Anders Kock talk

0:00 Organizing this event, I'm very honored to say that I would also like to express my thanks to the collaborators and I guess who helped out with the very difficult logistic situation in the last couple of days. It was really good to build up this way. Of course, I would also like to thank our collaborator, Marcel, who collaborated with us almost three decades ago. I would like to thank my scientific mentors, Bill Louvier and Andrew Israel. Then I also want to thank, also in the 70s I think, revived the idea of a parapsychotic seminar. And I'm very honored to also for the first time a parapsychotic seminar returns to a place of origin. The first place, I know it won't be the last. Well, not only parapsychotic seminars, but actual categories. Since I have called over, I and many of you have not been very familiar with geometry before I show some introduction to that. The main point is that it's modern revival. The important thing about an object is not what it contains, but the category in which it participates. The particularly good categories are the topos. This was realized by Gromendieck.

2:30 When you talk about, you can almost pretend that they are just sets. So you get a very, sets are axiomatic sets. Static differential geometry is an application of this and in particular on the, let's start with the most basic geometry. We would like to have the line in a suitable topos. There is a whole sub-channel which moves all maps from the category of truth to the category of truth.

5:00 It's just the case that all maps in E are smooth, so the word smooth, void, so that's all in E. And now we, among the smooth manifolds, we have the linear line, when we look at it in E, I'll tell someone just so they know that, well, depending on what topos you choose or the stocks, the maps between the points A or whatever will be the same because this embedding is, Sub-objects, sub-sets, are, or acquire, the kind of totals that are well suited for differential geometry, are those totals where the line acquires a sub-object across the neighborhood of zero, a flat just system. He consists of those elements. The square is zero. You can see the relationship to what André talked about an hour ago, because then you also divide it out by the square, or something in this way.

7:30 It is exactly what he described. The first label of the diagonal line is just the same as x to the y, such that x minus y is d. In other words, such as x minus y squared is zero. I'm not saying that x minus y is zero, squared is zero. More generally, in this context, any manifold n will acquire the first enablement of the diagonal. It can be described quite explicitly also in terms like playing geometry. Well, we had five geometers who invented this description of this first enablement of the diagonal. However, it also works for many books. I believe it was Malkan who followed up on this, despite his lack of love for the first paper. So in that sense, it's not a new object. It's just that when we see it from the topos theory, it becomes so easy to talk about the first paper. There is a much more direct way of talking about mathematics and they also explain this as a good basis for, for instance, a combinatorial theory of differential laws.

10:00 I only started to develop two years ago and I shall return to it. By the way, it reminds me of another picture which I also happened to show you. We also saw that now, namely, the intersection theoretic aspects of d, namely, the xy kind here, the unit circle of 0 and 1. So what is the intersection here? Well, if you calculate in coordinates, you will see that the size of the quantum of x is equal to 0, where x squared is 0. So that's the geometric, one possible geometric picture of the object d. And now that I've already mentioned the antiquity field. This was a matter of debate among the scientists because they wanted to compute geometry of Euclid by arguing that it's absurd that the line only consists of zero. We had to insist that there was only zero there, because otherwise Euclidean geometry where two different points determine a line and so on could not work. So they were refuting stating it's absurd that... Well, it's consistent to do that, but it's not so open to quantum geometry. We want a topos where this object actually is a flat version of zero.

12:30 So this is sort of an introduction to quantum geometry. And I've picked a particular that I would like to touch on. It's what we in elementary calculus often see in terms of slope. At each point in the plane, we have a little bit of a line segment. What we know for the moment occurs. The branch of equation itself is just this family of slope views, this family of small line segments, all together making up. That's an example of a distributional mental sense representing the branch of equation by a line equals. But why? Because I only had the patience to do a few of these. Now the point is that you can find the solutions. You can get sketches of the solutions just by, so to speak, taking these line equations as a geometric way of solving, in quotes, differential equations. This is an expression of the fact that one-dimensional distribution, one-dimensional distribution, small, affine subspaces that the line set indicates here are one-dimensional. And the line set so happens that one-dimensional distributions always are integral.

