Morning Discussions

0:00 It's a paper by Lazare and Kloppers. He presented it at Lazare's lecture. Actually, I don't know that name. L-O-E-S-E-R. He's also working with a younger student, Cloppers, and they're redoing foundations, and it's a bit rig-like. Thank you very much for your time. Flag-lattice as the subject of discussion in the last morning he's going to be here, i.e. tomorrow morning, because I hope we might discuss the pollution of broken deep points and his exposure in the bottom of the paper tomorrow morning. I was just saying to these guys, Bill, if it's a, you know, obviously, what's your input, but if we start off with a relatively short discussion of the... The algebraic geometry in the SGAs and the scenario as it led up to the presentation of topology to the categorists of La Jolla and then the increase in emergence and crystallization of the recognition of how the logic Fitted in, in terms of protective and injective limits and co-limits within this whole framework. And then how you, in collaboration with Tony, were led to the axioms.

2:30 I don't know if you think that would be a good way to proceed. Yeah, I mean we're making a rather natural transition from Pierre's presentation of the Did you ever have any contact with him? Did he catch you? There is some trace of it in something like the very early on. It's just, I mean, it's almost just, I mean, one of them. Yeah. Yeah. She's there anyway. There are some sheets of representatives. Yeah. Many models. Yeah. Yeah. They just come out. Not all sub-objects and representatives of our research. No, no, no, no. Right. Okay. On August 13th of June, and into the face of the discussion centered on the STA.

5:00 We have a very big seminar in the 1960s and early 1960s, and particularly to the emergence of the Grundy-Gerard Thomas, and perhaps more generally, we might lead off by asking Pierre a little bit about the field at the time of this, what I call this kind of, Mat-Matten-Treative-Tilsit, which was mentioned yesterday, this informal agreement to the division of labour, but very loosely speaking, the local aspects. And the global aspects of algebraic geometry, which was, as I went, formally a creed between the Lubacki group and Grobendieck, and then move on into a little more detail into the SGA, obviously we won't be able to treat that in anything like the detail it deserves, but focusing on the line of development that led through topos theory to the recognition of the logic in the algebraic geometry, and then of course on to the work of Bill and his colleagues. So, perhaps I can ask you to lead off from those lines. The tree was grown in 1938. The last night, as you thought, is 1960. The discussion of the problem that you thought about. Yes, but it's important. Accessible. But erotic. There was an intense collaboration between Sunge and Porter. There was, at that time, what actually was interesting was the promotion of geometry by John Healy.

7:30 And interestingly enough, I mean, I've been to some places, and because when we came to the, for instance, the approach of Milner, you know, Milner, and to where Milner was, he was going to find to, in the category, consider the category of, of, of, of, of, of, of, of, of, of,

10:00 And, well, it's an easy construction from one to the other one. I mean, if you have a principal bundle, a frame bundle, you can recover the cupoid and converse. I mention that because Ehlers-Mann knew that. Ehlers-Mann knew that as well. And recently, I mean, in a recent work of Bosman, Morten, Meisel, on the Galois-Floer differential equation, we resuscitated this idea. And it's implicit also in... Many works are on this. It's an idea which is everywhere, transparent in some books, especially his long paper, the three and minus three part, which is a wonderful paper. It's a book of seven seats, but I didn't mean it as a poor one. This point about, what do you call it, Milner's... It was presented in the book of Serge Lange also, I guess a couple of years later. Yes, but it's not by chance, it's not by chance, because at the time, Serge Lange was a member of Obaki, and he quit Obaki about the same time as Goethe did. And in a sense, I think one of his excuse, maybe, was that the book, it was out, and Serge Lange... The first digit is incredible, yes.

12:30 You support or despise a character as a phoenix inside of that.

27:30 So, you have evaluation and so on and so forth. And back then, and Sunwell participated in both projects. Sunwell was, but all of them. Sunwell on the other hand, with Zariski, where you can see the beginning of shift from I mean, if you go so far, the construction of local rings out of a primal day adds on restriction.

32:30 Well, when the rings are domains, when the rings are zero divisor in C, Chauvelet and Ouzkov had made some effort to extend them beyond, but it was not very satisfying, it was a little complicated. I mean, but we came with a general definition of what would be a waste. So, we said, okay. As a result, I guess it's often attributed to Cartan, I don't know, in topology even, you know, if you have a covering where the pieces are homologically trivial, then the whole homology, it only depends on the nerve, it depends on the axis.

