Afternoon Discussions, incl. FW Lawvere, Leo Corry, Angus MacIntyre, John L Bell, Colin McLarty (contd.)

0:00 ...in some sense to the 19th century. And they have some of them. There are certain uniformities you can detect more regularly than you can in this tremendously long inductor of proof of hermeneutics. I mean, there's one interesting thing there that, I mean, hermeneutics, I didn't spell it out, you probably knew it. I don't know if anyone spelled it out. You can detect uniformities of hermeneutics by pure logic after the fact. ...that varieties in characteristic zero can be de-singularized by the kind of process that you gain, a finite sequence of know-hows, then you can just use a compactness theorem of logic to prove even results in characteristic p. You can show that if somebody gives you a certain complexity of a variety, a certain kind of definition... Then, in all the trinity-many characteristics, you'll succeed in de-singularizing. Now, in some sense, one can, by a metamathematical argument, get out directly of the theorem. It's a kind of a lecture, that's why. At any given moment in characteristic zero, provided you're not using topology, you can really only be in trinity-many. So there, I mean, that's really the test. I mean, that's the thing that wouldn't accept us the hardest, and that is citation for a singularity. I think he said that explicitly at Nice in his citation for Hironaka. He certainly is incredibly competent. The algebraic geometrists could use it immediately. In Inge he used it. Sometimes they can in Inge get away. But he makes tremendous simplifications in algebraic geometry, because an average variety of units can lose the singularities, but there's enough factoriality in both intersection theory and non-singular to do. You can usually descend again to get some kind of answer. Does it have any parallels in question?

2:30 Again, thinking of the parallel of the development of algebra and geometry and number theory. Well, I mean, of course. I know, it's hard to separate something. Well, I mean, the characteristic P thing, certainly, if ever proved, will. Well, even in the integration theory, for example, there was, in Pierre again, you have, they're really, some, they've been studied by, and you, the solution is more p to the n, and then you form a generator, so you get this number a, and there's t to the n, and these things turn out generally to be rational functions, in simple cases like integrating down your objective state. First, the rationale is first proved by, similarly, at the way of saying it, which I mean, it's tremendously powerful, because you immediately, you know, if you're integrating some of that, you know you can sort of change variables and get into something very nice. There's powers of your variables in this, and then you can come back. It is fantastically powerful. But as far as arithmetic is concerned, well, it would be, it's only really going to be, I mean, several cases of the Bay conjectures for affine varieties of single varieties would probably be.

5:00 Something like in Delignan's program? Oh yes, certainly, this is relevant. This thing of Fujiwara's that I mentioned, the conjecture of Delignan, Delignan devised the conjecture, not that Delignan had this intuition about contracting maths, but Delignan knew that for the information they wanted in representation theory, because of the tremendous power of the eight conjectures, there was a sum of powers of the idea. If you twisted by Frobenius got the answer because you could come back using Hilbert's inequality. So this guy Fujiwara did this but it was radically new to me. But it had been done earlier by Pink using the unproved proof. So and that relates directly to the language. It's in the language for you. Questionably the theorem, it's an obvious condition, it's fantastically difficult. I'm sorry that I've been told about halting the theorem. I don't know how... I understand his original proofs state how cohomology, among many other things, and has since been brought to the level of Hartshorn. Approximation means using coherent cohomology. Well, that's probably true. Yeah, in fact. Yes, yes, yes, yes. That's very sophisticated. Yeah, no, that is actually true. So still extremely sophisticated. Yeah. But I mean, and they're more, the point is they're motivated more by it. Taking norm techniques in diaphragmatic approximation and translating them into the language of, you know, vector bundles and so on, and then positivity results.

