Pm discussions: FW Lawvere, C McLarty, L Corry, A MacIntyre, JL Bell

0:00 Okay, it's now the afternoon of the 15th of June, and after an extremely interesting, deep-going, far-reaching discussion this morning, we had thought that this afternoon we might, for an hour or so, look at the consequences of emerging overall visions of a structure from the category 3 view-point, On concepts in the theory of measure and integration. Perhaps particularly in the theory of integration, which is my impression, I may be quite wrong, but my impression is it's a relatively negative area in foundational studies and new philosophies of mathematics. And I think one subject I remember was the subject of a fascinating exposition by Bill to me a long time ago now, I think possibly in Cologne in 1990, was the Hennstock integral, which of the many theories of integration, as it were, presents itself as the one that perhaps contains the most attractive features from the point of view of the overall perspective on... Structure and foundations that emerges from category theory. That might be a topic for an hour or so, and then perhaps we might start on the chanel, conjecture, discussion, and then we'll continue tomorrow. Anybody got any other topics that they feel we should be discussing in this afternoon's session? I overheard Angus talking about intersection multiplicity. I've always wondered what we have there. Well, shall we discuss that first and then the answer? Maybe you could give an idea of the conceptual and problematic aspects of this concept. This is pretty difficult, in fact. I mean, to do this in full generality, let's see. Obviously, first of all, historically, it's a notion which was used systematically by the Italians, but certainly often with Eric D'Angelo, which in some cases we would like to contradict the results.

2:30 And when we were using some principles which, in fact, were not generally correct, one thing to say is that, I mean, of course, it's a concept, but ultimately, geometry, at least, at first. But in some sense they have an algebraic geometric inspiration. I mean, these are structures which are supposed to be like algebraic geometric sense that have dimensions and so on. But essentially, maybe it's best to say how this, perhaps the most interesting for our discussions here, is to say where it is now. I mean, what is the connection between cohomology and intersection? I think this is the reason. And certainly in the vague context, formalism is that you work on, let us say, to make it easier still, projective varieties, or even better, one should work on projected non-singular varieties. Projective? Non-singular varieties. Okay. I mean, so the point is you're going to attach to every projected non-singular variety in some factorial way, you're going to attach an intersection ring, and that ring will involve... The cycles are going to be linear combinations. Essentially, the cycles will be built out of sub-varieties of the objective variety, but the sub-varieties need not be non-singular. And the intention is that, so a cycle is a formal combination. Now, a cycle is, of course, ultimately something you can integrate over. I mean, this is the connection with other parts of mathematics, analysis and so on. And ultimately a cycle is something which is a graduality, a connection between integration and cycle.

5:00 But the ultimate intention, not so readily achieved, of the Italians was to obtain something of the following kind. They took a general variety of a certain dimension, d, and the matter is usually relatively trivial if d were 1. If d is 1, you're doing the curve, the cycles are, if it's an irreducible curve, there's a cycle of top dimension, namely just itself. And then you have cycles of, you have lots of points of dimension zero, and the cycles of formal linear combinations, and these are going to be everything we need. The whole analysis is done in terms of these cycles. In general, the problem is you have various intermediate dimensions. You've got the thing of sub-dimension n, and you can have varieties of every intermediate difference between them. Sub-variety, but typically one grades the situation. You have the varieties of... Intermediate dimension. D. And you essentially form linear combinations of these. Formal linear combinations. This will form, of course, a group. But, of course, you have such things for every single dimension. So you have a graded group. A structure of a graded group on the... What you get from intersecting to somehow, it's not really a set thing. I mean, for example, if I take two curves and look at how they intersect, they'll typically intersect in finitely many points, but sometimes they'll intersect transversely, the curves go through each other, and other times they'll intersect, you know, tangents will coincide, and in each case there'll be an intersection index, an intersection number, it'll be one when you go through transversely.

7:30 ...different, two, for example, like this, because if you move them slightly, in some idealized sense. Now, the notion of moving them slightly has no clear meaning in characteristic P. It has a clear geometric meaning for us in characteristic complex numbers, but not necessarily any obvious geometric meaning anywhere else. But of course, one does see that you can make some kind of preliminary analysis of this notion in terms of this commuted algebra that we're talking about today. You can use various numbers on the theory of ideals sometimes to define, in simple cases, give you correct answers, but it doesn't really set a hard thing to work. If the general objective is to obtain, to take this graded group, which as I say is an innocent And eventually, divide it out by some suitable equivalence relation, which will cause what is something of a general notion of home. You can move from one entity to another in a suitable family. Now, it will depend what it means, what the base space for the movement is. It may be one kind of variety, it may be another, and this will lead to different notions. But the idea is of moving things somewhere into a general position, in the sense of generic points along the family. With the objective that you'll eventually not merely have this formal, you'll have this formal addition on the cycles to begin with, you'll divide out by some relation approximately like homography, but it'll be an algebraic notion, and eventually you'll be able to define a ring structure as well, but you'll only get the ring structure once you've done some pseudomotor. The graded algebra will be collapsed and you'll eventually define a ring structure with the idea then that if you take two cycles, their product, which will be again an element of this collapsing, it will be a cycle or a cycle class, will be the intersection of these things, although very rarely anything to do with the saturated intersection.

10:00 Just doing things in terms of set theoretic illustrations gives you the wrong answers. I gave the example already, merely if you were to take two curves non-transversely of the count. As we know, it's the same thing as counting the number of roots of a polynomial. If we were to say that only because we count multiplicities that the fundamental theorem on algebra gives exactly n roots, and the whole issue is somehow to give a general account of something like this. It's not at all straightforward. And it turns out that there are several notions, there are several natural notions for collapsing, one natural equivalence for example for collapsing the string of cycles. There's a thing called rational equivalence which corresponds to sort of parametrizing over a line or something. There's an algebraic equivalence. That's the old one, right? Yeah. That's, I mean, but there are... Calgary learned to throw out Marie Lush's use of it. Yes, that's right. But then there's algebraic equivalence where you have a more general algebraic variety to move on. And then there are, ultimately, more subtle notions which relate up to what Grotendieck was after in Theory of Modus. But the general structure is that you want to assign two... I mean, usually you want to assign numbers, if humanly possible, but in fact it's unreasonable to expect the intersection of two, even two varieties, to have a number attached to it, because a number is really only something you can reasonably attach to a point. A pure number you can attach only, say, to a point. It's simple countering. If I have something of co-dimension, well, if I'm intersecting two curves inside a big, a very high dimension, there I expect to get points, but in general I only expect to get cycles of some other dimension. The multiplication is going to move the grading. You see? Now, is that because, I mean, with points you have simple multiplicities, but if two surfaces intersect, they might be tangent in one direction. Yes, yes, yes, and now they're all over the place. I mean, you really can only describe this by another cycle, a formal and a formal sum of things. I mean, even for the intersection of varieties, the beautiful thing is that sometimes the most important version of all is self-intersection.

12:30 You take a variety and intersect it with itself, which is no obvious meaning at all, except that you think it gives you back. It gives you back, generally, something very different. And typically, there's something in intermediate dimension, in the middle, intersecting with itself. I mean, this is where the big theorem is. Now, is it true in the topological case that often self-intersection means deform one of the copies? Yes, yes. I mean, this is almost the image you have. The problem is that in general fields, the notion of deformation is a very subtle thing. So I'm stressing at the moment the limitations. You're working in a projective case to make this work, and the ambient variety is non-singular. The creatures you're combining inside may well be singular. And their intersections are... But there are really several challenges depending on what you have divided out by. Rational equivalence, numerical, algebraic equivalence, and something such as some of the one most close to the ground takes the final formulation. Now, it is constructed by, it can be constructed by giving some idea of many books, even modern books. But it can be done in a totally different way. There's not enough points to get nice formulas. The typical thing is, you know, Bezou's theorem. If I take, you know, a system of, say, two equations with different degrees, I expect to get NM points, but I certainly don't get this unless I count things right and work in a projective situation. This is the reason, ultimately, one works in projective, to get a nice formula. You move everything into a place where there's no repeated roots in any direction. Yeah, that's what you try to do, you see. But the point is, the curve situation is relatively, it's not completely straightforward, but it's relatively straightforward. Intersection of curves and surfaces, the difficulty is coming in working higher up.

15:00 The products you get are only side ways. And there are all sorts of complicated things about, you know, the meaning of an intersection when there's a high dimensional component. And this happens often, this bedevils almost every serious piece of work in terms of counting. Okay, now how does this relate to what Ehrich, of course, used in destruction theory in at least one of his proofs. Ultimately, after the Wehmery's suggestions and people did this axiomatization, what would you require for, require altogether the Wehmery theory, the general Wehmery theory? We're saying something about the action of Frobenius and characteristic theorems. But what would you need even to get the theory for? You need to have, to attach to projective non-singular varieties, You want to be able to count. So you want to assign a cohomology theory with coefficients in something to count. It's interesting that what you first get is really... The characteristic zero in the coefficients is so that you'll eventually count. The integers will be in there. People had known before how to get something that looked vague like a cohomonic characteristic period, because that would only give you the answers you were after, modulus, and that's futile. One could axiomatize what one wanted, the usual properties of formal things that relate to what happened. The singular theorem of complexes would have some kind of...

