FW Lawvere, Pierre Cartier, Colin McLarty, Angus MacIntyre, John L Bell — Grothendieck 1950 - 1960 (contd.)

0:00 Thank you for watching this video. Yes, sure. And that gets to there as well. Yes, it might. I would suggest that they took it up. Yeah, yeah, yeah, sure. I'll take a look at this. The crucial point there, I mean, is that the big 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, so you cover it by some sort of pushing the problem with all continuity in that kind of a way. And that helps. I think, in fact, if you didn't have that, then this would not, this phenomenon would not occur. It's not just a sentence, though, it's an act that you're using.

2:30 Well, that's the amount of smoothness that remains in his mind. Okay, it's been mentioned that we were, this time, the session we were about to start now was just related to the topic of Greenleaf's work between roughly 1950 and 1960. That is to say an examination of how his early work in functional analysis, functional analysis, Related to his subsequent work in algebraic geometry and to the expansion of his vision for the description of structures throughout mathematics. However, before we commence on that topic, it has been pointed out that in the Sawyer's discussion we didn't say, we said very little about the role of Saunders and Quay, and that it might be an idea just to have perhaps a brief treatment of that topic first, perhaps for the first 20 minutes or so. So, who would like to lead off on the subject that's always been played? Well, yeah, I just happened to have... I think it was a rhetorical question. I just happened to have... It was a rhetorical question. Yeah, because we... I don't know, it led to the... It's an Italian... Oh, no, that's not a long thesis. I mean, actually, it was a long thesis. We're working on it. We're working on it. This is a journal... Something like mathematical intelligentsia, but all in Italian. I think actually better than intelligentsia. Okay. More. Well, I hope he stops taking it this morning. He will also, okay. Anyway, so I was asked to do something here, and I don't know, I feel slightly, I'm not quite sure if I'm doing the right thing, but the man is scarcely dead and I'm already...

5:00 Seizing the opportunity to develop my own ideological mind on the basis of it, so if anybody finds that offensive, please let me know. What date did he die? April 15th. April 14th. That's a few months ago. A few months ago, yeah. Didn't you have moves? No, no. They had moved to San Francisco. And it became really to continue. Thank you for your time, and I hope to see you again in the future. All right, then you knew OSA as well, probably. Yes, of course. I guess they lived there about a year, not much more, but it rapidly declined. Well, okay, so anyway, we already mentioned this business about the triple impact of the Hilbert-McLean spaces and how you would have cleared that part of this. Right, 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 it notices article comes out, is that he didn't start as a logician, he started as somebody who, as they often said later in life, was...

7:30 He was randomly interested in understanding, understanding. And so he, 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... Stone. Stone. No, no. E.H. Moore. Stone was much later. Oh, yes, yes. This was 1900. Well, not 1929 or something, something in there. So it was E.H. Moore who was the author of the general analysis. And in fact, it is said that Crechet took over this slowly from Moore, because Crechet also often used this. Great general analysis. In any case, apart from the name, I think that sometimes the spirit of Grishet was certainly embedded more 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. And as I... When I learned about this, 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. And now that we have 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, Noether and Hermann Weyl. Paul Bernice, who became his thesis advisor. So, I see a certain parallel with my own career here as a senior coach, and also with the career of my other mentor besides Allen Brigham and Fang, Mr. Lewis Truesdale, because, you know, trying to put it in general terms, starting from

10:00 A deep interest in the general understanding of the fundamentals of things and truth about space continually can be denied. 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 can be used, but on the other hand, that their vision is inadequate to the purpose that we have, so we are sort of temporarily logicians. The Church's book was written by Truesdell, after all. Which book? The Alonzo Church's, one of the most famous texts in mathematics. I've 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, it was a special example, maybe not typical, but anyway, so as far as I know, Truesdell didn't carry away much from this logic again. I did, of course. In my case, carried away quite a lot. So this is why I'll be volume one in the church. You know, it's cold in the mathematics. Oh, yeah. Volume one. Why volume two? Why volume two never appears? The assistant had defected. Exactly. So also, it seems to me that this general description applies well to MacLean. He also, of course, as you know, he...

