Morning Discussions incl. F William Lawvere, Pierre Cartier, Colin McLarty, Angus MacIntyre, John L Bell

0:00 Apologies for the late start. It was entirely my fault for keeping Bill and Pierre and Colin talking in breakfast. We would like to continue the discussion on the first phase of Grover Deep's career, 1950-1960, a little bit longer before passing to the discussion of his great work in algebraic geometry in the 60s. and we had a lastly exposition from Pierre yesterday I'd like to ask Bill to lead us off this morning by perhaps saying a little bit about how he sees probably clearly working functional analysis as carrying the imprint of, anticipation of the deep ideas in category theory which culminated in his later phase development and how he sees it as fitting into the overall picture of the theme as i said in an understanding of categories of space as he was developing it yesterday morning perhaps it would have been from there well i think we've already got over the what basically i proposed and pierre filled in a great many things that I didn't know, but it seemed that the basic outline was supported in a way that from the general functional analysis, one came to the example of the polymorphic dual spaces, which led one then to complex spaces, and to the Theta-Ga-Ga principle very rough very summary description but I think those were some of the main steps but some of them I'd like to ask more about well I had so much body man is perhaps the gaga a paper by Sayre. But the germ of the idea goes back to Riemann, I think. So it's a huge systematic

2:30 elaboration with the categories of vector models and categories of spaces over and so forth and so on and so forth. But in some sense the germ of the idea that there could be such a thing. in the so-called Riemann-Nilbert problem, to characterize an equation, to characterize the class of homomorphic functions through the singularity of the differential equation is satisfied. It's a basic example. Well, of course, Riemann-Nilbert program evolved in different ways. After that, I mean, it really is a basic idea, too. But the homomorphic functions should be characterized by this topological people start over the imat surface. But actually I was thinking last night, a connection I hadn't made or forgotten, I mean, but in some sense the Grauer Direct Image series. Oh yes, it was important. Connected with the um, um, um, what's it called, the relative speck of Sifu Yes, in the steeplechal range. Madame Monika King's thesis came after him, in a certain way this also was expressing this Vaga principle, because the fibers of this proper map are essentially algebraic. Well, I mean the starting point was a result of Charles, that if you have a, in the protected space, if you have a complex, a small, complex, analytic manifold, a clause embedded into the protected space, it's automatically an algebraic variety. It's a Charles film, I mean, that was the starting point. Right. Charles, yeah. And we wanted to interpret this and to extend it, and not only to the varieties, but to functions, and so on, and reformulation issues. So it's a big comparison between what you can derive by Schiff method using only a monomorphic function or using algebraic function, comparison. Well, it comes also to the question, to the whole question, I mean, of which hypergeometric

5:00 function, the algebraic function, which is Schwarz's work in the 19th century. All these things are connected. I mean, the Schwarz's work on hypergeometric function can be summarized in the following way. Well, hypergeometric functions are solved from a certain differential equation. So from that differential equation you generate, using a singularity, you generate a human surface. Human surface. By various methods, or by sheet or by doing it, but then whatever. And then, if Riemann's, I'm sorry, it's certainly an infinity sheet of Riemann's surface. But if, but of course it has, oh, it's a covering. It's a covering with infinity, many sheets, and infinite by one. But if you can go down to a certain finite covering, that covering space, then the function, so for some values of the parameter, the hypergeometric function descend from this huge infinitely sheeted Riemann surface to a finitely sheeted Riemann surface. Now, the point is, according to Riemann, that in this case, the finitely sheet of the Riemann surface is an algebraic curve, is an algebraic curve. And therefore, the hypergeometric function sitting on an algebraic curve is an algebraic function, but more or less. Of course, the technical point for which value of the hypergeometric function to get an algebraic function is a complicated algebraic problem. But this principle is expounded by Riemann, let's say. So the main point is that, well, of course, an open Riemann surface can allow many, many functions, which are certainly not of the bike in any sense. But if you can have a compact Riemann surface, by going down to the finitely sheet, to the compact Riemann surface, provided you know how to handle the finitely many points of course, singularity, then you create a Riemann cellophase. And a compact Riemann cellophase is an algebraic one, which is really the point in Riemann. And therefore, this is the first step

7:30 in the comparison between neuromorphic function and algebraic one. And so for the one-dimensional case, what Riemann did, and then Schwarz's contribution, such a completely is a problem. but for the higher dimensions that was the main challenge. And Chao Feuern was the first step in that. But Chao Feuern is a little different because it's already embedded in the projective space. So in a sense it's easier, well not so easily, but what they wanted is to have something like in the Riemann spirit, not something embedded. I mean that's a whole more, I mean to show that some curve is anti-brightage. that the project in plane is not too difficult, usually it's not too difficult. But when you have an abstract curve, it is not in battle, not an eight-point, and I think the steps that Seh took were still very similar. And of course he relied on the direct, which is a coward. I remember very well that he was closely associated to all his works at the time, and he had to go years ago, years ago was a goal between all these people and all of this, of this, of this Abayr's time being born, cannot be overemphasized, it was a place where all these people exchanged ideas and all these things. In a sense, that was, Abayr's time was more successful than the Bobakshi seminar, in a sense, quite different, but I think with something similar in the It was more successful in a sense to disseminate new ideas and to bring people together and so on. Opportunity I never attended in my time. Yeah, so then coming back slightly to this duality for spaces of all modern functions in which the, let's say, the distribution, if I understand it correctly, the distributions on a, the analytic distributions on a bounded domain are themselves represented by actual functions on the complementary domain, however, of exponential growth.

10:00 So, this is a kind of duality that's different from Pontryagic duality, let's say, or some duality in this kind of duality, where you, for example, the Pontryagic duality says you look at the distributions of one group and the functions on the dual group, so you pass as a dual group, not to any kind of complement. But as you were saying, in another situation, the space of departure may already be embedded in some simpler space. And a difference, a fairly different type of duality comes up where the distributions on one part are representative of a complement, sort of set theoretical, logical complement, you see. So, there is a similar result in topology, which is the Alexander duality, so I was wondering if one perhaps could deduce the Alexander duality from something at the functional analysis level by bringing in differential forms and currents of a similar sort as the functions and distributions and somehow passing straight down to the one-to-one duality might follow whatever. The qualitative one might follow from the quantitative one. This is just a speculation, of course. I think it could be known, I think it could be known, while that's always a stronger section. That kind of method that would give you the example of duality, it was the case of the so much important with embedded into the money but not the more general situation right yes and i think that can um i remember uh duadi gave one line proof of the prank area and

