Domain, image (Poincaré)

0:00 Not bad at all. But it's going to be a little bit impressive. I'll explain why. And precisely, in the course of the lecture, we're going to encounter ensemblist notions. But really, little things. We're going to encounter problems of domain of definition, of image, problems of intersection, problems of reunion, problems to know if a point belongs or does not belong to an ensemble, because it's important to know if it belongs, problems of injectivity, problems of surjectivity. So really, the ensemblist language, potatoes and arrows, that we can learn very early in... Do foreigners understand the arrow pattern? It's a Venn diagram. Venn? Venn, yes. Venn, yes, but Fathaf... It's a Venn diagram. ...or Fathafoid. In extreme cases. Yes, in extreme cases. Only when Fathaf is not a Fathaf. In particular, we will often encounter in this context the question that arises here. When there are problems, it's very difficult to solve them. It's difficult when there are problems. It's difficult when there are problems. So, the exhibition is built in two parts. First, a catalogue, in fact, a list of astonishments. My subject is not the emergence of the ensembleist thinking. I'm working on the emergence of the global local. But having been trained at the end of the 20th century, I tend to translate everything into assembly terms, so I don't have to ask myself what is the definition, what is the image, is it injective, surjective, and so on. And of course the author I'm talking about doesn't fit into this language at all. So as I read, I'm surprised, I say to myself, it's weird, I wouldn't have said it like that. Here, really, I don't see where we're working at all. So it's first of all this list of astonishment of me being a reader structured in assembly terms. This is a list of all the authors who don't write, so it's a list of all the authors who don't write, so it's a list of all the authors who don't write, so it's a list of all the authors who don't write, so it's a list of all the authors who don't write, so it's a list of all the authors These are local stable and homogenous situations of meaning, but what I want to show is that my astonishments are not only the astonishments of someone who was not trained at the end of the 19th century.

2:30 These are also points that work in mathematical texts and that will give rise to mathematical thinking, but not in order to describe a theory of the ensemble. It is really to study functions that things such as intersection, injectivity, domain of definition, image ensemble are at work. So, a series of astonishments, a little bit of context and comprehension, elements of rationality but internally, historically, and I would try not to make this artificially coherent because, in fact, when we see our authors faced with this, well, it's a bit of a mess, it's a mess, one doesn't understand what the others are doing, one chooses notations that are not coherent with the notations chosen by the others, and so on. So these are points of tension. So in fact, my presentation, I would rather call it intersectional image domain, points of tension, anti-geometric theory, functions of a complex variable. So after this list of cases, I will propose a synthesis, but far from being total. So, the first case of study is what we find, a well-known text, it's a correspondence, but it appeared in the other points squares. It's the correspondence between Poincaré and Fuchs, Lazarus Fuchs, in 1880, on the question of Fuchsian functions. Poincaré called them Fuchsian functions. So, a few very brief elements of the context before we see what we find in this correspondence. We have to study linear differential equations of the second order in the complex domain, in rational or algebraic cognition. Since we have linear differential equations of the second order, in each non-singular point, the set of solutions is a vector space of dimension 2. So in each point, we have roughly two independent solutions. If we have a regular point called f and phi of independent solutions, what Fuchs notes is that under certain hypotheses, if the equation has a certain form, if its singularities are of a certain nature, then by taking the quotient of two fundamental solutions, which we call phi of this function, By reversing, i.e. by considering that z is a function of xi, and not xi as a function of z, we define a uniform function.

5:00 And so this uniformizing z as a function of xi uniformizes the solutions of the differential equation, solutions which are multiform. We will see it in the correspondence between Poincaré and Fuchs, since Poincaré reads Fuchs' articles with great interest and does not understand certain points of the demonstration. So, a little bit of proof analysis. He writes a letter to him, he is a very young professor in Caen, to tell him, here, there is a point that does not seem clear to me, dear master, clarify it for me. So, my transcripts are dark, but I will clarify it anyway. So, yes, if the function is reversed, so the idea is that if we reverse that and we consider that z is a function of xi, we have a univocal function in the xi plane, with only one value of xi, which corresponds to a unique value of z. So, if it was true, and if... If this function was univocal in the whole Riemann sphere, it would mean that z would be a rational function, and therefore that differential equations would be rationally integrable. However, that, to a certain extent, means that it is false. So he says, well, I draw a corollary of the theorem that you mentioned, a corollary that is obviously false. So, in a way, we did not understand it. So here I can't draw a square, here it is a square, but it is a contradiction. What is this contradiction? The fact that the solutions are not rational. This is why, for example, here are our three sorts. First, those that can be reached by the function f of z over phi of z, by making the variable z on the sphere describe a finite number of times. Second, those that can be reached by the function f of z over phi of z, by making the variable z on the sphere describe a finite number of times. Third, those that can never be reached by the function f of z over phi of z, whatever the contour described by z on the sphere. So, Poincaré tries to explain the paradox in the lecture he has of the Fuchs theorem by saying that there are three types of things to distinguish, and so he is interested in the whole image of this function. He asks what are the values achieved by the function xi of z. Zeta.

7:30 Yes, it's zeta, of course. The function of the state of z. The first distinction is quite important, the measures that are reached and those that are not in the state plan. It seems quite necessary to distinguish, but apparently it is not. Yes, yes, there it is. We will need to know the topology of the image to know if the inverse is really what is proposed. So for now, we have the Z plan. Of course, my arrow is not the same as yours. The question is, when Z passes through the Z plane, what are the points that are really reached in the Z plane? We can't distinguish between three types. First, the criterion achieved by achieved can play a role. And second, a little subtlety, there are the values that are achieved by making an infinite number of contours in the Z plane. And in fact, it is assimilated a little further in the text, we will not have time to read it. So, possibly, if in the Z plane, the whole plane is not achieved. So, the set of points that are reached, one at the edge, and in fact it will be divided into outside, inside, on the edge, and this division will correspond to reached, or an infinite number of times, reached after an infinite number of circuits in Z-sets, and not reached. So, something a little more subtle than just reached, not reached, image set, not image set, but an idea of crossing the limit. These are all functions or annexations of the board for the whole image linked to a somewhat mysterious behavior for us of a variable z that makes turns. Let's say that you want to make a whole image that makes turns. And again, Poincaré is trying to explain what he is reading in Fuchs's text but which is not said as clearly in Fuchs's. Yes, yes. Yes, yes, but... When you write in 1907, you find the whole image under its feathers. Here we are in 1880. I'm not saying that it's the same as in 1907, but... So, achieved, not achieved, and on the side of what is achieved, which can be achieved by a strange behavior of the function z. In fact, it's a little more complicated than that. So, interesting.

10:00 I still find it a mess. Poincaré explains what he means by not reaching, not reaching. All the values of the plan of the stages could not be reached. But it's an ensemble, isn't it? Ah, well, that's what we call Poincaré. It's a good contour. No, that's the structure corresponding to the frame. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. This is a legend. I assure you that, by inversion, we will have a function, eindeutisch if we say it in German, so univoke. It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? It's a bit crazy, isn't it? So, I ask Foucault to explain a little bit and to justify his reasoning. A test like the one to frame the bed in the most naive way is wrong. There are some solutions that are not solutions. It is therefore to understand exactly what is happening that we must specify But before that, what does the whole image look like, not only in its distinction of atin, not atin, atin, board, but also in the topology of the whole image, i.e. is it simply convex or not? So, I come back for a moment to the question of atin after an infinity of circuits, because we need to understand the following. The function zeta of z, it is a horribly multivocal function.

