Discussions — Grothendieck's Functional Analysis Program

0:00 It is possible that you can place things in an L1 norm, so first the L1 norm of the first derivative, then the L1 norm of the first derivative, the L1 norm of the second derivative, and so on, or the L-infinity norm. So you could even interpolate to tell people that it's not so interesting. So the point is that at the level of Banach spaces, you have three tensile problems. But because a space of smooth function can be defined by family of norm, we belong to three families. The three tensor projects coincide. And after that, the candle theorem of existence of candles in the Schwarz, which is a duality statement, is nothing strictly. The real reason, I mean the true reason for this duality theorem is this tensor project. I have to admit that the tensor product of Hilbert was considered less important, in application it's a very important application, but Rottening did not consider it very serious the other time. But since you have the possibility of defining a sequence of norm for your space of test function, you have three notions of tensor product and they coincide. And the fact that they coincide says that this tensor product has all the nice properties. It's left and right and right. And then, I think that was the beginning of the functorial way of thinking, of course. He had his tensor product with nice functorial properties. I think that's really where he started. And then Danko Technik went to... his thesis is in two parts. The first installment is a general theory of tensor product of Banach spaces and Modener spaces. He was the one who really thought that a general locally convex space is an inverse system of Banach spaces, and by duality also a direct system of Banach spaces. So he was the first really to put emphasis on that point, that a general space is an inverse limit of Banach spaces. And sometimes you consider the inverse system itself not the limit. Well, for the analyst it means that you have a space of function with various norms and you deal with estimates of the various norms and you don't really care whether, I mean, there exists something in the limit.

2:30 Okay, so that's what he did. So the first part is a very well-developed theory of these various tensor products with various properties, exactness, and so on, and naturality, exactness, and so on. There was a very famous problem at the time in Banach spaces. It was a conjecture stemming from Banach. It's so-called the approximation problem. What people called at the time a completely continuous operator, what is called now a compact operator, so between two... Banach space is an operator which takes any bounded set into a relatively compact set. Of course, finite rank operators have this property for obvious reasons. And uniform limit of operators. This property is conserved by uniform limit. And the natural question was whether you could approximate any compact operator by uniform limit of finite rank operators. In many examples it's true, sometimes it takes some ingenuity to prove, but in most examples it's not. And at the time, most people expected it would be true. And the middle part of the thesis of Botany is concerned about this problem. And typically enough, he takes this conjecture and reformulates it in 47 different fashions. The hope was, no matter, the strategy is already there. He takes a very difficult conjecture, he reformulates it in many, many, many, many ways, with the hope that eventually one of the forms will be used to prove what he calls, I mean, it's a Joshua surrounding that, and with his trumpet until the wall would fall. So, Joshua. So, Goethe used this strategy, but unfortunately, I think he spent almost two years in seeking to solve that problem, and it took about 20 more years to discover counter-examples, and flow, and discover the counter.

5:00 The original example was very complicated but now we have a very short version and we can show that the most natural space, badass space after Hilbert space is the space of all bounded operators in the given Hilbert space which is a very natural space and in this space a proper Hilbert space and they are quite accurate which was written by someone else also and so on and so on. Gotenlich spent a great deal of effort, and I think part of his new methods were hinted to prove that, in that respect he failed. But then the second part is really the fantastic, certainly the best part. In the second part, he wants to put in an axiomatic basis the kernel theorem of Schwarz. So it's considering a category of spaces where the two natural, or the three natural tensor products coincide. You would say, okay, we have three natural tensor products, they are smooth, functional, they coincide, but is there a general category of spaces for which these tensor products coincide? And that's exactly the nuclear physics that he invented. Nuclear because, I mean, the kernel was called nucleus, of course, the kernel of nucleus. A nuclear operator is an operator which has a nucleus, a kernel, and so he wanted to have nuclear spaces, spaces where all operators have a kernel. So, and that, in the long run, is a better contribution. But it was a game from school, realizing. As soon as Schwarz published his textbook, I mean, Gelfand was very excited and a lot of energy was extended and they published six volumes, six, which is, the point is slightly different from the point of Schwarz and a little more concrete, so to speak, but Gelfand had a...

7:30 Gelfand has his limitations, but his imagination is fantastic. And he was the one who understood the connection with probability. What makes the Wiener measure exist? The Wiener measure is really the... A new step beyond a standard, ordinary analysis. It's a genuinely infinite dimensional analysis, and not real. The rest is just garbage, but the fact that there exists a winner measure, I mean, so a certain probability law on the space of all continuous functions was rather unexpected at the time, but we could develop a topology for function spaces for some reason. He spread an idea which was well understood in the 1920s. But the idea that you could do, you could do, I mean, integrate, a global integration of subspaces was brand new. And so, and then Gelfand discovered that the nuclear subspaces was a natural home to develop a theory of integration in infinite dimension. Min-Lohr's theorem. Galefront suggested the Min-Lohr's theorem, which says that it's an extension of the classical Bohr-Norff theorem to infinite dimension situation. The classical Bohr-Norff situation characterizes the Stiljes-Fouillé transform, the Fouillé transform of bounded measures as a positive definite function, or linear combination of positive definite functions, and that there should be something similar in infinite dimension. It was rather surprising and it required a great deal of imagination from Gelfand to guess that there would be something. Well, the work of Wiener and Levy, Paul Levy, are hinted to, but that there would be an abstract framework for the Wiener integration, I mean, was really, I mean, it's a Russian wonder.

