 Thank you very much. So first, I'm very happy to talk these special occasions. And when I was asked to give this talk, I feel very honored and I'm very happy about it. But I think this is an occasion for the Maxim Konsevich. So I feel I need to talk something about the kind of dream or something very much promising in the future. But unfortunately, for years recently, somehow we are more concentrated on something like writing down the foundations in very much detail rather than pushing the applications ahead. And it seems that it is not so much as Maxim's style of mathematics. But since we spend so much time to write up these details, so I feel that there is some excuse to write that kind of thick and heavy things. And these virtual techniques are first invented by several people, including myself and my collaborators like Ono Ota, or to study the foundation of group of fitness theory and the flare homology like a pseudo-phromoic curve. But then I feel that the technique should be somehow related to some people to try to use the quantum field theory physics. So it really can be used to study some quantum field theory, so it's a good excuse to write this thick volume. So that's something I dream. So in the application side, what I am going to talk, it does not go so much beyond what I talked in several other occasions. But I want to explain more about these foundation things. And in this talk, what a fundamental class appears for some simple modular space? Modular space of constant maps. And this is somewhat better for people who want to understand what is virtual techniques. Because if you start learning virtual fundamental classes, usually it is about pseudo-phromoic curve and you need to start learning nonlinear PDE and a lot of functional analysis are complicated. And if you learn a lot of functional analysis, then this story starts. But fortunately, what I am going to talk does not use any nonlinear PDE or functional analysis. It's just about finite mental topology. So there are some people who like this kind of virtual things, but do not like so much nonlinear PDE. So for those people maybe to explain this particular case might make sense to understand what this virtual technique is. Okay, so since the quantum field theory is in my title, I just say a few words about it and I'm not at all a specialist. So when you learn quantum field theory, we learn that kind of there are a lot of integrals like a five-man integrals, not yet a pass integrals, but still five-man integrals can be infinite in many cases. And what this is to is just to somehow regularize to this five-man integrals and to get some finite number. Usually they, for example, they cut at some energies. So many times this infinity arises from diagonal. So you cut energies and this infinity is a bit better. And one important point is to kind of regularize as much symmetry as possible you keep. And that's something similar problem we meet. And then we have this kind of system of numbers depending on some parameter, say, epsilon. And when epsilon goes to zero, you get kind of something which you want to really calculate. But in certain cases, especially what I heard from physicists that in the case of topological field theory or some other cases, there is something called a supersymmetry. And this supersymmetry implies that even if you kind of first kind of perturb it a bit, but this final result is independent and when epsilon goes to zero. So even if this diverges epsilon goes to zero, you can still get some finite things. That I think is one of the role of supersymmetry is useful. So I want to say some very simple example, but an interesting example. It's about addressing an index theorem. So it starts from this elliptic operators. And what you want to calculate is a difference of dimensions. Dimensions of this, so you want to eat a vector bundle. So this is the infinity measure space of sessions and the infinity measure of sessions. This difference, and of course, this is the infinite. And then what many people do is just to take this trace of e to the power minus epsilon p star p. When epsilon is zero, this trace is just this. And when epsilon is zero, this is just this. So you can just take a difference. And of course, when epsilon is positive and p is elliptic, this difference is finite. And what this supersymmetry means, in this case, that this difference is actually independent on an epsilon. So you can go to epsilon equals zero, then you get something well defined and that, of course, index of elliptic operator. And that I think is this particular, some example of this kind of story. But this is a linear story. So something nonlinear version of this kind of thing should be, I believe, is topological field theory. So I want to talk something similar about these different things. So this virtual technique started to, for example, main thing is to try to define Gromov-Witton invariant. And many invariant topological field theory is some kind of intersection theory. So intersection theory, you can see the following things. You have two cycles on M. And we assume that this is plus. I'm sorry. The plus of the degree is the dimensions. So in a lucky case, you take these intersections and to calculate the orders. And this one should be an intersection number. Everything works fine. But, of course, this works only in case they intersect transversally. And if they are not transversals, then you take this, so this is the infinite number. So you can just perturb them so that you get the finite number, then it is defined. But sometimes I cannot simply do this. That's it. And in a reverse geometry and differential geometry, there are some differences between the intersection theory. So in a reverse geometry, intersection of cycles is always defined. And if the intersection is somehow finite order, the edge is always positive. And there's that kind of negative intersection occurs only in case some component coincides. But in a differential geometry, there are something different. What's different is that in algebraic world, there is no manifold as a boundary. So complex manifold has a code dimension 2 strata only. So this is why this intersection number is always very defined. But in a differential geometry world, intersection number may not be very defined because the cycle, not cycle, but the kind of manifold, may have a boundary or a corner. That's one reason. And also, even in the case this intersection is finite order, but you may still have cancellations, the sign plus and minus. And in a reverse geometry, mostly the sign is always plus. But then somehow, how to handle this sign and how to handle this non-well-definedness of the intersection number, these two points is actually some kind of main thing. We need to work out when we do this virtual