12:30 Let's imagine that in the plan of Z, the differential equation that we consider has two singular points. For example, Fuchs does the same as Riemann, and he says that when we take a solution among the infinities of the solution of a point and that we extend it in the cut plane, we obtain a linear function. One of the singularities, we obtain a second determination of the function, and so on, and so on. So, at this point, an infinity of images is associated. If we associate one of the parts and extend it, we obtain another, and another, and another. So, in the space of arrival, of course, anachronism. The function, if it associates, once again anachronism, a whole image, to the private plane of the cut. Imagine. But then, when we cross the cut once, we obtain other images. And when we cross the cut again, we obtain other images. So the situation is not at all the same as we think in traditional assembly canons, assembly, application. We deal with a multiple function, in the sense that in each point of the value of z, there is the probability of the value of zeta corresponding. In order to know them, we have to do something in the plane of help. We have to do tours around singularities, expand culture. And as we do tours around singularities, we end up exhausting all possibilities. And while we do these tours, we see the whole image, anachronism, grow. That's why, among other things, the question of its nature is not the most obvious. We have nothing to do with a univocal function, with a set of departures, for which we ask ourselves the question of the set of arrivals. There is a domain in which we have to go round and round to really understand the function, and with an image that can be used.

15:00 And so, we still find a few potatoes in the square-point text. Square-point asks how it pushes. That's always a letter of square-point to Fuchs. A letter after. It's 1880. And the answer to Fuchs... I don't give Fuchs's answer. Fuchs answers with other arguments. And so Poincaré says, your arguments almost convinced me, but there is something I don't understand. Your arguments exclude this type of thing. As I push them, with what you have found, it is not possible to have self-recovering like that. But, he says, in your reasoning, nothing excludes... We are a self-recovering of this second kind, and a self-recovering of this second kind is part of those who would make the function inverse, non-univocal, because of the non-simple connectivity. So, the problem of self-recovering of the domain as it develops. So, two remarks. A remark on the inversion. We don't hear the same thing in the 20th century. It's a shame that Ivan isn't here. Yes, no, I agree. It's good that he has a good reason not to be here. It's not exactly the same thing as in the 20th century. In particular, it's not at all a problem of switching from local inversion to global inversion. Because I thought it was a lie. I'm looking for a global inversion. Of course, that's not how it's done. I'm not telling you which version I'm talking about. Adam will tell you that in 1906, but that's not the point. The question is that a function in the 19th century was not oriented from one domain to another. What happened? Not at the beginning of functions, which would rely on assemblies, but at the beginning of variables. They can be dependent. The general form of a dependent variable is this thing here. It's also called a differential equation. We find it in the same way in Bayer-Strauss, Chico-Chi. It's the same thing. When we have dependent variable vectors, this is an objective situation. They are not independent.

17:30 We can, and this is a conventional choice, look at one of the variables as the independent variable and the other as the dependent variable. And if we make this choice of dissymmetrization, but with a choice of change of point of view, of convenience, we create a function. If I choose to say that x is the independent variable and y is the dependent variable, I consider y as a function of x. I could also choose to consider y as the dependent variable. Inversing a function is not a problem of existence, on the one hand, because we don't have the constraint of universality, but it's not even that we don't have the constraint of universality, it's just that there is no existence of function. A function is a point of view on a dependence between variables, and so a change of point of view is not a theorem of existence. So here, they are not talking about the theorem of existence of the inverse function, they are talking about a property of universality of the inverse function. We come to this point because we will come back to it later. So we had Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, Atiyah-Witten, What Nukira says wrong is that, as it grows, you consider that we have two foyers, one above the other, but not at all crushed on top of each other. But indeed, if we consider that here we have two foyers, one above the other, and that my figure is just a projection of light, and that I should not have projected it, there is no problem of non-univocity of the reciprocal. The points of view are not yet fixed in 1880. So, in these contradictions, Fuchs chose to vote in touch, and that's very interesting, because he chose to change the functional language. At the time, we had a very geometric language, chosen by Poincaré. Poincaré chose to carry the demand for explanation he gave to Fuchs on the field, in quotation marks, geometric. We are doing a theory of what is geometric. We can see that it is drawn and there are potatoes. And then the question is whether it covers itself, whether it is in one piece, whether there is a hole in it, and so on, counts. So I call it geometry. I don't think we can call it geometry.

20:00 Yes, it is generally called geometry. Oh, but okay. Okay. Because the potatoes are rather the dancers. Yes, the people who pass. There is no geometric reality. Okay, that's what you said. So, this is a figure that I have never taken without education. There is no doubt that this is a schematic representation. Yes, yes. So, Fuchs, pushed into his crossroads, chooses to explain his argument in a totally different way. Fuchs answers that he has demonstrated... I'm not going to say that he hasn't demonstrated that everything is in order, that it's all in order. I have demonstrated that... This is written as it is in Fuchs's letter, so it's not at all the same kind of argument as the point square argument. Here, apparently, we are in a static point of view, we are no longer running on one side or pushing the other line, Apparently, we have an affirmation of injectivity. Of course, it's true, and at the same time, it's more complicated than that, since in this sense, it's a multivocal function. So, in fact, it's easier to read in two ways. At two different values of z, each corresponds to an addition of the value of zeta, but these two infinities change. So it's a kind of injectivity for a multivocal function. What is guaranteed is the univocity of the reciprocity. This is my claim, I don't know if you can see it. It's this one, this one. I mean, that's what we see on the inside. So, in fact, it's a number. So, the first case, where these stories of... What are the areas of definition? For example, to study it, you have to go through it an infinite number of times. What is reached? What is not reached? Is what is reached at the edge really reached? We really have the right to do an infinite number of times. I mean, to be the image after an infinite number of times. All of this has a role and apparently it is not clear in Fuchs's text.

22:30 I will continue the list of cases. It is interesting to see that even if the fact that in 1880 Poincaré explicitly with so much detail and the question tries to prove that it is not a standard way to express itself in 1880, we will have other examples. Other examples of a little strange functioning when we advance mathematics with a collective look. I'm going to focus on the interpretation of Kandon and Riemann in these lectures on the Riemannian theory of algebra and algebraics. But let's take a moment to look at Riemann. Here I am in 1851, the general theory on complex functions. So I'm going to show you a few passages that show a new ensemblist functioning, but with questions of a domain of definition. So, Riemann begins by giving an infinitesimal characterization of the functions he considers. When we have a two-dimensional variability, we can consider that it results from two real variations, C and Y, and therefore from U and V. The initial question is under what condition this function, we can say from R2 to R2, is a complex function of a complex variable. So the condition he gives is the partial equation system, the Riemann equation. So the functions they consider are characterized at the infinitesimal level and are interpreted geometrically in terms of similarity in small particles. Then we define them. So I read the beginning of paragraph 2. The Z magnitude, as well as the W magnitude, will be considered as variable magnitudes that can take all the values of the complex. The conception of such a variability, which is relative to a two-dimensional connex domain, is essentially facilitated if we rely on geometric intuition.

25:00 Let's imagine that each value of z is represented by a point O of plane A, whose rectangular coordinates are x and y, and each value of W by a point Q of plane B, whose rectangular coordinates are Q and z. Any relation of dependence between W and z will be represented as a relation of dependence of the position of point Q. Dependence, yes, the translation of logm is a little strange, but yes, something depends on something else, so what he calls the dependence between two magnitudes is what he calls function, a function is a dependence between two magnitudes, so you have to imagine that we have a plane in which we have a variable point W, the coordinate Z, So, knowing that there is an explanation of the approach to this subject, and a plan in which we have a point Q, and this is the W plan. And so, a function is two things, it is a dependence, as I said earlier, and it is above all a covariability. If these things depend on each other, it is that when the point O is variable in its plan, it leads to a variation of the lag position. The words play a role of the point of view, so we have to make the generic point of a plan, the generic point of a plan, the dependence between the magnitudes, the independence between the positions of the generic points. Up to now, it's not strictly like that, as you would say today, but there is nothing surprising about it, except that obviously we know that it is not that that we find Riemann, Riemann is not discovered to have found the surprises of Riemann in this text, and so we do not have fun doing complex analysis in the plan, not only. So, why go to Riemann's surface? When we read Riemann's justification, we have a hard time believing it anyway. Paragraph 5. In the following considerations, we will limit the variability of the magnitudes x and y to a finite domain, and as the place of the point O, not the domain of definition, the place of the point O, we will no longer consider the plane itself, but a surface that covers the plane.

