Discussions — Grothendieck's Program in Algebraic Geometry

0:00 Okay, well it's the morning of Monday the 13th of June. I'm going to move into the phase of the discussion centred on the work of the SGA, Greenwich Seminar, and perhaps more generally we might lead off by asking Pierre a little bit about the overall shape of the field at the point, as it appeared, at the time of this, what I call this kind of mathematical treaty of Tilsit. About which, which was mentioned yesterday, this rather informal agreement as to the division of labour between, very loosely speaking, the local aspects. And the global aspects of algebraic geometry, which was in a way formally agreed between the Volbachy group and Grotendieck, 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 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 colleague. So perhaps I can ask you to lead off. The agreement between Rotendieck and Bowen, you mentioned last night that you thought it was the 1960s. I don't remember exactly. As I said, Rotendieck participated to the 55 to 60s. I was six or seven years old. Yes, but some of the documents are not yet accessible. I mean, there was an intense collaboration between Samy and Grotelli and there was, at the time, Bobacchi was interested in promoting a project on differential geometry, manifolds and differential geometry. What came out eventually was this slim book, the so-called Facipule de Résultats, which is just a set of definitions, a set of definitions and constructions.

2:30 And interestingly enough, if you go through the table of contents, you will see Frank Tor at some places, and because when we came to the question of tensor product or vector bundles, we used the approach of Milnow, and where Milnow has devised... The general method, if you have a vector bundle over a certain manifold, and if you have a suitable phantom in the category of finite dimensional spaces, then you apply the phantom fiberwise to each fiber, so each fiber is a vector space, and then you apply this phantom which is... There is a certain continuity or smoothness condition on the factor which was described explicitly by Milner and we took this from Milner. And so you don't have to apply separatism to define separatism. In fact, there is something a little broader which was hinted by Goethe, and that has to do with Goebbels, and the idea which is more or less in the Arkansas, but not completely physically in the Arkansas report... Sheaths and non-commutative cohomology and the general find of spaces. It's an idea which was advocated by Everspan and I will summarize it in one sentence as a joke. A principal bundle has one leg while a groupoid has two legs. That means, the idea is that when you use principle bundle you compare the different fibres to a type fibre, to a fixed fibre, to a fixed model.

5:00 While in a groupoid point of view you compare the different fibres among each other. And well, of course, if you have a vector bundle, there is a certain groupoid. The groupoid, I mean the set of objects of the groupoid, has a point of the manifold, or if you want, a set of fibers, and you consider. So you consider the category of the various fibers with linear maps among themselves, between themselves, or octagonal maps, or in general. And, well, it's an easy construction from one to the other one. I mean, if you have the principal band or the frame band, then you can record the groupoid and convert it. I mention that because Erosman knew that. And recently, I mean, in a recent work of both Manon and myself on the Galois field differential equation, we resuscitated this idea. And it's implicit also in many works of Delisle. It's an idea which is, well, transparent in its approach. It's a book of seven seas, but I didn't read it before. This part about what you call Milne elders was presented in... Yeah, but it's not by chance. It's not. Because at the time, Saint-Jean was a member of Bourbaki. And he quit Bourbaki about at the same time as Gotenbeek. And in a sense, I think one of what is excused, maybe it was just an excuse. Was that Bobacki, I mean, when Gotenick left, it was out of a clash about what to do with categories in Bobacki. And Serge Lang supported strongly Gotenick. Supported strongly. It's review, isn't it? But Gotenick was the first CGA. He supported, he supported, despite the fact that his mathematical tastes were quite different. Mathematical tastes were quite different.

7:30 Yes, yes. But at that time, Cernan was there, Votendijk was there, and so we had this discussion about, first of all, we wanted to extend the treaties on differential geometry, manifolds, and so on, and as I said... Just what I wanted to say is that if you look at the table of contents of Sorbonne and Moghulbaki, you will see as a headline of a chapter. So, the hypocrisy of Wobacki, that he used the word factors and so on in the non-formalized part of the work. A title, in principle, a title of a section is not part of the formal exposition. But of course we wanted to show that we know what it is, and at least to give an insight for the reader who knows where we are going. So, the debate was about to start Manifold. Samy wrote a long exposition where he introduced so-called local categories, I mean he wanted to have a framework where you can glue things together, and he introduced local categories, topological local categories, which was a very ambitious program. There's a semi-published joint paper with Kai Kahn about it. Yes, that's right. That was the obsession of Gaubach at the time. Which Bobakir always entered to generality. Of course, we knew that we have real analytic manifolds, complex analytic manifolds, smooth manifolds, algebraic manifolds, and so many categories of manifolds, but there are many similarities. And one of the great progress of algebraic geometry at the time, before Kotanin, even before Kotanin, in the 50s, is the recognition of the method of differential geometry. The methods which were created for the real analytic case or smooth case work equally well in algebraic geometry. Once you have the definition of the tangent boundary, which is a crucial thing, then your rest follows.

10:00 And my observation is that methods of algebraic geometry can be used. Exactly. Well, just to show that it was a noble idea, in my thesis at some point I have to use the tangent bundles of my phone and I take two pages to really find it. Today, no one would do that. But then, of course, I was familiar with the ideas of differential geometry and Sway and Baudelic and all people were very familiar. As soon as differential geometry reached a certain stage, where the emphasis was on principal bundles, vector bundles, and various constructions among them, it was easy to extract this kind of construction and to To apply them in algebra and geometry. But as I said, in the 50s, in the middle 50s, when I wrote my thesis, it was not an obvious thing. At some point I needed something about the algebra of the group and I had to repeat the basic definitions and the basic constructions. So, there was this ambitious program. But Bobacki wanted to have unified things, so he said, well, we have all these categories of manifolds, can we make a unified exposition of the basis? And so Samy came with one solution which was local categories and so on, and Carton also helped him to develop that. But then I suppose that Chevalier wrote something or so, but I don't remember exactly, Chevalier wrote something or so, and then... Grothendieck, the part of Grothendieck was to supply the differential calculus and so he wrote a very extensive report, a very long report, 80 or 90 pages, about differential calculus in algebra and geometry or rather the conception of the differential calculus which would apply equally well to all situations. And that's where he introduced his ideas of infinitesimal neighborhood of a sermon or a scheme. The scheme did not exist yet, but it was one convincing, I mean, it was one of the real, one of the reasons we were all convinced that we needed so-called non-radial schemes, I mean, Nielbottom, Telemann, Nielbottom, and that's the schema.

12:30 So, Gottendieck, but the report of Gottendieck, which contained a lot of... Interesting thing, and still not completely exploited, has been repeated more or less in SGA III, in the chapters written by Gabriel. I mean, Gabriel, so Gottinger handed his notes to Gabriel. Gabriel was never a member of UBAC. He attended one or two sessions, but he was never, he was not interested. If he had been interested, he would have been, but he was not interested. So, too French for me. I extracted from the report of Grotendieck what I needed for the report. And about what's the date on that? From 55 to 60. I mean, it's a long process. So, but then... But I mean, when Grotendieck first wrote the 80 or 90 pages, because this is something people have already... Oh yes, oh yes, oh yes. Actually, I spoke about the origin of Grotendieck. Yeah, I know, but both Grotendieck and Tristan.

15:00 And Gordon came with a certain plan, I mean, so we will have to develop geometry and he came with an ambitious plan, so first volume, I mean, local category, second abstract, differential calculus, and so on, and of course it would be at least ten volumes to develop that. And then Chevrolet was assigned to continue the project. And Chevalier was a very systematic mind and he would enter, I remember, he would enter a tunnel, he would enter a tunnel which would be 20 kilometers long with no light and go boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, boom, He was a very, very principled man, which means that at first he would read the Gospels, or rather the Bible, because it was some kind of Old Testament, you know, and he would read the Old Testament and see what Moses said, and then, okay, so this is what God wants from me, and I'm not to say that he was a very difficult Protestant, he's been back over the years, but nevertheless he was a very difficult Protestant, at heart. Even if he was not an active church member, he was certainly a very Protestant at heart. So he would go there and he would decide on the principle. So, the Holy Grail or the Holy Land is there, what are the means to reach it? And he would decide on the principle and then go straight. But that's often when a project in Burbanki was given to Chevallet, it capsized. Because he would come out with 200 pages, very dull 200 pages, and in that occasion, again, the audit capsized more or less because of Chevrolet.

17:30 One would have to check that exactly in the file, but I think that's my recollection. At least, I just speak, maybe. So, finally, on the other hand, I mean... There was another fight within Bobaki, another project, which was the project on commutative algebra. This project has been there for many, many years. First of all, André Weil, a series of chapter or volumes, where he wanted towards the unification of number theory and algebraic geometry, according to the model of Dedeckin and Weber. Which was very much in line with German thinking in that time. So, the outcome is his book on basic number theory. So you can see his philosophy by looking at his book on basic number theory. André Rey was the driving force behind the project, the first version of the project. And André Wey and Chevalet and Samuel, who were the experts on algebraic geometry, I mean, wrote various data and the project went very far. And the project was aimed to what we call one-dimensional analysis, something like that, which was at the end a convergence of number fields and algebraic curves. And then the preparation, for the preparation there were many chapters of preparation. And I remember there was an extensive chapter by André Rey on elimination theory, or rather his lecture on elimination, and also, I mean, it was more or less the line of his treatise, Foundation of Algebraic Geometry.

