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

0:00 Yeah, we're at the beginning here. I mean, obviously, first of all, historically, it's a notion which was used systematically by the Italians, but certainly often with very dodgy algorithms, which in some cases we would like to contradict the results. I mean, different companies are using some principles, which in fact are not generally correct. One thing to say is that people are now trying to... But in some sense they have an algebraic and geometric inspiration. I mean, these are structures which are not in some monolithic sense that carry dimensions and so on. But essentially, I think 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 homology and intersectional theory? I think this is really important, certainly in the Bain context. I mean, the general, let us say, to make it easier still, will be even better once we work on a non-singular variety. What do you mean? Non-singular variety. I mean, so, the point is that you're going to attach to a non-singular variety in some factorial way, you're going to attach an intersection ring, and that ring will involve cycles, which may be either effective or general cycles, but cycles are going to be typically linear combinations over the integers. Well, essentially, the cycles will be built out of sub-varieties of variety, but the sub-varieties need not be non-singular.

2:30 And the intention is that, so a cycle is a formal combination, now a cycle is of course, I mean this is the connection with other parts of mathematics analysis, it's only a cycle of something which is non-religionality, a connection between integration and cycle, but the ultimate intention Not so readily achieved, but the Italians say what's to obtain something of the following kind is you have a general variety of a certain dimension, d, and the math is usually relatively clear that d is 1. If d is 1, you're doing a curve. The cycles are, if it's an irreducible curve, there's a cycle of top dimension itself. Then you have cycles of, you have lots of points of dimension zero and the cycles of formal linear combinations. These are behind everything like Riemann law and so on. The whole analysis of the geometry of curves is done in terms of these cycles. In general, the problem is you have various intermediate dimensions. You've got the thing of sort, dimension n, and you can have varieties of every intermediate dimension in general. So the co-dimension, the difference between dimension and sub-variety. But technically one grades the situation. And you essentially form linear combinations of these. This will form a quotient. And then, of course, you have such things for every single dimension. So you have a graded group, a structure of a graded group on the... At that level, to obtain, in general, some sense of, somehow... And it's not really a set theory. I mean, for example, a basic problem, if I take two curves and...

5:00 And look at how they intersect. They will typically intersect infinitely many points, but sometimes they will intersect transversely, the curves go through each other, and other times they will intersect. Tangents will coincide, and an intersection index or an intersection number, it will be one, and then they will be transversely, and it will be different. This is because if you move them slightly, it's an idealized sense. Now the notion of moving them slightly has no clear meaning in curves. It has a clear geometric meaning for us, it doesn't mean anywhere else. It does see that you can make some kind of preliminary analysis of this notion in intersections of ideals and various constructions of ideals. You look at the generatives of the ideals and issues of no purpose except for what you can make. You'll have ideals which are not acceptable. Things of this will come in. And you can use various ideas of ideals sometimes to define questions. And this will, in simple cases, give you correct answers. But it doesn't really say it's a hard thing to work with. The general objective is to obtain, to take this graded group. ...which is, as I said, an inner symbol, and eventually divided out by some suitable equivalent, which will cause one to suffer from general motion. You can move another inner, in a suitable... Now, it depends what it means, the base space for the movement is. It may be one kind of variety, it may be another, and this will lead to different... But the idea is of moving things somewhat into general... ...to find a ring, but you'll only get the range if you've done some suitable...

7:30 On the graded algebra, you'll eventually define a ring structure with the idea then that if you take two cycles, the product collapsing of the cycle or cycle class will be the intersection of these units, although a very, very medical doodle is a saturated intersection. Just doing things in terms of saturated intersections gives you the wrong answers. I gave the example already, merely if you undertake 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 the fundamental theorem of algebra gives exactly n roots, it's not to give a general count of something, it's not at all straightforward. But it turns out that there are several notions, there are several natural notions for collapsing, And that's the old one, right? Yes, that's right. And then there's algebraic equivalence where you have a more general algebraic variety. And then there are other, more subtle notions which relate to the theory of moments. There are a number of different kinds of numbers, but it is unreasonable to expect the intersection of even two varieties to have a number attached to it, because a number is really only something that can reasonably be attached to it. A pure number you can attach only to a simple quantity. So, in general, if I have something that I could dimension, a big variety that I dimension, there I expect to get points.

10:00 Dimension. The multiplication is going to move the gradient. Is that because, I mean, with points you have simple multiplicities. When two surfaces intersect, they might be tangent in one direction... Yes, yes, yes. 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. And even for the intersection of them. The beautiful thing is that, and sometimes the most important version of all, is self-intersection. You take a variety and intersect it with itself, which is no obvious meaning at all. It gives you back, generally, something very different. And typically, it's something in intermediate dimension in the middle. Intersected with itself, you get a number. This is where the big fear 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 always the image you have. The problem is that in general fields, the notion of declamation is a very subtle thing, so I'm stressing at the moment that the moment you're working in a projective case to make this work, and you're working in an ambient variety, it's non-singular, it may not be singular. And they're intersectional. New editor, of course, which is somehow the one most close to Grotenbeek's. It can be constructed moving into generalization. But it can be done in totals. Because, you see, you ultimately want to get a nice formula.