27:30 Why? How? So how is he going to tell us why we have to wait? No, a little later. We choose this mode of representation where there is nothing shocking to talk about superimposed surfaces in order to be able to admit that the place of the point O can repeat several times the same part of the plan. But in such a case, we will assume that the portions of the superimposed surface do not confound all along a line so that it does not happen that the surface is folded or that it is chopped into superimposed parts. So, in terms of motivation, for the moment, it's just that there is nothing shocking to consider that, there is nothing as a justification, we are waiting a little bit for the reason of this introduction a little bit extravagant of the fact that the place of the point O may not be the plan but a surface above the plan, I don't know what that means, which can cover the plan several times. For the moment, a little bit of mystery. The trick is that Riemann, in addition, doesn't know anything, there are still a few conditions, so we don't know very well what it is, we don't know very well what it is for, but it is still subject to a few conditions, so we have a little trouble understanding what it means. The trick is that in fact the explanation is in paragraph 15, so you have to go through the whole theory of I describe the surfaces of Riemann, I make circles, I move, I bend, I touch, I return, and so on, to paragraph 15, what do we see? It's unremarkable, so paragraph 15... We now conceive a function of Z which, in each point O of the surface T covering the plane A in an arbitrary way, has a determined value and which is not everywhere equal to a constant represented geometrically in such a way that its value W equals U plus BI at point O is represented by a point Q of the plane B whose rectangular coordinates are lighter. We will then have the following propositions. The whole of the points Q can be looked at as forming a surface S at each point corresponding to a determined point varying with this point Q in a continuous T way.

30:00 So, it requires a bit of explanation, but... I have a functional situation because I have a point O that is variable for this surface and I have a complex function of the position of this point. At the end, I have a plan B. Now the functional situation will arouse a surface above plan B. And this surface above the slope, it has a particular characteristic, it is that, if I consider that... Why is there a surface below the slope? The whole of the points Q can be looked at as forming a surface S. Yes, but it is not as marked as it is above the point B. This is the A plane. No, no, no. The whole of the points Q can be looked at as forming a surface S. For now, it is not marked that there is a surface S. At each point of the class? No, because it says somewhere that Q is in the plan B at the beginning. Yes. So F is not necessarily above B at the moment. I don't know, from the plan B. Yes, yes, yes. There is no word above it. But in fact... No, no. What I'm saying is exactly what I'm saying. Exactly. I'm learning the details of the paragraph to say that it is in fact a space above the plan B. So, in this functional situation, a variable point on the surface is a complex function for the moment of this surface. So, if I were to explain this, I would explain this. Imagine that you are going to do the sine function in the real domain. You have the right is a, the right is b. Here we are going to say that it is t, and here we are going to say that it is x. And you consider the relation x equals sine. So, when t travels, or the point associated, travels its domain to itself. The point Q that depends on it will also change its position.

32:30 But our point Q arm will do this. So, of course, the reciprocal function is infinitely multivocal. The reciprocal is defined by an antecedent infinitiveness. What doesn't work well? Well, this is actually this. We must consider that the location of the point Q is not in the plane, but is formed by an infinity of leaves above the plane, and that on this surface, the reciprocal function is univocal. It's because I'm in the real domain. That's it. If I'm in the complex domain, if I have a function whose derivative is... Sorry. If I have a function that is not constant, I have theorems that guarantee me that, precisely, I have no singularity in the real domain. So that's exactly what I was going to say. Not only do we see that Riemann's surface, which is introduced, we don't know very well why in paragraph 5, the use that is made of it in paragraph 5 is indeed the use in a problem of definition domain or image domain. The question is, to translate it in an anachronistic way, I have a function that is not injective, I change the nature of the image to make it an appropriate domain at a reciprocal level. And the conditions that he imposed on Riemann's surface are exactly the conditions that are verified when we consider the reciprocal of a function or element. This is a little clearer in paragraph 15. But of course, there is no assembly construction of Riemann's surface. Riemann's surface is a way of seeing, perhaps viewed as, which allows us to realize... There are many other motivations in 1957 for Hebelian functions, but this one in 1951 is the one I'm going to cover. What does Neumann do? I'm talking about Karl Neumann who, in the 1960s, decided to teach him the theory of Hebelian functions at Riemann. Riemann, himself, has been treated.

35:00 We understand him a little. Neumann gives a lecture. And so Neumann, what's interesting is that when it takes 40 pages for Riemann, it takes 400 for Neumann. So it's less obvious, but we have examples, we have explanations, we have explanations. It's really a lecture intended for students. And so it's interesting to see how Neumann tries to explain all this, really, seriously, to students, without anything. And of course, he can't explain it like Hermann Weyl, the French of the Central Union, he can't explain it like Hermann Weyl in his book. Who was the teacher? I have to say that I don't know. It wasn't Agotinian, but I don't know where he was. I don't know. I don't know. I don't know. Very good. It's for me, it's to try to situate things. Neumann's lecture had editions in 1965 and a second edition in 1984. But here, the passage I'm going to give, in translation, is unchanged in both editions, as you can see in 65. So here is a beautiful non-assembly flow. So, let's try to understand what he tells me. Either a ray from a point XC is free to rotate around this point. If we slide it along a given direct curve in space, it describes a cone table. If the curve is cut itself, it is the same as the curve of Côte d'Ivoire. I have a point of position, a variable radius. If the point of arrival is on a curve, it is on the Côte d'Ivoire. If it is cut, if the direct curve is on the Côte d'Ivoire. If the direct curve is on the sphere described by Centrocet-Brion, we know that we have the opening of the conical grid, the enclosed surface, on the sphere by its curves. That is to say, suppose that the curve is, as in the case of the 8, formed by a path that is cut once after a first loop, then after a second loop, a loop that runs in the opposite direction, in the direction of the starting point. We can call the point where this curve in the 8 cuts itself a double point, designated by d. Then we will see a multiplicity. I don't know if there is a point or if there is a d. Do you think that the point is always d? Let's follow the path of this electric current from the C-ray.

37:30 It's quite prolific, isn't it, Neumann? It's quite prolific, isn't it, Neumann? The naphthonics, just like the curve itself, recouple first after the course of a loop, according to the CD line, then after the course of a second, in the opposite direction, comes back at the start. This naponic has two openings, one corresponding to the lower part and the other to the upper part of the body. We have here in front of us a surface that cuts itself. We will often have to deal with such a surface in the future. This is why it will be convenient, as we are at the beginning, to make known from now on a fundamental idea, which, as much as possible, should be firmly preserved in the future concerning these surfaces. I don't understand what you're saying. I told you... You know, I don't think you understood what I was saying. Excuse me. Yes, of course. If you don't, we can ask Neumann, because he might be able to explain it to us. Let's try to see if we understand what he's saying. So this is the convention. He says that in a convention, we decide not to have an intersection. So, for example... Let's imagine in space two disks of equal size, cutting each other along a diameter and making one with the other a concave angle. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. One with the other. In this point, we arrive at the point where we have to go along this line. It's always the figure that describes us. It's a bit different. It justifies its convention. Here I have the double point. And here I have the line of auto-intersection of the surface. The one that tells us that... No, don't go there. Subtitles by the Amara.org community

40:00 In this line, between the anal that crosses like this and the anal that crosses like this, there is no connection. They don't meet for real in this line. What is the connection between this line and the anal? Thuzamangrang and Waziraj. Then we would have to explain exactly what we hear by Thuzamangrang. Thuzamangrang is the idea of ​​connectivity. It holds together, it is in one single element. So let's imagine in space, these two disks will be looked at as two pieces of surface, entirely separated from each other. That is to say, to look at two pieces of the surface that can change their position in space independently of each other, so transported into spacemakers, entirely arbitrary and arbitrarily homogeneous. So he says, I can give you that, but in fact you can't consider that we really have an intersection, you have to consider the proper intersection. That's what I'm trying to say. Are they always with a common diameter? All right, so let's go back to the topic of mathematics. Let's go back to the topic of mathematics. All right. All right, that's good. It's supposed to help us understand the origin. No, what's scary is that we can cut off the names of the masters and either be in the middle or not. I agree. We will continue to explain. When it is a question of a point that travels, advances or progresses on a given surface, we know that we always hear a point in the position on the surface that varies continuously. A point, therefore, always moves from a certain location of the surface to a location near it. You see that two black marks and two black marks are not the same. They can be in the same place and in the same place. I don't know how to translate it, but if you want to translate it into French, it's a bit difficult.

