Discussion — 0-Minimality program, tame topology, ramifications & other topics

0:00 Okay, and then just leave it up there, it just should work. Right, it's the 10th of June, 2005, and this is the session of discussion on tamed topology and the underdevelopment programme, its ramifications, taking place in Bruxelles, with the discussants being Bill Orbea, Angus McIntyre and Colin McLeod. I guess one of the first things that I'd like to develop would be a way of explaining this to category theorists, because I've been pushing it for 25 years, and it would be desirable to, I start 15 years, it would be a desirable thing, but I don't think any of them would actually learn what it's all about, except maybe what Ray is. And part of it is the terminology. I mean, O minimal means nothing to anybody. You have to explain it. Yeah, actually I would like to know why that term. I know why. I learned it. I studied it. So O refers to order. And minimal refers to the fact that the... The one-dimensional part of the model is the minimal complexity, namely the complexity equal to that of O. Again, minimal is unclear what you're minimizing over this. What title do people working in this and related areas actually use for it? I refer to the O-minimalists. Where it came from, I mean, the precursor is the term strongly minimal, which is directly connected with algebraic geometry, although it has never been something that algebraic geometry could have stressed. This was detected by logicians round about mid...

2:30 To late 60s, in the immediate aftermath of Morley's work on Categoricity and Power, and somehow the dominant example of theory categorical in uncountable powers was that of algebraically closed fields of a fixed characteristic. So somehow the setting, at least for affine algebraic geometry. It's a consequence of either Tarski's quantified elimination or Chevalier, the corresponding theorem that Chevalier proved, that constructible sets are constructible, is a consequence that the following is true. On the affine line, any set definable in terms of its basic language, with plus, minus, is as simple as you could have a definable set. Namely, it could actually be defined using equality alone. So, now this was called strongly minimal for the following reason. This is quite an important distinction. One then looked at a class, first of all, of strongly minimal theories, other theories that had the same property, that they're on their fine line, on the line of one dimension, with these trivial definitions, like algebraic geometry or something else. It can happen that a structure, an individual structure, is minimal in the sense that you have this minimality phenomenon on its line, but another structure won't have the equivalent to it. So strongly minimal means that you have this uniformity across models of minimality. There are examples known of structures which are minimal, but not strongly minimal. You've got the bigger one where things are different. Now, it turns out that algebraically closed fields are actually strongly minimal. If you take a definition, you get the same definition in terms of equality for all. So now, van den Dries was the first to detect the notion of all-minimality, at least as a general notion of logic,

5:00 although Tarski detected the phenomenon first in an interesting case, he just didn't, I think, see it as an interesting one. It was, I think, seen as a more interesting phenomenon by the group of people around Wojciech Tome and Hironaka. And they were working on a real one in the 30s and 60s. But Vanden Vees, you see at the time Vanden Vees was at the beginning of this, the dominant was to understand the structure of definitions which used the real exponential form. Classically one was trying to solve Tarski's problem by the decidability of this structure. But after a while, this was going to be the least of our concerns, one wanted to understand the nature of the definition. What was the topological structure, genetic structure? Now, the original case of Tarski for real pose fields, Tarski showed for real pose fields that they were strongly minimal and made no deductions from it. The class of sets he had identified, the quantifier-free definable sets, was of course taken up by real geometrists like Whitney and others in the 50s, and Whitney proved that the We've proved several things that these so-called semi-algebraic sets have finitely many connected components, their components are again semi-algebraic, and I think in terms of what Grotendieck says, the most important thing of all was that semi-algebraic sets in arbitrary dimension can be stratified, they can basically be broken up into disjoint unions of manifolds. And this is very important for people doing many connected things. A modern book on the topic is this book Stratified Morse Theory by Gretzky and Macpherson, which is done in the subanalytic category, which is the category that you're not characterized with. So, anyway, Van Den Roos was trying to understand what one might... I mean, if the theory of the reals with exponentiation looked like the theory of semi-algebraic sets, what kind of things might you expect?

7:30 One of them would be finiteness of connected components for... Now, the only other example at all, the only other setting or universe, I forget what Grothenby calls them, the tame spaces, the only other example of tame space or tame geometry known by the 70s was the so-called subanalytic setting. And it's important to see how different this is from what Tarski was doing. Of course, this is real geometry as opposed to complex geometry, although you can, to some extent, subsume complex geometry by looking at it as looking at n-dimensional complex geometry as two n-dimensional real geometry. It's a bit crude, but you can get something from it. They, of course, since you're dealing with analytic functions, your considerations are always now local rather than global. The set of a set being a set in n-space being a real analytic, a complex analytic, that basically means that you have to talk about points in the plane that are not necessarily in the set, so you have to quantify overall points in the space, you have analyticity if in every neighborhood of every point the set is described by... The vanishing of a finite set of analytic functions. These will vary as you move around. So you can talk about a set being analytic at a point even if the point's not in it. Then you can say a set is semi-analytic with a corresponding local definition. If the set is locally defined by, as well as equations, by order inequalities. Now, this would appear to be the natural analogue of Tarski's semi-algebraic. It was called semi-analytic. But it doesn't have anywhere near the nice properties of Tarski. It's not closed under projection, so it's not quantified. But what people did prove, well, what they did was the following. They realized that, and this is a phenomenon specific to the analytic case, By considering general projection or general quantification, you are just going too far. This isn't natural from the standpoint of this kind of geometry.

10:00 You should be considering proper quantification, that is, quantifying over bounded sets, over compact sets. So then you call a set subanalytic at a point if, essentially, and again in a neighborhood of this point, it can be described as the proper image of a semi-analytic set in some higher dimension. It means that it's the image of a compact set in a higher dimension. Essentially that's what it is, a slightly vague. And then, somehow in this situation, you no longer consider unrestricted quantification. Proper quantification is the only thing that you have. Then the miraculous theorem, proved by Gavriero in 1975, is that this class of sets is closed under negation, To say not there exists something in a complex set is equivalent to saying there is something in another complex set. This is the basic phenomenon that these guys discount. Now, you can make that into something more like Tarski's theorem, which you then do the following artifice. You say instead of working on arbitrary unbounded sets in these affine spaces, I work only on compacta. So I make my function, I somehow deliberately truncate each function to each compact which is defined, so I get, for each function I get a whole family of comps of it, traces of it, some of that, and now I simply start with some compact domain and my truncations of the functions, and I now go back to ordinary first order quantity, and the Gavrielov theorem then says that such a structure is all minimal. And so such an example did exist already. Van den Vries then tried to study this. To start, Gabriela was the subject, the big theorist of the subject, but the finer detail had been investigated principally by Wojciech Szewicz, who was a Pole and a friend of Tarski's, who had proved stratification theorems and things of this kind. Local finiteness of connective propulsion. To get from knowing something in dimension 1 to getting interesting information in higher dimensions.

12:30 Suppose you knew something banal in dimension 1, your definable sets are. What does it tell you about the shape of the sets in higher dimensions? Essentially, it's an absolutely standard game. Take, say, a two-dimensional set and, of course, you somehow use this uniformity. Eventually, there's some kind of cell decomposition. No matter how you rotate it. Yeah, that's right. And sometimes you may have to, to get things to work, you may have to do something in a slightly generic rotation. This idea goes back to Brown and Kuhn. There are various tricks of the trade there that these people use. I mean, generosity plays a certain role. Sometimes you need to use sarge, steer, and moon, basic things like that. But there was a technique, and Vannevich saw that this was a technique, looking for more examples. So he simply showed... Well, it's inaccurate to say he did everything. There were some issues with uniformity he didn't quite take care of, but these were mopped up. Then the problem was we didn't know any of these things. We knew now that if you prove something on the line, beautiful things follow that. Whitney stratification, all these many important, for example, the Weissel's inequalities are important because Hermann used them in PDEs, or estimates in PDEs. These things are quite significant. And then, of course, Milkey came along with the real exponential. Unrestricted reliance penetration revolutionized. That meant that all the sets you can define in reliance penetration have this nice structure. They are ten topological in working week sense. Moreover, some of these stresses, the sets can be, well, it's not that you've got a category so much. You do have a category, but you've really got just the definable relations, as opposed to something more like a... But at the end of the day, you can triangulate inside each of these all-minimal universes, internally, using maps which are already in there, you can triangulate. This is one of the things that Grotendieck wanted. He also predicted that there would be many components.

