Le Foncteur Sonne Toujours Deux Fois — De l'Analyse Fonctionnelle à la Géométrie Algébrique

0:00 The Institute is really here to help you for all practical purposes. The weather is supposed to be slightly better in terms of temperature for the rest of the week, which means likely to rain, so that's very comfortable from that point of view. So I hope you have to be careful that some parts of the premises at the moment are still a bit slippery, so be careful that there is some snow that means that the ground is still frozen, so be careful. Second point, very practical one, and it has to do with lunching. We had to put together quite a stated organization in the view of the number of people who showed interest in having lunch here. So please, the ones who committed themselves to have lunch, please have lunch. The ones who committed themselves not to have lunch here, please stick to that. We can still make some changes, but the changes made at the last minute are always difficult. If you would like to have lunch, they will try their best, but at some point there will be a limit for different purposes. There will be various breaks, which I hope you will have the opportunity to be joined up here in the way. The number of people registered is above the capacity of this amphitheater, and therefore we were forced to have the conferences, video, transmitted to any other amphitheater on the campus. It functions usually pretty well, and of course you are missing the presence of the printer, which is something quite important, but nevertheless we hope this will function. So, in a sense, if you want to be sure to be seated here, you have to come early for the concept. That's the only recommendation I can give you, because for sure, it's people, all the people who said we've done their specific time, it's going to be above the capacity here. And it's not possible to . So that was the second very practical thing I have to say about the arrangement for the Concerning all practical things, you can always ask the staff. They will certainly help you. Concerning how to get here, I must say that unfortunately the subject connection to terrorist is very unstable.

2:30 I'm using it very regularly and suffering from protesting against it on a very regular basis. I'm not saying that now it has reached a level where the various mayors of the cities along the line are really now taking a very formal protest. So he's there with us to do this. I hope it will be a good week, without too many interruptions from the point of view of transportation, which I know for many of you are very critical. So I wish you a very good conference, and I'm sure there will be many discussions further to the lecturers themselves. We are video recording the conference, but at the same time you can ask questions and you don't need a microphone. we have enough microphones to get the voice of people speaking. So, of course, the speakers, for obvious purposes, will have a microphone at one other moment. But when you ask questions, if you ask questions in a loud enough voice, there should not be a problem for you to get them recorded. I wish you a very good point. And a great thanks to Spencer there and Claire for all the work they did and also to the staff for all the cooperation. Thank you very much. Thank you. Thank you.

5:00 Thank you. Thank you. Thank you. Thank you.

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10:00 Thank you. Thank you. Thank you. Thank you. Thank you.

12:30 Thank you. Thank you. The protocol sometimes is harsh, the proof I accept will introduce Professor Gierke. In fact, everybody knows Professor Pierre Cartier in the majority much better than me than myself. So, Professor Pierre Cartier will give us a talk on the prototype And so for me it is a pleasure and I must add something because I could not participate in all the complexities of yet of these works, but I had the fortune to know his wife, Monique Thank you.

15:00 This is someone which is very dependent in my heart. It's memory. Thank you. Thank you. Thank you. I think everyone has understood the pleasantries. The actor is always two times, this film is the most famous. And the intention of my title was to explain that, according to what we often believe, Grottenwick a eu deux heures en mathématiques, sa première sur l'analyse fonctionnelle et je vous ai amené ici donc sa thèse, ce gros volume rouge qui parut en 1955 et qui contient laissant une partie de son travail en analyse fonctionnelle et puis ensuite il y a eu ce que nous connaissons, la géométrie algébrique. And we have a tendency to forget the first part, because if the first part was very brilliant, the second was even more brilliant, and had an influence certainly more durable. but it is my intention to show that there is no discontinuity between the two parts of the work of Grotendick and that the actors are always two times and that in his work of functional analysis the actors are already very present. Alors bon, si j'ai besoin d'une autorité, je cite ce que Deudonné a écrit dans l'introduction au Groteur-Diffrécherie il y a 20 ans. Il y abordait un sujet entièrement nouveau, l'étude des topologies raisonnables sur le produit tensoriel de deux espaces localement convexes.

17:30 Seul le cas des espaces de Banard avait fait l'objet de travaux antérieurs. However, in the study, as original and profound as it is, we already recognize its path. It is, even at this time, we don't even talk about catégories, it is already the spirit that dominates in the research constant of natural definitions, of the properties of the authorial, and what will become systématic in the works of the interiors. Well, everyone knows that Diodonné was the master of Grotendick, and that if Grotendick had an ancestor in mathematics, it was certainly Diodonné, for all sorts of reasons. It was already with Diodonné and with Laurent Schwarz that Grottennick had done his thesis. And then, in collaboration, it was mentioned earlier that the two first professors of this institute were Diodonné who accepted to come here at the condition that we refuse in the same time that Grottennick. And then there was this long collaboration that gave us the elements of the geometry of the algebra that Jean-Pierre Boutouillon has reminded all of them. So, Diodonné, if Grotendick wants to have a reference, it is Diodonné. Alors, Biodonné était un personnage assez étonnant, à la fois, bon, un personnage massif, avec une énergie vitale et mathématique extraordinaire, avec une voix tonitruante qui lui faisait souvent dire des choses qui paraissaient énormes à beaucoup de gens. But contrary to what we thought sometimes when we didn't know better, it was a man sensible, capable, a good musician, capable of changing his life when we had succeeded in a combat, which physically wasn't always very easy, but he was capable of changing his life. What is very sad, it is that the relationship between Grotendik and Dieudonné s'est well terminated in 1970, during the international congrats of mathematicians who was held at Nice, and Dieudonné was one of the main organizers at the time, he was a boyan of

20:00 the Faculty of Science of Nice, and there was, unfortunately, an episode a little and who has finished their relationship, but not Biodonné a conservé son estimate. Well, what I want to do today is to show that if there are already factors in the work of Biodonné as an analysis functionally, even as, as explained Dieudonné, the facteur was not yet as répanded as today. My thesis is that Wotendick has made possible to, in a certain way, the way which had been traced by Bourbaki and Dieudonné in particular. Well, then, I'll give you my plan quickly. Donc, je vais traiter la topologie et l'analyse fonctionnelle avant Grotendik, et j'examinerai comment elle s'est développée, et surtout le point de vue qui est celui de Bourbaki dans ces questions-là. the work of Gottendik in an analysis functionally, it is to say, for the essential, his thesis. And then, I would say, in two parts more quickly, the heritage, it is to say, what the school of Guelfand has been able to express in Gottendik, in perhaps that Gottendik would say in the devoyant, puisque les principales applications avec Guelfand Millos, c'est au calcul des probabilités, puis ensuite au fondement de la physique mathématique, la théorie quantique des chances, peut-être qu'au sens de Grotendik, il y a un dévoiement, mais en tout cas, c'est certainement le plus bel hommage que Guelfand et ses collaborateurs aient fait à Grotendik, c'est de s'emparer de ça. D'autre part, bon, il y aura d'autres, un autre héritage russe de Rothennik que nous connaissons, qui passe par Javarivis, Manin, Rinfeld et compagnie, et compagnie et beaucoup d'autres. Et donc, mais je voulais dire qu'en analyse fonctionnelle, l'héritage est important.