15:00 The two-dimensional distribution in R3, I adapted this picture from a book of Berkeley called Non-Integral Reel of Contact, Two Elements. I think most of you are familiar with what is a non-involatile two-dimensional distribution in R3. You can see that geometrically that here we have a problem of finding surfaces that collate all these plane elements. This is an example of a non-immoral system. Of course, this also represents a partial differential thing like z time. No, he said that the x is something like y, and he said that the y is zero, where when the partial differential, first of all, the partial differential equation is a field of line. In this case, of course, the reason why you cannot integrate it is because of the set of x and y. Well, I want to think geometrically and coordinately and to describe what involuntary distribution is. Well, first, I will have to describe what is a distribution. To that, again, I will exploit the control period set theoretically about me. So set theoretically, at least, gives a set with a reflexive symmetric relation, namely the neighbor relation.

17:30 It's a different way of talking about it. It's a reflexive-symmetric enable-relation. Well, I should say that the enable-relation is a refinement of the enable-relation in terms of the geometry of this purely combinatorial motion. To say that it's a strong enable-relation is a refinement of the enable-relation. Strong enables, I write mx for those y and m which are made. Strong monad is contained in the monad. m single means that I just write mx because it's done for enable-relation. The basic structure on the manifold. So that is what a pre-distribution is. And we return to the picture set, this one.

20:00 If you visualize the neighborhood around this point as a small circle around it, the strong neighborhood is just that part of it which lies inside. Besides this little line segment, that is the strong one. That's the geometry of... I'd like, by all means, a combinatorial notion in these terms, maybe an integral set or pre-distribution, which has the thought that we have two elements in that subset, x and y, which are labels, little and strong labels. This is a little smaller subset and stronger than what we would say is weakness or clear from very banal logic that the subset of an integral set is an integral. The main definition of an integral set is that the union of these three curves is an integral set. We would like something which, say differential equation theory, clearly we want integral sets, we want them to be all three of them belonging to one. There might not yet be a connectedness code that comes from that. Now comes the main definition of it, on my talk mainly, to describe what it means for a predistribution to be qualitative because it is complete.

22:30 The only theory I want to present is to give a relationship to this synthetic notion of volatilism and then the corresponding classical one. So here's a combinatorial synthetic. Free distribution is invol- involative if, whenever you have- And two of them, furthermore, are- Then the third, so in pictures, if I would double-bar as indicated, to say that all the strong monads are- This is almost clear, which is if two points y and z in n, which are neighbors, then they are actually strong neighbors, and the other is formulating monads. Strong monads are integral subsets. I apologize that I have to use a pre-distribution. This is essentially a matrix model. Ordinary differential geometry, but there are distributions with singularities.

25:00 Now, as I said, I have to assume a suitable notion of connectedness in order to be able to define what is a leaf. You don't want a disjoint union of several leaves. I assume that I have a notion of path connectedness. However, this is a rather axiomatic talk, so I don't want to say how we define path connectedness. So, suppose we have a notion of path. And if it is maximal among such integral, but it should also be connected, and then it should be maximal. Well, there's a very simple comes about just from the maximality. Maybe that if it's clear from the maximality that if there is a leaf to a point, then it is unique. So if there is a leaf to x, then x is a point from the leaf to y. This is almost evident from the maximality. Because qx is certainly, if y is q of y, the sum of y is y, and vice versa, the whole plane of x, because it's not an integral subset, if you see the picture.

27:30 Here I have a neighbor point of this point, which lives in the, is a neighbor, but it's not a strong neighbor. The whole plane of the point is not a strong neighbor, so that's in the context of... In synthetic differential geometry, we would like to have a notion of distribution which is stronger than that of pre-distribution, which reflects the classical notion of all, with no singularities, and in that case, we have in classical differential geometry that involuntary distribution through every point, that is, it would be, I am not proving it would be new theorem, then I would have to say I have not done anything. Finding topos is easy in this sense. I'm just giving an easy statement of it, namely, involutive distance throughout the motion of qx as before denotes the leaf through x, then, just because of the important thing that the point is contained in the leaf, rather than the two leaves and the two points of the point are equal, this means that the leaves give a partition of the involutive distance defined by the linear subgroup.

30:00 Oh, I think, yes, right, oh, explain that. Well, at least explain it in the case where the manifold is just a pole. This means then that the strong monad is of a monad with an affine subspace, in that sense they are non-singular. So they are strong monads or linear subsets in a variant sense. Right, as I say, the word smooth is void, so welcome to it. Yes, I already started out with this slide. We can also describe for Rn, Rn is a semantical. What is the monad around zero on Rn? Well, this is a set of those entries such that not only the square of each individual, xi is zero, but also the product, xi with xj is zero.

