 All right, so, so welcome back to the part two of the super seminar with the Chihong Chow telling us about flag varieties and middle symmetry. So please go ahead. Oh, so, uh, so I, yeah, so to summarize, so we call our goal is to compute PRS map when evaluated with the MV cycle. So we, but we know, um, so we can take a conversation and, um, the input is, uh, some matrix coefficients that that would be. So what we have seen is that, uh, some, uh, it goes to zero if, uh, the weight of that, uh, life is smaller than this, this, this particular weight. And when the weight is zero to this, uh, W's P, W's zero lambda that we get a quantum parameter, which is a unit. The one is important here because it's even a unit over integer. I mean, we have seen the, the intersection number is one. So we can, so we have a priori, I mean, the conversation of PRS map and why is that map is, is a map from the coordinate range of BET to a quantum commodity, but because, and because, because it sends this matrix coefficient to a unit, we can localize it. I mean, this map factors were localized, the localization. And because it sends, uh, this, the other mission is confident to seal this method of practice through the, the quotient ring and, um, and, uh, we look at the left-hand side, we see that it is, because we localize and take, I do. And geometrically the same as we take away a device and take a cross-sub-scheme. So left-hand side would be the, the coordinate range of the fiber product of BET, uh, with respect to some locally cross-sub-scheme. And we can show that if lambda is regular dominant, this sub-scheme is, uh, root-hassel, so we, so we can replace the domain by the scheme like this. And it has something to do with, um, the Peterson variety I introduced earlier, because we can, we have defined a map sending BH to, to B, uh, to, in this way, in a quite obvious way. And this map will induce, um, the core, the vertical arrow here, because we can take a fiber product with basically the same, uh, sub-scheme. And we have a five log, um, which I've defined in the last page. And the lemma says that we can show that, uh, uh, there's a map five. Making the diagram commute. And we can, and it turns out we can show that this map is bijective. And this puts the Peterson variety presentation. And using five and the localized version using five log. Yes. But, but I probably won't have time to explain why they are bijective. Okay. So let's go to mirror conjecture. Uh, we have rich mirror GMOP, so for any GMOP, uh, reached constructor, the foreign, the, and then that's been the Ginsberg model, uh, consisting of four things. Well, actually five, uh, uh, namely the volume, but we probably won't have time to discuss. So the space, so we have the mirror space, which is a fan. Which, uh, which should regard as a family of non-compact Caribbean over a subtaurus of, uh, the, the maximum torus of the land and deal. And, uh, and we have, we also have a super potential defined this way to receive. Well, I mean, supply is a projection here. And P is something, something neither only when we talk about the, uh, it could vary in mirrors. So P, we, we usually call it you could learn perturbations. And yes, he's a definition and the mirror conjecture says that, um, some, um, some, some connection associated to, to the rich mirror, which I will define, um, in a second is isomorphic to some connection associated to the, the, the conical emoji such that so it's an isomorphism, uh, identifying the two connections such that if we pass it to the semi-classical limit, we will get, uh, isomorphism of rings between the Jacobi ring and the conical emoji ring. And, uh, rich, uh, proves that, um, the Jacobi ring is isomorphic to the, to the quarter ring of a scheme I introduced before. So by the, by Pism variety presentation, more precisely the fact that five law being isomorphism by this fact, we, we see that, uh, this is true. And, uh, this conjecture is still open. But recently lemon Tom P a proofs the conjecture for all means skills. Gmo P. Oh, I, this is minus skill, um, prep varieties. And, um, I will try to explain to you, uh, why the conjecture is true for any Gmo P using the things, um, we have developed so far. But, um, before that, let me, um, give you some details about the two connections. We have a side connection, which is, um, which is conical emoji. And you see, there's this extra C side actions, but this C side action is just a trivial action. And, um, and, um, and the, the nation is, um, is the quantum connection. But, uh, we have a parameter H slash here because we would like to connect the case H slash equal one and H slash equals zero. So we just introduce this polynomial parameter at the center. This parameter is the equivalent parameter for this extra C section. And, um, the quantum connection, we also have this term, uh, which is the equivalent first-gen class of certain li-bundle on Gmo P, which, um, I associate, which is the li-bundle associated to the, the I fundamental weight. And the B model connection is more complicated. So roughly speaking, um, we have a module, uh, uh, which is roughly the top degree twisted D-round emoji, twisted by the, the super potential. But since we are considering the equivalent mirrors, we have this term, uh, coming from the equivalent third patient. And, um, so