22:30 Puis j'expliquerai rapidement comment s'est faite la transition vers la géométrie algébrique et essentiellement ce sera un commentaire rapide du texte dit de ton ou cul, c'est-à-dire sur quelques points d'algèbre homologique. Alors, je voudrais commencer par commenter ce que j'ai dit en disant que Dieu donné est le père de Grotendieck, le père mathématique de Grotendieck. Well, of course, the Père Naturel, if I say, of Grotendie, we will see the conference of Charles Laot, which will explain a bit of the origin of his family. It's a very complex story, as you know. And I signale that in the announcement of this conference, we have given a list of texts, texts more or less recent, which relate to the person of Grotonique, to his story, to his story personal or to his family, and that all these documents are which is téléchargeable in PDF on the site of the IHS. Just a little detail. So, the president of Rotendik, it's first to Nancy, as I said. I remember that Rotendik, after the studies at Montpellier, and by the next week, we organize a little bit of a conference which will be held to Montpellier for two days and which will be assembled on, essentially, the influence of Grotendieck in an analysis complexe, which is the influence of the complex analysis of Grotendieck. There is a difference between the complex analysis and the Grotendieck, which is a little bit of a satellite conference, and which was destined, our colleagues from Montpellier, tenait à remarquer que Grotendig avait été étudiant à Montpellier vers 1946-1947-1948 et qu'il était devenu professeur après sa reprise de 1970. Bon, donc, après une année à Paris, Grotendig a assisté au séminaire Carton,

25:00 c'est l'année 1948-1949, le premier séminaire Carton, And apparently the relations between Cartan and Grotonique have not been the most harmonious, but those who have known Cartan, who had all the abuse of an university huguenot, very strict, very firm, and the young fool who was Grotonique at the time, So, we can hardly imagine the relations precondent. It didn't have to have the style mutual. But the first contact was a bit rude. So, after a year, on the council of André Veil, Carton envoyed Grotendick to Nancy, where, at the time, Nancy was the chief of Bourbaki. There were no less than five members of Bourbaki at the time who were professors at Nancy. And Delsa, who was the lawyer of science at Nancy, had very great ambitions. And there was a series of books called Nankago, because at the time, André Veil was in Chicago, and the rest of Bourbaki, well, the most part of Bourbaki was in Nancy, so this contraction of Nankago. Un certain nombre de livres importants de l'époque ont paru chez Hermann, aux Éditions Hermann, sous ce cycle. Donc on l'envoie à Nancy, ce qui a été une très très bonne décision, puisque là, Delsa et les autres membres de Bourbaki avaient une activité importante, and they were able, but they were able to encadre the young, they were not submerged nor by the teaching of the students, nor by the number of students. It was the beginning of an enterprise, an adventure. And, of the other hand, they searched, there were a certain number of people who were all over there, and Wotendick was the one who was the best. a certain number of people have passed away. And then, the point of start, the point of start, it is in fact a theory extremely general of integration, according to the BEG,

27:30 that Grotendieck had reconstructed, without doubt, with some indications, but quite a large part of his own ideas, and as we can say, it's a very general theory of the integration of Alain Lebeck, and he was quite proud of it. In fact, he tells in Records and Semayes that during all his childhood, when he was at the university, then at Montpellier, he was facilitated by the notion of volume, of air, that's the theory of the measure, and and that he reprochait a lot of exposures that we would do, the manuals that he would dispose of, to give, for example, the means to calculate the volume or the air, but without explaining what was the concept of volume and air. And, as I said, as God has written it, it is not a path of a mathematician, so that he is able to pose questions where things are natural for those who are fouled. So, Diodonné et Schwarz venaient de finir un article extrêmement important qui s'appelle « La dualité dans les espaces FLR » sur lequel je reviendrai, et qui était destiné à fournir les fondements théoriques à la théorie des distributions de Laurent Schwarz qui se développait à la même époque. The first edition of the book of Schwartz is 1950. So, according to the legend, Grotonique is arrived, with full energy and imagination, and Biodonné considered that it was his duty to be paternal, to calm him a little. And he said, my boy, my boy, the theory of the Vec, she already exists, but it's not necessary to do it, but make you show what you are capable of doing. And at the end of the article on the space F and LF, which is reproduced in the works of the Dieu Donné, he posed 14 questions, quite subtly, on the problems of the space vectorial topology. Two months later, Dieudonné was at the University of Mathematics of Nancy, the Institute of Edicartan, and Schwarz who was on the second floor, had heard the voice of Stanton, of Dieudonné, say,

30:00 And, actually, that was the point of the start, and some articles are open. Well, then, he did his thesis very, very quickly, because if this thesis was published in 1955, in fact, it was finished in 1952, 1953, and, so, it was the achievement. Then, his career led to various places, essentially in Brazil, in São Paulo, A certain number of French mathématicians have visited São Paulo at this time. André Veil, Dieudonné, if I'm not mistaken, Delsart, and a certain number of others. So, he went to Nancy to go to... It is also to say that, as he was not French, because he was, in the history of his family, he was a patron, he traveled with what we called before 1939 the passport of Nelson, which is now the passport of the Nations Unies for the refugees, which is certainly a great progress for a certain number of people de disposer d'un tel passeport, mais enfin, ça pose toujours des questions, même quand on a un tel passeport, ça pose des questions pratiques, et Wotendik, à l'époque, ne pouvait pas être recruté en France, il aura, bon, il aura diverses bourses, et il faut dire qu'il a eu la chance, on a eu la chance que son, celui qui a l'administration de l'éducation nationale a eu, a regardé son dossier, qui s'appelait Magny, qui était un inspecteur général de mathématiques, L'ouverture et l'ouverture de se rendre compte qu'il pouvait peut-être aller à la limite des règlements administratifs pour permettre à un jeune comme lui d'avoir les moyens. Bon, ensuite, il sera pris au CNRS. Le CNRS était la seule institution qui permettait d'avoir des étrangers. Et il était pire qu'étranger, il était à Paris. So then he had several years where he went to Kansas, where he went to Brazil, and then in 1956 he returned with a post-scenario,