12:30 He carried away what he learned in logic, and he learned a lot more logic throughout his career, but he was never the logician, quote-unquote, in the sense that that was the main focus of his work. Although that's one point that I make here that 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 ordered it from Amazon as well, so I expected by the time we get back home, it'll be there, which is what we could look at. So, McLean ordered some of it from you? Yes, it was just published now. Oh, just by him? Just, just, uh, yes, by, by Klaus Pater. All of them? Yes, and it was... Oh, okay. It came out, I mean, two days after McLean's... So here I go on about this, about how this can make that sense. I forgot what that was in English, but it means a fierce need to understand. Many of his work led to deepening and amplification of logic, but the logic considered as a serious study of the general aspects of all exact thought cannot be confined to symbolic tradition, which looks only at presentation instead of reality, neither to the narrow tradition of Frege, who said that property is the best concept. So one of these tendencies were certainly gated by the relation between category theory and logic, which is flowing from MacLean to 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 had become possible.

15:00 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 relationships. By saying we can hope to satisfy the dream of Maclean and of Moore to transmit a more profound comprehension of modern mathematics to his students. ...completely uninitiated freedom of the working class, categories, naturalities, naturalities, and all. Anyway, those are, I could say a lot more about McLean, but those are some of the things that I thought of saying, which you probably have. 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 a standard 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. In the circumstances or such. I was so grateful that Columbia sent me. First of all, he accepted me in a completely unprincipled manner just because Chudel 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 threw sort of exercises at students the way they should.

17:30 Moreover, Stanley was chairman of the department that year, a period of years, and was very, very involved with that. I only had one or two even discussions with him. So a little bit ahead of the discussion, I had already sent to him a manuscript which I had worked out in Indiana about, essentially I had discovered the notion of adjunct hunters, basically, but you know, Sammy looked at it and we had exchanged a few words about it and so forth, and then a few months later, a fellow student told me, I was talking to him, namely Saul Lufkin, who was 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 punctures. You should call these things adjoint punctures. So I went to Sammy and I said, what is this about Kahn and adjoint punctures? I was told that this is... This is essentially the correct formulation of what I was trying to do, and he pulled out the drawer and gave me a copy of Don's paper, but all I'm saying is he never occurred again to do that before, but it's just because he was so busy, just because he was so busy. What year? 1960, spring of 1960, and then Peter Fry arrived as a postdoc in the fall, and so I actually... In some way, he was the most direct teacher in that category, even though he was not officially. 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. A year at Berkeley and listening to Tarsie and Dana Scott and Robert Roth and several of the leading group go crazy, leading logicians and learning as much as I could. I had a family, actually by that time two families that I had to support, so I got a job with the military industrial complex in Los Angeles for a year. During that time I wrote my thesis.

20:00 So, when I went back to defend it, Sammy didn't read it at the time. He gave it to Saunders MacLean on some plane that they were carrying around on the plane. And Sammy says, here, here are 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 automative theory. Using linear algebraic theories and so forth, and unfortunately not published, but he gave the four colloquial lectures of the American Man Society in the summer of, what's that, 67, right, 67, and one of the four was entirely devoted to algebraic theories and how he can apply this. If I'm not mistaken, it communicated your papers to me. The papers in the Proceedings of the National Academy of Sciences, am I right? Right. So after I got the degree, I was teaching at Reed College. This is reprinted in tact now, the commentaries on the parliamentary theory of the category of sets. I won't go into that, how it was just about me and Reed College, but in particular, I wrote up this abstract of the thesis, and the McLean team indicated that. McLean was a member of Mr. McLean at the time? Oh yes, yes, 1963. And then he suggested I apply for fellowships and so forth. I saw McLean at the Boulder meeting. It was the Boulder meeting of the AMS.