12:30 i think that could be accommodated to give their example duality. Duality is a fantastic idea. It says that Poincaré duality, well, there are two steps in Poincaré duality. First of all, you can use a Durand complex which is based on differential form, smooth differential form, to give you the commodity of the space of the man's ball. But on the other end, and it's known from the book of Durand, the book of Durand of currents, that you can build Durand-Tremplex using currents instead of differential force, of smooth differential force. And it's not too... using shift method it's easy to show that a true commodity coincide because all you need is a local primary and according to the what if you follow the proof by behavior let's say all you need is a local primary name I would say is that a closed differential form is locally an exact differential form and the same for current is easy to prove rather easy to prove and the method we I mean, the Tenso project method, for instance, that is what it is used for the Torbjolnir is easy. So, if you want to be sophisticated, but they are more elementary method. I might say that it's a short paper on an elementary explanation of such things, long So, then you have the comparison, and then Schiff's theory tells you that if you have two, you can calculate the same commodity using two different resolutions. Basically, that's the principle of that. of a shield chromoly. So you have a resolution by a resolution using smooth differential form, another one using white, while current is highly non-discontinuous, I mean the singular differential form, but they give you the same chromoly. But this is not yet what you want, because then if you want to compare, but of course differential form is created in duality, but you get into certain points of duality in the infinite dimensional spaces and exact sequences.

15:00 The point is that for you know, banner spaces or such, the notion of the exact sequence is delicate because you have very often you have a map of one banner space and another one and the image is not closed. So what does it mean, the exact sequence? So you have a very crude way of saying an exact sequence. The kernel of one map is a closure of the image of the other map. But that's too crude for many purposes. That's very much too crude. And of course, if you work within a category of banar spaces, that's a natural notion. That's all you can say. All you can say, if you stay within the particular Banner spaces, or more general spaces, that's interesting. So, but that's one difficulty. And if you try to apply this to the duality, so you have two complexes, one with smooth differential current, they can be put in duality, and then you can say, of course, the commodity of the dual is a dual of the card. But if you work in that setup, I mean, you are in Banner spaces, you are stuck by this difficulty of what is an exact sequence. What's needed is to change the functional analysis. Then, to what this idea was the following. So, I take the complex of smooth differential form. There is a topology on this functional space. And I take the topological to define the current. But what if I take the full antipide tool? Of course, in unantipal term it means nothing. So I take the complex of small differential form and I take at its level the complete antipide tool. All the linear forms, continuous or not, would be... Of course, formally that makes a new complex, but shape-corporal, shape-method enables you to show that this new complex, which is much bigger than the complex, so you have differential, smooth-differential, from current, and these, I would say, wide currents. Well, it's like Godemann's resolution. Something not necessarily continuous. Yeah, yeah, something like that. Well, I think it was... I think he was influenced by that. And then he said, well, when you think she is better than resolution, you show that this huge complex of a very wide object defines the same homology, but this time the duality is for nothing, because you have moved to not from the category of inner spaces to the category of vector spaces, whereas the duality frontal is pretty alright.

17:30 And it's, I mean, the real idea takes one line to explain, well of course there is some technique, I mean you have to rely on the technique, the standard technique, but the new idea takes one line, and that's a fantastic idea, fantastic idea. In a sense, one is working with the two books. It comes by the dimension. And then, voila, at the end of course then you can show what you want, you can show what you want. So you bypass this difficulty. There are other methods to bypass this difficulty, you're using Orch-fury and things like that, but it's much more than you think. Analytically it's much more than you think. You have to use Orch-fury and the existence of parametrics of differential operators, all these analytic techniques which are very useful by themselves, I mean, can show that, but it's a much more There's no economical way to put the Pankai-Guanity. And I suppose you could accommodate this idea for the Alexander-Guanity. It just comes to my mind. But somehow the analytic case, the linear dualization is one thing, but then the complementation is another thing, and in the analytic case, this is achieved by the Cauchy-Ternel. like that yes over z minus w basically so what's the analog of that in a non-analytic case you want to transport the linear dual of the punch's face on one semiway over it to you need a clear kernel of some kind Well, the Cauchy canon has been generalized in many ways, in many complex variables.

20:00 And I was really starting with Leroy. Leroy was a one-two person and then many people around him aren't going to develop these ideas. And I suppose in the first Real France book on distribution, There is a very general notion of residue, which is a real analytic version of residue, or not that complex analytic. Yes, we realized on the Stokes here. So I think the idea would be, let's see, the way I see it is that you have a certain manifold and the sub-manifold. Take a tubular neighborhood with a nice bounding which can be considered as a sphere bundle over the sub-manifold. Then using gazing exact sequence and a nice support you can do that. You can make explicitly gazing exact sequence by producing suitable camel and you could get the spy nation to find is kind of in the book by Gelfand and the distributions, because at some point he defines a real residue, which is just an application. The real residue of, I mean, the idea of Gelfand as a four-way, the Cauchy formula is really that you have one singular point into a re-conorment surface. What is the tubular neighborhood of one board? It is a small circle drawn around the board, a small cycle drawn around the board. And in the interior, the interior of the circle of the loop, so you have a one board to take a loop out and the interior of the loop is a tubular neighborhood. And the Cauchy formula the Cauchy formula can be derived from Stokes, the elementary version of Stokes,

22:30 I mean, one complex, two-wheel in a plane, I mean, Stokes. So the connection between Cauchy formula and Stokes, I mean, elementary calculus is well known. And the idea is that if you are one point in the n-dimension of 3, and apply Stokes in a proper way to that. But the interesting point is that this idea extended to some creation. So for the harmonic functions you have to use a Poisson character. Yes, for the harmonic function you can introduce Poisson character. Really? Exactly. And you can generalize this in various ways. You can generalize this in various ways. I mean for complex, for many complex variables this has been done by Loreheni, his diacipal Norge, Norge was a student of Loreheni, he developed these ideas and he has written a number of papers about these generalized chemists and there was also some Italian contributor, Maybe one of the first papers of the general army. There are some contributions by Italians. But yes, some people try to accommodate this idea to a real valuable situation. And I think, so I think you could derive, you could derive a proof of, analytical proof of the duality of the situation. But it's restricted, it's sub-manifold, non-singular-manifold embedded in the non-singular-manifold. So it's certainly much more restricted than what you can achieve by topology. Yes. And again, in the duality of homomorphic functions, I mean, the main point, the main point in the, in the, in the, what you need to see about duality theory is that in one complex variable, you take a compact, you have a Riemann surface, and you take a compact set, a compact tall set, but it's not a sub-variety. It's a, in a sense, it's, geometrically, it's It's compacted with a bound eye or even wider than the space with a bound eye, I mean with a smooth bound eye.