17:30 The cohomology objects would be finite dimensional graded algebras over anti-commutative over this coefficient field. You have a category there, you have a function. So that's to make it look like forms with the exterior problem? Yes, in the end, that's what it is. Although in the axiomatization of the big cohomology theory... There is never anything. There's no demand. It's just a problem. It's utterly abstract. But just to motivate why we want to do it. Absolutely. I agree. You've got these cohomology rings along with the Edward product. And you've got the intersection theory, which is a completely independent cohomology. That's already gone. It's a product. You then have these axioms that are given in many places, but most conveniently, perhaps in this book by Freitag and Keeler on etal cohomology, In the beginning of the paper, Diodonnet's article, is it dreaming about algebraic diplomacy, but he gives a history of this thing, and he gives the axioms of a very cohomology theory in this situation. Now they have all things like an abstract version of a Poincaré duality, quite straightforward to state in terms of linear algebra. Your spaces should be five-dimensional, you have a certain duality there. There are various cumulant formulas that are actually essential, so from the product, the natural product, you see the cumulant formula, and then a principle stated only for very simple things like cohomology groups of points and one or two other cases, from which you can deduce that this cohomology theory, which is a functor, will respect. We'll make the product of the cohomology groups correspond to the product of the intersection theory. I mean, just, it's an astounding thing how they get this out. It's not done, it's not done in this article, but it's spelled out in complete detail in several very useful articles by Kleinman on the standard conjectures. So the first one was written in this exposition.

20:00 I can't remember now. And then a much later one done in this, one of the two volumes of the Amass Group on Volubles. But he does just this magic of linear algebra to show that you don't say as part of your axioms that this author has got to make the cohomological intersection correspond to the cycle intersection. There's got to be a cycle map which attaches to the cycles of cohomology. So, cycle math, you make some very basic factorial problems and you say what it does to points and so on, and out of that you can teach this. And this is Verge's paper, the cohomology class associated to a cycle. Yes, I mean, of course, at the level, I mean, it's not trivial to construct, not completely trivial to construct the cohomology class associated to a cycle, but this can also be done very formally, as done by Tate. I haven't read that article, but a lot of people say that's when they first understood it. And this is spelled out, this is all spelled out in Kleiman and related literature. In Katz, Nick Katz's short study of mathematical homology in the in the Motivs book. I probably missed out some actions, but the spell out, you spell out the existence of this functor from Graded algebraic analysis was over some fixed characteristic zero coefficient field. Now the first interesting thing is that the coefficient field cannot be arc. It's known that from the geometric license derived from it cannot be the rationals, for example. It cannot be the reals. For reasons of representation theory connected with what one already expects to be true about the cohomology groups of the elliptic curve. The case, one of the cases that they had explicitly considered. They have to have a certain shape, and if the coefficient field was the rationals or the reals, you would have some sort of, you would have infinite dimensional algebras over these which don't exist, of a certain form.

22:30 However, algebras of these properties do exist over here. Does the Lambda-ring structure come into that? Not really, but the Lambda-ring structure is never very far behind with some curves. But at this point, I mean, these observations, I think, were made by Serre on the limitations of what the coefficient field could be. And it's still not really clear what the coefficient field can be, but it was known that at least, in terms of what we knew about the limitations coming from elliptic curves, it's left open the possibility that p-adic fields could be. But it's interesting, the reals cannot be, that's all supposed to be anomalies. Rationals are too small, the reals are in some sense too big, and you can go to the complexes and be different, but the reals are just not enough. Let me go back to the intersection theory. The intersection theory is developed quite independently, and it can be done in a number of ways. It can be done by pure commutative algebra. It's really rather hard to do it in full generality. And I guess what Serre did in this book is by far the most elegant. It is homological. What is the more general way? What do you mean? Well, in the method in Fulton's book, you see, the method in Fulton's book, Fulton's is a very long book with complicated arguments and the method was developed by Fersen. This involves a much more general machinery of sort of algebraic vector bundles and deformation theory and so on. It's beautiful but very tricky and it... It's simply deeper. It's got much more functoriality about it. It need not, however, involve... There is no cohomology whatsoever in it. There's some rather elementary commutative algebra, but there is... It's not fit at all in algebraic topology, but there is an algebraic homotopathy. Not at the level of Wojnowski, but there's an algebraic homotopathy. Scheme theory comes in. I mean, although you're only interested ultimately in getting the, you know, working with just straight varieties, the intermediate stage is to make these calculations work in full generality to get the functoriality of that.

25:00 Schemes need to be used. I mean, nothing very fancy about schemes, but nevertheless schemes are used. A subscheme, say, of an affine variety, need not be affines, so we sort of try to consider that as a structure of some kind. And you start to, you don't, you're probably, your ideals, and you're going to, you will get nilbotans and all the rest of it coming in. I mean, there's no way around it. You just have to do it. And this, I mean, the computational aspects of that, it's not, it doesn't really have very much to do, say, with Gruden or basis or something. I think I mentioned it. It's a very different game. It's the calculations of, calculations with length of our tinny and ring ones. Kind of stuff that's touched on briefly in Atiyah and MacDonald and so on. And it's delicate. This is quite delicate. The logical structure of that is really quite delicate. Can I just add a couple of hopefully provocative things? Yeah. So, anyway, these axioms that you mentioned, these are at least partly stated in terms of this already presupposed machine, like the cycles and... Yeah, that's right. The cyberbump is crucial, it's incredibly crucial. I'm really trying to back way, way, way, way, way up here. So I've tried various times to read Fulton's books and his other papers to try to make some kind of sense out of all this. And I find it extremely unsatisfactory because, well... It sounds like the solution to a problem that was never actually stated. This often happens in mathematics, you see that, I don't know, somehow the mathematicians have an intuitive feeling that there's a certain problem, but without ever actually stating it, they get into some technicalities, which would probably solve it. And along the way, they, of course, make precise definitions. So what's the general problem? So this typically happens with, you know, in a general way with, you know, adjoint functors and so forth. You have a nice universal property which defines the thing. So now you know what it is in one sense. Now you go and try to figure out what the elements of it are. You get into involved calculations and equivalent relations and so on and so forth. So all that stuff about the child brain and so forth always struck me as being... That, you see, is kind of that second stage, and the initial thing has not been actually stated properly. It seems to me, although I don't know if Rotenbeek ever said this, but it seems that this remark seems to be very much in Rotenbeek's spirit, you see,

27:30 that you should at least understand what the problem is, and then you can dive in, you see. That's right. I mean, it's not, this is not a problem. So the whole bit is about, you know, why should Poincare duality play any role? Why should cohomology play any role when it's really a homology problem? Of course, Connes theorem, as you mentioned, is crucial. That again suggests that there really should be a homology theory around, and the way that the homology is usually gotten at is simply by dualizing cohomology rather than any sort of direct idea of what the homology is. So that again clouds the whole problem. But now, another thing is, in passing you mentioned the intersection of ideals, but the intersection of ideals is not the same thing, it doesn't correspond. No, no, no, no, I said general operations of ideals also. No, no, no, but the thing is, you see, that maybe it's not intersection theory, but meaning theory. Yes, yes, yes, of course. In other words, it seems to me that the crucial... Difficulty or enriching aspect, same thing, in a way, is that in the community of algebra the intersection of ideals is not the same thing as the product of the idea. So if you take the product of an idea with itself, it's not itself, this has to do precisely with the nil problems, with the tangential intersections and all that sort of thing. So, somehow it's about the contradictions between product and intersection and ideals, don't we? Yes, yes, yes. That's a major part of it. And of course those things directly, if you look at the closed sub-varieties that they define, it's not about the intersections over there at all, but rather about the union. But of course union has a lot to do with intersection. Yes, yes, yes, yes, I agree. Well, I was hiking in the mountains of Italy one time, and I was thinking about this, and I was thinking, oh, the African plate comes up and lands into the European plate, or whatever it's called, and what you get is the Alps. You know, looking from above, what you have is the union of the African plate and the European plate, but you have more than that. You have this sticking up in the middle, which of course is also the intersection in a way, but I mean, so it's in describing this union, you have to take account of the fact that it's a push out over its intersection, and at that point...

