Am discussions: FW Lawvere, P Cartier, C McLarty, A MacIntyre, JL Bell

0:00 It seems that the basic outline was supported, namely that from the general functional analysis one came to the example of the polymorphic dual spaces, which led one then to complex spaces. And the Tita-Gaga principle to algebraic space. This is a very, 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 guess in a somewhat spotty manner, perhaps. The GaGa, I've asked you this before, GaGa was a paper by Sayre, but the germ of the idea goes back to Riemann, I think, so it's a huge systematic elaboration. Categories of vector bundles and categorizing 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... In the so-called Riemann-Hilbert problem, to characterize an equation, to characterize a class of robotic functions whose singularity of the differential equation is satisfied. It's basically that and it's basically something else. Well, of course, Riemann-Hilbert problem evolved in different ways after that, but I mean, it really is a basic idea, too. But actually, the holomorphic function should be characterized by the topological structure of the Riemann surface more or less. But actually, I was thinking last night, a connection I hadn't made or forgotten, maybe. But in some sense the Grauer direct image theory is connected with this and the relative spec of the chief of brains, the chief of local brains. But on Monikov-Kein's thesis, it came after, but in a certain way it is also...

2:30 I think that was expressing this Gaga Principle. The Gaga Principle was because the fibers of this proper map are essentially algebraic. Well, I mean, the starting point for Se was the result of Charles that if you have a, in a projective space, if you have a complex, a small complex analytic manifold in close, embedded into the projective space, automatically an algebraic variety. This is Charles' theory, I mean, that was the starting point. And he wanted to interpret this and to extend it, and not only to the varieties but to the functions. So it's a big comparison between what you can derive by Schiff's method using only holomorphic functions or using algebraic functions. Well, it comes also to the question, to the old question, I mean, of which hypergeometric functions are algebraic functions, which is Schwarz's work in the 19th century. All these things are connected. I mean, Schwarz's work on hypergeometric functions can be summarized in the following way. Well, hypergeometric functions are solutions of certain differential equations. So from that differential equation you generate, using a singularity, you generate a Riemann surface, a Riemann surface, by various methods, by sheaths or by gluing it, but whatever. But if you can go down to a certain finite covering space, then for some values of the parameter, the hypergeometric function descends from this huge infinitely sheeted Riemann surface to a finitely sheeted Riemann surface.

5:00 Now, the point is, according to Riemann, that in this case, the finitely sheeted Riemann surface is an algebraic curve, is an algebraic curve, and therefore, the hypergeometric function sitting on an algebraic curve is an algebraic function. That's more or less there, right? Of course, Technical points, I mean, for which value of the parameter hypergeometric function you get algebraic function is a complicated algebraic problem. But this principle has expanded by Riemannism. So the main point is that, of course, an open Riemann surface can allow many, many functions which are certainly not algebraic in any sense. But if you can have a compact Riemann surface by going down to a compact Riemann surface, provided you know how to enter the finitely many point correspondence singularity, then you create a Riemann surface. And a compact Riemann surface is an algebraic curve, which is really the point in Riemann. And therefore, this is the first step in the comparison between holomorphic function and algebraic function. And so for the one-dimensional case, what Riemann did, and then Schwartz's contribution, etc., completely is a problem. But for the higher dimensions, that was the main challenge. And Chauffeur, Chauffeur was the first step in that. But Chauffeur is a little different because it's already embedded in a projective space. So in a sense it's easier, well not so easy, but easier. But what they wanted was to have something like in Riemann's, in the Riemann spirit, not something embedded. I mean that holomorphic, I mean to show that some curve is algebraic, if it's already embedded in the projective plane it's not too difficult, usually it's not too difficult. But when you have an abstract curve which is not embedded, that's the main point, and I think the steps that Sehtuk took were very similar. And of course he relied on the direct invention of Coward. I remember very well, I mean, Coward was closely associated to all these works at the time. Through Yitzhigou. Yitzhigou was a golden ball. Cannot be over-emphasized. Cannot be over-emphasized.

7:30 That was, as I said, more successful than the Bobakir seminar, in a sense. Well, different aid, but I think with something similar in the spirit, but I think it was more successful, in a sense, to disseminate new ideas and to bring people together. It's an opportunity I never attended in my life. The distributions, if I understand it correctly, the distributions on a, the analytic distributions on a bounded open moment are themselves represented by actual functions on the complementary, on the however of exponential growth. So, this is, it's a kind of duality that's different from Pontryagin duality, let's say, or Stone-Scheck duality, this kind of duality where you... For example, the Pontiagin duality says if you look at the distributions on one group and represent those as functions on the dual group, so you pass to the 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 different, apparently a different type of duality comes up where the distributions are... On one part are representative of the complement, sort of set theoretical, logical complement, actually. 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 of distribution and somehow passing straight down to one-to-one duality. The qualitative one might follow from the quantitative one. This is just a speculation, of course.

10:00 I think it could be known. I think it could be known. Well, that was a kind of method that would give you the exact duality for the case of a sub-manifold embedded into one. And I think that kind of, I remember, gave the Poincaré duality. You can use the Derham complex, which is based on differential forms, smooth differential forms, but on the other hand, and it's known from the book of Derham, that you can build a Derham complex using currents instead of differential forms, of smooth differential forms. And it's so, and it's not to, well, it's using Schiff's method, it's easy to show that they're too common, because all you need is a local Poincaré and according to the wealth. All you need is a local Poincaré lemma which says that a closed differential form is locally an exact differential form, and the same for currents is easy to prove, rather easy to prove. The method, the tensor product method, for instance, that will go to NICUs for the Doral lemma applies easily. If you want to be sophisticated, but there are more elementary methods, I myself publish a short paper on elementary explanations.

12:30 So, then you have the comparison and then shift theory tells you that if you have two, you can calculate the same cohomology using two different resolutions. Basically, that's the principle of shift cohomology. So, you have a resolution using smooth differential form, another one using wide, one current is highly, non-discontinuous, I mean, singular differential form, but they give you the same cohomology. But this is not yet what you want, because then if you want to, but of course you have to reform the curator in duality, but you get into subtle points of duality in infinite dimensional spaces and exact sequences. The point is that for other spaces or such, the notion of exact sequence is delicate. Because you have, very often you have a map of one Banach space and another one and the image is not close. So what does it mean, I mean, the exact sequence? So you have a very crude way of saying an exact sequence. You take, I mean, the kernel of one map is a closure of the image of the other map. But that's too crude for many professors. That's very much too crude. And of course if you work within a category of Banach space, it's such a natural notion. That's all you can say. All you can say. If you stay within the category of Banach spaces. Or more general spaces, that's interesting. So, that's one difficulty. And if you try to apply this to the duality, so you have two complexes, one with smooth differential and one with current, they can be put in duality, and then you can say, of course, the cohomology of the dual is the dual of the cohomology. But if you work in that setup, I mean, you are in Banach spaces or generalization of Banach spaces, you are stuck by this. Difficulty of what is an exact sequence. What's needed is to obtain functional analysis. Then, Duody's idea was the following. I take the complex of smooth differential full. There is a topology on these functional spaces. And I take the topological dual to define the current, which is a standard definition. But what if I take the full algebraic dual? In analytical terms, it means nothing.

15:00 So I take the complex of smooth differential form and I take at its level the complete algebraic truth. All the linear forms, continuous or not, would, of course, formally that makes a new complex, but shift methods enable you to show that this new complex, which is much bigger than the complex, so you have the differential, smooth differential form, current, and this, I would say, Wide currents, I mean. Well, it's like golden mass resolution. Exactly, something like that. Something not necessarily continuous. Yeah, yeah, something like that, something like that. Which we interpret as moving to a different topos. I think he was influenced by that. And then he said, well, when using sheaf method and resolution, you show that this huge complex of very wide object defines the same probability. But this time the duality is from nothing, because you have moved to not from the category of binary spaces to the category of vector spaces, where the duality factor is all right. I mean, the real idea takes one line to explain. Of course, there is some technique, I mean, a standard technique, but the new idea takes one line, and it's a fantastic idea. Yes, and then at the end, of course, you can show what you want. So you bypass this difficulty. There are other methods to bypass this difficulty. You're using Hodge theory. It's much more delicate. Analytically, it's much more delicate. Well, Hodge theory and existence of parametrics of differential operators, all these analytic techniques which are very useful by themselves, it's a much more economical way to prove the practical humanity. And I suppose you could accommodate this idea for the Alexanderian. 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 kernel, 1 over z minus w, basically. So what's the analog of that in the non-analytic case?

17:30 ...to transport the linear dual of the function space on one side of the order into... You need a core kernel of some kind. ...has been generalized in many ways in many complex variables. And... So we're starting with Leray. Leray was the one to first... And then many people around him, around Leray, developed these ideas. And in the first Gelfand's book on distribution... There is a very general notion of residue, which is a real analytic version of residue, not a complex analytic, which relies on the Stokes theory. So I think the idea would be that, let's say, The way I see it is that you have a certain manifold and a sub-manifold. Take a tubular neighborhood with a nice boundary, which can be considered as a sphere boundary over the sub-manifold. Then using gazing exact sequence, you can make explicit gazing exact sequence by... By introducing suitable candles, and you could get the inspiration to find these candles in the book by Gelfand, because at some point he defines a real residue, a real residue, which is just an application, I mean, the real residue of, I mean, the idea of Gelfand is the following, the Cauchy formula, Cauchy formula is really that you have one singular point into, on a Riemann surface.