20:00 So, when he was writing this, this was it. These are very well known treaties. He supplied Bobakio with various drafts along similar lines. And so there was a very detailed project and at the time around 1953 or 1954 the project was well underway, almost ready for publication. And there should have been 700 pieces or 700 pages, well underway to be published. So with a great emphasis on fields. And what remains of that is the new edition of the chapter 5 in the first series in algebra, which is concerned about Galois fluid. But at the end of this chapter, you will find a rather detailed exposition of the function fields, not in one variable, but in many variables. The various class of extensions, separable extension, normal extension and so on. The point was that the chapter on Galois-Fury contains a very explicit account of something which goes beyond, which is not finite field extension but function field extension. And that was, I mean, that was finally extracted from these many projects, the part which survived and which could be accommodated in the first, I mean, in the exposition of field theory, in the so-called elemental exposition of field theory. So, but of course at that time the prevalent view was that algebraic geometry is more or less the same as the study of algebraic functions. And the algebraic version is that you should study extension fields, extension fields which are finitely generated as field extensions. And inside of that, so, and you have specialization, you have variation and so on. And back then, this was algebraic geometry within the framework of physics. And everything was aimed at that. And the project was quite elaborate. Of course, it came about at the same time, and Samuel participated in both projects, but again, there were slight differences in the train.

22:30 There was Chevalet and the Vey on the one hand, and Samuel on the other hand, and already Samuel had written these two volumes set with Zariski, You can see the beginning of shift from field to rings, at the beginning. And the thesis of Samuel, which was printed, I mean the most interesting part was printed as a booklet, A Gemma Locale, contains already the transition. Well, I will not read, redo the complete history of algebra in Germany, but the transition from field to ring was already there. And Samuel was able to make the transition. But then came Serre. And Serre came with his own contribution with Bunchbaum and Rosenberg, Bunchbaum and so on, homological method in the field of locality. And so, Serre... That was his own research, independent research. But then he came with a counterproposal. I mean, he said, but of course, and there was, I remember there was a heated debate between Weil and him, and so Weil was quite frustrated because the whole thing has already, the whole thing was already almost ready for publication. And I remember that Dix-May, Dix-May was more or less at the time. Geodony began to retreat a little bit from active life in Bobacki and Dixmy was a scribe at the time. Dixmy was a kind of Geodony-like guard. Maybe not the same strength but Geodony-like. I mean, if you order... Dix-Nier had to write 200 pages according to a certain class six months later who would have the draft completely ready with all the markings ready to be sent to the print. Okay, so and I remember Paul Dix-Nier writing seven chapters which approximately 400 or 500 pages

25:00 along the plan devised by Chevalier and Ray. And then I remember it was The discussion was supposedly to accept this, to send it to the printer. And then, Serre came with his new method in homology, homological method. And of course, it was so powerful, obviously so powerful. And that's where he invented the flat models. And he gave us, so I remember a discussion, and Sayer said, oh, okay, it's not the best way to do that. And I will make a counterproposal next time. And a few months later, he came with a counterproposal, which was already quite detailed, where he introduced the notion of flat module, phase fully flat extension, and so on. And a flatness idea is really Sayer's idea. Well, some here got some idea of that, but I mean, it was Seymour who really understood the importance of that. And then he came back, and then he came back and he offered an alternative to the plant by Chauvin and Ray. And it's more, it has been more or less, it's more or less what has been published, more or less, at least the first seven chapters, more or less what has been published. So it started with localization and at the time we discovered that so far the construction of local rings out of a prime line deal had some restrictions. When the rings are domains, when the rings are domains without zero divisor, it's easy to define the local rings. Chevalier and Ouzkov made some effort to extend that beyond, but it was not very satisfying, it was a little complicated. Well, whether it was just, I mean, the spirit of Bobacki, as we say. I mean, but we came with a general definition of localization. I think Gabriel helped in the first. And then, but then the plan was completely upside down. We began with local rings, localization, flat modules, basically flat extension, topohermeneur, hematic topologies, and so on, and then, and then, and then.

27:30 And, of course, what you see is that algebraic integers or integral extensions come only at chapter 6 or 7, and the device of evaluation theory comes only at chapter 7 or 8, very late, while in the previous outline they were at the beginning. And I think, I remember a discussion where people said, oh, after all, so much work has been done, well, valuations are not so that important, but so much work has been done, it should be a pity if all this effort would be a waste. So we said, oh, okay, we want to compromise. So we follow the outline of SAIL, and then we supplement it, but what remains of the ship would capsize before it capsizes. Already the sense of emphasis was very, very visible, very visible, and the trend from field to ring was already there, and that's really what Sayer gave us. And Rotelich was not yet playing a very active role. I suppose he learned, I mean, I suppose he learned much of the material from the discussion in Gaubach. I remember in the first, when he first attended, the first meeting he attended, I mean, he looked totally ignored. Equivalent. But his contributions were rather different. His contributions were more categorical. What he supplied us was, well, he took more or less as granted these things that were local and so on, and he supplied us with a plan to develop manifolds of various kinds according to some categorical lines. And finally, I mean, the thing... I think there was a point where things began to diverge. I remember there was also something, another point in the discussion, whether we wanted to have finite dimensional smooth reality or infinite dimensional smooth reality.

30:00 So this debate interfered with the other debates. And Serge Long was very vocal to include infinite dimensional work. And his booklet about, at the time, what he published under his own name was more or less taken out of a draft he wrote for Bobak. And Serge Long, as usual, was very impatient. He supplied us with a draft. We discussed it and we said, OK. We put that in order. And he was very patient as usual. And he said, well, well, well, well. And we said, okay, if you want to publish it on your own, do that, do that. As a result, his own book goes through several revisions. Publish first, revise later. As opposed to revise and then publish another book. Yes, exactly, exactly. But it was a question of temper. And so, but I think the debate was more or less spoiled by that. In retrospect, I think it was a mistake, because while Banach manifold, while I know more about calculus of variation now, I've been practicing calculus of variation extensively in my studies of mathematical physics, and I see that it's different. I mean, Banach manifolds are not relevant. Infinite dimensional, yes. Banach, no. What you have is really poor objects and not infinite dimensional objects. You have an inverse system of finite dimensional space and manifolds, but not really a real infinite dimensional object. That's the spirit of calculus of variation. But at the time, really in Bombaki, no one knew about calculus of variation. Myself, I mean, I've been working on calculus of variation after that, but much later. And no one knew, except maybe when Catton knew that his father did something, but... I remember when I was a student at Conor McCartan, I mean, Catton gave us a beautiful series of lectures about calculus, and at some point he said, well, I have to tell you something about the calculus of variation. So he took from the book of his father, Exterior Differential System, he took one section... Well, I mean, Élie Carton gives a certain set of formulas and he says it's a foundation of calculus of variation. It's true and not true. I mean, it's true that Élie Carton pinpointed a very important formula, very important formula which can be used in various situations, but it's not.

32:30 It's one tool for the calculus of variation. And Carton said, well, now you know everything about the calculus of variation, which was not true. And so, okay, and I think history is not you too late. I mean, if you don't take into account the clash of personality in this debate, I mean, you'll lose something, you'll lose something. And just, well, imagine, André Rey, Seyre, Grotendieck. Well, there is an account by McLean, I think in 54. He attended one meeting in 54 and he reported that, I mean, he wanted to put order into the discussion. And people laughed. So, and that was a joke for many, many years after what McLean did when he was doing his research. Okay, so, the situation is very complicated. I suppose that Grotonik learned about local means during this kind of discussions. And then, he had already this program together, but he did not know the tools. It's my hint today. This settlement easily was broken because it was not really conversant with the local algebra. He knew that Bobakir had many good experts, Sayer, Samuel, and so on, and that much work had already been done, and that for him it would be a waste of time to try to repeat. You say he had the time of Sayer's 1955 paper, algebraic coherence.

35:00 Yes. I think there is a report by Sayer at the ICM. There is a report by Sayer at the Edinburgh ICM. He has this statement, I will speak about algebraic geometry, which means theory of schemes, not the first sentence. And the sense which it has had for a few years now. I gave my account in Firenze. I gave my account in Firenze. As I said, I repeat that the word was catched by two Chevallet seminars, the one with the so-called Chevallet carton, but it's really Chevallet, on the foundation of algebraic geometry, still, and then he defined what is called a scheme, but in a restricted sense. The distinction between a scheme and a variety. A scheme is a scheme, a blueprint. A scheme is not a manifold, it's a blueprint for a model. That's important. It's a presentation. Yes, a presentation. And the idea is that the scheme is a collection of local names. At first, all local names within a given field, but that could be easily dispensed. But the real new idea was Chauvelet. Chauvelet wanted to understand the definition of abstract reality, how to glue together. So what does it mean to glue together a local model, I mean a fine model? And what Chauvelet discovered, and there is a separate paper on that,

37:30 It's that really the scaffolding, the scheme, the blueprint of your manifold is a collection of logarithms, which was, this idea was more or less implicit in Zaryski. And in Zaryski's, I mean if Zaryski and Samuel undertook this book about commutative algebra, it was with this idea in mind that Antonov... This idea of locality developed in the work of Zaisky about bi-rational transformation, blowing up and so on, and the search for minimal model of surfaces and then higher dimensional varieties. So, but for Chevalet, the scheme was a collection of locality. But even more, I mean, excuse me, I mean, there's a result, I guess it's often attributed to Cartan, I don't know, in topology. The fact that, 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 adjunct. Ah, it's heroic. It's heroic. Okay, the abstract scheme, 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, I think so. But the approach, the original approach of Say was rather different, rather different, because he did not put much emphasis on the local, he put emphasis on the scheme of regular fractions, but not on the local is per se. And so, but then, so there was this idea that variety, I mean, scheme. Excuse me, could you say that again, because I like that and I didn't really get it. The stairs shift. Well, in Sayer's approach, he did not put too much emphasis on the local rings themselves. He considered a space with a high-speed topology and a shift, but the shift is really a regular function. Of course, the local ring appears as a stalk of the shift, but they are not the prima facie object.