7:30 So it's an intersection theory, but it has to be an intersection theory in this arithmetic, algebra, and geometry sense. So notice it's a different intersection theory we've been talking about. It's one where you try to take into account the intersection theory of your spaces, something like the primes and points at infinity. But it's based on the same, there's a model for them, there's a common, in some sense, a common generalization. There's a Riemann-Roch theorem in this situation, and so on. These are figures, also. It's probably true to say that the cohomology has been stripped out of this to quite a large extent, but there is, maybe more, here the two unknown intersections. I would say that's probably the case. And there's no ambiguity for your, the self-cohomology, if you go a little bit, yeah, you do cohomology for the rest. For general sheaves, it wouldn't work well. For coherent sheaves, it does. It gives the long exact sequences you want. And Peary used that term. I've been faulted for using it by referees, but it's been used on the Oxford Tri-Post, it's used in several papers. Yeah, yeah, yeah. And the word is not in hardship. No, and I was surprised. But there's two things. Hartschorn develops the actual theory of this cohomology. And when you say brought to the level of Hartschorn, you could mean it really uses all of this cohomology. But lots of applications of coherent cohomology are really just calculating one, two, or three groups. And you can really do them in terms of resolutions, so the theory is there to guide you, but in some sense you know how to do it without. You could have given this proof without talking about derived functors. Other times, well no, you really, you had to use the derived functor apparatus. And I just, when they say this about functings, I don't know where they're even claiming it lands. You see this in the book on Fermat's Last Theorem, the 650-page book out of that Boston meeting where they explain all the paragraphs.

10:00 They have a chapter on Galois cohomology, and they say, look, this is the derived function of cohomology, and here are some things that tells you. But the truth is, we only need it in dimensions 0, 1, and 2, where we could have done it directly in terms of... So, in a sense, they're using to draw a functor perspective. It does organize their thinking and they tell you what should organize yours. On the other hand, they also tell you in the same graph that you didn't need to know that. Yeah, no, that I think is clear. What about the difference there between commutative and non-commutative cases? Is it a point at all? You mean in terms of geometry? Well, this is a bit, yeah, I mean, this is a point. I think... Yes, I think everything has changed on both sides of the equation. I mean, the ideal theory remains a messy business, and I don't think that's right, but... I had once the opportunity to... the first course I took in Israel was with Ofer Gabel. Oh, okay. He was teaching us this. It's a kind of business of localization. It has none in the commutative ring, and you take a module. It's a kind of just a ring of polygons, like that, something like that. It took me so much just to understand it. So I understood at some point that this was an important topic or something, but I think that somehow it faded away. Well, I don't know. I can't imagine that Gabbert would lecture on anything that was not ultimately fundamental. I mean, he is really one of the absolute, one of the authorities on this whole. He has taken this, uh, in the straight-to-the-air basis. No, I cannot say on that. I mean, of course you can give some notions of localization, but, I mean, any time you're dealing with non-commutative ideals, even the notion of crime, etc. I remember strongly that no commutative assumption was all around, and therefore it was complex, of course. It was very hard even to...

12:30 Of course, one puts on further assumptions in the kind of range you're dealing with. But these rarely... Yeah, I mean, the learning is rather nice because it's not too formal. It gives you some clear sense of the real difficulty in directly localizing. So, I guess there was no localization required at that point, it's a springboard. I remember very clearly, I guess, that in class he was completely avoiding any reference to categories or any point of view. So it was kind of green theory, go on with the idea of green theory. Of course, I mean, but this, I mean, this is another point. I mean, the... Certainly some of the things from the 1930s tradition can be done, only in a very limited way. You have to work with things like, you can do a certain amount of this for things like this via algebra, which comes out ultimately from an expression of differentiation. But that's just not commutative, and no more. You see that there's, it's not that there's anything like. There are arbitrarily long words in two variables at the same time, and that's not true in the game, so there you can deal with that, in that ring you can get non-linear diffraction fields, except in a situation like this, another case is for

