IHP Conference — Élie Cartan / Round Table Discussion Part 2

0:00 I'm sorry, I'm sorry. I'm sorry, I'm sorry. No, no, but it's the end of the day. It's the end of the day. And there, it's the reaction of the day. When we compare the three volumes of the book, and then we'll bring in a person and let's say the words to this book, and when we take the three volumes and the lyrics, it's the day of the night. I think it's the night. It's true. I've read the article in the 1980s, the first article. But it's true, it's true. But it's true, it's the present Engel, it's that all the thought, all the thought, all the thought, all the thought, all the thought, he does a summary of what he sees, and all the thought, all the thought. There's a plurality that Capron has worked for a year and a year and a year for him to arrive at his application. And again, he said that he wasn't complete. He wasn't complete. He was mobilized by the seniority of Einstein. I think, in this moment, he felt that there was a crisis because they were so grand-hommes. He wanted to know who he was going to arrive. So, let's go. Let's go. Let's go. Thank you. Thank you. it's interesting for us, it's interesting for at least one person. So, I'm going to return to the design, which I'm interested in, with the spaces homogènes which are the generalizations of the geométrie that we have.

2:30 For the georetically hippolydian, we have the georetically humanian, and the equivalent of the number of the georetically humanian, which are the generalities. So, first question, why, in the meantime, this is not a generalization, it is really an important construction. Why, in the research of the geometry of today, we cite very few of our time? In the research of the geometry of the geometry on the structure of the geometry, we cite, finally, very few of our time. And even when we don't learn from the universities, the connections, we have the impression that we cite very few of our time. Okay. The first thing, we said that it was very well written. Well, it was impossible. It was impossible. In my research today, Brian told me that he had a lot of pain, and that's what he saw in his research. It's very difficult for him to read it. And we have the impression, in fact, that it's difficult to access. And secondly, we have the impression, in fact, that the general space, it was done a little before that it arrived naturally, in which it was before his plant. Notamment, ce sont des choses qui s'écrivent bien en termes de fibrés, mais qui ne s'écrivent pas bien intrinsèquement sur une variété. Donc j'ai l'impression qu'en fait tout le monde, moi ce que j'en connais, ils ont vu plutôt R.S.M. including the article in 1950,

5:00 Connections infinitesimales dans les fibres principaux. So here, we can say that it's the code of Bourbaki, meaning that we have the definition of the fibres principales, we have the definition of what is the connection generalised of cartons, We have the definition of the procure and it is essentially this article, the RS Man, which has been repris in modern presentations, including pédagogiques, with the film, the Bujan, Kobayashi, and the Omidu. Yes, of course. Yes, of course. Apparemment, at the Est, it's a school of logistics, they have worked on Cartland, and their formalism. Well, they don't have a formalism, a little bit, a little bit proper for Baptiste. But they don't continue with the carton. Apparemment, but they don't have a lot of sense, they don't have a lot of development, an improvement. But the formalism is the same as the carton. If we take an article from this school, if we look at it, foreign And Cern a generalised these objects in the form of GX structures. I don't know how to present them like this, in the form of GX structures. It's essentially what he said. It's to say, for example, in the case of the geography human being, it's to say that the hyper-structural and the tangent, it's your own own. It's not expressed for part time. It was written by Tchern, because in the 1937-1942, he was in contact with the cartons,