40:00 And the shift in emphasis was done by Chevrolet. Who would say that the real object is a collection of local rings. But this was influenced by previous work of Zaisky. Zaisky is the first who really recognized the importance of local rings. And by the way, the Zaisky topology is not what we say. The Zaisky topology is a topology on the set of all local rings. Not a local ring corresponding to a certain model, but all local rings. If you have a field extension, and then you can say all locomotives, then there is a Zayaski topology. And sometimes it's called, I mean, it's Boudetouali in French. Yes, yes, yes. I mean, that Ironaca, Ironaca used that extended topology, I mean, generalized topology for various purposes. The advantage is that it's purely bivational. I mean, you don't have to choose a model of your own. And there are various models that innate Zayaski. But for Zayaski, it was... That's not exactly what we see. It's André Weil, I suppose, who understood how to use the high-skill topology. So, about schemes, one would have to check in the correspondence and say, I learned about the belief that he could do something. That was his aim from the very beginning. He did not have yet the tools, but his program was to prove that, at least to break what was necessary to prove. Since Gotenick has been a very active member in the Chevalier seminar, he learned from Chevalier's point of view. And then I developed it myself, and so I explained that it's a very interesting interaction between the two. Well, it's very difficult to dissociate. I mean, well, most of the things that occurred during discussion between

42:30 It's really global, translated by a collective of authors, which was, of course, a political connotation. It was a time, it was a spirit, it was a time. So, just to say that, but since he was not really conversant with the algebra of local things, I think he was very, of course, he pushed, I mean, he supported Svayr in the new plan. And I remember poor Dixmier, I mean, bringing in some sum of 400, 500 papers, and we had a preliminary discussion, and he said, well, we changed the plan. Next year, faithful Dixmier came with a new redacted according to the new plan. As detailed and as careful as the previous one. But he might be frustrating at the end. But the scheme, the idea of scheme was already well known. I think the idea of scheme developed from 55 to 60. And more exact timing would be different. So when Sarah makes this statement in Edinburgh, it may not really be exactly what we know now as scheme, actually. Oh, no, already, already. Already? No, no, already. We had already, I mean, we all knew at that time the definition. Okay. And you feel it's grown extensively in that. And maybe there were still a few technical restrictions, but basically everyone agreed on what a scheme would be, and that it was a proper foundation project.

45:00 And also arithmetic geometry, because they long advocated the necessity of devising algebraic geometry not over a field, but over the ring of integers, so that it makes them to specialize more P, to reduce more P, and Andreeva repeatedly made some efforts, and Keller published a long, long paper in Italian. Keller published a long paper. Proporting to establish that and so when I think in a sense Grottening was just fulfilling the dreams of Weil and it's interesting that at the moment Grottening was fulfilling the dream of Weil, it was probably a personal clash. I think it might have been neglected because of Andre Weil's review, which was very negative. It dismisses it as philosophy, not mathematics, basically. When did you notice that? I would like to... In the 50s. But the paper had been written long before. Yeah, it was a long time ago. No, but it was published only in the 50s due to the political problems of India. Cayley was a German, was a long friend, and he had many... But because of the fascist time. And he could publish it only when the peace came. You haven't seen the review of Andre Day? No, not. Ah, that's a very interesting document. What is that? It might be in the bulletin. Martina, Martina. It's a very long argument by Taylor. But indeed, I mean, there is an underlying complaint that Andre Day has. 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 a degree in many points, but not so...

47:30 But I just want to mention that Wade does say some generous things about Rotendieck on schemes for just this reason. Wade does not praise Rotendieck very often in print. But he does on this point of achieving an algebraic geometry of the universe. Yes, yes, that's true. And also about Keller. It's interesting that we speak of Keller differential, and they were introducing that paper. So this long paper of Keller has two sections, and one of the sections is about the differential calculus in algebraic geometry. And what Rotendieck did at that time, or in his Burbach Report, was more or less to repeat what was in Keller, but with no knowledge of the paper of Keller. The paper of Keller was almost published at almost the same time. Well, it was written long before, but it was published only in the 50s for political-historical reasons, but it's true that many of the ideas about differential calculus in algebraic geometry were already in Keller. We just retained the idea of Keller differential, but there was much more. It took enough time to really read this. I've seen it physically in the library, but never took time to read it, to study it. Very long in Italian written by a German. It's already on the website, isn't it? It's easier for me to tell you than in German. Yes, yes. Maybe we can have a seminar on this. German-Italian, German-Italian, German-Italian, German-Italian. So, but the point is that, yes, Andre Witten was in the revised edition of these foundations. He has a section where he intends to describe what a scheme is. His description is not very convincing, but at least it took pain to write 20 pages, about 20 pages, trying to explain what the scheme was. Unfortunately, he put some restrictions about integral clause and so on, which in a sense spoils the whole thing. But he took this effort to explain. And he says explicitly in the revised edition that, of course, The definition of abstract data is completely superseded by that, completely superseded, so on the other hand, you, on the other hand, you speak golden diction and I can't understand it.

50:00 Maybe coming back to this point about the fields and rings, it always seemed to me that this idea of putting algebraic geometry inside fields was rather like, methodologically rather like, Unbounded operators and 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... and instead of having lots of related objects, these things will not be unbounded in their own sense. So likewise, rational functions, putting everything... Fields are very nice, you can invert everything, but to force everything into it by taking the, you know, the limit of all partial localizations, it's putting it inside, and then you have to rescue it again by talking about valuations and all this, which is... And Grothen, you can talk about this in the introduction to the first DGA, saying that actually a prior argument of algebraic geometry, while it certainly has its value, has also its cost, especially in excessive habituation to the biorational viewpoint. Yes. And coming back to functional analysis just a moment, I mean, it's a... We spoke about Sobolev, but now we have a better view of Sobolev spaces were defined in a special situation, but now we have a better view of what is what is an unbounded operator and what came out an abstract term is the idea of a A scale of Hilbert spaces, a cascade of Hilbert spaces, it's one invariant density, the next one, and that's 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 Rodney but it was put in great emphasis by Gelfand and that You have, 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 the...

52:30 It's like a compact operator. Well, by a compact operator. Hilbert-Smith or... Hilbert-Smith or compact, it doesn't matter. And then when you take the inverse limit, you get a nuclear space. Yeah. So the nuclear space in itself... All of these appear just as the inverse limit, but what is the real object? The real object is the scale of Hilbert space. Well, I mean, it's like saying, what you're saying is like saying that a real number is the limit of a unique Cauchy sequence. In fact, there are different models for the same endpoint. Particularly the Sobolev space is, you know, it's quite... You could 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 that the presentation of the group is the group. Sorry. I thought about it in another way, see, the scale of Hilbert space. Does a very strange thing with language because he calls it a generalized Hilbert space. Oh, yes. But it could not be. A single Hilbert space is not an example. Yes, yes. Because if you take the 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 agree with you. I would also like to remind you that fields are very nice, beautiful people and fields are very convenient and for Galois theory they are enough except that of course if you want to define the Galois group you take a field exception but if you want to calculate it you have to use Frobenius and well it's very complicated to calculate explicitly in Galois group.

55:00 And you usually have to rely on the arithmetic property of the Galois group, which means that you are not really working over a field but over the ring of integers or algebraic integers or something like that. So, even in the Galois theory, even in the Galois theory. But, I mean, Stalin's exposition of field theory was the first, I think, really successful axiomatic presentation in algebra. And that serves as a model. And also there was the work of Dedekind and Weber. Well, I mean, all Riemann surfaces, the unification of Riemann surfaces and number, which was the dream of Riemann and which was accomplished by then and which remained a model for two or three generations. And they were, Dedekind and Weber, were they explicitly in fields? Oh yes, oh yes. Close to the ranks? Oh yes, explicitly. Because the word rank came only in 1913. That was called order in the beginning. He's explicitly in a field, but he's also, maybe some people give Connie that a theme, in some way. So, in 55 and 60, the idea of scheme was developed. And it was clear in 1960 that that was the starting point of the new algebraic geometry. As I said, on the other hand, Bobak has already made an enormous effort. To accommodate local range fields, extension and so on, and after the revision proposed by Sayer, which was almost immediately enthusiastically accepted, it was clear that he put things upside down, in a sense, that if you read in chapter after chapter the Bobacki Exposition, the fields come really at the end. In the previous plan, by the way, the field would be prominent everywhere. In Sayer's approach, it was adopted, and the field came as a corollary just very late, so it was already clear that the transition must be, and also the powerful homological method that Sayer and books bound into it.