12:30 It's just not, say, two equations with different degrees. I expect to get NM points, but I certainly don't get... I'm working on a projective situation. This is the reason I ultimately wanted to get a nice formula. You move everything into the picture. There's no repeated roots in any direction. Yeah, this is what you try to do, you see. The current situation is relatively the intersection of curves and surfaces. The difficulty is coming in what you're looking for. The products you get are only all sorts of complicated things about the meaning of an intersection when there's a high dimensional component in it. This bedevils almost every series piece of work in terms of counting the number of points and varieties over time. Certainly not singular cases. Okay, now how does this relate to what... What would you require for seeing something about the action of provenience in case of projected non-singular varieties? You want to be able to count something. I mean, this is because the characteristic is zero because that will only give you your data for modules.

15:00 That's futile. Formal things that relate to the singular give you the complex. The finite dimensional graded algebras are anti-commutative. Yes, in the end, that's what it is. There is never anything, there is no demand for it. It's just a problem, it utterly unstrikes. Why do we want to break it? Yeah, sure. I mean, it's crucial for everything. They want, you've got this intersection theory, which is completely independent of chromology, that's already gone, it's true. You then have these axioms, they're given in many, the most convenient, perhaps, in this book by Freitag and Kieler on italic chromology, because they give the beginning to be able to cook. Deoudanese gives a history of this thing and he skips the axioms of rate cohomology theory in this situation. Now they involve things like an abstract version of Poincare duality. It's straightforward to state in terms of linear algebra. The spaces should be tridimensional, you have a certain duality there. There are various cumulative forms. So for the product, the natural product, so the cumulative formula. Stated only for very simple things like points, which you can deduce that this cohomology group, which is a functor, will be sped, will make the product in your cohomology group correspond to the product.

17:30 I mean, just, when you're out of it, it's an astounding thing how you get this out. It's not done in Germany. There is no complete detail in some several very useful articles on the standard of conjecture. This one was written in 1967, 1969, I can't remember now. And then a much later one done on one of the two volumes of the AMS group on molecules. But he does just this magic of linear algebra to show that the variables correspond to the cycle. There's going to be a cycle map which attaches to a cycle that's a cohomol but one has, right, in the same order of theory. So a cycle map mimics some very basic factorial property and you say what it does to points and so on, and out of that you can read this. And this is Barry Ginnick's paper, the cohomological facts associated with a cycle. Yes, and then of course, at the level, I mean, it's not trivial to construct, completely trivial to construct a cohomol. This can also be done very formally, you know, done by Kate and others. I haven't read that article, but a lot of the others say that's when they first understood it. Well, this is spelled out. This is all spelled out in climate and related literature. In Katz, Nick Katz's short study on topology in the Moebius book. I've only missed out some actions, but you spell out the existence of this function. They tend to have similar varieties over algebraically closed fields, over some fixed characteristic zero coefficient field.

20:00 Now, the first interesting thing is that the coefficient field cannot be arbitrary. It's known that geometric light seems to be right, and it cannot be the rationals, for example. It cannot be the reals, for reasons of representation connected with what one already expects to be true about the cohomology groups of elliptic curves. In one of the cases that May had explicitly considered, they have to have a certain shape, and if the coefficient field was the rational or the real, you would have some sort of bound-primary-dimensional algebra over these which don't exist in a certain form. However, algebras of these properties do exist over here. Does the land of the ring structure come into that? Not really, but the land of the ring structure is never very far behind in some terms. 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, because of what we knew about the limitations of the curves, this left open the possibility of P-adic fields. Let me go back very much to the actual intersection theory. The intersection theory is quite independent and it can be done in a number of ways. It can be done by pure commuted value, generality, and I guess what Sarah did in this book, Local Algebra and Multiverse System. The method in Fulton's book is a very long book with complicated arguments, and the method was developed by him and MacPherson. This involves a much more general machinery of algebraic vector bundles and deformation theory and so on. It's beautiful but very tricky.

22:30 It's simply deeper. It's got much more functoriality about it. It need not, however, involve... There is no cohomology whatsoever here. There's some rather elementary commutative algebra. It's not here to call it algebraic topology, but there is an algebraic homotomy. Deformation as to the normal bundle and all sorts of stuff of this kind. And it is interesting that Schiff theory... Scheme theory comes in. I mean, although you're only interested ultimately in getting the, maybe it's safe to say that these calculations work in full generality to get the functoriality of that. Schemes need to be used. I mean, nothing very fancy about schemes. In other words, schemes are used. Just because a sub-scheme, say, of an affine variety need not be affines. That's right. So you sort of try to consider... And you start to, you know, you're talking about the idea that you're going to, you may well get Milbotons and all the rest that are coming in. I mean, there's no way around it. And this, I mean, the computational aspects of that, it's not, it doesn't really have very much to do, say, with Grudner Basics and stuff, I think I mentioned. It's a very different game. It's the calculations of, calculations with the length of a Tinian ring or something, kind of stuff that's touched on briefly in Atiyah and MacDonald. And it's delicate. This is quite delicate. The logical structure of that is really quite, quite delicate. So, anyway, these axioms that you mentioned, these are at least part of the state in terms of this already presupposed machine, like the cycles and... Yeah, that's right. The cycle map is crucial. The map, okay, but I'm really trying to back way, way, way, way, way up here. Sure, yeah. So, I've tried various times to read Fulton's books and papers to try to make some kind of sense out of all this. And I find that extremely unsatisfactory because, well, it sounds like the solution to a problem that was never actually stated. This often happens in mathematics.