10:00 And Gelfand, the contribution of Gelfand, is to feel that the nuclear spaces were the natural hope of the... And then, within 10 years, I mean, Galefront School developed a lot of things. That's partly explained in the textbook by Galefront. The property that bounded sets are compact. I'm sorry? Bounded sets are compact. Yes, exactly. Must go a long way. Yes, yes, bounded sets are compact. That's the main idea. Bounded sets are compact. And as I said, I mean, what we know now is that integration, I mean, in integration you can exhaust those space by compact subsets. Every bounded subset is compacted. A little more, you need a little more. But this is really... And then, interestingly enough, he was taken by Weidmann in the States in his so-called axiomatic field theory. Yes, well, which he did with Streeter. And Weidmann stayed there, and then all these schools in the 70s, and Edward Nelson, and so on. And they understood what the physicists could do out of these ideas, out of these Schwarz, Grotendieck, Gelfand ideas. And that was one of the high points in mathematical physics at the time. Unfortunately, I mean, it works very well in a space-time of dimension two, with some difficulty in dimension three, but in our world, which is four-dimensional, no one knows how to make it at the moment. And I think at the moment, the strategy would not be to try to extend it directly, we have to, and I think people like Kon, like many other people, I mean, I think we are in the right way, I mean, we first want to understand the algebraic structure before we do the analysis. And the algebra there is very complicated, and before we can come to terms with the algebraic side, I mean, it's too early to do the analytical side.

12:30 Well, maybe 20 years from now, 15, 20 years from now, we can go back to that. So, at the moment, it's... But this, at a great level, I mean, sort of this, if you trace back the influence of God in this thesis, it has been enormous, enormous, enormous, especially in mathematics. But it was the imagination of Gelfand, which brought together this idea with mathematics. And Grottenig himself, partly out of ideological reasons, was never interested in mathematics or physics. For him, I mean, he had an equation, Hiroshima equals physics. He had this equation in his mind. And so he would consistently refuse to consider problems, mathematical problems. He had all the expertise and all the imagination to do that. So if he wanted to do it, he would have done fantastic progress, but he did not. There was a particular question. I just remembered a textbook that I had in school, which was, again, J. L. Kelly's Topology. He refers to results of Rotenbeek on countable compactness, that there exists any bounded sequence as a convergent sub-sequence, but without mentioning longer sequences, only the countable ones. Yes, that's right. That's probably related to... Yes, that's related to his first insight into analytics. ...jurisprudentially, yes. Yes, yes. That comes together. So, if you want to understand, I mean, the contribution of Gottendieck to that period, and then we have to understand what was the influence of these results on his further development, which is another step. So, I finished with purely analytical part. Okay, so now, Gottendieck did not finish with that. I mean, when he left Nancy to go to Sao Paulo in 53, I suppose, 53, yes, he spent... Two or three years in Sao Paulo and then one year in the United States, what was it? Kansas. And so, what I remember, I was first acquainted with Boateng in 1953. He gave a talk at the Burbaki seminar, I mean, on some problem of functional analysis,

15:00 algebra of operators and so on. And then he disappeared for a few years and he reappeared in 1955 or 1956. Now, in between, I've been in Sao Paulo, and in Sao Paulo he gave a set of lectures, which was more or less intended to be a textbook in functional analysis. He gave a series of lectures which have been treated as seminar notes, lecture notes, and then at that time his new contribution, which is also important for what he did after that, And it's a very interesting paper. Gautam wanted to understand, I mean, more deeply the reason why for some spaces the two kinds of terms of product would coincide. But, systematic as he was, he began a systematic study of all possible tensor product functions on Banach spaces. Well, he soon realized that the core was Banach spaces, and then the rest, going to inverse limit, was more or less nothing, nothing. And so he understood that the core was... So, he starts more than developing a general theory of all kinds of tensor products that you can construct on Banach spaces. And so, well, we have a very systematic way, which is more or less along the line of what was known before by Shatner, because more or less it's closer to Shatner's point of view, that is, of his first work. And then, but he has, it's very interesting, there, I mean, the idea of natural transformation from one factor to another one is very obvious, even if the language is not there. And also, he had this idea, I think he was already influenced by the idea of resolutions, and he wants to show that there are factors, and he wants to show that if you have a certain tensile product, then out of it you can bring new tensile products with better conditions, that's it. Something like a projective resolution and injective resolution, or rather a projective curve or injective curve. More or less you can map any tensor product into an injector one or something like that. It's too good, but it's a spirit. And then, at the end, he ends up with, I mean, applying various general constructions,