22:30 By the way, I was the first person to apply category theory to so-called computer science. I presented a paper. There in the summer of 64, you know, behavior of the commons is a natural structure. I remember Church sitting in the audience asking, how do you use that? I don't know why I was in the audience. But the Sauter stage of colloquial lectures is that year, so that's actually published in the bulletins and is a known reference. So he advised me that 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 all personal things changed, and I thought, well, maybe I should go to Zurich instead and work with Daniel Ekman. So Saunders said, well, it's just as well because if you'd gone to Paris, Carton wouldn't have had the time for you anyway. What was it? 64. That's what we meant. You must have been at the ETH when we first met in 65. 65. No, this was 60. I think you were still at the ETH. I went to the ETH in 64. Yes, but in the summer of 1965, you were either just leaving there or just finished up there or... No, I was still there. You were still there over a period of time, because that's only three years. Right, so that's how I ended up with... When I was still as a captain, I knew what time it was, you know, and that was the end of the city, that was when he left the corner of Madagascar. It was a different time. And that was when I published two textbooks about the differential calculus. I mean, it was there from a distance and I went to a very good, very good, very good, very good, very good, very good, very good.

25:00 Very pure, 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 Sorinus had a very broad, I mean, you know, he seemed to take on all kinds of things from all over the place, you know. And again, it reflected some continued interest, at any rate, in logic as... Maybe not in this thing of general understanding, perhaps, because he certainly supervised, you know, a number of very, very outstanding logicians, or people like, I mean, I mean, Solovey, of course, didn't do logic with him. He jumped on the code, the code business, very fast when he went to Berkeley. I think he sees this as a... The Riemann-Roch theory. That's what it was all about. Then he got into logic very, very fast. Yeah, totally, yes. And he was a similar person. He was going to be able to learn a thousand maths. Right, right. He was going to be able to learn one of the other papers. Yeah, yeah. But then he got into logic. Yeah, he jumped very far. And Leroux was fully McLean's student. That's Leroux too, yes. Morley sort of broke with McLean. And Leroux is another magician. Yeah, a very, very wide-ranging one. He's made a lot of contributions on a huge range. And then Michael Morley. Morley, yes. But, I mean, Morley's case isn't so simple. I mean, Morley, in the end, I think we skipped to Berkeley to work with Bob Mott. On the other hand, Morley's thesis, which was the first time that beautiful, funtorial considerations appeared in classical model theory. This was a term, at least for me, I don't know if it was for my contemporaries. I would even be permitted to speak on categories, because I think Dana Scott is more or less the administrator of the seminar, so he let me. Well, you contributed to that model theory meeting, right? The one in Berkeley, the volume was published, you know. That was later, that was before that, when I was, I wasn't living in Berkeley.

27:30 Before that, when I was just a freelance logic student, I think he, well, Peter Fry claimed that he always was. Certainly, the spirit of his 1950s work. Yeah, but someone could prove Harvey Friedman was interested. What they meant was he's got ideas that he couldn't develop. Well, I think it's different with them. I don't actually recall me ever using a specific category with that long of it. No, no. It was something that we organized when I was here. Well, it was something in the spirit of category theory. Yeah, yeah, yeah. I don't recall me using the... Well, I think that I'm understanding that you meant, you know, that he started out with this kind of, I think that's right, which wasn't focused in any particular way. It was general understanding, I mean, but I remember my impression of data is he slowly, slowly, slowly, slowly, now he's 95 percent. But every one of those percents, that's it, it was ticking, it was not, it was just, you know, one more, one more. This slide was quite interesting. Yeah, we took a very good look. Well, not actively, but the Schoenfeld-McClain-Stober-Albert paper is a different nature from most of the papers of that time. Yeah, that's true. That's true. It is. It's a more reduced direct product, right? Yeah. Anyway, to return to me and McClain. Actually, 1965 was a very busy year. There was a meeting in Pahoya to represent the In true version of the category in my thesis. And then there was the Lester meeting of logic in the 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 to the National Academy.