25:00 I mean, somehow the mathematicians have an intuitive feeling that there's a certain problem, but without actually stating it, they get into some technicalities, which would probably solve it, and along the way, they, of course, make precise definitions, but what the general problem is, so this typically happens with, you know, in a general way with, you know, vaginal punctures and so forth, you know, you have a nice universal property, which defines the thing, but I don't know what it is in one sense. Now you go and try to figure out what the elements of it are. You have to do involved calculations and so on and so forth. So all that stuff about the Chao ring and so forth always struck me as being that, you see, because of that second stage and the initial thing has not been actually stated properly. It seems to me, although I don't know if Romney ever said this, but it seems to me this remark seems to be very much in Romney's spirit. You send us what the problem is, and then you can dive in. That's funny. I always thought this was another topic. 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? Yeah. Of course, couldn't there, as you mentioned, was crucial. And that again suggests that there really shouldn't be a homology theory. And the way that the cohomology is usually gotten at is simply by dualizing cohomology rather than any sort of direct idea of what the cohomology is, because that again clouds the whole code. But now another thing is you... In passing, you mentioned intersection of ideals, but the intersection of ideals is not the same thing. It doesn't correspond. No, no, no, no. I just said general operations are the same. No, no, no. The thing is, you see, that maybe it's not intersection theory, but union theory. Yes, yes, yes. In other words, it seems to me that the crucial difficulty or enriching aspect... Thank you for your attention. So, somehow it's about the contradictions between product and intersection and ideals, don't we?

27:30 At least that's a major part of it. And of course, those things directly, if you look at the close 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. Okay, so, I was hiking in the mountains of Italy one time, and I was thinking about this. I was thinking, ah, the African plate comes up and lands into the European plate, or whatever it's called, and what you get is the Alps, you see, so, 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. 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, you know, you get an intersection of sub-varieties as well, so those are obviously the things that 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 a total sub-variety, well, I mean, okay, if you, as I say, the ordinary lattice operations on the sub-objects have become an object. The logic in the narrow sense, so to speak, will, you know, you will see. The product of the two ideals gives rise to a third quotient range, 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 community development or equivalently in terms of affine geometry? I worried about this for a long time and never could ever find it. All the sort of naive things that you do, push out, push forward, you always wind that back in the same realm of your simple-minded logic in a narrow sense. Sure, sure, sure. Just various distributivities and so forth.

30:00 I think that's one problem that needs to be, you know, in order to in turn be able to state in some straightforward way what it is about how we all stick out, you see? Sure, sure, sure. So the lattice of subvarieties, it's a lattice, but moreover, you've got this product there as well, and that's somehow what it's all about. You have to resolve this contradiction. Now, the really interesting thing is that the universal algebraics, one of the new concepts that they came up with, they're going to do all sorts of incredible calculations in incredibly bizarre situations, and they like that. But one general concept that came up with this was a so-called commutator, which corresponds to commutators and subgroups in the case of, well, the normal subgroups in the case of the category of rings. It's an operation on equivalence relations. Well, no, it's an operation, it's a binary operation on equivalence relations in any category of universal algebras. 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 categories to try to understand, to formally understand what universal algebra has to have done. With this so-called commutator, and I told them, well, look, this is a desperate trying need in algebraic geometry. We 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 commutative range? They never do that. No, no, no, no, no, no. Because they have a certain mindset what the kind of problems... And so they talk about very general varieties. They know about groups. We talk about various extremely bizarre varieties, but to actually apply it to an honest, everyday, mathematical category other than groups, namely community branch, they never do it. At least, you know, you can see enough what they said that it does come out to be the commentator.

32:30 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-ups in that category, so it's just 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. Yeah, but people who are expert on that refuse to... There's a big problem we have with all the category theories in a way. They don't often enough focus on a particular example and apply these powerful tools and see what they mean. Yeah, I mean, that's true. I mean, obviously, in some sense, the heart of the intersection is the thing that leads to intersection. And then you take the diagonal and take the self-intersection of the diagonal, which is the direct form of meaning, leading to the form that the self-intersection is 2 and 2g, where g is the genus. The 2 comes from the top, cohomology is 1, the bottom cohomology is 1, and the intermediate one is 2. So, of course, the self-intersection can be negative in some cases. I mean, this is what played basic.