15:00 Now, basically, it's not just here finding some connected components. The kind of pathologies that he bemoans simply cannot happen. Measure zero, between measure and having interior. None of these things can happen inside this category and people subsequently found other even bigger ones and that goes on. In fact most of the classical functions have been shown to have at least large portions which live inside only, there are exceptions of course, an oscillating function cannot live inside, sine and cosine cannot. But aside from that, most of the functions, the Riemann zeta function, far out in the real world, the gamma function, there's a large variety of things for it. So I think in that sense at least, some of the requirements that go to the dimensions in here, but I certainly haven't answered your question about how to sell this category theory. I was not quite aware. of this what you call proper quantification. Yeah, it probably exclusively says it in the paper. Yeah, I believe it. I just didn't quite hook it up. I remember, well, Evandris's first paper. Yeah, yeah, sure. Because, yeah, what is it called? Bounded sub-analytic sets. Yeah, but it struck me right away that he wanted to make a category out of this. Yeah. You should not bring in that whole line at all as a generator, in other words, if you take a suitable category of compact objects only and look at pre-sheaths on that or general methods, certainly the line will occur in there, but you have, so that, if the... So, you know, sort of test objects where functions live is always compact. That would definitely simplify it as an issue of nothing else. That is a very, again, I might make a digression on that. I mean, the first time round seven you said the thing in sort of Tarski terms.

17:30 Well, I was going to just say, I realized that he talks about these Tarski systems. Yes, yes, yes, sure, sure. And I always thought this was really a very bad thing. Having gotten himself into the bullet, he wastes the whole paragraph explaining some stupid thing as if mathematicians didn't understand that the symmetric groups were generated by evolution. It's crazy. Anyway, but the point is that this Tarski system, that's a pejorative term in the sense we were talking about before. Precisely this insistence that there's one universe and you take its finite power, which is requiring me to talk about these qualifications that you simply took. Various universes, I'm sorry, which were, the basic ones of which were compact, had a category in which the compact objects were adequate. The same sort of thing that has been discovered to be necessary in topology itself, or doing algebraic topology and functional analysis and so on. You simply apply that basic idea here, then... And in fact he did later, you see there's a nice paper, I don't know whether you're familiar with it or not, a great improvement on that bulletin paper, it's by him and Chris Miller in the Duke Journal called, what is it called, it's got a kind of a bland title to it, I don't remember the title, it's in somewhere around 1997 in the Duke Journal. Geometric categories, or something like that. Not a very elegant term, but he does, they do, at that point they've advanced far enough to see that what he had previously been calling these finitely sub-analytic things, you may as well, in fact, what you're really dealing with is the real projective spaces and analytic, real analytic function. And that basically is what... So yes, he doesn't still take a very categorical point of view. I mean, the point you just made about this Tarskian thing, this Tarskian obligation that you start with a set, you've got to go to its Cartesian powers, as if this was the only way one operates in mathematics. For example, in projective algebraic geometry, you certainly don't operate that way. You don't take Cartesian powers. The distinction between projective and affine is precisely that, that you don't... So somehow or other you think, and this is still a moot issue in our discussions at the Newton and so on, and I keep telling you this, we can't have the right formalism here if we're always obliged to torture these projected spaces back into affines, you know, in some way or another.

20:00 But later on, this is one that's also connected with the Tame topology, but it's a complex version, you have to flourish. Zilber made a very nice observation. There is a relatively deep theorem of Riemert, which is in white, for complex analytics, which is different from what we know from reality, where Riemert is dealing with, again, the notion of a complex analytic subset of a complex manifold, for example, or maybe even more generally a complex space of some kind, and so these are again defined locally by equations. A proper image of an analytic set is analytic. And so some proper maps are the things that we really ought to be dealing with in this. And logic doesn't have the right means for, at least for saying this elegantly, paraphrase. At any rate, it was noticed, so he noticed this, and then he noticed, first of all, if you take an individual complex manifold, compact complex manifold, And you give it the language, you give it the structure generated by its analytic subsets, that this has quantified elimination, and it's not quite strongly minimal, but it has many three-dimensionality properties that were detected in Morley's work. But the format they now have for studying the model theory, they're saying basically you've got to sort for every compound constant. And they've also managed to get complex analysts to see that this formalism leads them to points of view that they never had before.

22:30 There's a guy, Campana, a French complex geometer, who has significant work on the structure of groups and so on, who uses this form. That's a little off to the side, but it is in some sense again a tame topology. You show that at least in the compact complex cases, there's nothing pathological you're going to be able to define. Because it's always different if you try to go back to the complex affine situation. That is definitely not tame geometry. Just another remark, I think, if you consider compact objects as generators of categories. More precisely, one difference with the classical approach is that the Groden-Dick topology that you're using is finite there, it's coherent. So in other words, the real line is not the union of an infinite number of intervals, it's only the finite coverings that matter. And that's going to force something to happen at infinity there too. See what I mean? I'm not clear with it. No, I know. I understand what you mean about coherent, not notarian. I thought a notarian topology. All covers have finite refinements. No coherence. Yeah, so, and then how do you get the line with that? Well, I mean, you don't get exactly the usual line. You get something that's the same points as the usual line, but which has a different sort of structure. I've never worked this out. It's all about, it's always all about, you know, when things are piecewise, they're a finite number of pieces, not an infinite number, as classically was commonplace. And that's, you know, as you look at it, that's all about sheath properties, you see, restricting to the pieces and extending again and so forth, but finite cover instead of arbitrary cover.

25:00 One of the dominant things, which is basically analytic things, break off analytic functions in many variables, maybe after a suitable generic change of variable, and you think of it as effectively they then behave like polynomials with analytic functions in them. He usually predicts that that would be used, and it has been used a great deal. Of course it's used in both areas. You may go and see that you can do this not merely in the real case, you can do it in the real case, there still is a tameness. If you have any ideas on this, this is something which we... See, the great thing, the remarkable thing about Wilkes, unrestricted exponential, is that there you don't need to restrict the problem. If you go to the complex case, it's wild, but if you go, if you work on the real exponential, it's tame. And I guess we always have the idea that there must somewhere or other be some kind of compact entity in which the action is really taken. Some related things, but because Wilkes remains the, well, it's not quite the only example, but it's the only example we really understand right down into the bottom. An analytic function, and in some way the arc tangent is different, because the arc tangent is a functional equation which involves the function one of them. So you can carry the arc tangent out of the unit circle to infinity in one swoop, you see, but the exponential, if you use the functional equation, you can, but you've got to do it infinitely often, principally, carry the exponential.

27:30 The exponential is the only, certainly the only, classical function where one has this tame topology, even allowing... It's not literature that's the only example of which one has this. It's the only example of which one actually knows everything. Yeah, but it is interesting to see how Vannebury's evolved a bit, I mean, to this very much more elegant formulation in terms of... He was driven by... I mean, that second paper was written in... It was almost written as a user's manual for Schmitt and Villeneuve who needed to use this material in the representation of the eclipses because they didn't really appreciate the classical model theory and they forced Van Der Beek to do everything on manifolds, whereas he wanted to do it in affine space before. That's okay. This sounds typically very, very, very elementary here. But when you say that the final sets have only a finite number of components, that's true and it's not true because you're working in a Boolean category, actually. So the notion of topological components is meaningless. So, in fact, you've got a relationship between two or maybe three categories, which probably all are inter-definable, but at least... Yeah, this is, of course, a good point. This is a real looseness of language when you say... Yes, absolutely. You're absolutely correct. I mean... I mean, the thing is, all of these theories, these are theories, not all minimal theories, they have different models, they have the... The real model, of course, we can just use the conventional notion of component and many different definitions, and the result is literally true. Of course, it does not immediately have a meaning, as you are saying, I think, for nor Archimedean nor De Fils-Royne because of topology. In fact, the reals are the only connected model of the theory.

