 Thank you. Thank you very much for the invitation. I should warn you that this talk is happening on ground zero. Nothing higher, I'm afraid. So I will talk about cohas associated to curves. And maybe it's worthwhile giving a very short introduction to cohas, although we've seen them several times already. So some heuristics about cohas was that you start with a, let's say, abelian category. Usually it's a small homological dimension. So for instance, coherent sheaves on some manifold M or also some interesting such categories can be built out of a quiver and maybe some potential. And suppose that this abelian category is nice enough that there is some modular stack of objects in A. So this could be very singular. But because this is the modular stack of objects in some abelian category, there is a, let's say, God-given correspondence between two copies of MA and itself, and MA itself, which goes through the stack of short exact sequences. So what is this stack? This is the stack parameterizing short exact sequences like this. And what are these two maps? Well, one of the map assigns the extreme points of the short exact sequence. And the other map assigns the middle. So maybe you get a better intuition if you think about what the fibers look like. So if I take an object here, then the fiber is the grass mining of sub-objects. And if I take a pair of objects here, then the fiber is somehow all extensions of these two objects. OK, and out of this, you can modulate some technical difficulties that you have to be careful about in all cases. Define a structure of an algebra on the cohomology of this modular stack. So this is some kind of associative algebra. So yes, I stress that you have to work to define it, because these stacks are usually singular. These maps are not regular at all. And also here, you have a lot of choices for what cohomology theory you want. You could choose a singular cohomology. You could choose Bohr-El-Moh-Homology, K-theory, mixed-hot structure, a lot of things. So well, there are many applications of cohomology. But I think the two main family of applications are to a numerative geometry. So this is the theory of Donaldson-Thomason variance, what Konsevich and Zoibelman relate to the algebra of BPS states, and so on. So this is really in the work of Konsevich and Zoibelman. The BPS was recently established. Toda or Toda? No, no, no, no, it was kind of a definition of more for some of the mathematics and things. It's a natural algebra, it's physics. It's original. OK. And then the other type of applications are to, say, geometric representation theory. So I think one good example is the cohas that were hidden, which can actually be hidden in a Hapchak's talk, and related to this AGT business. And also, there are some relation to quantum groups. So Yangian, say, Yangian actions on Nakajima-Kuiwa varieties. OK, so in this talk, I will really have the second type of application in mind. And so I want to consider the following type of settings. So here, I will consider two types of cohomological whole algebra associated to curve. So x is my smooth, projective curve complex. And then I have the 1D coha, which would be just take for a the category of coherent sheaves on x. So this is not Calabi-Yau. I forgot to say that nicest examples happen when the category is Calabi-Yau. So this case is a priori not Calabi-Yau. And so this, I will talk about this, is a joint work with Eric Vasrault. And so it's a priori not related to some quantum groups because it's not Calabi-Yau. So somehow, relation to quantum groups happens only for Calabi-Yau categories. And the 2D picture is when you take for a the category of coherent sheaves supported on a curve, but the curve is inside some surface. And here, typically, I will consider the Calabi-Yau setup. So category of coherent sheaves on t star of x supported on x. And this is the same thing as category of Higgs or nilpotent Higgs bundles or Higgs sheaves on x. So inside of here, of course, we have the category of coherent sheaves on x, but whose support is 0 dimensional. And this corresponds to Higgs torsion sheaves. And so this gives a subalgebra, which, in geometric representation, corresponds to modifying, say, a vector bundle on the surface at a point, just at a point. If I consider higher rank sheaves, then this would correspond to modifying a vector bundle or torsion-free sheave on the surface by a one-dimensional torsion sheave. So this kind of modification, if I understand, well appeared in the Rapture's talk also. So this is the real motivation to study this kind of things. And then recently, Kaplanoff and Vasro also considered coherent