 Thank you. So my work is about deformed random fields, so I will first introduce this model. All the random fields that I consider in my work are defined on the plane and they take real values. To define a deformed random field, you need two elements. First, a random field X satisfying invariant properties. So it is stationary and isotropic, which means that its law is invariant under any translation or rotation in the plane. And so I call X the underlying field and I write C its covariant function. And then the second element is deterministic application theta. So from the plane to itself, which is objective, continuous with a continuous inverse, and that sends 0 to 0. And I call theta a deformation. And to define, so now I can define my deformed field X theta as the composition of X with theta. So concerning this model, I ask myself two kinds of questions. First, when does a deformed field satisfy invariant properties? And second, an inverse problem, that is to say, in the case when theta is unknown and when we only have at our disposal observations of X theta, what can we say about the deformation theta? Is it possible to identify it? But during this talk, I think I will only have time for the first question. So here are two simulations. So a realization of a random field can be represented by a surface and here are the level sets of these surfaces. So on the left, you have an example of an underlying field. So here it's Gaussian with a specific covariance function. And so it is stationary and isotropic. And on the right, you have a deformed field constructed with the same field X represented here and with a specific deformation theta. And so as you can see on the image, the deformed field here is not stationary nor isotropic, so no invariant properties and no translations or rotations. So generally speaking, the deformed fields are not stationary nor isotropic, but in certain cases they are. For stationarity, it's not difficult to see that the deformations that preserve stationarity are the linear deformations from the plane to itself. And for isotropy, I'm going to explain in the following which deformations preserve isotropy. Just a word on some references about deformed fields. So they were first introduced in special statistics. In fact, they are precisely interesting because they can model phenomena or structures that are not stationary or isotropic. They are also used in image analysis and they can be used in certain fields in physics such as cosmology. Here I recall my hypothesis. So in the following slide, the only regularity assumption on the deformation theta is continuity and continuity of the inverse of theta. So my first problem is which are the deformations theta that make the deformed field isotropic for any underlying field X. So note that this problem is different from another question, which is if we fix an underlying field X, which are the deformations theta that make X theta isotropic. So for now, I focus on this first problem, but later I will answer to this second question. And there is an easy example of deformation satisfying this condition. I mean the rotations in R2 because X is isotropic. So X composed with a rotation has the same low as X. And so it is isotropic. So just to give an idea of how to solve this problem, I assume that X theta is isotropic. So its covariance function satisfies an invariant property and the rotations. So I've written it here. And the covariance function of X theta is linked to the covariance function of X, denoted by C, because X theta is simply X composed with theta. And so finally I have this identity. And I can choose a specific covariance function because my field X is not fixed for this problem. And so this way I get an identity that only concerns theta. And then using a polar representation, I deduce the answer to my program. So the deformations preserving isotropy for any underlying field X are what I call spiral deformations. And I'm going to define them now. So I just write hat theta, the polar representation of theta. And so a spiral deformation has the form written here, this expression. So here the radial part of theta is only a function of the radius r, so which exactly means that theta is a radial function. And the angular part of theta is the sum of a function g that is a function of r plus or minus the angle phi. So here I have two simulations of different fields constructed with spiral deformations. And no matter of fact you can see that there are isotropic fields. So on the left I have chosen a deformation theta with a function g here that is 0. And on the right function g is not 0, it's an identity function. So that explains why you have this spiral effect on the right and not on the left. But both deformations are spiral deformations. So I've answered to my first problem. Now I'm going to study different fields by using execution sets. So this is a second part but it's linked to the first part as you will see later. So I just recall what our execution sets. So if I fix a real u and a compact t, so in the following t will be a rectangle or a segment of the plane, the execution set of x theta restricted to t above level u is the set of points in t where x theta is above u. So here just an example. So in this simulation of the realization of a deformed field, the darkest colors represent the highest values of the field. And so I've represented here the execution set of this image above a certain level. And so you can see that this image corresponds to this one. And so I study execution sets, but