30:00 But there would be examples, not all the real ones, when you have this thing. So the question is, what does connected component mean in a disconnected... In a Boolean, you're using Boolean logic, so every sub-object has a complement, and this is... I think this is not... You see, I imagine various ways of doing it. You can start off with an intuitionistic formulation, and then you can always pass to... So let's see what this really means. I mean, let me say something much more naive and classical, first of all, and then, I mean, suppose I have a definition in a general real-world field, say, a real-world field carrying an exponential satisfactor. Then we now have a meaning of connected set here. But it means definitively connected, that you cannot break it using first-order definitions. It turns out that that notion, in the case of reals, is equivalently connected, but it has a general meaning. What does it mean? Well, I mean, it turns out, for example, that you can show that… First-order definitions permit you to insert logical negation. Yes, yes, of course they do, but you… Exactly. But you have the topology and you define the notions of closed and open. The topology is defined. So you can define what it means to be closed or open and so on. So you work with it. In terms of, so it's really in terms of the generators of theory. It's in terms of the order topologies or something. Yeah, that's right. Yeah, that's a good point. And that's the kind of point that Poisson has quite often brought up. That we don't really get that, quite often in these logical situations, we don't really get the... The topology is definable there, once you've got the formulas up in front of you, but the, yeah, there is a difference. Well, I mean, just to contrast, when people normally talk about real algebraic geometry, this is actually some totally different thing, logically even. From what they mean when they talk about complex, because the complex, you know, the schemes and so forth, these are topological, whereas real algebraic geometry, which is, you know, shorthand for semi-real, blah, blah, blah, in other words, things are actually, you use Boolean definitions, Boolean definitions are regarded as meaningful, which again...

32:30 Now, actually, destroys the invariant meaning of quantum mechanics. So, in fact, what's really going on, in a way, is that you've got this particularly positive theory, with no negation at all, lying at the core and generating, of course, the moving theory. But it's in terms of that, limiting yourself to that, that things like... This is a good point, and I mean, I've wondered about this. Let me see if I understand it correctly. There's already such a phenomena at the level of straight algebraic geometry. I mean, for logicians, of course, the dominant theorem will be the Tarski point of elimination of the Chevalier theorem, saying that direct images of constructible sets are constructible, but in fact, the fundamental theorem is that in a projective situation, or the deeper theorem, in a projective situation, the images of equationally defined things are again equationally defined, the positive result. And one doesn't somehow naturally get the logic to deal with, as the logicians immediately rush to pull in the negations as well, to disrupt the vocabulary. And this was the point, Hazard has made this point, this has been a problem quite often in serious investigations in applied model theory. By this passage of going to the constructible sets, the defined sets, you actually lose somehow any fundamental thing of what the Zariski clone says, the things that are really important. Now, the beauty of Rembrandt's theorem again, Rembrandt's theorem in the complex case, is the exact analog of this thing in the projective algebraic case. Take something defined by equations locally, take a proper map, what you get is again defined by equations. You don't need any equations. Zilder went on to throw in equations and to make it into a logic theorem, definitely with some loss of beauty. I mean, with some, there's a point to it, it gives you a dimension theory, etc., but it loses this point. Now, the real case is different. I mean, I don't know what the... and I've worried about this once or twice, I don't know what the... There are all sorts of real closed fields inside the complexes of co-dimension 2.

35:00 These are all very different. I mean, there's a maximum possible number of them. What's that again? This is something which I don't think everybody knows, and it cannot really be stressed strongly enough, but suppose you're just given the complex field. I mean, the complex numbers, after all, don't, you know, if one is working in some sense, they don't require the excellent choice to construct. They're just, these are parallels which are there, let us say. Fine. Now, we know that one way to get the complex is we're going to construct the reals and then join the square root of y minus 1, make the plane, or whatever you want to do. But I don't think it's generally known, of course, and now this is a Zorn's law or an excellent choice thing, that... So, inside the complex numbers, there are huge numbers of very different looking fields, k, which are real closed, There's lots of them with non-Archimedean orders and stuff like that. There's a huge number of them. I mean, it's more than just the fact that the reals can't be defined in the complex's algebraic package. There are more than that. There's a huge variety of such fields inside there. I mean, Shannon will get upon this question, I think, and of course now that it's over and I'm thinking about it too. If you go from the complex, the geometric solution to the complex... All of these terms range directly plus the exponential. You get a wild theory, of course, because you've got zeros of the exponential function, so you basically get the integers, which are two-by-items, the integers. One doesn't know if you can define the real ones. You can define the integers, so you've got the Gödel phenomenon. But you don't seem to have what would be a much more troublesome thing from the standpoint of a geometer than the real ones. And it's related to a question that Shaniol asked himself. If you take an automorphism of an exponential field, complexes with an exponential, it could be the identity and it could be complex conjugation.

37:30 Could it be anything else? I mean, there's a huge number of automorphisms of the complexes, but it's moving around transcendence. Now, I guess Shaniol knows it. There's quite a lot of strange numbers that cannot be moved. The reals must be more than the reals. It's unknown. So it could be that complex analytic geometry with exponential is a very different subject. But yeah, your question about the components, of course, is important. This caused Wokey a great deal of trouble in his original work, because he needed to try to prove, to work with general models of the theory, and he needed to give a meaning to connected combat. At that point, one didn't know that this, defining the connected world, embedded a non-standard model of everything, a non-standard kind of tool. It's actually the mathematical ideas in it are related to Morse theory, but he didn't know it at the time, to keep trying to be critical. Yeah, I, this point of views I think is a very important one. We don't, we are somehow ignoring the positive aspects of this. Yeah, there's a positive here. I mean, it's this fact, I mean, this is glaringly staring us in the face, the topos theory. Yes, yes, yes, that's true. It's actually the kind of theories that can be classified by topos. So it goes beyond the equational, but far short of the first order. And of course anything with classical first order can be expressed in that primitives to the negatives, less theoretically the negatives. So it seems probably in all these cases that there's a positive, I call it, there's four different names,

40:00 I call it positive, some people call it coherent, other people call it geometric, and I've seen that in REM they call it dynamic. Dynamic logic, again, Marie-Françoise, Coswell, for some reason. There are four different names for this logic, partly because it plays a central role in lots of different considerations and generates first-order logic, but not, of course, with the topos and the geometric morphism in varying ways. I guess one has to be in a complex situation to do it, and it's difficult then. I mean, we simply don't know any. Non-compact complex situations, apart from pure algebraic geology, where we have a... You see, with unrestricted... Of course, if one wants to say, well, look, I'm just going to work in the projective spaces or complex spaces, well, Rembrandt's theorem says that. Rembrandt's theorem says that you have this kind of finite, this phenomenon. You see, I mean, if you're dealing with projective spaces in a Boolean category, in a Boolean category, well, is there really any difference between a projective space? No, no, no, right, exactly, I agree with you. You're allowed to chop it. Entirely, entirely. You're allowed to chop it in pieces. I agree entirely. These pieces will no longer have this, you know, this remarkable property of projective spaces. There are no non-constant functions. Yeah, exactly. The meaning of being projected as opposed to something else seems to dissolve, and so it's just that there's no doubt, I mean, I think one methodologically, excuse me, one defines things by this positive, by a suitable positive theory, and of course you use any method you can, you say anything you can in order to go further, but in particular, that this, there are really three categories. You get one where... Where you have genuine analytic functions.

42:30 You get another one, the Boolean one, where you sort of totally explode that, where things come in jerks. But then there's actually an intermediate one as well, namely continuous piecewise analytic. This is the classical. This is the classical thing too. Yes, yes, that's true. It certainly plays a role in the intuition of that. In Grotendieck's consideration, when Grotendieck says, well, probably the answer to my problem has something to do with piecewise analytic, he does not mean pieces that don't match. No, no, no, exactly, yeah, sure. So there's a very important intermediate category between these. This is, I mean, this in general brings up a problem I have with some of the Ominomality literature. I mean, even Vandam's book, which is taught, I mean, they give... Well, I guess the idea may well have come from Shannon again and from a student, Szybonski. They define what should probably be called the all-minimal Euler characteristic. So, for example, you take a definable set in an affinite space, and you say, it has a cell decomposition. You make a number there, and you stratify it in some way, and you then just do, essentially, accounting. ...assigned counting of the number of cells of dimension 1 minus the number of dimension 2 plus the number of dimension 3, etc., and you get something which is an invariant of the defined set. But they stress the fact that it's an invariant for...Van der Gries stresses the fact that it's actually an invariant for definable general functions, where you're supposed to be doing still geometry or topology. What happens if you're... I mean, what are the various meaning for continuity? That's an extremely poor exposition. First of all, he got the idea from Shannonville, but then he totally negated the content of it. I thought so. I mean, I was shocked. You take the bird-side rig of a category and force it to be a two-rig. Yeah, yeah, yeah. Then you can calculate, this is the question, this is one of those situations where he's giving the answer without telling you what the question is.