57:30 So, suppose that... Grottenich was, I think Grottenich knew that he had enough to do and he did not want to spend too much of his time making again the foundation. So he was very happy that Bobaki did it. So the only point was that Bobaki, I mean, at the point, at the time, the Bobaki project was open-ended. Well, every Bobaki project is open-ended by definition. But, so the project was open-ended and we did not decide where to stop. And, of course, for instance, in the third chapter on the commutative algebra of Obaki, of an introduction to the spectrum of a ring by the set of prime ideals of the high-speed topology, the high-speed topology is introduced, doesn't play a very prominent role, but nevertheless it is there. So, the affine schemes are already designed. I mean, it's explicitly, if you have a commutative ring, you can take the collection of prime ideas, you take the collection of corresponding local rings. So, the affine schemes are already designed. At various occasions we consider introducing, and also at some point we develop a graded algebra, which is really the foundation for projective geometry, it's not really the foundation for projective geometry, and from what is proved, purely algebraic terms, the foundation of algebraic geometry can be provided very easily, very easily. And then you speak of dimension, which is really a geometrical notion. But, so we had to divide them a lot. And I think it was beneficial for both. Rotendieck would not have to invest in rewriting all these things, even if they were not exactly up to his taste. But, on the other hand, Bourbaki, this was his open-ended project in commutative algebra, We had other projects, and at the time we wanted to put all our energy on the legos, especially at the end, as I said, because at the time, I mean, if you put together Chevalet, Borel, Sayer, Dix, Croswell, Mice, Evans, all the experts, all the experts, the best experts in book theory were there, so we had to take this opportunity to bring all our efforts together to create a textbook on legos.

1:00:00 So, we were not very eager to continue forever along this. And then, of course, we had just to draw the dividing line, which was achieved by a very peaceful negotiation. It was a very peaceful negotiation. We just discussed freely and peacefully. Okay, we are not interested in pushing that forever. You want to start another project, where is the dividing line? And we decided that as long as it was purely local and could be expressed easily in terms of time, ideas, ideas in modules and so on, that was our job. But as soon as it begins, as soon as you have to glue together, consider proper maps and all things connected with proper maps, which are all these finite things, and cohomology, this is really different. And since we were never able to provide the foundations of an exposition of chief theory, of course, 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 chief theory. We had various projects at that time. And of course this interfered with the development of manifolds, because everyone understood that you cannot do the gluing of manifolds without at least a modicum of shift. And not only the shift of good, but shift per se. The very idea of gluing together local models requires the idea of a shift. A local category, which is more or less a substitute for that, or an abstraction for that, but it became clear that if we wanted to, and if we wanted to, and Rottening came with extensive problems, he said, okay, now my own project for myself is to develop the scheme, but in parallel, we will have Bobaki, and I remember this guy saying, so, schemes here are for me, WTT, Vangemar, it's already there, so it's, oh, come on.

1:02:30 So, I developed the schemes, but you, Bovaki, developed a general method of manifolds of all possible kinds, including local categories, and so on. Everything which was not properly algebraic geometry, but more or less in part. The project was, and of course he got support from Chabaret, but the project appeared so ambitious, so ambitious, that we would have to spend 10 years just working on that. And since Borel always urged, we are the best in the group, we have to do something, he was always putting that. We have to take this historical opportunity, which we may not repeat. It was wise to do so. We have this historical opportunity that we have in the group. We have all the best experts, or many of the best experts, and we have to do something. If we develop at length this ambitious program, maybe we will go somewhere, but we will not be able to do it. Very interesting, very interesting, because I always wondered why Grote Dijk had not taken a somewhat more general... I mean, he sticks only with ranks, you see, I mean, whereas there are other things, not even self-judgment, ambitious, but so he was actually limiting himself drastically in this disagreement. Whenever you talk, I'm amazed because there's an aspect that doesn't come out in the interviews that you've given or in Boyle's article. 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, Maclean's memory is completely wrong. It's not a chaos. No, it's not a chaos. At all. It's not a chaos. It's a volcano. Volcano. I would say it's a volcano. Full of steam and heat. Volcano. And, of course, when we had Serge Lang in the group, you can imagine that it was even more volcanic. A little too much for the taste of many people.

1:05:00 So, that's why I think, but you have to understand that the idea of topos was not in there. No, I know, but still. So, volcanic, I mean, the plans for Bobacky were, of course, Yes, you're right, you're right. I mean, he wanted to be in a special area and he gave the generalization to Bobacki. But topoi were not yet there and topologies and so on were not yet there. They were invented in 63, about 63, 84. But they, Goldenick was already out of Bobacki. So Serge Lang had left. And Serge Lang had left and leaving. This letter of resignation is Ad Maiorem Functori Gloriam, it's written, Ad Maiorem Functori Gloriam. Sorry to be late, we had to affect the change of room. And sometimes people question me, who wrote that? Sometimes the document has been seen by people. Ad Maiorem Functori Gloriam, it was written, it was out of the Penrose Journal. Everyone who has come to know Sertrongi immediately congratulates his time at Sertrongi. Such enthusiasm often reverses itself later. What I remember often said about Sertrongi is that someone is God, no one knows who is God, but Sertrongi is his prophet. And it's interesting that he shifted his allegiance from Weil to Rothenburg almost overnight. In the beginning you had one year of strong allegiance to Atiyah, then to Weil. When he wrote his book about a billion varieties, I mean, it was more or less taken from lectures of Antrim in Chile. It's not, I mean, he acknowledges that.

1:07:30 The methods are very similar to the methods. No one knows who goes there, but Serge-Auguste is more of an expert. In a sense, the turning point for Boubacar was 1960. That was more or less the last appearance of Henri Veyne. He still participated occasionally, but formally he retired at the age of 50, which was 1956. He sent his letter of resignation and there was a dramatic, dramatic meeting. And then, but it's really, he still participated for a few years, but it's only in 60 or 61 that, and not only he, but Cantor and all the so-called founding fathers of Oaxaca. Of course, they were quite reluctant to accept the rule of a very retired age of 50. Well, one can understand why. And there was a bit of fight inside, I mean, the young generation, one to two of them were free-roaming, and so there was a conflict of generation, a conflict of generation. And Doudanais continued. Yeah, and Doudanais continued. But then the conflict of generation was solved because once Lang and Gautam Dikri Tiger left the group and resigned from the group, I mean, then there was no more reason for confrontation. The new, the young generation and the old generation, they had, I mean, they did not want to have a direct confrontation, but both Lang and Grottenby put heat into the oven to continue the controversy. When they left, when they left, things became a little more restrained and the conflict of generation was solved. Much more smoothly. I mean, I think that it made at least part of the expository work, you know, rather almost impossible to, really to achieve. I mean, given the changes that have occurred even in the previous 20 years, and I would guess that the younger generation, I mean, we're very, the older generation is surely aware of that, but I wonder what the, just for a matter of interest, what did the younger generation think the purpose of the Burbanki group was at that point, you see? Since it clearly, it must have changed in some sense, given...

1:10:00 There were many, many discussions about it, and it's something to face that in the, I mean, look at an old edition, you will see that Boubaki was supposedly divided into two parts, structure fondamentale de l'analyse, which is foundation, and then the second, and for a long time it was accepted principle that the first part would be completely organized, linear, and so on, providing all the foundations. I would say advanced algebra and advanced calculus as it is usually taught and the graduate algebra. Logical order, you could rely on, I mean, when you make a quotation in volume 6, you would have to, if you only do volume 5 or volume 4, etc. So, very exacting set of standards. Then, it was considered that the so-called second part should be something different. But much more free, and there was always a discussion whether we could quote outside books. In the first part, it was a strict principle that Boubaki would not quote everyone, except in these footnotes, in these... But, I mean, logically, Boubaki was completely self-contained. But, of course... After already six sets of books, which practically means 30 printed volumes and about 6,000 pages, you cannot continue, you cannot continue.

1:12:30 And it was clear that, I mean, well, I remember the discussion, now we have to jump into the open sea. We are no longer in the Alamo, we have to jump into the open sea. I remember the discussion of that. And so, then we decided, so anyway, the standard to use for the second part, and what should be the purpose of the second part. And of course they were very ambitious projects, I mean, covering number theory, algebraic geometry, complex, I mean, and also at the work of Garton. We can talk about many complex variables and active mathematics of the time. But it was clear that we could not cover everything, so we had to make a selection, and on the other hand, the principle that we would refer only to ourselves was unbearable. So, before we came, we discussed about the origin of the project and how Bobacki and Rotending divided their tasks between, I mean, the local at the one hand and the scheme at the other hand. But then, we had these many ambitious plans. But also we had a Swiss national, Swiss character, Boel, who was always a practical person, who said, Wow, these are dreams, and we will never be able to fulfill them. Be practical, be practical. And so, since C was a motor, I mean, since C... I've been very instrumental in bringing to life this project on competitive algebra and since we had this deal with Grotonik which in effect was a decision to stop, a decision to stop, as I explained, Grotonik did not want to repeat what we had done already which was a great cost and on the other hand we did not want to pursue the project forever and so we came easily to a rather peaceful discussion. Grotonik doesn't want to repeat what it is good. We don't want to go forever. Okay, so just to draw a practical, to make a practical, I mean a borderline agreement, a border agreement or where the border, where to put the border, but peaceful discussion between two, two friendly countries. Peaceful, which is not always so. This time it was, it's fine. Despite the fact that Sèche-Lambert was there, putting the fire, everything was fine.

