Pm discussions: FW Lawvere, P Cartier, C McLarty, A MacIntyre, JL Bell

0:00 Yeah, I will certainly think it was in the university. Maybe, maybe. I suppose you know a piece of it. It's one of the key categories. Yes, sure. That is there. This is my friend. One suggested they took it up. Yeah, yeah. Yeah, that's true. But the crucial point there, I mean, that's what I was saying yesterday, that the grotesque topology is the coherent one. So any covering of the line by, you know, regular open sets would... It must necessarily have at least one infinite component. Let me say this, I don't know. So you cover it by... What are you covering by? So there's sort of pushing problems off to infinity in that kind of a way. Take care of it, the fact that it's a finite area. That helps, I think, in fact, if you didn't have that, then this would not, this phenomenon would not occur. It's not justice, it's the fact that you're using it. Well, that's the amount of smoothness that remains in this model. Yeah, in this case, I'm not saying, once again, I think this is intensely interesting and it's nowhere near understood once you investigate it further enough. It may still not be the paradigm case. No, no, no, sure. It's just the way these things interact. Okay. Yes, that's a very interesting paper actually. May I make a copy of that, Bill? It's on stage. That one I definitely haven't seen. The one on data types. Okay, it's been mentioned, but we were... This time, the session we were about to start now was to have been devoted to the topic of Grimby's work between roughly 1950 and 1960.

2:30 That is to say, an examination of how his early work in functional analysis related to his subsequent work in algebraic geometry and to the expansion of his vision for the re-description of structures throughout mathematics. However, before we commence on that topic, it has been pointed out that in this morning's discussion we said very little about the role of Saunders-McLean, that it might be an idea just to have perhaps a brief... So, well, I just happened to have... It was a rhetorical question. I just happened to have... It was a rhetorical question. I don't know that. It's an Italian journal. Nazi notices. No, no, we're working on... We're working on... Oh, yeah, definitely. This is a journal, something like Mathematical Intelligence. But all in Italian. I think actually better than intelligentsia. Okay. It's more... Well, I'll tell you something if you told us. You'll also, okay. Anyway, so I was asked to do something here, and I feel slightly quite sure of what I'm saying, but the man is scarcely dead, and I'm already seizing the opportunity to promote my own ideological mind on the last summit. So, if anybody finds that offensive, please let me know. When did he die? April 14th. April 14th. In Chicago? No, no. They had moved to San Francisco. Oh, San Francisco. They became really too… Mrs. McLean has a daughter. She, by the way, was the ex-wife of Irving Segal, so she's familiar with him. Her daughter is called Segal. I know, I know. I know the doctor. I know the doctor. But anyway, they moved there because Mrs. McLean had some help from her daughter.

5:00 I've known the daughter when she was a freshman and played card tricks on me. Alright, then you knew Osta as well, probably. Yes, of course. I guess they lived there about a year or not much more. Well, okay, so, anyway, we already mentioned this triple impact of the Eilenberg-McLean spaces, category theory, that's part of this. Okay, so it's a common statement that McLean, in his later life, divorced himself from his original interest in logic and my thesis, which I hope to be able to investigate much more in detail before the notice of the article comes out, is that he didn't start as a logician, he started as somebody who, as they often said, Later in life, he was rabidly interested in understanding, and so after his undergraduate degree at Yale, he actually went to Chicago for his master's degree to this department which had just been started up at the head of E.H. Moore. Stone was much later. Oh, yes, yes. This was 1900. Yes. Well, no, 1929 or something in there. So it was E.H. Moore was the author of the general analysis. In fact, as I said, Frechet took over this slogan from Moore because Frechet also often used this phrase, general analysis.

7:30 In any case, apart from the name, I think the spirit of Crescet was similar to that of Moore in the sense that there is a vision that there should be a more explicit greater unity of mathematics, let's say, or systematic understanding on certain levels and so on. I actually mentioned to Saunders a few years ago, I said, well, this was a great idea, but they couldn't possibly carry it out without category theory. On the other hand, now that we had category theory, why don't we try to take up that program and refine it and carry it out? But anyway, it was Moore who advised him to go to Germany to study with Hilbert, Nuerker, Hermann Weyl, and Paul Bernays. So, I see a certain parallel with my own career here, you see, because, and also with the career of my other mentor, besides Morgan McLean, who was Truesdale, because, you know, trying to put it in general terms, starting from... A deep interest in the general understanding of the fundamentals of things, in Truesdell's case, continuum mechanics, also, we search out the people who are the experts on foundations, and we find that on the one hand, there's lots of interesting stuff there that we can use, but on the other hand, that their vision is inadequate to the purpose that they have, so we are sort of temporarily logicians. Church's book was written by Truesdell, after all.

10:00 Which book? The Oanto Churches, one of the major texts in mathematical mathematics. Written in the sense that he took the notes and polished the notes and prepared them for publication, the whole course. Because Truesdell had taken from his studies at Brown University in mechanics and so on, he had taken off a year and went to Princeton, thinking perhaps to change his life and major in logic. And then he found after a year's experience that, of course I agree that was a special example, maybe not typical. Anyway, so, as far as I know, MacLean and Truesdell didn't carry away much from this experience. I never mentioned logic again, but I know it. I did, of course. In my case, I carried away quite a lot. So this is why only volume one of the church, you know, it's called, the volume one. Why volume two never appears? The assistant had defected. Exactly. So also, it seems to me that this general description applies. He also, of course, as you know, he, he, he... He carried away what he learned in logic, and he learned a lot more throughout his career, but he was never a logician, quote-unquote, in the sense that that was the main focus of his work. So that's one point that I make here, and I'd like to make more precisely when I get home. I'm hoping that his autobiography will explain this in more detail. Colin just got the notice from Amazon that his copy is on the way, and I order from Amazon as well, so I expect that by the time we get back home, it'll be there and we can look at it. So, McLean wrote some autobiography? Yes, it was just published now. Oh, just published now. Just by Klaus Pater. Oh, okay. It came out two days after McLean was gone. Oh, okay. I'll have to give a talk in the summer about what is mathematical autobiography. Oh, unless this should be an attempt at that. So here I go on about this, about the fierce need to understand.

12:30 Fierce need to understand. So his work led to deepening and amplification of that. But the logic considered as a serious study of the general aspects of all exact thought cannot remain confined to the symbolic tradition which looks only at presentations instead of the algebras presented, neither to the narrow tradition of Frege who says that property is a just concept. So both of these tendencies work certainly. So, yeah, so the geometric vision, the conscious vision of a guide for developing mathematics in a unified way, which Moore, a century ago, could only realize in a fragmentary manner, has become possible. By through the recognition of the unity of the principles of logic and geometry that the underlying principles of both have much in common and moreover the principles of their relationship so that I end here by saying we can hope to satisfy the dream of MacLean and of Moore to transmit a more profound comprehension of modern mathematics to the students. And here is my attempt to explain for the completely uninitiated theater what the categories of natural law is all about. Anyway, I could say a lot more about McLean, but those were some of the things that I thought of saying, which you probably have.

15:00 Well, actually, I'm wondering if you could say more how it is that you ended up working so much with him. I mean, the truth is I've always associated you with him, but you were Sammy's student. Now, some of that's just my confusion, some of it's an accident of, you know, which meetings I got to, but what... Well, I can't say I ever collaborated with either of them. Yeah. I mean, the circumstances were such that I was still briefly at Columbia with Sammy. I mean, there were many things. First of all, he accepted me in a completely unprincipled manner just because Truesdell said he should. As a result, I graduated without ever learning all sorts of basic things like how to compute pi 1 and stuff. I knew these things existed, but I just never went through the sort of exercises that students really should. Moreover, Sandy was chairman of the department that year. I was very, very involved with that. I only had one or two even discussions with him. It was so little that I had any discussions that I had already sent to him a manuscript, which I had worked out in Indiana. Essentially, I had discovered the notion of adjoining functors, basically. But, you know, Sammy looked at it and exchanged a few words about it and so forth. I was talking to a fellow student, and I was telling him this stuff. He said, oh, that's all in Kahn. There's actually a published paper by Kahn called Adjoint Functor. You should call these things Adjoint Functor. So I went to Sammy and I said, what is this about Kahn and Adjoint Functor? I was told that this is... This is essentially the correct formulation of what I was trying to do. Then he pulled out the drawer and gave me a copy of Connes' paper. But all I'm saying is it never occurred to him to do that before. I just forgot he was so busy, actually. I think I just forgot he was so busy. Which was the year? 1960, spring of 1960.