20:00 What is a tubular neighborhood of a one-point? It is a small circle drawn around a point, or a small cycle drawn around a point, and in the interior, the interior of the circle or the loop, so you have a one-point, you take a loop around, and the interior of the loop is a tubular neighborhood. And the Cauchy formula can be, well, it's well known that the Cauchy formula can be derived from Stokes' theorem, element derivation of Stokes' theorem in one complex. The idea is that if you have one point in the n-dimensional sphere, you take a small sphere and apply Stokes. But the interesting point is that this idea extended to some period of situation. For the harmonic functions we could use the Poisson curve. Yes, for the harmonic functions you can use the Poisson curve. And you can generalize this. You can generalize this in various ways. I mean, for many complex variables this has been done by Lorien and his disciple Norgay. He developed this idea. He has written a number of papers about these generalized curves. And there was also some Italian who contributed, maybe Gemona in the beginning, one of the first papers of Gemona. There are some contributions by Italians. But yes, some people tried to accommodate this idea to a real variable situation. And I think so, I think you could derive, you could derive a proof of, analytical proof of the actuality. But it's restricted. It's sub-manifold, non-simular, sub-manifold embedded in a non-simular manifold. So it's certainly much more restricted than what you can achieve by topology.

22:30 And again, in the duality of holomorphic functions, I mean, the main point in the... Grottenigda-Silva 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 row set, but it's not a circularity, in a sense it's geometrically two-dimensional, it's compacted with a boundary or even wider than the space with a boundary, I mean with a smooth boundary. The real point is that, by definition, a holomorphic function of such a space is a function which extends holomorphically to some neighborhood. That's the point. And, geometrically, the point is that when, suppose you have a certain compact domain in a complex plane, then, by definition, a holomorphic function is a function which extends slightly to some neighborhood. That's a holomorphic function. So, it infringes order on the complement. That's the point. And then, elementarily, what you do, so you have the holomorphic function which is on the compacted, it extends to some neighborhood, then in the ring which is around this compacted, I mean the intersection of this domain with the rest, then you draw a loop, you draw a loop and you apply ordinary Cauchy formula or something more sophisticated using a kernel, you know, a kind of Poisson kernel. And in the real situation you don't have this possibility because of course a smooth function on a sub-manifold extends to a smooth function in the ambient manifold, but far from being unique in this instance, even locally. The point is that in a neuromorphic case the extension is locally unique, at least for germs of function it's unique, but in the real case... Ah, so in the complex case sometimes you don't really need the whole complement. Yes, and you can restrict yourself. You can restrict the representation that you do. But then, I mean, the cohomology, when you go from one variable to many complex variables, then you have to use, the duality is now cohomological. And this was elaborated by...

25:00 By Mikio Sato and by so-called micro-functions and generalised... The starting point was Martino, who introduced so-called analytic functionals. And the work of Martino was directly connected with the work of... All the inspiration was more or less Laurent Schwartz in both cases, when Dasilva was a student of Laurent Schwartz, and I think that more or less Laurent Schwartz was the one who inspired these ideas, I mean, Martineau was a student of him. These are some of the key terms that we can use to understand the theory of Lorentz-Schwarz and what was still under the influence of Lorentz-Schwarz at that time. So I think if you want, I think it's really Lorentz-Schwarz who gave the impulse, the starting point to all these lines of development. But then it was developed. And on the physicist side, there are some physicists, especially in something Morse and other people, I mean the hedge of the wedge theorem, It's just an elaboration of that. But it's really concretical and I think a recent, I think the most recent exposition should be taken from the book of Shapira in Kashiwa. I mean, at least between the lines, there is a heavy homological mash in Shapira-Kashiwa, but the geometrical inspiration is that kind of thing, that kind of thing. All right, so I don't know if you read my paper. No, I'm sorry. And the Randy Crosses Seminario Matemático y Físico. No, the culturkulomathematiker, the Palermo. Yeah, Palermo. Palermo, Palermo, and Europa. Well, from a distance I can't come down to teach Spanish. So this is kind of a posthumous debate with Dieu Donne. You know, there is this two-volume history of the 20th century by Peer.

27:30 I have an article in the second volume, but in reading the first volume before that, I came across some of the issues which I'll address here, so there are at least two different things that you mentioned that are relevant to this. Du Garnier says that in a book, we must finally mention the first attempt at quote, functional analysis, unquote, of the young Voltaire in 1887, to which, under the influence of Hadamard, Has been attributed and exaggerated to historical importance. So this was DuVernay. And then he goes on, DuVernay, in connection with Volterra's notion of the derivative of a functional. Boudinet further states, with our experience of 50 years of functional analysis, we cannot help but assume that without even the barest notion of general topology, these ad hoc definitions were decidedly premature. So I remark that one might ask if calculus of variations was premature. Yes, exactly! So, under this, and then the Italian mathematician who was the head of the Academia d'Insee, Ficare, he answered this, you see, so, again, I didn't agree with his answer, but, so this is the origin of this paper, so I studied this, I tried to study some of these historical issues. So, well, for one thing, jumping ahead, I mean, you mentioned... Duody and Duran, utilizing the differential forms to obtain the occurrence, and you said topology, but actually they both use bornology instead of topology, which is, you know, for Banach's phases makes no difference, but for these things it's definitely similar, as Duody recognized.

30:00 So there's this notion, you used the term analytic functional. Already the Voltaire School, oh yes, that's right, another source of the debates. The authors, Russian names, have a recent... A very useful biography of Hadamard, probably you know these people, let's see what they're called. Anyway, while acknowledging Picarra's arguments, they had already access to this, acknowledging Picarra's arguments nonetheless described as, quote, naive, unquote, the theory of analytic functionals developed and studied by Pantapier, Pellegrino, Heffely, Suchi, Teichmuller, Silva, Voltaire, even Max Zorn and others. Yes, that's another language. I'm sorry, I forgot about it. More specifically, they say that the naivete of Pantapier's approach lies in the fact that he, like Hadamard 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 mathematicians were not in fact naive about the goal of continuity as demonstrated by the fact that they published many papers proving that analytic functions, in their sense, were in fact continuous, for grace, policy, and so forth. This was quite an interesting, actually, historical exercise for me because it's in some way the same general issue as the other realm. 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, it could be... There are various smooth and analytic things, but the analytic functions, functionals, like the, already in the work of 300 years ago, Calculus of Variation, were simply based on the idea that the smoothness of a functional

32:30 All of this is tested by looking at all possible paths or all possible figures of finite dimension in a function space and requiring that the results 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 inference of lambda calculus or in other words Cartesian closed categories, a figure in a function space is the same thing as a figure of twice the dimension. In another finite-dimensional space, you just bring the exponent down. So the same type of adjointness of hom and tensor, except in a criterion situation. So that was the concept of analytic functionals, which this whole Italian school, but they weren't all Italian, there was this Swiss and... But recently I mean there have been various attempts to find 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 the, the loop space in some, in a manifold. And we, he wanted to have a Duran complex, I mean of course the loop space, the cohomology of the loop space has been calculated by Young and Mayer, Maclean, Sayer and so on. And, but he wanted to calculate the cohomology of the loop space by using a kind of Duran complex. And now the chain was able to, was able to, to develop. To develop a calculus of differential forms on such loop spaces, exactly by seeing what is a differential form on a loop space. Of course, there is no difficult thing about the meaning of a map from some simplex, a continuous map from some simplex into the loop space. Which is just a slightly elaborate, well, it's the same idea as yours.

35:00 So, and then, well, usually in calculus of variation one takes one-dimensional deformation, but you can take infinitely many parameters of difference, exactly the same. So, for Chet, the definition is that a differential form, to define a different... So, the loop space comes with all these maps which, in a sense, describe the singular complex of your space. So a differential, of course, here you have a space which is far from being a manifold, but at least it has a notion of singular change in a good sense, and even smooth singular change, not only continuous singular change, but smooth continuous change makes the obvious sense in the spirit of calculus. And then what is a differential form of such a space? It's... Associate to every single chain a differential form on the chain, which is something finite dimensional, with some consistency, which means it's factorial. So that's the way you define a differential form on such an infinite dimensional space. And now you can, all the algebraic operations, exterior product, exterior differential, are given form. And then you calculate, so you have a very general definition. Then you have a Dirac complex and you have various sub-complexes and you show that the cohomology of this Dirac complex calculates the homology, at least the real homology or cohomology of your loop space or bar space and so on. And this has been very, very successful recently. I mean, the people like Sullivan, who developed the rational motor p-method, and then Richard Hine, Dick Hine, and then Deligne, and then Deligne, and one, and many people after him. And so, and now, of course, it's, when Deligne defined the Durham version of the pi-1. 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 the form, so that becomes a loop of tangent vectors over here, and you integrate over the loop? Exactly.

37:30 It's a very naive notion. What is an infinitesimal deformation of a curve? It's a vector at each point of the curve. Exactly. Naive. It does a good thing. But the point of what Bill, I think, was trying to envision was that it's a good thing. There are all these notions that somehow got... All of these were lost, if you like, in what was the mainstream development, which are then called naive, premature, unformed, raw, you know, and so on, which were actually always implicit in later developments that sort of got detoured by, if you like, mainstream. I mean, I think that's particularly true in… People were forced to come back to it. Yeah, of course they were. And I think that was the point. I think it was also the point of working, if you look at the Italian school of algebra and geometries, the point has 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, I mean, because it got resurrected, but really by later on. Through work in differential geometry later work, which was never really part of the mainstream of the set theoretic mathematics. I would say that the issue is really calculus of radiation. Calculations of variations and people invented them in the spirit of, let's say, well, there are various approaches. You can define functional spaces, functional spaces, various topologies, and then try to define infinite dimensional manifolds which are modeled on local, local model based on Hilbert spaces or Manard spaces. There was a great excitement at some point about that. And when Burbaki wrote his own sketch of differential geometry, I mean, the summary of results of differential geometry, I mean, more or less influenced by... By Lange at that time. Lange plays a role in the writing of these book lines and I followed the advice, well I was responsible for one of the drafts of the book and I followed the advice of Lange and then from the beginning we deal with infinite dimensional local model, the more local model are Banach spaces and the more valuable are these local model Banach spaces.