techniques. So write up the sign correctly and how to handle this non-well-definedness of the intersections. And in a kind of topological field theory, the situation I feel is rather closer to a differential geometry than an algebraic geometry because one of the important features of some topological field theory, especially that's the case of a flare homology, is you try to define various numbers or various quantities, but they are, in general, not well-defined. It means that if you put up what you kind of modify the problem so that you can count something, the number you get itself is not well-defined. And something you get well-defined is only you define everything as a four, and everything is well-defined only up to homotopy. So that happens very much repeatedly in recent years in this business, in this geometry or in some related fields. So this is the reason that this kind of differential geometric style of this intersectional theory is kind of related to these problems. OK, so that's what I'm going to say. So what I want to say about this topological field theory is we need to define various quantities which are not well-defined, and we define something only up to homotopy equivalence. But then what we do is just to kind of have some bad intersection theory. So we want to put up so that we get well-defined numbers. But then the number depends only on epsilon. It's a parameter you put up. And the number itself may depend on epsilon, but then there is some supersymmetry. And then this is independent on epsilon up to some kind of homotopy equivalence. So how to handle these kind of situations in general is a kind of theme of this virtual fundamental chain techniques. But rather than going to the directory, I want to explain one particular example of making system of intersections of various chains consistently to obtain a system of numbers. So the main issue is that each number are not well-defined individually. So what you need to be careful is not study each kind of one problem to get numbers individually. You need to study infinitely many problems at the same time so that everything you do is consistent. And what I mean by consistent, how to make it consistent, is a kind of a main headache for these virtual techniques. And that's kind of, yeah, so yeah. OK, so this is what these virtual techniques are doing. Yeah. And I will explain the case of constant maps. And that is something, and also the case of these maps from bordered Riemann surface. So Riemann surface is boundary. And I said that you don't need PDE, but you kind of finite dimensional topology. But still you have some systems of non-transbuff intersections which are interrelated to each other. So I just want to start explaining which kind of intersection problem you meet. And then I want to explain some kind of general machinery how to handle this kind of system of intersection problems so that you get a well-defined structure. So the case I want to explain is this Chan Simons. But I want to start a very simple case, simple things. That is wage product. So let m be manifold. And we consider triple power of m, m cubed. Then you have projections to first and second factor. It's m squared. And the third factor, this is m. So this is square. This makes things a bit non-linear. In a sense like linear partial differential equations, you have m squared, m and m. But they have m squared, m and m. So just linear things. But this is a multi-linear. And the thing we use is this small diagonal, m itself, x is x, x, x of this m cubed. And this kind of diagram is a kind of correspondence. And this correspondence gives you is wage product. Suppose you have two differential forms. So lambda m today is always a differential form. You have smooth differential forms, a pair of smooth differential forms on m. So you get h1, h2. Then you can just take exterior product. You get the differential form of m squared. Then you pull back by this map. And you take these distributions. It's a Poincare-Dewart diagonal. So this is m is the n-dimensional. This is the 3n-dimensions. And this diagonal has n-dimensions. So this distribution has a degree 2m. So you can wage these distributions. And you take integration of the fibers. And it is easy to see that what you get is this usual wage product. So this just means that you take these particular distributions. It gives some kind of Schwarz-Carnel of the wage product. So this is a, and this product is verified. There is no infinity here yet. But for the purpose of later use, I want to make these distributions t-diagonal a bit smooth to regularize these distributions. And the data we need to do is the following things. So this diagonal is somehow, this distribution is not supported on an open set. It's just on a closed set. That's why it is not smooth and it's distributions. So we take this open tubular neighborhood of the diagonal. And you have these projections of the tubular neighborhood. And you take a normal bundle of this diagonal named cube. And you pull back on this u. So you have this u and e. And you have a natural canonical section s. It's just a total logical section. Because u is an open subset of this normal bundle itself. And the property is that the zero set of this total logical section is just diagonal itself. So in place of starting diagonal, we consider this triple. u is a tubular neighborhood of the diagonal. And e is a kind of normal bundle. And s is a total logical section. So this triple is a kind of a simple, very simple version of the Krani structure. And actually, in this talk, the Krani structure appears rather in this simple way. Sometimes in general, we have only locally this kind of representation, presentations. You have a pair of spaces, bundles, and sections. And you represent your space as a zero set of the sections. And in general, you need to do it locally. But in this simple story of constant maps, you can do it globally. So you have just this kind of triples. And the reason that this is better is that this map from u to m3 is some margins. mq is some margin, because this opens it. So there we are to see that this picture has a diagram like this. So you have this disk with three marked points. And you have a map u. And the partial differential equation in this case is very simple. It's not even a differential equation. The condition is just u is constant maps. So you consider the modular space of constant maps from the border to the limit surface with three marked points. And border to the limit surface with three marked points is, of course, no moduli. And this u constant map is parameterized by m itself. So the modular space is very simple that m itself. And that is exactly this diagonal. And you have this separation map. And this separation map, of course, is this x go to x, x, x. So that's