32:30 This is actually a coordinate-free notion, meaning that it's invariant on the linear isomorphism, so this means that when I define a dimension, when I define them in 1m around 0, v, I call 1m around 0, then the labor population, in the first-order labor population, is just that, x minus y. It's real. What I meant by saying that, that should be sufficient to manage this constant situation. And that can be expressed in our days of quantum reaction, namely that any map into R, which takes zero to zero, extends uniquely to a linear map into R. The possibility that a map from zero to R, it's very easy to be actually quantified by that. The aspect of it, of reaction, is that a map from dB across dB into R.

35:00 It has the property that the value of the symbol in the second coordinate, likewise the value of the symbol in the first coordinate, extends uniquely to a bilinear map of the coordinate. That's about trivial. So in that sense, dv cross dv, dv itself is a linear map classifier. In this sense, d is a bilinear map classifier. For bi-constant reasons, it's on a subset of dv, d tilde, 2b. The bilinear alternating. Now let me explain the explanation of the classical distribution. For simplicity of exposition, I would consider only the case of an open subset of the finite dimensional vector space. An open subset of the bioradiator. And then, when we ask what it means that the pre-distribution, this means that to each x in this open subset there is a linear subspace. And then the constant. It gives rise to a combinatorial pre-distribution that is given by the fact that the square root of the monad around x is x plus the heat of the monad around 0. Here I have x, and here I have the linear subspace. Here I have the x, which explains for the linear subspace ux, and the square root, and I paint it here to correct the color.

37:30 An aspect of the couple reaction may be that the monad is sufficiently big to reconstruct or construct linear maps just by knowing their restrictions. This implies that as soon as I know this little wet subset, I can reconstruct the whole linear subspace. And that also explains the relationship between antiklassical models. One way of putting antiklassical distribution is on an open subset. This is that due to each point, you have a linear subspace, a contact element through that point. An example of such a nice distribution is when you have a differential one-form on M. Revening to each point X, you have a linear map of a bar X in a variable matching number. Then you may take Ux, that would be X. I've come to my main aim, namely, to relate the combinatorial notion of the involuntary with the classical one. The classical notion of the involuntary is a little complicated, and that's why I think this has some value.

40:00 The combinatorial sense was so extremely simple, two sides that are strongly related on the third side. As I said, the classical distribution notion is not the distribution notion. It is usually formulated in one or two ways, either through the round algebra of differential forms with an exterior derivative of the wedge product, or subordinates to the distance and the leap-racket of the discriminatorial notion of invalidity with the one that comes from differential form manifestation. I will make some elaborations on how differential forms manifest themselves in the context of the lecture. They also said that you have a notion. As soon as you have a neighbor notion, you also have a notion. If you can test the case in place, there is a k plus 1 rule of mutual neighbor points. And the differential k-form, I mean, is a law, omega, which associates, took such a k plus 1 rule. The requirement that the value is zero, that's the only requirement of the differential form. Nothing about linearity. Nothing about alternating. Why is it that there's nothing about linearity? Well, that's essentially again because of the reaction that things define one with respect to the cross-wall enablement.

42:30 That's not quite, and I won't prove it, Taylor series argument. But note that the notion of alternating here has a stronger meaning than a k2 or tan of x, y equals. I described the, how the Dirac algebra, in other words, exterior derivative and wedge product, appears in the I will only do it for one form. So given a one form, then I define a two form, B omega, by telling you what should the value be when I give it that. It's going to test for x, y, z. Then B omega of x, y, z is omega x, y plus omega y, z minus omega x, z.

45:00 So that is the definition of exterior derivative. I have two one-forms for them. I get two forms. Two one-forms for them. I get two forms. Omega wedge alpha. By saying what the value of this is, I get xy times alpha yz. I should say it differs from the... if I want it on the classical one, I should go down to two. I would not go... It's a nice formula because it looks exactly like the formula for coupled products. It's a very simple topology. Similarly for higher forms, as I say, going to higher forms. And the non-degenerate one-form or name defines that. Usually, maybe you say that X and Y are small neighbors. If they are neighbors, if the value of omega and omega times the neighbor pair, I want that in the sense that we want some distance. This is simple. I shall only deal with coordinates and what is the use of it. So now comes the theorem I present. Suppose I have given a differential one form, omega, on the manifold, and still I shouldn't know the subset of that finite space. Then the theorem says that the solution defined by omega is involuted in the combinatorial sense given on the integral.