25:00 But the real point is that by definition, a holomorphic function of such a space is a function which extends holomorphically to some even that point. And geometrically the point is that when suppose you have a certain compact domain in the complex plane, then, by definition, a holomorphic function is a function which extends slightly to a somnable as a holomorphic function. So, it infringes all the other complement. That's the point. And then, an element telling what to do. So, you have the holomorphic function which is on the compacted. It extends to somnable. then in the ring which is around this compact set, I mean the intersection of this domain with the rest, then you draw a loop and you apply ordinary Cauchy formula or something more sophisticated using a kernel, I mean, a kind of question, but in the real situation, you don't have this possibility because of smooth, well of course a smooth function, a smooth function on a certain manifold extends to a smooth function in the ambient manifold, but apart from being unique, even local. The point is that in the case, the extension is locally unique, at least for terms of function So in a complex case, sometimes you don't really need the whole complement, perhaps. Yes, and you can restrict that. You can restrict that if you want it to enable it. Representation of the dual, yes. But then, I mean, when you go from one variable to many complex variables, then you have to use... The duality is now homological duality. And this was elaborated by Mikio Sato and by so-called micro-functions in general. Well, the starting point was Martino, who introduced so-called analytic functionals. And the work of Martino was directly connected with the work of Gotenlich and Das Silva.

27:30 And all the inspiration was more or less Laurent Schwartz in both cases. when Darcylva was a student of Laurent Schwarz and I think that more or less good, Laurent Schwarz was inspired and inspired his ideas and Martino was a student of Laurent Schwarz and was still an influence of Laurent Schwarz at that time. So I think if you want, I think it was Laurent Schwarz who gave the impulse to the starting point towards the light of development, but then it was developed. And on the physicist's side, there were some physicists, especially in Sainte-Boros and other people, I mean, the edge of the wedge theory, the edge of the wedge theory, which was developed by many physicists, some theoretical physicists at that time, it was an elaboration of that. And that's, but it's really common. And I think most recent expositions should be taken from the book of Shapira and Kashiwara, I mean, a case between the light, the heavy homological machinery in Shapira and Kashiwara. But the geometrical inspiration is that kind of thing. All right, so I don't know if you read my paper. In the Rende Crocker Seminario Mathematico e Physico. No, the Cultura Matematica e Palermo. Yeah, Palermo. Palermo, Palermo. So this is kind of a posthumous debate with Boudinnet. There is this two-volume history in the 20th century by Pierre. Yes, I think they'll have something to do with it, which is not the best I ever thought. No, it's very selective. Actually, I have an article on the second volume. But in reading the first volume, before that, I came across some of the issues that I've addressed here.

30:00 So there are two, at least two different things that you mentioned that are relevant to this. One is, well, says that in the book, we must finally mention the first attempt that quote, functional analysis, unquote, of the young Volterra in 1887, to which, under the influence of Hadamard, has been attributed and exaggerated historical importance. So this was . And then he goes on, there's even a connection with Volterra's notion of the derivative of a functional. Udine further states, with our experience it's a 50 years of functional analysis. So according to him, it only started 50 years before. We cannot help even without even the barest notion of general topology, these ad hoc definitions were decidedly premature. So I remarked that one might ask if calculus of variations was premature. Yes, exactly. So, under this, on that, and then the Italian mathematician, who was the head of the academia, he answered this, so, again, I didn't agree with his answer, but, so this is the The origin of this paper is why I tried to study some of these historical issues. So for one thing, jumping ahead, I mean, you mentioned Duadi and Durand utilizing the differential forms between the parents. And they said topology. But actually, they both used bornology instead of topology. Which is, you know, for biox phases makes no difference, but for these things it's definitely similar, as do any recognized in this group.

32:30 So there's this notion, you use the term analytic functional, marginal, but already the Volterra School, Oh, yes, that's right, another source of the debates. The authors, Russian names, have a recent very useful biography of Hadamard. Probably you know these people. Anyway, while acknowledging Fikara's arguments, acknowledging Fikera's arguments, nonetheless described as, quote, naive, unquote, the theory of analytic functionals developed and studied by Pantapier, Pellegrino, Hathaway, Suchy, Teichmuller, Silva, Volterra, even Max Zorn, and others. Yes, it's another language. I'm sorry, I forgot about this. More specifically, they say that the naïté of Tamtapier's approach lies in the fact that he, like Adhemard before him, avoided the topology and assumed analyticity in the sense of his general theory, instead of using continuity. And then I say, well, that these repetitions were not, in fact, naive about the role of continuity is demonstrated by the fact that they published many papers functionals in their sense, were in fact continuous, were very positive. This was quite an interesting historical exercise for me because it's in some way the same general issue as the K spaces. Namely, the fundamental need is to have a general formalism for forming function spaces, supplying these function spaces with cohesion, in some appropriate sense. Now, it could be continuous, as in K spaces. the various smooth and analytic and all other things, but the analytic functions, functionals,

35:00 like the already in the work of 300 years ago, it talks about some variation, but were simply based on the idea that the smoothness of the functional is tested by looking at all possible paths or all possible figures of finite dimension in the function space and requiring that the the result of composing should again be a good map between finite dimensional spaces and that a figure in a function space and this is now precisely the rule of entrance of random calculus or Cartesian quotes categories a figure in a function space is the same thing as a figure of or twice the dimension in another finite dimensional space, just bring the exponent down. It's the same type of adjoinness of HOM and TENSOR, except in a Cartesian situation. So that was the concept of analytic functionals, which this whole Italian school, but they weren't all Italian. There was this Swiss, Portuguese, German, and so forth, But recently, I mean, there have been various attempts to strengthen calculus of variation. I mean, the most successful has been the one by Chen, the Chen calculus. And Chen was the one, the idea of Chen was take the loop space in some manifold. And he wanted to have a Durand complex, I mean, of course, the loop space, the commotion of the loop space has been calculated by Ivan Behrer, McLean, and so on. But he wanted to calculate the commotion of the loop space by using a kind of Durand complex. And now Chen was able to develop a calculus of differential forms on such spaces, exactly by saying, what is a differential form on a loop space? Of course, you will know, there is no difficulty about the meaning of a map from, say, some simplex, continuous map from some simplex into the loop space, which is just a slight elaboration. Well,