17:30 he comes with fourteen different tensor product factors. Fourteen. And he has a comment that they're all, why not the one you cannot escape? There are many more, but these 14 you have to consider them. And then the crucial labor is that an equivalent between two such factors. And it's written in a very cryptic way and he says that the identity map of a Hilbert space is a pre-integral something like that. Well, it's a completely cryptic statement. And when studying this paper, which was not yet one in a proof sheet, in proof sheets of this paper, and I reported at the Boba P. seminar on it, I discovered that what was behind was something very concrete, a very concrete calculation of norm of matrices. It's an elaboration of the old result of Hadamard. If you can bound, let's say, if you have a square symmetrical matrix, and if you can, well, not even symmetric, you can bound all the entries, all the elements in the matrix by a constant, let's say, between plus and minus one, okay, then the determinant has a uniform bound, depending on the size of the matrix, n to the n, n to the n, something like that, and which is... All the textbooks are very important for many things. It was used by integral equations. But the point is that this estimate about the size of the term depends on the size of the term. And then Godenick discovered final estimates which were independent of the size of the term. So about the size of the eigenvectors and things like that. But the idea is that you have a matrix and you bound every element in the real matrix, let's say, take a real symmetric matrix, every element is between minus one and plus one, and you want to discover the size of the eigenvectors, the eigenvalues, and so on. And you have estimators that are independent, properly formulated.

20:00 And then, of course, now the strategy to deal with infinite-dimensional spaces, as soon as your bounds vary in any finite-dimensional space with a bound which is independent of the dimension, you have a good chance to be able to go and develop something for it. It's a very deep strategy which has to be used repeatedly. And so, what I discovered is that the core of the proof, Gottlieb did not even give a proof of his reason. I mean, he had a huge machine, he had a huge machine, and out of the machine you could manage to build a proof. But he did not even make the proof explicit. And when I realized that it was, it could be done, it could be reduced to very explicit terms. And then, of course, I gave a proof, which was suggested by some of his remarks, and it comes out to some integral with a gamma function, so it was a good analysis, but not very deep, but a good analysis. And so, when I made the report at the Bourbaki seminar, I had two parts. In the third part, I said, well, I will explain afterwards the various factors that you can build between, and I used the word factor, that you can build between, it did not, but I used the word factor, between the tensor coordinates of vectors, of Habana spaces, and then I stated the finite of the main reasons, and I said in the second part, now, the main reason is so cryptic, I will give you an explanation. And Witten did not like it, and he said it strongly. He said, you have spoiled the things, you have spoiled the things. But I think my claim was substantiated historically because the people who took it after him mostly are the French people like Moet and Pizier and other people. And of course they use repeatedly estimators about matrices and norms of matrices and eigenvalues and eigenvectors and so on. But, I think, and, but, so... But, Urtenich is centered, it's only, I mean, I remember when I gave this exposition, the Boba Kiss, in many ways, it was... You spoil the scene. And he wanted me, when finally the lectures were printed, he wanted me to remove the second part, which I think was my contribution, his own contribution.

22:30 I reported on his contribution, I acknowledged his contribution, and then I said, this is my contribution. And I think eventually my contribution has been some more. I can imagine that later users might use your formulation rather than his. Of course, most people use my formulation, and his is the one. Did he write down his explicit bounds? Yeah, yeah. He wrote them down. Well, he gave... He found some concept which is a hyperbolic sign of pi over 2, something like that, and when I tried to understand how he got this, I mean, it's like when we're trying to understand this proof, well, it doesn't give all the details of the proof, it gives an interval. And then when I want it, I say, well, fine, it's good analysis, I can do that analysis, and people can take it. But the important fact is that already in this paper, it's clear that the spirit of tens of factors over something categorical, natural transformation between factors, I mean, derived from... In a sense, it's a theory of delight, right? It's a torsion, it's a tour for Banach spaces. It's not exactly the baton of the moment. And so the spirit of the category and tantrums is already very, overly present there. That's one important thing. So that makes the transition. I suppose that makes the transition. And so it means that already at that time, already in mysticism, Even if the language and the problematic of category theory was not there, I mean, the spirit was there. And it's not by chance because, I mean, at that time, Bourbaki finished publishing his textbook on topology called Vector Spaces, and there is a strong emphasis on duality and duality, strong emphasis on duality and also... And also under the influence of Schwarz, inverse system and direct system of spaces were considered. So, and if you want to understand, I mean, the origin of inverse and direct system is, I mean, I think the importance of the work of a functionalist, I mean, you cannot underestimate, you don't have to underestimate the influence of functionalists in the development of this idea of inverse and direct units.