40:00 But for the purpose of calculus of variation, it's completely irrelevant. Engineers continue to use the old form. They can actually calculate the derivatives of the functionals without ever thinking about topology. But what Chen produced is that, of course, a first order of variation in the calculus of variation is something well known. I mean, if you look at the elite engineering or physical literature, it's full of applications. Second order derivation is already a one. It's used in many places, Jacobi operator, and so on and so forth. The idea of having a calculus of exterior differential form is a new feature. But it can be given exactly as what Shen did was to do it exactly in the same space. So, a tangent, I mean, if you have a curve, what is a tangent, what is a tangent vector to a longer curve? It's a field where you are committed in, in physical terms, infinitesimal deformation, so for each point, for each point on 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, 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 machine of calculus of variation, the powerful method of elicator of exterior differential form. You produce a version of the Doral theorem which gives you the cohomology of the Loeb space, but the Loeb space is in the other interpretation. Interpretation of functional analysis. Look at the set of all continuous maps of your circle into space and all with a functional analytical topology. So there is a bridge, there is a deep bridge there between the two approaches, the 9-1 and the set theory. But, and a chance contribution cannot be underestimated. They are very, there was very, very, I must look into this because, you see, we have, we have a further step, namely that the notion of differential form itself is a function, it's a representable function, with the, with the representable and the other variable, right.

42:30 So the differential forms on the loop space, say order 3, are actually maps from the loop space itself into fixed representing objects independent of the space of order 3. It's sort of bringing this duality back to the more basic level of Euler. And there is one extension of this circle of idea, Souriau, the French mathematical physicist of Marseille, who is a rather exotic mind. He is a very original mind, and he was one of the first to publish a textbook on the new method of Hamiltonian mechanics. I mean, about the same time as... Arnold, a little later than 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 group of students working this idea of, well, that a differential form is a fine tool, etc. They called them deep theology. So, a diphological space is a space you don't give a topology or whatever, you just define what is a small smartphone, a simplex, into it. So you give a singular complex. The space is described as a singular complex. There are at least six different schools which have variations on this idea that communicate very little with each other. Exactly, exactly, exactly. I think it's a very good idea. That's why there are six different schools. But really, I keep discovering new examples of it in the literature which apparently we don't know about or at least don't refer to. Iglesias, I mean, if you want to name Iglesias, look at Iglesias. He is one of the students of... These people are motivated by problems in mechanics, but they tend to be a little too old. But can't be so.

45:00 Someday I will give a full explanation of my ideas about finite integral in such a way. I have a new idea, which is not in my book, which is in the retrain. Well, the book is here, you might have the proof she's here, but it's not. The idea is that in an infinite dimensional situation, you can build, well, the idea is that in ordinary manifolds, There are really two Durham complexes. 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, Durham is well aware of that, so he describes it. But the point is that Poincaré duality says that in finitely many dimensions, there is no real difference between them. The complex and its dual are more or less the same. That's exactly what Poincaré duality means. But when you have infinitely many dimensions, it's different. You have what I call the ascending the RAM complex. You start from the zero form, which are functions, then the function form would be one, et cetera, et cetera, et cetera, not a co-boundary operator in this. But if you have a space of dimension infinity, when you take the dual, you should descend from infinity to infinity minus one, infinity minus two, infinity minus three, et cetera. And I can define, well, I can define it by duality, but it's a little awkward. But I can define, I can define a descending complex. Of course, in infinity and dimension they will never meet. So there is no Poincaré duality, because it should compare the combination dimension P with the combination dimension infinity minus P. But you have really two complexes, one ascending, one descending. And they never meet, but they are in duality. So it's an extended version of quantum duality for infinite dimensional space. And now, of course, it brings us back to an old idea in differential geometry, which was well known in the 1920s, a distinction between a tensor field and a tensor density.

47:30 Fields, tensile fields and tensile densities. If you look at the book of Hermann Weyl on round-type materials, spacetime matter, it's fully explained. In any case, even 5.2 later for the densities and ordinary later for the speeds. I think it's in Spivak's 5.5, I'm sorry. Oh, maybe, maybe. Well, I did not check it. Well, it's not. It's fully explained in the book by Hermann Weyl, and more or less forgotten things, more or less forgotten things, and when you deal with physics in the four-dimensional space, I mean, it's good to distinguish between Hermann Weyl. It's interesting, but the point is that these currents, these currents, I mean, these currents make sense in infinite-dimensional case. And because, what is the difference? If you have a differential form of degree p, you have p indices, alternating indices, if the dimension is d, if you go to the corresponding density, you have d minus p indices, but you may have p indices below or d minus p indices above. But then, suppose that P is D, so you have a differential of a maximum degree, so infinitely many alternating indices down in 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 go down, you have one index, two indices, and so on, but I mean, a superscript. And while I developed this, I started from some people about divergence, the role of divergence, and I elaborated on that idea. And so you can define things, you can define things. So, I mean, but this is really in the spirit of the calculus of variation, a naive point of view of calculus of variation.

50:00 And I think this point... It's enough to substantiate most of the calculations done by physicists, and in a naive way it just amounts to saying that you are dealing with infinite dimensional integrals but you allow integration by part, in a naive way integration by part is allowed, which is a mind form of Stokes and whatever, but it's interesting to note that Feynman It's true that he was a master at making difficult computations with using only these rather primitive tools. Which in turn is nothing but Leibniz's rule of differentiation, which in turn, Heisenberg's principle is nothing but that. Exactly, exactly. Go back to Leibniz. So I fully agree with you that there is a naive line of thought, a naive interpretation, Mark, a line of thought in differential geometry, which is very powerful and very rich, and which is more or less at odds with the standard. This is my dream for 35 years or more. Basically, one can make a rigorous theory without foundations, that is to say, without a precise decision of various... These are all kinds of cohesion. Because physicists and engineers, they work that way, but there is this 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 some calculations, but should not be required for a rigorous... It's interesting also that there is another trend which is in principle unrelated.

52:30 But I may be one of the few people who is conversant with both, let's say, algebraic geometry and mathematical physics, well, I'm certainly not, but I think in some way, at least in France, I'm one of the few people who can be conversant with both mathematical physics and algebraic geometry. And in my recent work in algebraic geometry, I mean, connected with motifs and polynomials and so on, I mean, the fashion now, the fashion now is that, I mean, you deal... When you deal with this, the main 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 big dream is to prove that zeta of 4, which is 5 to the 4 divided by 90, is an irrational number, of course. We know that, we have known that for many many years. But of course I'm interested in a new method of proof because Zeta 3 has been known to be rational by Aperi but since Aperi 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 rough? All of this was very ingenious but totally obscure. One of the best explanations was given by Boyko's version of apirism. So, Boyko's idea in one sentence is the following. Let's start with Zetovsky. It's a triple interval. Now you want to find rational approximation to zeta 3 which conveys fast enough. Well, the rule is that in a transcendental number, if a number is approached too well by rational numbers, it cannot be rational. That's the basic idea. So you want to have rational approximation to zeta of 3, but the idea is to construct the form an over bn, which is the same to have linear combination an zeta 3 minus bn, with an and bn integers or rationals.

55:00 And which shows that an zeta 3 minus bn, you have control on the growth of an and bn, and on the other hand the difference goes very quickly to zero. So you need to have integral combination, control the growth of the integral, and the linear combination has a control decrease to zero. But then, Boyker's idea, Boyker's explanation of the proof by Aperin was to represent a n zeta 3 minus b n by another integral over the cube. So, the idea is that you take a certain rational function, raise it to the nth power, so you have a cube. You take a certain rational function, you raise it to the nth power, you integrate it over the cube. That's all about the decrease. 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. Obviously, well, it's easy, very easy. Nothing more is required from analysis. Nothing more. My new idea, which I'm pursuing, is that A-N and B-N, by looking at exactly what the people did in calculation, are interpreted as cohomology classes, characteristic numbers, I mean, periods, whatever, cohomology. And then, of course, the difficult way is to find this rational... I mean, Boelckes took the calculation done by Appel, which was... He gave the formula and then Boyker discovers that this formula represents the value of some internet. And his clever way was to find the corresponding rational functions. But in a more general situation, we don't know which rational functions. And now since I have a common liquid interpretation of the whole thing, OK, I can play. I can play my game. Well, it's not yet finished. But if I can extend Boyker's rule from three dimensions to four dimensions... I'm quite confident that I will have a very small hole in the wall and the mud and the water will flow out.

57:30 I'm quite confident that a small hole in a dam will provoke a flood and a disaster. Well, a disaster, not a disaster. You have a rationale. There is no reason for choosing this rational function. Exactly, exactly. A little bit of a coincidental reason. 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 these calculations, explicit calculations, integrals, indefinite integrals, and of course, to manipulate them we use integration by part, that means all the Stokes theorem, the Stokes theorem is behind all these calculations, integration by part, Stokes theorem, etc. Then we do analytic calculations, but at the end we have a so-called motivic interpretation of these things, which means that Motivic interpretation is that a motive 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 Durham complex. The Derham complex can be split as a direct sum of sub-complexes. You have interesting sub-complexes in the Derham complex. And all in all, I mean, splitting a natriubaric variety into a sum of two motifs means more or less taking the Derham complex and splitting it as a direct sum. That's up to, of course, quasi-osmomorphism in the direct sum of two compiles. It's more complicated enough to take mixed-source types, etc., etc. The technique is more complicated, but the basic idea is that. The basic idea of motifs from Grothendieck is that if you have a space, an algebraic manifold, which has cohomology in degree, it takes only the even part, the whole part is a little more complicated, it takes even part of the cohomology, H0, H2, H4, and so on. And according to Grothendieck and Atiyah, we know that H0, H4, at least with rational coefficients, can be calculated from K0, which is a Grothendieck-Atiyah theorem. Okay, so that's why I consider only the odd, even dimension, although I'd love to make suspension, but we're going from even to odd is not my suspension, it's not the input, but except that in arithmetics and algebraic geometry it's much more complicated, but then what you have, so you have...