42:30 Yes, I translated it literally, but in French, it's... Yes, it's true, but... But the fact that it's in French... It's in French, it's very good. It's of course not our concept of translation. Yes, no, no, but yes. By the infinitely French point of view, it's a bit of a literal translation. Ah, I see. No, there is not really a notion of distance. The successive positions of a point form a curve that is only made up of points of the surface. Only connected points of the surface. Two parts of the surface must be looked at as points systems. The points of the first system are all connected together. The same goes for the second one, but if the two pieces are cut off, then, according to the supposed representation, there is no connection between the two systems along an intersection. In other words, a point that would continuously exceed a surface like this, well, it cannot decide here to pass on the other. This is not a possible path for a... this is not a possible continuous path on this surface. So I finish, then he reverses, here he sums up his explanation, in a line where two pieces of surface are cut, the displacement of a point passing the surface is always what it would be if one of these pieces simply did not exist. The intersection should not be the same. It is a conception of cutting which is reversing. So precisely, we reverse, and I go back. So, these were the functions that would come from this point, and they are still there. Now, what is a continuous function on such an object? So, I cut the passage, but the conclusion is the same. In order for a function to be continuous on such a surface, in terms of its values on two pieces of the surface, As long as they take continuously connected values on each of the two pieces considered alone, it is indifferent that the values belonging to one and the other piece near their cutting line are different.

45:00 So there are a lot of cutting lines, there is no need to say it, but they decide not to talk about it. Why don't you talk about it? It's when you make a connexion. I understand. What I don't understand is the point of cutting them off. Let me explain. What I don't understand is the point of cutting them off. Everything you say would be perfect, but why do we need them to be cut off? We'll come back to that. Because there is a projection on the surface, it's an element, it's a projection. We're only at page 64, there are 400 more. So, what's interesting about this text... In fact, like Riemann, there is the fact that the geometric moment precedes the functional moment. Because what he wants to explain to us in his reading convention of intersections, which are not made as sigmas, is what one must understand by the notion of continuous function at value in such an object and what one must understand by continuous function defined on such an object, to put it in a simple way. This is a convention that gives meaning to the notion of continuous, vertical or double function. But before giving it, it presents it as a natural consequence of a lecture convention given in a geometric language. So there is a prealable geometric moment at the functional moment, which we also see in Riemann. Riemann who begins to introduce Riemann's surfaces and to describe them and to say how we can go through them and cut them before paragraph 5, where we see how they are aroused by inversion problems. So, of course, these are functions that arouse particular Riemann surfaces, but Riemann surfaces are still described before. There is an autonomous geometric moment in autonomy, and the problem is... You mean that here we are in the section, for example, 5 of the... Exactly. It's exactly the kind of explanation, but it's less clear in Riemann. It's less... Plagiaristic. It's less shocking in Riemann. In Riemann, we don't understand. Here, we laugh. I'm exaggerating a little, but... Chez Riemann, on Gras-de-la-Tête. Chez Neumann, on Spence pour y croire.

47:30 So, what is this situation? I'll tell you. First of all, the autonomy of the geometric moment, at least in the presentation, compared to the functional moment, which is of course entirely under the control of the functional moment. But what about the totally intuitive neighborhood? I think that neighborhood and connectivity, one by one, I think it's a block. First of all, it's not defined. Do you want me to tell you something? If you want, the neighborhood is the fact of being close to each other. There is no neighborhood. The connection is the fact of being articulated together. No, but there is one thing I don't understand. Which allows you not to have this path. I don't understand. You don't have this path. You can't go from one place to another. Do you have a measure? In other words, when we talk about engineering, we are talking about distance. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. We find that in Neumann we are not at all in topology. We are even before Weierstrass. It is a pre-Weierstrassian analysis. The notion of continuous path is a primitive, undefined notion. So we are not at all in there. Neumann, Barchardt, and Zsuzsanna, it is never defined. That's what I don't understand. I don't understand either. I agree that the two intuitions, the two paths, do not intersect, they are crossed, they are cut, they are crossed, and these paths can go together. In what way can you say that they are next to each other or paths, and suddenly they can't intersect? They can't intersect, that's it. They can't intersect. OK, we'll try to keep going, I think we'll give you the answers later. I'll bring you the answers, of course.

50:00 Go ahead, go ahead, I think it's over. You should have started with Socrates, I think it's over, for my explanations and so on. Do you want to finish? Yes, I know. So, what's interesting is that at one point there is a model of geometric recurrence of the independent figure of the position. And here, what's interesting is that, to explain what I want to explain, in a paradoxical way, to explain a understanding that nothing seems to justify, which is contrary to intuition, he calls it a geometric model. In other words, the geometric model is something that has properties independently of the position. And so, what's interesting is that this use of geometry, which we draw from a more direct, more common, geometric context, is the opposite of the general geometric point of view in the text, which is the point of view of analysis itself. I remind you of Riemann, Riemann in 1854, when he explains what is the point of view of analysis situs, it is precisely when we give up the possibility of building figures independently of the position. And so the point of view of analysis situs, the one in which is visible and must be taken seriously the fact of being included, to recover and so on, it is precisely the point of view where it cannot fail. So I quote Riemann in 1854 in the same way. Where the majors do not exist as independent of the position, but as chapters in a practical domain. So the point of the misticitus is when the majors are no longer considered to exist independently of the position, independent of the position, but as domains of a multiplicity, of a variety. So the general point of view will be to introduce analysis here, to give up the hypermobility of figures, but at some point, he needs, for his explanation, which is on the way, to resort to this kind of geometry and not to interfere. And I agree that we can't really understand what's going on. Otherwise, we see what it's all about. It's the definition of what we call continuity of a function on an object. So, in fact, what is he trying to say? It's always the problem of the inversion of the Z-plane towards the disk unit. In the W time, when we look at the unit disc, and when we apply this relation there, in fact, this disc there, 0 is equal to 0, but after that, when the point is in the 8th turn, the point is 14.9, and 14.9 is the free point, here, of the antecedent, which we find in the solution.

52:30 So, what interests me, in fact, is the multiplicity of the function of the square root of x. Multivocity except for zero. So what Riemann says, and what Longman says, is that this disc is actually a double neighborhood of the origin. It must be considered that there are actually two superimposed leaves. So we can consider them as not attached to each other. But where the model he gave us is not good, is that in fact they are still attached to each other. So his model is very wrong. The discs are not at all free to float with each other. They are still attached to the center. On the other hand, when we turn on a sheet of paper, we can't reach the bottom sheet of paper. And for how long will it stay the same? I don't know. So, its geometric explanation, with the independent geometric moment, is made to realize the geometry of this and the surface that will have to be aroused in the W plane to uniformize the complex square function. So, what's funny is that this convention of the invisibility of the intersection It is not at all unique to Neumann, for example in Darboux, in his 66 text on contours traced on surfaces, where it is said that the surface considered may have various nodes that could meet in some singular points, such as a cone, or cut mutually according to singular lines. We should not take into account these accidental links and not consider as contiguous on the surface two points infinitely adjacent, but taken on two different nodes. It's not bad, it's continuous. It's continuous. It's also the fact that we can imagine the loop of mobility. So if the intersection line can move, it's because it's there by accident. This idea of invisible intersection can also be found in Nobius, both in his text on the classification of polyhedra and also in the text on Nobius' book, which is officially on the volume of polyhedra. Here is an example. We will find the link between this notion of invisible intersection and multiplicity. Here we had invisible intersections because we had two foyers.