35:00 And that particularly, that was the kind of thing that foundations of algebraic geometry was designed to make absolutely perfect. Certainly new lashes transform. So when you get the self-intersection to come out to, I think they necessarily would know it as it was artistic piece. No, no, no. I mean, that's the point. I mean, that the base spotting that this was still true in the artistic piece was naturally... There are some pictures in the camera. Yeah. There is a cohomology theory, or there may be a cohomology. I mean, that wasn't the way he proved it. And it's just terribly difficult to make sense of this just in terms of scientific ideals and even small deformations. It takes quite a lot of work in any moment. It's a bit of a problem. I found it, I mean, I tried something maybe a bit perverse, but I had a very specific reason for doing it. I mean, okay, so the very cohomology theory is a functor. I wanted to get somehow, and at the time the axiomatization was done, one didn't know if there was F to the right anymore, so it was not uniform. I didn't know it then, but one would know it in the very early days of Keitar-Gomal. You could get a theory with Mozart in something like Z-module, on a torsion situation, and this had been known for some time. But not that you get this sort of characteristic zero coefficient. 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 it.

37:30 And they all seem to have the same property. I mean, the finer detail seems to be the same, the finer detail about the numbers can be the same in all cases, and they're related. I mean, what made one see that they were the same was the interception theory, so they're independent. Anyway, I decided I would take, you know, assuming one had a family of these commodities, I'd take some average. And sure enough you can prove it is a co-multitude, again, and a main variable. I mean, to show that this symbol, I assume, is somehow preserved, I mean, that is really rather hard work because it involves somehow, what Kreisel calls, unwinding the proof of the construction of the infrastructure, making it nowhere nearly as elegant as they do. I have also serious worries about the logical structure of infrastructure. One doesn't really, I think. I mean, you know what you want, one of those things, but... Yeah, I mean, just to give a very, very simple analog, the contrast between Chanwell's element definition of the collider grain and the dimension of the grain under that suitable category on the one hand, and the way Van Der Ries calculates it on the other hand, by never defining it in a conceptual way, he'll say, well, you have to drop it off this way, you have to solve the opposition, solve the composition, blah, blah, blah. That's what the definition he gives. Well, in some sense, the cell ring and all that stuff is much more elegant, much more, but still, the universal property has not been stated. Not really. No, I mean, the general insight is that in a general, I mean, the only real case one has is that, you know, when you've got things of complementary co-dimension, which formally you expect to. There are a number of different kinds of points, and 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 near the content of the model. Italians were working mainly on surfaces around curves, so that's typically not out of there.

40:00 There's going to be a cycle, the meaning of physics. So, this is a very long, very complicated... I've never even really understood the situation in topological manner. You've got the Laschet's fixed point theorem, and it's certain that if the Laschet's number of fixed points is not zero, then it's really a fixed point. It's not certain that the Laschet's number counts the fixed points in any apparent way. If they're nice and transversely... No, no, this is the whole problem. This is 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 different components of other dimensions in this formal way of doing things, unless we devil's everything. I mean, yes, there's quite a lot of things we have to do. I'm very aware of the pitfalls, and that's always the general problem. It worked very generally to actually give... I mean, 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, genuinely, from the foundations of this theory. In the sense of what one really needs to know. There's this general principle that you can...

42:30 You get a cycle, but the cycle will have a well-defined number attached to it, the sum of its coefficients. It's not clear that's going to happen, that's a bit blinding, but you prove that that is well-defined. So you invariably get a number out of the intersection of the product in such a way that it's two cycles. In the complementary dimension case, there are modest chances of having to do the number. You do get this number, and now you can define it. A cycle being numerically equivalent to zero, if its intersection with every other cycle of the right correlation is zero. What it goes in does not mean in our geometric picture. This is numerically equivalent, and this is in some sense the most important notion. It's completely devoid of any other picture. Here's why it's important. You have this, for any given vehicle model, the definition, as far as I'm concerned, there are many models. So you can ask yourself what, so it's a natural transformation. So you've got the, you take a given, take a given variety, you've got a cycle on it that's numerically equivalent to zero. Well you can then show, which means, so cycles induce maths and cohomology. This is part that comes out of the Poincare duality, again by general linear algebra, well known in this. The classical complex cases. But you can get out of this just with the axiomatics and the general case.

45:00 The cycles go into maps and cohomology, but then by reality there are elements in the complementary. In the complementary cohomology, quantum reality in this situation gives you some kind of fairly canonical map between H-I and H. Is this treating the cycle as an intersecter, sort of? And then, but then, so the natural question you want to ask is, when the, a point of duality is essentially, we get this picture, when is a cycle down on the, yielding your zero element, if this is so, it comes out in numerical, it comes out as zero, but the conjecture is that the, independent of the, of the homology, and certainly for things, for italic homology, so the conjecture is that. A cycle goes to zero if and only if it's numerically. I mean, let's go to zero and all come only if the answer is back in the intersection. That's not being proved and that is a concept, not equivalent I think. But the intersection theory is the intersection that all goes on. Yeah, I mean this is the picture. This is really what the motives came from. The motives came from this that you have the objects of the verb. So the varieties and then the thing is of course you and then the correspondences between so some varieties of products so this at least provide the morphism basically the more you want to define what is identical this is basically you've got important yeah um this can only really be you can only give a difference that this will work when you get an actual category what is this conjecture and this is not quite equivalent but it's the essential point And it will follow the Hodge conjecture, and the National Hodge conjecture, and the Kuhnian, and this is the other thing, in the Kuhnian theory, you take a cycle in a product, which in turn is living in a tensor algebra, there'll be products, the projections back down into the individual, the cohomology of the individual varieties, part of these things themselves coming from the intersection here, they come in cycles, or rational combinations of cycles, that is...