45:00 I was quite astounded when I looked at it. The question is to calculate, actually calculate the Burnside rig of this element, potentially with two, the Burnside rig of this calculator. And it turns out that one way of calculating is, like you said, you get cells, and quantum mechanics, and blah, blah, blah, which is fine, but they get conceptual content. And the thing that Shangri-La illustrated already in his first paper was that, in fact, the conceptual thing in a slightly different category leads to a different result. There is no a priori reason why dimensions... Such slides should be ordinary editors. They can be somewhat more interesting. Exactly. I was very surprised this morning reading it out in the garden. I mean, the book has been out of my possession for years. A student borrowed it and returned it today. And I was quite surprised at how, one might almost say, how devious Van Der Vries is at this point. I mean, there's no explanation of why you want to do this. Why you would stress... All definable maths, when you're really trying to do something logical, you know, and so on. Well, I mean, it is more invariant. Yeah, it is more invariant. The alternating sum of the ranks of a lot of groups is more invariant than the individual ranks, right? Yes, yes. This is true. It's also true that the notion doesn't quite agree with the classical notion that nothing stands above this either and the Witten. It's a bit odd. Of course, this then precipitated a great movement in logic and passing to other characteristics and general theories without giving... I've never mentioned the merge side of it. Yeah, certainly not. And I'm sure in many cases really arriving at results which are literally correct but must be... Thank you for your attention. It doesn't collapse. I mean, to me that is the deeper reason. That's somehow behind some parts of motivic integration and so on. Yeah, I think let's look more closely at a lot of... All sorts of quantities. I think that the lesson was clearly made by K-theory. If you ask, you know, what is the dimension of the vector space? Okay, that turns out to be a natural number. But if you ask, what is the rank of the projective module? Well, it turns out you first have to calculate K0. That gives you the entity in which the values...

47:30 The particular values are going to lie. It might turn out to be the natural numbers, it might turn out to be something else, which means that there are these two levels to the measuring process. This is a strange thing in logic. I mean, the only dimensions we have are those coming from the set genetic imperative, more or less. They are ordinal. They don't seem normally the right kind of thing. Whereas they ought to be somehow entities in some kind of formal structure. It's not even an iterative notion, you see. You might calculate it as something in the quidditch system. It's conceptual, but continuing with, not just van der Dijk's, I mean... Szymanski, you see, now this was, I kept telling Steve, it was kind of a delicate situation. He came to Steve as an advanced graduate student. He had these brilliant ideas in a certain direction, all about basically group objects, and the amazing groups of cardinality zero, which actually has a content. It wasn't fair. But to see the relationship was... I think that this guy was just totally locked into the logical terminology and Steve would tell him these things, would even go to the trouble of translating what he actually thought. The idea was he wasn't going to disturb this guy. This guy was really doing good stuff, so to call him aside and teach him a little elementary category theory might have deviated him. And therefore, Steve didn't do it. But that was bad, you see, because in the end, the student didn't learn any category theory whatsoever, and he developed an aversion to doing so, and so his results are, his results may be known to you guys, but they're not known to the category theory. No, exactly. I personally have remained very detached from that particular movement. There's a lot still going on, but they're doing it in a... There are some tremendously mechanical ways. You even find that they're beavering away all this kind of stuff. They don't even seem to understand basic things like Poincarean duality and so on, which would simplify a great deal of what they're about.

50:00 I was shocked at a meeting in Banff last year. I mean, they've pushed things quite far, and they do take significant notions and try to translate them into logic, but it's never done... Well, I've almost given up hoping. No, so it's all very literal and can be interpreted in a way that doesn't feel right. Go back to something you mentioned that might then feed back into this kind of thing. You mentioned a comparison of Chevalier's theorem on direct images of constructible sets to direct images of closed, projected subsets. Are these theorems equivalent or just the motivation? Well, this is a good... no, they're... It's not difficult to get the one for constructable sets once you know the one for projective sets. I don't think it's straightforward to do it the other way. I mean, you see, of course the projective set was known earlier, much earlier than the shared weighting. I mean, many people would have known this in the 19th century. But they knew it. Basically, to apply this formal machine to say, once you knew the theorem, it's been closed, then there was a game for getting a kind of a quantifier in a general situation and getting some nice dimension machine. The same, exactly the same, will work in other cases. It's an argument which is spelled out in Matsumura's book on community balance, but I noticed the number of bioscripts were not tremendous.

52:30 You need something on netheorianity, but you've got it in both cases as well. You've got netheorianity in these complex analytic cases because of the virus-stress preparation theorem, basically, that the conversion power series formed with the germs from the theory of the ring, etc. So basically, that's the point. There's a netheorianity thing hidden in it, and then there's images of closed sets of clothes. Then there's a general theorem. Then it will give you something called machines. But... Your question is a good one. I mean, I now remember when I first went to Yale, somebody sent me a paper to referee about the logic and projective space, and I must have rejected it, but I've often, I don't have the paper anymore, I didn't keep it properly, and I can't say it there. This guy really was doing the right thing, and we were doing the wrong thing. I mean, because obviously the fact that the direct image of a close-up is good. I mean, you don't require to introduce negations to deal with the... It's a theorem. By the way, if I just might mention that Juliet Elementary was the result, but... You know, in Bourbaki, any book on general topology that talks about proper maps, there is the, you know, the obvious condition that the inverse energy of a compact has a compact, and also the condition that the map is closed. Yeah, yeah. Now, the point is that the second condition follows from the first, actually, provided you work in one of these corrected categories, like Horowitz's k-spaces. Horowitz defined in lectures in Princeton, you know, they supported the notion of k-space, which was last in Kelly's book. Variants are used by Steenrod, Brown, Spanier, and all sorts of people. Simply because of the basic need for function spaces for arbitrary pair of objects. I think there are function spaces that have the basic lambda transformation. Well, it was Weiberg who pointed it out, I think, a well-known topologist. Rather late in this whole discussion, he had a short paper, I forget where. It's just the theorem that in the category of k-spaces... If the inverse image of compact is compact, then the map is closed, which is almost a tautology if you remember that the case spaces are characterized by the fact that a set is closed if and only if it's closed in an intersection with compact, or if you take a map from a compact space and the inverse image will be closed.

55:00 That's the condition of what a closed set is. The very definition of what a closed set is, in other words, is determined by the fact that you have this site for the big category consisting of compact numbers. It's almost a tautology then. It's a condition. Closed is automatically true. I think your question is a very good one, because another way of saying it is that there is no easy way back, perhaps no way back at all from the constructable category to the Zariski closed things. I don't see how you could deduce the projective version, gradually at least, from the affinite version. It can be done, but I certainly don't see any formal pattern whereby you can do it. Perhaps, maybe I just need to think about this myself, but you do see this as the projective and affine version of the same thing. Yes and no. It's difficult to say. I really think one should be in a situation as the algebraic journals are, where proper maps are somehow central stage a great deal of the time. Projective algebraic varieties, not single ones, as nice as because the maps are proper, I mean, you can do, there are theorems, of course, you then have for affine, for non... There are a lot of non-projective things, non-compact things. Of course, you can get cohomology theories, but usually you have to pass via a compactification or something like that, and you change the sheaf, etc., etc. It becomes infinitely more complicated, and you then only get theorems under strong assumptions that are proper maps. They have, in the cohomology theory, similar things. I mean, they're direct image under proper maps of various nice sheaf things that are nice. There are analogs of the Chevrolet thing and so on. These are two very important theorems. Yeah, exactly. I mean, this all seems to me unquestionably true.

57:30 Well, it isn't necessarily a projected theorem, but the theorem about probable energies of something nice being nice, these are always deeper. The logicians housed in the Newton are compelled to go to the negations, and it's not always clear what is gain. What they're doing can be done without their so doing. And yet I don't know, you see, I don't know how much of the dimension theory and some of the Morley created can be done in that. I just don't know. And it isn't that you don't use negation, it's just that you recognize, you see, that they live in the topos even if they don't live in the site. In fact, I was going to think about the evil themselves. There is no such object in the site if we have no minimal site, but we could always create a group of topos on that which will inevitably have such an object, which is even inevitably related to the first, but it's kept a bit outside, it's seen to be, I like to think it's seen to be a higher A higher level of idealization, and in the same way, navigation. It doesn't mean you shouldn't do it, you should just realize that it's another step. This is just a side remark, but it might be useful. It certainly relates to one of the more difficult theories on the subject. I think John and I had noticed that somehow the dominant thing that made it possible to get a minimality for the exponential function was its differential equation, the so-called Fafian equation. I mean, some of the Fafian chain of functions, you essentially start with some basic stock of functions, typically polynomials, and then you start closing under just the following operation.