37:30 Then, usually in calculus evaluation, one take one-dimensional deformation, but you can take many parameters of difference, exactly. So, for chance, the definition is that a differential form, to define a difference. So the loop space comes with all these bars which, in a sense, describe the singular complex of your space. So a differential outcome. Here you have a space which is far from being a manic form, but at least it has a notion of singular chains making good sense. And even smooth singular chains, not only continuous singular chains, but smooth continuous chains makes the obvious sense in the spirit of calculus of belief. And then what is the differential form of such a space? It's associated to every singular chain a differential form on the chain, which is something finite-dimensional with some consistency which is min-functorial. So that's the way you define the differential form of such an infinite-dimensional space. And now you can, all the algebra And then you calculate, so you calculate, so you have a very general definition of the ram-form, the ram-differential-form, then you have the ram-complex, and you have various sub-complexes. And you show that the ram-complex, the commodity of the ram-complex, calculate the homology, at least the real homology or commodity of your loop space or bar space and so on. this has been very very successful recently. I mean, the people like Sullivan who developed the rational motography method, and then Richard Hine, Dick Hine, and then Deligne, and then Deligne, and many people after him. And so, and now of course it's, when Deligne defined the Derag version of the Pi 1, we use exactly that kind of idea. Let me check if I'm understanding this. If you think of a tangent vector on that loop space, you want to know its value for this form. So that becomes a loop of tangent vectors over here, and you integrate over the loop?

40:00 Exactly. Exactly. I mean, it's a very naive notion. What is an infinitesimal deformation of the curve? It's a veto at each part of the curve. Exactly. Naive. And that's a good thing. Sure. Not the point, but the point of what Bill, I think, was trying to indicate was that it's a good thing. There are all these notions that somehow got lost, if you like, development, which are then called naive, premature, unformed, raw, and so on, which were actually always implicit in later developments and sort of got detoured by mainstream. I mean, I think that's particularly true in Johnny and Boris to come back. Yeah, of course they were. And I think that was the point. it was also the point in working if you look at the Italian school of algebra and geometers point have been made a number of times that differential geometry wasn't set theorized I mean it simply wasn't part of the mainstream development of set theory or what was going on in mainstream analysis if you like and then very much later because we got resurrected really by later on in in through to work in differential geometry later work which was was was part of the mainstream of the set theoretic mathematics, but I would say that the issue is really calculus of variation. Calculus of variation, and people invented them in the space of, let's say, well, they have various approaches. You can define, you can define function space, functional spaces, various topologies, and then try to define infinite dimensional money for which our model are local model being in those spaces or Banach spaces. There was a great excitement at some point about that. And when Bobaki wrote his own sketch of differential geometry, I mean there are some many of results of differential geometry, I mean more or less influenced by by Lange. At the time, Lange plays a role in the writing of his book rights. And I followed the advice, well, I was responsible for one of the drafts, and I followed the advice of

42:30 Lange. And then, from the beginning, we deal with infinite-dimensional local models. The more local models are banal spaces, and are many polar-based local models in banal spaces. But for the purpose of calculus of variation, it's completely irrelevant. It's completely irrelevant. Engineers continue to use the whole form. They can actually calculate the derivatives of the functions without ever thinking about the biology. But the problem can also come in sense. But what Chen produced is that, of course, a first-order variation in the calculus of variation is something well-known. I mean, if you look at the engineering or physical literature, it's full of applications. Second order derivation. Second order derivation is already one. It shows in many places. Jacobi, Obey, etc. The idea of getting a calculus of exterior differential form is a new reach. Well, it can be given exactly as and what Sheldin was to do it exactly in the same space. So, I mean, if you have a curve, which is a tangent vector along the curve, it's a field where you have, I mean, in physical terms, infinitesimal deformation, so for each point of the curve you point into some direction, infinitesimal displacement, and if you want to have a differential form, it's something which is obtained by integrating over the curve, something different. And then you have, but you have one direction, but if you want to do exterior differential calculus you just take two vector fields, three vector fields, an anti-symmetric combination, and so on. And this, I mean, but the deep result of Chen is that using this naive approach of differential calculus and calculus of variation and extending, I mean, footing into this matching of calculus of variation, the powerful method of elicato, of exterior differential form, you produce a version of the Duhalphian which keeps using the commodity of the loop space. But the loop space is in the other interpretation, interpretation of functional analysis. Loop is a set of all continuous maps of your circle into the space endowed with a functional analytical topology.

45:00 There is a deep bridge there between the two approaches, the 9-1 and the satirity. But sense-contributions cannot be underestimated. They are very, it was very, very deep. Well, let's look into this because, you see, we have a further step there, namely that the notion of differential form itself is a function. It's a representable with the representable from the other variable, somewhat like Eidelberg-McLean space. Exactly. So that the differential forms on the loop space, say, order three, are actually matched from the loop space itself into the fixed-representing objects dependent of the space of order three. It's sort of bringing this duality back to the more basic level of Euler's definition of intermitesimals. And there is one extension of this circle of idea, a Suyo, a French mathematical physicist of Marseille, with a rather exotic mind. He is a very original mind, but he was one of the first to publish a textbook on the new methods of Hamiltonian mechanics, and he has, I mean, about the same time as Arnold in Little Ages, and Arnold, he publishes a book about the Hamiltonian systems, which is very peculiar, well, very peculiar, because he has an exotic mind, but with deep imagination. And he had a line of, he had a group of students working this idea of what a differential form is defined too, etc. They call them dipheology. So a dipheological space is a space you don't give a topology or whatever, you just define what is a smooth spot from a simplex into it. So you give us the singular complex, the space is this kind of singular complex. There are at least six different schools which have variations on this idea that communicate very little. Exactly! Exactly! I think it's a very good idea. That's why they were still trying to have another such school. But really, I keep discovering new examples of it in the literature, which apparently don't know about.

47:30 Yes, Iglesias. If you want to name Iglesias, look at Iglesias. He is one of the students of the Suryo. But these people are motivated by problems in mechanics, but they tend to be a little too formal. And, uh, but can't be so. Well, now, my own, well, I will, someday I will give a full explanation of my ideas about how I need to go in such a way. And, uh, that's, uh, one more. I have a new idea of it. It is not in my book with Cecilia Witt, well, the book is here, I mean, you might have the proof she is here, The idea is that in an infinite-dimensional situation, you can wait... Well, the idea is that in ordinary manifolds, you are really two-derham complex. The complex of differential forms, where the differential operator increases the degree, and by duality, the currents. I mean, if you take the dual of this complex, If you have a complex with increasing differential, the dual has a decreasing differential boundary, boundary, co-boundary. So, Durand is well aware of that, so it is casual, it is casual. But the point is that precariduality says that in finitely many dimensions, there is no real difference between them. The complex individual, as I say, that's exactly what precariduality means. But when you have infinitely many dimensions, it's different. You have what I call the ascending Dirac complex. You start from the zero form, which are functions, then the function form of degree 1, etc., etc., etc. You have a co-bound variable. But if you use it, if you have a space of dimension infinity, when you take the dual, you should descend from infinity to infinity infinity minus 3, etc. And I can't define, I can't define a bituality, but it's a little awkward. But I can't define, I can't define a descending complex. Of course, in infinity