30:00 You know, the intersection of some varieties as well. So those are obviously the things you sort of see and are significant, but the intersection of the ideals has to do with the union. So, one technical problem which may be, again, behind the fact that all this machinery had to be developed... ...was that there is no easy description, no geometrical description of the intersection of ideals. In other words, if you construe ideals, of course, as, well, they're quotient rings, but then in turn they're closed sub-varieties, well, I mean, okay, if you, as I say, the ordinary lattice operations on the sub-objects of a given object, the logic in the narrow sense, so to speak, You will see the product of the two ideals. The product of the two ideals gives rise to a third quotient ring, and hence to a third sub-variety, but how do you capture that third sub-variety just in terms of maps to and from, either in terms of commutative algebra or equivalently in terms of affine geometry? I worried about this for a long time. I never could ever find it. All the sort of naive things that you do, push out, push forward, you always wind up back in the same realm of your simple-minded logic. So, I think that's one problem that needs to be faced in order to in turn be able to state in some straightforward way. What it is about how the alps stick out, you see. So the lattice of subvarieties, it's a lattice, but moreover you've got this product there, and that's somehow what it's all about. You have to resolve this contradiction. Now, the really strange thing is that the universal algebraists, one of the... A few new concepts that they came up with in the last years. I mean, they do all sorts of incredible calculations in incredibly bizarre situations, and they like that, but one general concept they came up with was a so-called commutator variety, which corresponds to commutators of subgroups, in the case of normal subgroups, in the case of the category of rings. It's an operation on equivalence relations. It's a property of the...

32:30 We know it's an operation, it's a binary operation, the only equivalence really in any category of universal algebras, but if you apply it to the category of consecutive rings, you see, it comes out to be the product. Now, it's very, very odd, you know, I've told these people, I mean, I've told universal algebra, there's a few categorists who've tried to understand, succeeded formally to understand what universal algebras have done. With this so-called commutator, and I told them all, look, this is a desperate trying need in algebraic geometry, you need to have a direct geometrical description of this, why don't you figure out, you know, precisely what's happening in the category of community brains. They never do that. None of them ever did that. Because I have a certain... So they talk about very general varieties, they know about groups, they talk about extremely bizarre varieties, but to actually apply it to an honest everyday mathematical category other than groups, namely commutative ranks, they never do it. But at least, you know, you can see enough of what they said that it does come out to be the commentators. And indeed it is, it was Maria Cristina Pedicchio, who worked out in some detail, and a couple of other people too, you know, what this means in terms of pullbacks and push-outs in a category. So it's just sort of... So it's sort of one step a little more complicated than the 90 things that you might do with pushing forward and pulling back, but it manages sort of not to collapse back into that. It really is just a product. That's interesting. Yeah, but the people who are expert on that refuse to... I don't know what it means in a particular example. That's a big problem we have with all the category theorists in a way, that they never, they don't often enough focus on a particular example and apply these powerful tools and see what they mean. Yeah, that's true. I mean, honestly, in some sense, the heart of the, I mean, nowadays, almost the heart of the intersection theory, the thing that really leads to the kind of magical results is...

35:00 When you take self-intersections, I mean, the most magical of all of them is the one that's in many generalizations, but the thing that they clearly understand very well, that if you take, say, take a curve, a very non-singular curve, and you take sort of a problem with it itself, and then look at the, so it's in there you're going to do the intersection, so inside this special surface, and then you take the diagonal, Meaning, leading to the formula that the self-intersection is 2 minus 2g, where g is the genus of it. The 2 comes from the top cohomology is 1, the bottom cohomology dimension is 1, and the intermediate one is 2. So of course the self-intersection can be negative in some cases, and it's never a characteristic. The reason for this is the life shape is transformed. This is what Weil basically spotted, I suppose, because Weil was in a position to... And so on and on and on and on and on and on and on and on and on and on and on and on and on and ...was more... I mean, they wouldn't necessarily... they would know it, I suppose, either from having really known it in a characteristic piece. No, no, no. I mean, that's the point. I mean, that they're spotting that this was still true in a characteristic piece, so it's naturally... Yeah, there's a picture of it in character. Yeah, there is a cohomology theory, or there may be a cohomology theory. I mean, that wasn't the way he proved it.

37:30 But this kind of formula is absolutely crucial, and it is terribly difficult to make sense of this kind of formula in any straightforward way. And just in terms of the theory of ideals and even small deformations it takes quite a lot of work in any development and it is the heart of generalizations of this scenario. I don't know, it's a bit of a problem. I found it, I mean, I was, I tried something. I had a very specific reason for doing it. Okay, so the vague cohomology theory is a functor. I wanted to get somehow, and at the time when the axiomatization was done, one didn't know if there were any models. If there were any such properties at all that would work uniformly, what one did know was that there were theories. I didn't know it then, but one would very early days. One would know that you could get a theory with most of these properties, coefficients in something like z-module p's on a torsion situation. This had been known for some time. But not that you would get characteristic zero coefficients. So I was interested in the possibility that I mean, the standard conjectures of Brodnick are basically about the fact that there are going to be, in the end, one found a lot of models of these axioms, and they all seem to have the same properties. I mean, the finer detail seems to be the same, the finer detail about the numbers is that they're related. I mean, what made one see that they were the same was the intersection, which was that they were independent of the logic. Anyway, I decided I would take, you know, assuming one had a family of these commodities, that takes an average. The probability is it takes a number of products or something like that. And sure enough you can prove it, but it takes really a lot of work with it again, and the main difficulties, to show that this cyclism is somehow preserved. I mean that is really rather hard work because it involves us in somehow, what Kreisel calls, unwinding the proof of the construction of data. Nobody nearly as elegant as they are. Also serious worries about what doesn't really, I think, I mean it's, you know what you want, but it's...

40:00 Van de Dries calculates it on the other hand, by never defining it, but he'll say, well, you have to drop it up this way, and these positions are solid composition, blah, blah, blah. That's the only definition he gives. Well, in some sense, the Chalering and all that stuff is much more elegant, much more, but still, general ilk, you see. The only real case one has is that when you've got things of complementary co-dimension, which formally you expect to be able to set in point, in a dimension zero thing, so that you have a discrete set of points, then if you have transversality, then this thing should be counting the number of points. I mean that's all you have, but that's nowhere nearly... The content of the modern literature may have been Italian's where I'm from, working mainly on services with the curve, so that's typically a harder matter there, but modernity. And so what should it be? An entity this thing is supposed to be, and certainly if you start with two varieties, the product is not going to be a variety, it's certainly going to be a cycle. What is the meaning of the coefficients? That's all. Fluence is a very long, very complex... I've never even really understood the situation of topological manifolds. You've got the left-chest fixed-point theorem. And it's certain that if the left-chest number of fixed points is not zero, then there's really a fixed point. It is not certain that the left-chest number counts the fixed points in any apparent way. If they're nice and transverse... No, this is the whole problem. It's the whole problem. I mean, even if... They are complementary dimensions. The number isn't really, in general, it's very hard because it can have components, hidden components of other dimensions in this formal way of doing things, unless we devils everything.

42:30 Yeah, I've run up against quite a lot of things that are starting to go up very well, and we have the pitfalls there, and that's always the general problem. You can be sure there's a point there if the... And there are these theories of topological degree that I'm not sure work very generally to actually give you. I mean, I just, there's long books and I never got to the end of the book, but what I noticed was they didn't say up front, we're going to give a theory of multiplicity such that. There's still a lot to understand about this, I think. In general, the foundations of this theory, in the sense of what one really needs to have, you know, there's this general principle you can, we understand that in the, that you can do some, in the complex sense, there's this deformation, deformation of a mineral is a great part, but then you get more mysterious things. Once you've got this mysterious theory, you get, you get, in general, for two varieties, you get a product which will be a science. And that's like it. The cycle then has a well-defined, this is plausible in the axioms, but it's not trivial, but the cycle will have a well-defined part attached to it, the sum of its coefficients, it's not clear that's what we're doing, that's a bit like topology, but you prove that that is well-defined, so you invariably get a number out from the intersection of the product in such a way that it's two cycles, but only in the complementary dimension case is the remotest chance this is going to have to do with the number of points of intersection. Well, the number of components in the intersection, that's one thing, but you do get this number, and now you can define, you define a cycle being numerically equivalent to zero if its intersection with every other cycle of the right co-dimension is zero. What in God's name does that mean in our geometric picture? This is numerically equivalent, and this is in some sense the most important notion.

