 All right, thank you. So today I'll talk about the Huber scheme of points on the plane and its connections to link homology, mostly contractual, but most partially proved. And before link homology, I just want to spend some time talking about the Huber scheme of C2. So many people here know what it is, but I want to give slightly maybe different perspective and relate it to joyous lectures very clearly. So what is the Huber scheme of C2? So you have zero-dimensional subschames of C2. So C2 is a fine. So the ring of functions on C2 is C of xy, polynomials in two variables. And any zero-dimensional subschames is defined by some ideal. So we're looking at ideals inside C of xy such that the core dimension of C of xy mod i is equal to n. And it turns out that this space is smooth of dimension 2n as a theorem of Fogarty. And it is a symplectic. So there is an algebraic symplectic form on it, which is non-degenerate everywhere. And in fact, to connect to joyous lectures, this is an example of symplectic resolution. And I would say it's my favorite personal example of symplectic resolution. So I think for joy, it was this Rp1. For me, whenever I think of symplectic resolution, I think about this example really. So what is the resolution of? So you have a hybrid show map, which sends an ideal to support of the corresponding zero-dimensional subschame, so support of the quotient of C of xy mod i. And this is n points on C2 with multiplicities. And so you have a map from the hybrid scheme of n points on C2 to symmetric power of C2. Again, these are just n tuples unordered and tuples on points. And this set of unordered n tuples of points is very singular and hybrid scheme is smooth. And in fact, it is a resolution of singularities of the symmetric power. And you can see more. So for example, this is a Poisson variety. It's a fine Poisson variety. And then this is not a fine, but this is a symplectic variety and the forms agree. And so one way to see it is to say that what is symmetric power of C2 really? So what are the functions on symmetric power of C2? Well, we have n points with coordinates x1, y1, x2, y2, x3, y3, and so on. So we have functions on those, and this will be just polynomials in x1 through xn, y1 through yn. And then we quotient by sin. So we take Sn invariance where the action of Sn is diagonal. We commute x's and y's simultaneously. And we take the spec of that. And that is clearly a Poisson variety because we can just say that the Poisson bracket of xi and yj is delta ij. Maybe I have to sign. And so this defines the Poisson bracket on C of x1, xn, y1, yn. It's an invariance. So it defines the Poisson bracket on the sin, yn functions. And so this is a Poisson variety. And this is the resolution of that, which turns out to be symplectic, I think, by result of boolean, if I remember correctly. And additional piece of data, which was very important for Joel's lectures, is that this is a conical symplectic resolution. And it has an additional torus action. So what are these here? So you have two cisteractions on C2, which they typically leave to cisteractions on the Hebrew scheme. So the Hamiltonian torus scales x by q and y by q inverse. There's an echo. There's a little bit of an echo, yeah. So there is a Hamiltonian torus T, which scales x by q and y by q inverse. So this Hamiltonian torus preserves the standard symplectic form on the plane. And one can prove that it preserves the standard symplectic form on the Hebrew scheme as well. So this is what Joel denoted by T in his examples. And then you have this conical structure. So you want some dilatancy's direction. And this dilatancy's direction just deletes everything. So you send x and y to Tx and Ty. So sometimes people use Q and T in slightly different sense. And it doesn't matter that much today. But this is the conical torus. So it scales everything, or it shrinks everything. If T goes to 0 to the most singular fiber, we're over 0. And it scales the symplectic form by T to the power of 2n, I guess. No, T squared, sorry, T squared. And another piece of data, which was also important, is that given a Hamiltonian torus, you can look at a traction sub-variety of this Hamiltonian torus. Or you can look at fixed points, first of all. Yes, maybe let's talk about the fixed points first. So the fixed points for the Hamiltonian torus are monomial ideals. So you have any partition of n lambda. So we draw it in French notation because we're in France. And we fill every box of this partition with monomials in x and y. So in the horizontal direction, we have x's and powers of x. In vertical direction, we have powers of y. And so in particular, this box will be x to the 6. This box will be x to the 5y, because it goes up by 1. This will be x squared, y squared, and so on. And this will be y to the power of 4. And then monomial ideal is generated by all monomials outside of this yan diagram in this yellow region. And in this example, the monomial ideal is generated by x to the 6, x to the 5y, x squared, y squared, xy cubed, and y to the 4. And one can check that all fixed points for the Hamiltonian torus actions are actually isolated. There are, finally, many of those. And they're labeled by partitions of n, as I just described. So we are in the setting of the journal selection. And final piece of data, which I skipped over, was Lagrangian sub-variety or this attracting sub-variety. So we look at the locus where the limit at few goes to 0 exists. And if you just look at C2, so we look at the action qx and q inverse y, the limit of this thing exists only if y is equal to 0. Because if y is not equal to 0, this q inverse y will blow up and go to infinity. And so if you have n points, then, and you have this kind of action, so you want all the points to be on the line when y is equal to 0. And so the attracting sub-variety in notations from Joeslack, this will be Hill-Bann of C2 plus. So this will be like y plus. And in more kind of algebraic geometric notations, is the Heber scheme of n points on C2 supported on the line C. And this is the