15:00 Which the very often uses as a powerful technique. I mean, you start with, say, a field, commutative or not, maybe with an automorphism or an endomorphism or something like that, and then you adjoin a formal variable T, and you make it act on the original field by T inverse, alpha T is 5 alpha, where 5 was the original endomorphism, then you form a formal power series thing which is somehow made this... This embedding inner. Right, right. And these things are perfectly fine. The ideal theory of the original thing was good. The ideal theory of the new one is good. You have materiality in many cases and you have this power of all the things. So you can squeeze out things there and it is important. I mean, this theory of d-modules that they work with now is around this kind of thing. And the abstract algebra is, although it's non-commutative, it's not non-commutative in terms of the kind of... Appalling non-commutative theory that people say the British school worked on in the 60s and so on, non-commutative in the theory of links and forges and all. I mean, this is a very hard line, and I don't think one can make any geometric sense to that. Now, again, you might be able to say non-commutative theory of functional analysis. But then, in those cases, you've got some extra structures, or maybe then you can do something. What do you mean by non-commutative function? You might be working with algebras. Yeah, non-commutative, but algebras and so on. But there are those other structures that may help you to understand it. But the pure ideal theory, say, of... Suppose you're dealing with something utterly free instead of commutative. I mean, you've got a polynomial ranging in two non-commuting variables. Very much. There are things that inspired people have done, I mean, there are things, I mean, you can embed those connections with three groups, and some of this is folks, folks, calculators, things, mouse servers, things, but it's, it's, this is, this requires trickier techniques, and I mean, so you just have, there has to be some, you can't be too far from community, to my mind. I mean, people are both like, whoop. In this case, the moment we saw the spec construction, we tried to find it for a general analysis. None of these specs really make sense. This doesn't really give you anything, that's the point. O'Rourke, of course, was a highly original mathematician. I mean, he did a lot of very, very important things that I think were not fully understood as important.

17:30 This is Ray Herring, but I meant to check. When you mentioned that one of the talents, and maybe it was Cremona, in connection with the non-Archimedean model, John O'Hazy, was it based on a formal power series construction? No, I don't think so. I mean, you define that model first of all. Really? You would get them, of course, by just, you know, doing your, taking a formal power system and putting your x or your t at either infinity or infinitesimal, and if you made the thing have real coefficients and rational exponents, you'd get a real-force field, so you would certainly get, you would get models of, I mean, the reason I ask is that, I mean, as far as I understand it, the serious study of these... Formal policies, from an algebraic point of view, even in the non-community case, did go back to Hilbert's time. This may be closer to what I think is Jew-Broadway Mall. Yes, he was working with me. He had this kind of analytic, this kind of analytic for most of us. We didn't know what Veronese was doing. No, I didn't know what Veronese was doing. Later on, after Hilbert, all the things converged and then they became quite a different thing. At the beginning it was from different traditions. I'm just wondering. I said we didn't know. The people in algebraic geomancy were broken. It was in another town, even, so it's not just about anything. Because it's rather straightforward now just to build on formal policies to get... No, no, it's just a deformation. But actually non-community brains don't exist. Clearly, there is this abstract algebra. Non-communicative polynomials and two variables are terrible things, you see, but in nature those things never arise. In other words, there's only one, actually there's one non-communicative ring, and only one, quaternions. Quaternions, you know, has its own existence and it just happens to, among other things, be a non-communicative algebra.

