 This name starts with A, I love A, it always brings me to become the first quarter. My quarter is also more in P as well as A, so the stock will be about A. So the plan is, the stock will be about bridging two topics, one in algebra and one in geometry. So on the algebra side, I will tell you about kubo, damas, and damas, kubo, nipin variants. And so I will tell you about our work from last year showing you how to calculate these variants using an ultra-cubo paper. And on the geometry side, I will tell you about council curse, so this is more what I have been doing. And I will tell you how to pass from kubos to torque varieties and how to set up a problem to count curse and torque varieties given the data of the kubo with some additional combinatorial information. And the upshot will be to tell you how kubo-DT variants are related to these cure counts which kubo saw from our written invariant and sorry for it. Okay, I will just briefly review the kubo-DT theory, so I will just tell you what our kubo saw from kubo-DT variants. A kubo is a finite oriented graph q, which may look like this. It has a set of vertices, in this case one to three, which we do not like q zero. And it has a set of arrows which we do not like. When you have a kubo, you can talk about talk of a representation of a kubo. A representation of a kubo is an assignment of a vector space to every vertex, like if you have this simple kubo, the three vertices, you can assign vector spaces to each vertex, and then add an assignment to a linear transformation for each arrow between these vector spaces. And when we talk of kubo representations, they usually fix the dimension vector gamma, which corresponds to the dimensions of the vector spaces you associate to vertices, like if you have c here, c to c, this will describe a kubo representation of dimension one to one. And this is an element of the positive lattice, and the n is the lattice c to the number of vertices of your kubo. So there is a natural notion of morphisms and isomorphisms between two kubo representations. It follows like if you have another copy of this, you can ask for some commutative diagrams and maps. So we will consider kubo representations or kubo representations up to isomorphism. And in addition, we will care about stable kubo or semi-stable kubo representation, so there is this notion of stability due to kink. It says if you have a kubo representation of any fixed dimension gamma, if you denote the dual space to your lattice n, which was a lattice c to the number of vertices by m and associated real vector space by m r, then any element theta in this space, which lives in gamma per is a stability per meter, and for such a stability per meter for an element theta here, you call your kubo representation stable. If for any sub-representation, the pairing of theta and the dimension vector sub-representation is smaller than zero and you call it semi-stable if it's smaller or equal to zero. So what we do is we will consider this notion of stability and we will try to understand the modular space of all theta semi-stable kubo representations of given dimension gamma. So we have we fix a dimension vector gamma, we fix a stability per meter theta per kubo and we look at this modular space. And nice situations, which will be very rare in this situation, for instance, when this modular space might be smart, we define the kubo DT invariants of the topological Euler characteristics of this modular space, which is alternating some of the data numbers of this modular space. And the most important thing about kubo DT invariants is there's a piece-wise constant dependence on the stability per meter theta. So if you look at your dimension vector gamma and gamma perp in the dual space, then there's a wall and chamber structure. So as long as you're inside one of these green chambers that I illustrated, your kubo DT invariants for any theta here will be the same, but if you cross one, there will be some change, which can be described using some wall crossing structure, wall crossing, new to consider. Here is just a simple example. If you look at the n-chromic kubo, which is the kubo, just the two vertices and arrows between them, then if you fix our dimension vector to be gamma to be 1,1, for any theta 1 minus theta 1, this theta is the gamma perp. And one can show that if you take theta 1 smaller than 0, you can verify the kink stability. There won't be anything semi-stable, so we take theta 1 bigger than 0, then the modular space of all kubo representations for any theta like this will be isomorphic to Cp n minus 1. The reason being you're essentially having a copy of C here, a C here, and a map between cmc is determined by a complex number, and you don't want all of them to be 0 to ensure some instability, so n-complex numbers which are not altogether 0 gives you Cp n minus 1. Okay, so let me talk about kubo. So far I gave just simple examples like here there were no loops or so, but I just want to point out generally we might have loops in this kubo, there might be cycles, so in addition you need to consider the path algebra of the kubo, which is the combinations of linear combinations of path. Anton, you enter the kubo with the choice of a potential function, potential in the spot algebra, which is just a formal linear combination of not paths but cycles, so whenever you have a cycle you can define a potential function. I will not talk too much about this because for our main result it won't be a crucial rule, but just it will be just to say the most general definition of what kubo dt invariants are. If you pick some potential, which is a formal linear combination of cycles, Cs, then for any cycle C you can define the trace function. It is a function defined by considering all the linear maps associated to arrows forming the cycle, and then you just take the trace of this composition, and then you define the trace of your potential, and then what