7:30 when he was in his cell phone, I don't think he was able to do it with the construction of the cartons. But the problem that was posed by Tchern, it was the classification of objects at a changement of contact, and it was still that. So the different structures that we put in place, it was a kind of feasibility in the 20th century. And when we look at it in a scenario, the structures, and when we try to implement the method of the performance of the part-time, I have colleagues who are realists, who are very important to the science, and the calculus is absolutely un-feuble. It is to say that we have the barriers of the group, which intervenes specifically in the dimensions of the interior and the sensitivity that appears, it is impressive. And I'm starting to have a bit serious on the fact that there is a whole generation for the conditions of cartons, the conditions that are in the space, with the structure, it's in the end of the year, with CHAP, and the group, and the Australian, and the Australian. The truth is that we don't know what is the best point of view for access to the authors. It's just a certain number of calculs on the videos and the method of the cartons and the structure that multiplies the calculus in a way exponentially. So in the meantime, I'm not at the point and when I look at the software, So on some of these questions, it's a very good question. The point is that the list of Carton, I don't know you, but it's important to make sure things. Well, for example, for a very simple, very common, Carton is very fier, in 1924, in the article, about the SMR, which I was cité, it's been exploratory on the connections collectives. The reality is that there are plenty of different articles that refer to the connections collectives, which give them the tenseurs. There are two tenseurs invariants, initially. when it comes to classifier the equations of the contour, it's just the equations of the contour. y is equal to x, y, and y. To classifier these equations at the changement of the length, you can't do it with the method of the length of the length. There are Cameron who has to do for the transformations not just the length of the length of the length of the length of the length, which we call the power of preserving. It's to say it's prime equal to x, y equal to x, y. And in passing by the method of the length,

10:00 because they are too difficult. So what did Trest in his case in 1996, it was a method of life, and it was also very low, but it was very low. So it's not very clear. We have a system, we have a certain number of correlations, but it's all that we have. It's not a problem, it's not a problem, it's a problem, but simple. In fact, I have just a question. Is it really an invention of Kantan? Because, in fact, I learned that it was an eclectic ruse. I tend to think that it is something that goes back to Gauss, etc. Or can we say that it is really an invention of Kantan? Maybe, maybe. Is it possible? Yes, it is. Yes, it is. Yes, it is. Is it true, it is not at all? No, there is no attention. Okay, good. Okay. and then we can see this angle of the rope rope and that's what we see all of them. When we see Bayes generalize what they say, and so on. They see it in another direction. And there is not a whole portion of it. Bayes explains very clearly why it is absolutely that the coefficient of Christopheel is symmetric. So very clearly. And I think that there is this point of view of RS-20 which is still much more than the imaginative. corrective. C'est-à-dire du fait qu'il remplace tous ces calculs avec une vision géométrique, notamment cette courbure qui mesure le fait d'être plate, d'être conforme au modèle, chez Hessmann est présentée comme un défaut d'intégrabilité d'un certain champ de plan, comme un défaut d'un certain champ de plan d'épanolition. C'est-à-dire que ces idées très géométriques, en fait, de dire voilà, j'ai un champ de plan, qu'est-ce qui n'est pas ? Et finalement, c'est aujourd'hui comme ça, c'est cette And I have the impression that this vision explains why all the curves are developed in the space model. Because the tension of the curve, we can say that it is two forms. So it is on the curve. And so there is no instruction to take our curve on our space generalised

12:30 and to put it in an space model. But in addition, it's visual to see that a field of light is always incredible and that there is no default, no problem to develop. All the dimensions of dimension 1 are flat. so this is the point of view of structures non non non and there is also I think an article Thank you very much. So there is an essential difference with the G-structure. Essentially, we are not homogène. So, for example, a field of quadratus which degenerates is a geométric to the ground, but it is not a G-structure in the sense that the group orthogonal is not always the same because, punctually, the metric changes the range. but there I would say that it's not exactly the connection, it's not generalised because finally he says, what is what is an object geometry? When I change the case, I change my form to the geometry geometry, but to a certain number of finies, the derivative of the change of the case. So they see what I'm saying as a function on a fibrel, a repair, a value, but it's not a parallelism, it's a function, but in fact it's this, I'd say, in the literature, For example, what has been written by the Chicago School of Chicago, the Zimmer,