32:30 This is the master of research research. He is, as we say, director of research in the second place. And it was in 1958 when the institute was created, when this institute was created, of course. We have 50 years of this institute, and I thank my colleagues who have insisted for that we do not finish the 51th of the DHS without mentioning that Grotonique had 80 years ago, and without rappeler, some of them have said no, just to rappeler his influence here and to show that she is always present. So, I've already talked about the rupture with the données. I would just say that, contrary to what often we say, we think that Bombaki was an opponent of the categories, and I have to ask myself in front of that. It is true that in the printed books of Bombaki, there was no chapter initial on the categories. There were several attempts to do it, it was an avorting for all kinds of reasons. Those who would be interested in reading a text that Armand Borrell has written on Bourbaki, which is approximately the title, where he explains, in my opinion, in a way all of a fiddle, all the internal debates of Bourbaki at the time, to know if it was or not to incorporate the categories. Just for a little bit of this, I have a list of the first generation of the first generation, that is to say those who, from 1935, have collaborated with this company. On the right column, I have seen those who quickly separated from the company. Le Ré du Breil, un algébriste de Possel, qui plus tard sera un des pères fondateurs de l'informatique et de l'automatique en France. Et Mandelbrot, non pas Benoît Mandelbrot, mais Solène, son oncle, qui a été professeur au Collège de France et qui était un spécialiste de l'analyse fine à la polonaise. C'est des fouillers selon les traditions polonaises. So, they were quite quickly separated from the rest of the group. They were all parted in 1939, they were all left.

35:00 Well, there is also one or two other ones, but it's less important. And then, at the left, there are five left to the end. But I added Rossmann, who was left in 1948, in circumstances that I had never been elucidated. J'ai fouillé les archives et c'était avant que je ne sois là et j'ai fouillé les archives. Je n'ai jamais pu comprendre pourquoi il l'avait quitté. Mais j'ai mis des petites étoiles à ceux qui ont joué un rôle important pour les catégories. and Rossmann has developed all the schools on the categories, in a way almost simultaneous with the Grotendick, and unfortunately, there were very few reports between these two schools, which were developed simultaneously, in the same year, and in the region Parisian, but Rossmann is the founder of the theory of the categories, and there is still a lot of things that we can learn from Rossmann, And then, well, there is Carton, who wrote with Hegelberg, homology to the algebra, so he was radiated very quickly. There is Chevalet, who, it's a little tardive, but when the geometry algebraic at the Grotendick had existed, it was radiated in the category. And then, Dieudonné, Dieudonné, who, well, Dieudonné, who was a pragmatist, I don't know if the categories were in the same way, But in any case, when he wrote the EGA with Grotonbeek, the categories played an important role, he les a absorbés. And then, of the other hand, Bloomberg and Grotonbeek represent the second generation of Bourbaki with Schwartz. with Schwab. Schwab was held at the beginning, but it was an analyst, and it was a bit, well, it was a bit in his mind. But Delsat, Delsat, Delsat, he doesn't care about it. Delsat is a mathematician, of course, more classic, who was interested in an analyst classic, but he didn't have a category, but he was not at all. Quant to Andre Veil, he He didn't want the category. He didn't want the category. He didn't want the category. He didn't want the homology. He didn't want the moment to use very finely, for example, these notions. In the homology, he had a very important result in André Veil on the deformation of the discrets, which is the good theory homology.

37:30 he used the gene homology, and so André Veil, when he had the need, he savait s'en servir. And maybe one of the most interesting contributions to André Veil, his demonstration of the theorem of Durham is visible onctoriel, there are more complex etc., and so André Veil, but it was part of his copy. Bon, alors, donc, le 19e, ne pas oublier la thèse de Grotendieck, bon, dans le Fechschrift qui est, c'est trois volumes qui sont parus il y a 20 ans pour les 60 ans de Grotendieck, on a, sauf la citation que je viens de donner, il n'y a presque pas d'allusion au travail de Grotendieck en analyse fonctionnelle. Et, bon, c'est... Mais enfin, Pizier nous parlera des prolongements actuels de cela dans quelques jours. Bon, voilà le... Voilà donc pour situer les choses. So, I will put a little bit on the topology of the function function before 1949. J'ai choisi 1949, c'est en gros l'année de naissance mathématique de Grotendick et en même temps c'est l'époque où paraît l'article de Schwarz et de Donner sur la dualité dans les espaces FLF et c'est aussi le livre de Schwarz sur les distributions paraîtra en 1950 et la recherche de l'écriture. Well, Bobakir, I would like to show you how Bobakir, in fact, he has not been able to achieve the category. He has not been able to achieve the category. Well, even if there is no book or chapter special on the categories, there is a last chapter of the theory of the ensemble, which is called structure, which is called sometimes the categories of poor, because in a logical system a bit arabiscoté, we try to define what could be the morphism.

40:00 And it is to say that the development is the following. What was the goal for Bourbaki, it was, well, the ambition of Bourbaki, it was to do for all the mathematics what Van der Varden had done for the algebra, the point of view algebraic of Van der Varden. l'algèbre dite moderne, puisque la première édition s'appelle Moderne al-Ghebra, et est devenu ensuite simplement al-Ghebra. Bon, alors, le modèle, le prototype de ça, c'est que ce sont les théorèmes qu'on appelait théorème de Mademoiselle Neuterre, je me souviens encore d'avoir lu ça, théorème de Mademoiselle Neuterre, c'est-à-dire Émi Neuterre. Or, dans la théorie des groupes, il y a deux assertions intéressantes, qui sont, que j'ai écrites, A sur C, on a A, qu'on fait en B, qu'on fait en C, of these groups. Well, we don't have to do that, we don't want to embarrass with these groups distinguishing. And so if I divide A over C by B over C, I get A over B. In other words, A, B and B are two groups, two groups of the same group. A, B divided by B, it's the same thing that A divided by A. Well, today, we replace this kind of argument by the six examples, and it's a way more flexible and more visual to say the same thing. Alors, ce qui est important, c'est qu'il y a l'unicité, c'est important pour les groupes, bon je prends groupe, mais on pourrait dire anneau, on pourrait dire d'autres structures algébriques. Ce qui distingue les structures algébriques parmi d'autres structures, c'est l'unicité du sous-groupe et du groupe quotient. Autrement dit, si j'ai un groupe et un sous-ensemble, si ce sous-ensemble peut être muni d'une structure de groupe telle que l'inclusion soit un homomorphisme de groupe, dans ce calendrier que c'est un sous-groupe, la structure de sous-groupe est unique. La même chose pour le quotient. Si vous avez un groupe qui s'envoie sur un ensemble, il n'y a qu'une seule manière possible, si c'est, au plus d'une manière de définir le groupe sur le quotient de manière que ce soit un projecteur. Another way to say that if we have, in terms of modernity, the group towards the ensemble, so we have in terms of modernity, the group, the group, the ensemble.