25:00 Okay, so now I come to the transition between this thesis and what came just after that and his work. The transition came out from the following. At some point, when you need a dilemma, you need a dilemma which is called the invertible matrices, or homomorphic invertible matrices, which is a sort of factorization. And also, you need, I mean, but at that time also, to make the connection between the she-fury and the Hodge-fury, which was more or less a contribution of Dolmo. Transition between the I and the I. I mean, how to reinterpret Hodge-fury in terms of she-fury. I will not tell you exactly the story. There are contributions from Dolbo, from Kodaira, and I'm not sure that I will give a few to get it to everyone. But it was easy. So you move into the analytic category. Yes, analytic category. Then there is a crucial lemma called the Dolbo lemma, whose purpose is to prove one very important fact that In Hodge theories, for complex, let's say for an algebraic complex manifold, compact complex manifold, then the cohomology group HL splits into Hodge decomposition, HBQ, where P plus 2 is L. And this is done using harmonic forms and so on. But then, I don't remember who was the one who first mentioned that. Maybe it was Say or Dolgo. HPQ should be the cohomology of some sheaf, and that HPQ in general is the HQ of the sheaf

27:30 omega p of holomorphic differential forms of exterior differential forms. HPQ is HQ of omega p. And I think, well, who invented that statement? I don't know, but at least they're using this kind of thing. And to prove that, Dolbo required a certain lemma about solving the so-called D-bar lemma. It's the existence of a solution for a system over D over DZ-bar. You have a smooth function in n complex variables and two n real variables, and then you have a smooth function in the D over DZ-bar statement about it. The D-bar. And this, I mean, Korn, Nirenberg, many people contributed. So it was a very good idea. But it was started by Dolbo. And Dolbo gave rather adult proof for that. And Rotendieck, in a book, in the Captain's Seminar, gave an exposition on that, using his idea about tensor coordinates. And using the idea that... Any reasonable species of function in n complex variables should be a tensor product of things in one variable. Then, using the exactness property of your tensor product, then you are reduced to one variable. In one variable it is. It's a fantastic one. It's a very good idea. But, of course, it requires to use his sense of order, which has all the exact same properties. And at that time, he developed a very general pre-net theorem. For that reason, he developed a very general kind of pre-net theorem, whose main motivation was to give a proof of normals. So, and the other paper, which is... Is it the Weierstrass Preparation Theorem in some way? Is the Weierstrass Preparation Theorem... No, no, it's independent, it's independent. It's just, well, it comes among the tools, but it comes among the tools with Weierstrass Preparation Theorem. It's slightly different. And the other paper, was it published in the Bobacki Seminar or the Cantor Seminar, I don't remember, but...

30:00 Botany has a paper about Kühnert-Fueren for topological tensors with application to problems in 10-complex theory. And Kühnert-Fueren, of course, now we begin to feel the transition. Kühnert-Fueren for what again? Kühnert-Fueren for topological vector spaces. So, of course, it exploits the exactness property of tensors. That was a very novel idea at the time, with the aim of reducing difficult problems for n complex variables to rather simple problems for one variable. One variable is easy to control, because you have control of the shape of the domain. That's the point. In the Cauchy theory you can deal with a circle, a disc, or a square. The shape of the domain is the root cause. When you move to n-complex variables, what I know, I'm trying to make analytic continuation of holomorphic function in n-complex variables. Woo-woo-woo-woo-woo-woo-woo-woo-woo-woo. It's very complicated. So, and that was, I think that was what changed the mind of Mozart. Or rather, the transition, the transition, let's say. There's another theorem that he did around the inequality. And that was the starting point of the so-called analytic functionals or, I mean, micro-functions. This has been developed later on as micro-functions. But this is, okay, this is also important. So, I mean, he begins by exploiting in depth the idea of duality in various analytical situations, non-orthodox situations. And then, of course, he had to read the idea of cohomology. And even in elementary terms, if you take the basis of automorphic function in the domain in one complex variable and then take the dual, you have to take into account the topology of the domain on which you have the cohorts and then the homology.