1:00:00 Goethe's idea is to take, let's say, a space which doesn't have all-dimensional probability, a projective space, a cashmere, a space like that, then the idea of Goethe's idea is that a different cohomology group, you can split your antimagnetic variety into pieces so that each piece corresponds to exactly one of the cohomologies. Let's say, take a projective space in three dimensions. It has cohomology in dimensions 0, 2, 4, and 6. My projective space, P3, should be split in four pieces. A piece which has only a zero-dimensional cohomology, a piece which has only a two-dimensional cohomology, and so on. And more than that, because we have Cartesian product, if you have something on dimension P multiplying by Cartesian product something on dimension 2, you get P plus 2. If you take the Cartesian product of this thing with its one and the corresponding object is called the tape motif, so multiply, take the Cartesian product with that, you produce something which has only cohomology according to Krennert-Fern in dimension 4. Well, notice that these objects do not have, I mean, for instance, these objects may have a cohomology in dimension 2 and no cohomology in dimension 0. So, it's not exactly a space in the ordinary life. It's a piece of a space. It's a virtual space. Virtual space. Larger category. Yes, in a larger category. So, I mean, so you take the tape motif. The tape motif is something which has cohomology dimension too. One dimension, cohomology in dimension 2, and nothing else. We multiply L by L, Cartesian problem, we get something which has the cohomology in dimension 4 and nothing else, etc. And now the projective space appears as 1, accounting for the cohomology in dimension 0. 1 is a one-point space. L is the take part of accounting for the cohomology dimension 2. L squared accounts for the cohomology dimension 4, etc. So the projective space P3 is 1 plus L plus L squared plus L cubed.

1:02:30 Now, one more thing. Suppose that you have your projective space over a finite Galois field with two elements. Count the number of elements. 1 plus Q plus P squared plus Q cubed. Take, so, if you count the number of points, 1 plus Q plus 2 squared plus Q, and you take 1 plus L squared plus Q, it's very simple. And you can do the calculation. So, practically what you do... You take your algebraic situation, you take all your algebraic models, you imagine they are over-finite, you count the number of points in a very naive way, so when you know how to control the number of points, when you subtract the sub-IT, when you make a blow-up, it's very easy to control, naively, then you get some expression, a polynomial in Q, and you say, okay, I place Q by n and have an expression in motivity, and then you have to substantiate this claim. But then the point is that all these complicated integrals... Finally, you pretend to do real calculus. You integrate from 0 to 1, dx, or whatever, xy, etc. You produce numbers. But then if you adhere to the moral that the only rule is the rule known to Feynman, integrating by part and differentiating under the integral side, So if you play that game, I mean if your calculation uses only that set of rules, or a little more, you can state explicitly the rules, then you are guaranteed that your calculation makes sense in this abstract category of multi-means. And I see it as so. So Goethe's next idea of multi-means was that so. Various cohomology groups are contributions 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 you have the excision-exact sequence of cohomology, space X, an open set U, and you write the excision-exact sequence.

1:05:00 You have in the spirit of motif tell you that each cohomology group has a motiving interpretation and that in this exact sequence, or is an exact sequence, you may, by using the exact sequence, you can transfer one cohomology group to another cohomology group but not necessarily with the same dimension. Because in the excision-exact sequence there are co-boundary operators, which is related to the duality of holomorphic functions we spoke about, exactly the same. Alexander duality, also this kind of theory, eventually rests on the excision-exact sequence. So the basic insight of Goethe was that the dimension of the cohomology group is not an intrinsic thing. You can migrate a motivic cohomology from one state to another by changing the dimension, but there is something more substantial, which is called the weight, which is the real dimension, the dimension which is kept around all the interpretations of the principle of migration of souls, but the souls have an inner power which is the inner dimension. Which they keep along all those transmigrations and reincarnations and that's called a weight and but then in the in the for the for the conjecture by conjecture by conjecture then that is which count so if you have something always w then you have to insert the factor qw in counting the points. Not diamonds and so on. The original version of Bale's conjecture was that in order to calculate the number of points, you have to take into account the various betting numbers and then to take the cohomology, but the discovery of, well, whether it's Deligne or Botany, it can be debated, but at least it's Deligne who took advantage of that, but whether it was inspired by Botany or not, it doesn't matter. But Deligne insisted very much on this idea of weight. But I think it's just a technical incarnation of these ideas of Grotendieck. And Grotendieck has this idea.

1:07:30 But Grotendieck failed on one point. He thought that in order to substantiate all these kinds of claims, you should prove first the Hodge conjecture. Not only the Hodge conjecture, but what he called the standard conjecture. Many people have reservations about the Hodge conjecture, despite the fact that it could bring you a million dollars. Already Andre Weym, long ago at reservation, and the present consensus among algebraic geometers is that the hot conjecture is almost true. That means that putting a few extra technical conditions, you can prove it. Well, no one is in a position to do that in a large case, but the consensus is more or less that you, in order to prove, let's say, suppose that your algebraic manifolds are the model over the field of... There is no doubt that in this case, but in a more general situation, people are reluctant to admit it. So I think it would be advisable, well I'm not interested in all the context of Atiyah, Witten, and it would be advisable at the same time to look for proof under very mild auxiliary conditions and for counter example in the general case. And that's my reservation about the clay prices. Because the clay prices, if someone proves the conjecture, it will be given over a million dollars. If you just find a counter-example, it will give you billions. I mean, to me, it's a scandal. But what do they say about that? Because people have complained about this before. What does the Institute say? Well, I rely on my objections, and many people did. I mean, the natural thing for them to say would be, oh, we said it wrong, you get the million for refuting it too. The same with the Poincaré conjecture. But now, since the work of Perelman, I'm more confident than the Poincaré conjecture. For a long time, I'm not an expert. Again, I'm not an expert. I can't say something on the Poincaré conjecture. I'm an outsider. Obviously an outsider. On the odd conjecture, I'm not completely an outsider, but on the other, on the practical conjecture, I'm an outsider. But then, the same, I mean, for a long time I was, I mean, when people discovered all these new invariants for three-dimensional money,

1:10:00 for the link invariants, the brain groups, and so on, I say, well, we have a lot of new invariants 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 more subtle invariants. Well, now after the work of Perelman, it's not yet the end, it's not yet the end. But nevertheless, one could discover in a quarter of a country. But somebody must have advised me, Clay. I mean, you know, any mathematician. They have to hear this all the time. Yeah, I mean, who suggested that they must have gone for advice. To mathematicians, I suppose. Any mathematician would have said, well, of course. I have not seen that petition said it this way. Oh, yeah, many, many. I'm not the only one. You know, it seems to me that... But somebody's put it the way the Clay Institute did. Exactly. That's outrageous. Now, I'm hoping this was in a moment of absent-mindedness. But they have not changed it yet, and they've heard that a lot. I'll tell you some insider information. They had the people at a dinner together. And they devise a scene after dinner. That's all! That's enough! That's enough! That's enough! That's enough! It's not so serious. But why they don't simply say, that's not what we meant? Yes, of course. when they presented the prices in Paris at the College de France and they presented I mean what did they present? He told me when I boarded the plane to come to Paris, I was instructed about what I would have to report. I was even amazed one hour before leaving Austin. They gave a good talk, no problem, they are good mathematicians and honest persons, but they did it very, I mean, without seriousness, they did it without seriousness, but of course they had occasion to re-read and to re-read even in face saving, they could do face saving, and this is not what we intended to say, but they did not. Well, I think, well, I will not enter into personal controversy, so... I have my ideas, but it's really not important. No, but the point is that, so, but let's go back to Hodge Conjecture and the Standard Conjecture.

1:12:30 Rotendijk, I think, made, I mean, made the wrong choice. I mean, he had the impression that in order to substantiate the idea of motive, one needed to first prove the Hodge Conjecture. And it's true also, and there's some written evidence, that when Dunning proved the wedge conjecture, Grotendieck was a little disappointed, because he had planned that we should, I mean, the order, the natural order would be first prove hodge conjecture and the standard conjecture. Then out of that, create a fear of motive. And out of this sphere of motifs, the very conceptual should follow immediately. That was for him the natural way. And he was a little disappointed by Deligne. Deligne was an astute clockmaker. I think he discounted his way of thinking. Deligne is a clockmaker. He knows, I mean, he has all the tools in his hands, he has good eyesight, good imagination, and he's very clever, and he knows how to use his hands to do clever, clever machinery, to do clever machinery. He's a clockmaker, or problem solver. He's a clockmaker. And it's true that, I mean, when he gave his lecture about his proof of the Bay Conjecture, unfortunately he could not attend it. He gave one week of lecture. And the people had the impression that at each lecture, well, they knew already everything. Okay, 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 screwdrivers and so on, and patiently he put one piece on top of the other piece. And, at the end he said, the clock is working. As I mentioned already, my ancestor was a clockmaker in the 80s. It's really a Chironian clock. So you spend many evenings and you assemble and at the end, not only the clock is ringing, but the engines are singing and dancing.

1:15:00 Now, I don't understand the standard conjecture as well, of course. And Gottlieb was very disappointed by that because he thought it was triggering, triggering. 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 etal cohomology proof of the vague conjectures? I think that's what Grotendieck thought was feasible in the 60s. The full theory of motives would be next. Yes, but nevertheless, I mean, he thought that we should prove the standard conjecture. Yes, yes, and not this elaborate calculation on Lefschetz pencils. Yes, exactly, exactly, exactly. But the motive... No, but I'm asking, because the standard conjectures don't really give the whole theory of mathematics. No, not at all. But they were a necessary step. Well, at least in the thinking of water. And the recent progress has been to bypass this difficulty. Because no progress has been made towards the standard conjecture for many years. But nevertheless, Morel and Voyevod were able, by using a huge categorical machinery, using a huge categorical machinery to bypass this. And to create a motive which has such a cost effect. Does their theory of motives prove that they exist? Oh yes, it's an easy outcome. No, it has been known for many years that if you have a reasonable theory of motives... Well, Wawowski doesn't have the whole theory of motives. Not yet. But he does have that. The status of a motivic theory. We have all these complicated constructions by Levin. Well, you should mention also Spencer Brock.