42:30 If we talk about a group, we don't have the elements, but we don't forget the multiplication. We have an ensemble. The group towards the ensemble, which is an extremely important factor. and this factor has the property that if you have two groups distincts on the same ensemble, so you have two multiplications on the same ensemble, if these groups are distincts, they do not communicate. It is to say that if you have two groups who have the same ensemble, or it is the same group, or it is no homomorphic of the one in the other. It is to say that the homomorphic would be identity. The identity is not a homomorphic of the one in the other. They do not communicate. Bon, alors, Bonbachy était très conscient, et c'est très net dans le petit fascicule de résumé et de la théorie des ensembles, qu'il faut généraliser ça à votre situation. Bon, alors, groupe divisé par un sous-groupe, on va remplacer ça par un ensemble divisé par une relation d'équivalent. Ensuite, on va, bon, sous-ensemble, c'est une notion bien claire, et la décomposition, bon, When we have an homomorphism of a group, the decomposition, which is now played by image and co-image, well, image and co-image, when we have an application of an ensemble E in Z, they decompose in a way canonically, what some call the decomposition of Bourbaki, and Bourbaki will be extremely, insistent a lot on it. Well, first, when we have E in F, there is an equivalent relation in E. Two elements are equivalent if they have the same image by the application F. It is R, this relation. And then there is an image, namely the elements of F, which are obtained from an element of E on the basis of the function. Well, we have this decomposition. And, well, in the theory of the group, in a category additive, we have this decomposition. And the important point is that here, the co-image, the quotient L sur R, and the image I, there is a bijection of the one on the other. And so, if we were not present in terms of ensemble, this would be the isomorphism. Or, in the structure algebraic, what is important is that this decomposition, which is ensemble-based, vu that if it is possible, there is one single structure of group on I and one single structure of group on E sur M,

45:00 eh bien, these structures are equivalent, C'est-à-dire que là, on a non seulement des dijections, mais on a une suite, et là, on a un isomorphisme de groupe. Bon, alors ça, c'est vraiment une des caractéristiques qui distingue sans doute les théories algébriques des autres théories. On peut le formuler pour nous en disant que le foncteur d'oubli a ses propriétés. Alors, c'est vrai pour les ensembles, pour les autres, mais c'est faux pour les espaces topologiques. C'est faux pour les espaces topologiques. Et ce qu'il y a, c'est que, par exemple, si on a un sous-ensemble du plan, on peut fort bien lui mettre deux topologies différentes. Par exemple, la topologie aiguille, au sens similaire, mais on peut aussi lui mettre la topologie discrète, pour aller à l'extrême. Et dans les deux cas, le sous-ensemble, avec cette topologie, se plonge continuellement. So, there also, it is to realize that in the point of view of the structures, it is a point of view heritage of the program of Klein, the program of Erlangen-Klein. C'est-à-dire que l'objet fondamental, c'est la notion d'isomorphisme. Symmétrie, c'est-à-dire isomorphisme. Et en fait, dans ce point... Bon, alors, la théorie des... Enfin, la philosophie des structures est bâtie autour de la notion d'isomorphisme, transport de structures, etc. Or, what is important is that in the group, in the group, we don't see how the notion of homomorphism was very natural, and the notion of the group of quotient was also natural, but what there is is that, in fact, in many situations, it is to go far away than the isomorphism, it is to go to the morphism, it is to build a category with a puncture of the body So, I said that on the same ensemble, there are several topologies that are comparables. The topology discreta is the topology usual, and even the topology dosaristic, if you believe, these three topologies are comparables. C'est-à-dire que, bon, l'application identique est continue dans certains cas et n'est pas continue dans d'autres. Bon, donc, il y a automatiquement, sur un même ensemble, une classification des topologies possibles, plus ou moins fines. C'est-à-dire fines si elles ont beaucoup d'ouvertes et moins fines si elles ont moins d'ouvertes.

47:30 Mais il y a des difficultés. Il y a des difficultés pour ça. Et par exemple, dans la première édition de Bourbaki de Topologie Générale, vous aviez un magnifique théorème qui dit que si on prend le produit de deux espaces et qu'on le passe au quotient, c'est le produit des quotients. Eh bien, c'était écrit dans la première édition de la Topologie Générale de Bourbaki. C'est faux. Et d'ailleurs, dans les nouvelles éditions, So, we have to give some additional hypotheses, and there are some examples that I looked at yesterday, which are extremely simple. So, there was also, it was also an internal comparison of Bourbacchi, the topology of Mandelbrot. Bourbacchi had also considered the infinite products of space. Well, it's Tycho-North who first introduced this notion with the theorem famous of Tycho-North that if we have a family of compact spaces, their product, even if it is infinite, is a compact space, which is one of the best forms of the action of the choice. Well, and then, but how do we define the topology on a infinite space? And then, in the first report that was proposed by Mandelbrot, which I talked about, he said, well, we take as an open space, we have an open space, in each XI, we take an open space, we take the space, we take the space, we take the open space, well, it doesn't work. We have to take who are almost all equal to the entire space. Well, of course, we can define this topology that we call between us the topology of Mandelbrot, but perhaps, for the little story, why Mandelbrot has quit Bobaki? Maybe he has well supported the Nazis on this topology, and also there is a story of concurrence for a chair at the College of France between Ville and Ville. We will ask the College of France to open his archives if they want to do it. So what it is, is that, in fact, why there were these errors?