sheaves on an arbitrary surface, so somehow including two-dimensional, I mean, coherent sheaves with full support on the surface. But it's OK to define it. I think it's really harder to understand this in this context. OK, and this is joint work with Francesco Salah. And it's also work of Minietz, especially in this case. OK, so most of my talk will be about this one-dimensional case, because I think it's related to a lot that's not related to quantum groups, but related to some interesting enumerative geometry on curves. But let me start by saying a little bit about the second case, which, in the end, should be the most interesting one. But unfortunately, at the moment, it's still quite mysterious. So let me just say about 2D coha. And again, this will be heuristics. So the definition is worked out by Salah and myself. But let me try to tell you what one expects. We don't know how to prove it yet, but there is some kind of heuristic to tell you what this kind of coha-mogical whole algebra should be as quantum groups. So let's take the following setup. Let's say that A is an abelian category. Suppose that it's homological dimension 2 calabiow. And let us assume, so it has a modulized stack of object ma. And let us assume that there is another category, which is homological dimension 1, which is not calabiow, but somehow ma is the cotangent bundle to mb. So typically, mb will be smooth, because this is homological dimension 1. And ma can be obtained by some symplectic quotient of mb. Cotangent bundle, cotangent bundle. OK. If mb is smooth, then means that ma is also smooth. No. But if you reckon it's cotangent bundle. Yes. This is a stack. Oh, it's a modular stack. Yes, yes, yes. Everything is a stack. At the very end, it will be a modular space of stable objects, but OK. So we consider another type of whole algebra. So here, I said that we consider the cohomology of the modulized stacks. But instead of considering the cohomology, one could, for instance, take fq points of these modulized stacks and take functions. And then we can define exactly the same thing. So let me denote this by constructible and over fq of mb. Everybody, do you want to consider the cohomology with the coefficients in constant sheaf or in sum? Yeah, constant sheaf. Yeah, but it's in terms of the three-calabiotes coefficients in the foundation cycles. It's equivalent to three-calabiotes. But for two-calabiotes, it's all multiplied by line. And then it's reduced to n. So then we consider this whole algebra. And in many examples, say, quivers, curves, this turns out to be the positive half of some quantum group. Let's say gb and gb, let's suppose that it is something like cat's moody algebra. And this cat's moody algebra, the character of this cat's moody algebra, is encoded by the so-called cat's polynomials of the category b. So character of gb is given by cat's polynomial. Let's say ab. So this is a polynomial in one variable. It is positive integral coefficient. And so this is, I mean, roughly, this is a number of indecomposable objects in b. Depends on the class in the k0 and so on. OK, then in this situation, you can guess what the cohomological whole algebra of ma will be. So of course, this is heuristic. So the coha, so let's say, so h star of ma should be isomorphic to the yangon of not quite gb, but some extension of gb, where gb certainly contains gb, but this is some extension such that the character of gb, so this is a graded, a z graded extension. And the character of gb, g tilde b, sorry, is given by the full cat's polynomial. So here I say this is given by ab of 0, the constant term of the cat's polynomial. And this is z graded character is given by the full cat's polynomial. So this is the heuristic that allows you to guess what these cohomological whole algebras will be. So I'll just run this for the case of coherent sheaves on a curve and see what this suggests. The prediction of what you can say the Camology or the Borelmore Camology? You can do both. I guess it's better here for this to consider Borelmore homology. Which one? I mean, only one of them will be an algebra problem. No, you can. So it's true that when you have a singular stack, it's easier to work with Borelmore homology. In the case of the smooth stack, actually I prefer the cohomology, but it doesn't matter. OK, so first of all, examples. Let's say the baby examples is supposed B is a representation of a quiver Q. Then A is representation of the pre-projective algebra of the quiver. I think this is more or less what occurred in G's talk. And then the constructible whole algebra, spherical is the B. This is the positive half of the smoothie