a simple case. So now I want to go to the next case. So you can increase the marked points. But in place of this increasing marked point, you put one loop case. And the simplest case is this guy. So you have this disk with three marked points. But you put the conditions that the first and second marked point goes to the same point. Of course, this is rather ridiculous, because your equation is u is constant. So this is always satisfied. But let us consider this as a kind of conditions. Then the modular space, so your modular space is this singular annular, so annular with one node. And you have a pair of this modular space. And a map, u to m, that is the constant maps. Of course, the modular space, so then you can look at this one as a foreign fiber product. So you have a diagonal. Diagonal is the modular space of this disk with three marked points. But then you equate that this z1, z2 go to the same point. This means that you take another m square and you take a fiber product. And the first factor goes to x, x, x, x. And the second factor goes x, y, and x, x, y. This means that you equate the first and second for the same place. So you take this simple fiber product. And of course, this simple fiber product said theoretically is again equal to m. It's just a constant map. And the source has no modular. But what's wrong about it is that this fiber product is not a good fiber product. Its dimension is not correct. I mean, this dimension should be zero dimension now. But it has m itself, because the intersection is not transversal. So you can kind of reflate it in a foreign way. So this is diagonal as a current is equal to this x, y, z, m. So you have delta functions and dx, dy, dz. These are two n forms. But then this m square is just to equate x is equal to y. So you have delta x minus y dx. So you want to take intersections and you want to count the number. The natural thing is to take a weight of these two current and integrate. But of course, this is not very defined because x minus y delta function of x minus y square is not very defined. So you try to do this by some kind of a ways, then you get infinity. And how to kind of remove this infinity is something related to this intersection theory. And if you think a bit about the intersection number of this delta m and m square, you will see that this is the Euler number. So somehow you want to regularize these integrations to get the Euler number. So this is the kind of first part of these things. And so now, in place of this taking this diagonal itself, you take this open neighborhood. So you take this u, the tubular neighborhood, of your diagonal. And you take E, that is an obstruction bundle. So you take this triple in place of this taking m itself. So what's nice is that this u, if you take this u, it's a neighborhood of the diagonals. Then you can take this fiber product. It is just a very defined fiber product. So this is, I think, n dimensional, no, no, no. 2n dimensional manifold. And E is rank 2n bundles. So you have a 2n dimensional manifold and rank 2n bundles and s. And so this manifold is 2n dimensional. And this is rank 2n. But s is something singular, something non-transverse. So this 0 set is again this m itself. So you can cook up this fiber product naturally, and E also. But s is not transversal to 0. So that's the reason your space is long. But still, what's nice is that you can take this fiber product. So by increasing somehow your space like this, you can always make your fiber product very fine. And you can make this picture a bit smaller. So this s is not transversal to 0. So then you can just reduce this. So the co-carnal of this s is actually tangent bundle. And then since this is the finite rank, so in place of this taking 2n dimensional manifold, that is something like a neighborhood of m. You just take m itself. So you replace this triple, this guy and E and s, by a bit smaller one. So you get m tangent bundle 0. And this is our triple. And what's wrong is that this s is not 0. So it's not transversal. But you can turn this 0 something like a non-genetic sections. And you get Euler number. So that's the story. OK, so now this is in this particular things. But then this thing is a boundary of some other modular spaces. So you can just take this general anulari. So in place of taking this modular space, just one point, you take modular space of anulari. And the constant map again. So now we consider this modular space m1, 1, 0. So this 1 means that I'm sorry, this is 0. I'm sorry, this is 0. Anulus has genus 0. Sorry, this is wrong. So you have a genus 0, Riemann surface. And you have this 1, 0. You have a two boundary component. And one of them has just one marked point. And the other has no marked point. So that's this. So this is m0, 1, 0. And you have a parameter u, that is m. So you have this modular space. And so this modular space is actually you can calculate very easily. So this is 0 again, I say. So modular space of anulari with one marked point is exactly interval. So you have this interval. And this modular space is interval times m. And this is a very nice, smooth modular space. But you are not supposed to regard this is a transversal modular space. And the equation that u is constant is actually degenerate equations. And in this case, you have an abstraction bundle that is h1 of your annulus. And h1 of your annulus is rank 1. And you have this kind of tangent bundle coefficient h1. That is our e. So you have this space itself, but you have this abstraction bundle. So the triple you have is this interval times m and tangent bundle and 0 sections. So this is actually this triple, which you get in this case. And what you can observe is that you go to one of the boundaries. So you have a boundary of this modular space of annulari with marked points as two boundary points. And one of the boundary point is just you get here. And the other boundary is something here. So this is the case when this inner circle shrinks to a point. And this boundary appears actually in a talk by Costello's and also Muhammad Abzai. So this appears in a similar way, but anyway. So you have two boundary of this modular space. But then on this first boundary, what's good about this is if you consider this boundary, then the pictures, you have this modular space of annulari is this interval times m and tangent bundle. And if you consider this, so you get by taking fiber product, you obtain this similar modular space. And this is exactly coincides much with this. So the story is something like this. You have a modular space of genus 0, two boundary marked points, and one boundary marked points. And then you put using H1, you put abstraction bundles. And you go to the boundary of this modular space. You have the same modular space as this fiber