20:00 Just, you know, they become community if it can become this grading or various, you know, something like that. Or, but in general, we see that in general... They're just objects in an additive category, you see. In other words, you could say, well, there are lots of rings. Take any of them in category and take the endomorphism of any object. But that's precisely the point. They arise in a context where there are several interconnected ones. The one that undergraduate students encounter is matrices. Well, of course there's 3x3 matrices, but this is stupid because there are rectangular matrices as well, they're just as important as square ones, and they all fit together into one category, whose objects are the natural numbers, and you can... So they're just the endomorphisms of a single object in a context where you have several interrelated objects, and that's really the typical way, or I think in some sense, except for the quaternions, the only way that real examples... Look, operator algebras, well, I mean, you have bounded linear maps between Banach spaces as well as around one of them, and that's very important, actually. That's part of the whole study of those things. So it's really a category, even if it's a very small one, even with just a kind of a number of objects or something. So you're saying non-commuter rings don't exist, but categories of non-commuter frames exist? No, no, not categories of non-commuter frames. Well, you're saying bunches of non-commuter frames with relations to one another. No, those are modules, if you like. It's not a category of rings. The maps are not ring homomorphisms. So, in other words, it's an additive category. Then the endomorphisms, you know, in any category there are endomorphisms in many objects. So, in an additive category, the endomorphisms in an object happen to form a ring. But the examples always arise as a category. Look, I've added the category and you should see the pieces of that. So you think you've got the non-commuter ring of three by three matrices. What you have is the category of... Non-commutative vector spaces, with the matrix transformations, the endomorphisms of the three-dimensional space, are what you thought were a non-commutative ring. And technically they are, but it's really a hom set in a category. But non-commutative monoids do exist.

22:30 Well, for example... Well, no, I mean, the composition just in the monoid, you know, a composition of maps. Well, no, no. Okay, that level. Sure, in any category, but again, typically. Again, the typical way that monomers arise, if you think about an example, it's really part of the category. Again, making only a very small category is relevant when you find that number of objects, even, or it's a common one for us. But typically that's how they, that's how they come up. So from that point of view, you look at all that research in non-communitative algebra or string theory, let's say, and you say, well, let's put it blunt, this was a waste of time? Of course not. Most of that can be retracted. Look, for example, the localization. In fact, it was Cohn, I believe. Remember, Cohn talked about division rings and so forth, you see, I mean, the localization, even if you start off thinking, I'm going to do this with rings, you find that you're doing it in an algebraic theory, which is a particular kind of category. You have to look at all possible sizes of matrices over the ring, and that's the structure that you localize. This is a very noble thing. That's a good example. Of course, a lot of work done by serious mathematicians can be sometimes in a small way, and other times it really is very important to make that difference, or you don't get a good fair, even starting from the beginning. I could also place that work back to Schutzenberger or something like that, I think, in formal language theory. Oh, really? So again, that's monoids. So in other words, you talk about words. People commonly say words are the three monoids, but actually the three category on the graph is a much more natural, natural in the sense that that's what tends to arise if you think you've got, you know, monoids here. You have the three category on the graph. It's embedded in that. But I mean, your general question may have been referring also to non-communicable geometry. I'm not really willing to talk about it, but we have discussed it in previous days. It's a pity you missed out on this, but I mean, in fact, even in connection with that, there was this Morita equivalence of rings and knots in connection with the way morphisms and so on.

25:00 Well, that's a special case for you. We'd like to come to these proportions. Thank you for your attention because of my page on metric spaces was an example. But really, it's not just the free modules which give rise to the matrices as such, but the prominently generated projective modules over a given ring forms the categories for containing the matrices as powers of the ring itself. But in some sense, that is really the thing, not the ring. Because in particular, You know, the functors between those kind of categories that categories are projecting may or may not be induced by ring homomorphisms, but they are the ones that give rise to, you know, the needed changes of invariance and so on and so forth. I mean, from the point of view of Cauchy completion, it means that there are contexts where, to speak in a sort of schematic way, you're trying to construct a homomorphism from one ring to another. But really, all you're constructing is a Cauchy sequence, and so the limit, you know, is you have to complete the ring in order to hit an actual point, where an actual point is defined to be a projected module, not necessarily the free one. And then there's the non-commutative algebra, where the algebra does become Galois-Cole. You don't just look at one non-commutative ring and its ideal structure. You look at actions that these were all group ranks, the one he's interested in, you look at the group, look at the category of group acts, it really does, so the work is there, the theorems are there, but they're put in this context. I mean this is also mildly, but it's not so much, I mean quantum groups, quantum algebra, general deformations, the definitions they use are close enough to the kind of things that people will do, so they're, I mean...