more precisely the topology of execution sets. And for that I used a functional which is the ULA characteristic. So this is an integral valued functional defined on large class of compact sets of the plane. And I will only give a heuristic definition of it. So if I have a compact set of dimension one in the plane, its ULA characteristic is simply its number of connected components. And if I have a compact set of dimension two, it's the number of connected components minus the number of holes in this set. So just for an example, in this image here to compute the ULA characteristic of this brown set, you have to count the connected components and you count the holes. So one hole, two holes, three holes. And so you make the computation. So I have to add some hypothesis on X and theta. Mainly that I assume that X is a Gaussian field and I have to increase the regularity of both X and theta. So I assume C2 regularity so that my field X theta is a Gaussian field of class C2. So I have formulas for the expectation of the ULA characteristic of extrusion sets of X theta. In fact I can apply them because there is a link between the extrusion sets of X, theta and the extrusion sets of X through function theta. And since the ULA characteristic is a homotopy invariant, then these two extrusion sets have the same ULA characteristic. And so I can use formulas that are proven for stationary and isotropic random fields, but that are really more difficult for not stationary and not isotropic random fields. So anyway, I have those formulas and so I apply them when T is a segment in R2. So in this case theta of T is of dimension 1 and you see that the expectation is an affine function of the length of the set theta of T. And in the case where T is a rectangle in R2, so theta of T is of dimension 2 and the expectation is an affine function of both the area of theta of T and the border length of theta of T. And you can see that the other terms, the coefficients are only functions of few, the level, and the random field does not appear in this formula. So you can easily express the red quantities thanks to the Jacobian matrix of theta. So here I've written j theta as the Jacobian matrix. And so you have formulas depending on the dimension of T that express these quantities as integrals of the Jacobian determinant or the norm of the columns of the Jacobian matrix. And so the idea is that if we have information on this expectation, then we can get information on the Jacobian matrix of theta and so on theta itself. So here I introduce a notion of key isotropic deformation. So deformation theta is key isotropic if it satisfies this condition. So here rho is a rotation in the plane. And so it is an invariance property. And first you can notice that if theta is a spiral deformation, then x theta is isotropic according to what I've said before. And so this condition is obviously satisfied. So theta is also a key isotropic deformation. So this notion of key isotropic deformation is, if theta is a key isotropic deformation then x theta can be considered as weakly isotropic. And so in the following I state a result saying that the only key isotropic deformations are the spiral deformations which mean that a strong notion of isotropy for x theta coincides with this weak notion of isotropy. So I don't really have time to prove it, but the idea is that the expectations that are here can be expressed in terms of the Jacobian matrix of theta. And so finally this condition is equivalent to this differential system on hat theta. So hat theta is the polar representation of theta. And this system is solved in a paper that I submitted with Mar-Brion. And the solutions are exactly the spiral deformations. So consequently to sum up what I've said until now, if I introduce different sets of deformations. So first as the set of spiral deformations I the set of deformations that preserve isotropy for any underlying field. I of x the set of deformations that preserve isotropy for a fixed underlying field x and capital X the set of key isotropic deformations. According to what I've said before I can say that all these sets coincide and so when a deform field is weakly isotropic according to the definition that I've given it is also isotropic in law. So I'm nearly finished but I will just have a word on the problem of identifying the deformation when I only have at my disposal an excursion set of x theta. So it is a problem that has been studied before but usually authors use the deform field on a wall window. They need to know it at each point on a wall window and so the idea is to use only the field above a certain level and so I propose a method based on the information given by these quantities to identify the Jacobian matrix of theta and to identify theta so which solves the inverse problem. Thank you for your attention. So we have time for one or two questions. Thank you. I don't understand how you, I forgot how you define the level set of x and t when t is a segment so can you recall? The dimension of t does not has no importance in fact. You just consider a set of points where a set of points in t where the field is above the level. How do you define the Euler constant Euler characteristic in the one-dimensional case? So for instance if I have a segment here so my extrusion set is a set of segments so I don't know. I think I see. For instance if I have, I don't know, here a segment, the extrusion set will be one segment. I'm not sure. I think so, yes, thank you. Is there another question? No? Thank you Julie.