47:30 I mean, that's basically part of the Hodge theory. So if you conjecture that, it's been proved in one or two cases. For a variety of reasons, I feel this is true because of the Bay conjecture. And another thing, if these are true, then in fact numerical equivalence is equal. Somehow the intersection theory is the intersection of all of them all. In some cases, I say it. The seminar I had with my teacher, he didn't very often make philosophical pronouncements or methodological ones, but one he said was, down with excessive double dualization. Yes, it was funny, but yeah, yeah, yeah. You see, who doesn't? Well, I mean, okay, you say there should be a map from cycles to cohomology, but if you've got Poincare UO, then cohomology would be a whole model. So you've got a map to a whole model. Yes, yes, yes. Yes, yes, yes, of course, of course, the projections are, I agree with you, but then you see, okay, there is a certain difficulty in this, not even carrying out this, I'm saying, I mean the difference between proper maths and non-proper maths, that's right, this is why we're in the projected non-singular case here, and that is crucial, the moment you try to do something like this, I mean, there's still an intersection theory on affine varieties, and it's local, but the appropriate cohomology theory in general characteristic P, then, I mean, sheaves then come in, essentially, you have to work with, you basically have to projectivize 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, and that changes everything.

50:00 And you need fairly deep terms to show that this is a cohomological problem. So it's, I mean, this is the, to make them a more bizarre field in that situation, and even worse, and if you started with a, that's fine if you started with, say, a non-singular alphabets, suppose you start with a singular alphabets, it's even worse. I mean, the course wants to work on when things are really only working perfectly. The point of the duality in there is between It remains desperately hard to get the right answers for affine varieties, number of points, technology even going beyond the means. I find that logic is still very... So this map from the cycles today now is called a homology. It's relatively a whole monody of a point, so you're taking the total of the distribution. Yes, it's the distribution where compact support, then you just take the total. Yeah, that's what it is, because it's the sum of the coefficients, he says, but typically, so that's going to come out to be zero and get the total, which is the total. Yeah, yeah, yeah, there's a total to that, but it's very, I mean... But the point is that since the models are not compact models, therefore the whole thing is... So you get, in my estimation, there's exactly the same problem as distribution theory. So, I mean, Schwartz made this, because it gets a little bit closer to his illusion that distributions are generalized functions,

52:30 he said, well, let's take functions of compact support to start with. Well, already you've destroyed the natural functoriality that the functions have, it's an intensive point, you can have arbitrary maps that change, but now it's only proper maps. And now you take the linear dual of that, and you get the distributions, which are not necessarily a complex, co-variably functorial, but only with respect to 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 come from a general map, for example, projection maps. And then the proper case is gotten by a sort of women's purpose. So that would make many of these things... You've got to take a dual of every point, so you're going to do the opposite thing. You've got to take a Poincare dual, and that's just... No, it's not even in climate. Climate, in many ways, is a beautiful article, just because you realize just how bizarre things can be obtained by a couple of applications of, you know, double duals. Especially when it's a product, as well as the additive stuff. I mean, it's kind of miraculous. Just how completely formally of these actions you get it. You get the cycles acting in sort of a cohomology. Of course, to give in any given case the action of the cycle requires a different, a different kind of punctuality and most... In fact, that's why you have a product, because you simply apply it successively to the operation, which is operating on the X-sensor. So, the way back means issues are incorrect. Absolutely, yes. In fact, he proves duality by more or less.

55:00 He draws a picture of two curves crossing each other like this and induces that if a curve is numerically zero, then it really is called homologically or homologically at that point. And then he gives you far less than a picture claiming this happens in complementary dimension as well. It's hard to even believe, 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 quite great development, which is correct, from this rotten argument, basically, that numerical zero is zero, he gets a very important corrective compliment. Well, of course, the algebra, just the pure abstract algebra, we avoid them all, generally, in, say, climate. It's not due to climate, I guess it's due to, it's a broken link under the boundary. I mean, you get the Lusher's fixed point, Lusher's trace formula out of all this. It's totally vulnerable. I mean, really, you wouldn't need to know much about the intersection theory. So when you, a couple of times you start to say fixed point theorem and change the trace. Yeah, well the trace, I mean, the number of fixed points, I mean, you calculate, you calculate, okay, So the degree of the intersection in terms of you take the action of something on cohomology and calculus, that's the general principle, and I mean that is not a climate, and of course what they wanted or what they did for the big conjectures is to take, you take a variety, a single variety and you take a diagonal inside the product of it, and now you want the provenience. The graph of Frobenius, and you want the intersection, you calculate, but 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 on the moment, and it's hard to perform that. And this isn't, this is nowhere near the level of A-conjectures, but that's the problem.