50:00 that we should be interested in what happens at the level of the ensemble or at the level of the group. At the level of the group, of course, this is true, if you use a product of groups by a product of groups, it works. But, just, it is one of the most delicate points, it is really one of the most delicate points, and in an analysis function, we will find it again with more difficulty. Bon, alors. relatively subtil. In fact, we can say that the development of the ideas of Grotonnik is going to be more and more distant in the view of the foncteure double. The foncteure double is an important role, of course, in many things, and the foncteure adjoints, but what happens is that, as the remarked that you gave at a certain point, when we pass to the category of schémas, At this moment, the product of two schemas has nothing to do with the product of the ensemble sous-jacent. And it's even one of the difficulties. This is explained easily because, geométriquement, I remember that the word Schema has been introduced for the first time in Geométrie by Chevalet. Schema was not synonyme of variety. Schema was the skeleton of a variety. And it was, we can say in several ways, the ensemble of all the various varieties of a variety, or, in the view of Zariski, it was the ensemble of the anneals for a variety of irreductors, the ensemble of the anneals in the core of the function rationally. So that was not the variety, nor the whole of these points, it was the whole of the sous-variities. Or when you have a product of variety algebraic, a sous-variity is reductive to the product, two times to the right, the product is the plan, is the product of something which depends on one factor or another?

52:30 No, in general, there is only the right parallel to one side or the correct coordonnées. So it explains that this point of view explains that as we have an accent in the schéma on the whole of the souverainty relative, it is not compatible, it is compatible in a way subtile with the product. So, let's say that. Now, let's move on to the analysis functionally. Now, the spaces of Bannard, which, of course, have also been invented by Norbert Wiener and Frechet, but, well, it's Bannard who developed it, and it's the book of Bannard who has done this theory. Bon, la théorie, elle date des années 1920. Alors, il y a un certain nombre de résultats clés. La dualité, quand on a un espace de Banach, on a son dual. Et ça, c'est en analyse, c'était une idée absolument fantastique. Les premiers résultats, c'est par exemple le théorème de Frédéric Kies, qui démontre d'abord que l'espace L2 est son propre dual, puis ensuite que L1 a pour dual L infini, for dual LQ with the relations that we know between P and P. And then also the theory of Ries-Markov, which identifies, say, on a compact space, the measures born A, the measures of radon born A, the element dual of the space of the functions continuous. So the point of view of duality is extremely important, and of course, it's this point of view that Schwarz utilizes for his theory of distribution. For the distribution, we will not enter into the debate about who invented the distribution. Sobolev certainly introduced the espaces of Sobolev, which is another way to consider the distribution. D'autre part, les distributions rendent compte d'un certain nombre de théorèmes classiques, d'un certain nombre de méthodes de calcul classiques en physique et il est assez étonnant de comparer par exemple les articles de Schwartz sur la transmettre fourrier des distributions qui sont de la fin des années 1940 avec les calculs que Dirac ou d'autres font à la même époque et les calculs d'une grande société. explicit donc on avait on avait une pratique de la transformer pour une distribution avant d'avoir une théorie mais ça sera tout ça joue un rôle important dans le développement de Grotten. Bon

55:00 alors je tous les toutes les théorèmes classiques, Banach, Tanaus etc mais justement il y a une difficulté qui est la même que celle qu'on a vu dans le cas des espaces topologiques c'est le The fact that when we have two groups topologics, or more precisely, two espaces of Banna, for example, one application continues the one to the other, even if it is, well, the one to the other, it is not necessarily, it is not necessarily, it is not necessarily a schéma that I had said, passage of quotient and isomorphism on the image. Well, and it was already in the book, in the Reveil, on the integration in the group topology, he had a choice to distinguish what we call today morphism, and morphism state, between spaces, between groups, and more generally, between, and more precisely, between estates, hectares and topology. Well, then, these spaces are all convex. It's a generalization, which is due to von Neumann in the years 1935. Au lieu d'avoir une seule norme, on a toute une famille de normes, ou de semi-normes. Et puis Dieudonné, bien entendu, avec son goût de la généralisation, a pris la théorie de Banach et l'a généralisé dans le cadre des espaces localement convexes, avec peu de motivation, quand Dieudonné l'a fait, il avait relativement peu de motivation. Because the spaces that were not in the banner at this time, were only, well, there were a few, but few, few, few, and few. Well, I say that in the, there, one of the games that was played, and in the thesis of Groton League, it's recurrent, it's a motif recurrent, diverse classes d'ensemble, on veut se définir ce que c'est qu'un ensemble limité. Or, il y a des tas de notions très voisines, bornés, faiblement bornés, qui continuent compacts, faiblement compacts, avec des tas d'inclusions et d'applications entre ces notions. And, just a principle of definition of a class well-faited space vector and topological, it is to say that two of these notions coincide, for example, in an space of the Montel, because it gave a theoretical justification to what we call the method of functions normales,

57:30 the classes normales of functions analytically complex, these are the ensembles in which all ensembles are relatively compact. All ensembles are relatively compact, which is always born, it's the intuition. Well, just to tell you what was the state of the vie in analysis function. The important point is the insistence that this school, around Bombachy and Schwartz and Zirnay, will put on the limits inductive and projective. And it will play the limits inductive and projective. This is perhaps one of the key points of the great article of Rottenberg on the algebra homology, that is to say on some points of algebra homology, the famous talk. The notion of limit projective, I looked at the book of André Veil on the topology, the book of 1940, he describes, he plays several remarks on the limits projectiles. He says that it was introduced by Herbrand and by Herbrand, in the language of today, a group of Galois is a group pro-finish. That's what we say today. And it's Herbrand the first who has noticed that it's a limit projectile of a group Finch. So Herbrand introduced it for the needs of the Galois. the fifth problem of Hilbert. The fifth problem of Hilbert was to demonstrate that all group local and anti-givian is a group of leaders, so if we don't do any hypothesis of analysis or differentiability on the multiplication of a group, it is automatically. It was one of the problems possible. It was extremely important at the time. It is to say that the solution which is a tour of force, technique, has not brought anything. There is only our friend Grombrow who, one of the first, has used the applications of the 5th century universe, but in the whole, it's a bit an impasse.

