Gustave Choquet recherche (last part) / André Revuz: Souvenirs personnels / L'enseigment de la geometrie

7:30 All of this in France remained prior to two books by modern mathematicians, Alguerra and Neutropoulos.

10:00 I looked at the list of teachers in CDI. In addition, I liked to leave the teachers in CDI. There was an exception, it was Paul Dubrecq, the term algebra. But the main difference that occurred was that this algebra did not serve as an analysis. It was really separated. In fact, the analysis of the technology in the modern world, well, in more detail, I'll send you a little article that I wrote, you have to know that I interviewed a person who taught in the province, Michel Baron, who said things like, how much a province wanted to be modern and it didn't dare, as much as Paris didn't dare to change.

12:30 Yes, Renaud too, he was called Renaud. And then, teaching. In 1940, there were tests of Cartan in Saint-August, in Rome, but they remained isolated. And Cartan was in Paris, but in fact, Cartan did not really teach at the university, he was with a friend. He did not have a direct influence on what was happening at the university. And then, his influence, his teaching, was essentially overshadowed by their position at the university. When we went to see him, we saw other things, we knew very well what he was doing. We knew very well that he was preparing for evolution. And he even talked about it with the teacher at the time. Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, Châtelet, And at the same time, the idea was to have clear and unbiased courses. So it's a bit difficult to teach powerful and abstract theories and at the same time to have clear and unbiased courses.

15:00 Well, it can depend. Essentially, it will depend on the starting themes that we have. In other words, what are we going to put forward, the problems that we want to solve and for which we are going to work. Where are we going to teach these tools that we put together? In any case, Chauquet did not hesitate to say which types of problems he wanted to solve. But the point is that we do not know much about it. Because we talk about it without saying it. It is radically transformed. So this is especially the second point that leads to the partialization of this theory. Especially with the partialization. And the question is actually the big problems for which we have built these tools. As a result, both the teachers and the students, in order to prevent the absence of motivation on a table that has been written or written by a critic who has just been taught,

17:30 we take the table and conclude that it is a little bit what is being written. And it seems, I admit, that the thing is getting worse. That's it. Well, listen, I think that's all for now. Thank you. Maybe before I give the floor, maybe some questions, some compliments, because there may be people who have a lot of things that you know, that I know, that I don't know. Do you want to comment on the questions? Yes, yes, if you want, you can ask the speakers. But that's it for this work of mathematics. You can start by... At the beginning? These are the steps of his career. Yes, that's it. It was done by observation. It was done by application. There are three parts. And there are other parts. These are the steps of the work of a mathematician. It's the ability of a mathematician to... Thank you very much. This one. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. We need citations from... ...observation, dematization, solutions, and... ...that's the one I wanted. So, I'll go back to you. ...citation and... ...that's the one I wanted. ...citation and... ...that's the one I wanted. ...that's the one I wanted. ...that's the one I wanted. ...that's the one I wanted. ...that's the one I wanted.

20:00 Thank you for your attention. Thank you for your attention. The man of science is a 90s writer, a document in which the authors ask this number of great French scientists to explain their views, their working methods, their fields, their beliefs, etc. By the way, he is called the man of science and there are two more women. I would like to make two remarks concerning shocked and lazy people. I knew him better than you. On the one hand, I remember his capacity. It must have taken him some time to do that. And secondly, a very personal memory. I needed at some point a theorem on convex compacts. I asked him. He answered me with a complete demonstration, which gave me a sense of service. So, I could... It's a productive barrette. He didn't do all of these.

22:30 There was a lot of things to do, and he couldn't do them all. Just to give an example, how many students did you have in 1958? It was absolutely full. There was a second on the door, there was a second on the door, and he had three meters left to move. There is even an episode of the two others that I have not mentioned, but which is also in the book, in the amphitheater this year, with someone named Philippe Courrèges. You know, I had a lecture at the beginning, at the beginning of the semester, in August 10th, and his theme was, you are not formal enough. Well, why not? And I'm glad that Philippe Courrèges, it's wonderful. I was shocked, anyway, I was shocked because he said... Ah yes, it's a good thing. You have to admit that at the time, he said, well, he thought I was shocked that there was a department of physics.