line y is equal to 0. So we're looking at all ideals such that support of C for x, y, and mod i is a subset of y is equal to 0. So all my points are supported on the horizontal line. And one can indeed check. And I mean, this follows from general theory, but one can directly check in this case that this is indeed a Lagrangian sub-variety. It has as many components as fixed points. So for each fixed point, you have a attracting subset, which is like all the points which flow into that particular fixed point. And all of them have dimension n. And you can study the singularities of the space and lots of other things, which we will maybe review in a second. And I think that's maybe most what I want to say about the structure of symplectic resolution. So there was some maybe any questions here. So I hope that most of you have seen this picture at some point or read an academic book where all this is explained very, very nicely and clearly. And maybe one last comment, which is not so important. So Joel mentioned category O for this action. So we want to quantize modules or sheaves on the hybrid scheme supported on this Lagrangian. And the quantization is known and well-studied by many people. And it's known as rational Trinic algebra. So maybe I'll just say it as a word. It's not important for what I will say later. It's kind of related to what I said last time. But just to connect things, so spherical rational Trinic algebra in the quantization. And in particular, there is a notion of category O where things are supported on this Lagrangian. And there is a lot of interest in studies. So you can look at, I don't know, Edinger's lectures on Trinic algebra and all this is discussed in detail. Question. Yeah, who's going now? OK. Right. So any questions? No questions? Good. So let's keep going. So we need a bit more structure there. OK, so a bit more structure is that we have, because it's not applied, we need to study some interesting bundles on the hybrid scheme. And the most obvious bundle is the doftological bundle T. So I think the research notation would be E, but I stick with T. So this is a rank n vector bundle over the hybrid scheme and the fiber over an ideal is just the quotient of C of x, y by that ideal. So this is really a vector bundle of rank n. And given T, you can cook up lots of other vector bundles by taking sure funters. In particular, you can determine of T or n is the exterior power of T. And this is known as O of 1. So this is a line bundle on the hybrid scheme. And in fact, it's an ample line bundle. So hybrid scheme is not a projective variety, because it's non-compact. But you have embedding into some projective bundle over something I find using this O of 1, as I say this way. Said differently, maybe another way to think about this O of 1 is to say that this is a resolution of singularity. This is actually a blow-up of some explicit ideal in here, in symmetric power. And then O of 1 is associated with that blow-up. Anyway, one has to change the underlying space to get trigonometric or elliptic. Yeah, so for trigonometric, we have to consider the hybrid scheme of C cross C star or contention bundle to C star. And for elliptic, I guess I'm not an expert. But maybe you need something like C star of elliptic curve. I don't know. I forgot. Anyway, so I think what I want to say in connections to JOS lectures just to close this general introduction is that, in fact, many people started from Nakajima and then later on the other side by Braverman, Finkelberg, and Nakajima identified the hybrid scheme of C2 with both the Higgs and Coulomb branch of the same theory. So in the notations of JOS lecture yesterday, we have to choose a group and representation. So the group is JLN. Representation is the Lie algebra of JLN or joint representation plus CN. And it corresponds to this quiver where I have one vertex with label N, one loop corresponding to this kind of operator in the Lie algebra of JLN. And this CN corresponds to the framing, which goes from the vertex, framing vertex with label 1, which goes to this vertex with label N. And then, again, it's quite classical by now that this is the Higgs branch of this theory. So if you take contention bundle to N and quotient by G, this, I guess, triple quotient, whatever, so this is the hybrid scheme. And this is explained, for example, in Nakajima's book. And more recently, Braverman and Finkelberg and Nakajima proved that if you do this Coulomb branch construction with the fine grass miners and all this stuff from last JOS lecture, you recover this quiver. And so for people who like geometric representations here, this is a fine A1 quiver with framing. And so it turns out to be a self-dual under this magical symmetry. And we will kind of see this, some instances of this in a little bit, although I mean, I'm not sure you can recognize these instances of being the Coulomb branch or being the Higgs branch from my top. Anyway, OK. So this is just the connection to symplectic resolutions. And this is always good to think about it. And one thing which I want to emphasize, which will appear a lot today, is that you have this Lagrangian sub-variety. So you can look at this hybrid scheme of points supported on a line. And it's not completely random thing. It comes from this very general construction of symplectic resolution and action of Hamiltonian torus. And we've come to the main conjecture of myself, Andrei Nygut and Jake Rasmussen. And by now, it is my understanding that it's mostly proven by, let's say, Blomkov and Lavrozansky. So they have five or six very long papers in archive. And they're working on even more. So I think most of the details are written out already. And then there are maybe some subtleties which they finish it. But I think mostly the conjecture is now proven by Blomkov and Rasmussen. So starting from a braid on N's trends, you build a sheaf, or strictly speaking, an object in the right category. And I don't want to talk about subtleties and homological algebra here. But for now, they just see a stark or a stark variant