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

0:00 I don't know much about that, but he was explaining how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, how to do that, Thank you for your attention. In fact, which is still on display in this museum in Oxford, where there's something I don't know about, but I'll leave it to you to get an idea of what it is there, and I'm going to take a look at the place, and I'm going to go back to the source, and I'm going to go back to it. And I think this is a collection of factors. It's a whole group of things, a collection of factors. It's a thought that is an honor, an honor that I have to go through, and I pray and all of it. There were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were, there were.

2:30 They have a great deal of knowledge, that's for sure, they have a great deal of knowledge, but my actual source of knowledge isn't great, but my actual source of knowledge isn't great, but my actual source of knowledge isn't great. That's right, that's right, that's right. It was definitely an unreal collaboration. It wasn't this united world of qualities. So that's how we lost it. I don't know. Some people said we were nostalgic. This is why I have to talk about that. I'm sorry. I was surprised. I don't know if I have to say that. Well, I don't know if I have to say that. Well, I know a little about you. I mean, I'm not saying that. I mean, he, he, he... I don't know if you can hear me. I think he's trying to describe it. He's trying to speak pretty clearly.

5:00 I think he said we should move more about it. I don't know. The total fact is still worth the time. This is worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation, worthy of an investigation There are a number of different types of physics. The first type of physics is called quantum physics. It is a type of physics in which you have two different types of physics. The first type of physics is called quantum physics. It is a type of physics in which you have two different types of physics. The second type of physics is called quantum physics. The third type of physics is called quantum physics. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. I don't know what you're talking about. You figure out, you figure out that I just did a long whiteboard, a long period, and maybe some sort of reading like this, and you look, and you saw it, and you say, oh, Bill! And like he grabbed, like he grabbed the scene, but then, that's the last word he said. Bill, then he said, and the motion inside of me was, well, I don't know if I can speak, but I can't write. The next rule was, I can't do math, so I'm going to say that it's just mathematics. Why is that not surprising? No, no, no, what I'm saying is that physicists are perfect, and they're good. Some of the lessons in physics are destructive. This is true, too. I think it's a very good thought.

7:30 There's another question here. The fifth goal is someone who's got a... Thank you for watching. There's all sorts of things, and I think that's a lot of fun to study if one is excited about it, and one is a little bit scared about this stuff, or like the way of asking, is it catching my breath, or is it catching my breath, or is it catching my breath, or is it catching my breath, or is it catching my breath, or is it catching my breath It opens the main Wall Street Institute, and it was led by this guy in the final, and I believe it was in the final, to the British government. And it continues to be run by the British party members in Ireland, not in Ireland, but in Ireland, in London, and so on and so forth. At least for the British people, you know, the Amarish state, the European state, were quite strong. For instance, Didolaire, yeah, they just fought at the end of which the British withdrew and ended up having, you know, a sovereign government. And yet, they continue to perform executions in Canada over and over again. They're just set across the river in quantum mechanics, including the kind of execution that was done over and over again. And the Royal Irish Academy continues to get to what it is to this day, the Royal Irish Academy. The role of a surgeon in the lab could take into consideration the role of a surgeon in the lab in the lab of scientific research and other activities. I'm not even sure that a surgeon can take into consideration the role of a surgeon in the lab in the lab of scientific research and other activities.

10:00 And so on and so forth. But our set of all things, especially in 1949, might have had to be treated with a lot of respect, which had a completely weird, ostracical relationship in that it wasn't, and British were not, We'll see, you know, total solar independence, and that was what, finally, I would expect would be the state of the arms of the world sport, and I think that's where it's going to be really important to start to think of it in the comments, which is very interesting, and I really like the session. And then that's why Hitler refused to recognize the state and wouldn't let it survive until the 1930s and then decided they would take it over, because one of the things they wanted to do was to make the state go to an order never to die, still have to take care of the beliefs in order to take it. There are a number of ways that we could do that, but eventually it might be a good idea to decide on how to do it, because they should do that after it, and then the question is really about the power of the Constitution, which set the whole connection between the power of the Constitution and the power of the Constitution. The first clause of which is, you know, Ireland was consecrated, but not only that, but by eternity, and separately consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary was consecrated to the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and the Virgin Mary and But not all of these are very valid and this is a critical part of the course.