45:00 You have this cycle map taking you from the cycles into the cohomology, for any given very cohomological, that's part of the definition that there's something factorial from the cycle structure into the cohomology, but as far as I know, those definitely are many cohomologies, so you can ask yourself what, so it's a approximation, so you've got the, you take a given, To be given variety and you can cycle on it, that's numerically equivalent to zero, well you can then show that it... The thing one wants to understand, you really want to understand is which cycles induce maths and cohomology. This is part that comes out of the Poincaré duality, again by general linear algebra, well known in the classical complex cases, but you can get out of this just the axiomatic in the general case of cycles and go into... But then by Poincare duality there are elements in the complementary, in the complementary cohomology. I mean the Poincare duality in this situation gives you some kind of canonical map between h-i and h-2. Is this treating the cycle as an intersecter, sort of? Yes, I mean ultimately... What it's doing to other cycles is intersecting. And then, but then... So the natural question you want to ask is when the, I mean, Poincare duality is essentially to get this picture, when is a cycle, on the original variety, yielding you a zero, well it has to be, I mean, if this is so, it tells us that numerically we come to zero, but the conjecture is that the independent of the vague chronology, and certainly for things, for italic homology, so the conjecture is that a cycle goes to zero if and only if it's numerically.

47:30 I mean, in other words, let's go to zero in all cohomology theories, if and only if the answer is back in the intersection. That's not being proved, and that is a concept, it's not equivalent, I think. But the intersection theory is the intersection of all cohomology. No, I mean, this is the picture. This is really where the motives came from. Motives, yeah. The motives came from this, that you have the science, you have the objects, so the varieties, and then the thing is, of course, and then you have correspondences between varieties, so subvarieties and products. So this at least provides the morph of terms. Basically the motives turn out to be... Well, you want to define motives in terms of components, so basically you've got things that begin with a projection, and this can only really be... You can only give a different... I mean, I can still get it formally, but this will work! I mean, you'll get an actual category of motives if this conjecture has got numerical equivalence and homological equivalence. And this is not quite equivalent, is it? It will follow, if they can prove, that something like the Hodge conjecture and the Kuhneth, I mean this is the other thing, in the Kuhneth theory, you take a cycle in a product, that goes into an element in cohomology, which in turn is living in the tensor algebra, there'll be products, there'll be projections backed down into the individual, into the cohomology of the individual varieties. Are these things themselves coming? From the intersection, the chemical cycles or rational combinations of cycles. That is unknown, but that is basically part of the Hodge theory. So if you conjecture that, it's been proved in one or two cases, for a variety of over-the-finite fields it's true, because of the wave trajectories. And another thing of... If these are true, then in fact numerical equivalence is equal to. Somehow the intersection theory is the intersection of normal cohomology.

50:00 The methodological he said was, down with excessive double dualization. I see. Well, I mean, okay, you say there should be a map from cycles to cohomology. But if you've got functuarine duality, then cohomology would be a homology. So you've got a map to homology. Yes, yes, yes. I mean, if you talk about the projection maps... This is clearly homology, not cohomology. Yes, yes, of course, of course. The projections are... I agree with you. But then you see, there is a certain difficulty in just not even carrying out this bay idea I'm saying. Maybe the difference between proper maps and non-proper maps. That's right. This is why we're in the projected non-singling case here. That is crucial. The moment you try to do something like this, But the appropriate cohomology theories in general characteristic P, then, I mean, sheaves then come in. Essentially, you have to work with cohomology with compact support. So you, I mean, you basically have to projectivize your... You've got to change your variety, but then of course you've got to change your coefficient. You've got to change, you know, you've no longer got the same sheaf here. You've got to sort of extend it by zero. That changes everything. I mean, and you need fairly deep synapses to show that this cohomology with confidence support has the functoriality properties you wish of it. So it's, I mean, this is the difference. To make them, then to get proper maths again, you have to projectivise, but really end up with much more bizarre sheaves than you had before. And even worse, I mean if you started with a, that's fine if you started with, say, a non-singular alphanumeric, because once you start with a singular alphanumeric, it's even worse, I mean, the transverse of the world is much more complicated, so it's, I mean, things are really only working perfectly in the Planckian duality, in the general case, it's a complicated tool, it's between a...

52:30 Compact. Compact. Sport. And that's right. And that also we did what you really want to do, but it remains desperately hard to get the right answers for affine varieties for a number of points, et cetera, because technology can go beyond the means. So I find it logically still very... So this map is a cycle, now let's call it homology. Yeah. Well, it's really a homology of a point, so you're taking the total of the distribution. So if the distributions were compact support, then you just take the total. That's what it is, because it's the sum of the coefficients, he says, so that's going to come out to be zero, and you get the total. It's the total. Yeah, yeah. It's the total. But the point is that since the models are not compact models. Yeah. Therefore, the whole business about... So you get, in my estimation, there's exactly the same problem with distribution theory. So, smooth functions, distributions. So, I mean, Schwartz made this, because it gets a little bit closer to his illusion that distributions are generalized functions, he said, well, let's take functions of compact support to start with. Well, already you've destroyed the natural functorality that functions have. It's an intensive one. You can have arbitrary maps that change, you know, but now it's only proper maps, you see. And now you take the linear dual of that and you get the distributions, which are not necessarily of compact support, covariately functorial, but only with respect to proper maps. That's right, yeah. So, one has somehow put the... I think that this properness aspect should be an application of the general theory, rather than the other way around. So the general theory would have it that both of these things are comfortable for general math, for example projection math. And then the proper case is gotten by a precise limit process. So that would make many of these things... You've got to take a dual at every point, so you've got to do the opposite thing. You've got to take a Poincare dual and it's just incredible.

55:00 Even in Plymouth, Plymouth's name is a beautiful article just because you realize just how bizarre things can be obtained by a couple of applications. Double duals, sensory specimens of product as well as the additive stuff, I mean it's kind of miraculous how about just how completely formally of these actions you get cycles acting on, I mean acting from the essence of the cohomology, of course given any given case the action of the cycle requires a different, I don't know, point of duality and most... In fact, that's why you have a product, because you simply apply it successively at the operation, but it's operating on the extensors. Oh, absolutely. In fact, he proves duality by more or less, he draws a picture of two curves crossing each other like this and induces... And then he gives you far less than a picture, claiming this happens in complementary dimension as well, for, it's hard to even believe, it's hard to understand why he wrote the next sentence, because nobody, it's never presented in any way of anything, you know, but he just, and yet he gets from this pi-grade duality, which is correct, from this rotten argument, basically, that numerical zero is zero. He gets a very important part here. The algebra, just the pure abstract algebra, devoid of all geometry in safe climates. It's not due to climate, I guess it's due to Grottenlich and to Voltaire. It's amazing. I mean, you get the... You get the Lachey's fixed-point... ...Lachey's trace formula out of all this. You wouldn't need to know much about the intersection theory. A couple of times you start to see a fixed-point theorem and change the trace. Yeah, well, the choice, I mean, the number of fixed points, I mean, you calculate, basically, you calculate, you can calculate the intersection in terms of, you take the action of, that's the general thing, you take the diagonal inside the product of it with itself, and now you've also got the Frobenius, the graph of Frobenius, and you want the intersection done with that, you can calculate that.

57:30 Since you're basically dealing with an intersection of the diagonal and something else, it turns out that this comes down to, since you know the trace of the diagonal, this comes down to the calculation of the trace of Frobenius, and it's hard to be formed. And this isn't, there's no anneal to it, but it's already surprising. It looks hierarchical, but I mean, I probably have never realized just how powerful this kind of dualizing is. But is it fair to say that if you had a really good intersection theory that worked just as you like, then the Lefschetz trace formula would be the Lefschetz fixed point theorem? Ah yes, yes, that's true. Just so I understand. No, I think so, yes, yes, no, that's true. What Cartier didn't say, like why I don't know, I mean he did refer to the fact that, as we saw, these things that broadly start by doing conjectures. Beautiful, in short, is the proof of the equations. I mean, this is done in climate. Climate is done independently by the current theory. Again, it's essentially at the level of some even more cunning abstract algebra. I mean, it isn't really just abstract algebra. I mean, vague sketches are really simple proof of the pictures, assuming a cohomology theory with integer coefficients and really nice, simple problems. We know that more than anyone. But this is sort of...

1:00:00 Almost the next best it could be given that you don't have any true coefficients. Essentially, if anything in character is zero, you've got a canonical embedding of the integers into it, and that is all that's needed. Of course, it leaves the mystery, and I suppose they still don't know the answer to this, as to which coefficients are possible. The p-addicts and then various bigger extensions of the p-addicts. I mean, the p-addicts are possible for the, or the l-addicts are possible for the l-addict. If you want L to be equal to B, you need this crystalline cohomology, and then you cannot work with QL, you've got to go to something very much bigger, this unramified, the Witt vector is over, there's reclosure, even beyond that, there are various rings in the trait, the flotans are needed to get, well, you see, you want to get, you want to detect other things, you want to detect p-adic analogs of two pi i and stuff, and they're not visible. To make any sense of what this means, we get periods, which are the Durand theory that they're aiming for, although you don't have any differential forms of them. But the interest, perhaps, of this crystalline, with more structure, but merely being a function of a divulgable, you've got the action of this, of Rubenianism. Of course, the Italian ones, you've got this with the Gallon group, which is again a great theory. So each, and there are other quantum theories discovered too, so they all, many of them have more structure, but merely this. And that's where you're saying you're not always over the field. It's only on that stage you disregard some torsion and you walk over the field, but there you are. And this also is a way of trying to do cohomology. At some point the category goes up and you can't achieve it at all and you're not working.