sheaf on the hybrid scheme of points. And not on the whole hybrid scheme of points, on this Lagrangian sub-variety, one-heeled band of C2, C2, C2. And so it is supposed to be a stark or a stark variant with respect to both conical and the Hamiltonian action. And the most important part of the conjecture is that you can actually recover back the triply-graded homology of your braid from this thing. So you take sheaf co-homology. So this is a coherent sheaf, maybe I should say this because yesterday was so somewhat rockable sheaf. So this is really coherent sheaf. And so you take sheaf co-homology of hybrid scheme of N points on C2 support and a line. And you take this sheaf and you tensor, you can take it by itself. That's already fine and good for most purposes. But if you really want to get the full co-homology, you tensor it with the exterior algebra of the teftological bundle. So unlike previous talk today, so this T is not the tension bond, this is really the teftological bundle. Maybe I should write it here explicitly. So this is a dual teftological bundle. And then you take, because the sheaf is a stark or a stark variant and the teftological bundle is a stark or a stark variant. You just take the space of sections. It has a natural or higher co-homology if there are any. So you take the character of C stark versus direction on that and that's your answer. So this thing is naturally triply graded. So a degree comes from this part from exterior algebra of the teftological or dual teftological Q and T degree come from the action of C star and C star on the plane on the hybrid schema and on the sheaf. And you have to be careful strictly speaking if you do have higher co-homology here. So if you don't have higher co-homology, this is a complete answer and that's well understood. If you do have higher co-homology, they're incorporated all together in a slightly weird way. And these homological degrees also is related to Q and T in a very subtle way which I'm not going to talk about but in many examples that we'll see today, actually you don't have higher co-homology so you don't need to worry about that part but that's the most subtle part here. And again, so then the question is, well, so why do we care as always is the question? So, and again, you can read it in two ways. So one way is to say, well, so you have an interesting sheaf on the hybrid scheme, can we interpret it as invariant of some link and maybe get some intuition or some kind of structural results from the link homology which might help us and we might see some example or at least I'll say some examples about this. And in the other direction, I mean, and originally this is like what's the main motivation for this conjecture is that we really want to compute HHH of beta or we want to understand what it is and how to think about it properly. And if we can guess the sheaf of beta, we have lots of different tools to compute this right inside. So we have a covariant localization to fix points. We have in under nice circumstances, we have homology of vanishing. We have other tools just directly starting to run through the hybrid scheme to compute this thing. So I will give a very, very explicit examples where we know the sheaf of beta very, very explicitly when we can compute the right inside. And so historically this was the main computational tool to make predictions about HHH and in many cases, like now these predictions are proven. And that's the structure. So what else do we know about this sheaf? So the action of, as I said a couple of times already there is an action of polynomial algebra on HHH of beta. And if we don't close the braid, we really have 10 different variables. And if we close the braid, they close up according to cycles and the corresponding permutation or components connected components in the link. And so these axes are just positions of the point somehow here. So we have hybrid scheme of endpoints on a line. We have endpoints, they're all in a line and their coordinates are xi. And so in particular, the action of these variables correspond to the choice of support of the sheaf of beta. And very concretely, if beta, for example, is a knot then all xi must act the same way on HHH of beta. And that we discussed several times but on the right-hand side, this means that my sheaf is not a random thing. It's actually supported on the punctual hybrid scheme of endpoints. So all points must be the same. And up to a shift, we can assume that all points are actually at zero. And so whenever we're talking about HHH of beta and beta closes to a knot, we're talking about the sheaf not on the whole thing, not on the whole hybrid scheme, not on the Lagrangian sub-right, but actually about a sheaf just on the central fiber on the hybrid scheme of C2 supported origin. And one concrete example, which was again, kind of motivating for most of things and developments that I've talked about in this course and most of the developments I was in last 10 years here is that if you have a torus node of type NKN plus one, then your sheaf is actually clear. So your sheaf is the line bundle of K on the punctual hybrid scheme of points on C2. So again, let me repeat that. You have this line bundle of one on the hybrid scheme of points. You just raise it into power K. If K is positive, let's say, but actually K could be negative as well. And maybe not right, not too confusing. So if K is positive, you know a lot about the sheaf and it's cumulose vanishing. If K is negative, this is still true, but it's much more subtle and I can't raise it, I'm sorry. And this is very explicit prediction and many people starting from Mark Heyman studied this answer and I will maybe give more concrete example in a second. But this is kind of one very specific instance of this conjecture, which tells us a lot. So in the right-hand side, we have very specific line bundle on the punctual hybrid scheme, tensor with a vector bundle flying. And then you want to compute it's space of sections or sheaf cohomology. You can do it by many tools and Heyman did it