product, which came from genus 0 with one boundary component. So what I want to claim is that this is a completely general picture. So you can take, I'm sorry. Yeah, the second boundary, there is a way to handle this second boundary, which I think is Costello mentioned a bit. In a particular case, you can just assume that the Euler class of m is 0. Then this 0 sections have a chain which bounds it. And you can eliminate this boundary using this chains. But that's not so much things I want to talk today. So now we can completely generalize these pictures in an arbitrary lemur surface of arbitrary genus and arbitrary boundary components and arbitrary number of marked points. So we consider this mg k1, k2, blah, blah, blah, blah, kL. So that's a modular space of bordered lemur surface. And this lemur surface is supposed to have a genus g. And we assume that it has an L boundary component. And on each boundary component, you have a k1, k2, blah, blah, blah, kL marked point on S1, which respect this cyclic ordering. So we consider this modular space. And this is actually a manifold with corners. So you get manifold with corners like this. And you consider this plus constant maps. So the modular space you get is this one. So I think this is not a manifold, but it's an manifold. But you have a kind of at least smooth manifold, and smooth manifold, and you have abstraction bundles. And the section is 0. So you have this triple, so that you have one manifold, one manifold, and one vector bundle on it. And the section is happened to be 0 in this case. So you have a system of this triple, or many z, k1, blah, blah, blah, kL. And what we know is that, yeah. But then I want to say something about this, because this one is a good manifold. But you consider a variation map. Then if the variation map just goes to diagonals, so it's not a sub-mergence. The problem is, if this is not a sub-mergence, you cannot take fiber product. I mean, as I say that in the case of this, going to 0 to 0, no, 0 loop to 1 loop, you have to kind of equate two points intersect. And you do not extend this modular space of three marked points disk. This equation is not good equations. So to make it transverse, you need to make this a variation map to be a sub-mergence. That you can do easily by just replacing this m to a kind of tubular neighborhood of this small diagonal in a big, big product. And you can just take normal bound and pull back. So you can just extend this a bit. And this is 0 is not correct. This is the tautological sessions. I'm sorry. This 0 is not correct. This first factor, the map is 0. But second factor, you take this tautological map so that the 0 set is again the same thing. So you have this a bit bigger modular space. You can just use this and this. And they are somehow equivalent as a kind of structures. So what we get is that you have this system of modular spaces together with this a variation map to the m to the power k. k is the sum of k i. And what's nice of this picture is you go to the boundary of this modular space. It means that you have a kind of border limit surface. You have some singularity. And then this singular things, in the case of this annular, can be written as a fiber product of something that you get in an earlier stage over m. So this is the general picture you have. You have a kind of each alpha a is some partial order set. In a special case, a is a combination of the genus and k1, blah, blah, blah, kL. You can order it by something like a lexicotic order. And then on this partial order set, you obtain this triple. You alpha is an orbital width corner. E alpha is a vector bundle in orbital sense on this. And S alpha is a section. And this zero set is your modular space. So you have these sections of this triple, or various alpha. And you have alpha from your alpha to this n to the power n alpha. And this can be assumed to be some margins. And what's nice is that you take the boundary of this, then it is a union of various fiber product of something you get in an earlier stage. And its fiber product is taken by using a part of these evaluation maps. So this is in the particular case of this system of the pictures, in the case of constant maps. So this is, what's nice is that it is rather easy to cook up these constructions. You have this modular space of Riemann surface. And you have just m. And you can just take this neighborhood of diagonals. And if you want to do this in a more general setting of kind of, in the case when you have a pseudo-hormoic cup, then you can cook up this kind of pictures using nonlinear PDE and a lot of functional analysis. But here, you're just finite-dimensional topology. So you can just get this. So the only thing you need is to understand what these generations of border Riemann surface occurs. So that's also, but anyway, so this is a kind of quite general situation. And this occurs, I think, in many stories, especially this occurs in pseudo-hormoic cup. But in this case, pseudo-hormoic cup is very simple, just a constant map. So then what is virtual chain technique is following. You start from this kind of systems. You have this system of all-before, system of vector bundles, system of sections. And these triple are related to each other as a boundary by fiber product. Then the virtual technique says that you can kind of systematically part up all these modular spaces so that it is kind of consistent to this fiber product distribution of the boundary. And so that is what this virtual technique do. And once you can do it, you get some system of numbers. And this system of numbers is actually a kind of Chan-Simon theory of your manifold. And I say that this system of numbers you get actually depends on the choice of perturbations. However, you can cook up some cobaltism argument so that if you make these perturbations consistently for all alpha, then what you finally obtain is well-defined up to certain homotopy equivalence. And what I want to emphasize is that this is a kind of complete general theorem. So you have these kind of general situations. Then you can always part up everything in a consistent way so that you get something well-defined after homotopy. So this is a kind of general procedure. And this is what this virtual technique are doing. And yeah, actually, so the virtual techniques, I think there are several group of people, including myself, ourself, to write down this in details. And fortunately, when we write down the details, that totality is rather thick volume. And I feel that it seems impossible to write down in complete detail less than 500 pages. But what's good about this story is even it is heavy, but it works always. So I mean, it means that you don't need any kind of special assumptions. So whenever you have these kind of things, it always works. So