50:00 dimension, they will never meet. So there is no precarious duality. Because it should compare the convergent dimension p with the convergent dimension i, infinity minus p. But But you're really too complex to have, one ascending, one descending, and they'll never be, but they are in Germany. So, it's an extended version of point-gagulity for infinity-dimensional spaces. And now, of course, it brings us back to an all-Ibrian differential geometry, which was well known in the 1920s, a distinction between the tensile field and the tensile density. If you look at the textbook, the book of Hermann Weybe on the run-time material, space-time matter, it's fully explained, and he has even a part two letter for the density is an ordinary letter for the world. I think it's in Spivak's part line, I'm sorry. Well, I do not check it. Well, it's not. It's fully explained in the book, and more or less 4.6. And when you deal with physics in the 4 dimensional space, I mean, it's good to distinguish between... It's interesting. But the point is that this current, this current, I mean, this current makes sense in infinite-dimensional case. And because, well, what is the difference? Let's say, you know, if you have a differential form of degree P, you have P indices, alternating indices. If your dimension is D, if you go to the corresponding density you have d minus p indices. So, but you have, you may have p indices below or d minus, below or d minus p indices above. But then suppose that p is d, so, so you have a differential of a maximum degree. So, but, so infinitely many alternating indices. Now, subscript. But if d is infinite, it doesn't make sense. But if you go to the corresponding density, the number of indices is d by d, which is zero. And then when you

52:30 go down, you have one index, two indices, and so on. But superscript, not subscript. And while I developed this, I started from some people by Kossoul about the whole of divergence, and I elaborated on the tiny lines. And so you can define things. So, I mean, but this is really in the state of the calculus of variation, a naive part of calculus of variation. And I think at this point, but it's enough to substantiate most of the calculations done by physicists. And in a naive way, it's just about to say that you are dealing with infinite dimensional integrals, but you allow integration by parts. In a naive way, integration by parts is analogical, which is a mild form of structure and whatever. But it's interesting that to know that Feynman, who was a little brash, would say, I know nothing, except integration by part. And it's true that in this calculation... Differentiating under the integral side. Exactly. I know a few of that kind. And it's true that he was a master of making difficult computations with using only these other primitive tools. Which in turn is nothing but Leibniz's rule of differentiation. Yes, Leibniz's. Which in turn, Heisenberg's principle is nothing but that. Exactly, exactly. Go back to Leibniz's rule of differentiation. So I fully agree with you that there is a naive line of thought, naive interpretation, by a line of thought in differential geometry, which is very, very powerful and very leech, and which is more or less at all with a standard uphorsion, self-economic uphorsion, topology functions. Exactly, that's very true. This is my dream for 35 years or more. Basically, that one can make a rigorous theory without foundations, quote-unquote, precise decision on various kinds of cohesion because physicists and

55:00 engineers they work that way but there is a story you see that their work is inherently non rigorous and ours is inherently rigorous and the only way to be rigorous is epsilons and deltas and charts and Cauchy sequences and all these things none of which all of which are obviously very useful for doing be required for a rigorous foundation in some kind of higher algebraic style but of course a closed category algebra but it's interesting also that there is another trend which is in principle unrelated but I may be one of the few people who was conversing with both let's say algebraic geometry and mathematical, but I think in some way, at least in France, I'm one of the few people who can be conversed with mathematical physics in algebraic geometry. And in my recent work in algebraic geometry, I mean, connected with motif and polylogy and so on, I mean, the fashion now, the fashion now is that, I mean, you deal, when you deal with the value, new idea is that the value of the zeta function for integers, zeta 2, zeta 3, zeta 4, should be represented as definite integrals. For instance, zeta 3 is a triple integral over a cube. Zeta 4 is a definite integral over a four-dimensional cube, and so on. And the idea is that all these numbers, which we hope to prove to show to be transcendental, my My big deal is to prove that zeta of 4, which is 5 to the 4 divided by 90, is an irrational number. We know that, we have known that for many, many years, but of course I'm interested in a new method of proofs, because zeta 3 has been known to be rational by Aperi, but since Aperi a very little progress has been made. And I have some idea how to prove it for zeta 4, if I go through, then the way is open. So, what is your idea? Can you give us a round? Well, the point is that, I'll start with,

57:30 Apenis' proof was very ingenious but totally obscure. One of the best explanations was given by Boykers. Yes, yes, yes. Boykers' version of Apenis' proof. So, Bollinger's idea in one sentence is the following. Let's start with zeta of three. It's a triple integral. Now you want to find rational approximation to zeta of three, which conveys fast enough, well, the rule is that in the transformational number, if a number is approached too well by rational numbers, it cannot be rational. That's a basic idea. So, you want to have rational approximation to zeta of three, but the idea is to construct of the form an over bn, which is the same to our linear combination, an zeta 3 minus bn, with an and bn integers of Archibald. And so that an zeta 3 minus bn, we have control of the both of an and bn, and on the other hand, the difference goes very quickly to zero. So you need two things. You have integral combination with integral control of both of the integers, and the linear combination as a control decrease to zero. But then, Boyker's idea, or Boyker's explanation of the proof by Appering was to represent e n zeta 3 minus b n by another integral over the cube, the tiny dimension cube. So, the idea is to take a certain rational function, raise it to the nth power, so you have the cube. You take a certain rational You integrate it over the cube. Now, elementary analysis tells you that our order decreases. Of course, if you integrate r power n over a certain domain, it's easy to control. You take the maximum value of r, or the models of r, let's call it alpha, and the integral will decrease as alpha to the n. Well, it's easy, very easy. Nothing more is required from analysis. Nothing more. But on the other hand, AN and BN. Well, AN and BN, my new idea, which I'm pursuing with one of my students, is that AN and BN, by looking exactly what the people did in calculation, are interpreted as commercial classes,