27:30 This has saved the lives of many British non-communicable linguists, you know. So you have two variables over some field. This is big. But the whole point of the quantum groups is the commutator relations, which mash things down a whole lot. In general, these commutators, you don't want to know about them. Right. You know, a source of... Yeah, well, actually, the commutators... Which means that the action itself... ...generates the commutative algebra. P times Q, Q times P is the commutative algebra that generates the algebra. Yeah, that's what I meant. I mean, you can't... You've lost, by doing that, you've lost the splitting of the action into. In the times configuration, if you could imagine splitting in a different way, you'd still have the same commutative algebra, but it is a commutative algebra. Yes, unquestionably so. I mean, it does belong to the... I mean, Levitsky was, in fact, quite a long time ago, yes? I mean, I think the work he did was really entirely natural. I mean, if I appear to sound hostile to the ring theory, it's only because I think it's a good example, at least in Britain, once for the same thing with functional analysis.

30:00 You know, there was people who tended to be too systematic and carry things far beyond the point that they make returns very rapidly. I mean, sometimes the math is very difficult, but occasionally, you know, something nice comes out. But there was never any real purpose in it. But what Velitsky did was very much more at the beginning of this, and it was connected with radials as well. I would like to say a couple of words about Ennis. He was a very distinguished biologist, because I've been prized, most prized in Israel. That one was up to me, and I was looking at his theory. I mean, I think he came from Nester. All of his works are something of room theory. I think he was proud of it. You see, the people, they were Americans, but these people, Kablansky suddenly knew where the stall, you know, and obviously Levitsky did too, I mean, Abraham Robinson, I think, worked, perhaps not with Levitsky, but all the things of Levitsky at the very beginning, his very first papers on, yeah, yeah, sure, yeah, he is a hard one now. He's dead, he's dead, he's dead. I started with a student of him. I knew Amazon quite well. Exactly, he's writing for a strong foundation. I mean, Amazon was a very, very important. Indeed, Amazon was a very important member. Again, he served with you. To call it a day. He was with the Nice International Society. Yes, yes, yes. Really, he was using adzoite functors Yeah, I mean, yes. to approximate. To the extent to which a non-commutative ring might be the 7x7 matrices over a commutative ring. See, there's a best, it's a best commutative ring, and it says a commutative ring whose 7x7 matrices are most closely mapped to a given non-commutative ring. So this gives a kind of... The two-variable measurement of the complexity of a non-threaded graph.

32:30 How big the matrices have to be and how complicated the commutative thing has to be. I used to see McGill in other places. He was much involved in serious cohomology in the beginning, and I think it still will be slightly relevant in the commons part. What about Jacobson? Well, Jacobson also, I would say. I mean, Jacobson, I was there. I mean, Jacobson wrote very well. He also did very serious things. He pushed Jordan out. I mean, that wasn't, it was never quite so clear, you know, how worthwhile that was. He hung on. I mean, you have very high regard for him. He had very, he had some extraordinary problems. Let me see if it's a different point of view. For example, classification. I don't know, any classification that you do there. Look at this, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, look at that, To find within this category of modules, over it, you know, a small, very small category, but not, usually not a single object, a category, which in some sense gives it, from the opposite side, generates the same category. So that's the mode of analyzing it. So it really is the study of representations of little categories rather than a single object thing, even in the application to single objects. So it's again this phenomenon, not just of the quotient completeness, but something else involving the... And do you get the... His definition of semisimplicity, not in terms of ideals, it's in terms of the simple modules.