55:00 In fact, we had a piece of paper to be counted twice, to be counted with multiplicity, because it is the only one on which there is a rational function. So, with Mobius, in this text on the volume of the polygons, Mobius begins by explaining the situation on the polygons, and he considers the crossed polygons. But he wants to define what we can call the volume of a crossed polygon. And so he tells us the area of a crossed polygon, like this. It is defined as a curving integral. You have to imagine a point that runs through the edge and a light beam that illuminates this point. And when we consider the surface integral, what we obtain as the result of the surface integral is what we call the area of the polygon. If the polygon is not crossed, we find the area in the usual sense. And if the polygon is crossed, it says, this is what I call the area. It becomes the definition of the area. But the question is, what is the link between this area and the new sense? What is the area of the two polygons that I see when I decide to see this point of intersection? The answer is the area of the new polygons. It is actually the area of these two, but actually counted with multiplicity, plus one and minus one. And when I have a crossed polygon, more crossed, Well, the link between this polygon, in which I consider that it is the intersection, the area of this new polygon, which I define as my integral, is linked to the area of the different polygons that appear when I decide to see the intersection, with multiplicities. For example, this piece is on the side of 3 times, this piece is on the side of 1 times, and so on. Because in fact, what is finished is the index of the path relative to 1. You see, intuitively, I still feel that the objects are different. So, I'm not saying that it's the same thing, no, no, it's not the same thing. Because here, it's configurations in the field, it's configurations that are produced by the effect of a quantum function, whereas here, it's the creation of an object that does not include the standard geometry. That's a question of point of view, because Klein will tell us, in fact, you are making a mistake, a hole. So it's the same as the standard. But if you want, that's not why, after this, it's going to be relegated to the standard. But from the moment we create things,

57:30 because we can, without having connections, etc., etc., it seems to me that it's exhaustive. That it's linked to problems in function. I have the impression, if you want, that it is not the same type of subject. I agree that the contexts are simply different. Here, the context is not functional. It is more caught up in classical geometry. But it must be said that even in a much more classical context of geometry, that is to say, the calculus of R2. We find aspects, I decide to see, not to see the intersections, and according to which we decide, by convention, to see or not to see, multiplicities appear, pieces of plan counted once, but also counted twice, counted three times, counted less than once. Anyway, what seems to us to be the case, it is not the same theory. It seems to me, I do not see the norm. Well, yes, what I show is a series of non-ensemble functions. Another case in Neumann, much more quickly, when we have something like that, of course, we do like in Neumann, we cut, but if we cut, the point here to which we arrive by there is no longer the same as the point here to which we arrive by there. So we end up with two points cut above the same point, Stell. So that's worth looking at in Neumann's citation. In fact, we had three different terms. To say, in a certain way, points, we had Punkt, Lage, and Stern. So, a point that runs before it progresses, that's Punkt. This point has a position that changes, and when this point changes position, it changes its location. When its position is variable, it changes its location. There is a point between Punkt, which is a variable point, and Stelle, which is a point in the substrate, which does not move. The conjunction of both is that the point is in this position. So there is the point as Stelle, and there is the point as Punkt, which is a variable point. And this is where we have to deal with two different spaces. We have the space of Stelle, which has no multiplicity, and in which the intersections are there for real.

1:00:00 And we have the space of the punct, of the variable point, but this space is a very different space. It is the place of the variable point. And this place of the variable point, it can be multiple. The variable point can travel several times this place. We have even Stellon, when I write an 8, we have a double point, and so on. I tried to look in the text to see if it was really systematic, if there was an underlying theory of the difference between Punkt, Lange and Steyr-Loeb, the answer is, I think, no. But every time he needs to differentiate the two, we see it appear. So here, for example, to say that there are two points above the same point, he has to choose two different words. So this notion that we have nothing to do with a Riemann surface as a set of points, the Riemann surface is the place of a variable point, and this place can be multiple. So, yes, I'm going to focus on intersections, because it's going to be a lot of work, so I would like to start by talking about the way the square point reads these things. In particular, in the comparison between the theorem of uniformization of functionalities given in 1983 and the reformulation given in 1907, and here it deserves to be exposed to itself, the comparison of the two texts of 1983 and 1907 shows an explosion of the ensemblist language at Poincaré, as much in 1983 at Poincaré as in this type of language. Even so, in 1907, it is almost a completely ensembliste writing. We are not yet at Hermann Waal in 1913, but almost. What happens when you do not write? It is difficult to say. Among other things, critics. In his theorem, a lot of people thought about it to believe it. So there was an attempt of criticism, at some points of the demonstration of the attempt, So, what is the use of a geosynthetics to make things clearer, more rigorous? Well, let's say that Poincaré uses it to deal with...

1:02:30 It's a bit funny because it's what he accused Fuchs at the beginning of his career, to deal with accusations of a blur, of we don't know where it's happening, of you're sure, of a time-and-time problem. The problem of what problem? The problem of the uniformization of reality. There is a demonstration in 1983 and one in 1907. The one from 1993 is 13 pages long, the one from 1974 is 20 pages long, and in fact there are two or three more with it. So, just at Neumann, no, at Schwarz, Schwarz invents a new mode of demonstration for the Dirichlet problem. The Dirichlet problem is not interesting, you know it. I give myself a surface with its edge. I give myself a continuous function on the edge. Is there a defined function on the surface that is harmonic on the surface and prolongs the continuous function on the edge? Something quite interesting, we saw that in the correspondence Fuchs-Point-Carré, the question of whether the edge exists or not was already a sensitive point. What is interesting is that in the Neumann Treaty, in 1884, in the Treaty on the surface of Neumann and on the theory of complex functions. It is on page 380 that for the first time, on the occasion of the presentation of this problem, Neumann distinguishes between a function on a surface, in the central sense, and a function defined inside a neural surface. So to know if the domain of definition is the interior or the interior and the edge, it only happens on page 380 of the treaty. So there is something that amazes me. Today, about Dirichlet's principle, with Kip Neumann and Jean-Bien Schwarz, Schwarz invents a new demonstration procedure, which is called the alternating procedure, the last of the four forms, and the idea is as follows, if I know that my problem is solvable for a domain, and that I know that my problem is solvable for another domain, Schwarz wants to demonstrate that it is solvable for that. Of course, the question is, how do we understand this thing? If we understand the human being, we can come to the human being with a certain kind of intersection, where we can see two superimposed lines, but which are not really connected to each other. Here, it's not that you want to do it. Here, he just defines our union, the 170th of the ELF, our ELF.

1:05:00 So what is interesting is that he is obliged to explain, by saying, I don't have a text under my eyes, but he explains by saying, This area must be counted only once, so neither zero times, as if it were taken off, nor two times, as if I put a layer, then a second layer, so it's neither multiplicity zero nor multiplicity two, I'm leaving, it's in 70, I don't have to specify, no, it's counted only once. That's how I build a new domain. And what's interesting is the way he notes it, it's that this domain there, he hasn't seen it. This new domain, U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' plus U' So, to designate this thing, in which this piece counts only once, he has to solve it by itself, and he uses it as a notation that seems natural to him, which means that it is not true, as we might think. On the other hand, what is alluring is that in another text on the principle of symmetry, when he takes a piece of the plane, which contains a piece of the edge, a piece of the right, and the symmetric, So, in this case, when he wants to talk about the intersection of the two, we have to paste the length of the edge. The edge can have a simple multiplicity or a double, it doesn't matter. When there is a real piece, we have to specify it. When it's just a piece of gold, there is no need to introduce a particular notation to say that this piece is a product of gold. So, I've only done half of my exposition, but I'll get to the conclusion. Ah! Conclusion! I'll be right back. I'm sorry, you've already used some of the technical terms, so you're already used to it. But it's because you always know what I'm talking about. So I'll just repeat it. You copy the big ones. I can't do the first part, I don't know how to do it. I would like to do it officially, but we won't invite you next time anyway, no problem.

