 So thank you very much to organize, thank you very much to IHS for the opportunity to give this small series of talks on one of my favorite topic and let me start with. So, Vladimir Vladimirovsky in six, so the number in is of papers and preprints is taken from the excerpt of my talk, so Vladimir Vladimirovsky in six and then to the category of frame correspondences is a hope to give a new construction of the stable material that is more friendly for computational purposes. Currently with Gankusha, we use frame correspondences to developing for a period of frame models. The latter theory allowed us to give in five and to give in five a genuinely local construction of SH. In my lectures, I will recall the definition of frame correspondences and will describe the genuinely local construction of SH. Provided that the base up is infinite and I will also discuss the application. And now let me properly pass to my first lecture. So here is actually one. The nearest thing of this lecture is this rich couple of smooth varieties of Y and X or K. We will recall the definition of pointed set frame Y to X of stable frame correspondences. Sorry. And pointed some crucial sets, these two, frame delta dot X tensor S1 and frame delta dot S1. Both of them are described in terms of algebraic variety as we will see below. Here is a question. What can we state about the formatopi groups? Here we have one. The same crucial set frame delta dot S1 has the formatopi type of the classical topological space omega infinity sigma infinity of the circle. That is, we recovered the topological space omega infinity S1, sigma infinity S1. And the same crucial set frame delta dot S1, which is described in terms of algebraic varieties only. This is one of the computational miracles of the frame correspondences. Here in one two, we take the mentioned space, the mentioned same crucial set frame delta dot X tensor S1, and take in termatopi groups, you can find a three-efficient zimotn. And from other side, we take the topological space X1 plus its third suspension and take the stable formatopi group of this first suspension with zimotn three-efficient, where X is a smooth complex of the prior varieties. And R is an integer at least zero. So the result is that these two groups coincide. The latter result is an extension to material stable formatopi context. And on the left side, we have the solution complex of X tensor S1 of the simplicial scheme X tensor S1 of the simplicial scheme. And we take its homology with finite pre-efficient zimotn. And on the right-hand side, we have again the first suspension of the space X plus smash of the space X plus. And we take its usual single homology with finite pre-efficient zimotn. So in a certain sense, as I will explain below in lectures, this frame correspondences, stable frame correspondences, is a very good replacement in the stable formatopi material context for the finite pre-efficient zimotn. This is core, which are due to VWOSC and which plays a central role in usual motivic business of VWOSC. So in 1.3, that may be an infinite perfect field, and it may be its field extension. Not necessarily finite. Then for each smooth variety S over K, and each arc, which is at least zero, we can take from one side this simplicial space, frame delta dot three capital, comma S tensor S1. And take is ordinary homotopy groups. And from other side, we can take the motivic space X plus smash S1, take its P1 suspension spectrum, take the A1 homotopy shift of way zero of this spectrum, and evaluate the shift on the field K. The theorem 1.3 states that the left hand side canonically coincides with the right hand side. So these are theorems I will take back over my course. But the in the first lecture, I would like to formulate this theorem properly. For that, I need to recall the definition of stable frame correspondences and some other definitions which are VWOSC. So let me do it. Firstly, I recall the notion of a tiny neighborhood of Z, sorry, a tiny neighborhood of Z in a scheme S, Z is equal to a sub-site in S. So this is a triple W pi S satisfying the following condition. Pi is a natal morphism. Pi precomposed with S coincide with the closed inclusion Z into S. That's S is a closed embedding. And the image with respect to pi of Z coincide with the image of Z under the map S. Let me draw a picture to make the picture more, this definition has more things. So here is S, here is Z, this is closed embedding I, this is my W, here is my ital morphism pi, and here is my closed embedding S. So obviously the diagram can be used. As I told pi is a tie, the diagram can be used means pi precomposed with S coincide with the closed embedding. And there is also this condition that pi in S of Z coincide with S of Z. So very informally speaking you could take P to be the complex numbers and replace ital neighborhood by the strong neighborhood. And in this case the picture will look as follows. Take it closed ZB, say closed complex of variety, and then take a rather refined neighborhood in strong topology. This will be your W. So I also like to say that if you have another ital neighborhood like this, then a morphism between W prime and