for you in some sense. And so you just have an answer on the right-hand side, but to compute the left-hand side, it took really a while in about like 10 years to confirm this prediction. And then another feature of this conjecture and of the sheaf is that what happens if you add a full twist. So I don't know, maybe I should, you can put T and N, I think later I will call it F to confuse everyone. So you have the full twist braid where you have N strands and you turn it around and then they come back to the same position. Okay. How is this conjecture related to braid varieties? It's in no way related to braid varieties as far as I know. That's a very good question, which maybe I'll give some comment in the end, but as far as I know, I mean, these are just different models. So you have three different models for lean homology. Lean homology. So maybe let me just summarize this before we go on. So you have lean homology is related to braid varieties and that is definitely proven. And that's essentially the work of Epster Williamson plus other work of Mellet, Trin and others. Then you have these hybrid schemes and single curves, which I've talked about last time. And that's a very different thing. And the relation between the two is also not understood. So maybe I'll, okay. Sorry, my computer is weird. So you have three algebra geometric models for lean homology. So just to put everything in context, so you have the braid varieties and then you have hybrid scheme of single curve and then you have heel band of C2. And there are three very different things. So here we have homology of some smooth, non-compact algebraic variety with weight filtration. Here you have homology of compact but a singular variety. And the question is what is the relation? So this is known to be related to lean homology. So this is done. Then this is still conjectural how this is related to lean homology, except for examples where we can compute things. And the relation between the two is supposed to be some kind of non-nibillion-hodge theory, which I don't know anything about, but I've talked about last time. And then you can ask what is the relation between this and the conjecture that I just said. And again, that's a very good question, which is not, I would say, understood at all. So physicists would say that this is some kind of geometric engineering. The relation here is kind of immersion. And I will talk about, but like just purely by terms of algebraic geometry, I don't know immediately how to see this. So maybe one way to get from braid varieties to the Huber scheme. So just for Joel, so there is a relation to character shifts here. And one can hope that there is a relation that in some kind of derived version of character shifts in the Huber scheme of C2. So maybe this would go through character shifts and I will say a couple of sentences later, but again, this is not understood at all, I would say. And the relation here is kind of immersion slightly better, but this is only for algebraic knots and it's still not clear. So I mean, all this is pretty much conjectural and that's why this is hard. So what Ablonkow and Rosanski did, I will say in a second and they did something very different. So they didn't relate to braid varieties and they didn't really use geometry here. They did something different. Yeah, sorry, I didn't really answer the question, but I don't have much to say. Any other questions? Okay, all right. So I'll come back to this question and maybe say more and confuse people. We have one more question Eugene. Uh-huh. Yeah, I just didn't quite understand in your hearing the second point, the link between the action of XI and the support of us. Oh, is this connection? Yeah, so if you have any issues, so you have XI or rather like symmetric functions of XI, there are global functions on this heel brand of C2C, right? So what are the global functions on the Hebert scheme? As I said, there's a just symmetric function and X's and Y's. So if all Y's are equal to zero, then symmetric function and X's are global functions on this thing. And so in particular, any module is, any shift on the Hebert scheme gives you a module over the symmetric functions. And this is supposed to match to this action of XI and the web. So that's one way to say it. Another way to say it is that you have a map from the Hebert scheme of C2C and you have just a map to symmetric power of C where you have end points on the line. And so the coordinates here are the symmetric functions that I talked about and they should act or you can just look at the image. And so these end points here are supposed to correspond to coordinates XI over there. Okay? Okay, thank you. Good. And so yeah, so in particular, if all XI act by zero on the left, this means that the shift is supported on like 0, 0, 0, 0. If all XI act the same way, this means that all points in the support of the shift are the same and or like push forward to SNC if you want are the same. And up to a shift, you can assume that the same point is zero. So this is to achieve. Okay, fine. And then another general property which is very useful in practice is that if I have a braid beta and I add a full twist to the braid, then the corresponding shift are just multiplied by the line bundle of one. And that is in some constructions, this kind of comes for free. In some construction, this is really hard to prove. And I'll talk a little bit about that if I have time. And so even more concrete examples, kind of unpacking. Sorry, can you see me? Hello? Yes, you can see me. Okay, yeah, sorry, my computer almost crashed and I don't know what happened. So even more concrete than this example, if you have a three-fold node, T2, 3, then the corresponding shift is O of one on the Hebert scheme of two points on C2 supported at the origin. So the Hebert scheme of two points on C2 is CP1, is P1. And so we're counting sections of one on P1. This is a two-dimensional space and we know how to compute it, we can do it equivalently. We get Q plus T and I just realized I changed my Q and T from the beginning