1:00:00 You take, it's better to say it in terms of differential forms, but essentially it comes down to you take that system of partial differential equations. In N unknowns, basically what you do, of the simple formula, you take your new function f and you express all its partials as polynomials in itself, an earlier created function. And this is the factory operation for creating more functions. And Havansky noticed that functions created in this way have good finite property in real situations, that they have only finitely many functions. In fact, he proved that the zero sets have only 500 mini-connected components, distinctly non-trivial methods, including Morse theory and so on. Now, one expected that if you took all the global Fafian functions on the reals, that this would be an O-minimal theory. But it does not respond to the usual methods from subalimentic theory or Wilkie's earlier work. And Wilkie eventually proved it to be O-minimal by... The following method, I'll just state it in an outline, but it's interesting because it gives negation a very special role. You start with the zero sets of neurons. You can also do positivity sets. Start with the zero sets of neurons. And now you allow yourself the following operations. You allow yourself direct image, and you also allow yourself closure. The logical closure of things you've got already. But you don't allow negation. And you start building up a hierarchy of sets this way. Well, there are some assumptions. You don't have to start with analytic functions. You'll certainly see infinity functions, but less even. At any rate, you just start generating sets in this way. So you're not using the conventional apparatus of first-order logic. You're using some subpart of it where you don't have negation at all, and the quantifiers, one is the direct image, which is a natural meaning, and the other is somehow the bunch of quantifiers that gives you closure. So he built up sets. Actually, the method had been attempted before by a Frenchman who got it wrong, Michael Charbonnel.

1:02:30 Look, he built up the sets this way, and he showed the following. If the original zero set she started with had the uniformity of the number of connected components, and she varied the families, then the same is true all the way up. And, moreover, that you will get finiteness of connected components. For all the sets in the Charbonnel hierarchy. Moreover, you will get cell decompositions and in the end you will then be able to prove that the class is closed down because you take a set, make it cell decomposition, but you can show you can get cell decompositions compatible with other sets and so on. You just get the negation out of complements of the cells. I was reading it and I noticed that I had done something. 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 and on and I like this very much, you know, that you just deviate a bit from versed ontology at the beginning and you get negation at the end, but this remains isolated in our culture for the moment. In real life, you have to really work to get to those negations, whereas this way of formalizing logic just makes it a cheaper thing. That's right. That's it. Somehow, concretely, you can sometimes... 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 and on and on and on and on and on and on and on and on and

1:05:00 In the construction of the constructable sets, he took much more delicate operations than Gödel did, to generate them much more slowly, to get much more information at the end. I mean, that's how he got the Sousslin tree and all the rest of it. You could never have seen the Sousslin tree by Gödel's way of generating the constructable set. This is similar. You just do it very slowly and you use the operations which are geometrically and topologically natural. But, you know, you have to use. You use Sar's theorem, generosity arguments, things that the topologists use on a regular basis. But that remains isolated. No one has pushed that. It's also been quite difficult to really extract. Yes, well, this is quite refreshing to me because I certainly have wanted to see the logic being done. You're not even talking about decidability anymore. Well, yeah, I mean the thing is... To what extent is decidability equivalent to a clear line of implication in one way by Goethe? Yes, this is the point. I mean, of course, typically what has happened, there have been some developments, I don't know what the philosophical significance is, but we'll mention them because they also connect the group. So, for example, the case of the real exponential. Once Wilkie had proved that it was minimal, one at least knew that there was no question of a kind of gross undecidability result of classical type by showing that arithmetic was centripetal there, because arithmetic couldn't be definable and it couldn't even be interpretable because of various ways of eliminating imaginaries. You have definable choice functions, basically. If someone gives you a A set defined using parameters, a definable function of parameters that pick something out. It's basically the very standard. Because it's the standard. So, one at least knew that the theory may still be undecidable, but it's not going to be Godelian undecidable.

1:07:30 For example, there are plenty of other, there are other examples in nature, undecidable but don't have an arithmetic. For example, they might interpret the word problem for group. There are some theories of modules which are known not to, it's known that no theory of a module interprets them, but there are many theories of modules of a specific means which are undecidable because they code the word problem for a general group or something. So that remains, had remained a possibility for the real exponential, and then Wilke and I suspected this from the ones I had understood with our heads together and we showed decidability assuming Shannon was correct. And the proof is quite constructive down to Shannon's country. What is interesting is that, of course, it's not the kind of decidability that would be useful to modern day computer scientists. The bans are certainly out. Just recently I came upon something which I think is potentially more interesting. This relates to the complex situation. Suppose you take, okay, so again the model theory, so this is another defect of the way that the current group of learning is that, as I said, they have this idea, they work with compact, complex manifold, but they typically don't, but they work only usually with predicates for their analytic subsets. They don't work with functions in these things, and they don't work, certainly don't work with the chiefs. I don't know why. Now, and this is really a glaring... There are many things to do because, for example, if you take a torus, it's got elliptic functions living on it, and more effectively, you know, something. Otherwise, there are more than functions in matter.

1:10:00 Now, the approach to compact complex mathematics tells you nothing about it, because you can't just talk about the function, you can't talk about the function if you only know about zeros, because there are so many zeros anyway. So, I took another look at this, and I reinterpreted, of course, I went through a real interpretation of it. I did various tricks from minimality and blah blah blah and eventually have been able to show, not for all the viastras elliptics that I'm expecting, I've been able to show for some viastras elliptic fractions, decidability, but only assuming a conjecture of brokenness, which in many cases is known to break the round of homology. I mean, somehow this is an issue still in homonymality too, because although you want to try to do the cohomology for the sets you've got there, you may need to do intervals to understand it, and these may take you out of your, so we don't, this is not probably understood yet. At any rate, there are all these conjectures in these, in the cohomology situations that, first of all, say a complex variety, very often will have many cohomology theories attached. But they usually turn out to be isomorphic. I mean, they may be defined in the singular homology, there may be DRAM, there may be the italic homologies, and there are others, crystalline cohomologies. Typically, one ends up showing that the groups you get are isomorphic in some way or another. And also, okay, that's one thing, one has known that for a long time, and also lots of conjectural things about this independence of L from the Hellenic theory, etc. But Rodney made a conjecture of a different kind. He looked at the comparison there between the algebraic theorem cohomology and the singular cohomology. So both have different ways of finite dimension or finitely generated entities. So there's a matrix involved in that.

1:12:30 And the matrix, of course, is not part of the basis, but it turns out that the field generated by the entries of the matrix is an invariant. He made the conjecture that somehow, although these two cohomology theories are the same, that the map between them is somehow, it's almost independent, it's kind of rigid in some sense, there's no hidden relations between the elements of this field that are creating the matrix of comparison. Now, he made it for number fields, so it's been generalised by Yves Andre and so on, so it's a vast conjecture now about one motives. Which they have. It's worked on with the French Transcendental Mathematics because one will not prove any examples of it. It implies Shannon's. But another version of it implies this thing about the decidability of the vast tracelyptic functions and many other things as well. Of course, it's one of these kind of systems in the same way as Shannon's is. One can hardly imagine. I can't really imagine, well I can imagine maybe somebody getting a proof that this would be fantastic to a diversion. I almost can't imagine what it would mean for a projection to be false. I don't know if we can discuss this, not so much false but as refuted. Do we know any case at all in the history of mathematics where numbers that were believed to be algebraically independent turned out not to be? Is there any possibility that I have a car key? I don't think you do, but I just want to know. You have a car key. No, Mimi has a car and we cannot find a car key anywhere. I'm really sorry to interrupt the discussion. Why did Bill have it? Well, no reason at all. I drove it the other day. So Mimi drove it and then... Mimi had a car key. Sorry to interrupt. No, no, no, that's a shame. We'll find something. Okay, sorry.

1:15:00 So I don't know how this fits into the schema. Of course, this is more a motivic thing, I suppose. He's looking to find so much behind all the terminology. So in this case, it's about the periods. Have you seen this paper by Konsevich and Zagier on periods? No. It's worth looking at. It's an awful lot, isn't it? This is almost all conjecture, but some fabulous formulas that got out of the pictures and I think Zaghi has verified some of them. I mean, the periods are basically the things that you get by integrating them. I mean, you just work in the semi-algebraic category or something like that, and you just do integrals, run semi-algebraic sets of definable functions. Of course, you create them immediately, your two pi i's and all the rest of it, but basically periods are defined as things you can get in this way. And of course, because you can deform much more general things into semi-algebraics, it turns out that this includes most of the classical numbers in mathematics. So they define the field of periods and make all sorts of conjectures about any identity between periods. There's almost a completeness theory. If there's an identity between periods at all, it can be derived using two principles, which we spell out. One of the weird things, the suggestion that neither one of them, pi nor e, definitely won't prove this, that you cannot get these numbers by integrating, say, algebraic functions around, or algebraic forms around the semi-algebraic states, but it connects then to deeper things too. But that's a definability. Absolutely, yeah, absolutely. I find it quite fascinating. It's the sort of thing I can't imagine any books can be made on time. The fact that people from that direction... The comparison between Dirham and Singular is for cohomology, but for homology, Dirham currents... Yes, yes, yes.