25:00 I'm sorry, I'm sorry. There is probably... ...at least 8 chapters. We have introduced it. You spoke about his tendencies in the 1920s. It seems to me that there is a chronology. There is first a demographist, and then he has explained it. Because one cannot think that the forms of social contradictions have always been shared between a tendency to... This is a combination of mathematics and geometry to combine with research, procedures, objectives, and so on. Hasn't this remained with you as two contradictory aspects? I don't know. The contradiction, I find, at the research level, is to say who is not competitive in France. There are not many ways to do them. What I described earlier is a little exercise in the dogma of mathematicians being like that. The difference between them is that one of them was very aware that they knew how to write and how to communicate. But as an academic researcher, I could not. In these terms, there is a theorization of quantum mechanics in which there is a re-authentication in the research of the two regions of the intuitive aspect, which seem to be in the mathematical realm, as in the case of the objective aspect, which is seen in our text on the theory of the kinetic mechanics that C.M.L.Y. wants.

27:30 Well, it's good that there is the collective aspect, not the whole skeleton of the chain. Thank you very much for your time. Questions? Thank you very much. I would like to give a few compliments to the students at SCATI, in Galski. Choke and I have had a friendship that lasted more than 70 years, which began very quietly at the normal school, which was under the same promotion. By the way, he was a very good professor. He had two little geniuses, I'm not talking about myself, but he had two little geniuses that I was very shocked by. Six honorable mathematicians from Basque, who had three remarkable physicists, with Elissi, Jean Lapierre, Coton. There was a fourth one who was a good one too. I'm going to finish with this kind of talk in Paris, with the general inspector. A small detail, the placement of entrée was completely out of the question. I think it was the last one. Entrée was, thanks to a deletion, the only one in the promotion. Choquet was in the middle. It is explained that Choquet was visualized, he lacked a bit of craft, like the theory of hydrography, which he did not like. The determinants in taube, it had it and that was it. So we found each other in this lecture, we got to know each other little by little, and then we saw that we had in common first of all a definitive passion for mathematics.

30:00 For us mathematics is nothing more beautiful, so it seems so. And then we had at the same time a lot to say about the teaching we received and that we were receiving. I will start with the one we received, although it was only later that Choquet and I agreed on the question, because when I was a fourth-year student, the day I did my first demonstration, it was a revelation. The way to conduct these thoughts is to find them nowhere else. This is the positive phase. But the negative phase, the beginning of the choreography, it sucks. Well, I'm afraid that at the end of the 20th century, we will have declared one second in total zero, it's going to disappear in one second in total zero by saying the same bullshit. So the bullshit, well, I noticed, shocked me, had spelled exactly the same words. Primo, demonstration of the state of quantum physics. This is a trickery. So, by the way, I may have been wrong, but in any case, what you do here gives an example. Let's take Moutamos' demonstration of the qualities of a triangle, that a spherical triangle and a Planck triangle, which have equal sides, are equal. Second bullshit is the division of the right. No, it was a property of the right, and not the definition, i.e. a characteristic property. And finally, to crown everything, it's nice, this magnificent definition, listen carefully, an axiom. It is a true proposition that cannot be proven. Well, I will be able to solve these questions later, but in the field, we knew very well the teachings of the Faculty of Science of Paris. And there, we were a little disappointed. Why? Because each of the teachings was not bad in itself, but there was no violence. Where did it come from? Where did it go? It may not have changed, but in the end it had deeply influenced us. I had in particular a certificate that was great. It was a higher geometry certificate made by Annie Cartan. The subject was the theory of Spinell. Spinell, as the captain told us, is a vector that must not be analyzed.

