 Hello everyone. My talk is the webs and the tropical coordinate on the surfaces. So it's a joint work with Daniel Douglas at EO. So a web, a tree web is oriented graph on the surface with trivalence interior vertices such that the tree oriented edges at each interior vertex point outward or inward. So the study of tree webs is started from 19th by Kuberger. He introduced tree webs or he called spiders to study the SL3 environment tensor products of irreducible representations of SL3. And it fits into so-called pivodab category. And I prefer to avoid these abstract mathematics but engage into more theoretical sense. So here is a picture of tree webs on this R square. So you can see there's only trivalence vertices in the interior and all the points, interior vertices points outward or inward. So you can think of these points pointing outward are black vertices and the the points point inward are black vertices. So it's like you move from one side to the other. You go alternatively the black vertices and the white vertices. So people study so-called networks or spin networks also have these similar stuff. There are many fields of mathematics relevant to this kind of picture, not only the study of representation theory. Here is another example of webs on the west puncture torus. You can think of the punctures at the corners and you glue the two sides, two opposite sides together and you get a torus and puncture at these vertices. And you can see there are circles and there are trivalence graphs. So Fountain and Cosmere and Kuberberg relate the webs to the so-called affine building in geometric representation theory. So first picture is a web on a disk and the second picture is the dual graph of this web. And this dual graph so-called discolored is a relevant to this affine building. And another research is a direction is by Fomin and Koliasky, they investigated cluster algebra structure of classical rings of Neuronal SO3. It's also run through the on the work of Kuberberg and then they found the basis of webs to parametrize the these tensor environments. And this is an example of Grassmanian 2, 4. And then because it's a graph, you can think of it's a skin algebra on the surface. So Shikoran from a study of SO3 skin algebra on the surface means that links graphs on the surface with some relations. So one, two, three are the relations. The minus in these relations equal to means that the miss term is equal to zero and the miss term also equal to zero. And when you have the two, this third relation will have two curves crossing each other, one on the top of the other, then it will resolve into the sum of these two terms in this way and in this way. So our approach to study the trail webs is actually inspirated by in the folk and the Gonshov duality and the so-called mirror symmetry on the higher tectonic space. So without knowing all this abstract work, I prefer to use in pictures to show what it means. So firstly, I give you a topological surface, this picture on the bottom. So this is a tris puncture torus and the genus is the number of miss holes and the holes is the number of miss circles on the boundary. And when I say it's a topological surface, miss holes can be filled by a casc region. So this casc region goes to infinity. So these two things are equivalent in sense of topology because we think about continuous motion of miss pictures in the space. So we study the modular space to parallel modular spaces was a variation of the repetition variety. The repetition variety can be think of you have some geometric structure on the surface, you put a matrix on the surface and this pair of modular spaces have a duality. It means that the ring of regular function of one space has a canonical linear basis, matching class group is currently parametrized by the top of point of the dual space. So without knowing this abstract word, there is only one word to see. It says that the observables of the one space are parametrized by the lattice at the infinity of this dual. So without considering the matching class grouping equivalence, the conjecture is actually so by Gross, Hecken, Keohen, Konsevich and also by Ganshaw and Shen in this case. So what I'm going to present our work is we follow a beautiful solution originally by Falk and Ganshaw for the simplest case G equal to Pdl2 using substance transversely made nominations. So for G equal to Pdl2 on the surface you just put a hyperbolic matrix which is a negative one curved matrix. So firstly we can think the question in your mind is that in our earth it's a sphere, topologically it's a sphere. There are many many cast means black dots on the earth and you want to draw the webs on the earth and you can move these webs around smoothly but you cannot cross these dots. So how many are there these webs on earth? How can we classify all these kind of webs? So this is the question in our mind. So here I give an ideal translation of this model. So here is a false puncture sphere and the ideal translation on this false puncture sphere is just the arcs connecting these dots and there is no two curves homotopy to each other. You cannot move you cannot move smoothly from one curve to the other connecting to these dots. For example you cannot move from here to here and this is ideal translation and then we flatten the edges. For each edge we flatten it to be two edges. So actually we double