15:00 all these schools related to the study of the group of the isométries and the structure of the geometry, they represent this definition of the norman. But in fact, it's the norman which is said. And then, for the general space, which means that they are exactly exactly G on H, today we will take the search term and we will take the language of the G structure, in the sense that there is an open case in the G-sur-Age where the changement of the earth, etc. It's not a problem, it's not a problem. I'm a mathematician, but when I was looking at the... Well, it's not a math technique. There's a question about the Kobayashi. Well, he's using the... I mean, it's not a general idea. Well, then, it was... Well, it was really referring to Kaki. It's a bit modern, it was... Well, in the United States, but it's not that today... The collection of cartons are the only one in the middle, but it's been a part of it. It's been a part of it. It's been a part of it. It's been a part of it. It's been a part of it. It's interesting. It's interesting to me. I think the concept of modernism has no kind of consensus on what are the questions and what is the approach which is, in fact, and I think that at least, in the case of January, that, for example, in the case, where, effectively, there are schools with different tools that pose different problems. I think that there is a reflection of this reality, the nature, the domain that I ask today, about the other domain in mathematics that we can consider as normal or non-indic. and it's just simply a reality that we have to do with the world.

17:30 I think there is also a place. Yes, I would like to talk about it. I think what I would like to say is a magnificent example, since we are in the seminar of the history of economics, where to introduce a historical context which would allow a good approach to introduce an object, for example, like the collection. I'm pretty sure that the Japanese Japanese Japanese are pretty close to the carton to re-écrire the formules and the formalism, it is to have, they will play for some with the different forms, and just like that. And then, to have a course of geometry, which would be like, and it's for that I love a lot of you, I read the chart, which would be an idea of the tension between the geometry human and the other side, of the space homogène and this idea of homogeneity absolute and of homogeneity infinitesimale and what do we do? And why are we made to ask this question to show the objects between the two spaces infinitesimals? It's the problem interesting to determine the calais which allows us to understand collection, and then we can adapt to the form of a novel, it's to say that we can look at the principle, we can look at the definition of the form, and there I think that we have a very very good example of the introduction historique, it is to say when I talk about the introduction historique, it's to say, it's to say, it's to make sense what are the problems in an other context mathématic, because today, nobody I don't know if it's a question, but to be able to understand, to be able to understand, to be able to understand, to be able to understand the tension that could be there at an time, on the people who did the geometry, and to be able to see the concept of the geometry, and to be able to see how Cartan composed of his own. to have a mathematical effect on how to understand what is a collection, and that

20:00 a collection, it's never, as he said, to find an infinite space in the same way to another. And I think that even, although I was not sure how to say that, because now we are completely in a form of an other, but this form of form they forget that they forget the idea of geométriques simple which allows them to have a good representation of these unique connections. I am absolutely agree with what you did. I think there is a lot of thought that has completely disappeared, even if, as I said, the curation of the concept in the previous year is very clear, there is a lot of information geométriques which practically has disappeared. There is a formalism with a little bit of the idea of the non-mengi, of the li, of the fibres principaux, and the complexity and the intuition that we met in the same way. And that's what it is. Well, I just wanted to be d'accord with just a certain point with what it is. because it seems that the real interest of history is not to say, ah, you have ATX who really made things very interesting, but rather to say, tiens, we have several sectors, because when we are not the only one, we are sure, we know that. What we observe now for the geometry, C'est plus frappant parce que, quelque part, la geometrie, pour des raisons historiques, bien sûr, est dans une position où les veilles sont plus évidentes, toujours, que dans d'autres domaines, peut-être. But what is interesting, because you have mentioned the book of Sharpe, is that Sharpe, in an introduction absolutely genial of this book, he talks about the difference between the germany defenseman who he learned as an undergraduate and the germany defenseman who he learned as an undergraduate and who is so enormous that he did not want to be able to write the same subject. Chan a fait le préface pour lui. Et Chan est étonné par cette réaction que tu dois ajouter. Chan dit par exemple que ce n'est pas clair en faisant l'agenturité des foncières au niveau avancé pourquoi une idée de domaine de bullerie est intéressante.