you do is you look at the critical locus of your trace function and start your modularized space, and again in my cases you can just start your kubo dt invariants to be the alternating sum of 30 numbers of your critical locus. So this is what you do to define kubo dt whenever you have cycles, and here is the most general definition. It's using quite some technical tools, so whenever you have a kubo with cycles, so you have this additional potential, and in this situation your critical locus is not smooth, then you need to consider alternating sum of 50 numbers where you consider the sheaf cohomology groups, and you take into account the sheaf, which is given by the intersection cohomology sheaf. It's not an honest sheaf, it's a perverse sheaf, but in situations your modularized space is smooth, the intersection cohomology sheaf will just be the constant sheaf with solid q, and then to pull back this intersection cohomology sheaf on the modularized paths to apply a vanishing cycle path counter to it, you obtain a sheaf on the critical locus, so these are all technical details that appeared on many people's works that I noted some here, Antoni Jupps and your kubo dt invariance with this formula, and generally one remark, because we will calculate these invariance using wall cross inches for computational convenience, we will not work with kubo dt invariance, but we will work with rational dt invariance, so the data of all kubo dt invariance is equivalent to the data of all rational kubo dt invariance, they're just defined by, you just set some additional factor in front of the kubo dt invariance which show up in a movable growth cycle. Okay, so this is the kubo dt site, it has a pretty technical definition, many ingredients here, so let me just say that there are many people among physicists who do calculate concretely these invariance, so here is a paper that I took this table from, if you look at the tree chronicle approval, which appears in this paper in the context of n equal to 40 SU3 superanx theory, something far beyond my expertise, but I just wanted to show you the numbers done for the kubo dt invariance can be calculated, so I'm trying to motivate that while their mathematical definition is somewhat technical, they do our important invariance, they appear in a lot of situations, so if you look at kubo dt invariance, they appear on the physics side on supersymmetric quantum mechanics, they correspond to counter-PPS particles or supersymmetric ground sites, and on more the mathematical side in some situations they correspond to geometric ET invariance, which have to do with counter-coherent chiefs in Columbia tree faults, which on the mirror symmetry is expected to give you the Council's special Lagrangian Sympylobiart tree faults, so they appear both in physics and on the math slide. Okay, so the question we asked last year with Pyrik is, is there a primitive set of dt invariance which we can use, which we understand well, a simple set of dt invariance from which we could determine all kubo dt invariance, and we showed yes, we can calculate all kubo dt invariance using wall crossing and floor trees that appear in the wall crossing or scattering diagrams from a simple set of attractor dt invariance, so I will first tell you how this works, it will be useful to relate the calculation of kubo dt invariance to counter-coherence. Okay, so to tell the main reason why I need to pick some notation, so for our letter sign I'm going to choose a basis as 1 to sq0, where q0 is the number of vertices in your kubo, I define a skew symmetric under form on the letters by the pairing s i s j will be defined by a i j minus a j i, where i j is the number of arrows from i to j, and similarly for a j i. For a fixed dimension vector gamma, there is a particular point in the dual letters in the associated real vector space of the dual letters, which is obtained by the contraction of the skew symmetric form that I defined with gamma, and this point in the dual letters is called the attractor point, and the chamber containing this point is called the attracted chamber. And for any small perturbation of this attractor point, which is generic in some suitable terms, one can define the attractor dt invariance as the dt invariance of this small perturbation gives you a stability per meter theta as the dt invariance toward a stability per meter theta. And these attractor dt invariance have been studied by various people, that I wrote some of their names here, and some nice property about these is that you might end up with your attractor point in MR to be on some wall, but you can characterize a little bit to lie inside the chamber, and the attractor dt invariance won't really depend on this perturbation. Okay, so the attractor dt invariance are nice through the simple dt invariance to study. There is a term due to John Gretchen, and that was generalised by Foucault. It says if you have an acyclic quiver, then the attractor dt invariance are all 1 or 0. There are just one, if you take your gamma to be one of the basic vectors of your letters, so of the form 0, 0, 0, just one at one place and 0, 0, 0 again, but it will be one. This is not very difficult to see if you have just one here, you imagine on some vertex you have just a copy of C and all maps are 0, so your modular space of your representations will just be a point, and the other characteristics will be one, but the difficult thing to see is for everything else or any other dimension vector you get 0. And it's a conjecture, which is proven for P2 now. The conjecture due to Gujarat-Masho-Piolin and Moscow-Piolin says that if you have a local d'Alpetzl surface, which is a canonical bundle of d'Alpetzl surface, then there is some recipe to construct a clover with the additional data of the potential function such that the right bounded category of coherent sheets will be equivalent to the category of representations of your clover, and then