12:30 And then, in 1949, they just settled the final limit of mathematics. But I assume they had a purpose. Yeah. They probably wouldn't put it on. Actually, of course, it's all changed now since we've been trying to agree. Yeah. They've changed their constitution. Yes, yeah. So, of course, there's still some sort of validity to that. I don't know what the last part of it is. But certainly, there's some sort of validity to it. So, yeah. I don't think it was much like a sovereign state to me. I didn't have a very good state of mind. Well, I understand that there's a fraction lately that said they'll execute their own law, but they won't turn it over to the police. Oh, well, that's right. That's right. That's what I thought you were going to say. Well, but they're claiming it. Fortunately, I... well, no, I mean, I always have a conscience. But they're claiming it. But they're claiming it. Well, that's the irony. Well, none of that goes in the IAEA. Well, there's no part of the IAEA? No problem. The IAEA was, the students didn't know about it when Jesus was born. The IAEA is on the headstone. I mean, they're, I mean, they're very poor. I mean, I've never studied them. I've never studied them. I've never studied them. I've never studied them. I've never studied them. I've never studied them. I've never studied them. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. I'm going to do a follow-up. I think it's a good point. Even if they don't have fuels, it's still a fact. I agree with that. No, I disagree. Okay, shall we give them a second? No, I'm just going to be careful. So it doesn't change the scale of the students? Oh, okay, wait a minute. Then I'll tell you something else. Oh, okay. Really?

15:00 Yeah. It's a great deal. It's why you are so important. I think I would like to weigh in on some of the stuff that was brought up. Thoughts set off that concern. Where is the whole thing? Oh, shit, yes, I'm sorry. We've wasted rather a lot of tape. The discussion of Grotendieck, and specifically the first phase of that discussion, was to discuss Grotendieck's 1950s and the relationship with his early work and his later work in algebraic geometry. What is the key resultant for Francis? What is it that he currently took advantage of? I didn't mind doing that. He first came to Paris in 1948, having graduated from the École Montpellier. He more or less covered the basic degree. He was given a degree and he made a big contraction of that. He has mentioned in a quote that his professor in Indian analysis told him that there was, I mean, he has repeated the question about what was, what is a land, what is an area. I mean, he was obsessed by that since an early age, 10 or 12, and he was always discontent by the fact that people took as granted the existence of lands, areas, and so on, and never came to that conclusion. And then he was told by

17:30 There was someone, by the name of Levesque, who solved all the problems in mathematics and then with a few hints, more or less reconstructed Levesque integration as already in a very, very general form. Okay, and then after graduating from... He went to Paris and he has explained that when he graduated his professor told him, his professor had taken a master's degree from what made him, of course, what we call the academic system now, which has to do with something like the different methods of science, it's a master thesis, about the same level of a master thesis, with any cap tone, about geometry, and then that, I mean, so he gave to him. When Rotendieck received a letter of support, he met Henri Carton in 1950 and he died about two years later. And when Rotendieck came to Paris with a letter to Henri Carton, he discovered that Henri Carton did not fix it, but he had a song. And so he entered the letter to Henri Carton. The two personalities were very special.

20:00 And it seems that, by all accounts, both Dundee and Baissé, in terms of witnesses, Baissé, as who never lies, always tells all. You can never pretend that they never lie. Things they omit. They omit the slim volume out of his paper.

22:30 When I was young, I mean, I did not appreciate it. Only later in my career, who he was and that. At first sight, I was a bit shocked by the external behavior. I did that in school.

27:30 He decided that he would reflect the light in a big place. So he started to win, maybe half a year in the French France. He signed a contract, I think more or less in front of the younger generations,

30:00 who developed the so-called protection method. There are inks in the book by him, in complete ignorance of what was done in Sarascia at the same time under the leadership of Gelfand.

32:30 It is only after the rules that they learn, and I remember, Godemont was really disappointed when he discovered that Geltrond has discovered many of the things, especially the thesis of Godemont, who is, I mean, there is a function of positive and negative functions, and so on, and he was the one who developed the beginning of the representation of Godemont. So I was a good expert in functional analysis, one of the best experts in functional analysis at that time, mostly under the guidance of Morse and Deuteronomy. And Deuteronomy is at a time where Deuteronomy shifted its mathematical intelligence to functional analysis at that time. Deuteronomy ought to have more or less invented the tow factor, at least the regular step towards the tow factor. He invented the 12-notion group for, I think, models over Didi-Kinley, something like that. But he was the first one to recognize that the tensor product is not an exact factor, and that he co-examined to some extent. Well, 42! And, as usual, I mean, Yosemite was not motivated by the application topology or whatever. It was like a 12.