algebra associated to Q. In a general set up, do you want to put positive part of the Yang-Gen? Yes, thank you. So this whole algebra is the positive half of a quantum group associated to quiver Q. And the Yang-Gen and the cohomological whole algebra of, let me just write, rep pi Q. Then this is the Yang-Gen of the extension GQ, G tilde Q. And this guy here is what acts on the cohomology of Nakajima quiver varieties. And at least conjecturally, this is the same thing as this Molikov-Kunkov Li algebra. So you should be careful that it is strictly bigger than GQ, some extension of GQ. You are right. Here I'm working equivalently. But yeah, so in this case of quivers, there's a natural C star action, because this is a cotangent bundle to something. So you have the C star action dilation along the fiber. And then this gives you this equivalent parameter. So I mean the definition parameter for the pre-projected algebra, if you state the form pre-projected. Ah, not deformed, no. There's no statement if you study deformed or what? Yeah, I think you can and you should do that. But then you should ask Ben Davison. This is more or less how Molikov-Kunkov define their Yang-Gen by considering pre-projected algebra, deformed pre-projected algebra. And then take a specialization to 0, something like that. OK, but let's see what happens for curves. So first of all, this case of dimension 0. This 0, torsion sheaves on the surface, essentially. So let's suppose that we start with B, this coherent sheaves support that 0 on some curve x. Then A is coherent sheaves support at 0 on the surface. It's a point on the smooth surface. And then here, the constructable whole algebra is the Heisenberg algebra. This is the classical whole algebra. So this is the Heisenberg algebra, which we can view as the positive half of gl1 hat. And then this cohomological whole algebra, this is the Yang-Gen of gl1 hat. So this is the positive half. So this is this affine Yang-Gen that appears in the AGT business. And now let's look at some real examples of corresponding to modifications along a curve inside a surface. So suppose that B is coherent sheaves on P1. So A will be Higgs, or if you Higgs sheaves on P1. And here, the constructable whole algebra is known by work of Kaphanov to be the positive half of SL2 hat. Actually, gl2 hat, if you work a little bit, if you add a little center to this. And so this gives you that the cohomological whole algebra of this case. I'm sorry, what's Q here? What? Q, because this is constructable over FQ. So this is not cohomological. This one is the constructable whole algebra. And this one is the cohomological whole algebra of the cotangent stack. So I am claiming that, essentially, you pass from one to the other by, first of all, extending the algebra to a z-graded algebra, and then taking the affinization of that. For Q, you can take the whole. You can do both, but if you work equivalently, they are the same, more or less, up to some grading shift. And so here, MA, you expect that it is going to be the Yangon of gl2 hat. So something like quantum toroidal, I prefer to say elliptic, because this takes into account the center algebra of SL2. Wait a second. The repliculist of Kevan and Kronecker puros. It makes sense, yes. OK, and then, for instance, just maybe one more example. So this should work in all cases. So if B is coherent sheaves on some elliptic curve, then the constructible whole algebra of B is this so-called elliptic whole algebra. And so you expect that the homological whole algebra of MA, in this case, will be something like the Yangon of the elliptic whole algebra. The elliptic whole algebra is some deformation. Let's say uqt of gl1 toroidal, in some sense. And so here, it's really crazy, it's the Yangon of gl1 toroidal. So it is something like positive half everywhere of gl1 triply graded. So I have no idea if this kind of thing appears somewhere, except in my imagination, but. Do you have definition of the right-hand side? I would say this would be a good definition of this side. No, I mean, do you have definition of this Yangon for gl1 hat hat? No, this is heuristic section. Yes. No, no. Yeah, but it looks like symmetric curve with some number of variables. And variables should be like functions of three variables. Yes. And then there are some three variables. Laurent polynomial, yes. And higher genus. So in higher genus, the whole algebra is kind of identified. It can relate it to some shuffle algebras. You can, so cat's polynomials exist in the context of any curve. And the funny thing is that if you take the cat's polynomial of the genus G curve, and you take the constant term, then