A Kock — Vinogradov School & archive

0:00 Thank you very much for your attention and I hope to see you again in the next lecture. Thank you for your attention. Thank you. They actually use the quantum space and the permutations to actually use the laws of time and space, but the only thing that the professor can tell us about which is absolutely there is the inverse of symmetry. But, well, the construction of all of it... So if you guys can explain how it's been called, because I know that they are in the same direction of what I'm understanding of it, so anything that we can think of would be very much interesting.

2:30 So, of course, I think that I would just... But at the end of the day, if you want this kind of space, you have to be able to analyze these objects, but there are two things in common. Thank you for your attention. This is not enough to solve every equation that we have. I mean, to really get there, you need something more. And the assumption for this, I would say, it comes from the level of judgements. Something more powerful than the gravity. Ah, yes. Well, this is always the problem. And the structure, the algebraic structure is easier to construct than integration. Yeah, I know. Yes. Going to Naples.

5:00 Thank you for your attention. Thank you for your attention. Thank you for your attention.

7:30 I suppose we can start the session where I would like to ask the Chairman of the Faculty of Mathematics, Patras, to give a few welcoming words to the participants of PSSL 1870. The Chairman of the Mathematic Department of the University of Patras. It is a pleasure to welcome you to our university. Mathematicians have the opportunity to get to know our department as a place where activity in their research area takes place. It is also a pleasure for our colleagues from the University of Patras as well as from other Greek universities who will get acquainted with the ongoing work in the domain of category theory and sheet theory. whose methods premade modern mathematics, and in particular geometry, that happens to be my own field.

10:00 Much more so, since some people that have led the developments in these areas are here among us. In particular, I would like to join the professor Andreas Koch, a very... So, informality was limited, but yet some extra care had to be taken. For that reason, the Ministry of Education took care to maintain the informality, and then the German government made sure that that happened. And then there is a slight catch here in the support of that.

12:30 Although I was assured the initial amount allocated was surprisingly small. So, it seems that it was classified as an internal Greek work, I should say, Greek work of the type. It seems to be the usual clear gallery, not exactly into dinner, but it is at some snacks and the Saturday dinner. I'm very happy to be here, to be the chairman of this, to see that there are so many bold,

15:00 And right now, he's in sabbatical from his university at the University of Barcelona. And so, today he will talk about the homotopy algebra. I am a very inspired person. It's always very nice to discuss with him. He has always new ideas and stuff like that. Okay, is there a question?

17:30 A long time before choosing the subject of my talk, I had two. This is why it was announced TDA. I had two subjects and I was thinking maybe I could start to talk about this one. I made up my mind. I'm going to speak about the third subject. This was my way of solving it. And I'm going to say something about commodity algebra. And I will not assume that you know much about commodity theory. So it would be quite elementary, so it may be more like a motivation for the subject, giving you an observer of the subject. There is some connection with the work of Andrews, and maybe I should start with this. In Wenders, the infinitesimal neighborhood of the diamond, which is a very important object, Wenders plans to write a book called The Neighborhood, the Relation, and Geometry. I'm not sure this is the same thing. Yeah, it's the same thing. So, I think the first physical neighborhood of the diagram was introduced by Ruben. Do you know how the scheme fits the diagram?

20:00 From the point of view, if the scheme is the spectrum of the graph, then... The dual function is that you take the co-diagonal multiplication, and this is a suggestion, and then take the denominator, j, which is defined by the kernel, this co-diagonal, you take j squared, you take the square, and this is a smaller idea, and you divide that. Why is this important? Well, one can use this to develop a three-dimensional form. So, for example, there are the complex. RHAE. RHAE. Yes, right. Thank you. You're welcome. It can be constructed in a sort of geometric way from this picture. Here we think that a pair of points, this is a scheme describing a pair of points, but two points can be very close, one close so to speak, close up to the third order, to use this to construct a synthesis, to look at x, y, and z, for example for two synthesis.