32:30 But it wanted to be the theory of physics. We didn't know anything about it. So, locally, we understood a total of zero, and the unfortunate Ebersmann who was in charge of conducting the papers told us throughout the year that he wanted to do some exercises, but that he really didn't see what he could do on this subject. The miracle is that the paper came up with a problem that was on the table and that was feasible. But that's not all. Next to the papers, the paper teaching, there was a teaching of what? Descriptive geometry made by an inspector at the University of Strasbourg named Thibault, whose name is Satie. He gave us very abusive lectures, and in the end, he managed to equalize the course of Cartan and the geometry of Zeta. I thought he liked to laugh about it. The descriptive geometry served me at least once. It allowed me to have a certificate in geometry of the universe. What did we do to protest against this? Well, we did not go to do demonstrations, but we tried to find something else than what we were told. So there was still a library at École Normale where we could find interesting things. So we saw the little books of the Borel collection. We attacked, we had to be brave, the Bair thesis. By the way, a small detail, the volume that I took from the library is a crumpled volume that was not cut. Well, let's move on. We have even slightly approximated the memory of Danjouas on the number of derivatives of functions. In other words, we have created a small universe. We found ourselves very happy, very happy, which was done with the work of Lebesgue, Borel, Baird... Is it marginal compared to the rest of mathematics?

35:00 On this subject, I read a Schwarz novel a few years later. Schwarz is good, but he is not in the mainstream of mathematical research. Then I notice a lot of Bach. It was a seminar that took place here in the room where someone had explored theories precisely in this area. There was in the auditorium in the awakening who declared it was pathology. I remember the head of Schottbeck, not me. He said nothing, but I have never seen Schottbeck in comparison to that time. So the problems of Bourbaki and Schottbeck, there may be some of that too. Well, who has a tendency to say, well, all this is very interesting, but it's marginal. In the post, you, you take care of everything, you direct the games, you tell us a case of immorality you don't know, the application of physics you don't know, etc. Anyway, it comes to the third year, we prepare the creation, and I and I, we put ourselves in the same place. So it seems, very amusing, our ways of solving the problem of the creation. The problem is that they go deep into stuff and they don't worry about details. So I, stupid as a pig, put the stuff in the right place. With the result that one day, Jean Tarmois gives us two copies and he says, no, you shouldn't write like that, it's superficial. He had gone deep into things, but... Pierre-Henri Marmureau-Gauzy is another one. I was a little embarrassed because it's not very fair. The lesson may be repeated. Because when infection came, Mr. Choquet came out of nowhere, and there was such an advance in writing that Schwarz, with two hands on his back, could not stop him. So we are all attracted. There is Marot, who in her youth was a polymedic who had cut a leg in the air. The woman who is a Quat, who was not yet an activist of the French, and Choquet. So all the others, to the co-practices of defense that we do in fact.

37:30 A decrease in the quality of the teacher and the teacher. The teacher is unbearable and has no errors. Chauquet puts a lot of written notes. Marot, whom I had seen one day in Paris, said to me, Chauquet takes written notes on the first foundation. I don't know if it's the first foundation, I'm telling you what I had said, Marot. On this, the war begins. September 1939, general mobilization, Choquet was mobilized and went to the defense practice that we all know, which was returned to Vizcaros. This is because it is a phenomenon that I have not seen too much detail, that I quote you with caution, but it seems to me still revealing certain things. One night or one evening, in the barracks, some official students sang the national anthem. Scandal! This is a statement that he had not sung. So what happened? Did he say he had sung? It's not the same as the fact that being a soldier, being in the IAD, he had the opportunity to do math rather than math, and he had jumped on the occasion. And the next time, oh my God, he said shit to the army. So this being the F40, historical data. And for ten years, we have been quite far from each other. First of all, I spent two and a half years in captivity, after which I was a professor, I studied a little, and then in 1945, south of the C-13 in Paris, I went to Turkey with my wife and children to teach mathematics at the Technical University of Istanbul. In France, what is Chopin? Chopin continued to count the notes of the world and then there were 46 where he passed his time.