you get the evaluation of the cat's polynomial of the SG quiver, quiver with one vertex and G loops at one. So somehow it means that if you take the top part, so if you take H top of MA for a curve of genus G, then what you should get is the Yangon associated to the SG quiver. This is the top part. OK. So I think I abused enough of your gullibility. And so I will move to dimension one case where we actually can make some more precise computations. OK. So D equals 1, so one-dimensional, Koho. So sorry, what does one-dimensional mean if it's not coming out? I mean, if you have stupid zero potential. So I will redefine it. But yes, I agree that it really looks stupid. I will try to argue that in the end it isn't. OK, so let me denote by Koho Rd. So X is fixed, so smooth, projective curve of genus G. So Koho Rd is the stack of coherent sheaves of rank R and degree D. This contains, as an open sub-stack, the stack of bundles. And what is the induction diagram in this case? So I take Koho alpha times Koho beta. Here, let me denote by tilde alpha beta Koho alpha plus beta. And here, this is the set of all short exact sequences, F, so F beta G H alpha, where F beta, say this. So F beta is a coherent sheave of class beta. H alpha is a coherent sheave of class alpha. So the fiber here is a quote scheme. And the fiber here, so everything is smooth here, I should say. Because we are in the one-dimensional situation, all these stacks are smooth. And this is a linear stack. And this is a proper morphism. The fibers are quote schemes. So here, there is really no difficulty at all in defining, let me denote by HX, the direct sum for all rd of the chronology of Koho rd, constant coefficient, has an algebra structure. Maybe I just say this is q, this is p. So m alpha beta is, so it goes from H star alpha. Let me denote this by H star rd, h star beta to h star alpha plus beta. It takes two classes. And you just pull back and push forward. And this defines for you an associative algebra structure. OK, so no slope on code tilde? No slope? If you don't require that the short exact sequence is some stability? No, no, no stability, no semi-stability here. No, no, no, I want everything. OK, so let me remind you the structure of the homology of Koho alpha. And let me first remind you what is the homology of bun, d. This is a classical work of Atia and Bot. So let me just write alpha is rd. So you pick the tautological vector bundle. So tautological vector bundle on bun alpha cross x. So the restriction to x at the point here is precisely the corresponding vector bundle. And then you can take the churn classes of this tautological vector bundle. And using the q-nest decomposition, you can write it as a direct sum. Let's say ci gamma e alpha tensor gamma, maybe gamma dual, where gamma runs in a basis, symplectic basis of h star of x. All right, so this gives you a bunch of classes on bun alpha. Just take the tautological class, tautological vector bundle, and take the churn classes, and decompose all the q-nest components. And the theorem of Atia Bot tells you that these classes freely generate the homology ring of bun. So then the homology ring of bun rd is freely generated by this ci gamma, where i runs from 1 to r, and gamma is in your basis. OK, there's just one subtlety. c1 of the c11 is equal to the degree. So this is not a scalar. But otherwise, this is a free super commutative algebra. So this is the Atia Bot theorem. And this was extended by Heinlert to the case of Koch. And here it's really amazingly simple. So you still have your tautological bundle, but now it's a coherent sheaf on Koch alpha times x. And if you look at the homology of this stack, so let's assume that r is at least equal to 1, then the homology of this stack is freely generated by all churn classes. So of course, if you take a churn class of cr plus 1, for instance, then this will vanish over the open sub-stack of Koch, which consists of vector bundles. But if you look at all of Koch, then you need all of these classes to generate and freely. So this relation c11 equal to constant. Also c11 equal to constant. Same constant. And then if r is equal to 0, then you can show that the homology of Koch 0d, this is the d-th symmetric product of the homology of Koch 01. And so this is the d-th symmetric product of h star of x bracket z. So this bracket here comes from the fact that I consider the stack of all degree 1 coherent sheaves. And so there's a BGM here. OK, and in order to make this a little bit more symmetric, let me just make a following observation. These ci gamma, since you have infinitely many of them and they generate a free algebra, they