 So this is a joint work with with Matthias Enrique Masson and Nick McLeary and it's going to be, as I said, it's going to be about real motion per equations, but I want to start in this concrete complex geometric problem. So we're going to fix generic homogenous polynomial of degree D plus two in D plus two variables. And then we fix a number s, a non-zero complex number, and then we look at the zero set in PD plus one of this, this equation here. So you see I used my polynomial f here and then I have this product of the coordinate functions as well. And I call this xs and the question I want to ask here is what happens then as s tends to zero. I have a so in some sense I was careful up here to say that series, non-zero complex number. The idea of packaging this is to say that we have some some some object x which really lies into in the product here of C star PD plus one. And this central fiber we have here is going to play a central role in what comes. The question is then what happens as this tends to zero and I in the in the one dimensional case I have this, this image of what's happening in the one dimensional case we have to have elliptic curves. So we have this Torah. And then as s tends to zero or at least as when s equals to zero we just have this union of three hyperplanes. And this is somehow the process but I want to I want to be a bit more precise about the question here. So, we have an embedding into PD plus one. That means we have a class, a particular class and all of my excess, which means, and they're also going to be collaborative. So each of these I can equip them with a collaborative metric. And then as the question what happens as s tends to zero in a metric sense. And when I say also metric only here. I mean, not just metric spaces but also in terms of the collaborative structures on these miracles. And there are two conjecture stand that predicts what's going to happen when s tends to sense to zero. So for the storm in the house of conjecture. And then it says that for small for small s these Calabria manifolds are going to be vibrations over something small vibrations over something smaller in some sense much simpler. So for small s these Calabria manifolds admits a special agrarian torus vibration. And if we look at the one dimensional case here, we just have elliptic curves, and then these vibrations are very simple. So, yeah, if we have our elliptic curve here, and this just corresponds to the fact that we have an s one vibration over s one. We can also draw this, of course, as a quotient of the complex plane. So we have. So each excess time is a quotient of the complex thing by the group generated by one and some generated towel towel depends on us. So something like this, and then we get our vibration here but I guess projecting down. So we have a probation over s one. So s tends to zero this is going to be generated in some sense and what's going to happen here is that one of the generators of this that is going to converge to zero, or at least up to rescaping. It's going to converge to zero. So one of these picture on the right here is that when we represent our elliptic curves in this way. The collaborative metrics are very visible. The collaborative metrics are sort of the ones you get by just inheriting the metric from C. So, if this generator here tends to zero that would somehow correspond to the these fibers collapsing. So, so there is. That's another. There's another conductor, which says that as tends to zero as s tends to zero. The size of these fibers is going to be much smaller compared to the base. So up to normalization if we normalize to make sure the diameter of the spaces is constant, then these fibers collapse. So, in this case we started with an elliptic curve and then in the limit we got yes, yes s one. And some more precisely than or did this is often called the conserved tribal man conjecture or the gross Wilson conjecture. So that says that up to normalization haven't written up to normalization but up to normalization, these metrics base will grow more house or convert to a metric of one pair type on the base. So let me say just a few things about what these things mean here. We're going to call the base here. So we start with something of complex dimension D and the base is going to have, it's going to have real dimension D. And part of, well part of the conjecture is that at least in the general case, the base is going to be topologically sphere. Moreover, it's going to be. So we're also somehow expected that this is going to have something called an integral fine structure with some more structure to it as a topological sphere. What do you mean by the base here. Yes. So, in this is in a conjectural sense. This is what this is in a conjectural sense. So this is what is expected them. So it's an addition. Exactly. There's also, well, more things expected about the base. One thing that's expected about the base is that it's going to be the dual intersection context of something, which in this case is just going to be this central fiber we had before. The dual intersection complex. Well, it's the central fiber, let's say we draw this in again equals one. Then we are at the central fiber we had these three hyper planes. And to draw the dual intersection intersection complex of this. I start by looking at all the intersections, the deepest intersections. I have them here. Sorry, that's not what I do for each for each of these devices. H1H2 and H3. I, I put, I put a bird is here. And then for each intersection. I add a edge between them. So in this case, then the this dual intersection complex it is the boundary of a triangle. In general, with the general dimension, we're going to get a simplex boundary of a simplex. Thank you. Moreover, then there's this word called motion pair. And by that, so we're going to have some, some space is going to be equipped with some kind of remanual metric. And this metric to say that is of motion pair type means that it's going to be locally of the form locally something which is the second derivative of a convex function. So with a convex function which actually solves them a real