 So I'm going to talk to you about two constructions of the Louisville theory on the sphere with three insertion points. Now you have so in the previous talk, when you have a measure of Louisville, you can have insertion points. I will explain what is the analogy of this in the map. Well, there's two constructions of random measure. One is by dk rv that you have just saw. They give explicit formulas for correlation functions. We must have more than three insertion points, and they're suitable for compact surfaces. So you have seen in the previous talk that we can construct this kind of theory for a lot of surfaces. Now there's another construction by DuPontier, Miller, and Sheffield. For example, on the sphere, they have less than two insertions with the same weight, but in the case of the pure gravity, in the case where for this specific value of gamma, they have a matrix instead of a measure. So they have something that is more powerful in some sense, and they have couplings with classical probability objects such as SLE, and they're suitable for non-compact surfaces. I hope that the reason why one is suitable for compact surfaces and the other one is suitable for non-compact surfaces will be clear in the end. So the goal of our work, I'm sorry that I have to present my co-authors, this is a joint work with Yuan Ahu and Sun Xin. So the goal of our work is to find a link between these two constructions. It was not clear in the beginning that, for example, on the sphere, they would give the same kind of limit. To begin with, we'll do something algebraic of geometry, of simple geometry. So we'll first talk about multi-transformations. Why we talk about multi-transformations? Because all the theory, they have some intuition from the large random maps that this should be the limit when you embed a very large random map onto the sphere. So when you embed a map on the sphere, you do it with a conformal map. So we'll start with the conformal automorphisms of the sphere of the complex plane. All the multi-transforms look like this on the complex plane. So we have four coefficients and one relation between them. You have three degrees of liberty. The first exercise is to find all transformations that fix three points on the sphere. So there are three degrees of liberty. You must fix three points. There is only one transformation. It's the identity. Solution to the first exercise, you only have one map, is the identity. Solution to the second exercise, now if you want to fix two points on the sphere, you have a family of applications. For example, all applications of this form, all applications of this form will fix 2.0 infinity. So I have a large family of applications and so that would be useful very soon. So we will talk about the conjecture of random map and the measure on the sphere. The conjecture is that if you take a very large random map and you choose three vertices uniformly on the map and you map these three vertices to 0, 1, infinity on the sphere, then in the limit you should get a random measure. So this is a simulation by Truyan. If you think about every one of these triangles, if you take every one of the triangles to have a volume one and you renormalize at the end, you would give some kind of random measure on the sphere. The point is because of what we have said before, if you fix three points, you have one random measure in the end. The random measure is well-defined because the embedding is well-defined. There is a unique way to embed your map on a sphere. So what happens if you want to fix two points? Now the embedding is well-defined because you can actually pass from one embedding to another embedding by this map. You can push forward your measure by one of these multiplications and you get another embedding. So if you want to describe the limits of what is the limit of the random maps, actually you must define simultaneously a family of measures. And two measures in this family are linked by some push-forward relation by this action. So in the construction, if you want to construct this kind of object, the construction must be invariant under this action. If you want to get equivalence class in the end, the construction must be in the equivalence class in the beginning. So we'll talk about two constructions now. The first construction we just saw and the other construction that corresponds to the k was two points. So this is the part that's a little technical. I'll just give you some formulas, definitions and formulas. I will not explain what it is. So this was the Louisville field in the previous talk. And the way of adding insertions, if those weights are gamma, it's like as if you've chosen three points. When you choose uniformly a point, it corresponds to insertion of weight gamma. So these are point of insertions and this lot singularities is the picture that you just saw with singularities as three points. So this is the Louisville field with insertions and you can define the volume associated to it by taking the exponential. Our work focused on the case of univolume measure. So somewhere you have to renormalize the measure and you have to do some shifting of the probability but I will not enter in detail. The point is you have something as well defined here and everything can be calculated. For example, moments at a certain region, for example, every estimation you can calculate in a very explicit form. We will have three points. Now we'll talk about the two-point