35:00 But the journey was mostly led by some encyclopedic view of mathematics. And so there was a natural question, there was a natural question, I mean, that Telson-Kornet and Telson-Kornet were just beginning to develop. Why? Telson-Kornet over here, Telson-Kornet over here, Telson-Kornet over here, Telson-Kornet over here, Telson-Kornet over here. We said, OK, we need some spaces more general than the Banach spaces. I mean, from normal, more or less invented, there have been, well, we don't invent the Banach spaces, but the so-called locally complex spaces were invented more or less by, from normal, at least, there are hints to that. And during that, we began to say, well, after all this went on throughout the world, with topology, duality in the Banach spaces, And I think the main motivation was that when you have a balanced space, you have two topologies. You have strong topology and a weak topology. But the weak topology is not defined by a no, so it's a little more general. And so I think the same kind of unification of functional analysis. 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 parts. So, it began with the systematic staticosite, and then it, with the strong department, called on. And then, later on, so Schwartz, Gaudemont, Dessert, and De Niro, the strong department, was in the main emphasis on analysis.

37:30 But then, quite independent from the liquid, much more algebraic and topology and geometry. Men in this, and then they began to put younger people. The evidence, of course, was that one of the most gifted men started with cattle. They would hide. And so, and then later they got from the new combined routine. First, he was first in CNN, then he wrote it. Then was appointed, stayed one more year at CNRS, and then was appointed as a professor in Nancy for two years, and then elected and collected funds. Very early age, I think you will know, very, very early. After the director of the Kornhof had ruled the world, disappeared during the war, I mean, because, I mean, the Nazis, there was some group of, I mean, underground fighters. And Dean, I don't know the father, the father of, I don't know, at least the acting dean, but the acting dean, and he protected these two. He was taken by the French police. And at the end of August, French physics at the time, I mean, his textbook, I mean, it's completely out of fashion now, but his textbook was a standard textbook, and then, I mean, did functional analysis, I mean, and he's the one who wrote with teeth in the fingers, I don't know, P.I.D. groups, okay, and readings. And then he has this, there was, the mother was, that's what he said, I mean, he said it all the time.

40:00 For example, Gardel. And François has a sister, Choquet, and was a famous poet in general in the 1880s. She wrote with Jean Choquet. And her work was under the guidance of Le Ré. She went for the non-linear equation of the system, but not the linear component system. And she was a really good student, and she is visiting IHES. And then students from a corner of the world to stay for a few months in Northeastern before they graduated and to have at least an idea of what they were and to give them special courses and so on. So all the people, I mean many people in my generation took the advantage of this school and so they spent a few months in Northeastern again and again and get lectures from here.

42:30 It's a standard university in Italy, pardon me. This thing, half of them are good people, but some must go. He constructed Levesque integration and he came, and when he came first to Nancy, he visited Dernier and he claimed to him, yes. And both Dernier and Sartre, they all had the same level of generalization. He said, my soul, I mean, well, we are. This is a kind of thing that, for various reasons, we have rejected, but you should do something about it.

45:00 And I thought, in fact, you know, this is dope. And just finished, I'm just finishing with Schwarz. Functional analysis, this is the paper which is called Duality in F and FH Spaces, where they are to extend the framework of functional analysis, local economic spaces, where they invented the, in that, in that time period, directly. But he wanted to base on the idea of duality. He did not invent the idea of topology, so taking the Ries-Marco frame as a definition. So you should define integration theory as a kind of duality. But of course duality of Hilbert space is not enough, or Banach space is not enough, so you have to put a larger framework. And because, I mean, Bragg takes as test function the function, the smooth function is compactible, I mean, of course, it's in a natural way a direct image, a direct image of a compact set. And so, you have to, well, of course, you are led to the idea of a direct image of topological spaces. Each space is a matter of space, but the limit is no more. And LF means limit of preciousness, and they are already realized that by duality, inverse limit and direct limit are inverted by duality. Even if it was functional, it is not categorical. It is 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 Captain Arthur, Ivan David MacLean,