40:00 So there I had the instructions because while we were in Istanbul, first I had been joined for two years by Jean Dufresne. We got together to invite Montel and Donjouan who had sung for us. And then Montel said Amen. I was the one who gave him his thesis, because he wanted to go to Poland, so I told him, OK, no thesis, no Poland. The problem is that he can do three theses. He was excellent, he went to Poland. Why Poland? Well, the Poles were still people who were still in the same sphere as the Danes and so on. And then came Professor Agournon on January 1, 1905. I come back from Turkey to be attached to research. I had still solved a little bullshit when I was in Turkey, which is by looking for conditions as general as possible for the existence of differential equations, differential systems. So, for pathology, it was pathology at the end. It was the goal of reconditioning, etc. Which, as we would say, had a lot of fun. From the point of view of practice, the funny thing is that it was published in the annals of the Technical University of Istanbul. The relationship with the technique of this theorem was completely marvellous, but Chogay liked it. So I come from the research of the CNRS, theoretically under the direction of Professor Arnaud Dornier. And then, automatically, with two declarations by Schottet, the first one, I did it for the research team. Well, we have a support that is precious to me. Second, you should study the growth functions.

42:30 A reflex in the same growth function, integral of the number, distribution function of probability, well, I tell him, I could maybe try to find a general graph, very good. I went there, and every time I found something, I said to myself, very well, let's keep going. I must say that at that time, I think there was a problem with the capacities. Because in 1950, they started the capacities. Well, I agree with that. So, in the end, they had only used me for the lighting section. So, let's see what's on the side of the controls, because the capacities are endless. I think I could demonstrate a few nice properties. Oh, you're crazy, he tells me. It's too beautiful to be true. Oh, we'll find a good example. We spent an afternoon at Sherbrooke. We spent it. He spent it. Because I have no moral to say, for example, Oh, he tells me, well, you keep it, we'll do it. You have to do a finer theory. So there, I follow myself. It's more my memories, perhaps. But it still had an interesting relationship with me. It impressed me. I have demonstrated the theorem that I am good at. When I go to the United States, we will learn everything. And then one morning I wake up, as I give you lectures all day long, but in the morning I wake up at 8 o'clock and I write the three demonstrations as if they dictated to me. I understood that day that the people who said that they had written under the dictation of God These were not necessarily charlatans, because, tragically, who knows who invented this? Who knows who invented this? It was me. When I talked about it with Joguet, he told me, well, yes, it's something that happens to me often,

45:00 we dry ourselves for a long time, and then in the morning, bam, it comes, and we say, well, where does it come from? And I said, there are several individuals, there is an unconscious who speaks of the code, and then there is an unconscious who says nothing. All of this, Choquet, I send it to him, and he answers me very well. It's funny, these years we've been working together, we've been working together in a certain center while being together. In the middle, I was telling Choquet what I was doing, but he didn't say much. But I was always reasoning according to the distribution and their capacities. To such an extent that a few years later, when I was alluding to these times, I would say, of the time of the capacities. Yes, of course, I had a little luck. The results I had given established a duality between my functions of distribution and these capacities, which allowed me to demonstrate things quite quickly. So I had my small contribution to the test. But what I would especially remember from this period is that in the end, when I arrived in 1950 in Paris, I was shocked. A professor at the Faculty of Sciences recognized my type. And I thought to myself, how many other bullshit are there in science? It could have been Examen or Havannes. We were on a totally equal footing. And I think only one benefited from it. Choquet did not like hierarchy, he did not like to be dominated, he did not like to be dominated. For him, every mathematician was someone with whom we spoke on an equal footing and it had to work that way. So, it was a delicious period, I can say, of academic work. So after that, there was the revolution of the Certificate of Differential and Integral Qualities.

47:30 I took part as an assistant to Choquet. Choquet had three assistants for this certificate, which were Marie Lechwarz, Jean-Pierre Kahn and me. So I have excellent memories of it. And the relationships with the students, as Rubot said, are in my heart. But the relationships with the students were excellent. And then we took exercises. And by the way, I'm going to tell you something else. It's that Choquet told me what he was going to do. What you have provoked, it's going to be terrifying. You... you... you're going to travel to Paris. The calm audacity, I would say, freedom. I mean, I think that, precisely, Choquet was the... an incarnation of freedom. We couldn't impose what he wanted on him. But if he wanted to do something, well, he would do it. So, that's all. And then I slipped away by occupying more and more questions. I was deeply engaged with the APM in many non-official national or international commissions. And then, in 1966, Choquet told me that they would introduce me to Paris. He said, we need someone who cares about teaching. But no, he called me a disciple, as usual. There were two candidates, Hervé and me. So they chose Hervé, and then for me, they said, it will be for the next year. And the year after, there was a more dangerous incident, it is that Schwarz, who was not there the first year, said, we have not taken into account the decisions of the last year.