47:30 In other words, the alpha at each point, x I can choose, I will add a linear one form, alpha. It will depend on which x I choose, alpha does not really define one form without further work, because since alpha is not unique in passage, which is what x associates, alpha x is not a function. I make a choice here, a choice that's not... I could do that, I suppose. Another thing which I think is what does classical is to pick a Riemannian metric on the manifold, in which case there's a canonical choice for the, in which case algebra does become a function, i.e. a smooth function. But let me neglect these and try to give a proof of them. Well, one implication is that omic and wedge algebra sides are strong labels. I claim that the third side is also a strong label. Well, that's the continuum from the wet form, because if d-omega, omega, which alpha, d-omega times x, y, z, omega x, y, times x, y, z, times x, y, z, and here I have omega, which alpha, times x, y, z, y, times alpha, y, z, is zero, because this is a strong neighborhood here. This is a strong neighborhood here. This factor is zero, because... This is strong neighborhood there, then the whole right hand side is zero, and therefore this is zero, and there, and this is strong neighborhood there. I have two sides, strong neighbors to the third side.

50:00 So that's the proof that the classical description of involutivity involving exterior derivatives and waves implies the combinatorial one which does not involve any. Multilinear algebra. There is no such thing as linear algebra as long as it does not involve further difference. In other words, I'm going to use the determinants here. So as usual, this is a finite convention of vector space. We have the vector space out, K, V, R, which is the same as K linear, alternating. And that becomes a Witten product, defined in much similar... Classical linear algebra, I would need classical multi-linear algebra, I would need a kind of a low-stone such as, maybe if I have a linear map, now I'm not talking about differential algebra, it's pure algebra, if I have a linear function from B to R, from rank 1, and the kernel ends within this 1, if I have a bilinear alternating theta bar. With the property that its value vanishes wherever u1 and u2 are in the curve of omega, then theta bar is of the form omega bar times alpha bar, or as usual, the linear function, because that's a kind of a Roskilde sense, a function that vanishes whenever, wherever omega vanishes in some theta bar belongs to the idea of generating that omega bar.

52:30 All we get divided, sorry, k-triple of the structure of the bar, which is what we need to have a k-coding issue and our explicit justice. So now, using that from multilineal algebra, then we get a combinatorial one-form on M, which is combinatorially involuntary. Well, then for each x, I get a linear form, maybe the one... Omega bar such as omega x, y on the label pair is omega bar of x on the difference of y minus x. That's the consequence of the fact that the labels of x are of x plus dv, linear mapping classifier, according to the real action. Similarly, given the combinatorial two-form, there is a unique bilinear alternating map that extends the given combinatorial two-form in this sense. Here, I need the fact that it classifies infinitesimal two-symptoms, zeroes, zeroes, in other words, alternating maps and the notion of, here I give the definition of, it's a subset of dv plus dv consisting of pairs, u and v, which has the fellow probability that the difference is also in dv, and it's an equation of condition.

55:00 So this third condition comes out as a subset of d. And now it's a piece of single algebra to see that function. It's not simple, but the point is that this d tilde 2 v, which as described is the set of, in the first of two, this is 0, u, v, able to 0, v, able to 0, u, and v, classifies all of them by their forms. Now I am applying the most instance to d-omega of theta and at the point x it gives rise to bilinear ordinary form. We like to see that this bilinear ordinary form theta bar is of the form omega bar wedge alpha. And by the most instance it suffices to see that it annihilates anything you want from both of them. By the construction of the omega bar, this is the same as the omega x-stake plus one x-stake plus one x-stake plus one x-stake plus one x-stake plus one x-stake plus one x-stake plus one x-stake plus one x-stake minus one x-stake plus one x-stake minus one x-stake plus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake minus one x-stake

57:30 You can see also the omega of that is something you want to use and that will find the most consistent form to which alpha. Let me make d tilde 2 rn a little more realistic. I already told you what it means for x, for an n-dimensional vector. But if you further assume that x, y, xj are at the end of x1 minus xj, let me take the simplest case. It is easy to see that the basis for this abelian group module I consists of six polynomials and that explains why determinants, i.e. multilinear algebra, comes in in connection with, if I may take one minute left, if I have two elements, x11, x12, x21, x22. Where this is in E2, it's in the further position that the difference of this row belongs to E2, which means that the product of the two coordinates are differences.