32:30 Again, by a systematic idea that once you have a result in one variable, you take a tensor product and you get it in n variables, he exported that many, many times. That's right, he has this paper and he has this paper. And there was someone in Portugal, da Silva, da Silva was a student of Schwarz, came back too. The first real mathematician in modern Portugal. He is really the founder of mathematics and during the hardship of the dictatorship in Portugal, he was the first one to really develop some mathematics when Portugal freed him, he was the one who nurtured, nurtured the mathematics. So all these ideas, I mean, so it's clear that in this At this time he was moving between Cato and Schwarz, not to mention Sayer. So he was moving in the middle of a triangle whose corners were, whose vertices were Cato, Schwarz, and Sayer making a bridge. Sayer was very good to make bridges, very, very good. As I told you yesterday, we had two fathers, Carter and Schwartz, and one Edmund Gross. My generation felt that way. So we are grateful to all. And Gross and Nick was a genius in the middle of that. I mean, the comment, the comment, the comment. So there was, yes, so that was the contribution. And so various contributions which were inting towards, I mean, he did with various... Duality could not ignore the Poincaré duality, of course. I mean, as soon as you would deal seriously with duality in various semantic situations, you would meet the Poincaré duality, and Hodge theory, and Durham theory, and the Poincaré duality. So I think that was a natural bridge, and of course, I mean...

35:00 Compared to the method of Cartel, the method of Cartel was not really analytical. They were more geometrical than algebraic, but using these very powerful analytical tools, he was able to do something. And Schwartz also contributed. I mean, there is a joint paper with Cartel, Sayer and Schwartz, I don't remember, I mean, about some... Some statements about which were used by Selim is proof of the finiteness of cohomology, Gagarin in the beginning, I mean Gagarin had to use some analytical tools which were given to him by, mostly by Schwarz, but Grotelli contributed to that. And Carton. Carton was good at analysis. Fras was good at analysis. Serre, maybe a little less, but then he was really the master of analytical tools. And then he came back and then at that time he understood the question and the problems. When he came back in Paris around 54, 55, he had matured enough so that he could understand the problem. And he had already, he was feeling already that he had done enough in functional analysis that he would have people invest in other programs. And then another step in the transition was his work in Kansas. About the GAGA, was this by Sayer just a summing up of things that were known or did he entirely come up with this himself? Sayer? GAGA. GAGA. It seems to me this is quite a crucial step also. Yeah, it's a crucial step. Because, potential analysis to the complex analytic case and Viagra. Yeah, yeah, yeah. I think so. How to break it down. Yeah, exactly. Viagra was an important step. I mean, when Sayer started to understand that the method which... I mean, I remember when Sayer wrote his paper, his fact paper. I mean, I was already very well-acquainted with Sayer at the time. First of all, when he showed me a draft of his paper, an algebraic variety is a topological space, and an abstract algebraic geometry variety is a topological space, I was really surprised, really surprised, and at the time I did not understand the meaning of the Zaisky topology, but very few people do.

37:30 He wrote in his lecture notes in Chicago in 1948 about Fiberspaces and he tries to imitate the method of Fiberspaces in differential geometry and he realized that he should use the Zayas-Kate topology to localize. But when he wrote his foundation of algebraic geometry there is not even a single mention of the Zayas-Kate topology. And even in the first seminar of Chauvalet about algebraic geometry, the Carton-Chauvalet seminar on algebraic geometry, then there is hardly a hint to the Zeisky-Tropon. It is only in his seminar about algebraic groups that he really took seriously. I think it's boring, really, the use of Zeisky-Tropon. I'm told that Zariski did not want his name associated with this. Yes, yes. And Kaplansky. Well, Kaplansky, in his small text about differential algebra, used it extensively, Zariski, topology, for good. But it's Boel who understood it completely. And after Boel, of course, it changed. But Sayer, I mean, I remember when Sayer introduced this idea. But Sayer, on the other hand, I remember Sayer was very much relieved. He said, it's so easy now. When you have Graven shift in algebraic, abstract algebraic, it's so much easier than with analytics, complex analytics. And of course, we know all the results, we know all the metals, but it's so much simpler. That's for Sayer, but more or less the things themselves. I mean, after climbing a difficult mountain, relax here by climbing a small step, one more small step, which was easy.

40:00 And so, but then, there was among the problems which were there, there was a problem which I think was overstated by Riemann, no, not Riemann. Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart, Gawart And various people contribute to the partial results to that. One person who unfortunately, I didn't know the recognition he deserved. I mean, there was Frank, who was a student of Carton and did his thesis and solved partly this problem. This problem is again mentioned by Carton in a talk in Mexico, but Jean Frankel, who was a student of Carton, I mean, contributed very positively, partially to that, but we can't take so much of it. So, and also Frankl contributed other things. The first calculation of the cohomology of the projective space, which is now the textbook exposition, how to calculate using the covering of projective space by half-and-sub space, how to calculate the cohomology, he was the first one in the analytical setting to do it. And Serre always recognized that his calculation must be the most to show to Serre that there was something actually behind it. I mean, at the disadvantage that he was a little too young, I mean, too old, which means that he graduated, I mean, he was accepted as a student in Economan in 1943, he was a Jew, he was in Toulouse, and at the time was a German invader, and one does the hardship of the situation. So, Jean Frankel escaped, like many other students.