17:30 Then Peter Fry had arrived as a post-doc in the fall, so I actually... In some way, he was the most direct teacher in the category. But then, precisely what I was just saying before, I got this idea that I've got to learn more logic, so I asked Sammy for permission to leave New York and go to Berkeley. Several of the leading logicians are learning as much as I could from these people. I had a family, actually by that time two families, that I had to support. I got a job with the military industrial complex in Los Angeles for a year. During that time I wrote my thesis and I went back to defend it. He didn't read it at the time. He gave it to Saunders MacLean on some plane that they were sharing a ride in the plane. And Sammy says, here, you're the outside reader. Here it is. And MacLean read it and liked it. And on that basis, I got my degree. A couple of years later, Sammy had read it and gone much further. He gave these lectures on automata theory and using linear algebraic theories and so forth. Unfortunately, not... He gave four colloquium lectures of the American Maths Society in the summer of 1967, and one of the four was entirely devoted to Algebras' theories and how he deployed this in the public theory. Is that, if I'm not mistaken, who communicated your papers to the, the ones on the theory of the category, the papers in the Proceedings of the National Academy of Sciences, isn't that right?

20:00 Right, so after I got the degree, I worked, I was teaching at Reed College, and this is reprinted intact now, the commentaries on the elementary theory of the category sets. I won't go into that, it's just about me. But in particular, I wrote up this abstract of a thesis and McLean indicated that. McLean was a member at the time? Oh yes. And then he suggested I apply for fellowships. I saw McLean at the Boulder meeting of the AMS. In a paper called Computer Science, I presented a paper there in the summer of 1964 about behavior and mathematics. I remember Church sitting in the audience, actually, for a hundred years. The Saunders gave the colloquial lectures that year, so that's actually published in the Bulletin. It's a reference. So he advised me I should apply for, you know, I should study in Europe, just as Moore had told him, so I said, oh well, I'd like to go to Paris and study with Cartan, and so this was all in motion, I was going to do this, and then, well, personal things changed, and I thought, well, maybe I should go to Zurich instead and work with Benjamin Eckman, so. And so on and so forth. I think you were still at the... Hey, 64. Where were you at... I went to UCH in 64. Yes, but at the summer of 65 you were either just leaving there or just finished up there or... No, I was still there. You were still there, that's right. Over a period of 23 years. Because that's where we met. I recall.

22:30 Right, so that's how I ended up with... I saw that character, I don't know what time it was. That was the end of his... Wait, one thing that struck me, if you look at the list of... There's an amazing range of students, PhD students, that Saunders-McLean had. I mean, you know, there are logicians, Robert Soloway, Michael Morley, and, you know, very... Yeah, I mean, there's amazing... Very few actually category theory. I mean, you know, the people who actually did category theory as such were mostly Sammy Allenberg students, at least that seems to be the case. And Saunders had a very... I mean he seemed to take on all kinds of things from all over the place. I guess it reflected some continued interest at any rate in logic as maybe not in this thing of general understanding perhaps because he certainly supervised a number of very outstanding logicians or people. I mean Solovey of course didn't do logic with him. He jumped on the Cohen. I would go into business very fast when he went to Berkeley. I think his thesis was in the Riemann-Rock Theater. That's what it was on. And then he got into logic very, very fast. And he was a plane steward there. He's got a paper on that. Right, right. Yeah. He jumped very, very far. And the road was fully... That's Nero, too. Morley. I know Nero. He's another magician. Yeah, very wide-ranging. And Michael Morley. Morley, yes. But Morley's case isn't so simple. I mean, Morley, I think he skipped to Berkeley. And Morley's thesis, which has appeared in classical moral theory, this was, at least for me, I don't know if it worked.

25:00 Oh, yeah. It's a beautiful work. I would even permit it to speak on categories. Well, you contributed to that model theory meeting, right? The one in Berkeley, and the volume was published, you know, what was it, 63? That was later. That was later, right. Before that, when I was, I wasn't living in Berkeley at that time. Right. But before that, when I was. Interested in category theory then? I think he, well, Peter Freud claimed that he always was. Certainly, the spirit of his... Well, I think it's different with Dana. The way Dana organized his work, I mean, what he perhaps started out with was his students, or something. I don't actually recall him ever using a specific categorical kind of knowledge, but of course he was seeing a tremendous use of dualities and so on. I don't know. The way he organized matters. It was something about the spirit of category theory, if you like. Yeah, yeah, yeah. Most of his work. I recall him using the... Well, I think of understanding that you meant, you know, that's what he was looking for. He started out with this, I think that's right. Yeah, exactly. Which wasn't focused in any particular way. No, it was general understanding, I mean, no question. My impression of Dele is he is slowly, slowly, slowly, slowly. Now he's 95% fun-sized. Oh, really? Yeah. But every one of those percents... It was kicking and screaming and resisting. Yeah. One more. Yes, that's true. That's true. It is. It's a more reduced direct product, right? That's what it was called. Anyway, to return to me and the thing. Actually, 1965 was a very busy year. There was a meeting in La Jolla where I presented...

27:30 And then there was the Leicester module later part of the summer, and then when I got back to Zurich where I was living, I wrote this paper on the category of sets, and McLean sent that also. Indeed, as you say, he communicated. Sorry. He indicated those two papers. But also, he invited me to stay in Chicago for the summers and so forth, so I wrote this longer version, which has now been reprinted on the tack by the help of Colin. And then, McLean, you know, he actually wanted me to, I mean, actually, given the hypothesis that he had a reasonable opinion of me, it was therefore natural he wanted me to come to Chicago. I was hired there. We were quite brash in those days. Because there were lots of jobs available. Yeah, that's true. So I said, the first semester I want to have a leave of absence. Because there was this gathering in Zurich, which became the Zurich Triples Book. Where really the whole business about monads and so forth was systematically fought by a number of people. I didn't want to miss being part of that. So I didn't arrive in Chicago until mid-year. ... students are not so happy. They could not put such conditions anywhere. No, I don't understand. It makes me wonder, I mean, you and I both had the idea that these were Samy students that did category theory. Yeah, I mean, that was... They were Peters. I mean, you're saying... Yeah, Fitch would be the... Peter is the reason that category theory is talking about. Who are you thinking of? If you continue this list, I don't know if I can do it right now.

30:00 Legend? Legend, yes. You will arrive at a list of ten. Yeah. None of these people were converted to category theory and then left as category theorists, they all had already been converted before they went there, they went there simply in order to deepen and broaden their knowledge. I don't want to say that he created them as category theorists, but it's of interest that I went there and gave up my study with Truesdell in order to go learn more about that. Yes, but it's of interest that I went to Islandburg rather than McLean. Okay, now this is true. There's a very simple reason for that. Very simple, stupid reason for that, at least in my case. At least in my case, I just don't laugh. I mean, don't forget I was an ignorant farm boy, and I still am. So I decided, oh, it was because I had been asked by Truesdell for a lecture on functional analysis, and I had developed some things about adjoint punctures and string spaces and all lectures, and I thought, oh, this is really the thing I have to get in on the ground floor in this category theory stuff. So I cast about for how could I do this. There was no one at Bloomington who was, you know, in a position to... Some professors knew a little bit about it, but, you know, I cast about to see. I'd heard of MacLean. I'd even studied with his algebra books. Yeah, exactly. But, okay, look at the joint authorships. Well, there's, okay, Eisenberg and MacLean, first paper, and there's... Eilenberg and Steenrod, very important. Cartan-Eilenberg, very important. And there were two or three others actually. So, you know, from a distance, I reached the conclusion, wow, this is the guy who's involved in all these things. He must be the one who really knows about category theory. So therefore, I decided to go… And then, of course, the fortuitous fact that the Truesdell happened to be very good friends with Steenrod, even though their freedoms were completely destroyed. It was because of their joint interest in art.