motion pair equation. So this, this is going to be defined on the base. The base is the dual graph or is expected to be the dual graph. So here what we have here then is a motion pair equation on the boundary of the triangle. Does that answer your question. That's a very good follow up question, like, or you mean in what sense can we make sense of this equation on this type of object. So this is, this is a D equals to one. So we have. So this is the equal to one so the boundary is going to have dimension D. Yeah, yeah, let me suggest a constant. More questions. Yes. Is there also. Yeah. So I guess at this point we forget about the vibration and just focus on this. Right. Yeah, you could you could do that. Yeah, at this point in this slide. Yeah. Yeah. So, one one more thing I should mention as well is that with respect to this conjecture about these vibrations is that the, the, it's expected that there is what will be singular fibers in these vibrations. So, in some sense what what's in the very strongest form of this conjecture you would, you want a special Lagrangian tourist vibration on some kind of perhaps a co dimension to subset or so. So, okay, so this is really the motivation that's where this thing. And then what I'm going to talk about the rest of this talk is the real motion per equations on this. Let me go back one one more time here. So we have the we have this kind of. We have a few different descriptions of the base here on the one hand we have a description of what the policy is supposed to be. There's also going to be something called an integral affine structure on it. There's also some kind of simplicial structure on it here. And I'll try to, I'll try to kind of play in a little bit about how these things fit together. And I'll also talk about the real motion per equation on this. And kind of the goal of the talk is to, I want to get is the following result here explained that the real motion per equation on the boundary of the unit simplex admits a solution. And this is going to be the data defining my real motion per equation is going to have to be symmetric that's a necessary condition. And I'll also say something about of stuff I will not talk about, but which is important for the context. So solutions to this equation. These are somehow solutions to an honest real motion per equation. They induce a nonarchimedean metrics which solve the nonarchimedean motion per equation and I'm not trying to say that the nonarchimedean motion per equation is not honest but there is still a description of solutions to the nonarchimedean motion per equation is quite significant. In particular, having such a description of solutions to the nonarchimedean motion per equation allows essentially allows us to prove a generic version of the of this sys connection. So if you say, so with this you can construct the generic special Lagrangian tourist vibration. So by generic, I mean that they are not, they're not defined on a co dimension to or then subsets but really, you have a vibration on a subset of 99% of the nonarchimedean motion per equation. So this, this last part here is really ideas and results by young Lee, in a couple of papers from 2020. And, well, yeah, so we have this image down here. So on the left side here we have this nonarchimedean world. On the right we have what I'll talk about now which is something called integral affine manifolds, or tropical manifolds. And you, what must he has talked about this Monday was the nonarchimedean world and motion per equations in the nonarchimedean world. And what I'm going to. So he kind of started building a bridge here. So what I'm going to do is I'm going to start in this world is integral affine manifolds. I'm going to start building a bridge in the other direction. And I'm not quite going to be able to connect these two connecting these two within some involved telling this story as well, which I'm excluding you know, but I hope to kind of illustrate how different flavor these two subjects are. And also the fact that somehow by connecting these two things you can, it can be quite quite useful to connect. Okay. So this, well, yeah, so what is the real motion per equation. So, when you talk about real motion per equation you usually start with some kind of with with a convex function to find an arm or a subset of arm. And, let's say that your current that your function is smooth. That would mean that, yeah, if you function is smooth then you define the real motion per just as the determinant of the second derivative of your function. And I'm adding volume form here because I want my real motion pair to be a measure. This is the classical definition. And then there is also the weak definition which is, in some sense, quite classical as well it's due to Alexander. So here we as need fight to be a lower semi continuous complex function. And then the motion per measure of five is defined by saying that if I take a measure will set and plug it into motion per measure then what I get is exactly the volume of the gradient image of a that should I say, should I say a person. Yeah. Yeah. So, so there could be an issue on the. Okay, you can think about continuous. Yeah, definitely. So in, yeah, in general if you have, you could have the clouds taking infinite values in particular on the boundary of your domain if you have a domain, you, you want it to be over semi continuous. I'm not going to say more about this. So there's one observation I want to make that is that this is SL SL SL DC invariant. And by that I mean that if I, it's invariant on the maps on this for this one. There's, there's, there's many ways to see that you could sort of get compute explicitly what you get if you apply a fine transformation here. You could also look at the week definition. And if I apply an affine transformation to RM, then it's going to correspond to applying the inverse transformation to RM duo. And if my affine transformation is volume preserving. It's not going to change the volume in the duo. So so this this is really what we have here is really. So we cannot really talk about real motion per equations just on smooth manifolds, we