1:07:30 You don't have to do that, I have to invite you anyway, it's not necessary. Very good, so I can finish by saying anything since I know I'm invited. These are the links I work with, which do not work specifically on the emergence of ensemblist thinking, ensemblist language and ensemblist writing. I work on the emergence of the global local couple. But the global local couple, explicitly at the meta level, at the level of what I call the local theoretical theory and so on, is a reference to a type of place. Here, I'm talking about the domain of definition of functions, I'm referring to the domain on which we have demonstrated that the function has this property. It's all there, knowing that it's true on the left, it's true on the right, it's true in the intersection, then it's true in the meeting, These are the places where it takes place, at the reference point of mathematical work and designations. So it is not directly a work on the emergence of English thought, in any case on the theory of the ensemble, but we see the interaction with this question of the emergence of English thought. It is inevitable, and this is what I find in the texts I read. So, of course, in the second half of the 20th century, An ensemble writing method of mathematics takes charge of this relation to the place. It takes double charge of it, first by allowing a standardized designating method, read and understood without ambiguity by all, when we see our authors being forced to make a convention every time, and secondly, the current writing method makes this designating of the place mandatory. If we talk about a function without talking about its definition, we go back to the rigor. We cannot write in a mathematical review like that. So it is both allowed, understandable by all, and mandatory. In this series of lectures, I looked at a series of works in which certain elements of what will become ensemblist language are brought to play a role. In each of these cases, the nature of the domains associated with a functional situation is the object of research or is one of the important elements of the exploration of the mathematical situation. Building the domain, building the original structure is the essential step in the Riemann theory. Collecting pieces of surface to extend a function is the fundamental gesture of the Schwarz theory, and so on.

1:10:00 And I had some nice things about Schwarz that I won't talk about any more. So in some cases it is a domain of definition, in the other a set of images, which is quite relevant. In some cases it is more numerous, where questions arise about both multiplicity and inversion. The distinction between a domain of definition and a set of images is less relevant, since we have a question of product. Some general remarks on the series of texts. First, the relative novelty of the attention to the place. We see it in the texts, whether the authors underline it. So here it is relatively to Weierstrass and to the theorem on the fact that continuous functions defined on a closed domain reach their maximum. We see the authors, when they cite this theorem, really say that this is the new analysis. They do not talk about the y-data and so on. This theorem, the fact that we work on a closed domain and that it is clear that the edge is in it, The authors insist on the novelty of this theorem. It's new either because the authors say the same thing, or because we can read this novelty in the contrast between the vast majority of cases where this reference to the place is missing in our eyes, where they refer to the place, but in the vast majority of cases, for us, there is a lack of information when we look at the reference to the place. So, we read this novelty in contrast between the large number of texts where the reference to the two is, in our eyes, insufficiently vague, and the few cases where it is prolifically explicit. I think Karim must explain very well to Fuchs the question he is addressing. Neumann is quite verbal on the explanation of what he means. As you can see, I can't help it. So, the second remark is the absence, at the time I consider, of general convention in the face of a certain type of situation. Since, in a lecture such as this one, what we call intersections may be counted according to the case 0 times, 1 times, or 2 times, and each time the authors are obliged to specify the convention they choose according to what they want to do. In any case, it is a question of revealing the true nature of things. The conventional character is explicitly assumed. It is a way of looking at things. We will agree, we will look at how, we can look at how, as described by Le Poitier in 1907. So, in a sense, I'm extrapolating a little bit, in a sense, it's still a little bit archaic, solidary of an era in which being a mathematician is actually being able to cast on mathematical situations an informed look by the metaphysics of an image.

1:12:30 So, being a mathematician is being able to see something other than just the points that are really in the whole, it's being able to see points with multiplicity, it's being able to see around a point the halo of the infinitely adjacent points, and so on. So, in a sense, everything I presented is a preliminary work by Weierstrass. Another interesting point, what are the resources of these authors, in the face of these problems, in the somewhat unexpected way? There is the resource of geometry, but here it is very ambiguous. Our authors talk about geometrization, about geometric representations, and so on. Geometry provides, at will, both the model of the figures given by positions, These are not given independently of the position, and so we don't have an analysis system. But geometry also provides the model of the figure that exists independently of the position. So, according to what we want, we can refer to geometry as we always prefer. But there are not only problems derived from the geometric tradition. There is a certain geometric intuition in history that allows us to put in place. We can act on the situation with problems of cutting, gluing, edge-to-edge gluing, piece-by-piece gluing, and so on. Of course gluing edge-to-edge and cutting is symmetrical, but if we can glue piece-by-piece gluing, we can also decolle two pieces. It would be very bad to say that you have to imagine several pieces, one after the other. There is another link with geometry. Geometry provides a material-material possibility to figure out the situations to give them a grip. I dare to say that there is neither an overall understanding nor a global understanding, and that's what questions them. So, in the plans of the ensemble thinking duo of geometric thinking, what is geometric thinking? I don't know. Because our authors focus on geometry. Each time, they can draw from something that suits them and differs according to the context. I had a good development on something more solid, but which for us has disappeared, which is the notion of variable grandeur.

1:15:00 Our authors have as a fundamental notion, not the notion of set or application, but the notion of variable grandeur. And so the function, as I explained, is a point of view on a dependence between variable grandeur. Exactly, the impetus... Yes, but... And so we will have as a resource, not to construct ensembles, but to write places of varying magnitude. Obviously, we have a fairly long development. But what we see in the examples that I have presented, and in the ones that I have not presented, is that, in a certain number of circumstances, And each time, the mathematical gesture is a little bit different, which is the surface of Rinalds, to demonstrate the time and distance that we analyze in this way, to consider the whole image to ask oneself if it contains its superior value or not at Weierstrass. Each time, we have a relatively new mathematical approach, a new way of showing things to you, which shows a certain relevance of an element of an ensemble, belonging to the edge or not belonging to the edge. Topology of the set of definitions, topology of the image. All of a sudden, the elements of the place where it happens become relevant, and the language of variable magnitude is no longer sufficient to say things. Each time, we have to tinker, set a convention, but a convention that will be used, not page 65, but that will be used the same way, page 540, and so on. In Riemann's book, Page 400, we don't see the intersection, he uses the word fusion to describe it. So, depending on what we have to do, we take geometric arguments to justify in a more or less convincing way. Depending on what we have to do, when we can, we use language of variable magnitude, and when we can't, we try to make small figures. Thank you. So what I wanted to present is, before the synthesis, all the points of tension between ancient paradigms, as if you were waiting for a variable, theories available, but whose meaning is not univocal, such as classical geometry, and problems in which the relevance of the question of the place makes it difficult.

1:17:30 Thank you. I'm masculine, like Mr. Dernier, even if I would like to raise an argument. Exceptionally first. No, no, no. Exceptionally first. No, not at all. But I said in advance that I would like to get José and Clérénon to interact on some things. So I would like to ask a question about them. But I agree. No, what bothers me is the use of the word game. Because in fact, when we talk about geometry, we are talking about objects, about a body constituted by an object. So here, I have the impression that we are talking about illustrations, ideas, rather than the derivatives of objects in which we make theories. My question is, why do we call it Geometrics? You see what I mean? It's not a mathematical object. It's more... It's more... It's more... It's more... It's more... It's more... It's more... It's more... It's more... So, when you say that pieces take off, it's a bit like there's a representation of a figure, according to which the square is a representation, whereas I have the impression that the construction of a figure is something else. So, it's true that I couldn't help myself from making a lot of anachronisms. It's true that the common term today is to re-glue, to re-glue, and that's the essence of mathematics. To take off, that's a big problem, whereas for them... These two gestures are possible, even if there are none, but they can be a problem to be cited in adult conventions.

1:20:00 So, I completely agree on the question of the figuration of geometric objects. So, two things. No, precisely, no. Say what you say no, and then we will try to understand your argument. If building a geometric object is to define the structure as a whole... No, not necessarily. The climate, when you build a geometric object, is not to define the structure as a whole. There are several ways to define it. I'll have to give you the title next week. I can't do it anyway. There are many ways to define it, but it seems to me that there is a question of the status of this thing. It's true that our authors never talk about objects or even construction. You might think that Riemann talks about the construction of Riemann's surface. No, Riemann describes it, but he never understands how to build it. On the other hand, the word that comes back systematically is the point of view. We can consider them as anchored by the point of view of geometry. That's why I said illustration. It's a functional situation that informs the gaze on a data, but which is still a data projected on a geometric support. There is still a plan. There is still the idea that there is the transition from Z to O. Very explicitly, we pass from a complex size to a variable point. There is no construction of a geometric object, there is a point of view that has a geometric corollary, but on the other hand, the link maintained with geometry, they still take from geometry this idea of variable magnitude, this idea of the place of a variable point. This is the idea they use, and in this idea that they take from the geometric theory, the most elementary at their time, they see the notions of multiplicity. The fact that in the 8th we have a double point is an elementary geometry notion. It is not elementary for us, this notion of multiplicity, it is very problematic. But I don't think that Riemann or Neumann have the impression of doing something extraordinary.