32:30 That helped to facilitate the actual move, but the motivation was, I'm sorry to say, just this fact about the joint, multiple joint authorships. Well, that may have had a similar effect on, at least something, a contributing factor to some of the other decisions to go and study with Ironman. I think probably it did. Ironman was a perfect go-between. I mean, he was, he has a talent. Yes, he was a remarkable concentration. Columbia, London in 1860, at least 10 people. But Barr didn't go there for category. Barr did get recruited into category by Beck, I think. Well, okay. Okay, but that doesn't change the point. I mean, the reason he went there, he went there a year later. He was already a post professor. Yeah. He went to do cohomology. Closely related, and he knew it before he came, I'm sure. He's told me that he was... Puzzled by a billion. My first encounter with a guy, before I actually got involved in an abstract with a serious physicist like me. Rejection. Tierney was there? Oh yeah, Tierney was there. I'm not going to finish a list of ten. Both Tierney and, he stayed longer than I did, another important influence.

35:00 He was very, trying very hard to be puncturized. And again, the other guys who stayed longer than I did, they had courses from... Special sets and so forth. Technically adept because of course... Another visiting professor at the same time. I mean, Sandy must have had something to do also with gathering these people, Dole and Soad, of course, from Helmetson and others, on Lie groups. Fantastic! And the third one was Peter Lacks, a co-committee member a couple of years ago. He was only at NYU, but he was teaching a course. In addition to the actual Columbia professors, there was quite an illustrious group to learn from. What about Colchin? Colchin was there. I didn't have any courses from him. Yeah, I mean, Stanley didn't have too much time for a graduate student. He certainly was doing a lot of very good work as chairman.

37:30 You wrote a paper about it, I think. Yeah, and there's some long notes in the Chicago album. Okay, so again, a longer version. Mostly just before the rebirth of Hamilton and Beckham. The book of Arnold appeared in 1963 or 1960. A few French books on the same idea. And then the emphasis on symplectic geometry began at the end of the 70s. Arnold and Kirchhoff on the Russian side. In France also, I mean, some of the people around her, they respond to it. Did anyone pursue it because of Raleigh-Thom and his hope that he could prove the stability theory? Maybe, but I mean, I had a discussion with Tom, I remember that Tom was not the dynamician. The basic idea of Tocque was to put everything into geometrical terms, and especially, I mean, one of the weak points in his catastrophe theory was that he thought everything was, and also what he tried to do in developmental biology and morphogenesis and so forth, he was looking at static picture, not at dynamical. And I don't think he had a real feel for dynamics. Well, he wanted to prove that stability was generic. Well, he could do it for potential systems. Then he wanted to do it for Hamiltonian systems, but it was actually refuted. It's not just that they failed. But how many dimensions? Oh, it's different, yes. But my discussion with Tom was never feeling for dynamics.

40:00 And I think that's what was on the weak point. He would say, okay, you have a phase transition, you have a curve. And he would always draw this picture and then you jump over that. But why do you jump? He would just say, there is a jump. For instance, I mean, Tom could very well discover the phenomenon of transition, which is delay, the delay transition. He's interested in this, about evolution. But the disadvantage of space time is that the dynamics become static. Well, there are many advantages, of course. But on the other hand, I mean, when you see a particle becomes a problem. So you have a line that is something which is not dynamical, and it took many years for people to understand that cosmology is something historical, it's a form of development, and how do you interpret a true development within a four-dimensional manifold, by definition, is given there. You are looking from the point of view of God. I mean, you're looking over the eternity. And it's very interesting in that connection that Einstein, of course, rejected, you know, the dynamic picture, the solutions of general relativity. He introduced the cosmological concept precisely in order to avoid dynamic solutions to the equations of general relativity. So, I mean, so, I don't think Tom was... But, of course, stability of Hamiltonian system is slightly different. It doesn't tell about instability of the development. It tells you about, I mean, if you perturb your Hamiltonian system.

42:30 Yeah, there's other kinds of stability. It's that his structural stability, which should lead to a simple classification of similarities, is actually refuted. It's known false. It's like, I mean, it's really the basic thing in calculus of variation. In calculus of variation, applied to mechanical one, one trajectory, which is a dynamical one, but a calculus of variation tells you how you modify this trajectory. But this trajectory is not dynamical, it's a curve, that you move your curve, you move your curve transversely into some space of curves, some configuration space. It's a different, and stability has two different meanings, whether you follow the long-term development of your dynamic. Or, if you just want to modify, to a small modification of the... And I think basically, we had many discussions. He made some mistakes. For instance, when he tried to... And then he would think in statical terms and, of course, it did not work. It's a very interesting... I mean, it's clear that having a four-dimensional space time is a great advantage. How do you understand historical development? In a sense, you are taking the point of view of God, who looks over the eternal world, I mean, all the things he has in his mind or in his sight, the whole development, I mean, it's a whole challenge of the philosophy, how can you reconcile historical development and also freedom of mind, let's say, and with the fact that you have a God which... Which sees all the time and all the places as I say. Phrase in theological terms it's a classical term.

45:00 But we have the same thing in Hamiltonian mechanics. I didn't know it was a course. You know, I mean it's a big part of mathematics. Yes, that's right. What's the right way? Of what way of doing Hamiltonian mechanics? Yeah, I remember that reflection in the book. I think David Eisenberg did it. I mean, they consider the situation. In the French tradition, there was a so-called rational mechanics, rational mechanics, which was a rather strange combination of various tools, but mostly it was mechanics before Jacobi and Haber, based mostly on Lagrange's idea, so middle of the age, but the people were teaching mathematics in a way which had brought what has happened with Jacobi and Haber, and it's interesting to refute Hardy's view. You have to remember that Jacobi who certainly invented zeta-fraction for two purposes, the dual use of zeta-fraction for both generating series in number of theoretical problems and a tool to solve differential equations because the zeta-series are series which are very rapidly convergent and if you express everything in terms of zeta-solution of your dynamical system and in terms of zeta-series, I mean, you have a very good numerical control.

47:30 Because the Theta C's are very fast. But it's interesting that Jacobi was, by my own consideration, a master of pure mathematics, invented, or at least understood, that Theta function could be. So, but old-fashioned to the point that it predated the development. But they decided that this kind of mechanic is dead and done. And the other end, classical mechanics, is dead and dull, and even revising it, Assange told them, who cares about trajectories? It was just the beginning of us. And people were out of the way. Calculating trajectories wasn't a very important job. And that even practical questions could lead to theoretical progress, like the Aaron Stokes discovery in 1964. The first example of a closed trajectory in a three-body problem, it's like, oh, it's like, oh! By the way, the trajectory would save the people from Apollo 30, when they're edging work. They weren't in the relative system, they were in the closed trajectory, which says that if they were doing nothing, they would come back to the Earth, which has been devised with Arab stuff.

50:00 In case the engine breaks, the people could come back to the Earth without any engine. And it made a great, great problem. The first bogus over a century. So, and then I remember this. But now for modern physicists, I mean the classical mechanics, who has tackled quantum mechanics seriously, knows that we have a deep knowledge of quantum mechanics. You should not teach them. And other reason, even today, even to the most elementary level, except what they learn in high school. After that, people who specialize in mathematics, they learn nothing about mechanics. And so, but the comment that, it seemed to me that the people who were, in some sense, close to Bobecki, like Maclean was also close. But the people who were a little out of Gourbaki, I mean, never lost his perspective and I mean the people were really in the core of strict influence of Gourbaki and said that, well, not only applied mathematics has no meaning, physical physics has no meaning and so on and so, and that has been a desire too.

52:30 I could take the best and forget the worst. I don't want to be in the middle of this. But just to say that my experience with Maclean, I mean, Maclean, in my opinion, never yielded to me. Well, the fact that he gave this course was to resist that trend whereby students know. I don't know now whether Chicago students now get it. Maybe the last time, I don't know. Well, since they got... Now we have the Russian influence. The Russian influence, yes, right. And the Russians are always big. They were never cut, not by chance, that Arnault wrote a lot about physics. Of course, the French were in a good position because Élie Carton, and it's ironic that, I mean, I heard many times Henri Carton, the soul, Hervé, they were the heirs to Élie. André Wey mentioned to me, not Dieudonné, but André Wey mentioned to me that one of the aims of Gauvaki was to put the discoveries of Élie Carton in geometry on a sound basis, And that's what one of their aims would be, substantiate all the claims of many of them, intuitive and not. So, but Elikaton was always there. Elikaton was always doing this exchange of later between.