47:30 that the idea of natural transformation came when they made a reflection about duality. And the fact that the space is the dual of its dual in some situations is not the case in other situations. And in function languages, you have plenty of situations where the dual of the dual of the space is not the space. And so, they have to invent a new framework. And they develop, so they develop, and just to substantiate the results of Schwartz for distribution, they have to develop a suitable framework. Data are equally aligned, 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, I mean, you can be snatched with that, I mean, you can work more concretely, but once you have defined the space of test function, I mean, everything comes to our sort of norm, equality between norms and partisan analysis. And it's true, back then, they were developing, I think, in the go-back history, they were developing these new things. 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 fourteen different questions. There is fourteen different questions. And Giordani would have something to be... Here is the paper which is a big print. At the end there are 14 questions. Make your teeth. Of course. But within a few months, all the counter is over. I'm developing you. So he worked from about 1950 to 1953. His thesis was defended in 1953 and then published in 1956.

50:00 In the Pambaydean American Mathematical Society, it was a very... Memoirs. Memoirs. What was then so? But his contributions are diverse. First of all, he answered all the questions by Dieudonné and Schwarz and developed a more subtle theory of duality. Yes, and he has, so he has a number of papers, where I know some are published in Brazil, and so, and where he develops his idea of duality, and he has the so-called, I mean, the so-called Schrotendieck duality theorem, which tells you that you have a space, in the dual of the dual, you can characterize the element which comes from the original space. It's a very beautiful description. Well, whether it's really helpful in general terms is very, very natural in practical application, but it's like an application of the Bayer set and so on, the Bayer-Briggs conversation piece. The Polish school was very fond of application of the Bayer idea to topological mathematics, but also in practice, in practice, you work with experts. It's good to know, of course, that you will have a solution. Knowing that there is a solution somewhere, finding a method to do it. So, gotending has a number of, it's a very interesting solution. And then, one supplement to the distributions that are developed as an integral.

52:30 You have to invert operator with a kind of kx and y, ordinary function but generalized function. Let's say the identity operator gives rise to the delta, the derived delta. This is exactly the purpose of... And then the physicists were very, very aware of that and still today. It's taken as granted by any practicing physicist that you write, I mean, every time you're an operator, you write, you enter, I mean, the, it's called Brian Kett, it's very convenient, because physics, I mean, I use, and even in elementary courses, you can't pretend that every operator, but you have to set some things up. And so that any operator while on condition, any operator can be represented as an integral operator, provided that a kernel K x and y should be not an ordinary smooth or continuous or measurable function, but a discrete. So, again, function of space is very different. So, if... I mean, that theorem is sort of totally untrue for Banach space, so that... Oh, no. So that the... It's not just the fact that they're kernels, it's the distribution, it's the fact that you're working in a world of completely different kinds of spaces than the ones which have been classical for a long time. Exactly. And the fact that your test bases are not continuous functions but smooth functions and so on. Yeah, they're analytic. Yes. And so, but then, this is a demo which will occur after the publication. So in the first edition of this book, this is not the first day. And I get excited.

55:00 There is a version for measurable function under which condition you can assume that an operator is a measurable function. So many people have thought of that and it made a great impression. Mainly because it gave a substantial foundation to this. And Schwarz was never far from it. Schwarz was knowledgeable in mathematics. Considers and they did. They did the whole work of Simpson's use. Then, what is behind this kernel? Those spaces, the first variable, the spaces of function, second variable, was more or less a natural idea, if you say, of polynomials. And that was beginning to be understood in geography by itself. That means that k of x, y is then supposed to be k of x is k of y, polynomial. It's valid in more analytical situations, not purely algebraic situations. And so, Schwartz repeatedly asked Wotan, is there a special property of the space of this function which ensures that? And so, he came with the idea that he should first develop a theory of tensor products, not function spaces. I think at the time he did not know that. I don't think he was really aware of the work of Schatten. But Schatten has a restricted aim.