this is the relative D-round complex because, uh, we are not just looking at one mirror, but a family of mirrors indexed by a quantum parameter. So we have, we take the relative, uh, differential form, uh, differential complex. And then we, right, twist the, the D-round differential like this, and then we take the top goal emoji, but this is not actually the, the, the correct definition because, uh, uh, the definition I gave you is just the case when we fix the equivalent parameter. So because of this term, I mean, in general, that would be the complex tensor with a seam of t-due h-slash, but if we fix the pyramid, equivalent parameter of it, this term is gone. But for simplicity, we do that here. And this term is that we pull back the certain one form on the dual torus. And the one form is defined by the parent with this equivalent parameter and the marker transform. So marker transform is like D-set over set. So we have this module, um, appear, we don't know it's where it is free or not. We just define it to be a, a cold corner or something. But we can define a glass-money connection for this, this module. This is the mid-b-model connection. And, um, also we, let us, let us look at the Chakumi ring. And again, for simplicity, let's fix the equivalent parameter and quantum parameter. By definition, um, the Chakumi ring. So the Chakumi ring would be, um, I mean, let's say this, the spat of this Chakumi ring would be a family. It would be a scheme over the, the, the space of equivalent parameters and quantum parameters. But since we look at, um, we fix them. So we look at the fiber of it and the coordinate range of the fibers. And by definition, it is equal to the, uh, the scheme theoretic zero class of this one form, which is the difference of two one forms, uh, over this space. And recall, um, our mirror is a family of, is a family over the, the space of quantum parameter. So we, if we fix one of, if you fix the point downstairs, we, we get a fiber. So that's why we consider this space. And there are two forms on it. One is the exact one form, dw. And the other is the closed one form defined by the, by p. And observe that, uh, the first form is a restriction of some one form from this. So I want to recall the mirror is the intersection of two space. One is the borough, the other is the, this space. It turns out that, uh, these two forms, one, uh, from one comes from the one form on one space and the other from the other space. And, uh, because, uh, these two spaces are sub, oops, are sub varieties of the length of g. And, um, we call, we call something from synthetic geometry. You will have a sub, sub manifold. I mean, if we have a sub manifold and we know the cold engine bundle of Y is a sympathetic manifold and we can look at, uh, we can have, uh, a grungian, which is different by the Coromo bundle where, but this is a Coromo bundle. But if the cement flow has a one form, close one form, you can actually consider the shifted Coromo bundle. Then we can take a graph of this form and then we, I mean, define how we define Coromo bundle. So then we get a Lagrangian. So for this two form, we get two Lagrangian of the Corangian bundle g, g2. And, um, uh, but we are not actually considering the Corangian bundle, but some Hamiltonian reduction, uh, namely the, the so-called total space and also the universal centralizer. Uh, and this is well known that we have two Lagrangian foray agents on this sympathetic manifold. One is induced by the projection to, to the Lie algebra. And one is induced by projection to the group. So here we have some identification, some obvious identification. And the first one gives us to the, the total system, and you know, each, for each of the pointed fiber is the, some centralizer subgroup. So that's why we call it universal centralizer. And, um, this total is also, it's basically the same as the BET I introduced before. It's just the base change along this ocean map. And there is another, and the second variation is given. Well, look like this, the, the base is actually the torus. So the space should be, I mean, the base should be singular, but it is stratified by some, some torus according to the parabolic type. So if we, if we fix the parabolic type, we have a torus. And, uh, for each of the point of this torus, the fiber is a smooth Lagrangian, which we deal by LH and L-Clairs, respectively. And, uh, here's, um, so we have, we only, we are only, only looking at the, the, um, a classical case. So, so we have seen this isomorphism, which is proof I reach, but, uh, Telemann's has an, another proof, which I'm going to tell you. Instead of, because the total space is a Hamiltonian reduction of the Lagrangian bundle, we automatically have a Lagrangian correspondence. And, uh, so I have introduced a four Lagrangian, two from the Lagrangian bundle and two from the total. And it turns out they're related by the so-called geometric composition. And, yeah, for, for, for the, for this Lagrangian, the, the, I mean, so the direction is, is a difference. So for this first Lagrangian, I go from the left to the right. And for the other, I go from the right to the left. So, uh, so for the first case, uh, we take the intersection with the, the, this is called isotropical manifold. And it turns out it is, um, it, the intersection progest isomorphically downstairs