55:00 I think it's also interesting that... Oh, yes. Yeah, so after I was at Chicago, I went to City University. While I was there, a series of ten lectures at IBM Yorktown Heights Research Center. And as I said, he's in one of his four colloquium of algebraic theories, and he had a collaborator, or two or three collaborators, actually, of the complex that we're pursuing. Yes, that's right, that's right. Jesse Wright and Calvin Elgott, Thatcher. So anyway, they wanted to have, so I gave them ten lectures once a week, waiting for my son to be born. So my son was born at the same period of time. So I talked about, of course, the categories in a more general way than just those of your natives inventing joint funders, cons, extensions, and that sort of thing. A long time ago, I had some notes either by you or on your ideas from the IBM Center that Ray Nelson gave me. I just don't even know if I have them or what they were on or anything.

57:30 But I went to my first year as a grad student in philosophy, I got my bachelor's in math about grad school in philosophy, and I went to Ray Nelson and I said, what kind of math do you think a philosopher ought to know? I don't know, give me a good program in math. And he said, well you should learn this lot of your stuff on algebraic theories, and you should learn the recursive hierarchy, and you should learn what this topos theory is. Well, the recursive hierarchy never really did take place. I should have said I also worked for the French military in a certain part. That's right. When I went to Zurich, I also... McLean had met this person at a meeting in Scotland, actually. For that reason, he told me to look him up in Paris. So I went there, and I met Jean Benabou then, who told me about the existence of something called topos, and I didn't know much more. But the person that I went to visit was Jacques Riguet, and he had some kind of a brand. I went to his country house and worked away, and there's a, it's not published in a journal, but a report inside the French military. I only did one stuff with some philosophical nature as well. Did Maclean do all the military work during the war? Oh yeah, he was the director of the Columbia University Center for Ordinance Research and Equality. He hoards up some people in the OTCC. The whole mathematical organization, he was the head of it actually. And he hired Eilenberg, I think. And he interviewed Truesdell. The young Truesdell applied for this position, but for some reason didn't go there. So many, many, many, many, many years later, I brought them together. I played Truesdell for the first time, really.

1:00:00 We sat down and had a beer together and now something comes to my mind, I mean, about Rota, Jan Karel Rota. Jan Karel Rota, yes. Was there some connection between MacLean and Rota? MacLean and Rota, what did I know of? I had some connection with Rota. No, I had myself had a connection with Rota, but I'm wondering whether there was a revelation between them. There is one relation between them in my mind, and not directly, but for many, many years, whenever I would meet either one of them, they would say, write, because I didn't write down enough. And McLean was always urging me to write more articles, but Rota also. Well, Rota was a prolific writer. He was a prolific writer, but he thought that... Now, it's because you mentioned this IBM 8, Yorktown 8 lectures, and Walter also had a connection with Los Alamos, and some of his work, some of his work has a connection with Los Alamos. And recently I was using his work on asymptotic expansion. It's a classical subject, but I needed, I needed some... Well, the point is that in asymptotic expansion, people take usually asymptotic expansion in a power, power of the solar wave. But in many applications you have also log epsilon, and you need to have good algorithms to deal with both epsilon and log epsilon. And I came to that already some years ago and I found that the motor and metropolis. But that was because it was not too far from it. So I mean the motor had a similar idea. Also in this paper with metropolis there is a... More or less, what is the definition of a real number, I mean, the automatic definition of a real number, what is an automatic number, I mean, and the whole time Metropoli studied problems of the nature of them. And that's, of course, it's...

1:02:30 California, back to New York, he was there and described the algebra based on a partial order set. Isn't it? Yes, Mervius-Rhoda function. Mervius, I mean... I immediately said, well, actually, you should do it for a category, not just a partially ordered set. So there are examples involving monoids and even a morphism between partially ordered sets, say, of natural numbers under divisibility, as opposed to the monoid of natural numbers under multiplication. There is a functor there which induces a map of these algebras. It's really based on the monoid, actually. It's injected into one there. And it's all done with this function. It's just one piece. But there's more to do with it. There's work, for instance, that's left here on one. It started as a very elementary problem. Take a plane, draw a certain number of straight lines, and count the number of domains, which is straight lines. Divide the plane. It's not an easy problem. We were talking a bit about it yesterday. It's curious you said both of these things, because both of these things are very much related to this whole minimality we were talking about yesterday. One of the main achievements in that area has been to construct a formal model of the real asymptotics with logarithmic and iterative logarithmic terms and so on. And one knows a lot about this. It's also connected to Shannon's conjecture, some of the algorithmics of this. When I knew Roda, too, he encouraged me quite a lot. It doesn't surprise me what you say, but this thing about the... I'm not surprised that he and Metropolis had been doing it, because I'd never seen there where I knew it existed, but eventually model theory and the same topology came to this too, and it's an instance of a very much more general.

1:05:00 Plane arrangement, I think it's called something like plane arrangement, the 17th. And it's still, well, it's connected with some Arnold and Briscoe. But what they developed, I mean, what they developed, I mean, that was, I mean, deceleration of real manifolds. Why? Arnold and Briscoe were interested in deceleration of complex ones. But we know better. We know that, of course. And then, of course, you have the real point and the complex point, and the tessellation in the real point, and the retrompterization in the complex point. They have to play together. And that's what I mentioned already, that the rebirth of combinatorial topology, mainly out of such examples. And it's striking that, and again we speak of fashion, and it's clear that this kind of combinatorial was completely out of fashion for many. So called serious mathematicians. Until they rediscovered that stuff. And also about Maclean in fashion. I remember when I gave, it was in the fall, mid fall, I don't think. Maclean was in Paris Porsche. And then I gave a seminar where I invented quadruple and things like that. Maclean was the only one who took it. Carter, I had some distance to one. Of course, a little over optimistic, I hoped something for the home. Yes, that was much too much, of course, of course. But nevertheless, when I formalized the definition, I invented this cobalt construction.

1:07:30 And then Maclean attended very, I mean, listened very carefully to my talks and gave me great encouragement. When Cantor was a little reluctant, so Maclean was, I think, much more open. Yeah, he was open to, he didn't ask immediately, what can you prove about this? He could recognize that a newly proposed structure might have implications even before any non-trivial theorems have been proven about it. This is contrary to the ideology of most of the tough mathematicians who want to know a hard theorem right away and don't care about the construction. That's a good point. But it's also a good question in the academic committees and when you have to hire someone. It's a recurrent discussion. Whether you should favor people who are able to invent new concepts and new methods, or people who solve so-called art problems. But Grotendieck expressed very well his ideas about the softening of the knives. And since Grotendieck has just cropped up, in other words, you know, the next item on our list, Take five minutes now, shall we, and start again at five and have an hour and I think we're not going to exhaust that subject in an hour but so we can at least make it work. Of course, I will have to make it work. Please, go right ahead. Go right ahead. No, go right ahead and use my phone, you don't have to. So I will have to give a... No, short of people deciding to read the U-Mumber tour...

1:10:00 And the what? No, please call my... Sometimes it's good to check the minutes. Sometimes you have to go out of time to find it. And then press the keyboard button. Just press the... Just press the... Okay. And then it's not. And then it's not. And then it's not. And then it's not. And then it's not. And then it's not. Yeah, the set wasn't really important. I don't know. And it's kind of the same, you know, because it's all because he didn't make a few off-the-cuff remarks. He was kind of a ten-year-old. That's my petition.

1:12:30 Yes, he didn't have the sit-spot. The motive is to come from a... Well, if you can have the energy to make some of the talk, it would be cool. I have an idea to try to keep going perhaps a little bit longer than that. So that's where I wrote the discussions.

1:15:00 It is why I am here today, because I have been a part of the track record of this seminar. Yes, we are going to be here on a school Tuesday, midday, so we will continue the discussion. We are going to play this thing one way or another. So the discussion is not going to be long. As I said earlier, their courage is going to be arriving on the Tuesday morning, so we're going to have to have the foundation starts this afternoon. The one that we were going to discuss about the distorting interpretation of the algebra theory is the one that was needed. And it's actually natural that what we're going to discuss will also be in terms of the discussion we're using now. But I'm not just going to put it once and now and say an apology for the repetition. We'll leave you on that, I'm sorry, while you're talking, I think I might just stop now.