1:02:30 Well, I guess that was his thesis of what he said, which we have to refer to. In the end, the numbers that you get are all coming, but typically the Euler characteristic is a self-intercept, certainly an insight one. Yeah, where elections do get it from, I mean, but certainly he's at pains to explain it. I've never really worked out how accurate this explanation turns out to be in terms of you've got your... You've got your manifold and then you deform it and you ask and you deform it into general position. Yeah, yeah. But this is the general position. Who is it that criticized some of the Italian different algebraic geometries? This is into generic, into the generic case where we have not proved that there are any instances of that case. There are other non-progress situations. I mean, the connections between intersection theory and cohomology can go on for ten years. I mean, this is one of the post-Staline. I mean, there is this general trace form.

1:05:00 It involves, for many it's actually true, you want some general result about the action of cycles. There is still an action of cycles in cohomology, and there is a theorem that says that they... The trace aversion on one side is equal to the sum of local traces for some kind of strange action and the problem is that this has been proved, but they really don't know what the meaning of the law here is situationally and it's not what you think it might be. There's a naive definition of what it might be, but this is not it at all. This is the key to getting decent results in the counting points of an affine variety and this has only been done in 10 years. He uses it, writes it. He shows that if you try to do algebraic geometry over the algebraic course of a finite field, they wanted to understand these affines. They need to understand the structure of the affine situation for purposes of representation. And they will only understand this if they can understand the local terms of the cohomology. Deligny had this intuition that if you twist it, you're trying to get the figure of the action of a cycle. The naïve local terms will actually begin to mean the naïve local terms, and this is a mere intuition. I don't know if they're linear, I don't know, but for Baynes the characteristic P behaves like some sort of contracting. Now how in God's name, what can that mean? Well, this guy Fujiwara, or Gabber I think, but this is just Gabber, figured out that what you should be doing here is working, instead of working in the characteristic finite field characteristic, working in the formal power series.

1:07:30 And you can actually, in the end, give a meaning to contracting here, so you go to some sort of what they call topos-associated to this thing, to rigid amyloid geometry, but you have to develop a cohomology here, you've got the sixth operation, the eighth, and the seventh operation, you've got this nearby cycle function, which you have to give a meaning to, of course. He does it all. He shows, in the end, he's got a slogan, a different one. The results, again, are completely out of reach by convention, but they can measure. So therefore, by iterating it, you approach something as simple as... That's right, yeah, yeah. I mean, that's the whole point. You can move... And what's interesting is, he had taken, it seems he took the idea for this from some fragmentary results that McPherson and others had done, kind of a stick zero, per se. But I was very struck just a bit. As Fujiwara did? Yeah, yeah, yeah. I was very struck when I took that in Cosmology course at Norbert, and I was watching the undergrad students. Sure, they were in a course on parts. And here's where they first met the term adjoint. Well, maybe not the first time, but they heard more about adjoint function. So they assume it's hard stuff, because this is a course on hard stuff. You know, it's been years since they had courses on simple stuff. And because they've got this mindset, you know, with no reflection at all, they think this is hard. And they just haven't the habit of learning simple stuff anymore. I mean, I know that myself when I read different books. You have to shift your mindset a bit. They would dream it would be beneath their dignity by that time. What? Oh, they have animals in the course of this course.

1:10:00 Dream of doing simple stuff. For one thing, graduate courses there didn't have work sections. This one had a work section. One hour a week work section. Must have had lots of humiliation. Which was expanded by popular demand to two hours a week. I mean, they threw themselves into it. Take a development like this. You sparkly mention categories and topos. Well, okay, you have cohomologies which has to be or is formulated in a categorical language, but I would try to think if without the, without, let's say, a strong... Categorical orientation, you could develop the same kind of ideas. To what extent? It's difficult to say. It's difficult to say because, I mean, the Sculpton and Versace theory, of course, it's constant stresses that must be carried out. You've got to reduce the simpler cases and deform, etc. Of course, he's not. He probably never uses a complex approach. It's just constant stresses. In that side, but you do need these, I mean, the functorialities that he spells out, they're very much influenced by the kind of the formulas in the room. And it's precisely that that he knows the top of all of these. Yeah, of course, it's a counterfactual question, so it's a problem, but I think it in the following way, for example. If we look at algebraic geometry before Van der Werden, so they did what they did and they had very good ideas, the Italian for example, but you know they could not systematize every result and some things remained very intuitive, let's say, and then obviously the introduction of the language of ideals, rings, all the commutative algebra held very much. But some people then could say well yeah but you know even with with it it adds clarity but it

1:12:30 doesn't really add insight. In fact you lose insight so one might say because you lose the geometrical insight. I think that's unquestionably true even in characteristic I mean characteristic zero the characteristic p was not such a big issue but even in characteristic zero I mean if you're doing if you're working higher dimensional complex geometry I mean you Be extremely careful about multiplicities, otherwise you won't get the right answers, and I think it's probably fair to say that, okay, so you do your ideal theory and you get the deeper results, and you won't find a detail about the structure of the ideals and dimension of these, but it is a bit hard to link it back to the job. Exactly. And what about the development, you know, using something from the beginning? Categorically, you can say, okay, a geometry, but it's not the geometry in the sense of, let's say, Sevelli or... But what I've heard from people, and I have not studied the Italians, and I haven't dug through this either, but a lot of people say, yeah, I mean, Van Verden and Zariski, they want to make the Italian stuff rigorous, and they, in fact, end up creating a somewhat different subject. It doesn't really go back to the... But, when Grotendieck's schemes come, then they can recover all the Italians. But with a combinatorial intuition rather than with a... Visual, somehow visual, geometrical. It's hard to separate. I'm not saying that you know when the intuition is visual or geometric and when algebraic, but... You're trying to picture singularities in complex two-dimensional surfaces, which is real four-dimensional. You're trying to picture these algebraic singularities. You're going to have geometric trouble anyway. I can't stand these people who say we can't picture things in four dimensions. I must say you've never seen anything in less. But of course, it's hard in four. It's hard in three. Some things are hard to picture in one dimension. So it's going to be hard, but certainly the impression I've gotten is that a lot of the Italian intuition did not... And then a lot of the, especially the resolution of singularities, was recovered.

1:15:00 Even Euclidean geometry is not the plane, you know, it's not really visualizable if you get into it. You know, if you try to apply that to designing a city or something like this, you get such a complexity that you have nothing else but, you know, a computer can use Euclidean axioms to help you figure out these things, but it's hard. So the real point is that visualization is always partial. And so at every level you have to struggle to get all the different kinds of partial visualizations that you can, and not complain that some particular type of visualization that's written in one particular case is not quite... It was very... I don't know if you remember, I think it's in Grotendieck's Esquisse d'un programme, where at one point he says, you know he's trying to arrive at some kind of tame topology, right? And he says, what we're struggling to express here is an ordinary, straightforward visualization of shapes. So, there's a partial visualization right there. He's talking about expressing this intuitive picture, but it's a partial picture from another angle. And when you read Zariski's Algebraic Surfaces, which is in the Old Italian, I'm glad to learn... Have you found this intuitive? No, those papers are talking about what I've seen. Well, it may be because we're in subsequent generation, but they seem completely impenetrable. And I think, yeah, when you hear these people have habituation, it's not in the sense of they had these pictures they could have drawn. No, of course not. They had extremely recondite habituation to these kinds of problems, not a spatial picture. If our grandfathers weren't supermen, we wouldn't even be here. I feel that about Broden, too. He actually could really deal with all these layers. You know, this problem, though, you see, this problem has expressed itself in another way, that this, you see, it was, at one point it was decided that the category of topological spaces was the default version of cohesion. That means that people struggle to see it that way. Topological spaces are not simple at all. I mean, they have incredible pathologies and so forth and so on, but you sort of swallow that and say, well, it's ordinary topology.