22:30 Et Chan ne comprend pas de toute cette question. Il est clair. Je ne comprends pas. C'est clair que la géantique n'est autre chose que l'étude des connexions collectives et principaux. Or, la géantique du clé est le premier exemple d'une certaine approche. Donc, évidemment, le clé est intéressant. So, this incomprehension of the notion of the geometry, allied with the idea of the regime of the agency, exactly like that and nothing else, it seems absolutely, in a way, more and more important, but illustrated by these schools, these different traditions, the way to read, to reformulate things in one sense or in another, which do not acquire a total of a total of a position in the world in the world. You know the two languages, a part of this one, which has transformed the things on which they have formed their intentions, and they have made a language more abstract, more accurate. So, you know, the most of the things are masked for the lecteur. There are things that are disparate from the pensée, and that's absolutely right. Yes, but... Because one of the biggest efforts on this notion of the geometry of the space of the image was in the production for diagrammatizing, because there are no figures in this book, all the identifiers, the identity of the Yankee, for example, the conservation of the function, the conservation of the groundwork, all of that, there is an enormous effort to think about the thinking of the problem. Well, when I read the book of the Covey-A-China, I'm sorry, I tried to read it because I didn't get it. When I read Cartan, I didn't get it, but I have the impression that there is a thinking of the thinking of the geometry. In fact, the image initials of Capcom, in which he presented in an espace generalised, technologically, it's more adaptable to some problems. For example, in the case of the system dynamical, of the variety of connections with, let's say, if we look at the titles of the articles, that would be a sense of space and connection to the platform

25:00 because it was the good point of view that we were losing, for example, the dynamics of G over H. For example, for the uniformization, the projection of the geométricization of the system, there it is not that it has to work, it is the analysis. And so, in fact, we don't have enough to do that. But it depends on the whole thing. Yes, it depends on the whole thing. our program, we're very specific, like Carlton, like Carlton and like the other Prince. It's absolutely a question of which questions you want to pose to your subject and which are the tools we can do to respond to these questions. And these questions change from generation It's not an important thing. It's easy, it's easy to do. There are, of course, things you can do. And things you can do. Well, I'm a mathematician. It's extremely difficult to transfer the trigonometry. It's an extraordinary difficulty. I don't know. In general, when you start to write the trigonometry, because it's not the trigonometry, it's extremely difficult to make sense what is happening in text because we are not in two spaces of communication which are totally extraordinary I think what I do is that I use the colors, I use the colors, I use the colors so I understand why we have to do it because we have to do it I think there is a big problem in the text and in the text like this I think that it won't. I think that it won't. Perhaps, you know, it seems that the writing of Cartland, which I said, is absolutely clear, I think you could read it. It's written, it's written, it's written explicitly, it's written to Einstein, it's written, which he wrote what he wrote, because it's not the word, but he wrote what he wrote, which he wrote, which he wrote in a language that he wrote. I think it's also a question of context in which we have a certain point.

27:30 I think that's why there was this exchange of letters between Carton and Weill, which is not the same thing. If I remember, there was an article in Armand Borel about this exchange of letters between Carton and Weill. And in fact, by the spirit of voice, he said, but when I have a variety of ways, as soon as I give the local coordinates, the space tangent in one point comes with a base, the d sur dx1, d sur dx2, d sur dx1. Why, as you write, the space generalization projectile, why, when I give the coordinates, I don't have a unique projectile in the project project, attached to one point. And then there is a debate where each one explains what they need to do. And in fact, what is interesting is that the application of Borrell is interesting, because Borrell explains the two solutions of the two authors, which were a little different and in the same time, which were similar, but... There is a chip that tried to make the right to a product of life, and he said that if you want to see Yes, exactly. Yes, exactly. But it's interesting to me, that's to say that Vague, he already looked at the point of view of fiber, fiber on a variety, fiber on a variety. He was already thinking more in a process. But he was also a project in the 20th century, in the 20th century, in the 20th century, and in the 20th century, Thank you.

