis of cluster algebras. Remember, what is a cluster algebra? Well, we've got some log Calabi our variety with some additional special properties. We just take its global functions That's the cluster algebra. And so, you know this canonical basis will be exactly this This base is described here. So in fact if you do this for all possible components and all possible positive integers M and you throw in The identity elements that would be called feet of zero these guys together form a basis of the ring of global functions Okay, but that's going to require quite a bit of work. So let me switch gears and start doing that So let's quickly review the notation that we Used earlier. So we have a cluster variety And we've got some toric model. So in the cluster language, that's what's called a seed. We've got the torus T that's the interior Of this toric pair That's a copy of c star to the n We've got the lattices. So n is the lattice of one parameter subgroup. So the first tomology of the torus Um, let's copy of z to the n n is the dual of n We have a holomorphic symplectic form on x Or strictly speaking on u with log poles along the boundary And what we sort of explained last time is um On the first day is that actually that becomes completely combinatorial thing if you just restrict it to the torus Um, let's call that sigma bar So that's a log two form on holomorphic two form on the torus And that's a very simple thing. It's just a constant coefficients given by some skew matrix And without choosing coordinates That's an element of wedge two of m tensancy So what was the story? So this toric, uh, I probably shouldn't write any lower. Should I go over here? So this is a blow-up Was a blow-up of a certain amount of data so It's defined by the following data So we had some centers z i they were given by a boundary divisor Let's let me write d sub vi so vi will be the Generator of the ray corresponding to the boundary divisor Intercepted with Zero locus of a character So let me write my character in multiplicative notation z to the mi equals lambda i So here the vi Is a primitive element of n Mi is a primitive element of n Lambda i of course is just a scalar And there was the condition which was for if you do this blow-up When does the the holomorphic form actually lift? And the condition Was so in these in this combinatorial fashion You take sigma bar you evaluate it At vi in the first argument that should be a multiple of mi That's just the condition for the holomorphic form sigma bar to lift to a holomorphic form sigma on the blow-up Okay So that's the notation that we had last time. I agree. It's a bit of a mouthful. So my apologies So now what I want to define is what's called a scattering diagram associated to this data so Let me first explain what's basically going on here. So we we understood that in the base of this Taurus vibration There were certain walls in the base that we had to attach some automorphism to Now one can certainly ask what happens when one when two of those walls collide Well, uh, if you were to go around a loop here, so there'll be some automorphism attached to each wall You know this This um composition of automorphisms will be some kind of commutator So it wouldn't be the identity unless these two guys commute and there's no reason to expect that in general So what one can think of is that the scattering diagram Is there's a canonical way to fix this So if you have these two walls, I'll draw some arrows to indicate the direction in which the sort of disks propagate So somehow this is the the direction in which the disks area increases So then what happens is it's going to be some kind of picture like this where you add a whole new bunch of rays coming out of this point Um, which fix this property that fetus should be the identity when you get around a loop And again topologically it's the same picture we saw before that if I have two incoming disks I can sort of glue them on To a cylinder like this at least topologically take, you know n times this m times this Glue them together and get some nearby holomorphic disk That's the meaning of these rays So basically, uh, yeah, what the statement will be is that somehow If you take the initial disks coming from the blow-ups and you apply this scattering procedure That will somehow tell you all the holomorphic disks On your on your manifold and and thus you'll be able to build the mirror Um using this scattering diagram, but the first step The technology of the scattering diagram is to just start with this data which tells you these initial Disks corresponding to the blow-ups. How does that data determine all the other disks? That's what's encoded in this scattering diagram. I should say Um, you know, this was discovered by consavich and soiblman In 2004 and it's still a kind of a miraculous object. We don't quite know why it works, but it does and it's very beautiful Okay, so let's try to say what it is So first of all, uh, I'll have a base ring. I'm just going to write down a polynomial ring These are just formal parameters which are going to control convergence issues Let's write n for the maximal ideal of the origin And we'll use hats for matic completion There'll be a certain amount of Care required to to to get convergence. So what's the scattering diagram? Well, it's a collection of walls So I'll use the notation b and f so d will be the support of the wall so this will be um rational polyhedral cone The vector space associated to the lattice n that's the lattice of one prime and subgroups of the torus where the Where the fan for a toric variety lives And it's got co-dimension one Okay, so given that so let m in the dual lattice be a primitive vector with um delta contained in m perp So that's just determined up to a sign just the hyperplane that this rational polyhedral cone lies in Um, so then we also have another part of the data. So what's this function f? First of all, I have a vector v In the in the lattice n this is going to be called