42:30 He had been accepted as a student in the summer of 1943, but that was the time that Germany invaded France, and then he went to Spain and joined the French forces, and spent the rest of the war fighting in the Gulf. So, like many veterans, when he came back in 45, he had already seen of war experience, and he was put with the people in his new class. Of course, there was a huge gap between such people who were veterans of the war and people who were put in the same class, but this has been experienced in many places. But, well, he was a very energetic man, so, I mean, he managed quickly to get his degree and to work on his thesis, and his thesis was defended in, I think, in 1958, so rather soon after that, I mean, he was accepted as a Connoman student, so he entered really Connoman in 1946 after leaving the army, and so within 10 years he had completed all the degrees and finished his thesis. But since, when he was in the army, I mean, he got a feeling for active life. And so when he came back into civilian life, he had many foreign interests, not only mathematics, but teaching and research and mathematics, many foreign interests. He was one of these, I mean, one. One of these catholic peoples that the New Pope does not like, the progressive catholic of the 50s and 60s, even the progressive, the leftist catholic of the 50s and 60s, and so and so and he devoted a lot of energy to various political and social action and I mean okay so More or less he did his thesis in his sleep time. When he had finished his teaching and with his moral obligation and his militant obligation, he would do some mathematics. But this was a very important problem at the time. I think the question had been raised, I suppose, by Coward. And then there is a report by Carton in the Mexico conference on geometry, geometry, and the geometry about it all.

45:00 And then Gautendieck considers this problem. And for that, I mean, and Gautendieck considers it in very general terms, and for that purpose he invented his theory of fiber spaces, fiber models. Where it takes seriously for the first time the idea of a sheath, a set theoretical sheath, I mean, so it takes this idea that you have the transformation, I mean, the fiber bundle transformation, so it really thinks that in order to define... A bundle over some space. You need to specify a certain sheaf, or rather group of it, you should say group of it, a certain group of it which has transformation compatible with the vibration. And he has a very, so not only a structural group, but a structural group of it. And this idea has been resurrected recently with great success. So, he had in mind this problem and he invented this very general method of shift used to define a certain category of fibromyalgia and out of this method he was able to prove some statement about, well, he could solve some particular cases of the coward problem. And he came more or less at the same time as Frankel. But of course, Frankel was more or less an amateur in mathematics. I mean, he did not want to invest his life in mathematics. But I think the contribution of Grotendieck and Frankel are almost on the same level in terms of the concrete result. I mean, Grotendieck invented new methods, but in terms of the concrete result, Frankel and Grotendieck came on a par. And Frankel is centered always that when he explained his research to Carter, Carter did not pay any attention until he came and also, in the process, Botanyk invented the non-commutative H1 cohomology group, which was developed by Giro afterwards, and Frankel had a very similar definition.

47:30 And for the same reasons, an H1, an uncompetitive H1. So, and so, and by then, now, I think the transition was complete, the transition was complete. And then what made the transition complete is a series of lectures that Bolton gave at the Carton Seminar, who were partly written by Husserl, partly written by Husserl, and about, I mean, Using Banach spaces in many complex variables. I mean a combination of topology, of shifts and functional analysis. And I think that is his last contribution. So this lecture, these many lectures that he gave at the Cato Seminar, which were taken up also by... That was before 1960? In the end of the 50s. Or early 50s. I'm familiar with his contributions. Yes, I think so. Well, I think it was more or less a coincidence, but his first real breakthrough in algebraic geometry was his proof of the Riemann Röpfer, which was written out in detail by Borel and Sayer. I've written a long letter type on purple, you know, a type of recto verso which shows the interval between the lines, spacing, very small spacing between the lines, and to save the price of the stamps. So he sent a 30-page letter, trying to save the price of the stamps, to say he was at that time visiting Princeton, and he sent a letter, and I remember it was very difficult to... And then I remember that Spencer, Spencer wanted to organize something out of that and I think Spencer urged me to do something for that so I remember I spent a weekend I mean that board with great difficulty from sales the original and we did not have the old marketing the old copy at the time so I I bought with great difficulty from there I mean there was a way for half a week and I said well

50:00 If you hand to me this letter, you lose nothing. And when you are back, I give it back to you. And I spent three or four days trying to see for you what was written, to understand the mathematics, and to type it in the normal form. And I typed it, and it took me three or four days, and then I handed my copy to Spencer, who made it for me. So, making 30 copies of a paper of 30 pages was quite an achievement, and you needed someone who has a position of a dean to have that, to use the facility of the university, you needed to have in your pocket someone with a high position in the university, and so, and then when I came back, I mean, we had finished, I mean... Spencer and myself had finished the job and we couldn't hand it to the participants in the seminar, I mean, at least the readable account of the manuscript. And this manuscript was reproduced more or less verbatim by Goldendieck in SGA 7 and more or less, SGA 6 and more or less, more or less. And I remember in this letter he went into the cave. And we invented the K-groups and not only the Riemann-Roch-Hertzog-Groffier and Riemann-Roch-Hertzog-Groffier but also the K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K-K And I gave the talks about Kézio, my point to explain Kézio, and so we had this seminar and then and from that on and then Gotany visited us for a couple of days, although it wasn't the year I did it, so and from that point on it was clear that he would become.