the good news is that something like this kind of things you can use without understanding. Because if you need delicate conditions, then you need to understand why this works. And you have to be very careful before you use it. But my propaganda about this virtual technique is that this is free. So it works whenever this situation appears. So you don't even need to read this kind of complicated things, and you can just use it. Because it works always. So what I want to do in the rest of my time is to state this theory of the virtual precisely so that you can use it. And something we are doing for this year is somehow to write down the statement which you can use without knowing the proof. Because I am pretty sure that not so many people want to read the proof. Because the proof is something like similar to Fatih Ram in a third or fourth year undergraduate course of manifold theory. And there are some people like Dominic Joyce. And I think Dominic Joyce wanted to write the story as close to the theory of schemes and stacks. But I am rather differential geometers than algebraic geometers. So I want to make it as close as to theory of manifolds. So it's at that advantage. I think smooth manifold is easier than schemes. And what I want to explain in the rest of the time is very much similar to Fatih Ram in undergraduate course of manifold theory. So I want to explain Fatih's integrals. And Fatih's a bit more precise. So you start from this station. So I want to consider individual space. You have a triple, u, e, s. So let me remind you that u is an orbital fold. e is a vector boundary in orbital sense. And s is a section. And maybe only assumption is S in bus 0 is compact. It may be very singular, but your assume is compact. u may not be compact. You have a kind of open neighborhood. And something you need some reliability. And also you have f from u to m. m is a manifold. And this map you assume to be some margins. So that's the situations you are with. Then the theorem is you have a sheaf on u, which you can associate to these things, the sheaf offset. And it has a following properties. So I want to explain which properties this sheaf has. And this sheaf is a kind of a sheaf which parameterizes the way you put up your modular space. So you have some sheaf which parameterizes the perturbations. So what is the first and first property of the sheaf? The first property of the sheaf is the following. So you have this triple. And suppose h is a differential form of compact support in some open set omega of u. And suppose you have sections of your sheaf on omega. Then you can define push out or integration along the fibers of these differential forms. You get a somehow differential form of m. But it depends on epsilon. That is positive. So this means that something I want to do is just to push out these differential forms on this space to m. Of course, you have a differential form on u and m with some margins. And compact support smooth form by some margin you can always push out, integration along the fibers. But the space itself is not u, but it's a kind of 0 set of s. And 0 set of s is extremely wild. So you cannot push out your differential forms on this wild set. But you have some parameter space which regularizes your 0 set. And if you fix this element of this parameter space and fix epsilon, you can push out the differential forms. And in a typical situation, I mean, it's easier situation if this push out is a following form. Suppose that this s, this s is a transversal to 0. Then it seems that 0 is a smooth manifold, compact smooth manifold. But f may not be some margins. But then you can still push out the differential forms. But what you get is a current. And the property of this push out is that if you go to epsilon, go to 0, then this push out is a current which is a push out of h. So you can write it in a following way. If you have any test functions, smooth forms on m, you pull back on these forms and you integrate over this 0 set of s. Since this is transversal to 0, this makes sense. This integration is equal to this push out and wedge sigma and take limit epsilon go to 0. So that's the property. So something we want to do is this is a smooth form which regularize the push out of this differential form on s in bus 0. That's something we do. But of course, in general, 0 set of s is not a smooth sum manifold. It is a singular space. And you cannot expect that this limit converge as a distribution. In the case, s in bus 0 is wired. So it can be a very wired thing. And I don't want to study epsilon 0 itself. But it's very close to 0. You get some number. And this number may depend on epsilon. This number may depend on these perturbations. But you still get something. So that is what the sieves do. But then there are several other properties of the sieve. The next property is this sieve is a soft. The soft sieve means that you have any closed set of an omega, then there is a kind of restriction of the sections. This is always subjective. So it means, so far, this is good. So because this property, you can use to prove the existence of the sections. I mean, what you want to do is you want to construct this parameter of this section of this sieve systematically. But then this is a soft sieve. It means that you define something over there. You can always extend to get something you want. So this is a soft sieve. Moreover, you can define partial unity for this sieve. And this sieve, unfortunately, is not a sieve of groups. It is a sieve of set. But it's kind of a fine type of thing. So you can define partial unity. Still partial functional unity. And you can define these extensions. So this is the main thing we can use to prove the existence of this kind of perturbations. And so since you are working on these integrations, the main theorem should be Stokes theorem. So it's something like we learned at the end of the course of manifold. It's Stokes theorem. So what does Stokes theorem mean? That suppose you have this triple, u, e, s. And suppose this u, all before, has a boundary. It may have a corner, but it may have a boundary and corners. So you restrict this h to this boundary of u. And you restrict this. So there is a notion of this. You have a restriction map of this section of sieves to the boundary. You restrict these sections to the boundary. And that's just the same. So you use this data to push out. But this is a kind of d of this push out and the push out of this d of h. So maybe minus. The sign you need to work out. So this means that if you go to the boundaries, this is related to this. This is actually a Stokes theorem. And to make it clear, I want to explain one special example. So suppose that m is just a point. So in this case, push out, you get a number. So it's a usual integration. So in