of the lecture but that's fine, it's still two-dimensional space. And this two-dimensional space matches the two-dimensional space that we saw before. And so maybe, yeah. And another example, the beta is T3, 4, 3, 4, torsino, that's kind of the first really interesting example. And the shift is still O of one but now the Hebert scheme of three points at the origin and this Hebert scheme of three points is well-known in classical objects. It's a corner or twisted cubic and you can resolve singularity to compute these sections of the shift and it's a good exercise for everyone which is really not so hard is to say that the dimension of the space of sections is equal to five. So you have Hebert scheme of three points on C2 at the origin, takes space of sections of O of one and that it's five-dimensional. And if you do it equivalently you can find the covariance sections and the character and everything. And you can check directly. It's also not so hard that this is the same as the link homology for T3, 4. So maybe for this particular example, just again going back to this equation of Joel and keep going back. So there are three different models for this T3, 4. So here you have again, Healt three of C2, 0, with O of one and you take homology of that shift. Here you would have this E6 cluster variety which I talked about or this with Detroit variety P37 I guess. So some open sub-variety in the Graspine N37 and you take its homology more or less with weight filtration. And here you would have this compactified Jacobian 434 that I talked about last time. So this would be this cone over P1 cross P1 or over here to Brook surface. So this is again some weird space. And so here again, you have a smooth and non-compact space but it's open in the Graspine N37 and you have the weight filtration here. Here you have a singular space which you can analyze. So this is this compactified Jacobian of X cube is equal to Y to the fourth. And you take its homology and again it's singular but you can compute everything. And here you would have this singular space cube scheme of three points of C2, 0, O of one. Compute its homology and get the same result. So all of them give the same result and the same as HHH. In this case, we can check everything. But again, the relation between them looks quite mysterious and still is. Okay, any questions? Okay. And another example which maybe I'll gloss over with briefly, that's fine. So if your braid is the torus node in general. So I talked a lot about torus nodes and the relation to the Detroit varieties, relation to this compactified Jacobians. So what do we know here? So here, this was studied in particular in the work of myself and Andreine Good. So the shift is actually tricky in general. And to get the shift, we need to consider the nested hybrid scheme or the flycuber scheme on C2, which also appeared this week. So we look at the space of all flags of ideals, I1 to the up to IN. And all of them are supposed to support the origin in this case. And on this thing, we have a lot of line bundles where LK is IK minus one mod IK, take K minus one's ideal, wash them by K's ideal. And this is a line bundle in the nested hybrid scheme. And roughly speaking, you take push forward of the product of this LI in some powers AI, where AI are this fractional thing. So you take integer part of IM over N, I minus one M over N, and then subtract this integer part. And then use this AI to build a line bundle on the flycuber scheme, and then just take push forward of this thing to hybrid scheme. And that's your shape. So that's pretty subtle and not obvious at all. And like why this answer is true is an interesting question, which we can discuss. But this matches, for example, these constructions of Oblong and Rosansky, as they checked. And I can explain where these numbers AI come from typologically, but not right now. And the main subtlety here, which is kind of the main obstacle here is that this is really, really singular space, as we heard from Richard. So here we use probably related, maybe slightly different construction, but we use some kind of digit structure on flycuber scheme to define this push forward. And so very roughly, you say that this is cut out by some explicit equations in some explicit smooth space. And then you just use that to define the virtual structure shift and everything. But again, like it would be nice to have a better understanding, for example, like how this structure that we use here is related to the one that Richard talked about. And I don't know. And like one reason why it's not clear because here we have C2, which is really non-compact. And then he worked on compact surface and that was really important for him. So I mean, I don't know if it is the same structure. In any case, you have some shift here and you can also use lots of tools and previous work of Andrei to compute the, at least all your characteristics of the shift on the hybrid scheme over here and check that it matches what we expect. And then you can ask, well, so all this was about notes. So when M and N are co-prime, what happens if you have more components and like the first basic question is what happens for identity braid? So if you have just all strands parallel to each other, how to think about it? And to explain the answer for identity braid, I need to introduce some auxiliary spaces. So I need to look at the following Cartesian diagram. So the hybrid scheme of N points of C2 projects to Sn of C2 and then you have projection from C2 to the N to the Sn of C2, which just quotient by Sn action. And then you take fiber product, which is denoted by XN. And this was introduced by Mark Heyman, who studied this space a lot. And in particular, he proved that XN is the blow up of the C2 to the N along the union of diagonals. So you have N points and we have diagonals where at least two points collide. And you take the union of all such diagonals, blow it up simultaneously, and this is XN. And then if you want to unpack this, then what is the ideal defining the union of diagonals? So if