1:17:30 No, I mean, my view, and Chanuel and I try to cultivate this, it's sort of a stance rather than a theorem, is that any distributive category, or more accurately, an extensive category, is a category of spaces. So you try to do what you can do. You can put these things together, you have figures, you have intersections, you have all the operations that you can apply in the continuous algebraic, algebraic, infinity, combinatorial categories. To a certain extent, the intuition of how these things combine is independent of those particular determinations of what cohesion is. Just as you add to the idea of the more extensive category the contrast between a more cohesive one and a less cohesive one. Well, this is a very simple axiomatic system, very general. There are lots of different examples, and at the same time... The intuition about it is straightforward and simple. It's Grotendieck's idea of shapes. Then you put more flesh into this, you get more detailed answers by making some particular determination. A sub-object being a neighborhood of a point, and a point being this, and all these things that make sense, direct sense, not with reinterpretation, but direct sense in a certain class of categories, perhaps the one that I best describe. So I think this is, I don't know how to, since I'm talking about the stance rather than the theorem, I don't know exactly how to promote it, but I think it's... This is helping me because, I mean, you often complain about taking topological spaces as a default case, and I'm thinking, well, I don't exactly, I don't certainly not take general topological spaces as my default case, I sort of take topological manifolds. And if something unusual happens, I make it generic so that it doesn't happen. And I'm realizing that's pulling towards tamed topology. And the difference that grows is that real insights never ever happen, you know. So if I'm thinking of topological manifolds, not general spaces, and they don't do anything very weird, you know. But that's still, that's still less general than what they're talking about. And when I say the default, I mean, okay, you have this, you start with this sort of vague idea of the topological manifolds,

1:20:00 and now if you want to do the precise calculation, what mathematical theory do you take to apply to it? That's what I mean by default. In other words, when you're confronted with that sort of problem... Which theory do you pull off the shelf, you see, and so there's this idea, well, it has to be first of all topological spaces in the general sense, you see, that automatically, in principle, invokes the whole morass of the whole category of topological spaces, you see, and then, you know, it's going to have extra structure, it's going to have a sheaf of ring, it's going to have this and this and that, triangulation, whatever it might be, but those are thought of as being superimposed on this. Default version, whereas in fact, what you originally had might be more direct if you just describe it as what you have in mind, or at least interpret it in terms of... Let's say in terms of topos. A topos is already a pre-digested theory to a certain small extent, but since it's so general, you have a choice at least. It's on a different level. You can choose which topos at a certain point, but the reason that a generic topos has much more content. And so what I'm trying to do when they talk about homology is I'm trying to learn to pull it off the shelf and attach it to my idea as a very special instantiation, but that's the one I want to know for this purpose. I have two more questions because I didn't know what I wanted to say. I was thinking of this foundational problem with you. In the first stage, moving from the Italian style to the algebraic really style, then the ideal theory or commutative algebra is providing a true foundation for algebraic geometry. You are understanding now things that you could not understand without. You could in very complex ways, it's not just a matter of exclusivity. Surely you understand better now, okay? And you don't want to give up. No one can come... You understand the local intersection. Yeah, okay, but no one can tell you, look, all what you are doing is saying things with other words. You are more than saying that, right?

1:22:30 Now, when you move to the next one, which is putting the factorial and categorical concepts in, still someone can come and say... You are just rephrasing, okay, okay, these are very nice words, or not. Or they can say it. No, okay, is here the functorial approach providing a foundation for the, let's say, for the algebraic, for the commutative algebraic approach that we had before? Is it extending it? No, I don't. Well, this is tricky, you see, I mean... What I think I've been getting from your remarks is, you keep saying, for the local theory, the global theory isn't just add up a bunch of cases in the local theory, and the global theory does need this context. Exactly. This is the whole point. We want all of this to behave as much as possible, like complex analysis with a very deep... Constraints between, you know, residues, intervals and all the rest of it. I mean, I think, you know, yeah. Where the global behavior is much determined by a few singular points. Yeah, that is some other point. The point, you see, the Vandervaarden-type analysis and maybe later what Serre did, I mean, it, you know, you can end up in, I think, it's hard to be wrong, but my understanding is that you can more or less... By developing the ideal theory, not mainly the polynomial rings, but the power series rings, etc., you can end up with a very perfect analysis of notions of multiplicity and so on. But that's, you know, one does much more than that. You have the singularities, but typically something global is determined by looking at what happens. And I think it's only at this level of the functuaity it really begins. It's advantageous nowadays to look at it. Beautiful, but I think it's really only when one comes to the geometry that, you know, once you're in the geometry, you've got all sorts of auxiliary things that are never visible in the outer plane sections.

1:25:00 Well, it may also be said that in some sense, every time some kind of progress is made in mathematics, an old term gets extended in some way. In some way, for example, the term geometry, the term space, it retains, of course, something like its, you know, something in connection with its original meaning. I think that's great. I think that's great. It's just, I mean, and then problems, what you would call problems, which probably before would not have been regarded, maybe you wouldn't have even had the language to express them. They then become problems about spaces, and then you stop calling them generalized spaces, you just call them spaces, and then become problems which have a new character, well, I mean, they may or may not have an entirely new character, but in some sense they are, otherwise it's very hard to see how mathematics progresses, it isn't simply dealing with exactly the same problems that Pythagoras and the ancient Greeks were thinking about, but I do think there's something There's something intrinsic, of course. Also, I mean, I think, you know, when you have this new language set of schemes, I think, you get what you might call interval problems, which are only even expressible in the language of schemes, even though they, these problems may or may not have any, some of them may have relevance to solving problems that could be phrased purely in the old terms. So, some, well, look, all right, I mean, a very good example of this, which I don't, proof I don't understand, is Lyle's theorem. I mean, it's a great example. I mean, it's an amazing. I'm not sure. I mean, no, why? Maybe not because it was a, but the type of mathematics that was used to solve it. Yeah, but he has a lot of internal problems of its own. He was solving the Taniyama-Shimura. Yeah, okay. It's so happy that it's applicable to the... Okay, I agree. Nevertheless, nevertheless, all those, it's true, this is a spectacular case because it happened that, well, it illustrates my point. There were internal problems, if you like, which then happened to have this application, which may or may not be, it's quite unusual at least, perhaps in the case of a problem of that kind, but that doesn't mean that the...

1:27:30 It's a contrast in that case between what you might call an internal development of all this machinery which was devised for lots of other purposes, and then amazingly it turns out, well not so amazingly, in hindsight perhaps, it turns out that it actually has its consequences. Yeah, but look, if you compare, for example, you take the Riemann hypothesis. Many, many new concepts around, and many new techniques and whatever, but the problem there is the same one as it was a hundred years ago. Here we are talking about something that has some parallel with, let's say, what happened a hundred years ago. Intersection and singularities and the dimension of it, but probably we are not in the same place, right? It's not just... Some techniques to solve the old problem. The problem may be, sometimes it is, some old Italian problems of resolution of singularities are first solved in mathematics school. Even though there's a tremendous amount of internal, you know, I mean, look, another example. Sorry. No, it's okay. I mean, I'll just say one thing. You mentioned, was it yesterday? He of course was determined, in some sense he was interested in the resolution of Singularity because he approached it essentially purely in terms of the ideal theory. I guess this is the kind of thing you might call purity in mathematics, but I mean, when Yamanaka did it, it certainly wasn't in terms of the ideal theory. It certainly wasn't. And what exactly did Hironaka prove? I always think of which resolution and what generality. He proved resolution and embedded resolution in characteristic zero, all for over and also in analytic situations. So he did detect, I mean, there's a very general pattern here, but his methods do not, have not worked in characteristic p, and in fact one has probably not really improved very much on what Abhyankar did. The Yonge was a tremendous step because it did enable them to, of course this was not foreseen by the Italians, but it did enable them to prove, to sort out various things that had been problematic in cohomology theory, but dimensions of cohomology, because typically the pattern is that you can...

1:30:00 If you can resolve something, then you can do your calculations up there and then use some factoriality in the Fulton type theory, to get back down. And de Young's method was good enough, good enough for that, not for everything. But it's still short, well, well short. The Italians want to say you've got a variety, there's a birational isomorphism to a variety with no singularities. But Dionne shows there's not actually a birational isomorphism, but there's some non-singular variety that's related to this as if it was. For many purposes. So it's not that he provides a de-singularization, but he provides a workaround to solve the same problems. Anyway, for many purposes. It hasn't got the full geometrical meaning of resolution, and that's never been done. I mean, I don't say that because it's just that Viagra was using the van der Waarden technology. I mean, probably. He thought he was doing it. And he had success in chemistry. But the scheme thing, I mean, the scheme thing is clear. We have to use it. But here and now they did provide what the Italians would have liked. Every variety, every complex variety, is bioradially isomorphic to a non-significant one, and this involves a lot of apparatus. Moreover, you can spell out the process, the kind of processes which are used, the Bloch processes which are very explicit and are studied algebraically and are used for something. Cathy mentioned this Vouté-Toilet thing yesterday. Everything is done now in terms of these... And the Italians already sort of did that with this quadratic de-singularization, which turned out to be a kind of blow-up. So there, I mean, I think it's fair to say that Hieronica certainly solved a problem that the Italians could have understood in 1920. There have been other proofs subsequently, independent of that.