place of writing this, I just want to write integrations. So you can integrate h of this space u with this perturbation data. This is a number depending on epsilon. And this integration satisfies this usual Stokes theorem. If you integrate the boundary, it's equal to this integration of d of h. So this is the Stokes theorem. Now, the other thing I want to explain is something like a Fubini theorem or a composition formula. So what is Fubini theorem? Fubini theorem is something related to how these integrations behave by product. And as you see, the story I want to cook up is you have a system of spaces so that one boundary of one space is described by five products of the other spaces. And something what this virtual thing, virtual technique do is you start from this geometric situation and you rewrite all these geometric relations systematically to algebraic relations. So this means that the equations we have on our spaces is something like this. Boundary of u alpha is a kind of sum of a fiber product. So I want to rewrite this equation something like a d of m equal to m. So this is a kind of algebraic relations among these operators. And this is geometry. So to go to this from this is the kind of main thing of this virtual things, chain-level virtual technique. But then you see that here you have this boundary. So you can kind of translate this boundary of your space to this operator d by using Stokes theorem. So this part you have a Stokes theorem. But on the other hand, the other part is a fiber product and you want to relate to the compositions. So that is a role of this Houdini or this push-out Houdini theorem. So I want to state this Houdini theorem a bit more precise. So we are in the following situations. Suppose you have a ui, i is 1, 2, ui, ei, si, i is 1 and 2. And you have a map f i to m. So you have this map. Then for these systems, you can take fiber product. And I say that this f1 and f2 are sub-mergeants. You can just take this fiber product. The space u is just a fiber product. And you can pull back these sections of abstraction bundles by this map. So you get ues, you have this u. And also, your shift has a kind of a compatible of this fiber product operations. So suppose you have this s1, u1. There's a shift here. And there's 2. u2 is a shift here. So you pull back this shift by this pi1, pi2. And you have this shift here, this is related to the structures. And you have these natural operations. So the shift operation means that you have a series of maps which are compatible with the restrictions. So you get this, you have two perturbation data here. Then you get this product perturbation data. So that's always possible. And what Houdini theorem says are following things. Suppose you have this h1 differential form on u1. And h2 differential form on u2. Then you do the following two things. And also, you have this s1 and s2 of u1 and u2. So first thing is that you push out differential forms by this using this data. You get a differential form of m. And you push out using this s2, and you get differential form on m also. You take the wage product. This is smooth differential form on this finite differential manifold. And you integrate. This is the second formula. And the first formula, you have a differential form here. You pull back here. And you have another differential form. You pull back here. And you take wage. And then you use this product perturbation data. And you integrate by this product perturbation data. So the Houdini theorem says that this is just equal. And if you think of a very simple case like u1 and u2 is 0, then this is exactly what you can prove by Houdini theorem itself. So you have these Houdini theorems. Now, this is Houdini. And I want to say that this is related to composition formula, something like a composition of smooth correspondence that I want to explain. So in a usual situation, you have this u. And you have f2. But not only f2, you have always f1. So you have this triple. And f2 should be a submargin. But f1 may not be a submargin. So this one is something like a C infinity version of the correspondence. So we use this correspondence diagrams. I want to script u. Then once we have this correspondence diagrams, you can associate a map from differential form of f1 to differential form of f2. It is very easy to do so. You have an h here. You pull back by f1. This is free. Pullback of differential form is very easy. You can pull back differential form here on the u. But then you prepare this data, which you integrate this on this u. Then you can push out using this. Pull back and push down, you get this correspondence. So once you are given this correspondence diagram together with this data script s, then you get this continuous map of correspondence. So that is something for this virtual things give in general. Then you can compose the correspondence in a following way. You have this correspondence from m1 to m2. And you have a correspondence. I'm sorry, this is 3. You have a correspondence from m2 to m3. So you take five products of this side. This one is submersion. So you can just take five products here. Then you can use this five product. And then you can compose this, you get this map. And you compose this, you get this map. This is the competition of the correspondence. Of course, it's usual. And then you have a competition. And also, since this one is on a submersion, you can just use this product of this perturbation data. And you get this perturbation data here. It is a competition. And what HUBINI theorem says in this case is that you take correspondence of this map associated to this correspondence. It's just a competition of this and this. You can just go to this one. That's what this HUBINI says. So HUBINI theorem immediately says that this process associating linear map of differential forms to smooth correspondence is compatible with the process to compose the maps. It's a completely factorial. So this is this HUBINI theorem. So you have these stokes of the HUBINI. And you can use these existence theorems. But then I want to claim another last property of these things. So you have this, again, triple UES and script U. Script U is this triple. And U, in general, has a corner. And supports that you have a boundary and a corner. So this is the kind of picture. So then this boundary can be decomposed into a union of finitely many components. And each two components intersects around the corners. There might be a higher dimensional corner, but the kind of co-dimension of two corners intersects. And this final property is a problem. Suppose you have this data which describes perturbations on this boundary one, boundary two, boundary three individually, this, and supports that they coincide on this co-dimension of two corners. Then you can extend this