30:00 Indeed, as you say, he, what's the word, transmitted? No, communicated, sorry, communicated those two papers, but also he invited me to stay in Chicago over the summers and so forth, so I wrote this longer version, which has now been reprinted on the tack by a couple of columns. Given the hypothesis that he had a reasonable opinion of me, he was therefore naturally wanting me to come to Chicago and I actually retired there. You know, we were quite brash in those days. There were lots of jobs available. Yeah, that's true. So I said, the first semester I want to have you leave with absence. Because there was this gathering in Zurich, which became the Zurich triples book. The whole business about monads and so forth was just about to be developed. I remember people I didn't want to mess being part of that, so I didn't arrive at Chicago until mid-year. I'm a graduate student. I'm not so happy. I could not put such content in there. No, no. In those days, yes. I mean, you're saying you got a lot of it from Peter. That picture would have been of Peter as the reason that Kat Reinhardt's ball came out of Columbia. Oh, they all went to Columbia. Who are you thinking of? Well, you and Mike Barr, one of the triples people. In fact, yeah, I have them. Well, if you continue this list, I don't know if I can do it right now. Linton? You will arrive at a list of ten. Yeah. None of these people at Columbia.

32:30 But Barry Mitchell, Peter Fry. Anyway, there are ten. 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 with Eilenberg, you know, as a center, but to say that he, you know, created them as category theorists would be inaccurate. No, no, no, because I went there, I gave up my study with Truesdell in order to go learn more about... Yes, but it's of interest that they went to Eilenberg rather than the... Okay, now this is true. No, but there's a very simple reason for that. Very simple, stupid. At least in my case. At least in my case. I just don't laugh. 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 Truesdale for a lecture on functional analysis. I had developed some things about edge-line punctures and stone spaces and all that in these lectures, and I thought, oh, this is really the thing. I have to get that 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 in a position to. Some professors knew a little bit about it, but I cast it out to see. I heard of MacLean. I didn't study with his algebra books. But, okay, look at the joint authorships. Well, there's Eilenberg 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 a 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 with him. And then, of course, the fortuitous fact that the truth-dealing would be a very good friend of Sanger, even though their appeals are completely destroyed because of their joint interest in art and so forth. But that helped to facilitate the actual move.

35:00 The fact about the joint, multiple joint authorships may have had a similar effect on, I think, some of the other decisions to go and study. I think probably it did. Well, I think there was a proof that Goebbels read. He wasn't there at the time to read. Yes, yes, yes, yes. So it was a remarkable concentration. At least ten people. Well, okay. You could say that. I mean, the reason he went there... He went there a year later. He was already a post-med student. Pretty technical. Closely related. And he knew it before he came on the show. Yeah, yeah. But he told me that he was puzzled by a billion. First encounter with a guy before I actually got involved This is way too abstract for a serious physicist like me. Injection? Tierney was there? I'm not going to finish a list of ten at this point. Both Tierney and, they stayed longer than I did. They probably got a more complete education. Not in California and Oregon, but it's an Albrecht Doolin to us.

37:30 I was a visiting public professor in those same years. He was trying very hard to be something that was very systematically categorical. But again, the other guys who stayed longer than I did, they had courses in semplificial sets and so forth, so they became much more technically adept. I mean, another visiting professor at the same time. I mean, Sammy must have had something to do also with gathering these people, Dole and Helgeson. So I had a course from Helgeson on lead groups and fantastic, you know, really great stuff. And the third one was Peter Lacks, a co-committee member a couple of years ago. He was only, you know, he was still at NYU, but he was teaching a course probably on distributions, so in addition to the actual Columbia professors, there was quite an illustrious group to learn from. I didn't have any courses from him, but yes, he was there. Stanley didn't have too much time for a graduate student. He certainly was doing a lot of very good work as chairman. McLean, so McLean and I gave a joint course. The one year that I was a junior mechanic.

40:00 Yeah, and then some more notes in the Chicago. Okay, so he had a longer version. But also mostly just before getting good so far. And then they let a few French people come out with the same idea. And in the end, I see some secretive geoscience, at the end of the sentence or so, I mean, some of the people around Erasmus. Did anyone pursue it because of the very term and his hope that he could prove the stability theory? I mean, I had a discussion with Tom, I remember, about the dynamism. Tom, I mean, the basic idea of Tom was to put everything in. 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 the biology and morphogenesis and so on and so forth, he was looking at static pictures. He loved that dynamic picture, 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. It was probably not possible. But how many dimensions? It depends on the dimension. But from my discussion with Tom over many years, I mean, I've never felt that I've been feeling pro-diagnomics.