52:30 He says that that was when he was first a little bit feared by others. And that's when Bourbaki became convinced he was serious. Yes, yes. But also, I think, it was the beginning of the misunderstanding, which ended up with his resignation. I mean, I see that happened around the same time, but did that work? Did you agree on rock theorem? No, not exactly that. I mean, it's true that... From that time on, Goethe came with a great authority in fields where he was not considered as an expert. I mean, so far people have said, okay, here's a student of Schwarz, he does some functional analysis, we don't take him too seriously, it's why functional analysis is not the first subject in mathematics, I mean, topology, geometry, and so on, it's not so important, that was a fact. And then, of course, Rotendieck was respected by the same people who had... There was a lot of content or misunderstanding for him when he first came to Paris in 1947 or 48, and that was his revenge, in a sense. And André Weil, who did not consider Grothenich so far as the competitor, said... Or, especially Weil, who did not consider so far what it means a competitor. He was doing functional analysis, but Weil never had any interest in functional analysis, and so while he is an expert in functional analysis, I can't ignore him. He will not come into my garden! So Vail never saw the imprint of these profoundly functorial ideas involving the punctual variant and covariate functoriality in this work on the kernel theorem.

55:00 He was just not interested in functional analysis at all. Vail? No, not at all. That is where it only came into focus retrospectively once. The pattern of Kroenig's achievement in algebraic geometry could be seen as a whole. But then, of course... But it is clear, I'm just summing up, since I think it's probably half of it. It was clear, it was clear, wasn't it, to Weil. I remember a discussion with Weil, or was it in 1960, 1960, something like that. Oh, maybe earlier, maybe earlier. And I, okay. And that was, no, just before I finished my own thesis. Where he was coming, at that time he was in Chicago, not yet in Princeton. I mean, he moved to Princeton in 58. I was there in the 50s, but okay. So, where he was still in Chicago. And he would visit France every summer. And we would occasionally give a few lectures at the convent. I recorded some of his records. And William was in very good terms with him, and he would, every time he would have... I was doing and at some point I remember just before finishing my thesis or maybe a year or a year later when I was working hard on my thesis and of course I completely absorbed the idea of sheaves, cohomology, etc. and then the beginning of the ideas of Goethe and then I was thinking in terms of cohomology, sheaves and so on and then I tried, I knew that... Wey was not very fond of these ideas, so I tried to explain to Wey, so I made a gated fool to translate things into a language which would be accessible to Wey. And at some point, I remember the nasty comment of André Wey, my friend. You know that I'm fond of foreign languages, I know a lot of them, including... Sanskrit. Well, you did not mention it but everyone knew it. Well, there is a new mathematical language, the Groton-Dicat. What? It's your native tongue now. I speak in your native tongue. And so, of course, I say, okay, let's take a look at this.

57:30 Where the idea of a double complex spectral sequence shifts and so on. It's not explained in this term, but all the tools are there. And still now, if I have to give a set of lectures, a starting set of lectures in differential geometry, I would prove the theorem by this method, exactly by this method, because it's very complete, explicit and very good. And of course, all the great ideas are there. So, Andre Weil, I mean, Andre Weil, one of his best contributions was a cohomology in class refueling. Very new, very new. I don't know why. So I tried to hide my major, but he said, well, it's a new language. It's your native tongue. I can translate for myself. Well, as you reported, that seems a very generous-minded comment rather than a negative one, but as you reported, on Weil's part, it seems a very kind of generous-minded comment, you know, come speak your native tongue, speak your native, I don't mind, you know, I can translate into mine. Yes, but report it, not me, but I did. They had a good feel for good mathematical ideas and I suppose my thesis there were a few good mathematical ideas and he wanted to learn about his ideas and he was not, I mean for him speaking one language and another one was not important. He wanted to understand the meaning of them. And it's true that, well, he was very fluent, he was fluent in his language. As he puts in his autobiography, both his father and his mother were German by the way. And he claims that sometimes the parents, while they both settled in France, that the parents, when they wanted not to be understood by the children, that means André and his sister, would speak German, with the obvious reason that the two kids loved German, which is what the parents say. Of course! So, I mean, in the home of André, right?