look like elementary symmetric functions in infinitely many variables. So I want to write this. So this is better. This ring is isomorphic to s infinity of h star of x bracket z. And actually, there is a canonical way to identify them. OK, so the picture. Is it a degree plus 2 here? Z is degree plus 2. So here is the picture for this algebra. So it is graded in this left half. I mean, it has components z to graded by r and d. Components are in this left-right half. And so here we have c. Here we have s1. Here we have s2, s3, and so on. So I'm skipping so this, yes. And everywhere else we have s infinity. What's the what? What is s infinity? S infinity, yes. It's not such a stupid question. So some typical elements in s infinity. So some typical elements in s infinity are some of, say, gamma i zi power l. So you should think of symmetric functions in infinitely many variables. But these variables are colored by homology class in x. And so when I multiply them, I have to also use the multiplication in x, in the homology ring of x. Sorry, but this is just a sum. So this is what I mean by s infinity. It's like restricted symmetric. It's like restricted, I don't know. It's the algebra of symmetric functions. It's an algebra. It's an algebra, yes. Just like the usual algebra of symmetric polynomials. What you wrote, should be symmetric in what? Symmetric. Is that? Symmetric here is this. So gamma, sorry. This is gamma. OK, maybe you prefer it like this. It's the same gamma everywhere. So it's gamma z positioned on the i-th component, gamma zl. Now it's positioned on the i-th component. Does this look better? No? So isn't it the same as you can see this? So it's defined for any vector space, right? Yes. So isn't it the same as you take this vector space. You can see there are Taylor series with coefficients of this vector space. And then the symmetric algebra of that. Isn't it the same as what you wrote? So you can look at the projective limit of sd h star x bracket z, where you fix the, so this is the projective limit in every given degree, co-mological degree. So is it important what you have? So you have s and f into something. So do you use anything about this something? Actually, no. It's always isomorphic to a freely algebra generated by these guys. You're running s and f into just a vector space. No, you should meet some type of plane and communicate with some fixed vector, I suppose. Is it useful? This is what I don't understand. I mean, are you using some structure on what you have inside the brackets? Yes, so if I write, so if I multiply two guys like this, so let's call this guy p gamma l. So gamma is a class in h star of x, and l is some positive integer. This is like the power sum function but colored by the class gamma. So if I multiply p gamma l by p gamma prime l prime, I will get sum for i, say different from j, gamma zl i times gamma prime zl prime j plus the sum for all i, gamma gamma prime zl plus l prime. Still, this makes no sense to you? Well, OK, go on. I mean, I can't understand this. I mean, it would be nice if you could just define each other in a linear algebra without using this z or anything. I mean, can you just say, is it defined for any vector space or for a vector space with some additional structure? It's a vector of many states. This is a, I mean, let's forget about the curve on the homology. So what kind of vector space can you talk about this infinity? So I would take a graded vector space with finite dimensional weight spaces, which is also an algebra. Graded algebra, positively graded algebra with finite dimensional graded piece. Commutative. You need an algebra structure on the vector space. Here I multiply. This is the cut product here. So the order of symmetric functions would be a corresponding case of x is a point? x is a point, yes. Makes sense. l is not net. l is positive, yes. Here. Branch, branch, cross, k. Positive, positive, positive. l has to be greater than 0, or gamma has to be non-zero. OK. So now the multiplication. What does the multiplication look like? So now we want to define something like m alpha beta, which will go from s, say, s infinity, tensor s infinity to s infinity. Of course, this will depend on alpha and beta. Suppose here that the rank of alpha and the rank of beta are all greater than 1. So that would be the goal is to get a precise understanding of these multiplications. And you also have some multiplication sd, tensor s infinity, and s infinity, tensor sd. And then you also have some sd, tensor sd prime. OK, so the ideal goal would be to get a real understanding of all these maps. So if you think a little bit about it, what