1:17:30 These are functionals. That's the general type of question I wanted to direct at. To what extent do we maintain anything like the... To what extent do we maintain anything like the... There are many good properties of old minerality when we pass to topology where we have actual exponentiation. Well, sometimes everything collapses because we have the truth value object. But what about just taking higher types? Let's take the Bernoulli, Hurwitz, Blabla principle. There is a unique notion of... The structure of an exponential object, namely the one that makes all paths smooth, there's a naturality in the pre-sheet of those. So if we just look at a few of those objects, is it possible that at the extreme, could it be that they don't have infinite components either? I would say there's sort of a naive reputation of the idea because you would think, well let's take the idea of polynomials and they would all, in a purely affine situation, we look at all maps from the line to itself, well that's a certain space, but after all polynomials have degree, so there's a map to the natural numbers we haven't seen. But is that in fact, is that in fact, you know, a morphism? You have to check that it really is, you know... So you're thinking of having... Spaces of functions... Spaces of functions... Spaces of distributions... Yes, yes, yes, yes, yes, yes, I understand. Currents and so forth. I mean... In a way, in a way that's closely tied to the basic global animal geometry of fundamental physics. Right, yes, yes, yeah, I mean... I mean, you can clearly do it, you see. You can clearly construct tempos, like strong, deep tempos, like that. Yeah, yeah, yeah, no, but I mean... What sort of specific properties do they have? What sort of good properties dare we expect? And so on. I guess you're mentioning a comparison theory that I reminded you of.

1:20:00 Right, yeah, yeah, no, I mean, this is something else. One of many, many examples. What does, what can tameness mean in some species of functions, are you saying this? Functional. Functional, yeah, yeah, sure. This idea of universal naturality gives you an idea of the structure of a function space, hence the idea of a natural world from which they back to the fundamental space. A special case of this, which of course would definitely come up in any attempt to do any systematic guarantee. Yeah, I mean you integrate functionalists depending on parameters against, and really we don't know the answer to this. Thank you very much for your time, and I look forward to hearing from you in the future. The parameterized period thing, if you like, I don't stick around closed curves, I just integrate function depending on parameters over sets. Now the question is, okay, to what extent can I create new functions, to what extent can I add these functions and preserve them? For sure the functions are not going to be in the original theory. So, on the other hand, we now know that the logarithm, combined with the semi-algebraic functions, is perfectly tame. And what is the actual domain of the logarithm? Well, that's another question. This is another question, you see. And this kind of question is beginning to come up. I mean, the Hovansky results sort of said, you know, a special case of Hovansky is integrating a function, but really between definite limits.

1:22:30 So, the Habansky method would give you a function of x by integrating, say, between 0 and x, a function f of t that you already had. That's a special case, that's a Fafnian operation, a fundamental theorem, and that is known to preserve momentum and relativity. It's the only case, I think, where we know for sure that it does, except for the following important case. Now this is the issue of which logarithm. Avansky's results and Wilkie's example typically say that you could really add all the quantities of a logarithm. But they've recently come up with other examples for other differential equations where you can add one branch or another, but you don't add them both. For the Euler equation, for instance, this is an interesting thing. What does Euler equation mean? Well, it's this thing where the single is like, what is it, like x squared dy by dx is... There's a big singularity. And they know a lot about this equation, and there's a whole family of them. The solutions are different from the point of view of all minimality, and there are two which are incompatible. I mean, they don't understand this. They had known the phenomenon before, but for somehow generic cases. For classically studied equations, this is only a very good example. That will have to be addressed, because until now, in the Fafir cases, they... You can just add all the solutions. You can add all the leaves and the foliations, but now we'll have to be more discriminatory. But what they do know is the following. You take the subanalytic, so that's the one that here in Africa, or take the projective subanalytic. And now you've got, say, you may have a subanalytic family of subanalytic functions, and you want to... Integrate those functions over a family, if you like, of subanalytic sets, and you'll see you get functions of the two sets of parameters, the parameters of the function and the parameters of the sets.

1:25:00 What can you say about these functions? Well, they needn't exist in the subanalytic category, but two guys in Dijon, they proved that these functions exist in this thing that Van Der Wies and I and Marker studied, namely The Tame Universe got by adding the real global exponential to it. So you can integrate out of the subanalytic category, but you don't have to go beyond the subanalytic plus exponential category. But there it rests at the moment. We don't know whether you can... And of course one will have to understand these things to have a decent knowledge. People are looking at it, but there may be some natural class inside here, some kinds of intervals. By integrating from 0 to x, say, another function of one variable. That is okay by Khovansky and then by things that Spicek and I have. You can't even take a function of two variables, say f of t and u, and create another function of two variables g of x and u by integrating from 0 to x a function of the two variables against t. We don't know whether that's a minimum or not, the potential or not. So that's a big gap. But generally, about decidability, well, I mean, of course, I suppose everybody's perception of these things is now a bit changed by computational complexity, because even if one proves decidability, it's rarely, it's rarely a useful bound. But it's still, I still think it typically, very often throws out interesting things if you really try to go all the way to show that there isn't any kind of Godelian form whatsoever. And one is rarely really passing some very deep.

1:27:30 Conjectures not yet formally of schanual type about Grottenegger some remark in here about not really ten spaces if I understood him correctly it was ten topology but I'm not sure what he had in mind it's well we can look at it later it's it's what's the end of his discussion uh this machine is acting temperamentally it's obviously no it's there but it's someone that yeah seems to come back about this one I took off your Grottenegger circle thing um I mean I had it before in French but I've lost it or something But it's also interesting getting notions of tubular, neighborhood, etc., etc., and also this idea of somehow working with what he calls isotopic categories. As far as I know, we haven't really gone that far in all of the... I mean, I don't even quite know what to make of this... Do you want me to look for Tane Topos in here? Yeah, yeah, yeah, sure. I mean, I don't know if he uses the expression Tane Topos, but he makes a distinction at some point, I think, between... Well, we make a huge difference when we're talking about so-called growth and so-called fatigue of spaces or some glorified version of the sheaves are in one space. You could imagine either one in this context. Yeah, sure, absolutely. Of course, unfortunately, in the O Minimal thing, the sheaves haven't really made any appearance. And this is a big defect also. In some sense, local to global, which come up there that we don't fully understand, I mean, we found, some of us have found these problems coming up in connection with the liberty functions, but also even in terms of understanding in what sense the theory of the unrestricted exponential is a limit of the theories of the restricted exponentials. These are theories of different kinds. I mean, the unrestricted is an example of that.

1:30:00 What they call a non-polynomially bounded data. Functions can be very rapid growth, because in a non-polynomially bounded, functions behave... When you say restricted exponentially, you mean just the domain? Yeah, on a compact. So I decided I'm going to just cut off my... first of all I'll do it at minus 1, 1, then minus 2, 2, etc. Just on a compact. Right. And so each of these theories have been known to be decent a bit earlier, also by Wilkie and others, but I haven't traced it on the method. It's not the case that, in any natural sense of topology or theories, that the theory of infinity is the limit of these, but for the important things it is, if you want. It's not a sheaf there. Yeah, exactly. Exactly. And I'd like to get students on that. That's the major defect in the culture of our students. I mean, they know an awful lot now. Yeah, I mean, there again it would be. Yes, exactly. This is coming up. There are a few of us who are looking at these issues. There are some guys coming also, who are interested in logical questions about, or modern theory questions about cheese. There's all sorts of things about alpha products, schemes, what kind of energies. You derive categories, how do they behave with alpha products. You have to work in a systematically sorted way, and you certainly have to work that way. You have to work with the sorting of the kinds of coverings you're using, et cetera, et cetera. I mean, the formalism alone is pretty ugly. I don't find a way out of it. Not finding tamed topos. No, I don't think that is my expression of it, but there's some distinction. If I remember towards the end of it, towards the end of his discussion of this, there is, I mean, topos. Can you locate the topos in there, I think? Ah, yeah. Oh, it probably was. I can't remember. Is it in French?