57:30 I mean, Schatten wanted to justify the various classes of operators in the first place. Space class operators and so on. So, but what was already visible in the work of Schatten, that at least for Banach spaces there were at least two natural notions of tensile momentum. More or less in duality with each other. And then Kotelnik, so Kotelnik rediscovered that, or rediscovered or learned from Schattenkassen, it's not really important, but at least he could rediscover that for Banach spaces there were, well for Hilbert spaces there was never any question. There is a natural notion of tensor product of Hilbert spaces which was implicit in the Fourier series in two variables and so on, Hilbert-Hitobel and so on. It was not formalized until quite late, but nevertheless it was completely used. And in principle, even if it was not put in a formal axiomatic way, it was completely used. And the so-called Fock spaces in mathematical physics are just a particular case of tensor product. But for Witten spaces, I mean, you have two spaces, you have two natural motions, and almost for which the tensor of order is right exact, the complete tensor of order is right exact, and I don't know what for which it is left exact. You expect by duality. And so, so Witten did discover that, but he proved it beyond what he made too important discovery. He gave the meaning of these two tensor problems, the so-called projective tensor problem. This is the one which is right-exact. And it has a main property which was if you take L1 space, the variable X, then so the tensor product L1 over Y is L of the function in today. So there is one tensor product, the one which is right-exact, which is adapted to dealing with tensor product of L1 spaces.

1:00:00 Hilbert spaces are really L2 spaces. So basically, any Hilbert space is a L2 space, more or less naturally. So, okay, so L2, so you have, at a level you have L1, which requires a special tensor product, which is right and exact. At L2 level, the tensor product is exact, not right and left exact, which is one important feature. L2 is its own. So if you have a tensor product which is right and exact, it is right and 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 at Westbrook we discovered many values of that. Is that the second tensor product, which was called Xenon in the lecture by Schwarz, behaves as a left-exact tensor product and behaves very well for the continuous function. Okay, so you have two, and so in the frame of the other spaces, I think, you also, I mean, the hierarchies, whether you can interpolate, and there was some work done through for interpolation, so LP, LP, and so forth, equity. So, in a sense, you have an L1, and so forth, an L2, and so forth, an LP, and so forth. If you deal seriously with function and analysis and so on, and the function, many variables, and the 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 important. Okay, but then Goat and Dick are very bright right here. If you deal with a test function of Schwarz, with a topology, a smooth function, The space of source fractions are two different, well, the basic part of the space of source fractions is that you are not one norm but a family of norms. Taking into account the value of the fraction, then the next one takes into account the value of the derivative, the second derivative, and so on.

1:02:30 And so the basic idea is that instead of having a space with one norm, we have a space with a family of norms. This is really the basic. But then, so... In ordinary spaces, you distinguish clearly between L1 and L2. But when you do it with a smooth function, whether compatible or not, when you do it with a smooth function, you have the choice. You have three series of norms, one which is an L1 type, one which is an L2 type, and one which is L8. And this is more or less the, I mean, L2 type more or less corresponds to the subordinate spaces. Which, at the time, I think it's interesting that the Sobolev spaces were not considered very seriously by the French school of quantum analysis for some time. It's only when Erwalder, I mean... They were known. Oh, they were known. I mean, Erwalder, I mean, Sobolev invented them in 1936. I mean, the controversy, why did Erwalder, I mean, Sobolev invented this solution or not, I think. It's like the controversy whether it's Poincaré or Einstein who invented relativity, both, both, with different emphases and with different matters. I mean, it's clear that he discovered relativity through the mathematical motivation, but that's something different. It's much better explained in terms of Poincaré, but he did not question the basic meaning of the physical concept. And so, for, I would say, for distribution, it's always the same. Someone has invented a beautiful technical tool, which is someone else's, which is still widely used. And the idea of duality is that he did not consider them as a universal tool. And the great advantage of duality is that it considers them as a universal tool, as a universal container.

1:05:00 What you call reasonable spaces are subspace of the distribution, a big space. And I remember the joke about hunting the lion, you know, it's a standard joke now, the very straight mathematical way to hunt the lion. Now, with distribution, what do you do? Well, in the distribution jungle there are many, it's very easy to catch someone in the jungle of distribution. So you catch a lion distribution. Then, angularization. How do you transform it into a thing, into an item? What was the way? Should the universe contain it? It's the same as with the set theory in the science of the art. It's the same idea. Andre Weil considers the universal domain in algebraic geometry. I mean, Schwarz considers the distribution as a universal domain. So the idea is that you have a very large domain. You have much freedom to do construction. 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 it's not a monster, and then you take it. So that's not a great idea. And I think, you know, Zobar had never considered, did not develop fully his ideas, because when the war came, he had to be involved with the Soviet side. I think it's important to first realize that you have, we have three, I mean, so you have, so you can say, I mean, so many spaces were considered L2 norm of the function and its derivative. And then, both of these, both of these discords that you can play the same game with L1 norm, so you first begin with the L1 norm of your function, then the L1 norm of the third derivative,