1:15:00 But then there was this other project. Now the project on Lieh group was supposedly dependent on the project on manifolds. But then again, when Gautenick and Chevalier conspired to propose another norm. And by any reasonable estimate, it would be at least 3,000 pages of foundations of manifolds in differential geometry. Well, if you had this idea that it was time to jump into the open sea, no more. In the swimming pool or in the harbor, I jumped into the open sea, repeating the same process of three or four thousand pages of preparation, foundation for the founding. So what we had was a foundation for the foundation, now it was a proper time to build the foundation. We were very impatient about that. It's clear that if we had followed this trend, the fight was very heavy at the time. If you had followed Cartan, Chevalier, Heidelberg, Rothenberg, and Lang, then we would still be writing this foundation book. It sounds like a project of a scholastic dimension. Yes, scholastic. A generation after generation writing for the common thread. It's almost parallel. That's all. But now, of course, everyone expected that. The goal was to write a textbook or a series of books about these groups and there was place for much innovation there. We had many new ideas. We were eager to bring to the public these new ideas. And on the other hand, I mean, our goal was also to devise Riemannian geometry, differential geometry. Symplectic geometry was not so important at the time. It was only in the 70s that symplectic geometry, and much more now, Poisson geometry, became important.

1:17:30 And so, well, at the time, symplectic geometry was not considered a very serious project because, well, in a sense, Riemannian geometry... The symplectic geometry is too flexible. As we know, locally, all symplectic manifolds, locally, look the same. Riemannian manifolds do not. There are many local invariants in Riemannian geometry. There is no local invariant in symplectic geometry, except that now. We understand better that symplectic geometry was not really the best approach, but Poisson geometry, Poisson geometry, and then Poisson geometry has many local variants, much more suitable, much more flexible, and much more interesting, both from a purely geometrical point of view and for application to various forms. So, and there have been extensive drafts about Riemannian geometry, more or less, I mean, written by Bohel and Kossoul. And eventually they were, I mean, Boyle published his own account, so both Boyle and Cossul published their own accounts separately about Riemannian geometry. And they asked, I mean, in the three years between 1960 and 1960, this project, these projects were going to be here? No, no, no. We left essentially in 1960. Occasionally he would participate, in a personal contact. He was no more a member. And also, he did not participate. Well, maybe occasionally. But the project on LIGUP was already so ambitious. It took so much energy. By the end of the 60s, and that's the time where Boel and Say retired from Boel.

1:20:00 He had read University of Paris 7, a newly created University of Paris 7, and he played a very, he had been vice, not never, well, he's a shadow man. He was never the president, but he has been vice president for 20 years. You know that, you know that. He was never officially the president, but he was the vice president for 20 years. You know that, you know that kind of person. He preferred to be the shadow man. Some people do that. And, well, he devoted a lot of, and also his own project with Stitts. So, he quit the group more. Well, we thought. But then, of course, again, there was, I mean, there was a change of generation. There was a change of generation, and there wasn't only one left from this generation. What's the, I mean, another lecture? At this point, Androides was needing to invent younger people. Uh, the time we hired him. The material? Yeah, the material. But not Gabriel. Well, Gabriel was offered. But the only thing, I mean, I mean, we're going to be providing... Demizer and Gabriel, for example, in group algebraic, which I actually translated some years ago, and Grotendieck's EGA, they're all written in the course of the Burbanki style. It's as if they're new outgrowths of the Burbanki program. No, that's an important point.

1:22:30 That's really what I was also saying. That's an important point. That the influence of Boubaki spread much farther than the limit of the book itself, and that many books, if you will look at the mathematical literature at that time, especially the French mathematical literature, you will see many books inspired by these books, in the same style, relying on the same foundation, and usually written by the same people! I didn't know Dan Mazur was actually an integral part of the team. Oh, no, he was. Well, he was for about ten years. Until he more or less lost interest in mathematics. And in the early 80s, well, that was a political change in France with the election of Mitterrand. And he invested himself very heavily into the new Socialist Party. And he played, well, he was a very... I am not an important member of the party, but a very active member of the party at various levels. First of all, in my own town, my small town, there was a deputy mayor at the time. And I was only a deputy member of the council, so not yet a full member of the council, according to the French rule. The French rule is that for municipal elections, this is a good rule, I think. We elect people on a ticket, and according to a proportional system. And so, let's say, we have a ticket of 30 people. The first 15 members, after the election, according to the first 15 members, the people from number one to number 15, are appointed full members of the city council. But the rest is just, I mean, a reserve. And so the idea is that every term of six years is a long term. In practice, It happens often that people die before the end of the term. They retire, they move to another place. There are many reasons why the people don't compete six years later. So the idea is that we don't want to have all the time elections, election and elections. So, on a ticket, you have two categories of members. One was being elected, and the other was the supply. And I was in the second part. So, it depends on the local conditions. Sometimes it's purely formal. It's like the vice president. As long as you are not promoted, you can be automatically at some point promoted to a full member.

1:25:00 But in other places, the mayor wants to extend his influence and to have better advice and so on. So he would include the deputy members, at least informally, in discussions without voting rights. It's usually so. We are included in discussion groups but without voting rights. If, by chance, you have promoted a full member, we know already about the job. So I think any wise man would do so, any wise man would do so. So, just to explain that de Maizure was very much involved in politics, both at the local level and at the national level. At the national level he was, one, he was a close advisor to Chevenmont at the time, who was a leader of one of the factions within the party. Maybe not the right one, but never the same. It's not another question. So, de Mazur was very, very involved in both local politics and national politics. He became one of the assistants to Chauvin-Mont when Chauvin-Mont was Minister of Education. And then, in scientific terms, at that time he was appointed a professor at École Polytechnique. And so, his mathematical life was more or less devoted to the teaching at École Polytechnique. So both politics. So just to explain that more or less. Then at about the same time, Verdi died too early. Verdi died too early. He was 44, something like that, and he died in a crash accident. He had a summer house somewhere in the mountains, not south of Damaville. And I remember the road to access to his place was quite dangerous. Oh, yes, quite dangerous. It's near Saint-Jean, Saint-Jean-de-Gare. I've been visiting him from time to time and he always advised me, be careful, the road is difficult and dangerous. And some evening, with his wife, we came from a party by some friends.

1:27:30 They missed, they missed the bridge, jumped over the bridge, and by bad luck, usually in the summer it's dry, but this time there have been thunderstorms before, and there was a small lake, left from the thunderstorm, usually it's dry in the summer, and the cald was upside down, like what happened to Shapashidi, the dead Canadian. We hope that Ted Kennedy was able to rescue himself, not his girlfriend, but he rescued himself, but Verdi was not able to rescue him. I suppose they were both wrong. We hope that they had a blow on the head before being drawn. We hope so, but we don't know. And, I mean, there was someone in the neighborhood who heard a noise. It was around midnight. Someone in the neighborhood heard a noise, immediately called on support from the firemen, but they came immediately. Within half an hour, the firemen were there, the rescue squad was there, and the rescue squad stayed for the whole night. You drive around there and you're on these cliffs and there's just a little... And that wouldn't be so bad except that the wall is broken through in many places. Yes, exactly, exactly, exactly, exactly, exactly. When was this? What year? Uh, 89. And just to explain why, I think there was never a new genie. After, after Goethe and Dick, Sayre, Borel, etc. left, there was never any really active new genie. For various reasons. And there was an accumulation of... Of three different reasons, I mean, the scientific climate, the involvement of everyone in the reconstruction of the academic system after 68, the change in mood, the change of political mood, the zeitgeist, that's why in 83, I advocate Bourbaki to decide that we won't stop in the vocab.

1:30:00 The divorce occurred. Not yet a plan for... schemes were there, but not the topology. And it's another step, which was taken independently of Bourbaki, except that the major actors, which were De Mazeux and Verdier, were incorporated in the book. But they never tried to eliminate them, never tried, both Verdier and... And de Mazur never tried to, I mean, were very clear of the bottom, of the division of work. And what they had been known is that they knew it was different from what they were doing for a moment. But on the other hand, I mean, Grottenley was, I mean, Grottenley was aware that they joined Womacki and made nothing again, etc. At the same time, they went to the IHS. The operation of the IHS started in 1958, but the campus where we are now was created in 1962, and in 1958 Motschan was a very strange person. A person who had an ambition in science, but was successful in business, and at the end of his life he said, OK, all the money I have earned I can put it in science, and maybe I'm not able to make a name in science by my scientific contribution, but I will make a name in science by creating this new institute.