W, between two neighborhoods which I have drawn is a morphine row between W prime and W, such that this row triangles commute. Particularly row is automatically ital. So with this in hand, I am able to give the major definition which is Dupoevo-Vivotsky. Similarly, a definition firstly of S V C prime correspondences of level N and then definition of A prime correspondences of level N. So, for case morph schemes Y and S and an integer N which is at the level, an exit prime correspondence pi of level N consists of the following data. Firstly, a closed subset Z, let me maybe draw this in a nice way. So the data are the following. A closed subset D in Y cross K N, this closed subset is supposed to be finite over Y. And an entire neighborhood view of this closed subset in Y cross K N functions pi 1, pi N, which gives us a morphine from U to A N, such that the common vanishing of this functions is the closed subset Z, in U. The subset Z will be referred as the support of the correspondence. We should also write triples pi like this or quadruples pi like that to denote exit frame correspondences. Also, one should say that frame correspondences, two frame correspondences like Z, U, pi, G and D prime, D prime side, D prime occult equivalent provided that the following condition holds Z coincide with D prime and the second condition is there is a common refinement of neighborhood U and U prime, a common refinement W such that G restricted to W coincide with G prime restricted to W and pi restricted to W coincide with pi with Psi restricted to W. In this condition holds we call explicit frame correspondences to be equivalent. The frame correspondence of level 10 is an equivalent class of explicit frame correspondences of level 10. I will give a motivation for this definition at the second half of my lecture. We let frame N between Y and X, you know the set of frame correspondences from Y to X of level 10. And it is a pointed set with the base point being the class zero N of the explicit frame correspondences with U equal the empty scheme. I mentioned that the set frame nodes between Y and X of the correspondences of level zero, with a set of pointed morphisms between Y plus and X plus. Next, we would like to define a composition of frame correspondences to get eventually a category of frame correspondences frame part of K. So to define composition of frame correspondences, take a frame correspondence of level M and take a frame correspondence of level M. And define the composition as a frame correspondence of level M plus M between Y and X. So here we go from Y to X and here we go from X to S. So the composition is expected to go from Y to S and this is the case. And we should take firstly Z cross Z prime over X as the support of the correspondence. Its neighborhood is supposed to be U cross U prime over X. And here there is closed embedding which is S versus S prime. And we have function or better say morphisms phi cross psi from U cross U prime to AM cross AM. And it's given by the family of functions phi, pre-composed with pre-composed so a string of function phi and psi. And what is morphisms from U cross U prime to S? It is given like this. Firstly we project to U prime from here we project to U prime and then we have morphisms to prime to S. So this way we bought what we call the composition of frame correspondences of frame correspondences which is the correspondence of level M plus M between Y and S. So it is not difficult to check that this composition of specific correspondences respects the equivalence relation on the M and defined associative maps between frame M, Y to X and frame X to S to frame M plus M, Y to S. And in this in fact we are ready to define our category of frame correspondences, namely its objects are those of smooth varieties and the morphisms are given by the sets frame plus plus Y between Y and X. It is the book here, the book here of frame and Y to X along all non-negative quintages. I recall that frame and Y to X are pointed sets and therefore the book here is redefined. So this category frame plus of P is called the category of frame correspondences. Also as a category frame not of P, all this category frame plus of P, its objects are those of smooth and morphisms are given by the sets frame not Y to X, which are pointed morphisms between Y plus and X plus. So the category frame plus of K has the zero object. The zero object is the empty scheme. A frame, but to say a frame pointed, is a point to set on the category frame plus of K. The components, the composition of two frame correspondences in certain couple phases is taken in a very easy way. Namely, if F is another morphism between Y prime and Y, and pi is a level and correspondence of this form, then pi precomposed to the set is given by this simple formula. So level and between Y prime and X. And also if you take morphism between two varieties, a morphism of level zero, then H composed with pi is given by this simple formula, namely we precompose H with G and the rest is the same as for pi. So the category is defined, and it gives a good aspectivity to define the set of stable frame correspondences, frame Y to X. But before that, I need to recall