1:00:00 characteristic numbers. I mean, period, whatever, comma. And then, of course, the difficult way is to find this rational, I mean, in, in, Boyker took the calculation done by Hapir, which was not zero-santiated. Well, I mean, he gave the formula. And then Boyker discovers that this formula represents the value of some integrals. And he found, and his clever way was to find the rational function. But in a more general situation, we don't know which rational function to choose. And now since I have a common legal interpretation of the whole thing... Okay, I can play. I can play something. I can play my game. Well, it's not a thing. But if I can extend the boy person from three dimensions to four dimensions, I'm quite confident that I will have a very small hole in the wall and the water will throw out. I'm quite confident that a small hole in the dam will provoke a shrew, a disaster, not a disaster. You have a rational reason for choosing this rational thing. Exactly, exactly, exactly. But, you know, the point, I come back to our point. Sorry, just a technical explanation. But the point is that what we do, I mean, so, of course, what we do, we do all this calculation, explicit calculation, integration, integrals, definite integrals, and of course, to manipulate them we use integration by part. That means the Stokes' theorem is behind all these calculations, then we do analytic calculations. But at the end, we have a so-called multiplic interpretation of these things, which means that a motif is, in a sense, a piece of an algebraic manifold. Well, you don't know what it means to take a piece of an algebraic manifold, but you know what it means to take a piece of the DERAM complex. The DERAM complex can be split as a direction of some sub-complexes. You have interesting sub-complexes and DERAM complexes. And all in all, splitting an algebraic

1:02:30 variety into a sum of two motifs means more or less taking the DERAM complex and splitting as a director, as you love to, of course, in a director of two complexes, while it's more complicated as the techniques or structure, et cetera, et cetera. The technique is more complicated, but the basic idea of something. The basic idea of motifs from Gottendik is that if you have a space, an algebraic manifold, which has commodity in degree, that takes only the even part, the odd part is a little more complicated, even part of the commodity, H0, H2, H4, and so on. to Rotendieck and Atziya, we know that H0, at least with rational coefficient, can be calculated from K0, which is at Rotendieck-Atziya's theorem. So, that's why I consider only an even-dimensional, although you have to make suspension. But when going from an even-to-odd, it's known by suspension. It's not the input. Except that in arithmetic and algebraic geometry, it's much more complicated. what you have. So, you have Gotenlich's idea. Take, let's say, a space which doesn't have all-dimensional projective space to cast mind and space like that. Then, the idea of Gotenlich is that the different commonality group, that you can split your algebraic variety into pieces, such that each piece corresponds to exactly one of the commodity groups. One of the commodity groups. Let's say, take a projective space in three dimensions. It has commodity dimensions 0, 2, 4 and 6. And then you say, my projective space, P3, should be split in four pieces. A piece which has only a zero-dimensional commodity, a piece which has only a two-dimensional commodity and so on. But more than that, because we have Cartesian product, because we have Cartesian product, if you have something on dimension P, multiply it by Cartesian product, something on dimension 2, you get P. Okay? Suppose that we have something which has commodity only in degree 2. If you take the Cartesian product of this thing with it, and the corresponding object is called the Tate model. Okay? So multiply it, take the Cartesian product with such, you produce something which has

1:05:00 only converging according to Kuhnet-Wern in dimension 4. Well, notice that these objects do not have, I mean, for instance, these objects may have a converging dimension 2 with no converging dimension 0. So, it's not exactly a space in the ordinary, it's a piece of a space. A virtual space. A virtual space. A larger category. Yes, in a larger category. So, I mean, so you take the tape motif. The tape motif is something which has commodity dimension 2, one dimension, commodity dimension 2, and nothing else. We multiply L by L, something which has commodity dimension 4, and nothing else, etc. And now the protective space appears as 1, accounting for the commodity dimension 0. One is a one-point space. L is a tape model, accounting for the common dimension. L squared accounts for the common dimension, etc. So the projective space pieces, 1 plus L plus x squared plus L2. Now, one more thing. Suppose that you have your projective space over a finite galaxy with two elements, Count the number of elements. 1 plus q plus 2 squared plus q, q. This is a very strong gesture. It's a very strong gesture. So, if you count the number of points, 1 plus q plus q squared plus q, and you take 1 plus l plus i squared plus l cubed, it is very similar. And you can do the calculation. Well, so, practically what you do, you take your algebraic situation, you take your auto-algebraic model, you imagine they are overfinite, you count the number of points in a very naive So, when you know how to control the number of points, when you subtract the severity, when you make a blow-up, it's very easy to control, naively. Then you get some expression, a polynomial, and you cue, and you say, okay, please cue my head and have an expression in multiplicity. Then they have to substantiate this claim. But then the point is that all this complicated integrity, finally, I mean there, so, you pretend to do real calculus, you integrate from 0 to 1, dx, 0, 1 plus x square, etc. You produce numbers, you produce numbers, etc. But then if you adhere to the moral that

1:07:30 the only rule as a rule known to Feynman, integrating by part and integrating and differentiating under the integral side. So if you play that game, I mean if your calculation is used only as a set of rules, little more, little more, little more, but you can state explicitly the rule, then you are guaranteed that your calculation makes sense in this abstract category of motif. So, most of the next idea of motif was, so, various commodity groups are contribution in an antibiotic manifold or some scheme, are contribution of some pieces. But you can isolate these pieces, and they live by themselves. And also, there is a principle of transmigration of thoughts, which means that suppose we have the excision-exact sequence of comority, space X, and open set U, and U, and L, and F. You write the excision-exact sequence. Then, you have, in the spirit of motif, tell you that each comority group has a motivic But in this exact sequence of excision exact sequence, you may, by using the exact sequence, you can transfer one commodity group to another commodity group, but not necessarily with the same dimension. Because in the excision exact sequence there are cobalt and the OPA, which is related to the duality or photomorphic function we spoke about. Exactly the same. The duality or this kind of duality eventually rests on the excision exact signals. So, the basic, the basic insight of Gottlieb, was that a homology group, the dimension of the homology group is not an intrinsic thing. You can migrate a modific homology from what is changing the dimension. But there is something more substantial, which is called the weight, which is the real dimension.

1:10:00 The dimension which is kept along all the interpretations. Those are the principle of migration of souls, but the souls have an inner power which is their inner dimension, So all those transmigators, migration and reincarnation, and that's called the weight. But then, for the vice conjecture, then that is which counts. So if you have something always w, then you have to insert a factor qw in counting the points. So, the original version of Bayes' objective was that in order to calculate a number of points, you have to take into account the various betting numbers and to take the commodity. But the discovery of whether it's Delinio of Wotelnikov can be debated, but it's Delinio who took advantage of that, but whether it was inspired by Wotelnikov, I don't know. It doesn't matter, it doesn't matter. But Berlin insisted very much on this idea of weight. But I think it's just a technical incarnation of these ideas of Gottelnik. And Gottelnik has these ideas. But Gottelnik failed on one point. He thought that in order to substantiate all this kind of claim, you should prove further odds conjecture. Not only the odds conjecture, but the odds standard conjecture. Many people have reservations about the Hodge conjecture, despite the fact that it could bring you over a million dollars. Already on the living room, long ago, had reservations. And the present consensus among algebraic geometry is that the Hodge conjecture is almost true. That means that putting a few extra-technical conditions, you can prove it. Well, no one is in the position to do that in large cases. But the consensus is more that you, in order to prove, let's say, suppose that your algebraic manifold has a model over the field of algebraic numbers. Well, there's no doubt that in this case, I mean, autoconjecture is true. In a more general situation, people are reluctant to admit it.