1:32:30 No, it's the English version. The translator, I see. Yeah, it was on, I got it off the circle. Oh, and that's not a word that's used a lot in this. Yeah, it's not stratified topos, for example. So he wants to, I mean, he stresses that he needs, he wants to get species stratified, like a nice stratification. So what is the notion of stratified topos? I wonder. This is basically the only context in which he mentions topos is this page. Yeah, right. Ah yes, true there is foundation, well he's talking about, of course, I guess in these stratifications you have to be able to move to a tubular neighborhood to do various constructions. So he says, the description of the daily science of a stratified topos is even considerably simpler in that framework than an attained topological one. True there is foundational work to be done here too, especially around the very notion of the tubular neighborhood of the topos. It's actually surprising that this work, as far as I know, has still never been done. I guess he refers back to the work of Artin Mazur, that early paper they had, and since the context of Ital topology had been more than 20 years, so they were dealing with homotopy too, because no one apparently ever felt the need for it. Surely a sign that the understanding of the topological structure of schemes has not been much progress since the work of Martin Mehta. I guess that would be superseded now by the work of Wojwoski and so on, but I guess maybe that's all there was. It was a notion of stratified topic. Once I had accomplished this more or less heuristic double work of refining the notion of daily science of a stratified space or topos, It actually appears that as far as these are concerned, one can actually take a shortcut for at least a large part of the theory via direct geometric arguments. Well, there's more, but I just wondered if it meant anything. It's somewhat in connection with moduli spaces and multiplicities, but I haven't understood it. Well, the motivation there is that moduli spaces, often all you need to know is how far the moduli set has collapsed at that point. Yeah. Multiplicities of roots.

1:35:00 Yes, yes, yes, to know about the root structures, to know about the possibilities of each one. Yeah. At this point, it was happening very much at the time of the cell decomposition, in the analytic situation, where the virus has preparation and the number of roots will suddenly change at some point. And that is usually a manifestation of non-problem. Draw, of course. We should be seeing that we have the Tarski quantifier elimination in a smaller number of variables, but with analytic coefficients or something. That's essentially how they get many of the others. The number of variables. We had this program many years ago, and there were a few non-trivial papers written about it in that society. Ah, was this Christiana Rousseau? Yeah, I've never met her, but I looked at these early papers. She became, like many, totally hostile to category theory. Really? She took up, you know, the study of differential equations and the plainness of it. A fine subject, but its hostility is hard to... I thought of this, this program is still not being carried out at all. I thought of it in terms of the Grauert direct image theory. This would depend on the real development of logic, internal logic, that one could give definitions of compact space and complex numbers and all of this in a good way inside a topos. So that, for example, the Grauert theory, which is about a proper map of analytic spaces, There are more general versions where neither of the spaces is an analytic space, it's just that the fibers are compact analytic spaces, which is the crucial thing, which in some vague sense lends support to this idea.

1:37:30 Namely, see there was the original theorem of Cartan-Serre, which was just about a compact space, and it said the cohomology of a coherent sheet is finite dimensional. Use Grotendieck's general idea of passing to a slice category in an even more serious way by using quite some of the higher order logic that's going along. So anyway, the slogan would be that if you prove the Cartan-Serre theorem... If you deduce topos as arbitrary topos, then you can deduce the Gower theory as a trivial corollary by applying it to the topos of sheaves on this base case. And some of the ingredients of this can be verified. You can give a definition of compact, so that if you externalize it, it does mean proper math, for example. So things like that. That's very effective. I would really like to see somebody... I discussed this thing with Grotendieck. It was interesting. I discussed it with Grotendieck and also with Husserl. Oh yes, yes, yes. Fun people said he had written the best paper. He wrote one of the big papers. No, a big paper, but he wrote a... Did he write a paper on the trace formula, was it? I used it anyway, it was something I was doing not too long and I found it very informative. I think it was around the, is it NSG 5? I can't remember. It's around the trace formula. I'm going to profess not to know who Zell is. It may have been just a momentary laugh or simply my ineptness at pronouncing the name. I don't know. Anyway, I don't think it's an SG. No, I would be surprised. I've seen Huzel in, there's something called the Seminar Banna, you know, functional analysts. But if it was five, I don't know five well, it's possible. Well, five, I mean, I had to put, and there's still enough to be, and I cared a huge amount of time on this paper. Anyway, I went to this because, well, there's John Gray's history of sheet theory, where he mentions various expositions, more and more general and more and more clear of the Grauer theorem, and he says Huzel has the best one.

1:40:00 Therefore, I looked at that. Now, Huzel's treatment was preceded by some... There is a functional analytic component to all proofs of this theorem, because you have some sheaves and you construct a resolution, and then you look at the sections of these sheaves, but the sections are given a functional analytic structure, like a topological vector space. And then you show that the homology of this complex would have to be simultaneously compact and open, you know, and therefore it must be multidimensional. This is the other thing. So there are these theories, not compact and open, but bounded and open. Well, Doody observed that this is all much better if you think of these as more logical. So he used the bounded structures, which in many instances is more or less equivalent, but it sort of works better in terms of the functorial variance, and Huzel incorporated that, among other things, into his expositions. So there's this big exposition, and he even uses toposes, but in another... You know, in another way, I mean, just in terms of the sheaves on a given space, you know. I see, yeah, I didn't know that. But, I mean, you said the topos theory. The topos and derived categories, morphologicals, everything is in there. Ooh, I should look at it. Some of it, I don't think it's really necessary. But anyway, so I discuss it with both of them. The problem was that Grodenbeek says, well, I don't quite understand logic, you know, he didn't know those words, this was the point, you see. So somehow bridging this gap is really needed here. Nobody's in a position to do it. I'm in a position to preach about it, it should be done, but I'm not in a position to do it. Nobody else even does that, but I think it would be marvelous because it would really be a vindication of Grodenbeek's general program. On a higher level, because you'd be using the internal functional analysis, whereas the internal algebra is one thing, but the internal functional analysis is another thing, because it genuinely uses...

1:42:30 It's a striking thing that we have like, what is it, 35 years of the elementary topos theory now. We've always known that exponentiation is one of the two axes. Nobody uses this. Nobody actually looks at examples of the functionals and applies it. It's just straight around it. Well, people look at it a little bit to see whether you could put the hard inverse function theorems into SDG. And he did verify that these exponentials naturally have the Fouchet topology. Yes, there have been a few series favored on this, but it's something that most people say is just the SDG. I'm not sure that was produced in paper. Right. I lectured at Huzel's seminar and again he said, oh this looks very, very interesting. But he couldn't quite judge, you see, whether it was crazy or not. Subcultures. It's not that anybody found out. Right. Nobody. Right. I don't think it's this thing in the... In the end, once you have these ten universal categories, there should be somehow only one isotopy type. I suppose it means that the stratifications of the triangulation should somehow or other be the same in all these categories. But I don't quite know whether this is true or not. I mean... I mean, it's true that if you can triangulate, as you can in all your mathematical theorems, it's true that in some vague sense, any shape of the semi-algebraic category, I don't believe that this is what Bohm is after.

1:45:00 I'm just trying to figure out what he could be after. I mean, he's clearly much more interested in these tubular neighborhood arguments. I don't know if logicians have really looked at this. I mean, what in the end is the difference between... ...stresses that, say, if you're on the real side, such that a function, a definable function, near a singularity is, and that's not true in the cases where you have the global exponential, that's not the only obvious difference between any two minimal theories. I mean, this theorem of Miller's, as you know, is such a beautiful theorem. You must get here. Yeah, that if you're not pollutantly bounded and you're in a no-minimal theory... You have the exponential power. And almost every you tell it to says this can't be true, the first time. I remember Grover Macpherson, I told him once, it's true, and it's, it's, I, yeah, the thing is I feel, it's not that I feel. It's a waste of opportunity, I just wouldn't have seen it, but it was almost staring me in the face. Chris and I had discussions, very often, we spent six months together in Southern California, and I was keeping encouraging him to study a paper of Rosenlich, a beautiful, formal paper on Hardy fields, but it's completely formal, and I love the algebra, it's sort of the algebra of L'Hôpital's rule of infinity and stuff. And Chris didn't actually know this material at the time, and I kept it. Gems are definable functions of infinity. Really, that's about all he used to get this theorem in the end.