42:30 And I think that's what was on the week. You have a change of pattern, you have a change of portrait. But how does it occur? And you would always draw this picture and then you jump. But why do you jump and how do you jump? You would just say, there is a jump. For instance, I mean, the top could have very well been discovered by the student of Riemann in Dax. And so it's a transition which is delayed, delayed, delayed, delayed. It was very well within this class because they had all the geometrical excuses, but because it was not interesting. Like when we spoke about evolution, I think the influence of Einstein who invented the space time. But the disadvantage of space time is that the time is because static. There are many advantages, of course, we all know. But on the other hand, when you see a particle becomes a world line, so you underline that it's something which is not dynamic. And it took many years for people doing relativity to understand that cosmology is something historical, is a form of development. And how do you interpret a true development within a four-dimensional manifold by definition is given. And it's very interesting in that connection that Einstein, of course, rejected, you know, the dynamic picture, the solution to general relativity. He introduced the cosmological concept precisely in order to avoid the dynamic solutions to the equations of general relativity. So, I mean, so, I don't think Thomas was the right person to do it. But, of course, stability of a metronome system is slightly different. It doesn't tell about instability of a metronome. It tells you about, I mean, if you perturb your metronome.

45:00 Yeah, there's other kinds of stability, but his structural stability, which should lead to a simple classification of singularities, 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, I try to make 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 curves into some space of curves. There are many discussions in the 70s and innate some mistakes were made, but one finds a way to try to play tectonics. Which is really tyrannical. And then we would, I mean, we would think in statical terms and of course it did not work. It did not work because of processes. I mean, it's a very interesting philosophical problem. I mean, it's clear that all-dimensional space-time is a great advantage. You understand the historical value of mathematics. In a sense, you are taking the point of view of God. The edge of the world, I mean, he has the challenge of the future and also freedom. And with the fact that you have a series of questions, what's the right way of what we are doing ever planning mechanics explains the perfection of the work.

47:30 I think David Eisenberg is in that book. Oh yeah? Yeah. Oh yeah. And he's in the pictures in that book. Uh-huh. I mean, they are considered as a situation. In the French tradition, there was a so-called rational, which was a rather strange combination of various tools, but mostly it was mechanics before Jacobi and Hamid, based mostly on Lagrange's idea and even before D'Arobert and Lagrange's idea, so middle of the 18th century. But there were people who were teaching about mechanics in a way which had to be complicated. We forgot about what has happened with Jacob in Habiton in the 19th century. And it's interesting that Jacob invented zeta function to solve differential equations because the zeta series are very rapidly converged.

50:00 And if you express everything in terms of zeta c, the solution of your dynamical system in terms of zeta c, you have a very good developmental control. Because the Theta C's are very fast, they converge very fast. But it's interesting that Jacobi was my master of Puma. I mean, at least I understood, I set up an exhibition, you know, the teaching of mechanics was very out of proportion to the poor, that is, pre-date, that had been developed in the 90s, not to mention Kovalev, which has something to do with stability. Mechanics is dead and done and so on. And the other classical mechanics is revising it as something from the mechanics, something wild. Who cares about that? That's just the beginning of a suspicion. But of course, cross-trajectory is a symbolic one, then to say it's like, oh! From a porous unity, because I mean, when the engine broke, and then the, I mean, the, this is, they weren't quite, they weren't quite, you know, related systems.

52:30 Come back to the, you know, the magic, I mean. I mean, the classical mechanics is dead because they have the quantum mechanics. I mean, we know better. And everyone who has tackled quantum mechanics seriously knows that the deep knowledge of quantum mechanics is an extension of quantum mechanics. He was invited to a few meetings and he was told that the people who were a little out of their minds, I mean he never lost his perspective in that and I mean the people were and the fact that he gave this course when the Chicago students maybe the last time I changed that one of the aims of Wobacki was to put the discoveries of A.D. Gatton geometry on a sound basis to provide a foundation for his discoveries.