57:30 And it's already, it's already... I mean, I probably never realized that there was 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? Oh, yes, yes. I think so, yes, yes. Quite why, I don't know. I mean, he did refer to the fact that all these things that couldn't be clearly hoped that they'd be started by doing the standard conjectures, um, I mean, Kantian didn't make it quite so clear just how short is the prune. This is done in the climate, in the climate setting, it's done independently by, again, it's essentially at the level of some even more chronic, I mean, it's just really just abstract algebra at this point. ...assuming a cohomology theory with integer coefficients and really nice simple properties. We know that all better. But this is sort of as it could be, given that you won't have any coefficients. Yeah, I mean, essentially if anything in character is zero, you've got a canonical embedding of the integers into it, and then that is all what's needed. Of course, it leaves the mystery, and I suppose there's still no big answer to this, as to which. I mean, the pianics, and then various bigger extensions of that. I mean, the P-idics are possible for the L-idics are possible for the L-idic, yeah, and then if you want L to be equal to the P, you need this crystalline cohomology, and then you cannot work with Q-L, you've got to go with something very much bigger, this unramified, the word vector is over the algebraic colonial, even beyond that, there are various rings in the trademark of Fontaine and others that are needed to get, oh, you see, you want to get, you want to detect other things, you want to detect. And they are not visible in the Piatics. You have to go to make any sense of them. It means you get periods, which is the demand theory you're aiming for, although you don't have any differential forms of them or how you're dealing with them.

1:00:00 So, the beauty of this, well not the beauty, but the interest perhaps of this crystalline case is that you've got an entity with more structure than merely being a finite dimensional planet. There's a great anti-commuter technology where you've got the action of this characteristic theory of Frobenius on it. Of course, the Italian ones, you've got the action of the Gaul group, which is again a great difference over the singular model. So each, and there are other cohomology theories discovered too, so they all, many of them have more structure than merely this. All you require for the very maximization is just graded anti-commutative over some field of privacy. I think Rotnik actually constructed it. This is complicated. I mean, you said you want to be in the air, and then you want to go over the ice, and then you want to disregard some torsion, and you want to go over a field, but the... And this also is complicated. It's a way of trying to do homology. At some point, the category you go is not going to be the category you achieve. I guess there was this thesis of what's... You also want to compete on differential equations, don't you? And I knew him once, but I don't come back. They have related, but they have typically the only characteristic is a self-intersection, an insight one doesn't have.

1:02:30 Yeah, we 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 deform it into general position. Yeah, yeah, yeah. But this is the general position, who is it that criticized some of the Italian differences? This is into the generic case where we have not proved that there are instances of that case. Yeah, and the biggest foundation for some of the design interview is the connections between the intersection theory and the cohomology become more and more tenuous. I mean, this is one of the, I'll say for the provenance, but it's actually true for some general reason of the action of, there's still an action of science, but there's a theorem that says The trace of the action of one set is equal to the sum of local traces for some kind of strange action, and the problem is that this has been proved. We really don't know the meaning of the rule, and it's not what you think it might be, this is not it at all, it's simply a deformed version of it, and you can't handle the deformation of it. This is the key to getting decent results in accounting points on applied varieties, and this has only been done in ten years or so,

1:05:00 There are a lot of prophecies around the whole time, and there's delineate coherence, coherent categories, and he uses it, writes it, because he shows that you're trying to work over algebraic geometry, over the algebraic culture of the finite field, an idea, they wanted to understand these affines, they need to understand the structure of the structures in an affine situation for purposes of representation. This is what they need to know. The only way to understand this is if you can understand the local terms of cohomology. Dunénia had this intuition that if you twisted, if you're trying to figure out the action of the cycle, if you twist the cycle by a suitable power, the graph of Frobenius, the local terms will actually begin to mean the naive local terms. And this is a weird intuition. I don't know if Dunénia had it or not, but Frobenius' Encapis Dei Pi behaves like some sort of contracting vector. 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 on The direct focus associated with this thing is rigid and linear geometry, but you have to develop a cohomology theory with a sixth operation and a seventh operation, because you're dealing with valuation theory, which is nearly two points in the spectrum of this nearby cycle from which you have to give a meaning to it. He does it all, and he proves, he shows in the end that, in fact he's got a slogan, that the Frobenius is really contracting that.