1:22:30 In geometry, the notion of multiplicity exists, the notion of variable points exists, the notion of given position or not given position exists, so there is no relation to the object of geometry, there is a relation to the figure, and there is a relation to geometric theories. Again, I would like to put in a parenthesis that geometry is not entirely dependent on figures. Of course, but what is the definition of geometry? The one that defines geometry, in the sense that it is a criterion, it is exactly the definition of geometry, which is the definition of the figure that is in front of it. There is no other way to define it than by defining the figure. The good idea, if you ask me, is that... My idea is that if there is something that is not elaborated systematically, constructed systematically, with gestures, operations, etc., but that it is another type of work with scientists, it is important to make the distinction. So maybe afterwards it will become georetical. To mark this position, it may be interesting not to treat it as purely geometric from the beginning, because maybe it will actually become a geometric. Yes, to become a geometric. I'm talking from a geometric point of view. I take it as it is and I spread it in the multiplicity of different senses, which is not necessarily coherent, even on the contrary. The subject I presented was more about the characteristics of decolation, painting, ad hoc conventions. That's why I thank you for bringing back the basics. In fact, the only explicitly pointed discipline, at least from a geometrical point of view, the only explicitly pointed discipline is statistical analysis. What I find extremely interesting is the object path to explore these things that have been discovered in space.

1:25:00 It is really how paths are a little energy to write the story. I find this very interesting because it's very particular. All traditions are like that. It's because we explore through paths that we become popular. That is, by temporarily banishing certain movements. It's been like that since Kochi. Yes, it's been like that since Kochi. Yes, it's been like that since Kochi. Yes, it's been like that since Kochi. I don't know what to say about something I've already said. You can ask Gauchy if you want. That's the question I wanted to ask. I'm sorry. You see, there's an evolution in relation to the language. Yes, there's an evolution in relation to the language, but it's not here. So, that's it. I don't want to see what I believe. I have a question about the square point that you talked about earlier. Maybe you can answer it. But what has changed from 1883 to 1883? I will talk about the point square in two other places. In uniformization and also in the field of race travel and the method of continuity and the road of gold. So in 1883, we see a square and a method of demonstration, a new canon. For a problem so general that no one would have thought that I would be able to describe it, but the mode of writing is really that one, the paths, the paths, and so on. And in particular, we have a complete blur on the edge, namely the singular points. Are they in or not in? The domain is the plane with the singular points. So the singular points, do we remove them or do we consider them in? Well, Hancard made a convention a little late in the text, it's at the end, there is a very long one, it's the last minute, he explains, of course, we understand that singular points are excluded. In this case, he has a construction in several stages, and according to the stage at which he excludes singular points, we do not obtain the same theory.

1:27:30 So we are really in the biggest blur when it comes to this question of the presence or non-presence of singular points at the edges, how do we build a space, from which right do we take off the leaves, and so on... So, in 1907, a few radical differences. In 1907, a square... To build uniform problems, we have multivocal functions. How do we build a surface on which they are uniform, a surface simply connected, on which they are uniform? So, Poincaré is very aware of the fact that if we want a uniform surface, we need a certain number of functions. And if we add a function, I did it in 1983, but I didn't ask him to do it. If we add a function, then this function can have a smaller domain. So, if we just construct the surface by using a function, we will have to construct a surface that is smaller than the initial surface. On the other hand, this function will add values, so we will have more sheets. For example, in terms of definition domains, and in terms of... In 1907, we clearly see this aspect of adding a function, which may be, in relation to the surface or the form, restricting the part of the plan on which this project is based, but inducing new ideas. So we see something that was completely invisible when it was the spring of the demonstration in 1883 to introduce a new function, but then what happened above... For example, when we introduce a new function, it can restrict the common definition to the family of functions, which is a natural consideration. It was invisible in 1983, it is essential in 1907. Are the points excluded in the surface or are they not in the surface? It was really written under the carpet very quickly at the end of the article in 1983. It was very explicit in 1907. There is also the question of the set of images. In 1907, we see the question of the set of images. In 1983, Poincaré shows that he managed to build the return surface. We can present it in accordance with the disk units of the complex plan. And there, Hilbert says to Hermi, I would like to know if I have to deal with all the disks or just a part of the disks. And so, a question that is totally untreated in 1923 and to which we answer in 1907.

1:30:00 In 1907, we talk about Hilbert's theory and a demonstration of Haute-Goude, as for all the plans of the disc, except for the problem of the antenna, and so on. In 1983, we can see that it has reached its peak. These are the questions of domain, borders, nature of the image. All of this in a rather radical way in 30 years. An international discussion. Because this theorem was so huge in its scope, the uniformization of all the things that are mythical. In addition to everything you can imagine, it immediately attracted a considerable international attention. Poincaré was already famous for his Fuchsian functions, and that was the cherry on the cake in 1983. We have seen this a lot in the lecture in 1986, which was published recently. Everyone started to look at this thing closely. It's true that when we read the equation on the square, it can happen anything we believe. It's great, but beyond saying that all the wheels are tight, even the people at the time, well, you know, there's still something to be done. Well done. Sorry to speak English again. I think part of the things I could say will come later, when you ask. So at that part I will need to... I think that you can imitate. We shall see. So I wanted to just reinforce the point that this idea of variable magnitude... This, in a way, is the opposite of the thinking. In several ways, you can see that there were several ways in which this was... In some people, it's simply that you have to stop copying this kind of language, eliminate it, and go to a completely different language. This is especially the case in Derrick. He's the most explicit in emphasizing that you need to get rid of that. But in other cases, it's more an evolution of the idea of magnitude. It's more difficult to understand. The evolution of the idea of magnitude can turn into a more, so to say, set theoretic kind of thing, like in the case of Grashman and the case of Diaspora.

1:32:30 So, but anyway, it's quite clear that when you are, if you are very... I am strongly immersed in this way of thinking about actions in connection with the values of narratives which are so-called simply the fundamental notion of failure and failure. Thank you, that makes me think of two things that you are going to say, two little things. On the one hand, it is true that it is this idea of variable magnitude. I like the authors of the first two, but they pose a problem. For us, they present two facets. When we have been trained in an assemblistic way, when we read these texts, we never know if we are dealing with the domain or the function. It is both the domain, or the domains, definitions, images, and so on, and it is both the domain. It's for us who are structured like that, I don't know, but for me a little bit anyway, I experienced the end of modern mathematics, so I tried, before I knew all the explanations, I was making potatoes, and we learned at the CP to locate the intersection, so I am a cable like that inside, so when I read these texts, now I'm a little used to it. The fact that this thing is both the domain and the function, I had a hard time understanding how it worked. The second thing about Weierstrass is the fact that in Weierstrass we see that this idea of grandeur eventually, little by little, transformed earlier. In the idea of domain, this was my third part, and I had translated a part of the Hauss-Geballte-Karney-Töller-Hauser, Funktionenlehr 86, the P-86 lesson, where, precisely, we see Mayer-Strauss define a few notions of ensemblist topology in the language of variable magnitude, but what he calls purely ensemblist. So, that was my platform. He called it, in German, .

1:35:00 In connection with the regularization of real numbers. It's a very nice way of looking at it. What is the question? This is actually a book on the history of analysis. It's translated into English. I'm sure they cannot translate it. You guys have your copy for us. Okay. We did. We did. Thank you. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. This is the name of the association. Do you know what we do by the people we put here and the one who wants to follow them? Yes, we put an example there and we do... Again, you know, I told you that I did it, and I know that I have both, but I don't know in which of my four libraries I did it. I don't know if it's in Paris, I don't know, I'll see tonight. There are paper distributors here. If there are two copies, we can take one, but if there is only one copy, it's a copy. It's there. Are you familiar with them? If there is no one else, I would like to ask my question quickly. As Karine is pushing me to do this, I would like to simplify and try to see the same thing, For those of you who say no, no, no. For those of you who say no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no.