need some extra structure, and what the extra structure we need is going to be related to this. So the extra structure we need is an integral affine structure. So if we have a topological manifold X, and these the real dimension effects. Then an integral affine structure is a special Atlas on this topological manifold, where the transition functions line is this group. Equivalently, it's a smooth structure together with a flat torsion free connection with all on all on a means. So, just one, there's one observation we can make them is that we have, if you have this kind of structure then you know how about then you have a connection which means you have a notion of your desix. You have a notion of parallel transport, but you have no notion of distance. So in particular these connections does not have to come from a Romanian metric. So it's a, that sense it's a much more general object and every man and manifold, but it's also special in the sense that your connection is flat and portion free. So this condition that is flat and torsion free is going to mean that the whole on me is going to be locally constant. But you can have global autonomy on your manifold. And the global autonomy could be sort of it could be outside of door talking on group, which means that if you're parallel transport vectors around. They might grow in size, when you return to the point where you started that can give some kind of disturbing phenomena. So, yeah, and so not quite so I'll get to I'll get to I'll actually as an example I'm going to write down an explicit integral affine structure on the unit boundary of the that's a that's a very simple. Yes. Exactly. Those properties. I don't, I don't know. I mean the. Yeah, I don't know. Okay, so right, we would also talk about single integral affine structures. So that's sort of loosely connected with the fact that the this special around some tourist vibrations are expected to have singularities. It's also expected and that the integral affine structure is going to have singularities so this particular atlas by mistake I've written tropical atlas but I mean integral affine atlas. It doesn't cover entire x cover everything except for a small set. And so, by small set here. I mean, co dimension to. So, in this general form that this is a very general object here we don't know essentially we don't know anything about the singularities more than the co dimension. Okay, and an important point is that every reflexive political admits a singular tropical. Admits a singular tropical. So tropical is a slightly stronger notion than integral, integral affine. So, so every reflexive polytope admits a singular integral affine structure on its boundary. So, so I'm going to reflexive polytope is a polytope, a lattice polytope whose dual is also a lattice. So, so I'm not necessarily smooth. Okay. And I talked before about the unit simplex. From the perspective of as a lattice polytope they're actually more than there are several units in places. I'm going to let Delta here be the, in some sense, this standard unit simplex or at least a standard unit simplex if you come from complex geometry. So Delta here is really the polytope corresponding to D plus one dimensional productive space. The simplex we're interested in is the dual of this. So it's the, yeah. Right. And then I promised you I was going to give you an explicit integral affine or actually a tropical lattice on the boundary of this unit simplex and I'm going to do this in dimension. When the dimension of the unit simplex is free which means the boundary of the unit simplex has dimension to this, this that's not going to cover the entire unit simplex is going to be some singular points and these singular points are going to be the midpoints of the edges here. So that's where we have our singularity would maybe you would expect that first glance at the singular point would be the vertical something like that. But in fact the singularities. So we're expected to always have co-dimension to so they cannot always be the vertices. In fact they're going to be something with sort of splits. Something that splits co-dimension one and just piece. And we're going to get one coordinate chart for each vertex of our holiday. So if you look at the vertex vertex V1, then my quarter chart is going to cover all the blue stuff here. So the interior of all the faces that contains V1 together with all these interiors together with half of these edges here. So we end up with this kind of shape. And if you do this for all the vertices, then you're going to exclude exactly the midpoints of the edges. So let's go down what the coordinate functions are. If you have a vertex, then you can look, then you can pick. Well, the vertex is going to be a lattice vector. And then you can take its orthogonal complement. And you take two generators for this or you take generators for this orthogonal complement, and this is going to define what it defines a function. It defines functions on this boundary and that does those are exactly the functions you're using. In some sense you're just sort of looking at it from the perspective we have here and just projecting it down onto R2. But you're doing it in a invariant manner, invariant with respect to SLDC. And the interesting thing with this is that this is completely explicit so you can compute the whole on them here. And as I said before the whole on the means locally constant. But as I circle around a singular point, I will get something non trivial. And if you compute the whole on me there, you get the following matrix 1401. And here you can actually hear you can see some of the complex geometry because this is this matrix to the power of four. And if you take four multiplied by the number of singular points, you get 24, which is the number of singular fibers in a generic key free vibration over as to. So in the generic case here, you get sort of 24 singular points or singular fibers in the vibration. And what's happened here is that they have come together in groups of four. And I did I start on time or did I start. So what we want to do then is you want to solve the most real motion per