one more way of definition, namely, given the case of scheme X, there is a distinguished morphism, scheme S, often called the suspension in pivot direction. It is a morphism of level one from X to itself, and it is defined by this explicit frame correspondence. So here X cross node is the support, and it lies in X cross A1, the italian Ibarcu is X cross A1 itself, T is written for the projection to A1, and projection to X is taken for G. So we need to take morphism to X, and the morphism we take is the projection to X. For each integer N, which is at least zero, defined using this sigma X define a pointed set morphism, sigma capital X, between level N frame correspondences to level N plus one frame correspondences. It takes a level N-respondence phi to sigma X, we can call this phi. Explicitly, sigma X takes the frame-respondence of level N of this form to the frame-respondence of level N of this form. So the support is equal to zero, its neighborhood is U cross A1, and there will be N plus one function, maybe the N functions are the previous one, phi one, phi N, and the last function is the projection to the last coordinate. The function G is replaced with the full N1, we add a morphism to G to X, and we recompose it with projection to U. Now we can give the definition of the set of stable frame correspondences between Y and X. It is just a collimit of this string. I should mention, which I did not, that this sigma X, sigma capital X, is an injection of pointed sets. This is an obvious picture. So roughly speaking, we could take, not roughly speaking, but exactly speaking, this collimit is just the union of mentioned pointed sets, and it's called the set of stable frame correspondences. What I didn't mention above, let me mention this right now, that for each frame-respondence of level zero between X and X prime, there is this equality, which I will draw as a picture of this form. We can take sigma X prime, recomposed with F, and we can take F recomposed with sigma X, and this equality means that the diagram comes. This is essential for what I will say right now. Namely, if F is a frame-respondence of level zero, then the assignment, which takes five to F recomposed with five, define a frame pre-shift morphism. A flow of star from the pre-shift frame delta bar X to the pre-shift frame delta bar X prime. Also, I would like to stress that this pre-shift and this pre-shift, they are all to, they are in fact mis-nevic shifts by a lemma due to we were asking. So, but what I would stress that there is this covariant morphism, there is this morphism, defined by a correspondence of level zero between X and X prime, particularly frame by bar X. So, as I told you, this frame bar X is a frame pre-shift. Since it is a frame pre-shift, therefore we can restrict it to the category smooth, and we get a pointed pre-shift on smooth. And since F flow of star is a frame pre-shift morphism, particularly it is F flow of star is a pointed pre-shift morphism on smooth. So, let me go one step. Next, namely, the nearest aim is to define these two simplicial pre-shifts, which I, about them, I use them to formulate here in the beginning of my lecture. So, this is my nearest aim is to define these two simplicial sets. For that, let sin flow of star, I will just pronounce in star, with the category of finite pointed sets and pointed morphism and pointed maps. For a scheme X, and define it non-pointed set A, I will write X under A for the scheme like this, which is the co-product of several many copies of X indexed by elements in A. I would stress that categories in star, smash product in the unit object one plus, and frame note of page, the Cartesian product cross, and the object point, the unit object point are the symmetric monoidal categories. And there is a fully faithful embedding in taking a pointed set K to the scheme of this form, where star is the distinguished point of K. So, taking the set K, the scheme spec K under K capital minus the distinguished point. And taking a morphism phi, I call this morphism, respects the distinguished point to the morphism of this form. So, I didn't do this in the file between spec K under K capital and spec K under K capital. Maybe I should say here, is everything okay? Yes, it's okay. Okay, I continue. Thank you. Yeah, so I should comment a little bit. So, what I need to define is a morphism enough phi, which belongs to morphism in frame note of K, between enough K and enough A. But this is equal to pointed morphism set between themes in of K plus and in of A plus. And I specifically written down that the left hand side here is exactly the in K plus. And the right hand side here is exactly the in A plus. So, what I need to do and so we have also this plus gives me this equality and this plus gives me this equality. So, what I need to take to specify a pointed morphism between this scheme and this scheme. So, what I need to do is to take this morphism identity number phi as the specification. It is a pointed morphism. So, this way, I defined this way, I defined fully faithful embedding of the category in star into the category frame note of K. And I would stress that this font in is strictly monoidal. The