30:00 I can't believe that, for example, I can't believe it. There is a tradition, I don't know, there's a difference between Cartan and Montaigne. But, by ailleurs, Cartan, what's interesting is that in his article, he explains and often justifies even his point of view. I don't know, for those that I know, I speak of the geometry, I know all the articles of the presentation of the group. So there, we have someone who tries to explain his points of view, and his buts, if not explicitly, are quite clear. And in fact, he has a reflexive, very hard, in 1924, and in 1922, Poincaré, par exemple, c'est quelqu'un qui intervient en film de film, par contre, il a assez peu de regard sur scène à pénalité. Ce n'est pas du tout le même... Je ne dis pas que Poincaré, je ne suis pas assez loin de lui, mais il y a assez peu d'explications. Je crois que le mot de Picard, quand on demandait une explication à Poincaré, that he is a piquet and he is a piquet and he is a piquet and he is a piquet and he is a piquet and he is even not explaining who is like that but in any way the tradition of what I call I think there's a lot of stuff that s'arrête quasiment. The last year, it's called Hermann Weig. I know a lot of people now, since... There are a lot of people who have written... But I don't think it's an interesting thing.

32:30 Well, the impact is... That's why it's someone who is very interested in passing their field at the same time. I have a question because I have a question that is simple on the notion of the variety, since you mentioned the list of the Fermat-Pak, the 19th century, which is the list of the surface of the Riemann, and my question is about the denominability of the Riemann, because in the terms of the uniformization for the surface of the Riemann, in a way, we demonstrate that the surface of the Riemann is the denominability of the Riemann, because it is true for the list, And so at a moment in the film, I imagine that there is a lot in the film, in the film, in the film, that there is no more than infinity. I don't know what this is going on in the film. I think that in the article of Pancaré of 1907, the uniformism is the functions, so the uniform. So the uniformism is in fact the surface of the image associated with the functions. There is an infinite number of features, but... Exactly. You are the number one. You are the number one. In your book, there is not that. In fact, we cite only the theory of Radeau for the number one. But, for example, if it is true in the superior dimension, how is it supposed to be the definition of the number one?

35:00 He says that there is an electric on it. Ah oui, si tu as une... C'est des endroits à l'infini. Une locale, peut-être ? Allez, allez. En 1913, il n'y a pas de même plus. Moi, je me souviens plus qu'en plus, car en fait, tel qu'il présente le cerveau de Riemann, c'est le cerveau de Riemann associé à une fonction, donc tu as des éléments de fonction, des endroits, et donc c'est des endroits à l'infini par construction en plus. Mais à l'époque, il y a ça. Il n'y a pas l'autre. Ah oui, à l'époque il y a ça. Oui, là on est bon, après on peut le généraliser, mais... Ben du coup c'est ça là. Mais pour les médecins, les gens qui interviennent, soit dick, clignent, soit vague, à la fin, c'est que des surfaces de l'Imane associées. D'accord, c'est un sujet de l'éthique, c'est d'accord. Mais alors à quel moment on a pensé abstraitement ? Je pensais quand même parce que c'était le livre de Goi. It's an abstract theory of this theory, of the theory of the animal. We have an atlas, but we are still talking about the fact that we have an empilement of disques. In these actions of atlas, there is no action of the human being, it's like the other thing. The theory of the radio, it's plutôt 1920. How do you say that? No, but... It's also to say that it happens to be human because of the fact that it takes place to be human, and somehow, if we do the wrong preuve of the uniformization, it's a consequence. Because once we have a uniformized, we know that the real universal universal, he is the infinite. And it's an issue of showing that it's the same as the universal universal universal. We, at a moment, we have to build... Once we have to build a green, we have to build a green on the surface. So there is a problem of the theory of potential.