22:30 A typical point, which I want those, you get, you can describe this, it will be another scheme, I don't know how to call it, let's say, it's two-dimensional, you have a little triangle, and you take the function on this, because it will be a time scheme, and then the functions of this can be described in terms of the theorem complex. I'm giving you this example just to tell you that this idea of the first liberator diagonal is very important. A related idea is that it comes from intersection theory. Curve can accept some versatility where, in this case, the number of points is unambiguous, but there's a problem counting the intersection between two curves if they are identical. The naive point of view is that there should be only one point to count, but the more refined point of view is that, in fact, the number of points to count is the tangent that curves do.

25:00 And this is called the multiplicity of an intersection, intersection of... And the section of this is really important because the Bisouk theorem, the Bisouk theorem says that if you take a curve of degree n and a curve of degree m, then the intersection, the total number of points should be n times n. And for example, well, the curve of course, they have to be... The first gamma 1 is a protective space, a complex protective plane, gamma 2, so they happen to be algebraic curves, and they have a degree, the intersection, the number of points in the intersection according to this one is the color of the degree times the degree of the curve. To be true, you have to add the point at the 50th of the bottom line. I'm assuming here that the curves don't have components in common, that it could be this length is a reachable curve. But you also have to take into account the intersectionality. For example, a fabric intersects a line of two points, not this one but this one, but if the line is extended, it's like this. Now, intersection theory is a very beautiful subject. There is a nice formula.

27:30 For the entire section, let me refer to one formula you just said, that suppose that you have some sub-varieties defined by an idea, sub-varieties defined by another idea, and you want to count the number of points in the extension, so... Of course, this number of points would be interesting, but assuming it is finite, you expect that the number of points in the intersection should be, if it's finite, the dimension of the variety defined by the two ideals, which is the direction of the two ideals, and I'm supposing that this is the case. The formula, this is not understood, all the generalized formulas are given by set, and this formula is that the probability of intersection is derived.

30:00 Now, where is this coming from? I mean, the tau of zero is related to tensor power. A over I over J is a tensor part of the symbol. And you see that there is a kind of correction factor. The first 1k is equal to 0. This is just a tensor part of power 0 over I. And so there is a correction factor that comes from the fact that there is 4, 1. This is the formula of CERN, and this is a very empirical formula of where it's coming from. How could it be that these four are carrying in the intersection of multiplicity?

32:30 Of course, there's a proof, but one would like to have a setting where this formula is completely natural. People really have developed a new way of looking at this formula, and a new way to incorporate the homogeneity theory, maybe one would say homological algebra, but it's really homogeneity theory, and what's the difference between cosmetic intersection theory and, well, the difference is that you look at the diagonal in a different way. The diagonal of the scheme in different ways. So, other ways of looking. In homotopy theory, in homotopy theory, if you want, if you have a space X, you can form, of course, X cross X, but you can also look at the space of X in the eye. The eye is the real X. X has two ends, source and target. And, of course, there's a map like this, and there's a map like this. So this looks like a map of my characterization of this target field.

35:00 Instead of considering a pair of points which are one-closed, infinitesimally closed up to the first order, you look at the pair of points here, connected to the path. Here, arbitrary length. And this sort of path diagonal... The amount of the so-called relativity put-back contains the intersection of two things is exactly the put-back. You intersect two things in its intersection, the put-back. In general, what they want to do is to just take the put-back of two spaces, let's say, The use of construction is to put back the diagonal of things. I mean, because one way of constructing the fabric part is to put back the diagonal, the diagonal, in other words, any kind of intersection is always obtained by pulling back the diagonal, the diagonal. Now, if you replace this diagram by the past diagram, which is here, you obtain a different kind of product, or putback, called the homogeny.

37:30 The homogeny putback, like a section, whatever, is obtained by, with that same diagram, but here, you put, you take, and the idea of Jekyll and Hull, The idea is that intersection theory should move, taking this kind of homology put back. And if you do this, the exact formula should come naturally. This is not entirely obvious, I must say, and I don't think I will be able to explain it here. But I would like to say a few things about this idea. The question is, to replace the replacement, to replace ordinary putbacks by, but not to do this, to use a kind of homogeneity, you need to have a past space. Where is this past space coming? In the case of Scheme 2, what is the interval? So you go to a more radical program, the more radical would be to replace ordinary limits.