the conjectures all attractor dt invariance will be 0 again unless again either if you're dimension vectors one of the basic vectors, in which case they will again be one similar to a ball, or if you're dimension vector is the multiple of the close of the points in which case your attractor dt invariance will be determined completely as the other characteristics of your surface. And this conjecture was proven by Piriq and his French collaborator. Okay, so yes, and here so your gamma lives on the lattice and here I'm identifying the cohomology of my surface with the lattice and I'm using some identification. Under this correspondence here, the close of the points you have all dimension vectors one, is this the question? Yes, I think so. Piriq here, you put the conjecture, maybe you can confirm. Okay, so yes, I think that's true. I want to tell you how do we use attractor plots to calculate the clover dt invariance. So how do we obtain general dt invariance? Starting from just a simple attractor dt invariance. And if you start, so if you start with the clover with the potential, if you've fixed your dimension vector gamma and the stability per meter and gamma group, then concivage Soebelmann already in their work dating back to 2013, showed a formula saying that you can obtain general clover dt invariance, a linear combination of attractor dt invariance, where there is a summation of all possible decompositions of your dimension vector gamma. So you decompose your dimension vector into several vectors, you take the sum of all possible decomposition. And there's coefficients appearing depending on the composition. There is an automorphism group, it's sort of permutations symmetries of the decomposition. And I will tell you what these coefficients are in a minute in a version of this formula, they can be combinatorially determined just as the sum of trees, which are called attractor fall trees. So, just to give a picture. So you have your space obtained by the dual to your lattice, which is V to the number of vertices you have for dimension vector gamma group. There live your stability per meters, you have your attractor point which determines the direction. So you can imagine the finding a tree starting at the stability point going in the direction of the attractor vector determined by the attractor point. And whenever you hit some whenever you decompose gamma say into gamma one plus gamma four into this your tree will buff your case so you can continue in this decomposition and the directions of these new attractor points in these new chambers. So we will imagine this as part of our wall and chamber structures so the lines where you hit will be the walls whenever you hit the wall, your flow tree will rotate. So this is an example of a particular flow tree starting at theta and flowing in the direction determined by attractor points. And here, as you see it could be of arbitrary valency here there's a poor valence vertex, it could be of arbitrary valency which makes it difficult to keep track of the comb not works and con such trees. So just a remark nonetheless attractor flow trees are very much related to tropical trees by their construction they satisfy a balancing condition at the vertices. And our goal was to sort of perturb them so we can relate to more standard trivalent trees and relate them to peritopsis. Yes the valency is bounded by so there if you so you the base of the composing gamma so you look at so here I decompose gamma into gamma one plus gamma four so you fix your dimension vector gamma so there is a maximal length of the composition that's bounded by that. Okay so we will turn this formula into a version of this formula that we proved where we say we don't need to worry about higher valence flow trees we can just look at trivalent things. So the theorem we proved is a version of the flow tree formula but we say if you have a quiver with a potential you fix the dimension vector you fix the stability per meter so you can determine all your quiver dt variants of the sum of all your decompositions of your dimension vector and you have your attractive dt invariance and you have some coefficients again again these coefficients have some combinatorial description in terms of this time binary trees so they will be determined by counting things over binary trees so this is a binary tree each time there is a flaw from the root each time you flaw every vertex is trivalent and we're considering binary trees whose leaves are decorated by elements appearing in this second position so that's simple version of the attractive flow tree formula comes to the soluble model it's conjectured by alexandra purely and we prove it's using flow structures around the same time there was another ice proof using operas which is due to most of it. Okay so I will tell you what these coefficients are how do we determine these coefficients here in terms of binary trees so these coefficients if our teta wants to have a decomposition are given as a sum of all binary trees with our leaves decorated by gamma one gamma a there are some horrible signs coming up epsilon t teta t r v are signs either they're minus one zero or one and I will be described these using some plugs I will not tell you how they are concretely described it's one needs to keep track of them it has to do with the realizability of your tropical trees these signs and then apart from the signs there it's just the pairing using your skew symmetric form you evaluate gamma v prime gamma v prime prime and v prime v prime prime for any vertex v are just the children of this vertex so here if you have a vertex v v prime v prime prime will just be the two children of your vertex so then you take the product of all v of this pairing where you evaluate the gammas corresponding to the children of your vertex yep yes that's correct and yes so that's that's that's a subtlety we handle whenever we have repetitions we need to they do