45:00 Ah, it's a lorraine. Lorraine? Lorraine. Lorraine. The abstract scheme, you see, with the diagram itself, whether you put at the vertices local rings or topological spaces with trivial cohomology is a, well, I mean, they are different, but the first approximation is the abstract diagram itself, which has the non-trivial cohomology. So probably this idea was there, too. But then the approach, the original approach of Sayer was rather different, rather different, because he did not put much emphasis on the local, he put emphasis on the scheme of regular functions, but not on the localness per se. And so, but then, so there was this idea that the idea, I mean, the scheme, so... Excuse me, could you say that again, because I like that, I didn't really get it. He did not put too much emphasis on the local rings, etc. He considered a space with nicely topology and a shape, but the shape is really the regular function. Of course, the local ring appears as a stalk of the shape, but they are not the prima facie objects. The shift in emphasis was done by Chevrolet, who are the real object of importance. And by the way, the Zayaski topology is not what we say. The Zayaski topology is a topology on the set of all, not the local in question, but all the extensions, and then you can say all local, and then sometimes it's called, I mean, in French, it's called Ibonaca, Ibonaca, Iusa, to make a standard, I mean, generalized topology.

47:30 The advantage is that it's purely bivational. It was not exactly what it is on the way, I suppose, who understood how to use ISD topology in the essential music. So, he said a quote, and he believed that he could, and so he judged it. It's very difficult to disassociate, I mean, there were most of the things that occurred during discussion, between Filsaert, Weiss, and Luzoni, Wiesel, and so it's very difficult to say, to translate it into a quote, the spirit of the time. But since it was not really conversant with the article of Locke Erling, and I remember proving in some summer that when Sarah makes this statement, it may not really be exactly what we need to say, it's interesting that this review, which was very negative, dismisses it as philosophy, not mathematics, basically, in the 50s, but we've been into it long before.

52:30 No, but it was published only in the 50s due to the political problems. Kehler was a German, was a wrong friend, and we have many good connections in Italy. But because of the fascists and so on and so on, and he could publish it only when the peace came. So you haven't seen the review of Andrade? No, not yet. Ah, that's a very interesting document. What is that? It might be in the book. It's a very long argument by Keillor. But indeed, I mean, there's an underlying complaint that Andre Bay has, namely he's changed all the terminology, so it takes a certain effort to grasp his use of the terminology, but then it's quite, you know, it's a vision, it's quite a vision of the whole subject, which is... But I just want to mention that Bay does say some generous things about revenue schemes for just this reason. But he does not praise Brodnick very often in print. But he does on this point of achieving an algebraic geometry. Yes, yes, absolutely. And also about Keller. It's interesting that we speak of Keller differential. And they were introducing that paper. So there's a lot of paper of Keller as two sections. And one of the sections is about the differential calculus in algebraic geometry. What was in Kennedy, but with no knowledge of it. The paper of Kennedy was published at Olmos. It was written long before.

55:00 Many of the ideas about differential calculus and algebraic geometry were only in Kennedy. We just contained the idea of clear differential until there was measurement. But I never took enough time to really discuss it. I've seen it physically in the library. Yes, yes. Maybe we can have a seminar on this. He has a section that he intends to describe what a scheme is. His description is not very convincing. But at least it took him to write 20 pages trying to explain what a scheme was. Only, unfortunately, he put about 20 very close. In a sense, he bombed the whole thing. But he took his effort to do it. And he says explicitly in the Rival edition that, of course, his definition of abstract art is completely superseded. So they were you and they were you. And I can speak what I can. You speak what a dictionary can and it's text. To this point about the fields and rains. It always seemed to me that this idea of putting algebraic geometry inside fields was rather like, methodologically, rather like unbounded operators in functional analysis. Trying to base functional analysis on unbounded operators and stick always inside this very, very nice Hilbert space, you see, but at the expense of creating a...

57:30 And instead of having lots of related objects where these things will not be unbounded in their own sense, so likewise, rational functions of putting everything, fields are very nice and invert everything, but to force everything into it by taking the limit of all partial. And then you have to rescue it again by talking about valuations and all this, which is, and Grover Deacon did talk about this in the introduction of the first DGA, saying that actually a prior acquaintance of algebraic geometry certainly has its value, has also its cost, especially in excessive habituation to the bimational viewpoint. We spoke about Sobolev, but now we have a better view of what Sobolev space is when we find this special situation, but now we have a better view of what is an unbounded operator and what came out and abstracted is the idea of a scale of Hilbert spaces. In modern functional analysis, people use that and also since we spoke about the nuclear spaces, I mean, there is a result, it's somewhere in the thesis of Rotundi, but it was put in later classes by Gelfand, and that If a nuclear space is usually obtained, you take a scale of Hilbert spaces, meaning a sequence of Hilbert spaces, each one being continuously, intensely, embedded in the next one, and if you take it... Well, you come back to Hilbert space to compact it a little bit. And then when you take the inverse limit, you get a nuclear space. So the nuclear space in itself appears just as the inverse limit. It's like saying, what you're saying is like saying that a real number is limited to a unique Cauchy sequence.