1:07:30 L1 norm of the second derivative, and so on, or the L infinity norm. So you have the whole, and you could even interpret it in 10 different ways. So the point is that at the level of Banach's thesis you have three tensile coordinates. But because the space of smooth spikes can be defined by family of norm, it belongs to the three families. The three tensile coordinates. The Kermel theorem of Kermel and the Schwarz, which is a duality state. Now the real reason, I mean the two reasons for... Here is this tensor product. So, now for binary spaces, you have three possibilities for tensor products. I admit that the tensor product of Hilbert was considered less important in application. It's a very important application, but they did not consider it very serious at the time. But since you have three possibilities of defining a sequence of nodes for your space test function, you have three notions of tensor products and they produce them. And the fact that they coincide says that this tensor problem has all the nice properties. It's left and right, it's right, and then I think that was the beginning of the functorial way of thinking. He had this tensor problem with nice functorial properties. I think that really worked well. And then Grotendieck went to his thesis in two parts. He is a general theory of tens of hundreds of Banach spaces and more general spaces. He was the one who really thought that a general locally-convected space is an inverse system of Banach spaces. He was dealing, and by doing, direct emphasis on that point. That a general space is an inverse limit of Banach spaces. And so far, you consider the inverse system itself. Not! And you play with it.

1:10:00 For the analysis, it means that you have a space of function with various nodes, and you deal with estimating the various nodes, and you don't really care whether there exists something interlimitably, it doesn't, it's not very important. So, that's his exactness and so on. T-exactness, oh, yeah, naturally, T-exactness. And that, he, but he used that to, he tried, there was a very famous problem at the time in Banach space, The approximation problem is a completely continuous operator when it's called now a compact operator, so between two parallel spaces, it's an operator which takes any bound that's set into it, and uniform limits, uniform limit of operator is concerned by uniform limit, and the natural question was whether you could approximate any compact operator by uniform limit of finite type operator. In many examples, in many droplets, 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 in the middle part of the thesis of Rotendieck is concerned about this problem. And typically enough, he takes this conjecture and he formulates it in 47 different versions. The whole, no matter, the strategy is already there. It takes a very difficult context and we formulate it in many, many, many, many, many ways with the goal that, eventually, one of the people will get to surround, I mean, it's a, it's a Yoshua surrounding that, you know, and with his trumpet until the wall would fall. And it took more years to discover it and throw it. The example was very complicated, but now we have a full version.

1:12:30 The most natural space, ballast space, after Hilbert space is the space of all bounded operators in the given Hilbert space, which is a very natural space. And in this space, the copper can be quite explicit, and flow into someone else's space. So, Rothernick spent a great deal of effort, and I think part of his new method was hinted to. And that is very different. But then the second pocket. He's really, I mean to, I feel he's a fantastic, I mean, certainly the best one. 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 types of products coincide. You would say, okay, we have three natural types of products, they're smooth, fine, so they coincide, but is there a general category of spaces for which these types of products coincide? And that's exactly the nuclear science he meant. Nuclear because, I mean, the kernel was called nucleus, of course. Nuclear operator is an operator which has nucleus and kernel, and so he wanted to have nuclear spaces, spaces for all of it. So, and that's in the wrong way, but it was again from school, realizing. I mean, when, as soon as Strauss published his textbook, I mean, the... So, Gelfand was very excited and a lot of energy was expended and they published six more.

1:15:00 It's slightly different from the point of the Schwarz and a little more concrete, so to speak. But Gelfand has his limitations, but his imagination is fantastic. And he was the one who understood. But if you understood the connection with probability, what makes the Wiener measure exist, what makes the Wiener measure exist, the Wiener measure is really a new step beyond a standard, an ordinary analysis. It's a genuinely infinite dimension of analysis and not real. The rest is just... The fact that there exists a Wiener measure, I mean, so a certain... Probability law on the space of all continuous functions was rather unexpected at the time, but we could develop a topology for function spaces for so far as to spread an idea which was well understood in the 1920s. But the idea that you could do, I mean, integrate, a loving integration of such. And then Gelfand discovered that the nuclear spaces were the natural home. To develop a theory of integration in infinite dimension. And soon after he started working on that, there was a proof by Milos, Milos theorem. I mean, there is a, Geltman suggested a Milos theorem which says that it's an extension of the classical Bonner theorem to infinite dimension situation. The classical Bonner situation characterizes the Stiebjes fully transformed. The transform of bounded measures as a positive definite function, or an accommodation of positive definite functions. And that there should be something similar in infinitum was rather surprising, and it required a great deal of imagination from people to guess that there would be something.