1:17:30 Oh, Christ! Did you make, did you cross-check that? Yes, I think I must have put some luminescent light in there just to remind me. I mean, I also tend to take a bit of blame at this point. The beach under a small willow leaf, quite from sunshine into the shadow. Yes, very easy. I mean, that's a way of finding out. Well, I'll do, uh, uh, yes, um, pedagogy. You don't have to get into that, you just get into the technologies, you don't know exactly how to do it, which you can treat, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it, you don't know how to do it.

1:20:00 Or whether the speed of light used to be different. That's the sort of damage we're talking about.

1:22:30 Oh, there we go. Well, it's a test time. It's a test time. ...rather than the metal thing, we'll speak. First, Christopher Webber's response to survive. Once the evidence emerged, it would be...

1:25:00 Where is the... You need to look at the index. Yeah. Sure, sure. Thank you. Sorry, I'm not safe at all. That was in 272, I understand. Yeah. What's right out there? Oh, that is interesting. So it was down there. So, as a matter of fact, it must have been José Corr, which of course was actually everywhere there. Are they? Right. They might be on the next page. This looks like a magnet-sized ball. Now, which I don't know if I can see. Oh, probably at about 60 meters. Oh, come on, look at the back there. Well, it must have grown a very, very guilty climate. There's no competition. Okay, it's just the problem of math, that's not much of a problem, but it's scary, it's math, that's not much of a problem. The terrain around there really is pretty insensitive. Yeah, yeah, yeah, yeah. This is where I visited during the 80s. Yeah, right. Oh, that was down in the jungle. Yeah. You have a ladder there. Is that it for the wall? Yeah.

1:27:30 Oh, yes. I think he's right. He gave me literature about that. He was not speaking. So what do you know about this? Well, I mean, I know another non-petition. No, don't show the tail of it. It's not funny to learn with you, but you just got some tongue. I don't think anyone from that time has ever been in Penrose, but you are a part of my experience, I guess. I rotate a lot in this wonderful place, and then the picture given is all the many great books that I have, all out of the room. I went somewhere and found out about Penrose. Yeah, Penrose is so, yeah, I mean, I still know about Penrose. I actually see her. I said, I don't know more than that. What's that? A TV show that happened a guy from the early days there, I don't know, this may be just one sec, but he was explaining how old he'd found it, he knew how to talk to angels, they'd grow each other out, and so the best you can be grown. I think that's been going on for quite a while. This is my understanding, it's like, what's happening around here? Yeah, what's she not, what's she not doing? That's what I'm saying. It's a test. It's a test. It's a test.

1:30:00 It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. It's a test. How they look back at the North. Exactly. Which is still on display in this museum in some sense. Where they took the Nile and Arbor around a few years ago and it was there. And then they changed it completely. So that's why it's a wonderful source to look back at. And I didn't mention the fact that there's a whole group of people at the NSF. This is part of the SSR, and I'll never receive automatic financial compensation on the basis of the investment in the whole business, which we've done. And I'll tell you, they have a great deal of knowledge, that's for sure. As I do know, they have a great deal of financial sources. All of these have been used for a long time, and now they are being used for a long time, and now they are being used for a long time, and now they are being used for a long time, and now they are being It was definitely under the auspices of the United Nations.

1:32:30 It wasn't this united world politics. I don't think so. I don't know. This is our... So the people at Connes were the most adept at actually seeing the fairies among the trees. Some people speculated that the work of Charles Gates was interested in seeing fairies, especially the vastness of spire, arctic plants, and also dark trees, for example. And who was very interested in the plants, and how they were produced. I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't, I don't. I'm surprised to you, of course, that there are so many responsibilities to the individual. There's a profound philosophical difference. I consider that the interaction of the collective and individual consciousness... The other thing was, well, everything is basically individual. An individual is a certain man, and together we influence each other in various sorts of things. If there were no such thing as collective thinking, why would we have education? Somehow, when it comes to arguments on particular cases like this, the fact is that not everything is the same.

1:35:00 You know, so, you don't think much about the genes or, you know, whatever. I don't think it's a real problem. That's how I came out of Australia. On what basis? I don't know what it is. I don't know if I'm going to Australia. I don't know what Australia is. Doing stuff secretly, against them, actually. Yes, yes, yes. I mean, he even talked to me when I went there. He came out of the, he came out of the Long White Grove, the Long Green Ridge. And they sometimes read it like this and you look and saw it and you said, oh Bill! Like he's glad to see me because that's a laugh when I said it. Because then he said it emotionally inside, but I can't write. And then the next rule was I can't do math. Two minutes later he said it's not perfect, but it's great. Nonetheless it's a teaching of destructiveness.

1:37:30 The Royal College of Surgeons of Ireland continues to be the Royal College of Surgeons of Ireland, even though Ireland has not been part of the British Crown for almost 90 years now, and continues to be the Royal College, and I'm not even sure that the observatory in Gonsignor is still the Royal Observatory, I'm not sure about that, but certainly the Royal College of Surgeons is still the Royal College of Surgeons of Ireland. Fischer, they left the Commonwealth in 1848. Fischer repudiated any class that stood in the middle of the US because this was a big event. Yes, yes, yes, most of the years that is wonderful. Most of the years, why? To make art better for the British Commonwealth. Yes, about five years ago. But at the time, he said, actually, the institution of the...

1:40:00 In 1949, they just settled at the time of the really big... I don't know, some of them have, but it wasn't totally obvious. Actually, of course, it's all changed now since the Good Friday Agreement, because they've changed their constitution. Yes, yes. So, as to restore some sort of linkage across part of the ISB system. But certainly there is this extraordinary vision of a new society involving new servants and advocates and everything to change. That's the development. And it doesn't look quite much like a sovereign state to me. Imagine an American state happening. Well, I understand there was a faction lately. Fortunately, I'm a workman, so I'm not watching out for much of the work that's going on at the moment. Six characters, who happens to reside in the UK, who have Irish citizenship, even if they don't have your citizenship, it's still a fact.

1:42:30 Okay, well, shall we assume that this is a really interesting topic? Okay, well, it's why you asked him before then. Well, I did ask him. Okay, well, I'll take this off then. Oh, so you want to take it away from the interviewers. That's all right, so I suppose that was a little bit of work. And now we've got a full set of that, can't we? Oh, damn, this has been hard. Oh, shit, yes, I'm sorry. We've wasted rather a lot of tape. Sorry, my mistake.

1:45:00 Well, it's OK. We may as well continue anyway. Right, well, we were at this point going to begin the discussion. The first phase of that discussion was to discuss Grotendieck's development in the 1950s and the relationship of his early work in functional analysis to his later work in algebraic geometry. So, on a personal side, I mean, he first came to Paris in 1948, having graduated from the university, and he more or less rediscovered Lebesgue's degree. He was given a few hints. He has mentioned in the courtesan eyes that his professor, I mean, he has repeatedly questioned about what is a length, what is an area. I mean, he was obsessed by that since the early age of 10 or 12. And he was always discontent by the fact that people took as granted the existence of Lang's area and so on and never gave a definition. And then he was told by his professor on analysis that there was someone by the name of Lebesgue who had solved all the problems in mathematics. Okay, and then after graduating from Montpellier, he went to Paris and he has explained that, okay, that's when he graduated, his professor told him, his professor had taken a master's degree from, well, made a master's, what we call DEA in French, academic system now, which has a term called something slightly different, but it's the same. It's a master's thesis. We are about the same level of master thesis with Elie Karten about some problems in differential geometry and then that, I mean, so he gave to me, to Bolton, a letter of support to Elie Karten when he would be a former student of Elie Karten, he did not know that Elie Karten was already old and now more or less out of his mind.

1:47:30 When I met Henri Karten in 1950, I mean, he told me that his father was already, he died about a year later. So when Groton came to Paris with a letter of recommendation to Eligaton, he discovered that Eligaton did not exist, but he had a son who was Eligaton. And so he entered the letter to Eligaton. Of course, it was a more unusual encounter because the two personalities were very, very different, especially as a young person. He was especially harsh and he had a very difficult life. He was an exilé in France and then Carton was typically high class. His father was a mathematician, his father was probably a physicist, and his behavior was really high class, Protestant high class. I wasn't there so I cannot give an eyewitness account of that, but he attended at least some of the lectures in the Carter Seminar, the first version in 48, 47, 48, 49, and the record is quite incomplete because Carter scrapped out some of the lecture notes and because he gave first his lectures in Harvard, Harvard lectures, and then he... It seems that by all accounts, Rotendieck and Carton did not go so very well.