these maps encode is intersection theory on the quote scheme. Because I'm just going to pull back. I'm going to take a pair of tautological classes here and here. I'm going to pull them back. And then I'm going to integrate along this map, which means precisely that I'm going to compute some kind of intersection pairing or integral of product of such tautological classes on the quote schemes. So let me just, just for motivation, give you some examples of information that all these maps contain. So I will be a little bit brief because I'm running out of time. But let's just take x equals p1. And let's just consider some multiplication like this, hr1 0. And let's assume that I'm going to consider some classes here. Let's assume that the degree, some of the degree of the classes that I consider, is equal to minus the degree of mr. So in some sense, I'm integrating a top class. Then essentially just by definition, c1 times the cr is just the integral over a flag variety of the corresponding classes cs. Yes, if you interpret this, you just look at the restriction to the open sub-stack of cr0 of p1, which is a trivial bundle. And then you see that everything here restricts to just the trivial bundles. And so you get this by restriction. So this is something like integration on the quote scheme of the trivial bundle. And this is the quote scheme of degrees 0 subsheves. So this is just the grass mania. So this is just to say that it contains, in particular, Schubert calculus. OK, so this was really very specific. I just take p1, and I take multiplication along a line. Now if I take still p1, but I take some more interesting products. So if I take still x equals p1, but now I take some e greater than 0, and I consider the multiplication like this, say r e. So I take h r1 e tensor h r2 minus e goes to h r0. So now I'm just I'm not looking for subsheves of the trivial vector bundle of degrees 0, but subsheves of the trivial vector bundle of some degree that could be negative. So then, so again, suppose that the degree of c1 plus the degree of c2 is minus the degree of the multiplication. Then c1 times c2, this algebra, is integral over something which we can denote by more e, p1 grass mania, of some corresponding classes. And so this thing here is the set of subsheves of the trivial vector bundle on p1 of possibly negative degree. And so this is the quote, let me write it like this, quote r r2 minus e. And this contains the set of maps from p1 to the grass manian, such that the degree of e is e. So there's some compactification of this thing. And so this is the, by definition, this is this quantum comology. So if you just take p1 and take product of two elements, then you get quantum comology of the grass mania. So this is quantum Schubert calculus. If it looks like you do, it's called quasi-Mexical interpretation, so it's like something like that. Yeah, something like that. So now if x is arbitrary, so I will just say a few words, so there are two types of interesting intersection theory on quote schemes. So there is intersection theory on the trivial vector bundle. So in general, this is not smooth, so you need some virtual class integration and so on. But this gives you essentially what is at least in some paper called Witten TQFT. So it behaves like some TQFT on the comology of the grass manian. So I still take similar setup, but now I take the curve of higher genus. And I look at intersection theory of quote scheme of the trivial bundle. So in order to do this properly, in order to integrate on this quote scheme, you have to use virtual classes and so on. But this is what this Witten TQFT does. But then there's another theory, which is somehow nicer, is when you take the quote scheme of a general stable bundle, and now this guy is smooth, and this gives what is called the weighted Witten TQFT. So here, again, on the grass manian. So here the reference is a recent paper of Thomas Gola. So it still gives you a TQFT. And it contains a bunch of interesting numbers, like Verlinden numbers. On comology of the grass manian. What is the word on? It means it's a Frobenius algebra structure. OK, there are other motivations to study this algebra. But I think I would definitely run out of time. So I'll just say very, very briefly. So there is also a relation to, say, geometric languages. So you want to understand the X-algebra of a certain category of Eisenstein sheaves on, say, Koch X. And this X-algebra acts on H, or, well, no. It acts on a big product of powers of H. But OK, so I just wrote this. If you want to know more, you can ask me. And then this X-algebra hopefully looks like a Kovanov-Lauder-Hucke algebra