1:17:30 There is a sub-category which are called the take-multi-pure, which correspond to the rational varieties, I mean, so it's not that, that's what I said, I mean... One-dimensional cohomology occurs, let's say, for an elliptic curve. And in the original work of Weyland, the Riemann Hypothesis for an elliptic curve, I mean, H1, so you have H0 and H2 which are trivial, and H1 which is an important step because what does Weyland say in the case of curves, which was already known by... For the elliptic curve, the way to the genus case, is that if you have a curve of genus, if you have a curve of genus G, define it over a field with Q element, finite field with Q element, the number of points, rational points, if it was a genus zero, it would be one plus Q. And there is a, because the projective line has, but then there is, An elliptic curve or curve of higher genus is intermediate H1, whose dimension is exactly the genus, and then you have corrections. So an algebraic curve of genus G over a field with two elements, the number of elements is 1 plus 2 plus corrections, and the correction comes from the H1. And the point is that the correction is a sum of a certain number of G numbers. These are the two G numbers, alpha 1 up to alpha 2G, but each one is of the order of square root of 2. That's exactly Bayes' conjecture.

1:20:00 So this number can be written, the equation can be written as alpha 1 plus alpha 2G, but each alpha i is bounded by square root, not only of the order of square root of 2, but bounded by square root of 2, which was the important step, which was known already to Hassell for elliptical genus 1. So, the correction is of order of square root of Q, and for all these applications to coding, cryptography and so on, that's an important thing, and to improve on this rough estimate square root of Q by moves, clever company. But then you have H1. But this H1 exists only in the case of higher genes, not Geno0. And what distinguishes genus zero from the other case is that genus zero is a rational variety, I hope so, so there is a general rule of thought, which means that if you have a rational variety, then the all-dimensional cohomology doesn't play any role. So while I'm simplifying a little bit, but so for if you have a certain suitable category of rational variety, only the even-dimensional cohomology will play a role. More or less, I mean, if you deal with this category of species, 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 the complex of rational, projective, non-singular varieties. A complex in a suitable sense, you have to define. It's not an additive category at first. So the first step is to transform these algebraic varieties into an objective of an additive category by using the old idea which is already presented in the proof that you replace a map from a variety X to a variety Y by your algebraic correspondence. In a sense, a correspondence which associates to any point in X a finite number of points. So it's a multivariate function from X into Y. With finitely many determinations So explicitly it means that explicitly in more abstract and more categorical terms you want to go from x to y you first take a finite unramified covering over x or maybe ramified we 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.

1:22:30 Yes, that means that you invert formally, you invert in your category of algebraic varieties, you invert formally the finite covering, finite proper maps. So, you take, you start with this category of, let's say, rational algebraic varieties, complete or projective algebraic varieties. And then you get an extended category. But now this category is additive. Because if you have a multi-valued function which is associated to any point in x, a zero cycle or more, this zero cycle may be additive, added. So you have a natural definition of sum of two such. So you have an additive category. Now if you have an additive category, it makes sense to speak of a complex additive category. You start with a rational, non-singular, projective variety. You make it an additive category by inverting formally these maps. Then you take a complex into that, up to, well, in the derived category, a complex up to quasi-hazard, and that's a motif. Well, it's a poor math definition. And the point is that... The point is that if you play correctly with, if you play correctly with, if you have a complex of variety, you may have a Derham complex, a double Derham complex, you have, let's say, variety, x0, y0, so over, then you take the Derham complex. Now you have a double complex because, first of all, each vertical is a dark complex with zero differential, and then the map from one variety to the other one induces a horizontal differential, and that's a double complex, you take the total complex. And you can even combine this with, you are really to use a triple complex, because what Goten discovered is that the natural commodity for an algebraic variety is not the shift commodity,

1:25:00 But relying on the method introduced by Say and others. So, take an algebraic curve. In shift theory, you have a curve of genus e. You have h0, which is the count field, and h1, which is g-dimensionally, which is the genus. That means if you calculate the cohomology with respect to the shift of local a. But this is not in the full cohomology. I mean, for a curve of genus D, you have something, you have the Betti numbers are 1, 2G, 1, and not 1 in D. Half of the cohomology is missing. Using Hodge theory, you realize, and according to the line we discussed yesterday, you realize that you have to add to this the cohomology, I mean, the coherent cohomology of Zeisky for this value in the shift of So now what you have, you have a complex of sheaves, omega zero, which are the regular functions, going into omega one. You have a complex of sheaves. And then you can, if you have a complex of sheaves over topological space, you can define the hyperthromology. I prefer to, in this situation, to call it the Chek-Durham cohomology because, I mean, it's known that the coherent cohomology, the search cohomology, can be calculated using a Chek complex by covering it with affine sub-varieties. That's all. If you want to calculate the cohomology, the Sears cohomology, you can do that explicitly with a check complex which was exactly the definition Sears used before it was shown by Bolton being that it fits into a general definition of cohomology of genes. But you remember the step. The Zayaski topology is highly pathological from the point of view of an analyst.

1:27:30 And at first, when in the capital seminar people concentrated, let's say, on locally compact spaces or far-compact spaces, but nevertheless outlawed spaces. And all the shift machinery was developed for outlawed spaces. And then came this shock that the science of topology was not outlawed. And that's one reason, I mean, people were afraid to use science of topology in the beginning. It's not outlawed, contrary to all intuition, well, analytical intuition. So, we had no guarantee of how the Czech cohomology worked and using a few tricks, a few tricks, Serre was able to show that the Czech cohomology had the right properties. But so we had another theory of cohomology for Schiff, which did not fit into a general pattern. And then, one of the strongest motivations for Gotenig to invent his general definition of chief cohomology. He wanted to cover both. He wanted, I mean, out of his general philosophical mathematical principles, he could not admit that there was, on the one hand, the theory developed by Rui, Boel, Carter, and Sayer for ordinary spaces, house of spaces, and on the other one, the cohomology developed by Sayer for Zeiss-Littroporty. He wanted to unify this. And that was one of his strongest motivations to develop two photo books. One of his strongest motivations. Now, if you want to calculate the cohomology, why do I call that check-durham cohomology? Because you have a double complex. If you have a check covering and a differential form, then you can have a double complex, which is exactly the complex introduced by André Weil in his proof of Durham cohomology. And when I have to give an introductory lecture in that field, I mean, I go back to this proof. It's a wonderful proof. And now what Goten did discover is that combining the idea of Sayle and this idea, one could define, well, it was more abstract. He said hypercognitive of a complex of Sheep. I want to be a little more concrete, and I say I have a double complex, which is a Chek-Durham complex. I could call it very complex because it was implicitly thought to be very complex, but it has two diamonds, Chek and Durham.

1:30:00 But now if you have a complex of varieties, you have a triple complex to calculate, you have a third dimension which is, let's say, suppose I have a simplicial variety, to be a little more specific, I have a simplicial variety which is often the case, a simplicial or co-simplicial, of course, but then you have a third differential which is a simplicial differential, so you have the check differential, the logam differential, and the simplicial differential, and you have a triple complex. And you have the full complex. So rather than one eliminating the others, all three must cooperate now. All three must cooperate. And so, that's a rather poor man's definition of a motif, at least a day motif, which is enough for many applications. And so, each object is really a complex of projective, rational, non-singular algebraic varieties. Well, let's say, take a simplicial variety. Very often it's a simplicial. Of course, if you can have an additive, well, you know, if you can add any additive category, a simplicial, out of a simplicial object, you can make a complex, alternating some of the functions. And so, but then, two commonly asked dynamics. You have a triple complex. And this is also called mathematics. Now, the game is forming. You first do these calculations, explicit calculations by integrating, and then you reinterpret everything in terms of a pairing between some homology and some cohomology. And all the calculations are done at a very formal level. And what distinguishes numerical calculation from the other? That's one of my discoveries. So at the end you have a certain cohomology. And all the objects will say that zeta of 3 will be an element in some cohomology group. But what makes it a number, a real number, is that at some point in each of these cohomology groups are vector spaces of finite dimension over the rational. But in each case you have a natural code in these spaces. And the point is that, if you have a natural chord, you can speak of a linear form which takes positive value on the chord.

1:32:30 And now, let's see, just to take a look. What is square root of 2? I take a two-dimensional vector space over Q. I'm drawing the line x is y multiplied by square root of 2. This line is irrational. It doesn't contain any rational point except... But it divides my rational plane into two half-planes. I take one of them as a positive core. Now, on these vector spaces, there is exactly one linear form with value in the real, which takes positive value on one of the half-planes. Explicitly, it means that you are looking for inequality of the form a minus b square root of two positive. So the dedicating, if you want to have a geometric idea of what is a dedicating cut, you take the rational plane with two rational coordinates and you draw a line, a straight line. It doesn't contain, it doesn't contain any rational point except the origin. Divide the planes into two and these two half planes correspond to the two lower half and upper half. I mean, if you want to understand what is a continuous fraction expansion of, a simultaneous continuous fraction expansion of more than one number, let's say one has square root of two, square root of three, then you move to a three-dimensional space and you cannot accommodate. And a brain, I mean, there are some algorithms for extending a continuous fraction. And so the idea is the following. So everything is purely algebraic. I mean, it doesn't have any mention of the real number. But if you start from this idea that the real number is a cut among the rational numbers, if you give a geometrical interpretation, instead of a cut you say it's a rational plane with a division into two half planes by a certain line, then you have the whole information.