1:00:00 But there was an important point, it's that when we have a compact group, the theorem Peter Waugh and the company says that it's a limit projective of a group of Lee compact. And a group of Lee compacts is also the same thing as a group linear compact, and it is also the same thing as a group algebrite. So we have seen a group compact as a limit projective, that we know from 1940, as a limit projective of a group algebrite. It is explained in another form in the book of Chevalet on the group of Lee, it is called the duality Tanaka. Alors, maintenant, limites projectives, c'était, bon, il y en a beaucoup, et les espaces de Fréchet, une classe d'espace qui avait été introduite par Fréchet, qui est un peu plus générale que les espaces de Vanas, mais on peut le voir comme les limites projectives d'une suite d'espaces de Vanas. Alors la topologie, elle ne pose pas de problème, parce que, par construction même, la limite projective, c'est une partie du produit, and even a part fermée in the case of the end of the product so how we know what is the topology on a product infinity of space well on the limit projective it doesn't cause no problem and when we say a group compact is a limit projective of a group of the compact it's to this topology that we would refer for the limit inductive it's a little more subtle it's a little more subtle then of course when the face will be there we will see And for us, the main place where the limits of inductive intervient is in the fibres of the vessel. But at the time, it was not even that. And the limits of inductive, they also intervient at the 5th problem of Hilbert. There are some allusions to the limits of inductive. And by the way, he did well the symphony between the limits of inductive and the limits of projective. Mais le problème de la topologie sur la limite additive n'était pas, il ne se posait pas chez Veil et il n'est pas résolu. La première utilisation sérieuse, en dehors de ce que je disais, des vaisseaux bien entendu, c'est dans la théorie de Schwarz. Parce que Schwarz introduit son espace de fonction test qu'il appelle D et qu'on appelle plutôt aujourd'hui C infini. c'est, fonctions de classes C-infini que c'est ici à support compact, c'est par définition obtenue en gonflant des parties compactes de plus en plus grandes jusqu'à obtenir tout l'espace,

1:02:30 et on prend des fonctions qui sont ici de compact, sur chaque compact on prend des fonctions de classes C-infini et on grossit le compact, et ça forme bien entendu une limite individuelle. Or Schwarz définissait des distributions comme des formes linéaires. Rappelons-nous notre maître Schwarz. Une distribution est une forme linéaire. Bon, alors, il définissait des distributions comme des formes linéaires. Continu, mais il était embarrassé parce que ce n'était pas un contenu sur un espace bien défini. and it is with the pseudotophogenes, etc. And it's Dieudonné, who, with his mind very rational and systematical, he said, but after all, we will introduce the limit inductive. So, in the article, duality of the space F and LF, LF means limit inductive, the space of freshet, F is for freshet, there, Dieudonné and Echva se definisent precisely what is the limit inductive. So from there, the notion of limit inductive, this definition is a bit ad hoc, the topology is made to have the good properties, but in fact, it's really there that the categories point their necks. And the definition of the limit inductive, it doesn't take its force or the limit projective that if we formulate it in terms of category. And then, we have the chance that in these cases, the limit, it's what I say here, we have the chance that the limit inductive calculates in a category coincide with the limit inductive calculates in the ensemble. In this case, as often, we call it the fact that we respect the limit inductive. Alors je disais, bon là, la méthodologie de Bourbaki pour ces choses-là, c'est, bon, Bourbaki de plus en plus, si vous suivez l'évolution des diverses éditions de Bourbaki, vous verrez qu'il y a de plus en plus d'insistance sur la notion de sous-espace, espace social, limite inductive et limite projective. And the limits inductive and projective are going on, and in the last edition of the theory of the ensemble, there is an enormous section on the limits inductive and projective.

1:05:00 Tout ceci a un sens catégorique, mais en fait un parfum d'infini, parce que dès que vous arrivez, c'est pareil, vous allez plus loin de tout ce que c'est parce que c'est conscient, vous avez des familles infinies. Et je dis, c'est ça qui va servir à votre vie dans le talk. Well, I'll come back to my second part, it's to the work of the group in an analysis function. I made a panorama, I told you how was the function function, avant l'arrivée de Grothendieck, et maintenant, je vous dis ce qui en est. Alors bon, le point, la thèse de Grothendieck s'appelle « Espace, produit tensoriel d'espace vectoriel topologique et espace nucléaire ». Et alors, ce qui est très frappant, c'est que là, vraiment, on est dans une situation catégorique. Le produit tensoriel, c'est une histoire tout à fait functorielle, et les espaces nucléaires, c'est en gros les modules plats. Bon, alors, rapidement, donc, la philosophie des produits tensoriels. Pourquoi nous intervençons-nous aux produits tensoriels, tout au moins en analyse ou en géométrie ? Pour deux raisons. First, if I have two spaces X and Y, and I say, of a very vague function of X, function of Y, a certain class, the product tensor of the spaces of function on the core of the base corresponds to the product of these spaces. Let's say, the polynome has two variables. It's the product tensor of X and Y, it's the product tensor of the gene of the polynome in X, with the polynome in Y. Well, all this should be done with a grain of salt. And of course, in the theory of schema, when we have a schema affine, which is by definition the spectre of an anneal, or of an agile on a core of base or on an anneal of base, Eh bien, le produit des schémas correspond exactement au produit personnel des années, par le morphisme, par la transcription par le spectre.

1:07:30 Bon, deuxième point, deuxième point, les opérateurs de E dans F, au sens intérieure, par exemple si on a des espaces vectoriques de dimension infinie, l'ensemble des opérateurs linéaires de E dans F, c'est le produit personnel du dual de E avec F. these two principles are true, for example, if x, y are finished, we take all the functions of x, y, the first assertion is true, if e and f are done, it's true, it's true, the second is true. But in these cases, we have to take a real theory in a very limited case and see the spirit of this theory and what he does in other situations. And that's why we are interested Well, there was a lot of examples of situations where there were products tensors. The calculation of the Levit-Chivita, of Levit-Chivita, obviously, is tensors. But it is not formulé comme ça. En général, ça n'est pas formulé comme ça. En algèbre, on parlait du produit chronécanien, par exemple, de deux représentations d'un groupe, le produit chronécanien. Mais là aussi, la notion prétention et l'état victorien n'est pas très nette. Bon, extension d'escalaires dans les systèmes hyper complexes. Système hyper complexe, c'est le nom qu'on donnait aux algèbres. Extension d'escalaires, c'était un point important. Mais bon, ça n'est pas... C'était des choses dispersées. Il faut se rendre 1949, l'Algèbre III de Bourbaki, où le produit tensoriel, dans la première édition, c'est là qu'on introduit le produit tensoriel, mais pas apparu. Et qu'en 1938, c'est dès qu'en 1938 que Whitney, pour des besoins de topologie, définit le produit tensoriel. Le produit tensoriel de deux espaces vectoriels, deux dimensions finies sur un corps, on peut considérer que ça fait partie du faux corps. but if we replace the core by the name of the entier, it's Whitney in 1938 for the needs of the archaeology for defining, for to demonstrate a theorem of Clunet for the archaeology entier and then, above all, the changement then, this is Cartan which, I can't find out in the dates, in the discussions interne of Bourbaki a characterised the product sensoriel by this formule which is