for the category of coherent sheaves on X. So there's hope that this can be used to construct some categorification of quantum groups, for instance, of the elliptic whole algebra. OK, so I need to tell you something about the structure of this cohomological whole algebra. So essentially there's some kind of presentation for this algebra, structure of HX. All right, so first of all, you can always multiply any class by a tautological class. So I want to introduce some kind of universal tautological ring, which I denote by H like this. And this is just a free ring generated by some class of C i gamma. Let me maybe underline them. So it's just a free polynomial, super commutative polynomial algebra. And this acts, so it's a commutative HUBF algebra, where I just use the usual rule for taking the co-product of a churn class. So delta of C i is the sum of C k tensor C l, k plus l equals i. And here C l is the sum over all gamma. OK, so this formula defines for me co-commutative HUBF algebra structure on this polynomial ring. So I mean, it does to the churn classes what you expect when you have a filtration of your vector bundle, for instance. And what I say is that HX is a H-module algebra in the following sense. So H acts freely, actually, on each H alpha by multiplying by the corresponding tautological class. This is an abstract ring, but if I fix alpha, then each of these can become, I can evaluate these, if you want, on the tautological coherent chief. And this is compatible with multiplication. So p applied to u1 tensor u2 is p i1 u1 applied to p i2 u2. So usual compatibility between the co-product. And this comes from the projection formula. So this is now nothing difficult or mysterious. And so now let me describe the subalgebra, which corresponds to the zero dimensional sheave. So if you want the vertical subalgebra, what if it were a quantum group, you would call the carton subalgebra. So let me denote by H0 is the direct sum of the homology of Koch 0L. So this, yes, so here's the proposition. This algebra, this is the easy part. This can be totally explicitly determined. So H0 is a shuffle algebra. So remember that this is the direct sum for all L of SL. And the multiplication goes like this. I have p of z1, zL. So these guys are colored by co-mology classes. And I multiply this by q of z1, zn. Then this is the sum over all subsets of 1n plus L, such that the cardinality of i is equal to L of, so the kernel of this shuffle multiplication. So this is somehow the important ingredient is like this. So i is an i and j is not an i. It is 1 plus omega ij zi minus zj. And here it is p, where I apply to the variables corresponding to i and q to the variables corresponding to not i. OK, so here omega, this is in S2. And this is the class of the diagonal. OK, so this really looks like a Yang-R matrix, something like a Yang-R matrix. And indeed, indeed, there exists. But I'm not going to write it, because I'm really running out of time. But h0 has a presentation as a Yang-Yen, so with Yang-Yen relations. So it's generated by this p1 gamma, sorry, no, p gamma L for gamma in pi and L greater or equal to 0, satisfying some relations. OK, so this is somehow the rank 0 part. So this would be the cohomological Hecchi algebra. And now I have to tell you how it acts on a higher rank part. So I need to take this was the Hecchi algebra, and now I need to define the Hecchi operators. So the Hecchi operators. So let me take pz1 zd. So I need to define a map from sd, tensor s infinity to s infinity, and this is hrd. So this map depends on the rank of the bundle to which I'm applying it. And so p of z1 zd applied to q of an infinite family of variables. This is the sum for all i and n. So this is a shuffle multiplication where I insert d elements inside infinitely many elements, countably many elements. So this makes good sense. And again, I have the same type of kernel, p of zi, q of z different from i. And there's a similar formula in the other direction. Should I worry about the convergence of the product now? No, you should worry about the possible poles. And it's true, I'm cheating a little bit here. But you should not worry about convergence of the product. Because here I mean this restricted product. So you only take finitely many of these guys at a time. But can we find the number of j's? I have an infinite number of j's. And I have finitely many i's. But I only take finitely many pairs. I take this term in only finitely many pairs ij. So omega is nilpotent. And omega is nilpotent, actually, yes. But no, no, no, omega is not nilpotent. It's a class of diagonal. It is a class of the diagonal, yes. So it's