35:00 So in other words, the category, we have the category of all modules. Why does that have the category of all simple modules? And it's that category which is the real invariance. The real thing's properties are being analyzed and cataloged and so forth in Jacobson's program for analyzing variance. So it's something like that, a similar sort of thing in the algebraic case. So in the textbook by Jacobson, the newer one, there are two chapters on categories. That's where I learned it. But I think that then he doesn't go to complete the work in terms of categories. No, he doesn't go far enough. He uses for remodeling. They were written quite late in his life. He was a very hard worker. The book is fantastic. Oh yeah, I think in the end there's three volumes in all sorts. It's got a lot of importance. This is one of the clearer, you know, many people actually were in that situation of having written books and then realizing that you had a chapter on categories. I think he did the best job out of many of these, making sense of that. But in terms of actually, I mean, Nikolay and Birkhoff, for example, didn't that, right? And it didn't come off too well, I think. Another version that starts with categories? Yes, yes. It's called algebra. Oh, that's the one I mean. It's really just tacked on. Well, at the beginning. Okay, tacked on to then. Fine. I mean, but it's still sort of tagged. It isn't really systematic. Well, yeah, I agree, I agree. That's what makes it complicated. A lot of big surveys, I would say. Yeah, in a way. I mean, in certain places it's not trivial, but... You know, the categories should be a simplifying tool instead of a complicating tool. You have to rework all the different stages a little bit more thoroughly than is done there. You know, I should say Jacobson should have done that, but he didn't. I mean, of course he did. He did a lot anyway. I think he used it just for cohomology, for explaining what is topology and things like that. Yeah.

37:30 Five terms. Tor and Ek. And he has this little chapter on theta functions. He still uses his code sets like he wrote to us a long time ago. And X in some way existed for four categories. Tor did not exist and X wasn't anything like it. It's clear as it becomes. And then how have you seen the word for... They probably... Oh, whatever, I haven't read that book. They must use Tor in the commuter development. And I'll bet they don't call it a functor because they don't think the word occurs. I've never read it on the web. But I do say, it took them a yearbook that was 26 years out of date in that year. They knew going in that the Tohoku paper was right. And they knew going in that their strategy would be known to use those things. But they saw it as in terms of this division of labor. Okay, we'll let Grotendieck do that version. We'll do the part that doesn't need whatever. That doesn't really need categorizing. For the benefit of readers' conversion with homological algebra, and they show how the theory of flat modes is related to that of the Tor functor. But they have not defined it. You know, it's one of those rare cases. It's one of those cases where they introduce something that has not been named. So, this is because they couldn't do without it. So, instead of just starting to go around... And they do introduce Tor as a certain quotient. And they use that as a criteria for mathematics. But it's not. They can't call it in one direction.

40:00 Yeah, yeah, there's a group of certain members that mainly click one. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah.

42:30 Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. They know there's a group of extensions for one group by another. Right, oh yeah. So when did Bayer become active? 1930, I don't know about that. He edited, sorry he didn't edit, he advised Jermael on the re-crediting of Cantor's selected work. Which was 1931, I believe. He also, I think, also edited the Steinitz paper on the fields. And it was in 1930. His wife told me that he attended the Hegel conference. It was re-edited by Edmund Bayer. Does he recommend your book in any way? I'll tell you something. He doesn't really come, but it's mentioned in a footnote. And I have written something about that. I don't know what I did with that. There are two people that appear. It's Sebastián Silva, who you mentioned in some way, and Krasner.

45:00 Someone told me, do you know this guy Newton da Costa, a Brazilian? I knew of him. I never met him, but I certainly knew of him. It's very interesting. He told me, look, all these things you say about structure, the first person to do something like a general idea of structure was Sebastián Silva. Because he went to Rome to work with, who was at that time there, Locotero, perhaps. Where did he go? During the Second World War. Ah, Castelnau. Castelnau, Castelnau. He went there, he came from Portugal, and he had this idea to define a very broad concept. Let's say abstraction, let's call it like that. And he devised some kind of Galois theory for this. But the Italian guy told him, look, this is very nice, but you cannot waste your dissertation on that. You have to do something on functional analysis. And then he did it. But still he published. And also Kastner had a similar article. Again, with the Galois theory, the area was very similar. I mean, Krasner had a big effect on Poisson, one of the leading moral theories, and this is still around. The places of Krasner are still around and have made a difference. It's une généralisation de la notion de corps. It's 1938. And then he has généralisation abstraite de la théorie de Galois. It's 1950. That's Krasner. And Sebastian Silva, in 1945, studied the automorphism of a mathematical system, Qualunque, to give a very general formulation of whatever. And then it's translated in English. So it's a kind of... that's what Acosta told me. He's before... I have written something and for some reason I decided not to include it and then I wanted