1:35:00 So, the number, the ration, you create the square root of 2, zeta 2, zeta 3, etc., they are first of all purely algebraic objects, except that at the end, they are interpreted as static incomes by this check. And this trick is in Quine, and in fact he points out you can use the integer plane. Yes, of course you can do it. I don't claim it's a Quine trick. But the zeta function thing is not. I don't make a claim for that. Interpreting the continuous fraction expansion of a number in terms of a straight line in the integral plane or rational plane, it makes little difference. Except that it makes little difference. Except that in our situation usually you have a rational space and not a relativist, and maybe a preferred relativist, but not always. It's irrelevant to choose from, it's irrelevant to choose from, the rational side to choose from. So I'm making no claim, I mean, about novelty, I mean, just using, I mean, my claim is, I did not want this interpretation of Dede Pinkert is, well, no, but what my claim is that you can do all the calculations, what is delta of 3? Zeta of 3, there is a multiple zeta of 3 which lives in the certain cohomology group. And just at the end, you have the map into the real number. And the real number do not have to pre-exist. They do not have to pre-exist. And so, we really create explicit real numbers, let's say, and then to prove that they are transcendental or whatever, you don't need pre-existing reals. I think that would be interesting. This is Chanuel's construction of the reels. You mentioned Tate because Chanuel claims to have been motivated by Tate. He called it the Tate reels for a long time, and then the Chanuel reels, so now he says, no, it's the Eudoxus reel, because Eudoxus did exactly this. But I have a categorical account, which you might read me. Once again, strangely enough, it's bornology which comes with it. Because, say in the case of the relation of the, you start with the integers, as Colin said, you start with the integers, and you ask, there's the notion of homomorphism of the adequate group of the integers, which of course is nothing but multiplying by a constant, but now there is the notion of a bornological homomorphism, an approximate homomorphism.

1:37:30 Which means that you ask f of x plus y is f of x plus f of y. You take the difference of those two and you ask to be bounded as a function. I've heard that already. You told me such and such. So you have the approximate things and then you can take the equivalence. So basically you start with the category of monological abelian groups and then you define new hams by this method. You get a new category. Such that the, for example, the endomorphisms of the original object Z has now become the reals. And so on into the vector spaces and so forth. And you're essentially just giving us a non-linear version of that, because in a sense you could use some algebraic equation like x cubed root of... x squared. x squared, in order to also to have these... The difference between your approach and what you're mentioning and Quine's is we've got this line that has no points. How do we specify that line that has no points? You're doing it by this novel homomorphism. Quine just says subsets. Just take the subsets with these properties. So that's where the difference is coming in. But the point is that, I mean, of course, I'm well aware of the philosophical implications of all these reasonings, but my aim is a little more technical. I want to have an approach to prove the validity of numbers. And it's exactly that. I mean, you say, how do you know that there are no points? And there are no points on the line. I mean, how do you prove that the number is irrational? That's my main concern. Bolker's proof of zeta of 2 and zeta... well, Apery's proof of zeta of 3 is irrational, but in the process he re-proved that log 2 and zeta 2 are irrational, which was known in the past. Irrational has been known already in the 18th century by Lambert. And log 2 is irrational, I think, what? I suppose Leibniz could prove it if he needed. I mean, I certainly already knew that.

1:40:00 So log 2 and zeta 2 are not new, but zeta 3 was a known thing. Bering gave proofs which are quite similar to these three cases, and Borkers reformulated them in terms of integrals. So log2 is a first-order integral, zeta2 is a second-order integral, I mean a two-dimensional integral, zeta3 is a third-order integral. And in my Japanese lecture once ago, I explained at great length all these strategies on these examples. How do you prove that log2 is irrational? And of course I took a very sophisticated... At every step in my lectures, I follow this example, but of course my aim is to, if I can extend this method to Zeta 4, I will reproduce something which has been known for many years, but with a totally new method. It's very interesting actually in this connection, the sort of historical thing, the fact that when Cantor and Dedican used their definitions, And Weierstrass too, I think. He was the first really to attempt to give the definition of the real numbers. What they did explicitly was to introduce irrational numbers. I mean, they actually took rational numbers as given, and then somehow you feel that, you know, what you're doing when you add the irrationals, it was really a construction of irrational numbers that these people were doing, not the construction of the whole lot, because the rationals were already there. And of course, they didn't actually have to give an account. Well, I mean, all these people, Camdor and Dedican, were well aware. There was a real difficulty in deciding for any real number whether expressions were irrational, but the point is that their construction gave you all the irrationals without even knowing exactly which ones were, right? It was sort of implicit in the way that they defined it, that they simply added all the irrationals without knowing exactly what they were explicitly, and that problem remained. I'm distinguishing what really new things were being added in this construction. That problem evidently still remains because it's kind of getting damaged in the construction. There is an idea, there is an idea which is in the mind of many people that between the full set of real numbers and the rational numbers there would be some, there would be a certain hierarchy and that there are, in terms of naturalness, there are classes of irrational numbers, first of all algebraic numbers, but beyond algebraic numbers, beyond algebraic numbers, of course algebraic numbers are the first step, but beyond that, and...

1:42:30 The idea of motif and motivic cohomology and motivic Galois rule is to get control of these numbers and to have a Galois theory for this extended class. So what we aim at is a Galois theory for a suitable class of irrational and transcendental. Of course, this, but then what I discovered over the years, since I thought about this idea for about 10 years, and I'm already four, five years in complete, and we're building step by step all this, and what I've discovered is that at the time you build a class of reasonable numbers, you have to build at the same time a class of reasonable functions. And a reasonable function, a reasonable number is a special value of a reasonable function. And you cannot have a reasonable number without building at the same time a hierarchy of reasonable functions. That's a philosopher because it's not reasonable. Why is log 2 a reasonable function? Log x is a reasonable function. Why is log x a reasonable function? Because it's a primitive of 1 over x. It's derivative is 1 over x. Okay, now log 2 is your reasonable number. If you specify your reasonable number in log x, you get a reasonable number. But the point is that we discover, and that's why I've been so interested in the Galois theory of differential equations, because, I mean, the Galois theory of differential equations gives you an entry into the world of reasonable functions. Adverse approximation, a reasonable function in the function which satisfies at least one differential, a reasonable differential equation.

1:45:00 Of course, everything has to be built together. You have to build... So, you want to build reasonable numbers. Reasonable numbers are special values of reasonable functions, and reasonable functions are solutions of reasonable equations. And what is the difference? Well, for instance, the equation dy is dx over x is by all means a reasonable equation. Intuitively, you have to control that. dy is dx over x over dx. xdy is dx. It's an isoneval differential equation. And so you have to build, at the same time, a class of differential equations, a class of solutions. There are various interpretations. What is a solution to a differential equation can be interpreted in different ways. It can be interpreted in a purely formal way as a power series expansion, for instance. Or something more complicated, as an asymptotic experiment. I mean, in the spirit of Riemann, if you have a singular point of your differential equation, you should control the asymptotic behavior near the singularity, and which Delisle translates as a tangential base form. Just one interpretation of tangential base form, but it means really asymptotic behavior of your solution of your differential equation around the singular point. The notion of solution of a differential equation is not a unique one. There are many possible interpretations, and you have to play between the various interpretations. In terms of logic, I mean, you have various models. I mean, you have a world of, I mean, you can formulate a formal language to deal with the solution of a class of differential equations. Formulate a formal language. And what you have, there are different models of this. But so, building a class of reasonable irrational numbers cannot be done in isolation. You have to take all these steps to guess. And to think in two categorical terms. Motives are more or less one way to interpret all these things in categories. It's not the only possible way, but at least it's one way. And then, of course, if you have categories, you have homomorphism and you have automorphism.

1:47:30 These are more or less what he got in mind in his long time. It's rather faithful. In my Chicago lectures in 1967, I proposed, published about ten years later, essentially what I'd learned from Gabriel about the foundation of algebraic geometry. I wanted to have a foundation for continuum mechanics, actually, but as a step towards a way suitable. And algebraic geometry over that. So one algebraic theory is of course the theory of commutative range. Another one is C-infinity functions, but I pointed out there are many theories in between, any number of theories in between. A second ingredient is that in algebraic theories we talk about the arities of operations, you know, binary, ternary, and so on. Well, certainly in terms of the general theory of monads, any object could be an arity. This is sort of obvious, but in particular, within this setting of an infinitesimally generated topos, there are sort of preferred arities which are So, for example, instead of a binary operation on a space X, which is a map from X squared into, say, X, you could have a map from X to the T, where T, this is really the tangent button. So, you see the maps, but the differential operators are themselves... There's a form of algebraic theory which is intermediate between the classical one with discrete finite varieties and the, you know, the extravagant one of monad theory where any object can be, which is precisely tailored to just the situation you described. But I think, I think the usual descriptions, you say, you say there's a way to... You know, we code classes of solutions of differential equations, differential rings and the like, but the differential rings are really not a very accurate expression, what you have in mind, because you want to be able to compose these maps as well.

1:50:00 So it should be a category, not a ring. Just because when you would join some solutions, you then want to substitute those things, you know, arbitrarily in other things that you worry about. Just the basic operation of categories of composition. In differential rings, there is no chain rule, but you should have a chain rule. They captured Leibniz's rule, but you need also the chain rule. And this is inherently something about composition, about... There are lots of algebraic theories in between here. For example, if you could join the exponential function to the polynomials and you could, you know, and then people say, what are you talking about? You know, this is of no interest. It's not because they're all people, I guess. Well, I would suggest that I mean, at the technical level, I've been dealing extensively with the category of so-called differentially simple rings. That's a quite convenient category, sorry. And then using not one differential ring, but a category of differential rings is not completely what you want. No, it's a bit different. It's a bit different, but at least it's some step. But I insist on the point, on the methodological point, is that, to have a very simple example, on the definition of pi, and I would not favor any definition for two, and I suppose, I mean, in order to understand what pi is, you have to have more or less an idea of all these possible definitions, for a simpler definition. But then, many, many definitions. Which is a perfect definition is Leibniz formula, pi over 4 is 1 minus 1 third plus 1 fifth minus 7 and so on, which is beautiful. Leibniz was rightly proud of that, and he claimed he had solved the problem of squaring the circle by that. Not in a scientific way, compared to the simplicity of some infinity hidden there, but nevertheless it's so simple.