1:17:30 Well, the work of Wiener and Levy, Paul Levy, are hinted to, but... But there would be an abstract framework for the Wiener integration, I mean, was really, I mean, it's a fashion. And Gell-Fond, the contribution of Gell-Fond is to feel that the nuclear spaces were the natural hope of the future. And then, within ten years, I mean, that's partly by Gell-Fond. So, uh, sorry. Bounded sets are compact. Yeah, yeah. Must go a long way. Yes, yes. Bounded sets are compact. That's a big idea. Bounded sets are compact. And as I said, I mean, what we know now is that the integration, I mean, in integration, you can absorb those space by compact subsets. Every bounded subset is compacted. A little more, you need a little more. But this is a little... Mid-north, all over. I mean, they all develop these. And there's a vision to get one through. And then, the analysis was taken by Weitmann in the States. In his so-called constructive, constructive, axiomatic, well, which he did, white man state, and then all these schools in the 70s, and Edward Nelson, and then understood what the physicists could do out of these ideas, again, for white, as far as mathematical physics at the time. Unfortunately, I mean, it works very well in the space-time dimension too, with some difficulty in dimension three, but in our world, it is four dimensions. No one knows how to make it. And I think at the moment, the strategy would not be to try to extend it directly. And I think people like Conn, like many other people, I mean, I think we are in the right way. I mean, we first want to understand the algebraic structure before we do the analysis.

1:20:00 And the algebra there is very complicated, and before we can go to Turkey, the algebraic side, I mean, it's too early to meet the analytical side, so it's, well, maybe 20 years or 15, 20 years from now, we can go back to the analytical side, so at the moment it's, I don't know why I say that, but this has a great, I mean, so this, if you trace back the influence of God in the thesis, it hasn't changed. He knows, he knows, he knows, especially in mathematics. But it was the imagination of Gelfand who put together these ideas with mathematics. And Gottlieb himself, partly out of ideological reasons, was never interested. He had an equation, if Hiroshima equals physics. He had this equation in his mind. And so he would consistently refuse to consider problems, but the practical problems increased. He had all the extra days and all the imagination to do that. He wanted to do it, but he didn't know. I have a question. I just remembered a textbook that I had in school, which was again J.L. Kelly's topology. He refers to results of Grotendieck on countable compactness. On what? Countable compactness. Any bounded sequence has a convergent sub-sequence, without mentioning longer sequences, only the countable ones. Yes, that's right. That's probably related, too. Yes, that's related to the first insert into it. The first insert into it. Yes, yes. But Comstock has a sub-sequence. Yes, yes. So, if you want to understand, I mean, the contribution of Goethe and Dick to some period... And then we have to understand how, what was the influence of these results on his further development. So I finished with a purely analytical part. Okay, so now, Gordon did not finish with that. I mean, when he left North Sea to go to Sao Paulo in 53 years, of course,

1:22:30 yes, he spent two or three years in Sao Paulo, and then one year in the United States, and what was it, it was... And so, I remember, I was first, I came to this boat and he gave a talk at the Bobacky seminar, I mean, on some problem of functional analysis, algebra of operators, and then he disappeared for a few years, and he reappeared in 55 of it. It really happened in Sao Paulo, and in Sao Paulo, he gave a set of lectures. ...which was more or less intended to be a textbook in functional analysis. He gave us a series of lectures which have been treated as seminar notes, lecture notes. And then, at that time, his new contribution is also important for what he did after that. His new contribution was a paper which is published in the Summa Brasiliense and it's a very interesting paper. Rothenbeck wanted to understand, I mean, more deeply, the reason why for some spaces the two kinds of tensors would coincide. But, systematic as he was, he began a systematic study of all possible tensors for the fact of balanced spaces. Well, he soon realized that the core and then the rest going to inverse limit was more or less nothing. He starts more than developing a general theory of all kinds of tens of words that you can construct on Banach spaces. And so, and, well, he has a very systematic way, which is more or less along the line of what was done before by, by Chaton. More or less, it's closer to Chaton's point of view, that is, of his first book. And then, but he has, it's very interesting, there, I mean, The idea of natural transformation from one factor to another one is very obvious, even if the language is not there. And also, he had this idea, he was already influenced by the idea of resolutions. And he wants to show that the high factors, and he wants to show that if you have a certain tensor for that,