57:30 And that's what my material would be to substantiate all the claims of many of them are rather intuitive and not too much. Oh, but Elikato was always there. Elikato was always true. The connection between mechanics, geocontrol, geometry, and mathematical physics. And there is this exchange of letters between Dr. Torrey and Einstein, which have been published by his advisor. Yeah, so after I was at Chicago and I should go to New York City, Eilenberg arranged that I should give a series of ten lectures at IBM, Yorktown Heights, and Thomas J. Watson Research Center, because he had been associated with them, and that's, and as I said, he's in one of his four open, on-the-branch years. We had a collaborator, or two or three collaborators, that were pursuing this, Jesse Wright and Calvin L. Goss, and Thatcher. They wanted to have, I gave ten lectures once a week, waiting for my son to be born, so my son was born in the same period as I was.

1:00:00 So I talked about personal categories in a more general way in the first half of the year. You're made as a bedding and anyone comes with some kinds of extensions and that sort of thing. I had some notes either by you or on your ideas from the IBM Center that Ray Nelson gave me. I have no idea what they were on there. Grad student philosophy, I got a bachelor's in that. Grad student philosophy. What kind of math do you think a philosopher ought to know? I don't know. They gave me a good program in math, and he said, well, you should learn this lot of gear stuff, and I don't know the correct theories, and you should learn the recursive hierarchy, and you should learn what this topos theory is. But Ray Nelson, he was a computer scientist and philosopher at Higgs. The recursive hierarchy never really did take place. I should have said that I also worked for the French military at a certain point, because that's right, when I went to Zurich, McLean had met this person, I mean he was Scotland actually, and for that reason told me to look him up in Paris. So I went there. 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 brand. So I went to his country house and worked away, and there's a magazine published in a journal with a report inside there, in French. I only did one session. I didn't go back again and wanted to do stuff with some, you know, philosophical nature. Does McLean do all of your work?

1:02:30 Oh, yeah. He was the director of the Columbia Center for... There's a whole mathematical organization. He was the head of it, actually. And he hired Eilenberg, I think. And he interviewed Truesdell, the young Truesdell, who applied for this position, but for some reason didn't go there. So many, many, many, many, many, many years later, I brought them together in McLean, New South for the first time, you know, sat down and had a beer together, and now the only thing that comes to my mind, I mean, about the whole town, you know, the town of Rota, you know, the town of Rota, you know, and was there some connection between McLean and the whole town? McLean and Rota, um, that's what I know of. I had some connection with Rota. Well, I had my say about McLean, so that's what happened. One relation between them in my mind, but 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 the brain was always urging me to write more articles, but wrote to us. Wow, that was a bunch of writing. He was a prolific writer, but he thought that... No, I think what you mentioned is that he and his, his lectures and the author also have a connection with Los Alamos. Oh, I know that. Some of his work, some of his work has a connection with Los Alamos. And recently I was using his work on the asymptotic expansion, which he wrote with the two parts. Which is, I think, well, it's a classical subject, but it has a good insight. I needed some technical points and a lot. Well, the point is that in the asymptotic expansion, people take you to the asymptotic expansion in the power, power of the subpar matrix, you know. But in many applications, you have also long epsilon. And you need to have good algorithm to deal with both epsilon and long epsilon.

1:05:00 And I came to that already some years ago, and I found that the Routine metropolis gave good algorithm, too. And that was one, because it was not too far from the ideas of Okumata, and so I mean about that, similar ideas, of course, about Okumata. And there are also, in this paper, this metropolis, there is 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 the metropolis studied, I mean, more or less, the basic products of the nature of numbers, real numbers, and that's, of course, it's connected to this complexity. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. So, 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 the partially ordered sets. Under multiplication there is a hunker there which induces a math of the algebras, so what he calls the Dirichlet algebra is really based on the monoid, actually, but it's injected into the one based on the concept by this particular math, but he never quite accepted the fact, you see, so he was always... Consisting that it's a partially ordered set, and it was drastically distorted in various ways. And there's quiddance. Quiddance of construction of the narrative of the category.