40:00 Pullbacks are special cases on physics, but how can you do this? How can you use a procedure for doing this? And one way is to use derived, called derived algebraic function. Understand how this is constructed. You have the notions of derived scheme, but how is it constructed? One way is to use a time that is not by starting with...

42:30 Let's start with semi-cell. You have, let's see, a semi-cell ring is just a delta. Is the semi-cell can you read? I recall that the objects of the delta are useful. Semi-cell can you read? I'm not sure. Semi-cell can you read? Semi-cell can you read? Semi-cell can you read? Semi-cell can you read? Semi-cell can you read? Semi-cell can you read? But in order to work well, this theory has to be a bit more radical. The next thing is to replace by a more difficult one. Where are the candidates coming from? It's because the schemes are developed, the apparent schemes, by doing, by the equivalent process. So, for example, objective spaces are planned by doing a certain number of apparent spaces. And this is a coherent process. You do two different things. And this coherent process is well taken care of with the sheet. If you have C, a category, any category, for example the category of a fine scheme, then you can complete C under the coordinates.

45:00 This C-hafta, the very nice description, is the category of free sheet of C, or free sheet of C. You can do your test and you will work at the pre-sheet on it, and you can do your pre-sheet on it, and this is, yeah, this is why you can do your pre-sheet, that's what I'm talking about, when you're a subject of mathematics, mathematics, physics, mathematics, it's because you can do it, you can do it, you have to do it, that's what I'm talking about, this thing is the solution of the pre-completion. If you have an English C and you want to add the co-products to it, you do it in C-act, and C-act is really tricky. So, you should do this, but not using co-limits, but a lot of the co-limits. One way to do this is to use a simple shell. Suppose that C is a simple shell, that means that C may be hung. The onset is a synthesized set. You have a synthesized set of maps between two objects. For example, the category of the semi-cell community is a semi-cell. There is, between two semi-cell communities, there is a semi-cell contract. And you can call it a semi-cell contract. So I'm going to write S for the category of a semi-cell set. And the combination that I want to look at...

47:30 Now there are two things that we are doing. We replace commutative rings by semi-self commutative rings, and then we get a semi-self category that's called a C, but in the opposite. And then we take the pre-seed, unseed, but we just take the punctures, preserve the average, the punctures should be seven, that's kind of the underline of the construction, of the classical construction of the natural agency, the book of the classical construction, the punctures. Yeah, so it's a rather involved construction, and what are we really doing here? It's a kind of a problem, because why simply subsets? I mean, what is this? And this derived algebraic geometry works. I mean, this formula of serivivics is meaningful there, and it's solid, obvious.

50:00 There are many theorems about that. But what are we doing? So the question is, what is... We know what a limit is, and we know what is the product of the object is, etc. What they do there is they tell you, well, if you have a diagram of space and you want to compute some other piece of it, you follow it, you follow the following procedure, you give a recipe for constructing it, and they say that's the definition. It's a description of the kind of adduct, and so what it is. It's one of the problems that's understated by the notion that the quillen model structure explains after a certain extent. Again, you have some kind of recipe, but there is more, it's more rational. There's no better understanding of quillen model structure. Maybe I should be wrong about what quillen model structure is. The co-cooperation is that of co-cooperation and co-cooperation.

52:30 And it should satisfy axioms that could include the mathematical algebra, which I can't explain without the axiom, which I can't explain without the axiom, which I can't explain without the axiom, which I can't explain without the axiom, which I can't explain without the axiom, which I can't explain without the axiom, And if you are in mind or are in the middle of the group, so in particular, the definition is a total of a composition. So we have the inference that a composition is a weak acquisition system. We call it a weak acquisition system. It is a weak acquisition system.

1:35:00 In some sense, also, I can see that because of the lights, it was a bit wrong. It seems to him to get a right natural level of generality. You're going to have to interpret it, as I said, by just a special meaning. Yeah, they didn't have the right... They didn't have enough generality, they didn't have the right natural level of generality. Which, in some sense, is right. Because there are many situations where you're not going to be involved in generality out there. That's not the point. On the other hand, since I said... Sometimes we want to move away from them, but the center of all of this...