the edges and then we have triangles ideal triangles in the interior and then we have bi-gones in between the two ideal triangles and then we put the webs on this surface and we can actually put it in good position respect to this ideal translation. So how we do that? So when we have these blue eyes represented the web, when we have this kind of picture because the intersection number of this blue line with this bi-gones fold we can move these dots smoothly to here to get intersection number only two. So we decrease the intersection numbers in this way and actually we can do it so that the intersection number is minimal and all over the good position also means that the restriction of the web to each bi-gones is a ladder. The ladder is something like this. So we put sin plus or minus on each side of this bi-gones and there is a way to connecting the two sides uniquely by a single bi-lider and then all over the restriction of the web to each ideal triangle is a oriented honeycomb with oriented arcs in the corner. It looks like it looks like the picture below. You have a lot of honey-gones in the middle and you have a corner arcs here and actually generally you have some arcs here and the restriction, the good positions is that we put edges, this edge into the bi-gones and then push this honeycomb into the triangles and topologically we can do that smoothly. So here is a picture in the in a square how the picture looks like generally. So each triangle is a honeycomb with a corner arcs and in the middle is a ladder and the way we parametrize these kind of geometry object is by assigning numbers for each ideal triangle. We assign seven numbers and the way we assign it is by putting, if we have a arc going from here to here, we put one on the left and put two on the right and if we have a junction, a trigonal vertex in the middle if it goes in, then we put one on the left and two on the right and the middle is three and if it goes out, we put one on the left and two on the right and middle is three and then we glue all these numbers together on each ideal edge is identical and actually we assign this a web because we can represent this kind of honeycomb by this trivalent vertex, simplified version trivalent vertex with an integer number x in the middle. So if you need to go out, we put x equal to the number of intersection numbers of this edges go out with this edge. Here is tree, we have the tree arrows going this way and x we put tree. If it goes in, we put minus tree and then we count the number of corner arcs then we get the seven numbers corresponding to these seven numbers and actually the assignment here is bijective with this assignment here. So next WS be the space of reduced tree webs up to equivalence. Here reduced means that you cannot resolve by the skin relation mentioned before anymore. These webs are the basic ones. The other webs are just the combination of these basic webs and we show that given an ideal translation, there is a well-defended bijection from these webs to a polyhedral convex cone. So the polyhedral convex cone is studied very frequently in the computer science in linear programming. So here the example is just a lattice in the polyhedral convex cone is sub-ordinary to a family of inequalities, so called polycentile inequalities. So this lattice can be understood as in the tropical positive real points of a modular space. So actually you can think about this WS observables of the modular space and the city at the top of a lattice at the infinity of the dual of the modular space. And also this equivalence is very natural in the sense that it is a mapping class group recurrent. So how to describe these inequalities? We can use this Ramby's drawing inside the each triangle and we can put a polynomial respect to this Ramby's. And then we can obtain the collision tau inequalities by taking the tropicalization of this function. These are just the tropicalization just by taking the the plus with a minimum and the multiplication with a plus and division with a minus. So to prove the mapping class group recurrence we're using actually the new unique here about the basis of the lattice in linear programming such that the lattice is spanned by this basis with no negative coefficients. So we found actually we found a big digmization of this lattice of this convex polyhedral cone with 14 two sectors. So each sector is a linear isomorphic to a cut-tron. Each sector corresponding to 12 webs. Thus we will define the total type of the huge sector by these 12 webs. And actually the interior of the sectors are disjoint. And the sectors are separated by co-dimension one walls. The topological type of two sectors differ by only one tree webs if and only if we're sharing a co-dimension one wall. So here is a picture for the reason why there are exactly 14 two sectors. We don't know the deep reason why 14 two but we we get 14 two by computation of these geometric pictures respecting these geometric pictures. So actually it's unknown the relation among the web basis and the concha machine spaces through geometry, sataki and also across hacking, kill, conserving, data function bases. There are many further work to do after this. Once in relation to the application mathematics is that maybe it's mostly written to so-called optimal transports. So you can see the details in the archive version of this work. Thank you.