omega is sum of gamma, tensor gamma dual. No, yes, omega is nilpotent. But for instance, you can have omega 1, 2, omega 1, 3, omega 1, 4. This is never 0. OK, so there's a similar formula, S infinity, tensor SD to S infinity. And the final formula, I have to stop after that. OK, this is somehow, I would say, the easy part. Easy part, because you're multiplying rank 0 by rank 1. Now you want to understand the relation between, if you think about COHAs, two modifications along we have dimension 1. So here we don't have a general formula, but somehow we have enough. Enough to give a presentation of the algebra. So let me just say higher rank products. So for any R and D, I have this, 1, 1, D. This is the one element in S infinity. And you should think that this is rank 1 in degree D bundles. And I multiply by 1 R, RD plus 1 minus G. If I stayed like this, this would be a degree 0 element. But I also need to understand what happens when I take higher rank, higher degree classes. And this is equal to R plus 1 power G times the sum i is equal to n of the product for i and i of D plus 2 1 minus G omega i minus Zi. So this is some explicit guy. This is some Euler form. And you might think that this is the important part of the data, but actually this is the hard thing. So this is, for this you need to use some theory. So this is the number of points of a finite quote scheme. And if you do this for more general ranks and degree, here you would get some Verlinde number. So this is a very simple Verlinde number. But if you were to write more complicated products, not rank 1, rank R, then here you'd get some really complicated expression. This is due to Hola. I mean, the value of these numbers is due to Hola. OK, and finally the theorem, and I will stop here. So theorem is that H is isomorphic to the H algebra, so H-module algebra, generated by all these 1RDs, R and D, modulo relations above. So all these three types of relations, the structure of the cohomological Hecke algebra, the Hecke operators, cohomological Hecke operators, and just this multiplication, right? So this is like a, yeah, this multiplication. And this is enough to, in principle, determine all these intersection numbers. Now I'm not claiming that it is easy, but definitely the information is contained in here. And I don't have time to talk about the application to computation of the cohomology ring of stable vector bundles. I'm going to have a modular space of stable vector bundles. So I will stop here. Can you incorporate a stabino somehow into this computation? Is there anything? Is it, I didn't get something interesting with you. I mean, you and Eric had this work when we had ellipsic curve and then somehow. Ah, OK. OK, yes. So everything here depends on this gamma. So I would say the promidius is kind of seen. So for me, in the complex setup, the promidius means that I need to work not in the category of vector spaces, but in the category of GSP2G modules, in the monogonal category of GSP2G modules. This would be. Is there anything more to just say forward? Yeah, it's not very clear in my mind. But so, I mean, this promidius is all these cohomology classes, the gamma. Question about your early examples when you had the Youngian of G1 to hat versus the Youngian of G1 double hat. Yes. Did you use the same cohomology theory to get those? Yes. I mean, this was heuristic. I don't know how to prove it. But yes. You didn't use elliptic cohomology to get to the Youngian of G1? No, no, no elliptic cohomology here. So if you use k theory, then you should get not the Youngian, but the quantum affine algebras. If you use elliptic cohomology, then you have to look at the work of Yaping Young and Gu Fang Zhao. And here, they say it's something like the Felder quantum groups. But I don't know this well. It's just strange that from P1, you got a GL2. But from elliptic, you already got a GL1 hat. Not strange. Because P1 is this, elliptic curve is this. And so this is the diagram, dinking diagram of SL2. This is the dinking diagram of GL1 hat. This is the affine dinking diagram. And for a genus G curve, then you should have to look at this guy. And here, somehow all the interesting information is hidden in this extension. Because this is just a free, if you look at the Katsmudi algebra associated with this, this is just a freely algebra. But then you have to look at G tilde. And then G tilde really depends on the genus. And so somehow here, the subtle difference between the Youngian, so between Youngian of G is really different even as a graded vector space from G bracket T. And the difference between the two is important, especially in higher genus case.