1:07:30 So, therefore, approach something that's simple. Yeah, that's right. Yeah, yeah. I mean, that's the whole point. You can move. Exactly. And what's interesting is, he had taken that. It seems he took the idea for this from some fragmentary results. My personal analysis was going characteristic zero, per se. Uh-huh. But I was very struck, just a very casual one, by his constant use of words. Yeah, yeah, yeah. So they assume it's hard stuff. This is a course on hard stuff. You know, it's been years since they had courses on simple stuff. Now that they've got this mindset, you know, with no reflection at all, they have a habit of learning simple stuff. You have to shift your mindset a bit. They wouldn't dream it, it would be beneath their dignity. Oh, they have had enough of it. In the course of this course. Doing simple stuff. The courses there didn't have work sections. With the loss of humiliation. Which was expanded by popular demand to two hours a week. I mean, wow, they threw themselves into a category for use the simpler and simpler cases in the form except of course he's not he probably never he probably never uses it's just a constant stress in that side but you do need these functorialities that he spells out they're very much influenced by the kind of the formulas besides the map in order to talk yeah of course

1:10:00 I see it in the following way, for example, they did what they did and they had a very good idea of the Italian, for example, and I think it would not systematize every result and some things remained very intuitive, and then, obviously, the introduction of the language of ideals brings all the commutative algebra, helps very much, but some people then could say, well, yeah, but, you know, even with... It has clarity, but it doesn't really add insight, in fact, so what I say, because you lose the geometric... I think that's unquestionably true, even in characteristic zero, I mean, characteristic P was not such a big issue, but even in characteristic zero, I mean, if you're working higher dimensional complex geometry, I mean, integrations, you already have to be extremely careful about... And I think it's probably fair to say that, okay, so you do your ideal theory and you get the deeper results, you know, and find the detail about the dimension of the reason.

1:12:30 But it is a bit hard to link it back to the geometric idea. You know, you seem so strongly to say, okay, a geometry, but it's not the geometry in the sense of, let's say... But what I hear from, what I've heard from people, and I have not studied the Italians and I haven't talked through this either, but a lot of people say, yeah, I mean, von Gardner and Zariski, they want to, they want to make the Italian stuff very grisly. In fact, they end up creating somewhat different subjects. It doesn't really go back. But, when the experience comes, then they can recover all the Italians. Yes, that's right. But with a, let's say, with a combinatorial vision rather than with a visual, somehow visual. It's hard to separate. I know I'm not saying that they don't know when they should 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 saying we can't picture things in four dimensions. And I'll 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, I mean, yeah, so it's going to be hard, but the impression I've gotten is that a lot of the Italian intuition did not, it was, Zariski and the founder of Arenal aimed at it, didn't reach it until scheme theory, and then a lot of the, especially the resolution of singularities was written over. Even Euclidean geometry is a plane. It's not really visualizable. 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 a computer that can use Euclidean axioms to help you figure out these things. 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 is right in one particular case and it's not quite... I don't know if you remember, I think it's in Grotendieck's Espice d'un Provence where at one point he says, you know, he's trying to arrive at some kind of tame topology, right?

1:15:00 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 that he's talking about. He's talking about expressing this intuitive picture, but it's a partial picture from another angle. And you read Zariski's Algebraic Surfaces, which is in the Old Italian. Well, those papers are talking about what I've seen, maybe because we're a subsequent generation, but they seem completely intangible. And I think, yeah, when you hear these people have intuition, it's not in the sense of they had these pictures they could have drawn before. 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 Bruton, too. Yes, it is. This problem sets itself in another way. You see, 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. No, I mean, my view, and Chandruil and I tried to collate this sort of a stance rather than a theorem, is that any distributive category, or actually more accurately, any extensive category, is a category of spaces. So you try to do what you did. You could put these things together, you have figures, you have intersections, you have all the operations that you can apply in the continuous algebraic, analytic, infinitive, combinatorial categories. To a certain extent, the intuition of how these things combine is independent of those particular determinations of what the program of cohesion is.

1:17:30 Add to the idea of an extensive category, the contrast between a more cohesive one and a less cohesive one. Well, this is a very simple asthmatic system, very general. There are lots of different examples, and at the same time, the intuition about it is straightforward and simple. It's grown these ideas of shapes. Then you put more flesh into this, you get more detailed answers by making some particular... Determination, you know, the idea that a sub-object being a neighborhood of a point and the point being this and all these things that make sense, direct sense, you know, not with reinterpretation, but direct sense in a certain class of categories, perhaps the one that I just described. So I think this is, I don't know how to, since I'm talking about a stance rather than a 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 the default case in Africa, and I don't exactly, I don't necessarily 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 same topology. The difference between real insights and how it kind of happens. So I'm thinking of topological manifolds, not general spaces, and they don't do anything very weird. But that's still less general than... When I say manifolds, I mean, look, you have this... You start with this sort of vague idea of the topological manifolds. Now, if you want to do a precise calculation, what mathematical theory do you take to apply to it? That's what I mean by default. And when you're confronted with that sort of problem... Which theory do you pull off the shelf, you see? 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 an extra structure, or it's going to have a sheaf of rings, or it's going to have this and this and that, a triangulation, whatever it might be.

1:20:00 But those are thought of as being superimposed on this. These are the default versions. Whereas, in fact, what you originally had might be more direct if you just describe what you have in mind, or at least interpret it internally. 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 you want to go to, but the reason that I quote generic topos is much more, has much more content. And so what I'm trying to do when they talk about cohomology is I'm trying to learn to pull it off the shelf and attach it to my idea. And it's becoming, you know, as a very special instantiation, but that's the one I want to know for this purpose. 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, you understand a 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? Of course. Putting the factorial and categorical concepts in stigmas, someone can come and say, you are just rephrasing, okay, okay, these are very nice words. They can say it. Is here the factorial 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?