categories of simpatic objects, frame delta of simpatic objects, and in categories in star and frame note of K, asymmetrical monoidal in the standard way as well. So, the font in is a fully faithful embedding between these two categories, which are the categories of simpatic objects in star in frame star and in frame note of K. The left of the font is strictly monoidal too. Now just couple notation. Let X dot be in delta op frame note of K and let K dot be in delta op of in star, a pointed simpaticial set in this category. Write X dot tensor K dot for the object X dot cross in of K dot. I recall that in of K dot is already an object in this category. And in this category, the cross product is monoidal. So explicitly X dot tensor K dot, and we like to take its n simpatic, it will be X dot tensor K dot. So, continuing, it will be X dot cross, sorry, n, and this is a pointed set, this is a pointed simpatic. So this will be, no, this is just a simpatic. And I also will write K dot for the object in of K dot in delta op frame note of K. Particularly for the simpaticial circle S1, which is a simpaticial object in the category in star. In the same X, which is small, we have objects, X and the first one, and this one in the category delta op of frame note of K. This is very good. So, since the pointed set frame of stable frame correspondences between Y and Y prime, is a covariant, is covariant factorial with respect to Y prime with respect to morphine in frame note of K. So, we get pointed simpaticial sets frame Y comma X tensor S1 and frame Y comma S1. Due to this covariant functoriality in the second variable, functoriality in frame note of K, and our objects X tensor S1 and S1, they are simpaticial objects in frame note of K. So, these two objects are simpaticial sets. So, replacing Y with the standard posse-impecial scheme delta dot K, we get pointed by simpaticial sets, frame delta dot comma X tensor S1 and frame delta dot comma S1, respectively. Eventually, applying the diagonal, we get the simpaticial sets, we get the simpaticial set, that we denote also frame delta dot X tensor S1 and frame delta dot S1. So, all these simpaticial sets are well defined eventually and frames 11, 12 and 13 are stated properly now. I would say that this is the end of first part of my first lecture, and I have about 15 minutes, little bit more for the second part of my lecture. So, the second part of my lecture has subtitle, namely, the material function of the sigal theorem, and the meaning of this simpaticial set of this simpaticial shape, which is indeed a simpaticial shape. So, just a second, consider now the pointed material spaces P1, pointed by infinity, and the standard object, the pointed material space T, which is A1, a1, minus naught, regarded as the same shape. For a pointed material space M, you will write like this for the end T suspension of M, and you will write omega m P1 naive of M for the inner form from the end smash code of P1 with itself to M. So, there is a remarkable VW dilemma, which is a very key for this topic, which I decided to formulate in this form, which is a little bit unusual, but it is technologically equivalent to the original VW dilemma. Namely, this magic shape introduced on smooth varieties of a, introduce a bar, frame n bar comma x, just inside, economically isomorphic to the omega m P1 naive of sigma n T of x plus. Similarly, the shape of frame corresponding to this. Like this. Canonically coincide with omega infinity naive of sigma infinity naive of x plus. So, on the right-hand side, we could take the collimit and so on the right-hand side, the right-hand side is defined as the naive code limit of omega infinity and sigma n. So, due to this limba, it is not surprising that the motivic spaces of this form, which are the spaces of this form due to this limba plays a fundamental role in the stable motivic rheumatopi theory. Right below, I will make these stages a bit more precise. But before I would like to stress one principle. Namely, the phanta x goes to him or delta dot bar comma s, makes all the stable motivic rheumatopi theory local. This principle will be specified in the second lecture. This is quite the same as the phanta x takes to go to form delta dot bar comma x. So, this is the motif of x. Makes the viva theory of motives local. So, in certain sense, this simple shell shift in certain sense, not in a very precise sense is a substitute for m of x. At least this construction is very close to this one. Let us consider now the following picture. Take on the left hand side. Take on the left hand side points of motivic spaces and on the right hand side take p1 spectra. Not rheumatopi category, but just the category of p1 spectra and the category of point of motivic spaces. The right two funtons, one is the infinite suspension part of the phanta, with respect to p1 and another one is the naive omega infinity p1 loop phanta. They are two, we joined each to the other. This is the left one, this phanta is the right one. These two funtons induces the corresponding direct funtons, a1 direct funtons. Between the pointed unstable motivic category h of k and the stable motivic category sh of k. The