1:10:00 and oh, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a, I'm a carton, gentilman de carton. Mais c'était, c'est une caractérisation qui cette fois est franchement catégorique du produit tensorique. Bon, je ne sais pas, après vous connaissez tout ce que ça veut dire, je n'insiste pas. Et, mais c'est très remarquable qu'en 1947, Dieu Donné publie un article où il démontre que si on a un anneau d'Edeckine et un module sans torsion sur un anneau d'Edeckine, So, it's the first example. So, it's to be aware of how the theory of the produs sensoriel was primitive at the end of the 40th century. And when you compare what happens at the end of the 50th century, when Grotonnik manipulates, Grotonnik S.A. manipulates the produs sensoriel, the tors, the exes of the fecese, etc. It's to be aware that the progress has been absolutely... It's a cataracte that has been reversed. En analyse fonctionnelle, il y avait le produit tensoriel Hilbertian, sans doute fait en 1935, dans le livre de Von Neumann sur les fondements de la mécanique quantique. En mécanique quantique, les états d'un système, c'est un espace de Hilbert. Si on a un système composé de deux morceaux indépendants, on fait le produit tensoriel des espaces de Hilbert. Bon, les opérateurs d'Hilbert-Schmidt sont liés directement aux produits tensoriels, et là, pour les opérateurs d'Hilbert-Schmidt, on a vraiment ce projet écrit là-haut, c'est-à-dire que les opérateurs d'Hilbert-Schmidt dans un espace d'Hilbert-H, c'est le produit tensoriel, puis H, le produit tensoriel est complété au sens hilbertien avec H-bar. I think it's a vector, and you can't forget it. And then, in fact, it's what the Ecclissian is, the notation of Dirac. The scale of two vectors is like this. And if we separate A and B, that is an operator. Because if you apply C, if you apply C, you will have A multiplied by the scale of B, C. Well, a little bit before Rottenbeek, there was a great memory of Schatten. Schatten is a class of Schatten, which is a class of Schatten, which is a class of Schatten, which is a little more large than the class of Hilbert Schmitt.

1:12:30 En gros, Hilbert-Schmidt, ça veut dire que les valeurs propres forment une suite de carrés SOMA. LPH, c'est en gros les opérateurs dont les valeurs propres font une suite de puissance TX. Voilà à peu près tout ce qu'on savait avant Grotonnik. Bon, alors, maintenant, alors je décris ce que Grotonnik a fait, ce que Grotonnik a fait, il y a un certain nombre de points importants dans sa thèse, alors un certain nombre de points importants dans sa thèse, First of all, there is a remark that the universe universe that Carton has given, to say that to envoy a product tensor in the third module corresponds to functions bilinaires, it is one of the points that is essential. It is one of the ways to formulate, It is that the operators of E, an espace of Bannard, in a duality, are in duality with the product tensor of E3M, and that the norm which is the duality is the norm E3M. So, the universal problem of the disappointments that you search for, it's theki tensor which is designed by Schwartz, It's not the concept of Grosni, it's the one from Schwartz, it's the variant Pi. It's the result that when you bring E F into G, you have to renounce uniformly and continuously EAM, F. Alors, je dis ça pour l'espace de Bannard, mais le point fondamental, c'est qu'un espace localement convex est toujours, d'une certaine manière, une limite projective d'espace de Bannard. Et ça, c'est une stratégie qui est à fond, à fond, dans tout ce qu'il fait. Donc, une fois qu'on a les notions, les bonnes notions pour les espaces de Bannard, on peut passer à la limite projective, et c'est facile. Mais la grande nouveauté de Grotonique, la grande nouveauté de Grotonique, c'est qu'il y a un deuxième produit tensoriel. D'habitude, le produit tensoriel est compatible à la dualité. Dual du produit tensoriel, produit tensoriel est dur. Mais là, il faut introduire une nouvelle sorte de produit tensoriel, le produit exilose.

1:15:00 And that is really the importance of Grotonique in this subject, the importance of the fundamental. And it seems to be, according to what Charles told me the other time, that when Grotonique discovered the need of a second product sensoriel, it was rather a disaster. He said, the theory does not work, the theory does not work, because I need two products sensoriels. Then after, of course, there will be more. Bon alors, ensuite, le point c'est que tout ça c'était pour justifier, enfin toutes ces constructions c'était pour justifier le théorème des noyaux, qui est un théorème du même genre que ça, tout opérateur continu qui envoie les fonctions à support compact, les fonctions aussi infinies, dans les distributions, est représenté comme si c'était un opérateur intégral, mais en appuyant à un noyau qui est une distribution. So that, it justifies largely the practice in physics, particularly when we call the notation of Iraq, and it justifies a lot, and it has been, it has been, it has been, it has been, it has given the rigueur to a certain amount of calculations in physics. But, what there is now, are the properties of the function function. And then, the two first lines show that there is a need of products tensorials. Because if you take the products tensorials with continuous functions, well, you have to take the product Exilog. If you take the products tensorials with an integral, you have to take the product tensorials. Et puis ensuite, si vous vouliez, pour ce cas intégral, il faudrait le troisième produit tensoriel, celui qui est spécifique aux espaces de Hilbert. Donc, il n'y a non pas un produit tensoriel dans l'espace électrophologique, il y a plusieurs produits tensoriels. Bon, alors, maintenant les propriétés d'exactitude. Il y en a un qui est exact à gauche et l'autre exact à droite. Ça, c'est assez facile formellement à partir des propriétés d'adjonction, comme vous diriez. But, of course, Gottfried Dick's pose a problem. Is there a theory where I would have a product of science which would be important, exact? And of course, we have to restrain and we have to take the nuclear space.

1:17:30 And the nuclear space are, in general, the analogies of the flat modules. And then, of the same way that the modules that allow for the exacts of the surface, Well, Grothendieck is to have a theorem of tunnel. Grothendieck demonstrates a theorem of tunnel vectorial topology with these notions, where we put the space in the air, and there are applications, for example, that allows us to demonstrate quite easily the theorem of Dolgo, on the barred topology, where we bring a variable, the strategy, when we have several variables, we take a point of tension and space to a variable, to a variable, to a variable, to a relative element, etc. Bon, alors, maintenant, voilà. Voilà, en gros, je vous ai résumé ce qu'il y a dans la thèse, et ce qui conforte ma thèse au sens philosophique, à savoir que l'esprit, et ce que je disais au départ, l'esprit fonctoriel est très présent, très, très, très, très. Well, it's just, well, the Assuma Brasiliansis, it's an article that Pizzi nous parlera, which is released a little bit later, where Grotonik studied systematically all sorts of products sensoriels that we could do, that's to say all sorts of topologies of the norm on the space of the banan or the space in general. Pardon? Ah, oui, bon, c'est plus si c'est normalement. Oui, oui. the space nuclear, for example, these are the I will not have a question. These are the functions of the car, the functions of the car, the functions of the morgue, the space of the Schwartz, the space of the dual, So, a large part of the space function, usual, which, if they are not the space of Banas or of Huber, are the space, well, many are the space nuclear. And it is very important, it's a theorem that Grotonique demanded at the beginning, that if a space is nuclear and of Banas, it is of dimensions.