1:00:00 The German tongue was a native tongue. Both French and German tongue were used extensively. I have known some families, you know, a Jewish family from Exeter, close to my family, one family here. We shift, I mean, people shifted between the two. And so, but then he learned English. Well, I remember he spoke good English, but with a bad accent. But he spoke it perfectly. And then I heard him speaking in Italian and according to my Italian friends his Italian was perfect. A little out of fashion but perfect. I mean, it's quite closer to, let's say, to the classics and to what you speak on the street of Rome, but nevertheless, I mean, he spoke perfect Italian, and also, of course, he learned Latin and Greek in school, and then Sanskrit, and he had some knowledge of Russian, not very much, but some knowledge of it, but he never learned Hebrew. And he was part of this, I mean, his family was like an ancient family in the beginning of the 20th century in France. They did not want to be Jewish. They wanted to be French. My own family was in the same place. Super assimilation. Yes, super assimilation. I don't believe that Emmy Noether thought there was anything to being Jewish. I mean, it didn't cross her mind. She didn't want to be or not want to be. It just, she was just, she was German. Just a little bit indifferent. I mean, it is Hitler who made these people discover they were Jews. It's Hitler. And Thaissemitism. It's a German... No, but these people did not want to be Jews. And about the sister of André Wey, I mean, Simone, I mean, there are some... It's very, quite anti-Semitic, anti-Semitic at a superficial level. And certainly, I mean, well, Andrei told me that he was once visiting Armenia long ago in the Soviet system, in the Soviet system, and that if some old Jew there had engaged him with shalom, and he was so surprised, he told me that many times, shalom. Simone Weil converted.

1:02:30 But you know, about Armenia, I remember I was invited by Drinfeld's conference in Kenya many years ago, and among the circle of friends of Drinfeld, there was a young lady who was not a mathematician, but Everyone understood that, I mean, they are in the archip of the system, I mean, they could have some rest and use his companionship with mathematicians, you know, for a few days. She was a medical doctor. And at some point, I mean, there was an evening of Russian Jews, Russian Jews, and an evening of Russian Jews, and this lady was next to me, a very charming lady, and I asked her, I mean, are you Ashkenazi or Sephardi? It was not an obvious answer because the people, the Jews from the south of Russia are mostly syphonic. So the question was a serious question. I thought, I said, but everyone obviously was Jewish, is he? And I asked, are you syphonic? Oh, sir, I'm Armenian. But then she explained that the early 2000s doesn't move to Israel. Well, in Russia you knew if you were Jewish. But for many people it didn't mean anything. Many people told me that. And because of the various things. Many Russian Jews were brought into Cleveland, and that the Jewish community in Cleveland was hard-fought. Can I just say, very briefly, before we take a break and time dinner, tomorrow morning we'll resume at about 9.30, if I'm okay with you. You know, it's very... and I think a brief... obviously it's been fascinating to hear about this kind of thing in sports, but I think we should have a short response to some of the things that you've said.

1:05:00 You said more or less the same thing, but I read it as it will be. I don't think I want to spend the morning on this thing. Well, we do have also to make room for the whole of this session. It's a pity that they never have the opportunity to come up with a new talk series. No, no, I think we'll do both in Princeton, but you know, the impact... I think it's going to make it rather difficult for us to be able to do this in a way that's not going to happen. No, no, I agree. What I'm saying is that I agree with you. I think it's going to be a lot more difficult to get that bolt on some of the things you were saying earlier. I think it's going to be a lot more difficult for us to do this in a way that's not going to happen. Thank you for your attention. Very fantastic, very fantastic. Thank you for your attention and see you in the next lecture. Of course, the fact that he was the target of the Nazis must have made him Jewish. But apart from that, let's see. The resistor was famous. I don't, but not to name so. I don't... Well, I think he was not active, but he was much sympathetic to the actions of physicists. No, I didn't mean to say that. No, I did not mean that he was active. But, of course, he was very sympathetic to the actions of the physicists.

1:07:30 And I have a good reason to believe that. And so there was a world that we saw in India, and we actually met Gandhi, and at the end of his letter, he said, we've got a revolution, and he considered Gandhi as a revolution, and I think he had much in common with that, much in common, and I don't think he was really… In the political action, it was more interesting, but it was much simpler, and it had to be surprised, because when it was... In the fall of 1939, when he went to Finland and was arrested for a few days and considered as a Soviet spy, his recollection is mostly exaggerated. But nevertheless, what has happened, at least in this case, is that the one who tried to call on the French ambassador for help, the French ambassador did not want to support him. I said, if the French ambassador had given any sign of support, everything would be settled in a matter of a day. So, we might not talk about the forces of gravity, but in general, especially because of this system. And also he's one of the very, very, very, very, very, very, very, very, very, very, very, very, very, But nevertheless, the fact that he is important, oh he was, and he was very much in the circle, and you know that I'm not sure about the A.M. or so, but in the whole business of it.

1:10:00 So, interestingly enough, I'm out of time. I know I'll do as I have to, but I can't speak for the style of the publication. I mean, there was a chairman, an architect, who made a play about quantum mechanics, and then he did an opera by, and then, and then, and then, and then, and then, and then, Because from the start, we were just trying to find a way to get rid of this information.