1:07:30 In his case, I mean, all that is covalent. He said quiddance of construction of the narrative of the category. In the case of the category associated to the whole section. Also, but I didn't have success. But I... We are completely out of the discussion because one of the best ideas of Ota was to bring Hoppe, Algebra and Homology within combinatorics and Ota's function is just one instance, but more generally I mean the idea of Homology and his work for instance with Saslacki and... We started as a very elementary, how can paper take a brain towards a certain number of straight lines and count the number of domains which these things divide into. We were talking a bit about it yesterday. I mean, 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. I mean, one of the main achievements in our area has been to construct a formal model of the... 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 index. But I knew about it too. It doesn't surprise me what you say. But this thing about that it eventually becomes so exponential. I mean, this is, of course, it's very much connected with randomised algorithms. And I'm not surprised that he and Metropolis have been doing it, because I've never seen them there where I knew they existed. But eventually... Model theory and stem topology came to this, too, and it's an instance of a very much more general. I wrote for the Boba pieces. I mean, I must say, there is going to be some play-arrangements, I'd call it, something like play-arrangements. And it's still, well, it's connected with some very developed, I mean, sort of complex things.

1:10:00 But we know better. We know that book. It's got an algebraic manifold defined by all of the real algebraic equations. And then, of course, you have the real world, the complex world, the destination in the real world, the parallel from the real world to the complex world, they have to play together. And that's what I mentioned already, the rebirth of combinatorial, mainly out of such examples. And again, we speak of fashion, and it's clear that this kind of combinatorial is completely out of fashion. Until they rediscovered that MacLean in fashion. I remember when I came to the Ford, MacLean was in Paris-Torch. And then I gave a seminar where I invented co-algebra and things like that. And MacLean was the only one. But never the less, when I formalized the definition of quantum, I invented this Cobalt concept. And then, Maclean attended very, I mean, listened very carefully to my talks and gave me great input. 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 theories were proved about it.

1:12:30 This is contrary to the ideology of most of the tough mathematicians who want to know the hard theory right away and don't care about the construction. That's a good point. It's also a good question in academic colleges when you have to hire someone. It's a regular discussion, whether you should pay for people who are able to invent new concepts and new methods or people who solve so-called art problems. And since Grogan-Deak has just cropped up in conversation, I suggest that after this session, let's take a five minute break, and then move on to the discussion of Grogan-Deak's work in functional analysis, and particularly his next movement through mathematics. The next item on our agenda is the next item on our agenda is the next item on our agenda is the next item on our agenda is Thank you very much for asking. So I will have to give the short of it, the short of it is I agree to do another talk So we'd be able to say that. And you can take it, but if you want to make it in private, you can take it. And I'll get to the part where it's, you know, it's revoked. It should be able to go anywhere. Oh, you can have it.

1:15:00 Yes, I was going to be able to comment on this, saying that this is an over-conceited interview of the kind of art that you do, and that you obviously don't know how to be kind about the act of politics and mathematical research. I think the new ones have been great as well, I think the new ones, you know, that I was able to. Speakers include mathematics, geometry, algebra, analysis, quantum mechanics, algebra, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, Yes, he didn't have the sit spot. He would have done more, he would have done better. Which is probably a very fair point.

1:17:30 No, it's because the motor is temperamental. For once in a while it does. Yes, we go into your, well, if you have the energy, if you want to stop, well, don't push it to, say, 6.15 or 6.30, so that you can make a start on the very good notes, and then have the rest of the morning, which is, in fact, exactly how we do it. And then, of course, the challenge we get, which is obviously then to come up with a new one. I'm actually going to suggest that in order to... And then tomorrow after lunch, I will take you to the parking lot. There's a lot of stuff going on in mathematics, but we need to have something like that, you know, that's a hundred times better than what we're doing now. It is fine. We need a lot of capital work to see what we're doing. Yes, there's going to be a school Tuesday, midday, so we're going to continue with that.