1:47:30 And he told me, I thought. Didn't exactly kick myself, I thought. God, you know, how stupid can you be, really? The theorem was just too super polynomial growth. There's another limitation that means anything mathematically at all, but we don't know any example, much around the reals, where there's a function of transexponential growth, a function that grows more rapidly than any iterated exponential. And it's known that that is not... In terms of the Hardy field stuff, that's to say the growth rates of functions of one variable, this is not ruled out, this is kind of Bozhanitsyn found examples of Hardy fields where there are functions of trans-exponential, but we don't know if they fit into O-minimality, of course there are no such functions around in little experiences, but somehow some limit of these things... You have got e to the x, which is the summation of x to the n over n factorial, you interpret over n factorial as the modulus of the symmetric group, and so that's the free commutative monoid. The concept of rig is simply what you get by applying the Beck distributed law to a composite, but in this case the composite is the thing with itself, so that's e to the e to the x, that's the free rig. That's the only time that E to the E ever occurred in my life, actually, and I noticed it was the three red. That's interesting, yeah, that certainly is interesting, because normally, if it occurs at all, it occurs in some sort of derogatory statement. I mean, towards the end of Siegel's book, Volume 1, I think, of his series on complex functions, it's a nice thing.

1:50:00 I hadn't been aware of it until I started working on the elliptic functions. Elliptic functions are much more beautiful objects than the exponential function. And he points out that you can get the exponential function from the elliptic functions by... You see, the high-stress elliptic functions depend on a... Essentially, a high-stress function is given by... Take the number one as one of the... scaling. And then the other one can be an arbitrary thing in the upper half. So, a vastness function depends really on a tau and then on the variable z. So it's really a function of two variables. And Siegel shows that, and it's recently been rediscovered by some model theorists, that if you let tau go to infinity, you can pick up the real exponential function. But he points out, so he somewhat takes this as, first of all, a justification of the fact that the elliptic functions are... You know, the elliptic functions have this nice algebraic generation, you know, you can get everything from the Bachelors function and it's derivative, you know, for a given lattice. There's no relation connecting E to the E. I don't think so. I mean, I don't think I could produce one offhand, I mean, I could do it artificially, but I forget how much Anitzen did it, it's the papers and the transactions, quite a long time ago. I don't know, it's the kind of thing that Lee Rubel would have been a specialist in, producing a natural example of one, but I can't think of one. They certainly don't occur in...

1:52:30 Yeah, I mean that's the first connected with the story of Rotendieck from 1950 to 1960, if I couldn't do mine. Calculating the dual space of the space of analytic functions on a bounded set. That turns out to be analytic functions on the complement of exponential growth. So there's a natural way that that limitation to one exponential only comes up just in the idea of taking the dual space. It's a funny way to get the dual space in some sense. You take the complement in some sense, in a way, in a specific sense, but it's just, it's sort of funny, it's like Alexander duology and homology, cohomology, homology on the inside of the sphere now is in terms of the cohomology on the complement. I was wondering if one could deduce the latter from the former, by sort of the wrong method. Yeah, I don't know. There was a lecture at the Newton last week, last week I was there, by Andrei Gavrielov, who has made, well, I mentioned his theorem with a compliment, and he's, he now has built a lot of logic theorems. But he, this lecture was much more charming. He, it was about Maxwell and the problem of Maxwell's, which is still not solved, a problem of a critical point in a certain charge system. In three spaces. And the number of kinetic components. And he claims, and Gabriele was certainly a very scrupulous individual, that in this paper basically Maxwell anticipates Morse to be his antidote. I don't know if it's true or not, but it's sort of clear that these, I mean, these do typically have a long-term intention of things that we're eventually going to get right. Oxford to get a honorary degree, but it's 20 years ago now. And he began the lecture by saying, this is joint work with Sir Isaac Newton.

1:55:00 And there was an argument, you see, that in Newton there was some anticipation of a wheeling function. It's mentioned in this book of his own. It's almost ascending to gossip, but I wish philosophers would read that stuff. Yeah, yeah. I read it, but I haven't read it. But if more people would read it... Did you say again about the... I mean, just citing this theorem of Grotendieck, it also occurred to have proven about the same, more or less the same times. In the early 50s, he was considered the greatest functional mathematician. Yes, yes. No, I remember this because I was a number demon. In the 1960s, of course, they didn't know there what he was working on, but they knew very well just how dominant it was. It wasn't just a functional analysis. So I claim there's another day reserved, an hour or two, that this is really how Goethe-Dick actually moved from functional analysis to algebraic geometry. And that's the first one, because that's... That's a natural question when you study dual space. What about these concrete spaces and analytic functions? So if you see, if you could only freely use the topos, which envelops the category you're interested in, you could immediately write down stuff like this and see if it's true. The space of functions, the space of distributions on the human pieces and so forth is all there. In the same way that the external natural numbers, you can use them to do induction on formulas and all this stuff, but you can't do that inside the geometrical network. This, of course, is more geometrical than that. Well, there were a number of other things I wanted to ask you. That was the first question. The week is young. This idea that dimension one governs the higher dimensions, this recurs in other ways.

1:57:30 I call it the Hadamard principle. Hadamard, in some lectures, gave differentiable or smooth function of n variables just by the fact that So, in other words, the only real information is the monoid of endo-maps of the one-dimensional space, and the rest is just general objective logic, taking up function spaces and quantifying. So the real information is just in the one-dimensional space. And that turns out, of course, this is the correct definition, you can always make this definition, it doesn't take any more than that, you can immediately do it, so I call that the Audemars Functor, actually, but the thing is that the Audemars Functor actually agrees with the known correct one in so many cases, that it's also for continuous functions, continuous functions on a cube, for example, and even polynomials, the only... The only, the only, let's see, you have to be a, you have to have a ground field, I think, which is not countable. Can you be either finite or uncountable? In any case, in any case, if you consider, you know, a function of n variables, sort of an abstract function of n variables, which is compatible with every polynomial in one variable, then in fact it's a polynomial of n variables. So this is a little more surprising because now we're talking about actual concrete coefficients and stuff. So, you know, this is, again, this is a remark of Shannon. I think it might even be useful to you, I don't know, because it's just, you know, you don't have to really check the definition of enjoyable. It's just sort of a... Reverse of the usual point of view, because you also want to feel, well, the interdimensional space is important in itself. But this is a criterion.

2:00:00 Let's see, what are some other examples? Because this is kind of opposite of what I was first thinking. You're talking about maths and I was thinking you were talking about vibrations. The one-dimensional. No, that would be another approach. You build up things by repeated vibrations where the fibers are one-dimensional. Yeah, so it's not, yeah, it's a different, I think, and, yeah, and Euler's observation that real numbers are just ratios of infinitesimals is even, it's pushing that down to an even more basic level, which is rigorously verified. Let's see, well, there are many examples, like, just take the category of partial derivative sets for example. Again, a map of... In the category, if and only an abstract function is border-preserving if and only a Murphy-Math 2-elementary set. Unless it's too obvious, but still, if you think of it in those terms, maybe those higher-dimensional things aren't entirely so complicated after all. If I can get at them in this fixed level. In other words, it happened there often enough to be considered as a non-random phenomenon, I think. I have no idea of any common proof of those things. It's trivial, it's a common statement, but since they're in different categories that have totally different properties, it's not clear what the common reason might be. This notion of Vakhnik-Chervenenko's dimension, which is true in animal theory, so it's roughly speaking the divinable families of sets in, say, n dimensions, but they may depend on a lot of parameters. It's about the independence of sets, too many subsets of a given set using... There's a dual property which Schallach had studied. So this first notion was discovered by the statisticians.

2:02:30 Schallach was working with a dual notion where somehow the role of the variables and the parameters is flipped round. This had been around in the modality of the 1960s. And Schallach proved with considerable, by passing into non-standard model of set theory, where a particular property of Derrikin completions was true, doing a calculation and dropping it back out again, he showed that if this dual property held in the dimension n, and then you can conclude from that. That if you can verify Shalek's thing in dimension one, then you get the Svatnik-Chervenenko's thing in all dimensions. The probabilists hadn't known all of this. I don't think they could have proved it straight forward. I think they could have done it, but Shalek did it by this. That's probably the most difficult example, and of course, something that you would expect to need to know in all dimensions separately, actually, from all constructively, really. Then an interesting geometrical example, you can sort of see it constructively. You understand that there is a result there. Yeah, I wanted to quiz you. Thank you for your attention.