1:12:30 So I think it would be advisable, well I'm not interested in all the conjecture, but there are many people who have, and it would be advisable at the same time to look for a proof under very mild CDI conditions and for countering somebody to know how it gets. And that's my determination about the clay prices, because the clay prices, if someone If you can't prove the conjecture, it will be even more million dollars. If you just find a counter example, it will give you a piece. I don't think how he's doing it. But what do they say about that? The Institute said it, because people have complained about this before. What does the Institute say? Well, I rely on my objections that many people did, but I don't know what the outcome is. The natural thing for them to say would be, oh, we said it wrong, you get the million for refuting it too. Yes, but the same with the Poincare Conjecture. But now, since the work of Perlman, I'm more confident about Poincare Conjecture. But for a long time, I'm not an expert, again. On the Poincare Conjecture, I can say something. On the Poincare Conjecture, I'm an outsider. Obviously, I'm an outsider. On the old Conjecture, I'm not completely an outsider, but on the Poincare Conjecture, I'm an outsider. But then, the same, for a long time, I mean, when people discovered all these new invalions for three-dimensional money for the link invalions, the break groups, and so on, I said, wow, now we have a lot of new invalions for three-dimensional money, why should you imagine that they're all encoded in the Taiwan, no one can really imagine that they could be most of the invalions, well, now, in terms of the work of Perlman, it's not yet the end, but nevertheless, one could discover in the corner of the country, but you suddenly must have advised me, Clay. I mean, you know, any mathematician. They have to hear this all the time. Yeah, I mean, you suggested that they must have gone for advice to mathematicians, I suppose. Any mathematician would have said, well, of course. Some mathematicians said it this way. Oh, yeah, many. I mean, you know, it seems to be... But somebody put it the way the Clay Institute did. Exactly. That's outrageous. Now, I'm hoping this was in a moment of absent-mindedness. They've not changed it yet and they've heard that. I'll tell you, I have some insider information.

1:15:00 They are, the people have a dinner together and they divide a thing after dinner. That's all. That's all. That's enough. They thought it was enough. It's not the serious work. But why they don't simply say, that's not what we meant? Yes, of course. And when they presented the prices in Paris, and they presented, I mean, what did he present? I mean, the hot conjecture and something similar. He told me when I boarded the plane to come to Paris, I was instructed about what I would have to report. I didn't even be made one hour before living was in... Well, it's not serious, but they gave a good talk, no problem. They gave a good man, I'm sure, an honest person. But it's very, you know, I mean, without seriousness, without seriousness. But of course they are located to retreat, or to retreat even... Not even much. They could do face-saving, and this is not what we intended to say, but it is not. Well, I will not enter into personal controversies and such. I admire ideas, but it's not important. No, but the point is that, so, but, let's go back to Hodgkin's conception and the standard conjecture. Gautendijk, I think, made the wrong choice. He had the impression that in order to substantiate the idea of motive, one needed to first prove the odd conception. And it's true also, and there are some written evidence that when Dering proved the Vege conjecture, Grotanik was a little disappointed because he had plans that we should, I mean the order, the natural order would be first proved odd conjecture and the standard conjecture. Then out of that create a fear of motive. And out of this fear of motifs, a very conjecture should follow immediately. That was for him the natural way. And he was a little disappointed by Doreen.

1:17:30 Doreen was an astute clockmaker. I think he just can't be the way of thinking. Doreen is a clockmaker. He has all the tools in his hands. He has good eyesight, good imagination, and clever is very clever. And he knows how to use his hands to do clever, clever machining. He's a clockmaker, or problem solver. He's a clockmaker. And it's true that when he gave his lecture about his proof of the vague conjecture, I mean, unfortunately, they could not attend them. He gave one week of lecture. And the people had the impression that at each lecture, they knew already everything. That means he did not invent new tools or new basic methods. I mean, so what he did was to assemble all the pieces. So he took all the pieces from his stock, and all the screws and the screwdriver and so on, and patiently he put one piece on top of another piece. And at the end he said, the clock is working. As I mentioned already, my ancestor was a clockmaker in Chihuahua. So, it's really a Chihuahua clock. So, you spend many evenings and you are saying, well, at the end, not only the clocked medium, but the angels are singing and dancing. Now, I don't understand the standard to do this. The Germanic was very disappointed that he was... And Rotenig was very disappointed by that, because he thought it was trickery, trickery. And Rotenig has a bias against trickery in mathematics. Well, but isn't there a shorter, a less grand route that goes through Hodge to the standard conjectures, not really yet a theory of motives, but still a simpler et alchol homology proof of the Bay conjectures? I think that's what Botnik thought was feasible in the 60s. The full theory of motives would be next, and be an even simpler route. Yes, but nevertheless, I mean, it's so that we should prove the standard conjectures.

1:20:00 Yes, yes, and not this elaborate calculation on Lefshund's consoles. Yes, exactly, exactly. But... So the standard conjectures... Well, you may not... No, but I'm asking, because the standard conjectures don't really give the whole theory... No, not exactly, but they were a necessary standard. But at this in the sinking of water. And the recent progress has been to bypass this difficulty. Because no progress has been made towards the standard conductor for many years. But nevertheless, Morel and Royer-Walt and Levine were able, but using a huge categorical machinery, using a huge categorical machinery, to bypass this and to create a few mortgives which are all the features which were expected. It's sort of a huge cosmological machine. Now, did they have a motivic proof of the Veig conjectures? Anyone? Does their theory of motives prove the Veig conjectures? No, it's an easy outcome. No, it has been known for many years, actually. If you had a reasonable theory of motive, the Veig conjecture should flow out of that. Well, Gorgonsky doesn't have the whole theory of motives that he wants yet, right? Not yet. But he does have that. And at the present time, the status of Maltivic Führer is the following. We have all this complicated construction by Levin, Morel, Wojewódzki, and you should mention also Spencer Block. He took the initial steps. Spencer Block took the initial steps. And so, and you are now a very elaborate homology furium, which is the chief furium in a complicated, over complicated site, and which substantiate most of the claims. Not all, but most of the claims. But there is a subcategory, which I call the take motif, pure mixed take motif, which corresponds to the rational varieties. I mean, so it's not that, that's what I said, I mean, one-dimensional carbon monology occurs, let's say, for an elliptic curve.