1:32:30 And also in analytics. In the same journals? Yeah, so they're published. Now these are perhaps a bit closer in some sense to the... I mean, and they have some of that. There are certain uniformities you can detect more readily than you can in an extremist, immoral, conductive world. There's one interesting thing there that Yarnaka didn't spell out, he probably knew it. You can detect uniformities by pure logic after the fact. You know that by the kind of process that you gain, a finite sequence of blow-ups, then you can just use it, it's not so trivial, but you can use it to compact the scheme of logic. Even results in characteristic P, you can show that if somebody gives you a certain complexity of a variety, a certain kind of definition, then all but finitely many characteristics, you'll succeed in de-singularizing. Now, in some sense, one can, by attending my class, get out directly of here and there. It's kind of precious, but at any given moment in characteristic zero, provided you're not using topology, you can really only be excluding finitely many. So that's really the test here, and that's the theory of the citation. I think he said that explicitly in his citation for Hironaki and we certainly said that. Algebraic geometers could use it immediately. Sometimes they're going to link it. But it affects tremendous simplifications in algebraic geometry because average variety means it's got loads of singularities

1:35:00 but there's enough factoriality in both intersection theory and cohomology that if you can... If you want to blow up something to be non-singular and do nice intersection theory and homology up there, you can usually descend again to get something. It is a fundamentally fundamental and tremendous difference. Are there any parallels or questions in the algebra theory? That's an interesting question. Again, thinking of the parallel of the development of algebra in the 20s, algebraic geometry and number theory, and of course, yeah, I mean, this is the... I know, it's hard to separate sometimes. Yeah, I mean, if you, well, I mean, the characteristic P thing, certainly, is a certain kind of problem. Well, even in the creation theory, for example, there was, in P-adic integrals, you have, they're real value P-adic integrals, and they're important. You've got some more generally-constructible terms that you'll find out about at the end. They've been studied by Panese and by Serre and so on. And you get some kind of generating function that counts the solutions mod p to the n. And then you form a generating series to get this number a times t to the n. And these things turn out generally to be rational functions. This is not at all. Except in simple cases like integrating directly over a projective space. And it was first, the rationale was first proved by, similarly, Atiyah. People use rest of the signal and it's in connection with the problems of distributions. That's the way you say it. I mean, it's tremendously powerful because you immediately, you know, if you're integrating some of that, you know you can sort of change variables and get into something very nice that's, I mean, tolerable. It's just a product of something.

1:37:30 There are various powers of your variables and this kind of thing can be integrated readily, etc. Then you can come back to get information. It is fantastically powerful. But as far as arithmetic is concerned, well, it would be, it's only really going to be... I mean, certain cases of the vague conjectures for alphabet varieties are simple. Something like in the L'Alliance program? Oh yes, certainly. Absolutely. In fact, this thing of Fujiwara's that I mentioned, the conjecture of Deligny. Deligny devised the conjecture not merely for the reason that... Not that Deligny had this intuition about contracting maths, but Deligny knew that for the information they wanted in representation theory, because of the tremendous power of the Bacon-Jackson, some of the powers of the eigenvalue, Deligny knew that for what they wanted to do, it wouldn't really matter, twisted by Frobenius got the answer, because you could come back using Hilbert. So, this guy Fujiwara did this by inventing this radically new technique, but it had been done earlier by Pink, a German. Using the unproved. And that relates directly to the language. In the language, we probably need to know. Unquestionably, the theorem, like I said, makes an obvious conclusion, fantastically difficult. There's a story that I've been told about Fulton's theorem. I don't know how accurate it is. The Mordell conjecture. The Mordell conjecture. He shows that... As I understand, his original proof used a talk of homology. And then it's since been brought to the level of Hartshorn. Well, which to a first approximation means using coherent cohomology. Yes, that's probably true. So the first proof is very sophisticated.

1:40:00 No, that is actually true. These proofs are still extremely sophisticated. The point is they are motivated more by taking known techniques in diaphant and approximation and translating them into the language of... vector bundles and so on, and then positivity and so on. So it's an intersection theory, but it has to be an intersection theory in this arithmetic algebraic geometry sense. So in other words, it's a different intersection theory from the intersection theory that we've been talking about. It's one where you try to take into account the intersection theory where your space is something like the primes and the points are infinities. But it's based on the same, there's a model for it, there's a common, in some sense a common generalization. There's a Riemann-Roch theorem in this situation, and so on. These are very, very different areas also. Yeah, so it's probably true to say that the cohomology has been stripped out of this to a quite large extent, but there is, maybe more, a hitherto unknown intersection with it. And there's an ambiguity... What is Hartshorck? You do cohomology for the Zariski topology, for general sheaves it wouldn't work well. Sarah discovers for coherent sheaves, it does, it gives the long exact sequence as you want. So this is an... and Peary used that term. I've been faulted for using it by referees, but it's been used on the Oxford tripos. It's used in several books. So I... Krzyzewski uses it as well. Yeah. So then, as a word, it's not in Hart's book. No, and I was surprised, because I... In fact, when I was first... When I was first offended by something by saying Hart's, I thought it was about... No, you talked about it without calling anybody. Yeah, I think that's... But there's this ambiguity... But there's two things. Hartshorn develops the actual theory of this cohomology, and when you say brought to the level of Hartshorn, what you could mean, it really uses all of this cohomology, but lots of applications of coherent cohomology are really just calculating one, two, or three groups, and you can really do them in terms of resolutions, so the theory is there to guide you, but in some sense you know how to do it without. You could have given this proof without talking about derived factors.

1:42:30 Other times, well, no, you really, you had to use the derived from your apparatus. And I just, when they say this stuff, I don't know where they're even claiming it lands. You see this in the book on Fermat's Last Theorem, the 650-page book out of that Boston meeting where they explain all the apparatus. They have a chapter on Galois cohomology, and they say, look, this is a derived functor cohomology, and here are some things that tells you. But the truth is, we only need it in dimensions 0, 1, and 2, where we could have done it directly in terms of... So, in a sense, they're using the Dreyfus-Hunter perspective. It does organize their thinking, and they tell you it should organize yours. On the other hand, they also tell you in the same breath that you didn't need to know that. Yeah, not bad. What about the difference there between commutative and non-commutative? Is it a point at all? You mean in terms of geometry? Yeah. Well, this is a bit, yeah, I mean, this is a point. I think, yes, I think everything has changed on both sides of the equation. I mean, the ideal theory of this is a messy business and I don't want to extract much. I had once the opportunity, the first course I took in Israel was with Ofer Gaber, okay? He was teaching us this business of localization. It had non-commutative rings, and you take a module. It's a kind of just a ring of polynomials like that, something like that. It took me so much just to understand. So I understood at some point that this was an important topic or something, but I think that somehow it faded away. Well, I don't know. I mean, I can't imagine a Gabbard would like to learn anything that was not ultimately fundamental. He is really one of them. Absolute world authorities on this whole thing, because he doesn't publish. I mean, he has taken a certain... No, I cannot say on that. I mean, of course you can give some notions of mobilization, but I mean, anytime you're dealing with non-commutative ideals, even the notion of crime, except for our very names, I mean, it's...

1:45:00 But he was just trying really to do the... I remember strongly that no commutative assumption was all around, and... Therefore, it was complex, of course. It was very hard even to see what is... Of course, if one puts on further assumptions in the kind of range you're dealing with, there will be problems. I mean, but these assumptions are restrictive. I mean, things like all remains. I mean, there is a... But these rarely occur. They sometimes do. I mean, they do actually... I mean, in connection with derived categories, I mean, in the Galvan Manning, he explains them. Maybe that's what Gamow was actually doing. It's in the... I mean, Mannion is rather nice because it's not too formal. It gives you some clear sense of the real difficulties and it's only really in a... So I guess if he was doing the localization required at that point, it almost surely is what's in his spring. But I remember very clearly, at least in class, he was completely avoiding any reference to categories or any point of... So it was a kind of ring theory. Go on with the idea of ring theory. Of course, but this, I mean, this is another point. I mean, certainly some of the things won't be done, only in a very limited way. I mean, you have to work with things like, you can do a certain amount of this for things like this vial algebra, which comes out ultimately from an abstraction of differentiation. But that's just not commutative, and no more. You see, it's not that there aren't like arbitrarily long words and two variables that go with that, I mean, that's really the game.