1:32:30 He had some claims of some deep insights in the Sousslin problem. Yes, now he worked for years in the Sousslin problem, but of course after the work of Coyne... No, Solovey. Solovey, Coyne, and so on. Tenet and Baum. Yes, Tenet and Baum. But he eventually defended the thesis. But there was no bar in allowing him to defend the thesis. But Molchan created this institute out of nothing. And out of bluff. Out of bluff! And really out of bluff. And so, but in the beginning, the Goethe-Nicks seminar started in 1958, and they are not yet their own place. So, they are the foundation de France. Fondation de France is a cover organization for many foundations. And then they have near Gare Saint-Nazaire, there is a building which belongs to this, and they give some fellowship to some students. Every year they give about ten fellowships, usually to Normaniens, and for two or three years, in order to start their PhD, at least in the old family. Thiers is a former president in the 19th century and the Fondation Thiers. I think that Thiers, after retiring from political life, created a kind of, how is it called, the French house in Rome for artists, Villa Medici. It's more or less the same idea. To give a fellowship, a residence fellowship, to a few young people, promising young people, whether in art, in literature, not often in science, but sometimes in science, most often in literature, it's a kind of small villa medici in Paris, and they give the same thing, fellowship, residence fellowship.

1:35:00 And so, but then, I mean, the Grotonik seminar was hosted in this place, I suppose due to social relations, of course. But IHC did not have a campus. And in 1962, for a few years, I mean, a mutual maneuver, until he discovered that in Bure there was an estate which was, oh, it's made from the war. Warner have been too friendly to the Nazis. They escaped to Argentina. And so, I mean, there are no... The French state was, I mean, taken by the state, I mean, put sous-cellé, which means the state takes control, but then, in the 60s, the French state was impatient to get rid of Saint-Exix, and of course, I mean, the people have definitely disappeared or were dead, and it was time to, and De Gaulle, De Gaulle wanted, I mean, to erase all scars from the war, these were scars from the war, and De Gaulle was at the time... And he wanted to hear. So the state was ready. Bochan, using his social relations, knew how to buy it for people. It has been said, sometimes people say that, but it has been said to me by witnesses, that he knew that there was some zoning board restriction. That a piece of the land should be, I mean there were plans to expand the highway system, and that a piece of the land should be taken when the expansion of the highway system would take place. There's a term for it, but I can't remember what it is, yeah.

1:37:30 And it has been said, which I swear not by, as I know we say in French, on a pret-covish. Okay, it has been said that... Molchan knew that when the time would come for this highway project to be completed, the state would pay a compensation. That the compensation would be twice as much as the original price, or at least pay back for the original price. I don't know whether he met some, but they would have to speak to each other. It's a lot of business, good business. Okay, so then the institute moved in 1962 there, and the Goethe-Mixeminer moved. And at that time, there were, in the original property, there were three buildings. There were the so-called commons, There was a common, there was a main building, and there was a so-called Pavilion de Musique. And there was also the tower which was in the neighborhood called the Lover's Tower. But then the main mansion has been destroyed in June 1944. This area was heavily bombed in June 1944 by the U.S. Air Force. The U.S. Air Force, I mean, they were heavy bombing in the south of Paris. And many casualties, and many deaths. And so, the main building was destroyed by a blow from the U.S. Air Force. And that's where now we have the main parking place. And when the gardener, I mean, the caretaker of the gardener of the park, I mean, the do-do-do. There is a hole. He finds the remnant of the building. So, and then there was some additions, so-called provisional building, office building.

1:40:00 And until the reconception of the past few years, I mean, that was still the existing place. Which means that for five years we had... We lived on a building site for five years. So, yes, and because formally, I mean, the seminar was run by Eugenie Engholt and I attended every meeting and the two... No, because at the time, well, first of all, the three years I had to spend in the army, three years in the army was a long time. At the time of the war in Algeria, it was another set of equalities, not for four years. And then I moved to Strasbourg. So I would come once a month, usually, once a month. So the SGA must have been one of the first seminars held at that location. Yes, exactly. Therefore it's called Lebois Marie. Yes, that was the name of the estate before. No one knows anymore about Lebois Marie. No one refers anymore to this place as Lebois Marie. We had some dedication and I insisted that we should invite a lady. Do you know this history by David O'Banth that he wrote at Princeton? Yes, yes. What do you think? Do you have any thoughts on it? I think it's all right. I mean, he got information from, he had first access, he had direct access to much information.

1:42:30 And what he wrote is very, very scholarly. It's impressive, but in fact I'm not asking. I started the seminar of Tom, because when Tom joined the Institute of History, I suppose, then he started his own seminar. It was one of the successes. They weren't parallel for many years, despite the fact that they would hardly exchange a word. I mean, Tom and Gordon would hardly exchange a word except to pass a glass during lunch, but except for that, they would hardly speak to each other. He says that he was not interested in formal mathematics and that the multi-tweeting mathematics is why O'Donoghue is making the irritations go back to 1 to 12. He may have exaggerated, but he said much more extreme than Tom. He had these colleagues who actually studied Lie groups and functional analysis and all these things, but he doesn't need any of that. He doesn't need any of that. Tom, Tom. Tom, because it falls directly from the sky he said. Oh! Well, I mean, what has happened, I have, well, I joined IHS in 71 and already it was no more there. And Tom was in his last phase, already in his last phase, and I attended occasionally his seminar, but usually I was disappointed. He exaggerated his time. Over the years he exaggerated more and more.

1:45:00 He also had a pretence to be able, by his own method, his own insight, to speak about all possible sciences. So he began with biology and embryology. At some point he moved to plate tectonics and he gave a series of lectures about plate tectonics. It was a disaster, a disaster because he did not take time to learn the basic facts about it. I suppose the plates sometimes they come to a line. Yes. I remember his explanation that the earth was flat and the radial surface nothing deep. And of course, plate tectonics is at such a global scale that you have to take into account the curvature of the Earth. And on the other hand, it's known that you don't understand the subduction phenomenon. Subduction phenomenon is a very important thing, a very important part in this model. And that is, one plate goes under another plate. And really, I mean, the region... I was mentioning, I have a friend watching a piece of the California crust which has almost reached the core. It dove down and it's just approaching the core. It will get there in like 10 million years. Okay, okay. But it's closer to the way there now. Do you think Tom was isolated in some way in French mathematics? I mean, since most of Chang certainly was in French mathematics, he's very much to be alone. Well, let's let them go brush these lines. Tom Farmer claimed later to be a rather beleaguered figure. He said he was later on claiming he was a philosopher. He wasn't really part of the trend, the formalist trend that was prevailed in... Ignore of the Italian mathematics trend either. No, no, sure. But I mean, he seems to have become rather isolated at least. Isolated from both. Yeah, yeah. That was my impression. But to be isolated at the IHES, you know. Now, I would say, I would say there was not much praise by the mathematicians, by his fellow mathematicians.

1:47:30 But he had a group of, who are usually not mathematicians, a few mathematicians, but also many non-mathematicians. And he had a high praise in the public at large. I mean, at the same time, he had... Mathematically, he would consider it his own reservation, but on the other hand, he acquired a name in the general public. His name was known everywhere. I observed that people, many people, he wasn't isolated, many people came to see him. I meant within the mathematical community. But to the extent that I could observe, these people were journalists, they were philosophers, historians, in short, ideological propagandists of one kind or another, rather than... And that's where I was different with Aubin's dissertation because he sees this confrontation between Grothendieck and Thom and he writes it as how Thom really had the vision of mathematics that would carry on and Grothendieck had this sort of, you know, it was a dead end. Really he says that? Well, not that sharply, but that's the direction he wants to take. No, of course, Aubin is a supporter of Tom. I have my own reservations. When it comes to the fact, he's quite accurate. When he describes the fact, he's quite accurate. But in his judgment, I don't agree with him in his judgment. He's really a supporter of Tom and he claims that the main asset of IHS was Tom. And he says enough of the facts of Tom's work that I think you could conclude this wasn't the future. He does say it, but it's not his direction in saying it. Which book is this? I don't think I know. Oh, it's a dissertation. It was never published. A Princeton dissertation in history of science. But it's good scholarly work. It's good. It's terrific. Can't one get held of it? Yeah. The Microfilm Center? Yeah. I wrote to him and I said, first of all, you need to make it two books. You need to make it a Grote and Dick Tom book and then a Ruel book. Because the whole last half of it is on Ruel. But they were both good enough work. Could you write down the title? But also, just to finish, since you mentioned Ruel, that's right.

1:50:00 I mean, you have to also include all of them. Very, very good experts. He had this bright idea which came more or less in contrast to the idea of Randall and Kolmogorov, but which was, I think, in itself the ideal one. It was just, I mean, I think, a rebirth of the ideas of Adamach, chaos in deterministic chaos. And there was at the time that was, but the subject developed very well and now it's very good. Not only at a theoretical level, but practical application in self-formation and so on. It's a good model, it's a good model. And of course, there was a propaganda of Benoit Van der Waal, because there were factors and so on. But the idea, I mean, basic ideas are very sound and still very food for thought, very serious of physics. Ruel himself, I mean, I think, became more and more, I mean, first of all, he got some fame. If you want to be known to the general public, you have to invent a few catchwords, and he was good at inventing a few catchwords, but unfortunately he became more and more isolated, so he continued for many years to publish some contributions.