appear and we need to go to larger space to be able to do these perturbations and describe this line so I will tell you in a minute how we do that when we have repetitions that makes it whenever we have repetitions that makes sense more technical and let me just assume in the simple case it's more upset if we if we may not have any repetition um okay okay if we may have repetition if gamma one gamma r is not a basis we need to work with a bigger lattice and then we need to do these perturbations in a bigger lattice uh where we define a new skew symmetric form in the bigger lattice and then the signs the description of signs will depend on this new skew symmetric form because on other repetitions we cannot do the initial perturbations we describe in the usual letters okay so there's some technicalities here I will not spend much time on this let me just give an example in the simple case van your gamma is one one just one one so there is only one decomposition to one zero and zero one and in this case there are only there is just this trivial tree and one binary tree corresponding to this decomposition where you decompose one one into one zero and zero one and then uh if we again look at the cruelty t-ingrarians for the anchor and I put quiva by the formula I just showed you so for the trivial decomposition the coefficients will always just be one and for this decomposition into gamma one gamma two you calculate your coefficients by there's some sign times you evaluate your skew symmetric form at the two children of gamma in this case which are one zero zero one if you calculate in this example the sign to be minus one and if you evaluate skew symmetric form at least two uh vectors appearing in the decomposition and the n form I could pull out by the definition of the skew symmetric form you have n arrows you get n so eventually you find your f2 to be n minus one times n and as expected as the Euler characteristic of cpn in this case given this as expected you're getting your cruelty t-ingrarians in the simple situation okay so this was just illustrated so we do obtain these quiva d t-ingrarians just by counting flow trees which are binary and using this skew symmetric form associating the number to every such flow tree uh for every decomposition okay so the main goal of this talk that I promised that I will tell you was the correspondence how can we relate uh quiva d t-ingrarians to concept curves and torque varieties so more concretely I will tell you how to relate these coefficients appearing in the quiva d t formula which tell you how to obtain your quiva d t-ingrarians in terms of attraction variance to concept rational curves and a torque variety obtained from the quiva so here's the setup of the counting problem so I'm going to tell you how to construct a torque variety out of a quiva of given dimension vector gamma so I fix any decomposition of them as in the thought-free formula for any decomposition there will be a curve count and then I will sum our all possible decompositions similar idea so for any decomposition I look at a toric fun in mr I want a smart projective toric variety whose fun contains all of the rays determined by contracting the skew symmetric form along gamma i where gamma i is anything that appears in the second position so you decompose your gamma like in the previous example if your gamma is 1 1 gamma 1 could be 1 0 gamma 2 could be 0 1 and if you use if you contract your skew symmetric form you obtain a vector from both 1 0 and 0 1 and the dual and you want to construct a fun containing those so you obtain a ray uh with primitive direction these and you want a fun which contains both of these rays after your fun might not be smooths competes but then you can always add rays to it to turn it into the fun of a smooth projective torque variety and I think additional rays it's not going to change our result in the curve counting because the counts we will describe will be invariant under barational volos okay so I described the historic variety out of little for quiver which picks the mansion vector gamma and my counting problem is stated here I'm interested in counting genus zero stable maps to the historic variety with r plus one marked points r is the number of vector set appear in my decomposition and I impose these conditions that I want all p is marked points to back to the hyposurface hi where hi is the hyposurface inside the toric boundary divisor given by this concrete equation I take c to the gamma i the primitive vector for gamma i equal to punsten and I restrict this equation to my divisor c i so I get type of surfaces here so I want all my ps mapped to these surface surfaces for all but the last point and I impose the concept order for the image of all these p is to be equal to the divisibility of my vector gamma i of my vector obtained by the contraction of the skew's magic form along the line okay so this is a counting problem I just wrote in dimension two in general we will be at arbitrary dimension and dimension two this counting problem is easy to handle because if you look at your generalized space it will be zero dimensional but your toric variety will be having dimension as much as like whatever your number of vertices are at least appear skew's magic form is non degenerate and in higher dimensions this counting problem it's challenging because you will have higher dimensional modular spaces you can imagine if you have one dimension higher if you multiply this with p1 you will have a family of such curves one-dimensional family of such curves and what we did is try to make sense of this counting problem when you have higher dimensional modular spaces and obtain numbers which relate to the way that you think I am yes for every choice of dimension vector I define a different toric variety a different counting problem and eventually similarly to while defining these coefficients are summed up over all possible decompositions