1:50:00 Now, of course, we have some accounts by Sayer, but in terms of witnesses, Carton and Sayer are... The kind of witness who never lies, but not always tells all the truths. Well, that's a very typical one. You can never pretend that you never lie, but some of the things they omit. They omit it, according to their own. And so, Fehr claims that, contrary to what Wotanik said in the Spektral Sequences, 14 seminars, which I doubt a little, because when Borel came the year before, I mean, in 1947, I mean, studying very carefully the work of Florent, he knew very well about spectral sequences, and Borel was among the first people who really understood what to do with spectral sequences in his thesis. I mean, I remember the joke, I mean, Armand Borel is the only person who climbed to the E4 level of the spectral sequence to calculate the cohomology of the group E6. That was just a bad joke. We like that joke. And, well, it's clear that Borel was certainly the best expert in spectral sequences. And he was there and so doubted that there was never spoken about. I myself learned spectral sequences from lectures that I remember gave at a conference. It's reproduced as an appendix in the book. I mean, it deterred me from studying the classic method for a number of years. It was not the best approach. And CER also used extensively. But after a year, I mean, Carton was not very happy about it. And in the summer of 1949, my prehistory, I'd say, came to the imagination.

1:52:30 At the time, Nancy, there were three persons. There were three persons, Nancy. One of the founding fathers of Boba Key, Jean Delsarte, who deserved to be better known. I mean, they published a slim volume out of these people. And Delsarte was very, very, very fascinating. And I have to confess to my, I have to confess that when I met him when I was young, I did not appreciate him. Only later on did I realize who. At first sight I was a little shocked by his external behavior. He pretended to be cynical. He was not cynical, but he pretended to be. At that time I did not like that kind of behavior. I think it was partly because... On the other hand, he has been a very, very important figure. I mean, he was not... Well, eventually he became, but also he was the one who established connections between the university and the various engineering schools, you see. There were various engineering schools, from mining, from metallurgy, it was a huge industrial part, a very active industrial part, and Dersart was the one who managed connection between the universe. Not only he founded the institute of pure mathematics, but he also managed connection with local industrial and local engineering schools. It's quite different if you read it carefully. And when I was giving lectures to the students at Economa on this program, how to teach, I always recommended to my students, read Whitaker and Watson and read this account of Bobacki. And I thought at first, I reported that to Andre Ben, I said, well... It may seem a little surprising that I recommend both of these volumes, but don't forget Jean-Denis Sartre, who was the driving force behind the publication of this series, and the most classical part was also Veil.

1:55:00 I am very fond of Whittaker and Kratz, so no contradictions, and I still like Whittaker and Kratz, and while you know in French mathematics special functions are not very fashionable, except when they are called elliptic curves, not elliptic functions, elliptic curves! Automorphic, ford, etc. But they are special functions. In Perseus, I don't consider them. So, when I had to recommend, of course, when I had to do thoughtful exposition of hypergeometric functions. So, Jean Dessart was a driving force in us. And his influence is still to be felt. Still to be felt. He has a very strong and long life. At some point he decided that I will not do what most academic people do in France. I mean, after some years in exile in smaller towns, I mean, we want to join Paris University. He decided that it would be better the king in a small place than the count in a big place, in a big empire. Well, some people like that. It's better to be the king in a small place than to be the count or the knight in a big place. And I think his influence has been there. And then, so he started to build something. And so he first called on Jeudonie after the war. And Jeudonie, so Jeudonie was appointed a professor in Nancy. But after the war, he spent a number of years in exile, first in South America, and then in Northwestern Europe, I remember meeting.

1:57:30 So he first called on Germany, and then he spent a few months, or maybe half a year, in Groningen, and then moved to London. Wonderful years in his life. So, and then there was, there was, he inclined to apply to Matema. And, but he was also the manager of the book. I mean, he was the one who took care of all the, I mean, so, I mean, yeah, he hired a secretary two times. He took care of the, of the proof sheets. He took care of the, of the way to, to his expenses. He signed a contract with the publisher, you know, et cetera, et cetera. He was a, he was a real manager. He was a manager. And I remember when I joined the book-making group in 55. He was no more active in promoting mathematics but he was still a manager and did that well. And then after that he decided, so he called first, and then after the work Schwartz was called when he was, when Schwartz was in Nancy that he discovered. And then he started to build a school and I think more or less influenced by the action of Marshall Stone. In Chicago, I mean, he had more of them. And so, then he called, and then after Schwarz, he called on a younger generation. So, who came after?

2:00:00 Gonneman was a student. Carton, and did his work on, well, Carton at the time was interested in potential theory. And so, Carton had two students at the time. Delis, who did his theory, potential theory, and he invited them in there. One developed the so-called projection method. And then, Gaudemont, who was an account of the duality, the Pontryagin duality in the group and the transformation. There are hints in the book by André Veil, hints at both Cato and Poisson, and then Gaudemont tried to develop a similar idea for a non-committed J-book, just at the beginning, in complete ignorance of what was done in Sarasciato at the same time under the leadership of Gelfand. It is only after the wars that they learn, and I remember Godemont disappointed when he discovered that Gelfand has discovered many of the things, especially his thesis, and Fouilleton was the one who developed a good expert in them, one of the best experts in functional analysis, mostly under the guidance of Bohr's and Diodonis at the time where Diodonis, I mean, shifted his mathematical interest. About that time, all the papers he put into were more or less invented the talk factor.

2:02:30 At least it's the very first step to Mulder's photography. He invented the Tau of Gaussian book 4, I think Mulder was dedicating something like that. But he was the first one to recognize that the tensor product is not an exact factor, and that you should correct that to some extent. What did he do? What did he do? And as usual, I mean, Bivoli was not motivated by whatever, you know, like Heineberg, but which was a mandatory. But Giordani was mostly led by some encyclopedic view of mathematics, and so there was a natural question, there was a natural question, I mean, the tensor product, and the tensor product was just beginning to develop, I mean, the tensor product over a field, the number of years, the classical tensor characters, I mean, 50 years, but the idea of having a tensor product over, first a commutative image, not a remote of the field, was a brand new idea, and first he studied the ocean world. But then he shifted his interest and he started to develop the ideas which are in the Banach text. And I think the influence of the paper from Norman said, OK, we have to understand, we need some... The space is more general than the Banach spaces. I mean, von Neumann more or less invented them. Wiener invented the Banach spaces, but the so-called locally convex spaces were invented more or less by von Neumann, at least there are hints to that. And Dürer and Lee began to say, well, after all this went on for him about weak topology, duality in the Banach spaces, we would like to have them in a more general pattern. When you have a Banach space, you have two topologies, a strong topology and a weak topology, but the weak topology is not defined by a knob, so it's a little more general, and so I think the same kind of unification, I mean, of functional analysis, I mean, but it is, in functional spaces, the new feature is that the same functional space carries many topologies, carries many topologies, which is one of the interesting and annoying facts in functional analysis.

2:05:00 So he began a systematic study of that, and then Dersaert wanted to build a strong department called on Goldemont immediately after he graduated, and then later on, so Schwartz, Goldemont, Dersaert, and Diodonnet, that was already a strong department, but with the main emphasis on analysis, on analysis, functional analysis. Quite independent from the development are on Qatar with much more algebraic and topology and then geometry. Okay, then in this and then they began to recruit younger people. And so, well, of course, they were ready to accept Gotenick as a student. There were all the evidence that Gotenick was one of the most gifted students. And since he was not very interested in topology and he did not come to terms with that and with Kato, they would have him there. And then later they hired people from my own generation, first there, for a very short time. He got the field middle and very, very good. But other people were included. Brouhaha, whose father was the director of a conglomerate during the war. He disappeared during the war, I mean, because there was some group of... And the dean, the father of Brouhaha, was the dean of the school at the end of almost 14 years.