1:52:30 Very technical, very good at the technical level, but there was a contradiction between, I mean, his fame outside the circle of mathematicians and his production. His production was still good, I mean, no doubt, I will not claim that it was not good. But the point is that he worked in more and more isolation, more and more isolation. And not only isolated, but I mean, he doesn't want to, to, to... He has developed a kind of path now of disappearance, more or less a drama in a dramatic way. So, is there something, I mean, it's clear that there is a common thread to what, I mean, in a sense there was a psychological breakdown in all three. The drama of Gotenick has been highly publicized and Tom is less, but even so, I mean, at the end of his life, Tom was quite isolated and quite paranoid, quite paranoid, and also, but he became, you know, the last five years he was out of his mind, really out of his mind, but when he was there, he would be nice to me. He would not follow anymore and he would not recognize people and so on, but he would forget about the time, but he would come back. Well, I think it's less publicized and is no more a public figure, but I see him almost every day and at least I see him physically. I don't think we discuss it every day, but we see each other every day when he comes to collect his name.

1:55:00 But it's a similar tragedy. Is there something which explains that? That's three people who contribute to really new ideas to science. Each of them really contribute to a new idea, despite maybe you can be antagonized by the heavy propaganda about catastrophe, but there was a new idea, there was a new idea, obviously. Jorgensen gave me the explanation. Of course, it applies to himself too, perhaps, but in particular with regard to Tom, since he's a very special man. Because of Deligny, hiring Deligny, he said that there is this, basically there's a state-organized machine for transforming geniuses into gods, and it's so powerful that they may falter and believe it themselves. Once they believe it themselves, they're lost. This is his, roughly his explanation of Tom. You know, because Tom was not always crazy. In the 50s he did very excellent mathematical work, still very intelligent, and then into the 60s he was suddenly this public figure, you see, as well, a public figure who was allowed to make very irresponsible statements to national TV, and he was praised for that, you know, that kind of thing, because he believed it himself, so Grotendieck was saying, well... I was attacked by this state-sponsored machine, but I didn't, I believed it at first, then I didn't quite believe it again, but then, after that, we know what happened after that. But there is also something different. This work of Tom in the 50s, his work in topology, we know that he had the insight, he had the imagination, but Cato asked him to rewrite his thesis.

1:57:30 So Cator was the moderator. Cator was the one who could transform wild ideas into written mathematics. And when Cator was his thesis advisor behind him and forced him to rewrite things and so on, and there also Cator played this role to control the fantasy and the imagination of time. And then when Thomas acquired his own fame, he was no more under the control of anyone. Fortunately for Grotendieck, for about ten years, as I said already, I mean there was this team of three fantastic personalities, de Jonay, Sayer, and Grotendieck. And there is much evidence from the correspondence between Sayer and Grotendieck. And the hard-working capacity of Doudouny made the success of them. Otherwise, maybe left to himself, Gautami would not have been that successful. This really, the real success came from the combination of three outstanding and very different persons. And what is interesting is that they could cooperate, they collaborate, despite the fact that they were... Yes. They somehow exploded. And you can see, I mean, when Gotendijk retreated to Montpellier, I mean, I visited him from time to time. I mean, obviously, he was lacking something. He still had the imagination. He still had the impulse to do mathematics from time to time. But he had a few students, but there was no milieu. I mean, Montpellier University is... It's a good university, but it's not the first place. I mean, I don't mean, I would not insult my, I mean, infuriate my colleagues there, but they are good mathematicians, they are honest people, they do their jobs, they teach, they write papers, but it's not the first place. And Grotendieck was, I think, Grotendieck needed a milieu to nurture him, to nurture him. And maybe the admiration, maybe the admiration. And in Montpellier he was considered, people were a little unhappy about him because he was a little too exotic, I mean, when it came to give, to deliver routine, undergraduate lectures, he was of no help, of course, he was of no help, when every time he tried that it was a disaster, of course he had fantastic ideas, I mean, how to teach geometry and so on, but of course it did not work because, wait, let's say we...

2:00:00 In a first-rate, in an elite school, like Harvard-type school or Econormal School-type, I mean, in a first-rate university with first-rate students, you can be adventurous and imaginative, but with a very, very ordinary undergraduate, you cannot do that. And so Gothenburg was of no help, of no help. And the people, well, I mean, they had their duties, and they had to organize an undergraduate teaching for large classes and so on, and it was no help. It did seem a strange place, although I know he started there, he was an undergraduate there, but it seemed kind of a strange place for him to go back to in any, you know, well, I don't know, at least that was my feeling. People go to Las Vegas. They go to the University of Nevada in Las Vegas, light teaching load, buy a nice house 20 miles out of town. Yes, Montpellier, because he graduated there and he lived his youth in the neighborhood and he's still in the neighborhood, not far. I know why, because he's a Pomano to this crowd, you know. So, but you are, well, about state marching, you are something, but a state marching promoted Goldenick and then, I mean, as soon as you follow the established trends, I mean, the marching will promote you. But as soon as you come in the transverse way, let's say, for instance, why did Goten go to it? When he left IHS in Renko, there was first, I mean, there was a debate about why there was some modest founding from the defense department. There was, in IHS, some modest founding from some, I mean, some office of the...

2:02:30 And I think it was just a personal relation of the Minister of Defense at the time with Olshan. Well, this was a modest, modest contribution of the idea. Well, if you believe in conspiracy, sometimes I do. No, not too much, but sometimes I do. Conspiracy, yes. Conspiracy. I think the idea has been sometimes advanced. A number of prominent colleagues disappeared from religious universities in the United States. and any good German intelligence service. Would realize that Mr. Bater is no more in the corner, Mr. Fermi is no more there, and any good intelligence service, any intelligent intelligence service, which is not always so, would realize that, okay, you draw a map, you draw a map of the state and the state university, American university, and you see, okay, this man was there, he disappeared, this man was there, he disappeared, where are they? And you compare, you say, of course they are winning, they are coming back. The idea has been that, don't take it too seriously, that at some point NATO was afraid that some similar occasion could occur where all the Western nations would have to mobilize its scientists for a new world effort. And if you just draw the people from various universities, it would be visible to the Soviet intelligence. Let's have a network of institute. Let's say the initial motivation for Oberwolfach, according to what we know now, was a secret research institute sponsored by the German Air Force for fluid dynamics to help in their program of organs, L1B2. Okay, and if you believe in the conspiracy theory, which I don't, but... At least not too much. You can draw a map of Europe with various institutions which could be sheltered for secret research.

2:05:00 It has been suggested that IHS was part of this. And that's the reason why they got funding from it. I have no evidence, there is no historical evidence, but it's a little paranoid. There is no historical evidence, but it makes some sense. And certainly there was a general trend for even for the high-level research to be dispersed into more and more state universities in the Midwest. So, and I think, well, Grottening is not explicit about that. Maybe he felt something like that. I think Grottening was too naive in political terms to imagine that kind of concept. But when he discovered that there was some military from New York... I think it was a little too naive politically to make this step in imagination and say, well, after all, what does it mean, etc. It just took the first evidence. I refuse to retire from it. And then the confrontation with Mo-Chan, but Mo-Chan was too clever, I mean, too... He knew how to resist provocation. Well, Goldschmidt had his own temper, but he knew, he was a very clever person. He knew how to, I mean, to provoke, I mean, how to cope with provocation. And Gottlieb thought, I mean, when he felt that the cause would be to lose Gottlieb, of course he, and he said, of course, I will refuse this. Which was very more in terms of the budget, and it could easily be. This is the military thing, right? Yes, yes. And I said that, well, I explained it. If you believe in the conspiracy, I explained that. But Goldenig was too naive politically even to imagine such a thing. He just discovered the bare fact that there was, on the budget line of the institute, there was some money coming from the Department of Defense.

2:07:30 It was called at the time D-R-A-T, Direction des Recherches et Etudes Techniques, something like that. Well, it's a non-office in the Department of Defense. And they support some research, not very much, but some research in the industry and in the university. But Grottening was too naive to really understand the meaning of this. If there was a deeper meaning, he did not realize. So he went to a direct confrontation with Mo Chan. And Mo Chan was clever enough, well, of course, he was temperamental. He liked confrontation, but he was clever enough not to antagonize his goal with confrontation. So he was ready to yield and say, OK, if you are not happy, I will send back this money to you. Well, sorry, what exactly what year was this? Ah, yes, yes. Long before I met him, yeah. That. Yeah, we're close. It's interesting. But the rest of the faculty was a little hesitant. The rest of the faculty was a little hesitant. They were hesitant to the extent for Bouchard or for Godin. They wanted a piece. They wanted a piece. The rest of the faculty wanted a piece. Five years. A truce was signed and the faculty urged Moshan to give and to send back this one to achieve a peace in the city. They did not want a confrontation in the city. So, everything seemed to be okay. And then Gotendi came to this ICM-70 which was in Nice. And the local president was Diodonnet and Lorenzo. Lorenzo and Diodonnet were local for the organization. And then there was this confrontation between Gautenich and Diodonnet during the conference. This confrontation which ended with Gautenich wanting to be put under arrest.