37:30 I don't know. I don't know. I'm going to ask you a question because I thought... I thought it was... I thought it was... I thought it was a generalist. I thought it was a variety with a structure of geometry. And, for me, an espace homogène, it's a variety where there's a group of dimensions that agit positively. So I'm going from a concept more abstract to a concept more particular. But these spaces-là, since they are structured like that, they are enormous. And I'm going to ask this question, because what I'm doing here is that this year, I'm going to ask a question about the question, what is it that is a variety of people who use the word or the concept? And I check a little bit but I'm sorry but at that point, we don't have a variation to see it as a reason, Thank you. just to put in the matrix. In general, it's a commonality. The matrix is a commonality. It's a commonality. In fact, it's normal to say what is the most important thing to do? I think it's a fascinating thing. In the case of the Vague, in the case of the Vague, there are many parts of the Vague.

40:00 Thank you. for me this question, the term uniformity is very clear, but I think that for uniformity of the surface, we have a lot of spaces generalised, which are in fact the spaces rimanian of the courbure sectional constant, but for uniformity of the dimensions 3, we have a lot of spaces generalised, where G sur H is a space rimanian, not necessarily the biosexion of constant. It's the 8th century of search form with 3 biosexion of constant, 3, which are the products of biosexion of constant, and also groups of lits with a variety of different parts. So, if you want, for me, in other words, these spaces-là, these places, it's a natural I think what is innovative here is not to give the list of the 8 geométries

42:30 to make a realisation compact, but it's also to think that it's possible that all the different varieties of topology can be admitted to the structure. We don't have a position... Well, we don't have a position... But we don't have a position, it's Pianti, in fact. Pianti, perhaps, for the group of the U.S. Yes, yes, yes, yes, yes, yes, yes. Pianti. Pianti. Yes, but when you say group of the U.S., in fact, You can't tell it, tell it. It's a structure in some way. It's a forage and identity. No, no, no, no. No, no, no, no, no. Thank you very much. He is in a problem with geometry, and I think that when he says that I'm going to build a case in which I know what I know about the GCH in geometry differentials, there are at least a large part of it that is concerned. But once he introduced the torsion, he introduced a certain number of fundamental elements, and that's why I said that in his article on the fin parallaxinus, I have the impression that he used to serve, essentially, to the formulas. And that everything he put in place before, to interpret what is the subject, there is only the development of a group, a series of formulas, etc. That's why I say that in these articles, in this correspondence with Einstein, there is no geometry. It is essentially a work, I don't know how to verify, It's not an analyst or a relativist, but, yes, all these ideas juridiques that you have in class do not serve as much as possible. Well, that's just a point. Well, it's a very profound question about the philosophy, and because I would like to say that if we want to think about the geometry,

45:00 and if we want to think about the best possible way, it should be that each act of calculus is, for example, by the French and the United States. And it's not that the act of practice are made. But it's Gauss himself, the most grand calculateur of his time, who says that. And, actually, the reality is that since we are in the transfer of Riemann, most of the time in all the books we don't have to appear the fundamental form of Riemann-Christoffel which is the point of the matrix on the stratage of two forms. And after, when we start to do the calculations algebraic when we look at the cost of Einstein, we are not able to do this. It's a real problem. It's a real problem. It's a paradox. It's a paradox. With the basic intuitions, it's a problem. It's a real problem. So I think it's ignorant. But it's a real problem. It's a real problem. It means that, in the correspondence, at least, Einstein pousse Carton after a few things. He doesn't know anything about the change, because Einstein is not very interested in the change. He is interested in the change. He is interested in a question, which is how to determine the degree of determination of the system of the equation. He wants to determine the degree of determination of this equation. And it's a question for which the captain has already developed in the beginning of the year. It's a technique that you can do. And it's that what Einstein can do. And it's that what Einstein can do. You can do it for me. You can do it for me. You can do it for me. Thank you.