1:22:30 Well, this is tricky, you see. I mean... What I think I've been getting from your... You keep saying simple notebook theory. No, it's the whole point. And the global theory does need this context. Exactly. I mean, this is the whole point. I mean, we want all of this to behave as much as possible like complex analysis where there are very deep constraints between residues, intervals, and all the rest of it. I mean, I think... The idea is much determined by a few singular points. Yeah, that is somehow the point, you see that the Van der Waarden type analysis and maybe later what Sarah did, I mean, you can end up in, I think, what I'm understanding is that you can more or less, by developing the ideal theory, not mailing the poem to the powers that you bring, by looking at various things around the jet species, you can end up with the very functions of multiplicity and so on. You know, one does much more than that. I mean, 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 functoria that you really begin some of the changes nowadays to look at the functoria. It's beautiful, but I think it's really 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 algebra. Well, it may also be said that in some sense, every time some kind of progress is made, in fact, an old term gets extended in some way. For example, the term geometry, the term space, it pertains.

1:25:00 Of course, something like its, you know, something in connection with its original meaning. A lot of you were originally in that time. Yes, well, who's right? I think that's great. Yeah. I think that's great. It's just, I mean, and then problems, what you would call... We're thinking. Yeah. But what you would then call problems which probably before would not have been regarded, maybe when you've had the language to express them, they then become problems about spaces. And they stop calling, then you stop calling them generalized spaces, you just call them spaces, and then what then become problems which count a new character, well I mean they may not have an entirely new character, but in some sense they are extended, otherwise it's very hard to see how mathematics progresses, it isn't simply dealing with exactly the same problems that Pythagoras and even the ancient Greeks were thinking about, but I do think there's something that There's something intrinsic, of course. Also, I mean, I think, you know, when you have this new language, say, schemes, you get what you might call internal problems, which are only even accessible 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. I mean, it's a great example. I mean, it's an amazing... I'm not sure it's... Well, I mean... No, why? Well, I mean, maybe not, because it was... but the time of mathematics was used to solve it. It's all internal problems of its own. He was solving the Tamiyano-Shimura. It so happens that it's applicable to the... Okay, I agree. But nevertheless, all those... It's true, this is a spectacular case because it happened, that caused, it illustrates my point, that there were internal problems, you know, that you invite to the development, which then happened to happen while I was solving it, and I couldn't turn down this application. Which... Maybe, 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, you know, I mean it's a contrast in that case between what you might call an internal development of all this machinery which was devised for lots, perhaps for other purposes, and then, you know, sure, then amazingly it turns out, well, I mean not so amazingly, in hindsight perhaps...

1:27:30 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 100 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, resolution of singularities, our first problem we wrote in high school. Even though there's a tremendous amount of internal, you know, I mean, look, another example, sorry. No, that's okay. I mean, let me just say one thing. You mentioned that, was it yesterday, Abiyan Faraday? Yeah, he, of course, was in St. Christian. I mean, he, of course, was determined. I mean, in some sense, he was interested in the resolution of St. Goliath because he approached it essentially purely in terms of the ideal theory. Yes, there's this kind of thing I call purity of method, but I mean, when Hironaka did it, it certainly wasn't, I mean, ideally it was relevant, but it certainly wasn't. I mean, you really had geometric pictures and scheme theoretic language. What exactly did Hironaka prove? I always think, which resolution in what generality? He proved an embedded resolution in characteristic zero, all over, and also in analytic situations. 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 Abiyangar did in characteristic P, for that to have been an announcement. Well, depending on what? Well, it depends on what you understand. The young, but the young is not, and the young is fantastic. It's not distinctivization. Absolutely, and the young was a tremendous step because it did enable him to, of course this was not foreseen by the Italians, but it did enable him to prove To sort out various things that have been problematic in cohomology theory about dimensions of cohomology groups, because typically the pattern is that if you can resolve something, then you can do your calculations up there and then use some factoriality in the Fulton type theory, in the right now.

1:30:00 And the 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 singularity. What de Jong shows is 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. For many purposes. It's not, 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 Avianca was using the Van der Waarden. He had success in chemistry. The scheme thing, I mean, the scheme thing is clear. There's nothing called that, and I think they're not, they don't believe. But here in Aachen, did... In fact, what the Italians would have liked, every variety, every complex variety, is biretically isomorphic to a Nazi. And moreover, but on the process, you can spell out the process, the kind of processes which are used, the law processes which are very explicit and are studied algebraically and are used, for example, in, well, I mean, Katya mentioned this bute, toile thing yesterday, the thing before. And the Italians already sort of did that with this quadratic de-singularization, which turned out to be a kind of blah-blah. So there, I mean, I think it's fair to say that Hironaka certainly used post-Thunderdark. And solved a problem that the Italians could have understood in 1920. But how many other proofs subsequently of independent of Hironaka?