2:10:00 And there was some incident that was there when the chief of police of the town came. Nasty discussion. The chief of police was a clever person, and he knew that if he arrested this world star of science... But was the couple with Giordani over military spending? I'm sorry? Was this couple with Giordani over military... Well, Giordani was no more in the faculty of IHES. He had moved to Nice. He was out of that. But he was about that general... In 1965, Giordani moved to Nice to be the first president of a newly created university. But then there was a confrontation between Gautam Nick and Giordani at the ICM, which had nothing to do with this question of military affairs. Ah, something independent. Something independent. What is propagandamaterial for survival and what is material for survival and what is material for survival and what is material for survival and what is material for survival and what is material for survival He eagerly wanted to be put under arrest, so he did all the provocation he could imagine to be put under arrest, but the chief of police of the town, who was a clever person, knew that if you put under arrest a world-star in science, you're doing the World Congress of Mathematicians! That would be a political disaster. And he had instructions from the mayor not to yield to confrontation, to provocation. You were there, weren't you? Indeed, I had my own. He tried to give a point of interest here about achieve low cycles, about what you were on. So it wouldn't say on the street, but there is a connection to that. It's a formal poll that's right. Goethe was on the street, within the responsibility of the city. And if he accepted to retreat, he would be under the responsibility of the city.

2:12:30 But, well, the whole thing was very ridiculous. The whole thing was very ridiculous, except that if the deep meaning that Grottenig wanted to be Martin. Grottenig wanted to be Martin, and the local authority were clever enough not to yield to that. Not to yield to that. Not to satisfy his desire. Well, I mean, even though this is very interesting, and I was also involved in it, what I would like would be back to mathematics. Yeah, okay. Sorry. No, no, no, please, it has been extremely interesting. It really has, but I think if we can come to the issue of the SGA, it's been merged around... Well, I don't think I merged those things myself, because I was in Samoa at the time, I was not a regular attendee of the seminar, and I've seen the outcome, but I followed that from a distance. But I do have inspiration from some of these papers, if you don't mind what it is. Yeah, but about the topos, I mean, Giraud, I guess, through some process that I don't know, they had arrived at this set of actions which Giraud presented in May 1963. So this was, in a sense, the public birth of toposes and sites and so forth. It was in May 1963. It's that early, 63. Oh, yeah. Yeah, which is, I pointed out, it's the same year I wrote my thesis.

2:15:00 It's the same year that, almost the same year that Robinson came. That's right, that's right. Non-centered analysis. Kripke. Kripke. I'll give you a small spreadsheet. All of these are probably part of some... even if they are not aware of each other. I know Miyabi and his homology kind of do this. But when Giroud presented the topos axioms, or when Giroud and Grosvenor were thinking about schemes, he says it was simultaneous with schemes, because it was the idea of covering. The original form was the notion of covering. Actually in the later edition of EGA he refers to Kehler. Even though that isn't quoted, it probably played more of a role than Andre Bey would have liked. Because, I mean, it's not just covering, it's the idea that you can have infinitesimal isomorphisms that complex analysis would give rise to by some implicit function theorem, but do not. And so with that whole amazing necessity for generalized space, which is... I mean, the site of all the etal maps to a given space, it's not the lattice of open sets, so it's really a generalized space in that sense. We've had a lot of debates about exactly what is generalized space, how general is it, how special is it, and so forth. So this is, interestingly enough, it's only recently that Colin actually wrote down the proof of the key fact that...

2:17:30 The Petit et al.'s site is indeed an example of a generalized space according to some appropriate definition. The obvious motivation was Riemann surfaces. I mean, the very idea of Riemann, that the function, multivariate function, is a function of some et al. I will recall it now, et al. bar. And at that time, I mean, people, I mean, especially in the school of Grauert and Rehmert and Benke and Carton were concerned about what is true for many competencies and the very idea of just time manifold was to try to axiomatize. But something to do with Goff-Tate in Galois theory, his work on profanity, I mean, cohomology of profanity, and profanity, so he saw that when you deal with the cohomology of profanity, you really have a tower of finite quotient. Since Gottenbeek, I think Gottenbeek wanted to devise cohomology in such a way that suitable Galois group would act on that, so he saw, by combining this idea of Riemann surface, multi-sheeted. Function defined on the multi-sheet. And the tower of finite group defining and the cohomology and the way, the way of dictating. But there was also another motivation came from the work of Serre in the Chevalier seminar. When Serre realized that if you have, well in topology it's a familiar idea, is that you have a group, a Lie group and a subgroup, then you have a final bundle.

2:20:00 Where the base is the homogeneous base and the fiber is the subcrop. But this one, this has been very developed by Le Serre, Carton. But the crucial property is that when you have G map over G over H, this is locally treated. And when you want to explain the technology by groups, Serre knew that, Zayaski topology with the explicit purpose of mimicking the method of differential geometry to algebraic geometry introduced in 501 of what his Chicago notes are the first introduction of algebraic 501. But Selye immediately realized, out of his own experience with Liebhoop and his joint work with Boel, that if you wanted to transfer this kind of knowledge to Algebra and Goethe, it did not help. Because the definition of a principal bundle, a low-collectivity principal bundle that you get, is not satisfying. It's a rather extended class of fiber bundles. Yes, exactly, exactly. So, as long as people were interested in vector boundaries that did not appear, because they are locally trivial, but then when you have an algebraic group and an homogeneous space, it doesn't work anymore. The paper by Atiyah, which was also used by Chauvelin, is that if you have, let's say, if you have G mapped over the homogeneous space, you don't have a local cross-section, but you have a multivariate, a multisheet.

2:22:30 You don't have the implicit function theorem in algebraic geometry. That's the point. That's the point. That's the main difficulty. Moving all these considerations from the analytic point of view. How do you... Well, it was all right. Well, it was no theorem. It doesn't function. It functions with power series. And that's why people introduce, you know, completion, you know, to deal with... We would say Taylor expansion. If you deal with Taylor expansion, the implicit function theorem is perfect, even in algebraic geometry. But at the cost of introducing not really algebraic functions, but formal power series, which do not represent really algebraic functions. So, Sayer knew that, and he made an effort to imagine a system. Not a single value cross-section, but a multi-sheeted cross-section, and then he developed that idea in one or two papers, and then Grotendie immediately called Aminger, and he immediately said, well, I take Riemann surface, I take this idea, and I take date calculation, it seems that within ten minutes he had the idea of it, but then he had to develop it. But I think the initial motivation... I mean, Serre works in terms of isotrivial fiber bundles, not covers. I'm taking it that the people who were here with the seminar, it wasn't a big jump from fiber bundles to covers, to see that these are tools for the same kind of... Serre did not introduce systematically. Serre looked at these special types of fiber bundles. We have this grimy fiber, we have this multi-sheet, but covering.

2:25:00 But he did not introduce the set of all possible coverings. The idea that you should consider the class of all a total map over a given scheme is really important. It never occurred to me, but it's really important. And the idea that you should consider the technology used in carving, this is really important. But it came very, very quickly. The whole development has been very dramatically fast. I mean, if you follow from 55 to 60, you can see that all the major ideas have been obtained, everything found. It was fantastic. So the notion of covering, you could probably call it, emerged then before the notion of topos. Of course. How did the notion of topos emerge? I mean, is there a clear way of describing how the notion of topos emerged from the notion of covering, or was there some other input from... Well, presumably there was a category of sheaves, but it's an interesting question I've never, something I've never quite understood. A process of freeing oneself from particular presentations and making explicit the invariant essence of them. Which Sammy had already been doing. We need a more general theory of patching, which we will now apply.

2:27:30 So Grotendieck says in Recall a Seminar, of course this is a memory somewhat later, but what he says is that the idea of topos was born simultaneously with the idea of Scheme, but the Schemes were articulated to some extent. So then in some way the notion, the idea of covering, enabled, if you like, this... He sees he's doing some kind of fairly new and general notion of coverage for schemes and he quickly sees this is not special to schemes, I think. But the details of how is never made clear. I mean, I recall, but I didn't really understand his focus, which I heard in 1971 when he gave at this Muldoon meeting. But I do recall that Grotendieck spent most of his time talking about sites, rather than the topos as such. It seems that most of the actual calculations and the nitty-gritty of it was actually at the level of a site, and the topos was a kind of space which enabled you to see the whole thing as a whole. The arena within which all this stuff's really going on, right? I mean, as I say, you have the same phenomenon of commuting of rings with groups, everything else. You have the invariant notion, which is helpful for certain general constructs. But to get at a particular case, you need presentations. Unnatural presentations, in a way, but then you can calculate them. On the other hand, it does say in SGA somewhere that... The solution to a problem may be just to construct the appropriate topos. I think it's an interesting statement, you see, because, well, dramatically... I think there's some content to it. Once you've constructed it, you've already got lots of things because there's so much involved in constructing a particular example that you're simply, you're really encoding the solution at the same time as you're giving the invariant statement of the problem. The way I've put it sometimes, I don't know what you would think about this, is that you create a universe in which virtually all that exists is the problem you want to solve. We'll stand out here because virtually all there is is a problem, a contradiction between problem and solutions, and they're resolved in the medical speeches about university terms, and they connect with heuristic perspectives.