47:30 to publish it on a separate piece, the article is available, someday I probably will do it. I think he died in the 70s. He probably died in the mid-70s. There was a meeting in Clermont-Ferrand in, I think it was from 1975. It was quite near the end of life, and I spoke to him quickly. No, no, 1912 to 1995. Was he possibly at this 1964 meeting? I wasn't there, but I think it was quite late. He was almost the world's biggest algebraicist. He was huge. I think he was on the same plane with you. Not sitting next to you. What? Not sitting next to you. I like to talk a lot. Oh God, yes, yes. It makes more sense to me. Because he was a fantastically original mathematician. He did a lot of very important things around the analytic function because the issues are very delicate. Really delicate. And he was quite an inspiring character, I must say, and he was quite a, I have some, they came into my hands from what gave me some, they were in Greece by some disciple, not very good enough, but they had quite a lot of interesting information about Krasner's life and so on. I remember, I was so young, I was still a bit nervous and shy, I remember this Clermont-Ferrand thing. Before, I met him later at a garden party or something like that, but he came to my lecture and he, as I say, it was cool, but he went quite far up, I think it was, I don't know, a few things, and he had, he's binocular, and he's, who knows, he'd be the old guy. I found it unnerving, honestly. But then to talk to him, it was very, it was very nice. I mean, many things he did. Krasnoy's lemma is almost the fundamental type of the Gallaudet, beginning of the Gallaudet, but he had it.

50:00 If you know one big thing... I mean, it's French. No, it was born in Odessa. Yeah, I mean, it's a big thing that I didn't talk about. ...quaza, which you truly knew of. Quaza, have you ever heard of quaza? It's an interesting... How do you spell his name? P-o-i-z-a-t. He said Leon. I mean, one of the most imaginative. He writes his papers in a rather unorthodox way. His library style, I think, is probably very good in French, but it is very annoying. He comes up in big trouble once many years ago. It was stupid given his general sophistication, which is considerable. He published a book on the theory of groups, modern theory of groups, and he had illustrations for things like chain condition. He had a naked woman with a chain around her neck. There are six or eight pictures. In the index, you have the names of various female mathematicians attached to these pictures on the push of a button. I didn't imagine it. I mean, the AMS, I think, refused to ever review them. It was taken off the shelves. Correct! I mean, it was an act of stupidity from someone who is, in fact, a very sophisticated and interesting individual. I can never understand why he did that. But he has a model video, which is brilliant. He's still active? Oh yeah, yeah. He's late 50s. He's still active. He knew Krasner, I think, and he certainly was very strongly influenced by him. I mean, he took this Galois theory of Krasner's and he developed this theory of Galois imaginaire later on. But essentially you only get a proper model theory of physics if you include action. And this is very nice. This is why he was led to get the Galois theory of physics.

52:30 Now, he's one of the most original of us, maybe the most original of us. Trusovsky, but he is very, very, in a strange, strangely forgettable way. I think that also Newton-Dacosta told me that some of the ideas of Krasner and Sebastian Silva are retaken now, it was in the 80-something that he told me, that are retaken by him. Yes, yes, no, I first only saw, I mean, there's quite a lot of fairly serious Galbraith here, you know, I mean, there's... There are sometimes attempts to recapture the category of models of a theory, in some sense, from one gal, Daniel Lascaux, who worked mostly for Mackay. It's essentially rather nice, really general theory. How about taking a break? Yeah, I think we can take a break. I can remember we did Italian, but it's not very normal. It wasn't very severe. Yeah, why was it? It was severe. The problem was still around at the time of the war.