1:00:00 In fact, there are different models for the same endpoint. Particularly the Sobolev space is quite variable, you can choose. Many different expanding domains and many different ways to norm the higher derivatives and so forth, so it's really, I think, to say that that's the real object is like saying, you know, that the presentation of the group is the group. I'm sorry. I thought about it in another way, see, this scale of Hilbert space. Gelfand does a very strange thing with language. Because he calls it a generalized Hilbert space. But it could not be. A single Hilbert space is not an example. You cannot memorize two. Because if you take a constant sequence or something, then of course the bonding maps are not these completely continuous operators. So it's a strange use of the word generalized. I don't know. I didn't think it was going to be like that. Diamonds. Diamonds. Diamonds. Except, except, except, but if you want to define the Galois group, you take a field, except, except, but if you want to calculate it, then, well, it's very complicated to calculate the complexity of the Galois group, and you usually have to rely on the arithmetic problem of the Galois group, which means that you are not really working over, you are not working over the rate of integers or arithmetic integers or something like that. So, even in the Galois theory, even in the Galois theory, I mean, Steinitz's exposition of field theory was the first, I think, really successful axiomatic presentation in algebra, and that also was a model.

1:02:30 And also there was the work of Dedekind and Weber. Well, I mean, all the Riemann surfaces, the unification of Riemann surfaces, a number of it was a big model. All of these have been accomplished by them, and which remained a model for two or three generations. And they were, Dedican and Weber, were they explicitly in fields? Oh yes, oh yes. Oh yeah, explicitly. The word rain came only in 1913. No, that was for older people. It was for older people. So Dedican is explicitly in a field, but he's also... Maybe some people give Conacher some credit for this, the first to pick out the order of integers in that, to make that a theme. Yeah. No, I'm not, I mean the people have, in some ways, more of that. Okay. So, I think between 55 and 60, the idea of scheme was developed by... And it was clear that that was the starting point of the new algebraic geometry. As I said, on the other hand, Boballeghi made it very normal, and so on, and after the revision of... It's upside down. It seems upside down in the sense that, chapter after chapter, the book might take position. De Fitt came really as he had. In the previous plan, by the way, De Fitt would be in Steyer's approach, adopted. I mean, De Fitt came rather just very late with Steyer and books, but he knew that he had enough to do. He did not want to spend too much of his time making games. He was very happy.

1:05:00 So the only part that decides where to start is the science, the topology. Science, the topology is introduced as a collection of fine ideas, a collection of go-go's, go-go's, go-go's, the science, the topology, the skill, how it is considered, introduced, and also at some point we did it in geometry, and we, I mean, from what you have told me, the foundation of algebra and geometry can be very easy. And as I said, because at the time, if you put together...

1:07:30 We decided that as long as it was purely local and could be expressed easily in terms of time, ideas, and so on, that was our job. But as soon as it begins, as soon as you are glued together, consider poplar maps. All things connected with poplar maps, which are all described like this, and cohomology, this is really different. And since we were never able to provide the foundations of an exposition of GFU, that was also the dividing line, because according to our standards, we would have first to devise a set of chapters on the foundations of GFU. We had various projects at that time. And of course this interfered with the development of Manifold because everyone understood that. You cannot think of a manifold without at least a modicum of shift, not only the shift of work, but it requires the idea. And so, sadly, it produces this doubt on local category, you know, to hold as a substitute. But if we wanted to, if we wanted to have, Rottening came with extensive work and said, Schemes are for me. I've already told you these are here. So, it's all common. Common good, common heritage.

1:10:00 Matter of which was not properly activated. He was so ambitious. Many of the best experts had refused. You have now taken a somewhat more general. He sticks only with ranks, you see. I mean, whereas there are other things. Not even so much of ambition. So, he was actually never doing himself. There's a fact that doesn't come out in the Orwell's article, namely the fact that at every stage it's a conscious conspiracy. Within a conspiracy, of course, there are struggles and all this, but it's all planned. I mean, McLean's memory is completely wrong. It's not a chaos. Not a chaos at all.

1:15:00 All of this was solved because it was long and ready to achieve, given the changes that occurred even in the previous 20 years.

1:17:30 And I would guess that the younger generation, the younger generation, think the purpose of the Boba King Group 1, you see, since it clearly must have changed in some sense. There were many, many discussions on one point. And we see that Boba King was supposedly divided into two bodies and then the second. The principle that we would refer only to ourselves was unbearable, was unbearable.

1:20:00 So, the foundations of a manifold, such as geometry, this trend, and we should follow

1:22:30 the carton, Chevrolet, I don't get the sense of it. It sounds like a project about some scholastic dimension that's going on right now, generation after generation running through the common ground. It's almost parallel. Yes, that's all. Poisson-Giometty and then Poisson-Giometty as many locals.

1:25:00 He was around for about ten years until he became friends with the election of Mitterrand and he invested himself in the election according to the first fifteen members.

1:30:00 The idea is that every time they move to another place, there are many reasons why people don't discuss them without voting rights.

1:32:30 It should be so. We are in class to include it.