I send J's points collide, this is XI minus 8J and YI minus 5J. This is a co-dimension two hyperplane. And if I have the union of diagonals, I take the intersection of ideals and this is this thing J. And so if I blow up, I just take the approach of direct sum of J to the K. So this is very, very explicit blow up construction. And the main thing which Heyman proved is that if you push forward the structure shift of XN down to the Hubard scheme, you don't get, oh, in fact, we have a remarkable bundle called Preciase bundle. And this has rank and factorial. So kind of this map on the right is generically and factorial to one. So the left-hand side is generically and factorial to one. And what he proved is this is actually flat. And do you get a vector bundle of rank and factorial? And so this Preciase bundle plays a very important role in the study of the Hubard scheme. And we will see it in a second here. And another thing, so kind of connecting to the nested Hubard scheme, which is important. So you can look at the nested Hubard scheme of endpoints on C2. And you have a map to the usual Hubard scheme because you have a flag of ideals. You just forget everything. You just project to the last one, to the last ideal. Or you can fix support of each quotient. So we have these quotients, but we can look at the support of these quotients. There's a end different points. And so we have something in C2 to the end. Where these quotients are really supported, this is ordered set of points. And so you have a map here. You have a map here. And so by general nonsense, you have a map to the fiber product, the XN. And so you can define Preciase either by pushing forward all from here from the XN to the Hubard scheme or by pushing forward all from the flag Hubard scheme down to the Hubard scheme. And this is actually the same, roughly speaking, because for this map, push forward of all is called. And this we actually, I don't think Heyman proved this, but we proved this like, we needed this for something. And this is in our paper with Sundry and Jake. And so again, like, if you don't like this kind of weird singular derived DG schemes like flag Hubard scheme, you can just think of this. If you like it, then you can think of it as push forward of all from here down to the Hubard scheme. And somehow this is more natural for me to think about this, this way. But if you like kind of classical algebraic geometry, maybe you need to push forward all from XN down to there. There was some question, no? Anyway, so there is some extra bundle Frankenfactorial, whatever it is, who cares? And so now the answer is that, what do you associate to the trivial grade? You associate just this pre-chase bundle restricted to Lagrangian sub-variety, Hill-Benow, C2C. And if you have TNKN, which appeared a lot, then you, so this is the power of the full twist. And then you associate to it pre-chase bundle times of K, restricted to Hill-Benow, C2C. So yeah, I mean, and this is no easier way to describe the shape for the trivial braid and for the powers of the full twist. And in general, for kind of braids associated to links with many components, you always expect some kind of pre-chase bundle or some kind of restricted version of pre-chase bundle to show up. And this is an interesting feature of this construction. Okay, and I wanna note something because that will be important for us in a second. So how do you actually like, okay, so we believe in this and suppose that we believe in the conjecture, what does it give us? So we have this hybrid scheme of C2C with this vector bundle P times O of K. So the space of sections here is the same as space of sections upstairs on XN of C2C, fine, of O of K by projection formula. So this is just the projection one. And XN of C2C, well, or XN, at least, we can think of it as a blow up of this C2 to the N. And so we know how to compute the space of sections on a blow up of O of K. This is just the Kth power of the ideal that we started from. So maybe I'll write it as a separate line that if you just look at the sections of XN with O of K, this will be J to the K. And if instead we have the C2C, well, we have to quotient by Y's and that's what we do here. So we quotient by maximum ideal and Y's. And well, yeah, I wrote it here already that H0 of XN of K, J to the K just by this blow up formula. And then J to the K is actually free over polynomials and Y's. That's not obvious at all, but that was also proved by him. And so to sum up, so if we believe in this conjecture, this said, the conjecture says that this J to the K mod Y to the K should be the invariant that was associated to NK and torus link. And one theorem from lecture two is that this is actually true. So maybe this is the HHJ. HHJ of TN, KN by lecture two. So we prove that this is actually true. And so in this case, conjecture is true. And that's a very non-trivial check because we have this very non-trivial ideals and stuff. But of course, this particular computation was a motivation for that theorem. We wanted to prove that HHJ of TN, TN is given by this weird formula. And if you remember, so I said that this J to the K mod YJ to the K. So the way I sketched the proof, I said that you need to deform your homology. So you need to introduce Y's and then kill Y's like this. So in this sense, you have a shift on the, you want to describe a shift on this Lagrangian subvariety, but it's much more natural to describe a shift everywhere and then restrict to Lagrangian subvariety. And so in some precise sense, this verification business corresponds to extending the shift from Lagrangian subvariety to the whole hybrid scheme of points of shift. And so, right. So let me sketch at least some ideas why this might be true and what is like the relation to other stuff that we know. So the first approach that I certainly don't have time to talk about, and that requires like their own course of lectures by Blomkopf and Rosanski, is that what they do is they define in Julian Converry. So they say like, forget about HHJ. They use matrix vectorization