1:20:00 finies. Donc, les espaces nucléaires, c'est vraiment la généralisation naturelle des espaces de dimension finies, puisque l'intersection entre espaces nucléaires et espaces de Bannard, ce sont ceux de dimension finies. Bon, alors, justement, bon, j'enchaîne. Donc, Grotonique, je disais, dans Souma Brasiliensis, explote, tout autre, mais ce qui est important, c'est qu'on voit très présente la stratégie de foncteur dérivée à gauche, foncteur dérivée à droite. Well, I'll let you talk about that. Now, the heritage, to finish. The heritage, it's that, now, I've said, what are the spaces nuclear, since Max reclamed a list of the spaces nuclear, so a large number of spaces of the analysis are nuclear, and that's why we can apply the theorem of the way of Schwarz. Like, the command that had been made by Schwarz and given a great idea, it's to explain the theorem of the way. and Brothenick will implique it with this theory, and, of course, we need to find out the theory. Well, today, we would like to introduce these nuclear spaces as the limits projectiles of the space of Hilbert with the importers TN, which are Hilbert. And that is the strategy of the space of Somoet. In fact, today, rather than having a space of distribution, we prefer to have a scale of the space of the Soboleth. For many problems of analysis, it's much more useful. And the definition that we would have today is rather than an space nuclear, it's this one. Limit projective of an space space of Hilbert with the applications intermédiaires of Hilbert Schmitt. Now, I talk about the heritage of the Ecole de Guelfand. The fundamental point is the terrain of 2011, which has been of Xavier Fernick, a little bit simultaneously, and that was simply because the communications between the Soviet Union in the 1960s were difficult for us and that we had worked at Strasbourg in knowing what was happening at Moscow at the time. We have recognized it very quickly. So, the point is that we can do a theory of distribution. We can define correctly On peut définir ce que c'est qu'une loi de probabilité sur un espace tel qu'un espace de fonction ou de distribution. Si l'espace est nucléaire, ça marche particulièrement bien.

1:22:30 And we have a theorem of 1100, which is the generalization of the theorem of Paul-Lévy, or the theorem of Paul-Lévy, or the... which caractérise the probability of probability on a group by the transformation of Fourier, which is a function continuous. which is positive. Well, then it's a term classique in the analysis of Fourier and which is in dimension infinity and which gives a flexibility extraordinary because, generally, to build a measure on a distribution space, it's complicated, but to define a transformation of Fourier, it's relatively easy. Well, then, that, there were important applications important qu'à partir des années 1970, toute l'école de... toute une école américaine autour de Glenn, Jaffe, etc. a utilisé ces techniques-là en s'appuyant sur les travaux qui avaient été faits par l'école de Moscou, et à s'appuyer et à développer une théorie constructive des champs. C'était le summum de ce qu'on savait faire rigoureusement par in 1970. Malheureusement pour la physique, c'est limité à un espace-top dimension 2 à la rigueur 3, ce qui relève une partie de son interview. Bon, alors, maintenant, dans la thèse de Grotendieck, il y a aussi ce qu'on appelle le problème de l'anthropologie, le problème de Banach. Le problème de Banach, c'est un problème d'approximation. It is that, in all the constructions of polytonsorials, we are going to consider these operators which are the limits of norms of operators of remplies. So, two types of operators are compacts, that is to transform a part of bourbon in a part relatively compact, and Barnard has asked the question of the reciprocal, is that a compact, in this respect of Barnard, is the limits of norms of operators of remplies. approximation of Banach. Well, it sounds like Grotonique

1:25:00 y'a cru. Y'a cru tellement que toute la page de sa première partie de sa thèse, c'est une typique de la stratégie du Grotonique, c'est retourner le problème dans tous les sens. Et il y a 30 ou 40 propriétés dont chacune est équivalente à la propriété d'approximation de Banach. Ils savent bien qu'il y est cru. Well, he would not have developed a... His strategy, which we have seen in many others, was just... Well, some would say noyer the fish, but no, he would have explained it to him, to put the oil in the water until she would be able to do it, and that the fish would be able to do it. Well, he would have done that. Malheureusement, malheureusement, contrairement, to what everyone thought, the problem has a negative solution, And there are spaces of Banas for which the proprietor and approximation is not available. And there was a first example, in 1970, which is quite tordu. It was explained in the seminars of Charles in analysis at the University of Polytechnic. So there is a first example which is quite tordu. And then, surprise, surprise, surprise, the second one is a little bit more natural. What is the space of Banach méchant, fautive? On prend a space of Huberth, on prend a jet of all the operators in this space of Huberth, which is really a space of Banach a little bit more classique and natural, and they violate the proprieties of the approximation. Well, it will also arrive several times at Grothendieck to do very natural and then that, those who know the Grothendieck who generalize the concept of George, may be afraid of time in time, I imagine, perhaps. Now, just in conclusion, how does the transition happen? Well, the transition is that Grotendich, Grotendich first gave several applications of these techniques for problems of the algebra homology, etc.

1:27:30 Well, there is a theory of predominant, which serves as a function of the state, and things like that. There is a theory of vector and topological theory, which I have already talked about. There are techniques that are used in the theory of duality. Alastair, bon, ça aussi, c'est tout ça. Il y a une interaction très forte entre les méthodes analytiques, celles de Schwarz ou de Grotonique ou des données, et des problèmes de fonction de plusieurs variables complexes. Mais bien entendu, le point fondamental, c'est l'article dit qu'on connaît sous le sigle de Thauphine, c'est-à-dire sur quelques points d'algéromologiques. Et si on regarde le point crucial dans cet article, d'ailleurs c'est celui qui fait problème, c'est l'emploi des méthodes infinitaires, limite inductive infinie, limite projective infinie, dans les catégories, ce qui n'est pas surposer des problèmes logiques de fondement, parce qu'on est très près du paradoxe de l'ensemble de tous les ensembles. Et donc, c'est ça qui a été le coup d'éclat de Grotonix, ce qui est la transition. Ce qui est la transition puisque à partir de la Grotonix a pu montrer qu'on peut faire des résolutions injectives de faisceaux et donc on pouvait engouffrer toute la transition. Thank you.