1:22:30 And in the original work of the Riemann hypothesis for attribute curves, I mean, H1, so you have H0 and H2 which are tubular, and H1 which is an important step. Because what does the survey say in the areas of curves, which was already known by Hasser for the elliptic curve, which was extended to the higher genus case, is that if you have a curve of genus G, yes, if you have a curve of genus G, and you define it over a field with QN, The number of points, rational points, if it was a genus 0, it would be 1 plus q, the number of points. And then, because the projective line has a commodity with a 0 and a commodity with a 2, it went to 1 plus q. But then, there is an analytic curve of higher genus, there is an intermediate H1, whose dimension is exactly the genus. And then, you have correction. So, an algebraic curve of genus geographic risk element, the number of elements is 1 plus 2 plus correction. And the correction comes from the h1. And the point is that the correction is a sum of a certain number of g numbers, or 2g, 2g numbers, alpha 1 of 2, alpha 2g. but each one is of the order of square root of the cube. That's exactly Bayes' conjecture. So this number can be written, the correction can be written as alpha 1 plus alpha 2b, but each alpha i is bounded by square root of the cube, not only of the order of square root of the cube, it's bounded by square root of the cube, which was important then, which was known already to hassle for any continuous one. So, the correction is of order square root of Q, and for all this application to coding, cryptography and so on, that's important here. And to improve on this rough estimate square root of Q by clever kind of conditions.

1:25:00 But then you have H1. But, okay, but this H1 exists only in the case of higher genes, not genus 0. And what distinguishes genus 0 from the other case is that genus 0 is a rational variety. So, there is a general rule of thought, which means that if you have a rational variety, then the all-dimensional commodity doesn't play any whole. So, while I'm simplifying a little bit, but if you have a certain suitable category of rational variety, only the even-dimensional commodity will play all. And more or less, I mean, if you deal with this category of spaces, you have a very economical definition of the corresponding motif. And the idea is that a motif is now a complex of rational varieties. So, you take a complex of rational, projective, non-singular varieties, A complex in a suitable sense, you have to define. It's not an additive category, actually. So the first step is to transform these algebraic varieties into an object of an additive category. By using the old idea which is already presented in the pool by Henri Vey, that you replace a map from a variety X to a variety Y by an algebraic correspondence. In a sense, a correspondence which associates to any point in x a finite number of points, so it's a multivariable function, from x into y, with finitely many determinations. So, explicitly it means that, explicitly in a more abstract, more categorical term, you want to go from X to Y, you first take a finite unamified covering over X, or maybe unamified, you take a finite covering over X, and then to go from X to Y, you just climb from X to the covering and then down. Yes, that means that you invert formally, you invert in your category of algebraically, you invert formally the finite covering, So, you start with this category of, let's say, rational algebraic varieties, complete or projective algebraic varieties, you invert formally these maps, the final, the final

1:27:30 like a proper colouring, or how we find all, how we find just this, just a technical point. And then you get an extended category, but now this category is additive, not the point. Because if you have a multivariate function which is associated to any point index, a zero cycle in y, I mean a finite number, boom, this zero cycle may be additive, added. So you have a natural definition of the sum of two such parts. So you have an additive category. Now, if you have an additive category, it makes sense to speak of a complex in an additive category. And really, the poor mass, the poor mass definition of the gate-mortem, you start with a rational, non-singular, protective variety. You make it an additive category by inverting formally this mass. Then you take a complex into that, up to, well, in the DI category, a complex of two, up to quasi, and that's all. It's a Poors-Max definition. And the point is that if you play correctly with homological algebra, If you have a complex of variety, you may have a DRAM complex, a double DRAM complex. So you have, let's say, variety X0 going in X, X, X, X. Then you take the DRAM complex here, DRAM complex over each one. Now you have a double complex, because first of all, each vertical is a DRAM complex x0 differential, and then the map from one variety to the other one induces the map, horizontal differential. complex, the data total comorology of the capital complex. And you can even combine this with a, you are really to use a triple complex. Because what Gauten did discover is that the natural commodity for an algebraic variety is not the sheep comorology. Well, there is a total comorology on one hand, which was his greatest discovery, totally unexpected. but relying on the method introduced by cell and others. So, take an algebraic curve. Take an algebraic curve. In shift theory, you have, in shift theory, you have a permutivacy.

1:30:00 You have H0, which is the count field, and H1, which is T dimensionally, T is the T. That means if you calculate the common deal with respect to the shift of copper rings, regular functions. But this is not in the full carbology. I mean, for a curve of G and Z, you have something, you have, the veggie numbers are 1, 2G, 1. And not 1 in G. Half of the carbology is missing. Using Oitch Fluid you realize, according to the line we discussed yesterday, you realize that you have to add to this, the carbology I mean, the coherent cohomology, or Zeiss-T cohomology, for the curve is valued in the sheath of the differential form of the d1, omega-1. So now what you have, you have the complex of sheaths, omega-0, which are the ordinary, well, the regular functions, going into omega-1, you have a complex of sheaths. And then, if you have a complex of sheaths over a topological state, you can define the hypercharmology. I prefer, in this situation, to call it the check-deramcharmology. Check-deramcharmology. Because, I mean, it's known that the coherent cohomology, the stairs cohomology, can be calculated using a deramcharm complex, a large check-complex by covering with a fine sub-variety. Now, if you want to calculate the sales commonology, the Raven commonology, you can do that explicitly with a check-complex which was exactly the definition of sales. it. Before, it was shown by Groton being that it fits into a general definition of common theory of genes. But you remember the step. The Zeiss-Litopology is highly pathological from the point of view of an analyst. And at first, in the seminar, people concentrated or locally compact spaces, or compact spaces, but nevertheless, house-door spaces. And all the shift machinery was developed for house-door spaces.

1:32:30 And then came this shock that the size-speed topology was not house-door. And that's one reason, I mean, people were able to introduce size-speed topology into the game. It's not house-door. Contrary to old intuition. Well, analytical intuition. So, then these are applicable. We have no guarantee of how the Czech commodity works, and using a few tricks, a few tricks, Sam was able to show that the Czech commodity had the right properties. But so we had another theory of commodity for SHIELD. We did not fit into a general pattern. And then, I think it was one of the strongest motivations for Gauternic to invent his general definition of Czech commodity. He wanted to cover both applications. He wanted, I mean, out of his general philosophical mathematical principles, he could not admit that there was, on the one hand, a theory developed by Leroy, Borel, Carton and Sey for ordinary spaces, house of places, and the other one, the comedy developed by Sey for Zeiss-Petoparty. He wanted to unify this. And that was one of the strongest motivations in Europe for talk. One of the strongest motivations for talk. Okay.