2:07:30 But this is a great family. This father was a very important person in French physics at the time, I mean, his textbook was a standard textbook in physics. And then, so, François, my friend, I mean, did functional analysis, I mean, and he is the one who wrote with Titz, the famous accountant of the Piatty Groups, okay, and Wittings. And then he has a sister, well, the mother was the dean of one of the... The best lycée in Paris, the Lycée Montaigne, near the Luxembourg Garden. And François has a sister with the wife of Choquet and was famous for her work in general relativity. She wrote with my friend Cécile with a textbook about geometry and physics. This is Yvon Choquet. Yeah, Yvon Choquet. She was the first to really, under the guidance of Le Ré, to prove very deep existence theorem for the nonlinear equation of Einstein's gravitation and that is hyperbolic system but nonlinear hyperbolic system and proving existence theorem is very difficult and she was really the first to prove it and just to tell you she is I think 83 and I hope she is visiting IHS. A shelter in my office door! And she still liked him. So, a great family. But the father disappeared at the end of the Second World War. But, well, François Bouin was in his prime. And then the idea of Delsarte was to select a group of students from Economat to stay for a few months in Nancy before they graduated.

2:10:00 And to have at least an idea of who they were and to give them special courses and so on. So all the people, many people in my generation took advantage of this opportunity. And so they would spend a few months in Nancy and get lectures from Durigny, from Schwartz. So just to say that there was a very strong spirit at the time. It was a very strong department. Today, Nancy is wild. There is a standard university in the department. The colleagues in the department are good people, but it's not a first day in the department, and they don't claim to be. So Goten was, I think, in a very, very good atmosphere. That's important to know. But I think he was in a congenial atmosphere. He had an affair with her, and he had a son, and this is the elder son, and this is the one to whom he is still today, he is really the one child to whom he is still today, well his daughter also, five official children and maybe a sixth one, never know later sixth one. Redisco more or less reconstructed Lebesgue's integration and when he came first to Nancy, he visited Durnanet and explained to him, yes, how the very general field of integration and Durnanet went to his closet and took a long manuscript he had written for Bobak, he proposed about the same subject and about the same level of generality.

2:12:30 It's in my thought, I mean, while we are, we go back here and discuss what it is, that kind of thing, and for various reasons we have rejected, but you should do something better. And I think, I suppose Goodland was a little, a little, and I think more or less as a retaliation he told him, well, but you know, this is done, this is known, you should do something else. And just that to, to make your teeth. Let's think. I'm just finishing with Schwartz, a paper on functional analysis. This is a paper which is called Duality in F and LF Spaces, where they had to extend the framework of functional analysis, locally common spaces, when they invented the direct limit for spaces, yes, under the influence of, well, I mean, Schwartz had created this theory of distributions. But he wanted to base it on the idea of duality. He did not invent the idea of explaining it in his book, so taking the Ries-Marco theorem as a definition. So you should define integration theory as a kind of duality. But of course, the duality of Hilbert space is not enough, or Banach space is not enough, so you have to go to work. Schwarz takes as test function the function the smooth function is compatible. I mean, of course, it's in a natural way a direct limit, a direct limit of increasing the compact set. And so you have to, well, of course, you are led to the idea of a direct limit of topological spaces. Each space is a Banach space, but the limit is no more. And LF means limit of fresh space. And they have already realized that by duality, inverse limit and direct limit are inverted by duality.

2:15:00 So, you know, even if it was functional analysis, not categorical, it's the same kind of idea. I mean, the point is that duality played a very central role. And, of course, we all know from the first paper of Heidelberg and MacLean that the idea of natural transformation came when they made the reflection about duality. And the fact that the space is a dual of a dual in some situations, not in other. And in functional analysis you have plenty of situations where the dual of the dual of the space is not the space, original space. And so they had to invent a new framework. And they developed, just to substantiate the results of Schwartz for distribution, they had to develop a suitable framework. Later on, people realized, and mostly in the exposition of the Russian school of Gelfand, Gelfand has some nasty words about the irrelevant fields of functional analysis and topological vector spaces. It's true that you can dispense with that, I mean you can work more concretely, but once you have defined the space of test function, I mean everything comes to a sort of norm, inequality between norms, and the practicing analyst knows. But then, they were developing, I think, in the Bobacki spirit, they were developing this new framework. But in the process, I mean, both Schwarz and Giordani, they were certainly well-trained people, but they met a certain number of questions. At the end of their people, there is a list of 14 different questions. They raised 14 different questions. And Giordani wanted something to be... Here is the paper which isn't in print. At the end there are 14 questions. Well, make your tea. He did not know what the outcome would be, but within a few months, Hortendieck has found the solution or the counter-example to all questions and developed a new framework. So, from that on, Dieudonné and Schwartz had great respect for each other. He has proved himself and so I come to the contribution of Golding. So he worked from about 1950 to 1953. His thesis was defended in 1953 and then published in 1956 by the American Mathematical Society and it was a memoirs.

2:17:30 But his contributions are diverse. First of all, he answered all the questions by Diodonnet and Schwarz and developed a more subtle theory of duality. He has a number of papers published in Brazil where he develops this idea of duality. And he has the so-called, I mean, the so-called Schrottendieck duality theorem, which tells you that you have a space, a topological hyperspace, and its dual. In the dual of the dual, you can characterize the element which comes from the original space. It's a very beautiful description. Why? Whether it's really helpful. In general terms, it's a very, very nice theorem in practical application. It's like an application of the Bayer, Bayer set and so on, the Bayer principle, the Bayer principle, and the Polish school was very fond of application to topology called the Ballard spaces, but of course, in practice, in practice, I mean, we work with explicit norms. It's good to know offhand that you will have a solution. People are not interested in knowing that there is a solution somewhere, but finding a method to... So, Grotendieck has a number of very interesting reasons about the duality, which he published, and solutions to the question by... And then, at that time, there is one supplement to the book on distribution that Schwartz had developed, a so-called Camberfield. The idea is that we should borrow it from physics.

2:20:00 If you use the method of Dirac of the Bride against, I mean every functional operator appears at least in the formal way as an integral operator. That's the basic idea. But of course you have to invent not only well- So, you have an integral operator with a kernel kx and y, but of course you have to use not ordinary functions but generalized functions. Let's say the identity operator gives rise to the delta, the Dirac delta function. This is exactly the purpose of the Dirac delta function, to represent the identity operator as an integral. And then the physicists were very, very, of course, very well aware of that. And still today, I mean, it's taken as granted by any practicing physicist that you write, I mean, every time you have an operator, you write. I mean, the formalism of Brian Kett is very convenient. For physics, I mean, I use, and even in elementary courses sometimes. But it's very convenient, and you can't pretend that every operator is an integral operator. But you have to substantiate that claim. And so, Schwartz proved that any operator, while under suitable conditions, any operator can be represented as an integral operator, provided that the kernel K x and y should be not an ordinary smooth operator, but a distribution. So, again, functional space is a great one. I mean, that theorem is sort of totally untrue for Banach space. Oh, no. It's not just the fact that the kernel is a distribution, it's the fact that you're working in a world of completely different kind of spaces than the ones which have been classical for too many times. And the fact that your test space are not continuous functions but smooth functions and so on. And then Schwartz, I mean this is a development which occurred after the publication of his book, so in the first edition of his book this is not common.

2:22:30 But that started a great excitement. I published my book, which is called Measurable Function, under which condition you can assess an operator's integral. So many people worked on that, and it made a great impression. Basically because it gave a substantial foundation to... So, Schwartz was knowledgeable in mathematics. He did not always have the best state, but at least he was knowledgeable. He always considered that his distribution would have end in D, end in D, end in D. So, that both indicate, I mean, Schwartz asked, what is behind this kernel? So, I mean, everyone at least has the idea that Spaces of function in two variables should be the tensor product of spaces, the first variable with spaces of function in the second variable, was more or less a natural idea if you think of polynomials. And that was beginning to be understood in the algebraic science. That means that k of x, y is the tensor product of k of x is k of y. So, what is valid in a more analytical situation, not purely. Schwarz repeatedly asked Gottendieck, is there a special property of the space of this function which ensures that? And so, and Gottendieck came, I mean, came with the idea that he should first develop a theory of tensor products of function spaces. I think at the time he did not know that Shatner had already taught it. I don't think he was really aware of the work of Shatner and the so-called core spaces of... But Chaton has a restricted aim. I mean, Chaton wanted to justify the various classes of operators in Hilbert space, trace class operators, Hilbert space operators, and so on.