1:47:30 I think it is commutative, in a bi-gradient sense. So there, you can deal with it, I mean, at that range you can get non-linear diffraction fields, except in situations like this. Another case is for... Which, to be honest, is a powerful technique. I mean, you start with, say, a field, commutative or not, maybe with an odomorphism or an endomorphism or something like that, and then you adjoin the formal variable t, and you make it act on the original field by t inverse, alpha t is phi of alpha, where phi was the original endomorphism, then you form a formal power series thing which is somehow made this... And these things are perfectly fine if the ideal theory for the original thing is good, the ideal theory for the new one is good, you have materiality in many cases, and you have this other thing, so you can squeeze out things there, and it is important, I mean, this theory of d-modules that they work with around this kind of thing, and the abstract algebra is, although it's non-commutative, it's not non-commutative in terms of the kind of... Appalling non-commutativity that, let's say, the British school worked on in the 60s and so on, non-commutative theory of main central generality, I mean, this is a very hard one, and I don't think one can make any geometric sense to that. Now, again, you might be dealing with, say, non-commutativity and functional analysis, but then there, in those cases, you've got something, you've got extra structures, also, maybe then you can... You might be working with algebra, but there are those other structures that may help you to understand it, but the pure ideal theory, say, of, suppose you're dealing with something utterly free instead of two non-communic variables, you're not going to be able to do very much, but there are things that inspire people have done. That started with Nathan, by the way. This requires trickier techniques. You can't be too far from commutative, to my mind. People are going to have looked at the general case. The moment they saw the spec construction, they're trying to find it.

1:50:00 Very important things that I think were not fully understood as important. I was interested, this is a red herring, but I meant to check. When you mentioned that one of the talents that I met was Cremona, in connection with another comedian of ours. Veronese. Veronese, excuse me, Veronese. Was it based on a formal power series construction? I don't think so. I think it's a finite model, first of all. Really? It's a finite geometry. Because you would get them, of course, by just, you know, doing your, taking a... Formal power series, putting your x or your t at an infinite or infinitesimal, and if you made the thing have real coefficients and rational exponents, you'd get a real close field, so you would certainly get, you would get models of, I mean, the reason I ask is that, I mean, as far as I understand it, the serious study of these formal power series did go back to the... Yeah, this may be closer to what, I think, is Jubal-Raymond. Yes, he had this kind of athletic theory. This is not what Veronese was doing. No, no. Veronese was closer to Peano, to trying to create these people. Later on, after Hilbert, all these things converge. It's quite a different thing. At the beginning it was from different traditions. I just wondered. I simply didn't know. No, no, no. For example, it's not connected with the people in algebraic geometry program. He was in another town even. I didn't know. Because it's rather straightforward now just to build a form of power cities. No, no. They didn't mix up this. It's just an information. But actually non-communal brains don't exist. No, really, there is this abstract algebra. Non-commuting polynomials and two variables and, you know, terrible things, you see. But in nature, those things never arise. In other words, there's only one, actually there's one non-commuting ring, and only one, the quaternions.

1:52:30 Just, you know, they become community if you're going to use grading or various, or, but in general, you see, but in general, they're just objects in an additive category, you see. In other words, you could say, well, there are lots of rings. Take any of them in category and take the anamorphisms of any object. But that's precisely the point. They arise in a context where there are several interconnected ones. Well, of course there's three by three matrices. But this is stupid, because there are rectangular matrices as well, just as important as square ones, and they all fit together into one category, you see, whose objects are the natural numbers, and you can, so they're just... So they're just the endomorphisms of a single object in a context where you have several interrelated objects, and that's really the typical way, or I think in some sense except for the quaternions, the only way that real examples, look, operator algebras, I mean, you have bounded linear maps between biomath spaces as well as around one of them, and that's very important actually, that's part of the whole study of those things. So it's really a category, even if it's a very small one, even with just a kind of little number of objects or something. So you're saying non-community of rings don't exist, but categories of non-community of rings exist? No, no, not categories of non-community of rings. Well, you're saying bunches of non-community of rings with relations between them. No, those are modules, if you like. It's not a category of rings. The maps are not ring homomorphisms. It's an additive category, and then the endomorphisms, you know, in the category there are endomorphisms of any object. So, in an additive category, the endomorphisms of an object happen to form a ring, but examples always arise as a category. So you think you've got the non-commutative ring of 3x3 matrices, what you have is the category of finite dimensional vector spaces with the matrix transformations, the endomorphisms of the three-dimensional space are what you thought were a non-commutative ring, and technically they are, but... It's really a homoset in a category.

1:55:00 Right, and you have 2x3 matrices, and 2x2 and 2x3 just as much. But non-commutative monoids do exist. Well, for example? Well, no, I mean, the composition just in the monoid, you know, a composition of maps. No, no, it's the same. Okay, that's what I mean. Sure, in any category, but again, typically. Again, the typical way that monies arise, if you think about an example. But they're also parts, again. That's really a part of the category. Again, maybe only a very small category is relevant. Only a finite number of objects, even, or a countable number also. But typically, that's how they come up. So from that point of view, you look at all that research in non-commutative algebra or ring theory, let's say, and you say, well, let's put it flat, this was a waste of time. Of course not. Most of that can be retrained. Look, for example, the localization. In fact, there was Cohn, I believe, remember? Cohn talked about division rings and so forth, you see, I mean, the localization, even if you start off thinking I'm going to do this with rings, you find that you're doing it in an algebraic theory, which is a particular kind of category. You have to look at... Make all possible sizes of matrices over the ring, and that's the structure that you localize. That's a very novel thing. So that's a good example. Of course, a lot of work done by serious mathematicians, it can be sometimes in a small way, and other times it's really very important to make that difference so you don't get a good theorem, even starting from the point. We'll also trace that work back to... I should have said something like that, I think, in formal language. Oh, really? Yeah, yeah, yeah. City. Groups. Makes sense. Makes sense. Again, that's monoids. Yeah. So, in other words, a monoid. You talk about words. People commonly say words is a free monoid. But actually, the free category on the graph is a much more natural object. Natural in the sense that that's what tends to arise if you think you've got, you know... You have the three categories on the graph, which are better than that. But I mean, your general question may have been referring also to non-convergent geometry. I'm not really willing to come up with some further, but we have discussed it in previous days as a video, Mr. Cartier, on this, but I mean, in fact, even in connection with that, there was this Moide equivalence of the fact that rings came up in connection with morph, they have, like, morphisms and so on.

1:57:30 Well, that's a special case, really, of what's coming to be called coaching. And so on and so forth, and so forth, and so forth, and so forth, and so forth, and The functors between those kinds of categories may or may not be induced by ring homomorphisms, but they are the ones that give rise to the needed changes of invariance and so on and so forth. From the point of view of Cauchy completion, it means that... There are contexts where, to speak in a sort of schematic way, you're trying to construct a homomorphism from one ring to another, but really all you're constructing is a Cauchy sequence, so the limit, you know, is you have to complete the ring in order to hit an actual point, where an actual point is defined to be a projected module, not necessarily the free one. You don't just look at one non-commutative ring and its ideal structure, you look at actions that these are all group rings, the one she's interested in, you look at the group, you look at the category, it really does, so the work is there, the theorems are there, but they're put in this context. I mean that happens nowadays too, one has this quantum group stuff and so on, I mean this is also mildly... But it's not so far. I mean, quantum groups and quantum algebras generate information, and the information they use, you know, are close enough to the kind of things that people learned in the last century.

2:00:00 So they had, I mean, this has saved the lives of many British non-commutative ring theorists, you know. So, yeah, it's, if you look at the non-commutative polynomial where you can do variables over some field, this is big, but the whole point of the quantum groups is the commutator relationship. We've actually mashed things down a whole lot. In general, these commutators, you don't want to know about them, but then they all become a source of some certainty. You know a lot more about these things. Yeah, well, actually, the action is going to take the multiplier of the momentum times the configuration of that. The commutator is a constant scalar, which means that the action itself generates the commutative algebra. The Q times P is contained in the commutative algebra generated by science, you see. Yeah, that's what I meant. I mean, you can't... It's as close as it could be to these things that are... You've lost, by doing that, you see, you've lost the splitting of the action into, I mean, the times configuration, so you could imagine splitting in a different way, and you still have the same commutative algebra, but it is a commutative algebra. By chance, do you know any work of Levitsky? No, no, no, no, it's only because I think it's a good example, at least in Britain, one saw the same thing with functional analysis. You know, there was people who tended to be too systematic and carry things far beyond the point of dimensional returns very rapidly.

2:02:30 I mean, sometimes the math is very difficult, and occasionally, here and there, something nice comes out, but there was never any real purpose in it. But what Lelitsky did was, of course, very much more at the beginning of this, and it was connected with radicals, and so it was much more fundamental. I have to go to someone who asked me to say a couple of words about that. He was a very distinguished biologist. He got a big prize, a gold prize in Israel. So someone was asking me and I was looking at his theory. I think he came from Neter. Yes, yes. He immigrated into Israel. And all of these words are something of ring theory. I can't just browse it and figure it out. It's just that they're myths. You see, the people, they were Americans, but these people, Kablansky certainly knew when to stop, and obviously Levitsky did too, I mean, Abraham Robinson, I think, worked, perhaps not with Levitsky, but on things of Levitsky at the very beginning, his very first papers. I started with a student of him when I was eight. I knew him quite well. He's right, it's a strong tradition. I was always aware of him. Me too. He's a very important one. Again, he's something you went to call it a day. He was also at the Nice International Conference. Yes, yes.