equation of this. So we want sort of for each chart, we want to have a local convex function, which solves the motion per equation in this chart. And then we want, when I take these local convex functions and put them together, I want them to sort of the second derivative should define a remaining metric, these metrics of motion per side. So they in some sense there's a compatibility right here on the overlap of the charts these local complex functions need to need to agree up to a certain man. So again, I wanted to say something. Something before I sort of state the main theorem. I think the main theorem is really that we can solve this equation. And then there is some technical aspects of how to actually state the theorem. So I wanted to take the last couple of minutes just to say something about how we solve it, because I was talking about these two words before these two banks of this river we had a nonarchemedian world, and we had the this integral affine world. You could say lacking a better word you could say that the nonarchemedian world is a bit non explicit form is serious. There are powerful theorems there that tells you that you can always solve the motion per equation, but it's not quite clear if the motion, the nonarchemedian motion per equation is for example, locally defined partial differential equation or something like that. On the other hand, in this integral affine world things are more concrete and the motion per equations are actual real honest motion per equations, but this integral affine world is somehow too loose, somehow too little structure to work with. You would expect to be able to solve these problems you have to inject some kind of algebraic information, and a priori it's not quite clear how to do that into the integral affine world. And then but this with this slide and I want to sort of speculate or say something how we did it and also, you know, maybe that can provide a speculation about how it can be done in general. So, I think this has this this functional we've seen it before. Yes. So, or. Yeah, that's a great question what is my measure you on this simplex that I'm talking about. On the one hand, we have this integral affine structure so I have the transition functions in my Atlas, their volume per second. That means there is up to constant there is a well defined measure on this. I just take the big measure and then I need to normalize it. I just take the big measure in one chart and it's going to be compatible. On the other hand, if you inject is not an Archimedean perspective. This is also the. In a sense it's a weak limit of collaboration volume forms. Right. So we looked at this functional before. In some sense, this is a way to find the solutions to the real motion per equations on RM. We, it's a function, the dependent on fire here. We asked if we want to solve this equation here. We integrate fire against me and then we integrate five star against. I'm calling it new P here. So, so new P here is the. If you're for example in the torque setting this is the uniform measure or the big measure restricted to a polytope. So this is how you solve this, how you solve motion per equations in in a torque setting. I understand, I understand the question. There was not, not intentional though. This is, you should all be D and and yeah, and I'll keep this as an M. Okay. That's solving the motion per equation on RM. Okay, so, so we want to solve the real motion per equation on the boundary of a simplex. So, so then it makes sense to replace this with just the boundary of the simplex. More precisely, I think I noted delta dual where we wanted to solve it. But on the other hand we have these two objects which is sort of unclear how to replace. How do we replace the dual. And how do we replace the parent here, which we need to define this, this. Legendre transform, which is essential here. This, this, if we ask somebody. Many people if you ask them what this is they would not say that this is this comes from Toric geometry anything like that they would say that this is something connected to optimal transport. And this is the contour of its dual optimal transport problem. And if we look at that we can sort of get some inspiration because the optimal transport problem I'll just quickly say what kind of what what what data defines an optimal transport problem. A source space together with the probability measure and a target space together with the probability measure and some kind of cost and a cost function on on the product here, then the optimal transport consists of sort of transporting the measure new to the trans the measure new somehow minimizing this cost. But the details are not so important here I just want to make an analogy. Because the problem we had up here was that we had no way to we couldn't figure out like, what is the view of this integral affine world. And also what is the pairing in this integral affine world. Well of optimal transport these have really natural answers. The dual that's the target space. In other words, this why. The pairing that's just the cost function. So, optimal transport give this way of thinking of RD and are the dual as something more. As something more general. So, what we will do here is we will, we're gonna. Yeah, so the question is then what is a suitable target space, and it turns out that the suitable target space in this case is just the boundary of the. The units, well, of the of the other units. So, I'm solving my equation on the boundary of. We have two simplexes. We have a standard simplex, which is delta and we're solving the motion pair on this dual delta and the target space is just going to be the boundary of delta. And the pairing where we have a natural pairing between these two, so we can just keep this here strictly speaking is sort of in the language of optimal transport you have to, because you don't want to take the supreme over all of RD here then you just want to take the supreme over this space and then then it's usually called a C transform in optimal transport so I'll have to replace this with a C. But apart from that, this is really. When we're solving this equation.