I will sum up over all the grammar written in the range of different counting problems the boundary divorces are in the stored variety are corresponded divorces corresponding to the spray gamma i are open for you thank you for your question okay so are there any other questions I will go ahead I will dive into safe photos of log geometry which will enable us to understand how to count such curves okay so to count such curves we need to do log geometry this is some set up it's a generalization of a theory relative from our written theory of juni developed by our grandma with shine gross fever juni was interested in counting curves whenever you have a divisor we impose some tangency conditions along the smooth divisor and the set up of a promo with shine gross fever this this count is set up set up is slightly more different it uses something called log structures so and the divisor d is not necessarily smooth in the setup it's can be simple normal crossing for more general divisors and I will tell you how to define these counts and with more general situation using log structures so to tell you what's up from our written theory is which is which is the count of such curves with tangency conditions I need to tell you what log geometry is by definition a log structure on a scheme is a sheet of planets to get over the map to the structure sheet which restricts an isomorphism on the invertible elements of the structure sheet in a minute I will give you an example there can be many log structures on a scheme and it has discreet and the non-discreet part of the log structure the non-discreet part which we denote by m bar it's just this sheet of monouns where you quotient of the invertible elements the isomorphic copy of your invertible element and only this non-discreet part discreet part which is the ghost sheet is gonna mostly be used while telling you how to do tropical geometry and in particular how to describe the tropicalization be described the tropicalization of a scheme and those with such a structure as the complex which is a unit of cons from the ghost sheet that's called eta for any generic point eta to r0 greater equal 0 and then you need to identify these cones using some generalization maps here is an example the typical log structure we will work with is called the divisorial log structure in this case when you have a scheme x and the device the d in it you can embed the complement into x the map j and you can look at define your log structure to be the sheet to find this way it is the sheet for regular functions on x which are invertible away from d and as an example if you just look at the f-line with the point zero in it all such regular functions invertible away from zero or from h times t to n where h is an invertible element if you look at the ghost sheet you just forget the h you just have t to n and you come up t to n just by considering its power and by a map similar to the logarithm so that's why particularly it's a log structure you will always use such identifications to work with non-monument generally and in this case if you look at tom and to r greater or equal zero your tropicalization will just be r greater or equal zero and in general whenever you consider a toric variety with the toric boundary device and this work structure the tropicalization will give you the toric boundary whenever you have a log structure like in a one if you have a sub scheme you can pull back your log structure and then for instance if you pull it back to zero you get a log structure on the point and here is the definition of a stable log map so whenever we have stable map c to x we end up both c and x with log structures and we we both we we both of these log schemes now over some log points and the log points over the target is based on if you were if you're working on a per meter one per meter family in a deterioration situations or not and the slope point is determined purely by the combinatorial type of your map so I will tell you which is the combinatorial typist whenever you have a map the combinatorial type is just determined by the dual intersection graph and the image of the vertices adjacent flex and the dual into the dual intersection graph together with the contact data of the marked points and given a combinatorial type here is the way to construct a monoid just as for every node of your curve you get a copy of and and for every irreducible components you get some additional mono ends so the combinatorial type gives you a mono ends purely based on this combinatorial type which we call the basic monoids in this theory a stable log map is basic if this base over the curve is equal to basic monoids just determined from the combinatorial type abramovic changro seabed through that modular space of all basic stable log maps is the delin-monford stack moreover it's a proper delin-monford stack and there's a virtual fundamental plus so the degree of the virtual fundamental plus gives you what we call log promoviton invariance okay so it's a result of nishinov and seabed you can come to you can obtain log promoviton invariance by council tropical curves and this is like sabrina we'll talk in the afternoon about some of the correspondence due to me collecting in dimension two this is a higher dimensional setup so in any higher dimensional torque variety he tells you how to count log promoviton invariance in terms of tropical curve counts what we want to do is to generalize this theorem so in this situation of nationality but your modular space is zero the emotional everything gets rigid what we want to do is generalize this theorem where we count maps whose tropicalizations are not rigid but one dimensional families we want to trace out the whole chamber by deforming our attractive flow trees so our combinatorial types will not just be the tropicalizations you work with will not just be trees but families of trees and so we need to make sense of such account so our