1:25:00 then out of it you can renew tensor for that with better conditions, let's say. Something like a projective resolution and injective resolution. And then, at the end, he ends up with, I mean, applying various general construction, he comes with 14 different tensile products. And he has a comment that the one you cannot escape. There are many more, but these 14 you have to consider. And then the crucial thing is that there is a difference between two such. And it's written in a very cryptic way. That the identity map of a Hilbert space is a pre-integral or something like that. It's a completely cryptic statement. When studying this paper, which was not yet well written... I discovered that what was behind was something very concrete, a very concrete calculation of the norm of matrices. And so, if you have a square symmetrical matrix, and if you can, well not even symmetric, you can bound all the entries, all the elements in the matrix by a constant between plus and minus one. Then the determinant has a uniform balance, depending on the size of the element, n to the n, n to the n, something like that, and which is why it's called the textbook and very important in many things. It was used by the integral equation.

1:27:30 ...which are independent. So about the size of the eigenvectors inside. But the idea is that you are in the real matrix, but it's a thick, real, symmetric matrix. Every element is between minus one and plus one, and you want to discover the size of the eigenvectors, the eigenvectors, and so on. And your estimations are independent. And then, of course, the strategy to deal with infinite-dimensional space. You had a good chance to be able to develop something for infinite dimensions. This is a very deep study. I discovered that the core of the brain, Godden did not even give a proof of this. I mean, he had a huge machine. He had a huge machine. Not a machine you could manage to really prove, but he did not even make a proof explicit. And when I realized that it was made, and then, of course, I gave a proof, which was suggested by some of his students. It comes out to some integral with the gamma function and so on. Well, gold and I'm gold. And I made a report at Wacky's seminar. I had two. The first that I said, well, I will explain after gold and these various factors that you can build it with. And I used the word factor. That you can build it with. It did not. But there were factors between them. And so, for example, it was about Habana spaces. And then I took the main results and I think about now the first... The main reason it's so cryptic, I will give you an explanation. And what that means, it's not like it. It is saddening. My claim was substantiated historically because the people who took it after me were the French people like Moet and Pizier.

1:30:00 And they use repeated re-estimates about matrices and laws of matrices and eigenvalues and eigenvectors and so on. I remember when I gave this exposition to Bobak, he said, you spoil the scene. And he wanted me, when finally the lecture was printed, he wanted me to remove the problem, which I think was my problem. This is my contribution and I think eventually my contribution will bring some more. I can imagine that later users might use your formulation rather than his. Of course, more people use my formulation than his own. Did he write down his explicit mountains? Yeah, he wrote them down. Well, he gave them to me. He found some concept which is a hyperbolic sign of pi over 2, and when I tried to understand how he got this, I mean, it's my time to understand that it's true, well, it doesn't give all the details of what we give, but then when I want it, I say, well, finally, it's good analysis, and I can do that analysis, and people can take it. But the important fact is that already in these people, it's clear that the spirit of functions, tens of functions, tens of functions, factors of a certain category of mathematics, natural transformation between factors, I mean, derived factors, in a sense it's a field of derived factors, and in a tor, it's a torsion, it's a tor for mathematics. It's not exactly the same, but all the people. And so, so the spirit of the category and terms and factors is always very overwhelming. That's one important thing. But that made a difference. I suppose that made a difference. And so it means that already at that time, I mean, already in this thesis, even if the language and the practice of category theory was not there, I mean, the spirit was.

1:32:30 At that time, Bobacki finished publishing his textbook on topological vector spaces, and there is a strong emphasis on duality, and also under the influence of Schwartz, inverse system and direct system of general spaces were considered. So, if you want to understand, I mean, the origin of inverse and direct system is, I mean, I think the importance of the work of a functionalist. I mean, you cannot underestimate, you don't have to underestimate the influence of functionalities in the development of this idea of human beings. So now I come to the transition, the transition between, the transition between this thesis and what came just after that and his work. The transition came out from the following. Invertible matrices, volumorphic invertible matrices, which is a super factorization of volumorphic...