1:37:30 How do we do transverse analysis? That was my question. And the answer that I believe, for a long time, is that there is no R, but there is the right. And we don't need an ensemble because there is still a way to talk about the totality of R. The way to talk about the totality of R is the right and the variation of the plane on the right. We don't have functions. I agree that these are two very important things. So we have the idea of the size of the human body, and the idea of the size of the human body, which is very similar to that of the human body, and there it is written, and we have objects that we can call geometric, not really geometric, in the same sense that they are figurative, although I think that in geometry there are many figurative objects, so they would not all agree that we project so many things, but hey, I can't go into that detail. But the thing that made it clear to me at the end when I gave the solution is that this is a totally different thing from what has happened and what will happen in space. And what are the changes? It is much more profound than the movement of modernization, the movement of the introduction of competitive algebra, things are there, but a much more profound change is the introduction, on one side of the notion and on the other of the notion, of the assembly and not of the creation of the ensemble. And that's what Dina said. Everything I've said confirms the same impression on another aspect of the matter, so declassified and desensitized, because we are talking about complex analysis, but in the end it's the same impression that I've obtained, it's the same... So the thing that I want to... It's a question for Philippe and for you, is that on this point of view, maybe I didn't read it, but I don't understand how... I hope that you have understood that the notion of variable variety is the first in its history.

1:40:00 I find that the two things are totally opposite and I don't want to find a way to justify it. Probably I see that there is something wrong, because I have the impression that you agree with me on a lot of things that I just said, the things you said this morning, the question you just repeated. It doesn't seem to me to be completely the antenna of my reading. So, obviously, there is something that I don't see. So, the question is, what is it that I don't see? His famous paper on the oblivion function in 1767. The dissertation was not published at the time, but it will be published later on. So these things that may be available to some people. And what I reconstruct, at least my reconstruction, and I can be involved. As his evolution in between 1851 and 1857, I think there is a very interesting evolution of his thinking, and this is not visible in, as far as I know, any of the papers he publishes then. It will become... Here, invisible, little by little, from the time when the geometry lecture was established, then it becomes a little bit, but in a very difficult way, and then with the Baroque in 1876. So the process of getting to know this is a very complicated one, and it's perfectly possible for people like Neumann or other people in other places. To have read this work and be completely unaware of the possibility of reading it from a more abstract point of view. So what is the evolution? The evolution is, I think... I mean, basically, I think we have the right to understand what he says and shows us about the room and surfaces in the dissertation as a very, kind of, as an illustration, I would say, of the behavior of functions and so on.

1:42:30 He had very interesting reasons to be, I mean, to pay attention to that, that way of illustrating. Maybe we can talk about that, because from the methodology he uses, it's quite interesting. But still, it's quite clearly a language of just giving some kind of geometrical, intuitive illustration of some of the aspects of the functions we are interested in. So it's a very kind of unclear language. One doesn't really know this. To what extent he is really serious about that notion or not, I think that's that. Now, now comes an interesting evolution that we know... I'm not sure I'm going to illustrate things, because there are properties, mathematical relevant properties of objects that you cannot describe. For example, continuity. The regions are continuous regions. How do you want to describe continuity without the set of theory? Otherwise, then, why not? Why not? Because it's not so much the contrary. The continuity would be easier to live with the geometrical... Exactly! This is my point. This is my point. It is that the geometrical property of continuity is the... The fact that this object is continuous is a geometrical evidence. So the figural property is not simply a representation. The property of the representation is a property of the object. I don't know what to say. You have the property of the mathematical object that you cannot describe otherwise. That are property of the physical object that is represented in the mathematical one.

1:45:00 And these properties can be represented by Nielsen's constructs. Yeah, because what we read is a continuous relation with respect to each other. That's what you read, yeah. But anyway, what I wanted to emphasize is now the evolution that comes, These were published a few years ago and I understand that they give evidence that Riemann was worried about having presented function theory this way. Because what does it mean? Well, for one thing, the Riemann surfaces are geometrical objects and they live where they live. They cannot be living in a three-dimensional space. So, and also the kind of players they are going to have in analysis, I suppose that this is partly at least because, well, they're all students of the university. He knows very well this tendency towards a more arithmetical presentation of analysis, but in a way it seems that he is going against it. What do you think about this? He becomes interested in how one can make sense of domino surfaces. And this is the starting point for an evolution which actually, in the manuscripts that he's writing, takes him to start speaking about a manuscript. And the idea is complicated to discuss, but anyway, the basic idea is, he goes to the idea of a manifold as amorous to a counter-idea. Any kind of concept. You will have the associated class of points, maybe something else, and it will have a structure, of course, which is introduced by the kind of concept we have, so it may have a continuity. It doesn't really try to, doesn't at all try to analyze or reduce the theory or anything like that. This continuity, this is something that is given.

1:47:30 Anyway, so now from this point of view he is able to reconstruct what is going on with the human surfaces and he even says that this is not at all an introduction of geometry into analysis. It could be understood as a proto-subtheoretic kind of understanding that you consider in these classes. So the point of presenting it in a geometric way is not at all an essential point. The essential point is, it's only the way he did it in his dissertation, and it will be the way he does it in the 1857 paper, but that is, in my interpretation, this is because it's a type of expository research. He doesn't want to get involved in too many complicated things. He has enough complexity to discuss, and so he decides to present things in a geometric language. He says, something like that, which already shows that this is not essential. What is essential? The essential thing is to understand and analyze the topological purpose of these continuous manifolds. Even before the geometry lecture, he has written a manuscript on basic ideas of topology in a dimensional manifold, which is very abstract, much more abstract than anything you can read in the partial theory papers. So my understanding is that he has these evolutions himself, which he made. In a way, he got into complexity and difficulty with the dissertation, and he found some way out, not completely rigorous and pure, but he found some way out, and along the path, by the way, he comes to realize that now if you have this manifolds with a topological structure, then you can have all kinds of different electrical structures, and so you can develop a dimensionality in the way that you do.

1:50:00 So this is my understanding. And I assume that there could be... We got a good idea of the kind of things he had in mind because, for one thing, we went to his courses for a couple of years and we had a lot of time to discuss with him. But with many other people it was not at all like this, and even after having published the Mathematische Werbung, it takes a lot of work to really go through connecting these things and reconstructing. So, most of the people, I suppose, didn't pay attention to this. They didn't want to proceed very headlong? Yeah, it was a long answer, but... No, but the role of geometry remains high-fiving or not? Yeah. Well, that's a good question. In the text, Tungeski said, and in fact this is perhaps a way of speaking, because it was necessary to find one to make himself understood, and perhaps afterwards he made remarks and said that this was not the best way. He chose a way of speaking. In a way, it's geometric, but it's not a geometric of other geometries, it's a geometry that is not assemblistic. It's the geometry of old variables, it's a geometry in which the notion of multiplicity is a notion of a first evidence, so it's a geometricalization that is not at all a geometricality of assemblism. But that's the impasse, it had to be said like that. But what he wants to do, it's not in paragraph 15 that I presented that it's 10, it's in paragraph 1. So the fundamental question is the question of the mode of donation of a particular analytic function and the question of the necessary and sufficient conditions, the minimum, for a particular function to be given. So the Riemannian strategy is what he calls the Grenz-Wunsch-Kreuzung-Kreuzung-Kreuzung-Kreuzung-Kreuzung-Kreuzung. In other words, the geometric aspects are in fact ways of presenting the interdependencies between the elements of a function.

1:52:30 When we have a function, the values that it takes at different points are independent of each other. And so, in order to have access to this function, we need to know the totality of the values. But when we are dealing with a function of n, the number. It's not like that at all. Values are not independent. So they present this non-independence in two different ways. On the one hand, when we have the real part, values in a domain depend entirely on the values on the outside. In a geometric way, we can see these two types of dependencies. Secondly, the real part and the imaginary part are not independent. They are very straight forward. And the degree of connection between one and the other is the degree of connection of the surface. So we are free to choose the number of constants that are introduced when we talk about the real part and the imaginary part.