left hand side phanta is still the infinite suspension phanta and the right hand side phanta is the a1 direct p1 loop phanta. So, shortly I prefer to write on the infinity p1 for this direct phanta. One of the major tasks of the stable motivic or motivic theory is to compute the motivic space like this, where x is the smooth variety. A similar task in top logic has been solved by the single machinery. The motivic version of the single theory. Let's be an infinite perfect field. Then there is a canonical morphism between the simple shift frame delta dot cross bar comma x times rs1. Omega infinity p1 sigma infinity t of x plus rs1. Such a canonical morphism exists due to the fact that on the left hand side we take the naive omega loops and on the right hand side we take a1 direct p1 loops. So, it's naturally to expect that there is such an error and this error exists and it states that this canonical error is a local theorem. Particularly, let k capital over k be a field extension not necessarily finite then this morphism we evaluate on the field k capital. And on the left hand side and you'll get a weak equivalence of official sets on the left hand side. We have the same official set frame delta dot k capital comma x times rs1 and on the right hand side we have. Omega infinity p1 sigma infinity t of x plus rs1 evaluated on k. So, this is a motivic space. So, this is a shift you can evaluate it on the field k capital and get a simple set. So, this error is a weak equivalence of a simple set. This theorem has a very nice and strong theory. Namely, if you take k to be the complex numbers and we take x to be the point. Then the homotopy group, homotopy groups of this simple set which is frame delta dot comma s1 coincide with the stable homotopy group of the classical circle s1. Let me derive this category from the theorem. So, firstly, this simple set is weakly equivalent to this simple set by because this error is a weak equivalence of simple sets. So, this is the true equality. The second equality caused by the very definition of the infinite loop p1 loop phanta a1 homotopy loop phanta or by the very definition of a1 homotopy group of this vector. So, what is on the right-hand side? On the right-hand side are the following group. We take the suspension, the t suspension spectrum of s1. We take the a1 homotopy shape of weight zero, weight zero. This is a shift and this shift we evaluate on the complex numbers. So, this equality holds up to by the very definition as I told of this function. And the last equality is a very deep theorem due to my theory. It says that a1 weight zero, a1 homotopy shape of say s1, particularly of s1, evaluated on complex numbers coincide with the corresponding stable homotopy group in topology. So, and eventually I would stress that this corollary has a stronger form, namely, that the space, the simplicial space frame delta dot s1 is weakly accurate to the topological space, omega s1 infinity, sigma s1 infinity of the usual topological circle. So, a kind of conclusion is this, that the usual topological space omega infinity, sigma infinity of s1 top is expressed as this simplicial set, which is defined in terms of algebraic varieties only. And also, since I have a couple more minutes, I will take back to my thesis, this one. Saying that this construction plays a central role in the stable motivic rheumatography theory, and it plays a central role due to the fact that up to some extent it makes all the stable motivic rheumatography theory local. And in the second lecture, this principle will be specified, but let us say clarified. So, we will make this principle quite precise. Thank you very much. This is the end of my first lecture. Thank you for now. Thank you for a very nice lecture. Maybe I can post the first question. Okay, that's the second ideal. Yeah, so, yes. Let's, let's go to theory, theorem 110. 110. Okay. Yeah, yeah. Okay, go. Yes, I should put here, as I already have done, I should put here smash as well. Okay. Can you get the maps in that theorem? I would like to say, only very few people earlier, and then a bit roughly. So, as I told, this is on the left hand side, we have a delta dot bar. And here is omega infinity naïve of sigma infinity. Okay, so from the infinity naïve, sigma infinity naïve, we have pretty obvious map to here. Just a transformation of omega naïve to omega infinity. Yeah, it gives an up. So, is it okay? Yes, thank you. Let's see, are there any other questions? Use a mic, sorry. Can you explain a little bit why you need k infinity in theorem 110? Yeah, in theorem 110. So, this theory of, I would say, this theory of frame motives, which is behind of the theorem, is written down in published papers for infinite field. And for field which is finite, there is, there are some reference due to the Georgianian and Paul, help me please. You're the form of PhD student. Jonas. Yes. And due to Jonas. Yeah, but which is preprint steel, preprint steel in the preprint form. I mean, surely one can replace k by, I mean, eliminate in and say, take any difference. It doesn't seem to be any other questions. One else. I think it means that your lecture was accepted. Thank you.