system to the whole systems. And let me remind you that this sheaf is actually a soft sheaf. So if you extend this system to a neighborhood of the boundary, they actually are done. Because it is an open set. So you have a kind of closed set, and it extends it to its open neighborhood. Then it extends to everywhere. But this boundary itself is not an open set. So you need some property to extend this system to a small neighborhood of this system with boundaries. But you can always do it. This is what this theorem says. So I believe that these claims are kind of simple thing to understand. I don't want to explain what is this sheaf itself, because you don't need to understand what it is. It is not so much complicated. But you're going to let it to make it complicated. Yeah, yeah, yeah, yeah, yeah, yeah, of course, yes. Yeah, yeah, yeah, yeah. So not only soft, but also jam is non-trivial, yes. That's important, yes, you're right. So you know that all this property is then kind of what I claimed about this. You can do this perturbation in complete generality. It just follows from all the property that I just explained. So this is the situations you start with. You have a system of these triples. Some obfault vector bundles and sections. And what you know is that this is parameterized by partial order set. And you go to the boundary, it is a private product of something you get in an earlier stage. So now, something you want to do is just to associate this S of alpha for each of this U alpha so that it is consistent with this identity, with this kind of identities. And you can do it just without knowing any proof, just all the things I claimed. You start from these inductions. So your alpha is parameterized by some partial order set A. So you start something smallest. There it doesn't have a boundary, and Maxim says that this is no empty. No empty at one point, this softness implies that global thing is no empty. But then you go to this induction hypothesis. So you want to construct, you have this construct S of alpha for any beta, which is strictly smaller than alpha. Then you want to look this diagrams, and you take this boundary here, they have this S of beta, one S of beta, two everywhere here. And actually, I didn't say that somehow this kind of stratifications are designed so that these descriptions are automatically compatible in co-dimension two corners. So then the assumptions that everything which is each boundary component very defined and co-dimension two corner consistent extend to the interior is satisfied. So you get this system of this data of perturbations. So that's the kind of free you can use those properties I craved. So now you can use this data to integrate differential forms, which you have on M. You can just pull back the differential forms to integrate. So you get the kind of system of numbers for each alpha, or kind of system of linear maps between the differential forms on each alpha. So you get this huge family of operations. And what's good is that, as I said here, you have this equality between boundary of one modular space by other modular space via fiber product. But then you can use the Stokes theorem so that this one gives just D of these operations. And this composition formula means that the second one is just a composition of these two morphisms. So you get these equalities you want immediately once you have this cook up the systems. And the typical example is if you just restrict to this tenus zero and one boundary point, then you get infinity structures immediately from these constructions. But that's what you need analysis because, yeah, you should have a curve. But in this particular case of this constant maps, you have this system of constant maps. So you get this system of some operations of differential forms on M. And that basically is the Chan-Simon theory. And I think this satisfies certain locality properties that I can use because every evaluation map lies in the neighborhood of the diagonal. And you read this Costello's book about this part of the Chan-Simon. And he's constructed, you kind of prove some kind of uniqueness. You have some system of these numbers so that it's supported in the other diagonals. Then it is unique. And it should be something coincided with what you did. So this implies that what we produce in this way exactly coincides to this part of the Chan-Simon. And also, so this is this part of the Chan-Simon in some particular case. But you can say that here we don't use any functional analysis. You have this just M itself, constant maps. But this story, you can immediately generalize. You know a bit about the functional analysis and the pseudohormic curve analysis. And you can generalize to any diagonal sum manifold in symplectic manifold. So you can cook up this modular space. So this is a kind of reading on the time of the modular space. It is a constant map. You can include non-constant maps. And you have these similar systems. And you can do the same process to get the whole kind of quantum version of part of the Chan-Simon theory. And as I said, to that generalities, we need a bit more. Because here we have these descriptions of each modular space by just single, or before, and single vector bundle and single sessions. And if you want to do this functional analysis, you can obtain similar pictures, but only locally. So you consider this modular space of pseudohormic curves. You have similar descriptions by this triple. But it works only a neighborhood of each point of your modular space. So a neighborhood of your modular space, you can construct this kind of triple. That is this, I think, goes back to Kulanesh's work on deformation of complex structures 50 years ago. And that's Kulanesh pictures. And something which is a bit more recent is we can glue. There is a notion of coordinate change of this kind of local pictures. And so what we can do is just to have a kind of system of this kind of neighborhood at each point of the modular space so that they are compatible. Then we can write down, we can work out the stories I just explained in a simple case in a complete general setting on this Kulanesh structures. And you get system of this perturbation data. And you get system of operations among differential forms, which satisfy this property which is the Stokes theorem and this Fubini type theorem. So in that way, you can do this should form a curve business. And I believe that in that way, kind of almost most of the modular spaces, you can use the same techniques. And the output should be that you can usually kind of sync the modular spaces. And you forget any mathematical details. You can kind of frankly sync what is the boundary of this and what it should be looks like. And you write down this kind of formulas. And that formula should be