first result shows that if you have some general hypersurface journal constraints whose tropicalization is a d minus two dimensional family where d is the number of vertices in your approval you start with to define your an American problem then if you look at all the look maps whose tropicalization is this family of tropical curves we show this is um finite number if you count appropriately we count all maps whose tropicalization whose combinatorial type is fixed and if you count this appropriately and if you generalize some classical results in algebraic geometry to the log setup you can show that zero strata in your modular space is finite and on the main theorem is saying this finite number of your plane by counting all these block maps whose tropicalization of the family of curves tropical curves is equal to the coefficients appearing in our flow tree formula the proof uses the generation arguments the degenerates there is a gluing formula to describe gluing for plot maps and toric varieties the main technicality of this proof is to relate the numbers appearing from the gluing formula to numbers that we calculate using our skew symmetric earlier other form and the river dt theory and just some final minutes i will just make a remark in progress we're currently working on our correspondence relating directly to the dt variance of block curves and cluster variety not only to the coefficients appearing and the flow tree formula as such you can obtain a toric variety with the data of hyper surfaces from the data of the pillar if you blow up these hyper surfaces and look at the complement of the strict transform of d that gives you a cluster variety and then this cluster variety you can look at counts of curves which touch the boundary at the single point what you're currently writing up is a correspondence between all the counts of such avankers and river dt variance directly and the situation when the attractive dt variance are trivial this is a heuristic picture your cluster variety this is my last slide admits the map to the space of stability conditions you can imagine the projection of your avankers into this space and these have to do with the walls appearing in the stability space of fluidity variance so heuristically these were expected to be related such counts of curves with fluidity variance and i will just stop here thank you very much for the for your attention you get the toric variety from the data of the curve gets for each decomposition of your dimension vector so you fix the decomposition of your dimension vector and once you fix this decomposition so or for instance if you have two one you can fix the decomposition one zero plus one zero plus one zero and this will be a situation where things are not this strong and now you need to go to one hard dimension and modify these vectors a little bit but if everything is this strong you construct the toric variety whose rays are generated by these vectors that appear under the composition so once you fix the decomposition you construct the toric fan whose rays contain these vectors appearing under the composition and it might not be smooth if it's say or complete if it's say one one you have just one zero plus zero one that's not corresponding to a project of toric variety so we need to add more rays in general to these things but i think more rays because of some invariance results of flochromobit and invariance under birational transformation it's not going to change the result so fixing the dimension vector determines a bunch of rays in the fan of the toric variety and be at some things to determine the exact thread. Yes, that's true actually so you need to consider all possible decomposition so okay so as you said if you look at local pt for a quick summation vector gamma there are like pt invariance for some stability and you're saying that we can obtain these pt invariance as a sum of all possible decompositions so for all possible decompositions there are for any decomposition there are coefficients and for any coefficients we are relating these to a particular toric variety be obtained from this coefficient and the gromobit and invariance of this particular toric variety and be sum of all these all these different gromobit and invariance and different toric varieties to obtain the original pt invariance which in this case be also not corresponding to the pt invariance yeah so for instance while drawing the sara correspondence we do know that by by a paper of friction these pt variants are calculated by law forcing conservation stability diagram and yes so on the left hand side the the question is not going to depend on this generic stability parameter teto so fixing teto just brings you inside one chamber and the left hand side it's not it's not going to depend on this particular stability for me so you start with um maybe i can tell you more about that later yes so yeah so you need to keep track of signs that's correct yes like the numbers so i'll show you for instance are just obtained from on particular teto when things are not reveal and if you change the teto to another chamber then you get a different counting problem but we fixed it up priori and we keep track of signs to yeah that's a very good question so we are yeah to show this main thing we're showing that you can relate the combinatorics appearing in the different fit ups of wall structures so the main thing is to relate these wall structures and different fit ups so okay in this question we do match the wall structure appearing in the British and stability condition with the wall structure appearing probably count a one curse and the main result we used to be able to do this matching is that are in our paper with mark cross we showed that you can purely from a combinatorial scattering diagram calculate these and that combinatorial wall structure we can match with this side um that can also be related to a more nice scale wall structure