construction. This is a very interesting construction in the sense that they use some kind of encoding of your surface. You can construct your surface by using a basal process somewhere. So the idea behind this is that if you take the function that is defined on a complex plane, you can decompose it into two parts. First, I will tell you just the average of this function on each of these circles. If I only tell you the average of a function on each one of these circles, you don't know much about this function. If you want to know entirely the function, you must add fluctuations of this function on every one of these circles. So if you combine these two parts, you will get your function or distribution. Now the point of this construction is that you can sample actually these two parts in an independent way. And we can give explicit constructions, well, explicit ways to sample these two parts. Now I will not talk about the fluctuation part, because it's given by some abstract manner, but it's not very difficult. And I'll talk about this part, the average of this. And we will see why the construction is invariant under this action. So what you do, well, this is a little bit hard to read, perhaps at first view. But when you select the basal distortion and you take a lot of this basal distortion, it will look like something like this. So this is on R. It will look like a process like this. It will look like something that we call the two-sided drifted Brownian motion. It will look like a Brownian motion, but with one drift here and one drift here. Now the rule is that we will parametrize this so that it will look like a Brownian motion. But if you think about it, I can translate my picture by a constant. And it does not change the fact that it has quadratic variation. So actually, when you do this, you are defining a process that is invariant under this kind of translation. So if you take this picture back to the sphere or the whole plane, you are defining an object that is invariant under this action, because the law here does not depend on this special choice. So our theorem is the following. We can pass from the two-point definition to the three-point definition, or we can pass from the three-point definition to the two-point definition. How we do it, we'll use, well, to explain this, I'll use some intuition from the random map. So I recall that a two-point case should correspond to the case where you pick two vertices on your random map and you embed it to the sphere. And this three-point case should correspond to where you choose three vertices and you embed it to the sphere. So what we do is that we take the two-point, we take a measure in the equivalence class of two points, we'll pick a third point according to this measure. It's like as if you're picking a third point uniformly among all vertices. But this will give you something that is still in the equivalence class. If you want to pass to a measure, we must use this kind of push-forward operation to fix these three points. We'll fix it to 0, 1, and infinity. You push forward, you measure by this action, and you should get the measure with three points. So this is the intuition behind all that we do. But the difficulty here is that this conformal map is a random map, actually. So it's a little bit non-trivial to characterize this law to describe it. So a consequence of this theorem that you just saw, from passing to two points to three points, is that you take these three points, say with insertion points at 0, 1, and infinity, you take away one of those points. So you forget about the fact that I have one point at 0. You just say that I have two points at 0, I have 1.1, and you just say that I have two points at 0, infinity. Forget about one point. You apply this kind of lateral shift, or you apply this action to your measure so that you pass the equivalence class. And then you get the equivalence measure of the two-point construction. So that's about it. I hope it's not very long. And this was a part of this work was done at the Newton Institute. And I want to thank them for that, and thank you all for listening. Thank you very much. Any questions for Richard? And you can just say a few words about the proof. Do you take the explicit construction and show that it satisfies some actions? Maybe I'll be... Something about the proof? Yes. Yes, OK. If you wish, in the beginning, what we wanted to do is very simple in the beginning. What we wanted to do, you take this field, you do this decomposition, and you see if you have the same part for the radial part. OK. But the only thing is that this thing is well-defined when you only have two download insertions. Well, you should become clear because you don't have one measure, but you have several measures. So we can't do this for two points, but if you do this calculation, actually we'll find something like this. So about the proof that we do it, we actually use another description in the DMS paper. They approach this kind of field by approximations. So imagine you're on a sphere, you have three insertion points. So you can take away a little region. You can take away a little region from this. And you can consider some kind of GFF defined on this domain. The thing is, if the boundary condition here is well-chosen, it will converge to its free point. So this is the crucial lemma in our proof that we use to identify this true process. Any other questions? In this case, thank you again.