2:25:00 But what was already visible in the work of Chaton is that at least for Banach spaces there were at least two natural notions of tensor parameters, and were more or less in duality with each other. It's not really important, but at least he first rediscovered that for Hilbert spaces there was never any question. There is a natural notion of tensor product of Hilbert spaces which was implicit in Fourier series in Hilbert-Chrysler-Dubéry. It was not formalized until quite late, but nevertheless it was completely, in principle, even if it was not a formal axiomatic way, it was completely. And the so-called Fock spaces in mathematical physics are just a particular case of tensor products. So that was not the question. But for Banach spaces, you have two natural notions. And of course, we know now that there is a natural notion for which a tensor product is right-exact, a complete tensor product is right-exact, and another one for which it is left-exact. And as you expect, by duality, you can bring them one to the other. And so Grothendieck discovered that, but he proved beyond what was known to Schatzen. He made two important discoveries, which gave the meaning of these two tensor problems. The so-called projective tensor problem, which we call now pi, after some lectures by Schwartz, is the one which is right exact. And it has a main property, which was discovered by Guttendieck, that if you take L1 space, space of integrable function, L1 in variable X, then so this tensor product L1 over Y is L1 of the tensor. So there is one tensor product, the one which is right exact, which is adapted to dealing with tensor product of L1 spaces. Hilbert spaces are really L2 spaces. So basically any Hilbert space is a L2 space, more or less naturally.

2:27:30 So at your level you have L1 which requires a special tensor component which is right-exact. At L2 level the tensor component is exact, not right and left exact, which is one important feature. But then at the bottom you have the dual, L2 is its own dual. So, if you have a tensor product which is right-exact, it is left-exact. Now, L1 has a tensor product which is right-exact. So, by duality, the dual of L1 is infinity. So, there should be a tensor product which matters for L-infinity. And so, what Gordon did discover... There are many variants of that. Is that the second tensor border, which was called epsilon in the lecture by Schwarz, is a left exact tensor border and behaves very well for the continuous functions. Okay, so you have two natural... and so in the framework of Banach spaces, I think, the hierarchy is well understood. And what has been discovered afterwards is that you can interpolate. And there was some work done for interpolation, so LP, LP, tensor product, everything. So in a sense you have an L1 tensor product, an L2 tensor product, an L3 tensor product. And if you deal seriously with functional analysis and tensor, and function many variables and operator, you have to meet them and you have to use them. And the interpolation, well at some point there was some excitement about LP interpolation, but it did not prove so. But then Goethe and Dick. Goethe and Dick had a very bright idea. If you deal with a test fraction of Schwarz, with a topology, a smooth fraction, then the space of such fractions has two different... Well, the basic fact of the space of smooth fractions is that you are not one known but a family of knowns. So, a norm taking into account the value of the function, then the next one takes into account the value of the derivative, the second derivative, and so on. And so the basic idea is that instead of having a space with one norm, we have a space with a familiar form. This is really the basic idea.

2:30:00 But then, so, in ordinary spaces you distinguish clearly between L1 and L2. When you deal with a smooth function, you have the choice. You have three series of norm, one which is an L1 type, one which is an L2 type, and one which is L-infinite. And this is more or less the, I mean, L2 type more or less corresponds to the Sobolev spaces. Which at the time, I mean, it's interesting that the Sobolev spaces were not considered very seriously by the French school of functionalism for some time. It's only when a model, I mean... They were known. Oh, they were known. I mean, Sobolev invented them in 1936. And that's always, I mean, the controversy whether, I mean, Sobolev invented distribution or not. It's like the controversy whether it's Poincaré or Einstein who invented relativity. Both. With different emphasis and with different methods. I mean, it's clear that in the discovery of relativity... Poincaré was good at mathematics, but Einstein was there to make the conceptual, and it was much better explained in the terms of people, but he did not question the basic meaning of the physical concept, Einstein was bold enough. And so, I would say for distribution, it's more or less the same. And the idea of duality is there, but he did not consider them as a universal tool. And the great advantage of Schwarz's approach is that Schwarz considers them as a universal tool and as a universal container. That means you have a variety of various functional spaces, but all reasonable spaces are subspace of the distribution. So they all sit into a big space. And I remember the joke by Schwartz, I mean, it's hunting the lion. You know it's a standard joke, the various mathematical ways to hunt the lion.

2:32:30 Now, with distribution, what do you do? Well, in the distribution jungle, it's very easy to catch someone in the jungle of distribution. So you catch a lion distribution. It's quite a cheat. Then you use regularization, you transform it into a change, it takes time. That was the way Schwarz was formed. So the real great idea was that there should be a universal continuum. More or less the same as with the set theory in the sense of... It's the same idea... Andre Weil considers the universal domain in algebraic geometry, I mean, Schwarz considers distribution as a universal domain, so the idea is that you have a very large domain, you have much freedom to do constructions, of course what you get out is sometimes monsters, but then after that at least you have an object, at least you are able to have an object, and then... Maybe it's a monster, so you put it away, you put it in the garbage, or if it's not a monster, then you take it. So that's one great idea, and I think Sobolev never considered it. And also, Sobolev, for various reasons, independent of Marx, did not develop fully. Because when the war came, he had to be involved with... Okay, so, but then, I think it's Grotanik who first realized that you have... You have three, I mean, so you have, so you can say, I mean, Sobolev spaces were considered L2 norm of the function in this derivative. And then, Gottendieck discovered that you can play the same game in L1 norm, so you first begin with the L1 norm of your function, then the L1 norm of the first derivative, the L1 norm of the second derivative, and so on, or the L infinity norm. So the point is that, at the level of Banach's basis, you have three tensor problems, but because the space of smooth function can be defined by family of norming belonging to three families, the three tensor problems coincide.

2:35:00 After that, the kernel theorem of existence of kernel in a Schwarz, which is a duality statement. Now the real reason for this duality theorem is this tensor problem. So, now for Banach spaces you have three possibilities to define tensor product. I have to admit that L2, the tensor product of Hilbert spaces was considered less important. In application it's a very important application, but Gordon did not consider it very seriously at the time. So, but since you have three possibilities of defining a sequence of norm for your space in 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 exact. And then I think that was the beginning of the functorial way of thinking. He had this tensor product with nice functorial properties, really well. And then what he went to is this is in two parts. The first installment He was the first really to put emphasis on that point, that a general space is an inverse You consider the inverse system itself, not the limits. For the analyst, it means that you have a space of function with various nodes, and you deal with estimates of the various nodes, and you don't really care whether there exists something in the limit. So, that's what he did. The first part is a very well-developed theory of these various tensor objects with their various properties, exactness, and so on, and naturality, exactness, and so on.

2:37:30 There was a very famous problem at the time in Banach space, his conjecture stemming from the so-called 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 operator. 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 Rotendieck is concerned about this problem. And typically enough, he takes this conjecture and reformulates it in 47 different fictions. The hope, the method, the strategy is already there. It takes a very difficult conjecture, it is formulated in many, many, many, many ways with the hope that eventually one of the people will get to approve what he calls, I mean, it's a Joshua surrounding, and we just stomp it until the wall would fall. But unfortunately, I think he spent almost two years and it took about 20 more years to discover counter-examples and flow in their discovery. The original example was very complicated but now we have a very soft version and we can show that the most natural...

2:40:00 Banner space, after Hilbert space, is a space of all bounded operators in the given Hilbert space, which is a very natural space, and in this space, a cooperative space, and they are quite explicit examples. So, Gottenick spent a great deal of effort, and I think part of his new method was hinted to approve of that, in that respect he failed. In the second part, he wants to put in an axiomatic basis a kernel theorem of Schwarz. So he's considering a category of spaces where the two natural, or the three natural tensile products coincide. He would say, okay, we have three natural tensile products, for smooth factor they coincide, but is there a general category of spaces for which these tensile products coincide? And that's exactly the nucleus. Nuclear, because, I mean, a candle was called a nucleus, of course, a nuclear operator is an operator which has a nucleus, a candle, and so he wanted to have nuclear spaces, spaces for all operators. So, and that, in the long run, is better, but it was again from school we realized the impact. I mean, when, as soon as Schwarz published his textbook, I mean... Gelfand was very excited and a lot of energy was there.