1:52:30 So that sends you stuff from this series. ... of dx over 1 plus x squared from minus infinity to plus infinity to come by symmetry you split the interval of integration minus infinity to plus infinity to four pieces, minus infinity to four, minus one, minus one to zero, zero to one, and one to infinity. And by simple manipulation the four contributions are equal. So it's just an instance of... Symmetry group or rather the transformation of an equation and the change of any or functoriality of integration of differential form if you want. Then you are reduced to four times zero to one while just repeating something. So now dx over one plus x squared, one over one plus x squared is one minus x squared plus x four and so on. And you integrate term by term and you get exactly like this formula which is more or less a way like this dv. You have this, so you have this interpretation. And we started from a series which is explicitly, which has a lot of explicit ad infinitum, ad infinitum, as Leibniz would write, plus ad infinitum, explicit reference to it. When I write the integral dx over 1 over plus x squared, well, of course, it's a reference to minus infinity for the integration domain, but it's not a real infinity. No, not the same. Not the same kind of infinity. And if I split, if I take the integral from 0 to 1, it's certainly no explicit difference to infinity, except the definition of, I mean, now, if I take the integral as given, God given us a certain mathematical object, and I'm not on a certain operation, I'm not questioning how you define it, it's perfectly finite mathematics, perfectly finite mathematics. And what I learned from...

1:55:00 If you are reading carefully Bull while preparing my Japanese lecture, they will be available on my website at HCS. I have to admit that I was reluctant to have a website because people just bomb us and I don't want to put... To pull to the picture of my girlfriend and my dog. It's avoidable, actually. I've noticed some avoidance. They're one of two self-indulgences. We do have our dogs on our website. No, no, no. I'm not blaming anyone. I could do it already some years ago. That's a good thing to be doing. So, that's going to Google. This integral process is a purely formal process. Now you have a divergence. You consider all these integrals as purely manipulable. This is a symbolic, a British symbolic line of development, which we trace back in doing breakfast, I mean, which goes from bool to hevis iron to diark. After diark you find... This is a certain line of thought. But it's really Boole who started this. And Cayley, you could say also Cayley, or an algebraic system. And he said, oh, by the way, and when Boole introduces calculus, or logical calculus, when he says, which is a symbol for, it's an operation. Predicate for Boole is an operation. It's a selection operation. Yes, that's right. It's an important part which has been forgotten. When he writes x as a symbol for predicate, he means that our certain universe

1:57:30 X is selecting out of the whole universe a certain subclass and X is not the name for a class, it's the name for the operation. A very important point vis-a-vis the Frege to say that everything is a property. But I was very surprised, of course I've been reading the law of thought of Bohr many times, but I did not realize his point. Because it's explicitly in his book on the finite difference equation. The combination of two, which is an intersection of course. It means that you have to make two operations. Out of the universe you select according to the selection function when you select a certain subclass and then out of that subclass you apply x. And why is xy equal to yx? Because these two operations commute, yes. That fails in the quantum case, which is from one point I've tried to make the operation of selection. Yes, but I never realized that for both x and y, and the operations are just operations on operations. I mean, when you write x, y, it's a combination of operations, and y minus x, etc. Very often people misunderstand that. They never even don't mention it. But they're actually functions. Yeah, functions. This happens to be true science. Let us recall these remarks when we come to the day when we'll have our logic discussion. Yes, I guess so. So, you can follow more or less Boole's way of dealing with differential equations, difference equations, and so on. And as I stated during the break, first, I mean, there is something fantastic which I read in the book of Boole about differential equations. At some point, he says that when he compares two different differential equations, and in modern terms, he says that they correspond to modules over the via algebra, and these modules are isomorphic, so any result I can obtain in one situation can be translated. Could give me an explicit transformation from the solution of one equation to the other one, it's important to know that, but I can bypass. If I know how to, by abstract algebra, just rephrasing this.

2:00:00 If I am rephrasing, if I am by pure algebra showing that the module, in modern terms, the module of the binary algebra in one situation is isomorphic, without producing an explicit isomorphic, what? Producing an explicit isomorphic by analytical means, I can do algebra to show that these two modules are equivalent, by bypassing the Laplace transformation, which is very important, to me, an important remark, because after every side, symbolic calculus. People devoted a great deal of effort to substantiate it by Laplace transform. So-called justifying. Justifying by Laplace transform. It's not necessary. It's not necessary. That's a problem. That's a problem. Yes, of course. And Heaviside was faithful to Bohr. Well, I did not... Well, we should have to make a historical study about explicit influence of Bohr on Heaviside. I just read a book on Heaviside. I don't remember you mentioning it. He doesn't. He doesn't or he does? I don't remember. I do not remember any reference to it. But I think people devote to it. Many efforts to justify the Evisai calculus, first by Laplace transform, and then in a more direct way by using distribution and convolutional distribution, but it's not necessarily, it's not really necessary. And Bohr was very well aware of that. And these terms are so close to, I mean, I cannot imagine that Evisai was known to a historical study. Okay, okay, okay, that suggests that someone should look at the history. But, so, we have, so we have algebraic manipulation of differential equations. Now, if I say that pi is the integral of dx over 1 plus x squared, I can do a great deal of calculation with pi using only that definition and the formal rules of manipulating integrals. The so-called Fubini theorem to multiply to integral, change of variable which is the Jacobian formula or differential form if you want, the factoriality of differential form, integration by part which is stored through n, and well, about that I mean that.

2:02:30 And by the way, this is exactly what Zaghi and Konsevich advocated in their paper about periods. Well, I've been reading carefully the paper of Zaghi and Konsevich and Konsevich. All these manipulations can be done with dispensing not only with an explicit integration theory, so we can completely dispense with an explicit analytic theory of integral, of course it's a restricted class of integral, but you can even dispense with proof of real numbers, that's my point. We don't even need to know what a real number is. That means that the machinery, let's say, suppose that I want, I start with this, I start with the definition of phi as an integral. Just take this as an example. Okay, start. So now, I define this as a real number. What does it mean? I want, out of the calculation rule, to give, the point is that I don't have any a priori theory of the De Kinkert and the concept of the real number. I have an explicit number, a certain symbol, as a symbol in a certain calculus, and I want to associate by purely algorithmic means a certain dedication. I want to find, to produce a dedication. Now, using the idea of infinity for a moment, I sketch how to transform this integral definition into line matrices of pi over 4 is 1 minus 1 third plus 1 fifth and so on.

2:05:00 Now it's an alternating series. So by elementary analysis, we know that an alternating series produces alternatively an upper bound and a lower bound. So, I mean, this series produces a cut, obviously. And that's okay. So you can say, you can say what? The standard way of looking at it is pi is defined by an integral. You know nothing about pi. If you do the geometric series, you produce a series. This series is an alternating decreasing series of rational numbers. Therefore, it's a convergent series. Convergent series means, if you look more carefully, what it means that an alternating series converges, it means that it produces alternatively upper and lower bounds and they concentrate and, in effect, they produce a dedicating curve. Okay, now we would like to have a final explanation of this rule. Okay, what do you do? As usual, 1 over 1 plus x squared is an infinite geometrical series, truncated. 1 over 1 plus x squared is 1 minus x squared plus etc. In terms of inequality, it means that 1 over 1 plus x squared is between 1 and 1 minus x squared, truncated series. And if you go one term further, that means instead of producing, you have an alternating C, but not of numbers, but of polynomials. And you truncate it, because it's alternating, and if you truncate it at each term, you produce an upper bound and an lower bound. But now, in the world of polynomials, so you get the C. 1 over 1 plus x squared is between 1 and 1 minus x squared. Now integrating a polynomial over a given interval is algebra, a finitary process. By integrating, you get that the integral over dx over 1 plus x2 from 0 to 1 is between integral of 1, which is 1, and integral of 1 minus x2, which is 1 minus 1 third, exactly what we have. And once it's done, instead of dealing with an infinite alternating series, and put this series producing alternatively upper bound and lower bound,

2:07:30 There is a set of estimates on the geometric series, and you know what you control when you have a geometric series, and it's pure algebra, again, it's pure algebra, to show that 1 over 1 plus x squared between 1 and 1 minus x squared can be known by pure algebra. Then, what you apply is the principle, okay, you have, so now you have a certain, you are integrating a rational function. But you have polynomial bounds. You can define the rational function as a dedicated cut in the polynomial. At least if you restrict everything to the given integral. So you produce a rational function as a dedicated cut in the polynomial by purely finite means. Then, if you integrate a positive function on an integral, the integral should be positive. Now, inequality, I mean, if you know what it means that an integral is positive, you can manipulate inequalities. And finally, what you have is that you are produced, finally, by purely algorithmic means. First among the polynomials, then... So, I manipulate integrals not only without a theory of integrals, but without a theory of integrals. That's the point. I'm more or less paraphrasing the paper of Zaghi. And of course, motifs, what is motif in this paper? It's just a multidimensional extension of this line of reason, and with good control. So the abstract numbers are replaced by objects. Yes. But the crucial point is that you have a category, but somewhere you have a notion of positive homomorphism.

2:10:00 That's what you have. And of course in geometry what you have. I mean, you have the distinction between virtual severity and, well, a positive cycle and an effective cycle and a virtual cycle, which was well known to the Italian geometers. And so the positivity is really the geometrical notion of positivity. That means that you have a linear combination of sub-varieties. It should be deemed to be co-positive is the question, not positive. I have a question I'd like to get to. I don't know if you want to pursue much more on this. I don't want to cut this out, but there's a question. If we can leave sort of the motivic world and get back to just coherent cohomology. I think that when you calculate cohomology in his way, you can do this by check, but this should be regarded as a secondary fact and it's very important not to take it as the definition. And I'm wondering if that relates to what you were saying about him being sort of, he couldn't stand that Serres calculated his cohomology differently. But the point is that as you do. He wanted a theory that would apply to all. At the time there were two known situational forms, small differential forms, many complex variables, and then the new discovery of Seve was imitation, mimicking. Gottendieck, I mean, was, I think, highly pointing out to the necessity of devising a general framework which would encompass both, with dozens... That in specific situations, you have to not to use specific means of tackling. I mean, it's a different thing.