1:00:00 So let's multiply out these four terms. Two of them vanish because it's one-one times one-two. Likewise, the product of these two vanishes. So what is left? X-one-one. Minus X11 times minus X21. So in other words, X11, X22 plus X21, X22 vanishes, which means that every symmetric bilinear function vanishes on D tilde, which is generally bilinear functions. Now every bilinear function has a sum of a symmetric bilinear and a skewed bilinear, so only the skewed part is left. In a sense, two symphyses have a multilinear or bilinear force network. There is a question. What is I? I is the ideal. Could you move down? I is the ideal defined by an equation of condition for each I. This is an equation of condition for each Ij. And these equations define an ideal in the algebra of a matrix. Why are you in Witten? Why are you in Witten? Because I want to say that this is indeed, and this is indeed, and the difference is indeed this. I want to say that the two vectors in our tool are formed and accessible in two sentences. The neighbor of zero, this is the neighbor of zero. The difference is the neighbor of zero. It gives me some equations. These are matrices two by n.

1:02:30 Yeah, in the general case, I have not 10 by n, but 2 by n. 2 by n, okay. Of course, we have a similar d tilde k n, which would be 2 by n matrices. Well, if you want the full theorem, I take the completely general case with general l to k, and then draw up certain l by n k matrices to define d tilde. A basis, a fuller basis, d tilde k, determines 0 by 0 submatrix. Here, the 1 by 1 submatrix is then determined. And here's the 2x2, generally b tilde k. I define that k times n determinants modulo the ideal i defined by the equations and basis modulo i consists of all determinants in square submeters of any size. Well, I think the Kluber coordinates are only the maximal square submeters. Here I take all square submeters. One of the difficult things in classical real analysis, as opposed to complex analysis, is that you have functions that are smooth in the sense of being infinitely differentiable, but don't have Taylor series representing them. Is that something, I mean, it could appear to be difficult to keep up with?

1:05:00 No, no, no, just, I mean, there are functions from R to R that are infinitely different from all that they don't have Taylor series references to at every point. So, when you're saying smooth and it's synthetic on that, I'm not sure, should I be thinking of functions with Taylor, with power series references, or should I be thinking of Taylor series? Of course, if they are flat, the Taylor series happens to be the zero series. I'm not, in any way, claiming that function. So it is, this is what happens in Taylor series of functions and functions are constantly detracted from the things we have to think of when we think about planets, like I said, when you're talking about the three of them. So the Taylor series are really not interesting, and in particular for, since I've only been talking about the first member of the class, it's the Taylor polynomial of degree one. I started describing the fact that not just x i squared but also x y times x to the j means that if you take the standard of Taylor expansion, all terms, total degree, are really equal.

1:07:30 You should ask the chairman. No, I don't know much about it. It's unknown whether, well, maybe you know it, but I don't think that neither you know it, or the book. It's something connected, not quite. There is an existence of solutions. Unique solutions. As far as I know, this is a question that is open in all the documents because there is a proof of that that was published by a book as found by Mankai to be correct. So the questions about HOPOS, you know, questions like this and questions about analysis about HOPOS will be widely open as far as I know. I think that the monad of Geometric, as you have described Geometrically there, is the point that the 13th of the x-axis, whereas in Ronsonian the 13th is the whole y-axis, the monad.

1:10:00 Well, I'm talking about the meat of the x-axis with the y-axis. I mean, in Robersonian infinitesimals, the x-axis is the monad of zero. The whole x-axis is the geometric representation of the monad of zero, the Robersonian infinitesimal. At least one... the whole ring of the Robinsonian-Nagel relation is transitive. It's destroying all theory we carry on today. I'd like to add also that the Robinsons, they are quite different in their spirit because they are in general, if you guess not, hard. So this allows you to... There are many real numbers and so on, but as this is the decimal, they're a little bigger, they're a little harder, so they cannot be inverted. That's also why the linear algebra that we have to use is a little more delicate, because it's not linear algebra over a field.

1:12:30 R, the basic thing, cannot be a field, because then either it's zero or it's invertible, so there can be no element square zero, except zero. I guess we need, tomorrow, a dance service. Guys, do you want to tell us what time to gather in the lobby to come tomorrow? Do you want to tell us what time to be ready to come over? Nine o'clock. Okay, I would say so too, but that might be a good idea. Let everybody know that. Thank you for your attention.