1:20:00 And so on and so on and so on and so on and so on and so on and so on and so on and so on and so on Is that very much as different as data? I mean, it's also not as important as it's possible, mostly because it's not all the different ones. It's not all the different things. I believe in all of that, but I'm sorry, I want to go to school. Oh, Christ! Did you guys, did you pass my exam? No, I didn't, I didn't. Thank you for your attention.

1:22:30 I'm actually underneath. And I was counting from time to time after the show, and suddenly I didn't see the topology, but it was more funny. Oh, I didn't know about that. That would be great. It was more, uh, similar to music. Correct. But this, well, it's not actually, it's a bad instance, and look, there's not much I can do about it, because that theme, and that's what I'm going to be happy to do, but it won't. Well, it's a lot of it, but it's a good theme. And the common point is that... You can learn more about these topics at www.magnus.ac.uk Well, I guess, uh, pedagogy, yeah, that's right, pedagogy. Uh, but I have been, uh, given the values of pedagogy and all of this, you know, I'm going to try to have a little bit of information, you know, the three big books, uh, the three of its forms are cut down, but I'll be doing, you know, like, yeah, because we might want to keep some of them on the timbers, and then I'll be, I'll be able to talk to the performer.

1:25:00 Oh, yeah, I was pleased to talk to him. There's a picture of a soldier or something there over about 5,000 years ago leading to these holy camps with himself. So what we're really doing with the article was they only unearthed one half, the image of that half. Yeah, yeah, yeah. And they've done that for many years. And they've done that for many times. No matter who you are, you have to have some kind of creation system to have that kind of development. I don't know if that's what you're talking about. Actually, not many of them have that kind of project. The intelligent design people are happy to say, well, we don't design with your class, we don't use your tools. We're just, we're not talking about that. But that's what they're selling. So, nobody's really arguing for a six-year contract. Well, I hope somebody did. Well, yeah, but not with any public evidence. And as for the millions of years, well, you know, we've all heard of it. The evidence on a radioactive decay is pretty constipated. Yeah, well, I was actually wondering if you wanted me to point out a metronology that doesn't devolve science and strategy. Yeah. There's a great debate about whether God made the distant stars with sunlight probably to us from love, or whether the speed of light used to be different in the past. Well, how it looks is a different story. The facts are so bad in this position. Yeah. It's a very, very... Yeah. It's a very, very... Yeah. It's a very, very... Well, I was the position of... I'm sorry, I'm a bit on file, so you know, I am Peter Voss's father, who wrote the first Prism of Mathematics, which belongs to Mark.

1:27:30 I'm not sure what that is. Thank you for your attention. Well, I've heard you say something there, Julia, you know, it's a very hot area. I thought you were going to say she didn't know that. She didn't hear as much about that. That's important. Well, yeah, Webber made the build-out, and so, you know, we're trying to shop. If I were to shout about a film, I might not know the rest of it, but most people think it was Dawson, but there is a minority view that it was not Dawson, that there were five of them who were present when it was first published, but Dawson was in the series of special editions. Well, I was waiting for the time to get out of there, and we wanted to do something like that for the rest of the time. But what I guess is, you know, it's time for us to come up with something else, and that's what we have to explore. Would you probably be able to find that? Yeah, okay.

1:30:00 I'm sorry, I'm not on set. I'm still open. No, the 272. Oh, it's right down there. Oh, so it doesn't mean the... No, no, no. Oh, that is interesting. So it was down in Hilbert Hall, not very high, I just thought it was up in the opposite one. It was just falling. So 2% of that there is probably 2% there. So that must be the record, which of course is actually anywhere there. Thank you for your attention. Thank you for your attention.

1:32:30 So what do you think about this? Well, I would say, I mean, I'm not a mathematician now, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. I mean, I'm not a mathematician, but I'm not sure I can tell you what these are. And so on. It should give us motivation if we go from there. We're all rather liberal, sometimes, but sometimes I'm not going to do it. Yeah, that's so, you know, I mean, that's the only knowledge I have. All of that is writing. Yeah, yeah, yeah. Actually, it's very... I'm not quite, but I'm still... I've already seen it. That's true. It's a human sensation. It's important. It's a piece of... It's not... It's not... It isn't. It isn't.