2:12:30 To use, it's a question whether you use a certain definition or just as a trick. And I think in a philosophical perspective, it's very reasonable to assume that you want to do it with very bold faith. And then, when you do a specific situation, you use all the tools at your disposal. Which is perfectly and I think it's I think any good mathematician knows that he has to compromise I mean the mathematical compromise is exactly is it you aim at a concept which are as general and broad as possible but when you have to deal with a specific situation you don't tie your heads you don't tie your heads out of ideology unfortunately some mathematicians do that. I can understand what is a general conceptual framework, but then when I teach a course in differential geometry, the first thing I tell the students is that there are various approaches to differential geometry and various formalisms. For a while, on the one hand, you have principle bundle, vector bundle sheets. On the other hand, you have Gij in beta and zeta. And if you are too dogmatic, you will ignore one of them. So, in differential geometry, we have a left hand and a right hand. The left hand, well, I mean, because, of course, under the influence of the papers of Einstein, when I was young, I was a kid, 15, 16, of course, I learned the beginning of Einstein's concept. Of course, I remember. I remember I was going to the science museum and that was printed on the wall delta of integral of g mu nu dh squared equals zero. Where does it be? And there was the next, in the science museum there was another one. In this way, I know the display, and there are gamma, mu, nu, with explicit formula. How do you go from that to that? It took me maybe two months to understand one other way. I instructed myself. Of course, I got some hints, and I said to myself, I have to go from that.

2:15:00 Then, from that, of course, I was not afraid of writing gibbous, and when I do calculation, I can do extensive calculation, I remember at some point I had to calculate with a tensor with 12 indices. I never really did know what a tensor was, though. I mean, when you do general relativity that way, you do get manipulation, right, and you could raise and lower a tensor index for the best of you, but on the other hand, I was not quite sure what a tensor actually was in a rigorous mathematical sense, which is actually what pushed me for what it's worth. You know, into things like abstract algebra. They're right out of that area, although beautiful as it was. I confess, I didn't really understand what these objects were. They manipulated beautifully, you saw them formally, but it was very difficult to give a definition that was mathematically satisfactory. Exactly, exactly. But I mean, a complete, in my opinion, a complete theory. All of these are considered both sides, given the means of manipulation and calculation as a conceptual one. And if you ignore one of them, you are partly right. Well, I think that was a mistake. You are good or blind if you ignore one of them. But this is what I think... Heaviside, for example, Heaviside pushed both. Heaviside used pure symbolism, but he was also exponent of theory where others were opposed to it. But I think we're at a specific point here, and maybe you'll know this better than me, but I think a specific point here was of course you're going to use check resolutions to calculate these things, but... Serre's approach couldn't get started. You had to very early on prove the cohomology of an affine variety is zero, and Grosvenik's approach doesn't need that until much later, when you're actually concerned with it, so that he felt that by doing it in this more conceptual order, you also, you made what should be simple look simple, and you didn't use hard facts until you were dealing with hard facts. To insist that we should have a conceptual framework to encompass once. But it doesn't distract us from using tools, specific tools, when we are dealing with specific situations and in differential geometry I am arguing the same way. I mean, I learn quick, I learn very early to manipulate, especially at the example of...

2:17:30 Because of Einstein I need to learn to manipulate these tensors without knowing what the tensor was and then I learned what the tensor was and I'm very relieved and very happy to know what the tensor and now of course and when I do differential geometry I'm always I'm not called colorblind I mean I'm not colorblind I know that the red means explicit calculation and I know that the blue means conceptual but there is one more thing now I mean And I think it's basically the discovery of Penrose that you have a diagrammatic representation, a third approach. I think we can credit Feynman first in that. And at one point I was dealing with a tensor with 12 indices, which was a nightmare. I had a page long calculation to show some identity. Then I said, well, gee, I know about Penrose's way of formulating things by diagram. I rewrote my calculation with dialogue, just moving one string across another string, and it was not. And it was not. So, and also that's one answer to the objection of the people who don't want to calculate with indices. We have an explanation of the calculation with Penrose diagram. And Penrose diagram convinced us that we tend to use code. The regime you knew is not. You and you do not refer to explicit coding. I mean, if you use Penrose's representation, G is inside a box, and there are two strings. One has a label mu, one has a label nu. It's just a label to this. The label itself has no meaning, except that when you want to do what is basic in tensor calculus, the contraction. Well, you have a tensor which has one string below and one star below it. And then, if you have upper mu and lower mu at the same convention, it just means that you join one lower string to one upper string, so you join them.

2:20:00 And, of course, when you have joined them, the level is just to show in which way you sue, I mean, you sue the upper string to the lower string, but when it is done, I mean, well... What it is known you can forget about it. This idea of suing, I mean, suing strings. I like to say that, my friend. So, an objection, I mean, it's non-intrinsic because when you write Gmu, you select the coordinates. No. If you really believe in the diagrammatic approach, a tensor is a box with strings attached. And the calculation with stencils is just sewing suitable strings and curving them and permutating them. It's purely diagrammatic. Well, and if you think of G as a diagram instead of as short for a number, it would have depended on your coordinate system. I mean, it's just another graphic presentation. Exactly, exactly. That's my point. So, but I said that, I mean, in differential geometry we have two possible ways. The coordinates, the abstract way of the tensor, I mean the tensor product of vector bundles and so on, the covalent derivative and so on, but then you have this sort of push now, this is diagrammatic, and the mirror, I mean, and I think anyone... One to really understand what is differential geometry and what is at stake should be conversant with the three dialects. But they also show a tensor as a machine. It's not just a box of strings, it's a machine with chutes going into it. But what you end up with, it makes the actual bit of calculation, in many cases, look to exercise, in a way much harder to do. Because you have this machinery that you're required to understand, which is admirable in one sense.

2:22:30 But on the other hand, it was much easier to do these calculations even when you didn't know what a tensor was. You see, I mean, in fact, I know, I mean, both of us know, right, that, in fact, you didn't have to know. You should be able to retain that facility. And I don't think you actually get that facility from, from, uh... I think I did get some of it from that. Well, I don't know. I never... I've looked at it and I never felt that I... As a parable, I lied to... To speak about the modern driver, just referring to my own personal experience. While I commute daily in Paris, the area, I'm driving about 25,000 kilometers a year, which is a lot in every traffic. Okay, well, I have the advantage of living far away from the town, but the disadvantage is that I have to go. But so, what I'm asking for my car, and when I came here, what am I asking for my car? The small department who I trust, who doesn't cheat too much on me, and when I come to him, I come to him, well, George, I mean, this doesn't work well, I take care of that, come back at noon, okay, and he charges me reasonably well, okay, and I know that he takes good care of my car. Press the button of the radio if I want to listen to the radio and commute and do my daily transportation. Which enabled me to see Macy's in Africa, because in the 80s, they all volunteered to, for the time French people had an option to legally escape the military service by volunteering to certain programs. You had to be accepted. And the price you had to pay is that instead of serving for one year in the army, you had to serve two years.

2:25:00 Which, I think it's a reasonable bargain. One and a half year or two years. I think it's a reasonable bargain. Okay, while all these programs were helpful and useful, that can be debated, but never the less. And so, many of my youth, or the husband of my niece, took this option to go to Africa to serve in various places. And that gave me an opportunity to do business in Africa, not as a tourist, not as a businessman, but as a visitor, as a visitor who has a home, which is, you know, I mean, I don't have to explain. So I remember I visited one of my nephews in Niger, not Nigeria, Niamey, and by the way there were French mathematicians there who were at the university, Marie-Françoise Roy and her husband, but they are still there, engaged into that, and now that Marie-Françoise is president of the Mathematical Society of Niamey, Okay, so but I visited Niamey and with my nephew I decided to go around and he had a four-wheel car and then but then I spent a full day with him going to the Vegas repayment, buying extractor tires and learning some mechanics. Of course, when I go for 3,000 kilometers in the desert with a four-wheel car, but I better to be at least slightly acquainted with mechanics and to have the necessary supplies. Okay, just a parable, just to show, just to show that, of course, I mean, for science it's the same, the tools. We use the scientific tools the same way, let's say, in terms of calculus. It's good, at some point, to ignore it. My parable is that when you do science, I mean, if you have efficient tools, the efficient tools are the tools you can use even without understanding them. Go back to the meaning, go back to the meaning. You can ignore what is the real number, you can ignore what is the thing, sir, and do good work without knowing explicitly the meaning of this concept, but at some point, when things become delicate, when you are alone in the desert, when you are alone in the desert, better be your own mechanics than able to repair your own car, the concept of your car.

2:27:30 Mathematics comes with the introduction of a good notation that you don't actually have to think about. Except, of course, later when precisely you argue situations when you want to make new events. Or when you discover that the good notation is thinking about it. Well, the use of it is unconscious thinking. No, it makes you unconscious of the stuff that wasn't really helping. You didn't need to think about it. It makes you conscious of the stuff that was really helping. Well, a good notation cannot be invented, John. No, it encapsulates what was necessary in order for these concepts to actually be used. Traveling through the desert is a terrible idea. A defense of Grotendieck. To believe that there's only one right way of doing something. This is, of course, wrong. On the other hand, it does give one certain enthusiasm to push on through the deserts. Yes, yes, yes, indeed. Certainly Groten-Dick had. But also the general conceptual framework that he was determined to clarify, even if it didn't, even if it wasn't necessary to solve a particular problem. But he wasn't saying, don't think about these things. He never suggests not using these resolutions. He does say, here's a format for organizing the subjects. And it is for organizing the whole subject. In fact, at this stage, I think we should try to press on a little bit before lunch. Have we as well in the chronology left behind the 1950-60 phase? Because it seemed quite fruitfully the topics of our next two phases I think have blended again now in the general discussion, both the overall shape of Grosvenor's legacy and what it suggests, mathematical conceptualisation, and his achievement in algebraic geometry, and to pursue those as it were.