rigorous rather immediately. So you don't need to work out any artificial or ad hoc methods. You can do always that kind of things. So that kind of thing I wanted to work on, we are now writing one more details. And so I don't know how much we can use this to quantum field theory, because this is a chance I'm on is very special. But something like this, do you have this kind of integrated diameter and to regularize it and to make it completely consistent with the kind of symmetry? You see a lot in a textbook of quantum field theory. So I don't know if you can do this, for example, in a Yan-Mills theory or something like that. But if you can do this, then you may try to use this kind of techniques for quantum field theory. I don't know, but it's a kind of dream. I stop here. Thank you very much. Thank you. Thank you. Maybe I missed something in the beginning. Sure. AkaZ. Pardon? What you are saying is not three dimensions, akaZ. Yeah, so in that case, you have a kind of operations between different forms. So in a usual 3D-measure chance I'm on, you assume something like a cohomology is zero. So in this way, you have a kind of operation between cohomologies. And you also get the family version, so that makes it non-terrestrial. So when you say it's chance, what is the group? Groups, so here we are just using different forms. But you can include this vector band. Yeah, yeah, yeah, these are trivial, but you can include groups on them. Surfaces, yeah. Yeah, yeah. Another question? So you have two unions of fiber products. Yeah? So are those unions like plumbing? I mean, you have this fiber product or two, and they are glued in a kind of triple fiber product. But it's like plumbing, that you actually plumb vector bundles. No, no, it's just a direct sum. Fiber, and you? Triple fiber, no, this is just mindful of these corners. Yeah, but then you get corners, so the union is double-forty-eighted by being. Yeah, yeah, yeah, yeah, yeah, yeah. So there is something like a kind of, you know, this set itself, boundary set, it's not mindful of these corners. Because you can just kind of pull back, push back. So it's something, yeah, yeah. It's something, yeah, normal, yeah, yeah, yeah, yeah. Here. If you have the solution, the master equation, you produce some kind of coaching complex in secular homology, right here. Yeah, yeah, yeah, yeah, yeah, exactly. Yeah, can you do this locally in the manifold? Please, like a factorization over there. Locally on manifold means? Locally on the three manifolds. Yeah, probably it's local, yeah, because your perturbation just works on this, on the diagonals. But you have epsilon, yeah, epsilon, maybe needs some kind of. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Can you describe extra magic support? Yes, yes, yes, yes, yeah, yeah, that's also what this later, we are doing, so-called stillness appears. If you fix the epsilon's, you can just do this on a finite many, many stage. So you have to do this kind of inductive limit things. Yeah, yeah, yeah. And so the strategy is that you put the structures up to some level, and then you fix epsilon's, and then you take this homotopain inductive limit. That I think is similar to what you appeared in your books. And then we do it all for 950 things. So that's always what we need. Actually, I have no question, but some kind of tangential remark, there was a lot of these questions. But in the case we don't have corners, suppose, in a complex situation, we have two complex cycles intersect, not transverse, it should be zero dimension, yeah? And then it's maybe, in fact, it seems to be dimensionless, which dimension is zero. And then physics should give something, depending on small parameters as well. And then I think you just apply heat current with time epsilon, very smooth. And that was made conjecture that you get smooth volume four, which will be concentrated in the six. Make conjectures that will have limit as a distribution of the intersection. Yeah, I think in a physics situation, yeah. It's kind of fructose equation in the code telegrammetry, for example. Yeah, in algebraic geometric type situations, or complex geometric type situations, you can still expect that the connection goes to zero. This kind of thing converges at least at the distributions. But I think in different geometric situations, no. It's it's wider. So is there actually an axiom that describes what happens when you change epsilon? Yeah, some behavior, it's epsilon, yeah. I mean, you can use this statement itself. You have epsilon one, epsilon two. These two things are kind of covalent. Satisfying all these systems. So this covalent gives you homotopy equivalence. So you have this corner extension. And in the corner extension statement, there was an equality at the double intersection. So should I be worried that that was an equality instead of just an isomorphism? Yeah, actually, in this case, we have a kind of global pictures. We can make it exact equality. But in an actual current structure case, it is better to say it's isomorphism. And in that case, it's not co-dimensioned two corners. We have to deep, arbitrary higher corners. We wrote down this property precisely, but it's longer. What was the reason for having your punctures on the boundary rather than in the interior of ribbon surfaces? Oh, I mean, you can include the interior boundary, interior also. You can just analyze the story. But the boundary is, I mean, we want to cook up these operations on a differential form, on M itself. And if you put the interior, it's not so much natural, because if you imagine the kind of a serial markup, then the boundary lies on the Lagrangians. So you want to construct operations on a differential form on Lagrangians. On the interior, you are supposed to put differential forms on an ambient impact manifold. In this case, cotangent bundle. So you want to plug in something on M. M should be zero sections. M should be Lagrangian sum and full. It's unnatural to put it for the boundary marks points. Because non-constant map case, it's a kind of. But of course, you can include the interior. In that case, the story gives some kind of two spaces, Lagrangian and ambient impact manifold, evaporation is the kind of open-closed story. So what if your three-poor system is some kind of group acting on it, such that section is like equivalent? Yeah, I think we can do everything in an equivalent way. You do it in an equivalent homology. I didn't write it up, but I mean, that's something interesting, because in a case we have a kind of group action on a manifold, we can try to do this in an equivalent versions. And this part of the process, I explained, is not so much difficult to generalize. What is more difficult is to cook up that kind of structures. But that I think is one can do it.