This is related to what's called the direction of the wall Again primitive We have our usual condition on the Relation between these two guys. So if I take sigma bar of v And that should be a multiple of m The same condition we had over here and our f is now a function So we've got this coordinate. We've got this coefficient ring a So just take the polynomial ring in this variable z to the v and that lives inside um This ring here, maybe I should just say so what's this? This is just the coordinate ring Of the torus the dual torus actually so it t dual uh cross um a They are we've Or that's t1 to tr So these are just global functions on the dual torus um But we've introduced these formal parameters t1 up to tr and um I'm going to have to complete here. So this won't actually be an element of this ring It'll be an element of the completion. So let's put hats So what's sorry? I should have said so t dual. It's the obvious thing, of course My t is n tensor c star. So t dual will be n dual tensor c star n tensor c star So it's always confusing. I've been doing this for a long time today. It's still get confused So n is the one parameter subgroups of t, but it's the Characters on t dual. So this z to the v is a character on on the dual torus. That's our patch of the mirror variety Okay, and so this guy should satisfy some property. So it should be congruent to one Modulo the maximal ideal and also its constant term should be one. So it'll be So it's in this polynomial ring. It's congruent to one mod the maximal ideal and the constant term is also equal to one We have a finiteness condition. I just can't hear you n times this actually So every term will have a non-trivial You know exactly what I said, this is correct So the finiteness condition so for every Positive integer l there are only finitely many f congruent or not congruent to one mod m to the l And so so okay, this is a whole load of junk. What's the point here? So I've got a wall crossing automorphism associated to a wall So here's my wall or two-dimensional picture is m greater than zero m less than zero if I cross this wall In that direction so from m greater than zero term less than zero I get an automorphism of the ring Roughly speaking an automorphism of the torus the dual torus What does it send? So it sends z to u z to the exponent u goes to z to the u times some power of f and the power is given in this way slightly Pairing between u and m So this is again should look familiar. It's something like what we were just doing in the symplectic discussion translated into this framework No, so um Sorry, finally many walls Such that So what it means is that I mean, sorry, let's say in terms of this So attached to each wall is an automorphism But at any finite order, I only want to have finitely many non-trivial automorphisms, right? That's the that's what it means So now what you get But now for any path in the scattering diagram So the ambient space is n r um So I don't want it to go through the singularities of d the the places where it's not a manifold with endpoints Again, I don't want endpoints on a wall Maybe I should write vertical lines for the support of these things We'll get an automorphism in the obvious way you just compose all the automorphisms You get by crossing the various walls And you know if you sort of this is why this sort of completion is kind of Crucial remember all that means of completion, you know our hat. I'm just taking the inverse limit Like this So what I can do at any finite level this makes sense because there's only finitely many walls And then I pass to the inverse limit. I get this This automorphism, which in general could be kind of gnarly Okay, let me let me go in for one minute because I want to state the theorem Of consavich-soyverman and gross seabit gross seabit The consavich-soyverman did this in dimension two In a slightly different context and gross and seabit Translated it into algebraic geometry and did it in arbitrary dimension So if I take Let's call it d in this is an initial scattering diagram Consisting only of hyperplanes So each wall the support is a hyperplane then unique scattering diagram d containing this Initial scattering diagram with two properties so such that so first of all The new walls are all outgoing and secondly we have this consistency property that theta D gamma is the identity for every loop gamma and it occurs to me. I forgot to tell you what an Outgoing wall is so let me do that now very quickly So in this data, luckily I haven't No, I have erased it Oh, sorry. Here it is. Uh, that's right. So This vector v. So what's an outgoing wall? The direction of a wall df The same notation as before is minus this vector v And again informally, this is the the direction that the disk Propagates The direction in which the area of the disk increases And we say it's incoming if this vector v is in the wall and outgoing otherwise, okay, so so the um again the heuristic here is sort of we're constructing this varieties a blow-up They will have some disks Coming from the exceptional divisors, which are they're going to be the incoming walls Then we'll have a whole bunch of other disks Which are sort of generated by this process, which will be the outgoing walls. And so this theorem Tells you that somehow there's this sort of magical construction Um, I mean it was basically just amounts to a computation in some uh, lee algebra associated to this A group of scattering automorphisms Which tells you that you know given any initial data you can construct Uh, a unique scattering diagram with this consistency property So unfortunately, I'd had wanted to talk about some examples, but uh, I better stop so actually, um Lofa, am I supposed to talk this afternoon as well? There's some sort of exercise session or I just thought well Yeah, well what I wanted to do is to discuss discuss some examples of this And so which might that might be appropriate to do it this afternoon. I don't know anyway. Thank you very much