so over some very, very complicated space roughly related to the flag hybrid scheme of points or the hybrid scheme of points, but it's like a bunch of groups and the algebras and you have matrix vectorization of some complicated potential in there. And then what they prove is that first of all, this is a Lincoln variant. And secondly, more recently, they prove that this is isomorphic to HHHH. And so then how to get from their shift on the hybrid scheme, you just kind of push forward from a heel to heel or from whatever they're set up, they're using the critical locus of their potential to the hybrid scheme. But like their theory is intrinsically related to the hybrid scheme. And so it's kind of natural to get a shift from there. But the hard part is of course to prove that their theory is isomorphic to HHHH. And that's what they recently proved. So a different approach, which is like, there are two more approaches which are not completely finished and mostly contractual, but there is a lot of progress there. Is one work of myself and Bedrich and myself and Hohkenkamp and Bedrich. So we want to understand lean homology and the solid torus. So this is the same thing as to have a variance of beta up to conjugation. So we don't require any mark of moves, but we require a variance and a conjugation. And it turns out that like if you have seen, I mean, it doesn't matter, but it turns out that this technology, the right homological algebra language to work with this is the language of the right categorical traces and host of homologies of categories and things like this. And so we do this for the category of Zorgel bimonials which governs HHHH, we compute something. And so that is supposed to be related to this arrow between braid varieties and the Huber scheme. And roughly speaking, when we close the braid and we look at this braid link in the solid torus, you can associate to it some kind of derived categorical version of character shifts. And that is supposed to be related to the Huber scheme, but maybe I don't have time to answer that, but you can ask me later if you're interested. Like why this might be related. And the thing that I really want to talk about in the last eight minutes is the approach that we initially took with Andre and Jake. And I think this approach still makes perfect sense except that again, you need to overcome some homological algebra difficulties. And so what we said in the original paper, we consider the graded algebra. So what is the graded algebra? So you have a full twist braid corresponding to TNN. And so you have the powers of this full twist braid. This corresponds to NKN torus braids. And you just take the sum of them all over K where K goes from zero to infinity. And so this is an algebra because if I tensor FT to the K, if I multiply FT to the K and FT to the L, I get FT to the K plus L. And so by general nonsense, you have a map multiplication map. If I have a home from R to this guy, and if I have a home from R to this guy, we can multiply tensor these homes and get a home from R to FT to the K plus L. And so this gives a graded algebra structure on this direct sum. So in addition to all this like QT and A gradings, we have this extra gradient by the power of the full twist. And by lecture two or three, the homology of the Kth power of the full twist, the homology of this guy is precisely J to the K mod YJ to the K. So we know this answer and which I discussed 10 minutes ago that you can prove conjecture in this particular case. And not only prove this conjecture, you can, and J is again this ideal of the diagonal in C to the N. But this by the virtue of the proof, this agrees with multiplication. So you have this multiplication on the left hand side where with abstract homes from R to FT to the K and you have multiplication on the right hand side because you have parts of the same ideal. And this means that this isomorphism agrees with tensor structure with multiplication. So maybe let me, this isomorphism agrees. There's multiplication. And so not only will you know this thing as a vector space or as an X-module or X-y-module, we really know this thing as a graded algebra. And given an arbitrary braid, we can construct this complex, so it's over by modules, whatever. But we have naturally a module over this graded algebra because we can take home from R to TB at times F to the K. So this home is a module over this algebra because again, I can add an additional FT to the L in here and get FT to the K plus L. And so this is a graded module over a graded algebra. And so this gives a coherent shift on proj of A and proj of that algebra. As I said, this is exactly this XN of C to C. So that's why I talked about the space XN because you can write it as a proj of some very explicit algebra and any graded module over the algebra gives you a shift on this XN. And so in some sense we're done. And again, the real question is to lift it from this kind of naive computation to the level of DG or derived categories. And this is still not done, but I think like this is one way to go. And so what do we expect? And we write very clearly what should be true that we have some kind of DG funter from the format of the category of certain modules to derived category of hybrid scheme of C to C. And it has a lot of interesting properties which released quite carefully in that paper. Okay. And so maybe I have a couple of minutes and before saying thank you, I want to comment again on Joel's question. So I didn't really explain very much Joel's question here. So I didn't really explain what is the relation here and what it has to do with character writers. But here we have a single curve like again, how you would expect to have it here. And so the idea is to look at the same graded algebra. And so the same graded algebra, maybe as a note, I'll put it later. I don't know, let's find input here. So note, and this graded algebra, A, such that the approach of ASXM can be seen and can be identified in the offense story, in the Coulomb branch story. I think Joel briefly mentioned this