50:00 Clash! It was barely necessary, because I was in principle in the middle of the two. Very different years already, because with Choquet, etc. Mathematics, and the teaching of mathematics, which had not been there before, for Schwarz, it was quite the opposite, everything was mathematics. So, Schwarz, Maïtras, we have a letter, worthy of the four ancients of the friendship of the commentators. In the end, I was in Paris, I was already in the commission. We called Iremes, we made amusements to what was there. The naivety of Schottet tends to what we can do in the second-degree training. So, I must say that here we should be a little careful, because many of Schottet's missionaries, not exuberant but serious as you can see, putting their brain cells where they should be, we accompanied an experiment on hundreds of places with voluntary professors. Now, who says volunteer professor, says volunteer professor. A volunteer professor has a lot of experience, he gives it his all, and he has all the qualities that you would like to find. The result is that it worked. Well, another stupidity, I must say, is to ask ourselves the question, how will the students react? The result of the experience is that the students react well, so we can go there. We haven't asked the right question, how will the professors react? Of course, it was hard to react. I understand very well. We were twisting their habits. It implied that they did not know what they were doing. So it was a... The only positive thing that resulted from the commission of scientists, it was the iras.

52:30 Well then, Choquet, well, Choquet has disappeared. It is one of the pains of my age to see all the... We didn't see each other very often, but every time we did, it was as if we had left the day before, and then we really felt like we were at home. And Choquet had qualities that Raphael didn't have. First of all, freedom. He was a free man, but he respected the freedom of others. I would say that he was more than respected. He provoked it, he helped it to develop, and on the other hand, his sense of equality. It's a bit funny, but freedom, equality, fraternity, in a sense, shocked, it was a... Yeah, I got lost. An incarnation of freedom, equality, fraternity. So, I... I found it to be an absolutely exceptional lecture, not only from a mathematical point of view, but also from a mathematical point of view. And what I would like to do is that it is an elucidation and that Jean is the same, an inverse of Jean. Yes, maybe a comment anyway? I'm very satisfied with your explanation of the failure of the Michnet reform. You say that the professors were not disposed, that they were not ready, but I'm not quite sure. First of all, I'm not quite sure that it was true, and I think that it's less of an explanation anyway.

55:00 There may be the fact that they were not trained enough to do it, but that they reacted badly. I think rather that they had a good reaction to that, but listen, as soon as the IAEA was created, it was said that it would be a continuous training. Who came to continuous training? It was not a GC. So, experience, please, because I had taken advantage of it. I am from a generation that benefited from what you had told me, and especially since I was in a college, I have the majority of... He seemed to have a terrible taste. He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... He did not want to... These were people who were about my age, that is to say, I wouldn't say they learned it, but they found themselves in something that made them happy, that was familiar to them, and where they saw a little bit of pedagogical application of the media when they read, and it wasn't... I'll give you a hypothesis. One of the reasons why it didn't work... It is true that there were some problems with some of the experiments, but it is also possible that there were psychological conditions. And all the internal cultural aspects have not been measured in the same way, or precisely in the same way that I would like to see. If I had been a student, I would have said, those who want to do the new programs will do it, those who want to do the old programs will do it. And I am sure that the new programs will be answered. Only in France, where it had to be the same programs, I was worried that it would not be enough.

57:30 Thank you for your attention. This was the same problem that existed in the other countries than in the rest of the world. But for the PEGC, in Paris, where there were only 500 PEGC, they came because of the fact that they were happy, because they were going to give back their houses. And besides, the result was that some of the competitors, but the others, they were rejuvenated. I'm still remembering them, aren't I? But I'm not asking for that. It's going to be with a radiant smile, because it will help you understand the calculation that you have to do in the first place. I have to tell you in such a fundamental way that I was at the time, I had the opportunity to talk about the difficulties of the first years, after his reform, and he told me, I don't remember, it was normal, a generation had to be sacrificed. So here is the conclusion of the lecture he gave me for the course of the lecture. Thank you for your attention.