last time that you, I mean, you can build your Coulomb branch as a pack of some complicated algebra, but you can also build a resolution of Coulomb branch in nice cases as proj of some algebra. And this is exactly this algebra, except that we don't want to get the hybrid scheme, we want to get XN, but this is kind of minor difficulty, which can be overcome, or like we think we know how to overcome. And so one concrete thing about the graded algebras and graded modules we should follow up from this is that you take some gamma in GLA, I think I called it Y last time, whatever. So we can consider the spring of fiber for gamma, we can consider the spring of fiber for gamma times X, the spring of fiber for gamma times X, clear, and so on. And then we take the direct sum over K is equal to zero to infinity of homologous of S P gamma times X to the K. And again, this is supposed to, this is a graded module over the graded algebra. And so this would be a shift approach. And one can ask lots of questions, like how to at least to construct the action and how to quantize this and how to quantize this graded algebra to some kind of Z algebra that other people started. But there is some work on this direction and mostly done by Webster and others. And also there was work in progress by myself, Oscar and the Blanco. And so in some way, I would say one of the key ideas is to identify this graded algebra and to prove that you always have a graded module structure over this graded algebra. And we see it here in the school branch story and again, to answer Joel's question, it would be great to see this algebra on this side as well. I can have some kind of multiplicative structure for starters to have to say that you have a great variety for beta, you have a great variety for beta prime. How do you get to the great variety of beta, beta prime? And that's already not completely obvious to me, but maybe that's my ignorance. Anyway, but if all this is done properly that here from a curve, you not only consider the curve, you consider kind of the curve with additional full twist. And this corresponds to graded module of a graded algebra and so coherency here. So I think this is kind of more or less directly from here to here in nice cases, which is still yet to be completely worked out. And I'm over time. So let me say last two things. So first of all, thank you very much everyone for coming to these lectures. I think it was great and there were lots of great questions. Great thanks for organizers of the school. I think it's awesome. And I personally learned a lot from Richard and Joel's lectures and I hope to learn more from other lectures. And as an advertisement, if you want to learn more about link homology and some connections to topology, to algebra, geometry, miniaturics, algebraic geometry in particular, I'd like to advertise AIM research community, which you can find by this link. So this is sponsored by American Institute of Mathematics in San Jose. And this summer, where we have a lot of reading groups and other activities specifically for grad students. So if you're grad students or postdocs who want to learn more, what is a link homology? Why do we care about this? What are the brave varieties? What are other structures? There are five or six reading groups specifically aimed at this problem. Like how to compute link homology? There is another group. So if you're interested in this, please register here or write me an email and I'll give you all the contents. So thanks a lot and see you all later. Thank you very much Richard. Please give me a quick answer. Any questions online offline? So that in the P equal W conjecture for curves, you have three spaces. You have this doble modular space, basically modular space, doble modular space. But here it seems that it's different from the three spaces you have. Here you have just two of them, I think. So you have- You also have a hillb-NC2. Do you have any- No, hillb-NC2 has nothing to do with like the bore, the rammer. No, no, no. This is completely different story. So like the braid variety and the hybrid scheme of single curve, they are, and I always forget, like there are two of the three spaces that I mentioned, that you mentioned. And somehow the relation to hybrid scheme of C2 is very, very different. So for kind of her character varieties and for P is equal to W, I think the closest analog is the work of Haussel, Lettelier, Rodriguez Villegas and others. So they say that you can compute one correprenominal of character varieties with weight filtration or with this diverse filtration or the kitchen modular space with some extra data. And so all these things are expressed as some complicated formulas with monotonic polynomials. And so somehow the hybrid scheme of points keep track of that. And it says like all this, you can interpret this on shifts on the hybrid scheme of points. And Prechesibandel also appear in this work of Haussel and collaborators. And so if you've seen that, like that is the closest analog, but this is not part of the, it's not part of this non-ambient holds thing. So the relation to hybrid scheme of C2 is very different. And like physicists, I think say that this is some kind of genetic engineering, whatever it is. I don't know what genetic engineering personally, but that's what they say. But I want to emphasize, this is a very, very different thing. So there you're talking about kind of related varieties. Here you submit the jump from like studying homology of a variety to the homology of a shift on some completely different variety. And so this is very, very different thing to do. Okay. Any other questions? To continue on the same question, but sort of, you know, if you don't push theory, you have like space in the middle. So like, I guess, drum or U.S. space. So is there analog of this, like something between like great varieties and, that's a very good question. I don't know. Any other questions? So I just put the link to this AIM program to the chat if you're interested in. Okay. So let's thank Eugene again for being here.