through the, the Lagrangian in the total. And for the other direction, we will start with the Lagrangian downstairs. And then we take the pre-image, so that will be the fiber bundle. But it turns out that it is, it is the stricter column bundle. So we see that from this picture, we see that the intersection point upstairs identify with the intersection point stands there. So that we get this isomorphism. And we've seen this isomorphism before. So by composing it, we get the isomorphism, the Kubi ring and the coordinate of the intersection of the Lagrangian in total. And then we, by some straightforward argument, we can actually generalize the arguments to the family versions. Yeah, this, this isomorphism is nothing but the gen, a family version of the isomorphism in the bottom. So so far we are, we are looking at, let's say, a macauncle case. But we can, so here's the big picture to our, I've introduced all this map. So to get the, the, the original non-semi classical case, which, oh yeah. But before we do so, we, because we are going to contact it, but this is not very convenient to contact. So we look at, I mean, we just consider some other form. Well, we've seen that this is localization and then to the ideal. But it's not still convenient enough. And, you know, because the localization can be written in this form, we can take the tensor product first and then we take it like this, generated by this element. Now we can contact everything. And what this, something, let's start with something easier. So this is the B model we've seen before. And this is the map we are looking at. And the C star, the extra C star action is the trivial one, I told you. And as I said, this C star action is the loop rotation. And it is something that I need to tell you. And this idea is also something, because it's no longer commutative, the idea will not be the two-sided idea, but will be like right idea. But we have to replace the original matrix coefficient by some other, some other elements. We also have to replace this map as well. So it turns out that I'm not able to show that this map are objective. I can show that they are subjective. But it's enough for me to induce a map. But I have something more to do because it's not injective. I have to do something to show that the final map is objective. So let's start with the quantization of the BET. As you kind of probably think about, construct a ring isomorphism from the, so we first have a deformation quantization. Well, this is really the standard notation left, right, the minority. So Toyota is a simple metaphor. We can consider deformation quantization. But it is defined by quantum Hamiltonian reduction. And they show that the ring, so this is a non-commutative ring. And they show that it's isomorphic to the convolutional algebra. And we, I mean, when there's no c-sciation, this is just a point-jargin product. But if there's a c-sciation, this becomes a non-commutative product. And by extension of scalar, we get a modular isomorphism. And now, so here, we define it to be some extension. And we call, we have a matrix coefficient, which is, we can show that it is a pure, not an animal of this ring. But we can, and because OT is a free module for OT multiple, we can take a basis such that the first term is one and the other terms have higher degree. So we can express the matrix coefficient such that we also have a leading term. And we will use it to define the ideal. So here is the definition. But we are now looking at the, I forgot, left or right, it should be right ideal. Because it's now a module, I mean, it's not, it's not written by the module of Pula. And this element is the animal of Pula. So we can add it from the right. So we can form this ideal. So I told you this map, this map, this map. So what remains is this map. And we have a long-term shift operator. Because when we define a southwest side of the map, we can actually look at C-star action. So if gamma is C-star covariant, which is the case for posthumism resolution, the C-star actually adds on the fire bundle as well. So we can consider a moderate space of step. And then we, when we define the homomorphic environment, we look at T times C-star or G times C-star equilibrium integral. So then we get such a map. And one new fridge is that before, I mean, I mean, we look at, you know, when we define a southwest side of the map, we look at one map point, hitting some cycle over infinity. But now we look at the second map point, which is you try to hit an input cycle over the zero. Yeah, the way I draw this input side like this, because where is topological retrieval? I mean, this device is topological retrieval. But it has nothing to do with retrievalization away from zero. So that's why I draw it like this. And one important thing, a lemma is that they're kind of a border map. And more importantly, they commute with the A-model connection. And we define the deformation of PRS map by x going to x acting on one. So you see when the input is one, it's just the whole device itself. So the definition is just, it's basically the same as 5SS, except that we are looking at t cross c star, the carrier integral. So we are able to compute it like how we compute this map. And also, we are introduced, after we introduce the C-section, there are some new properties, and we can use this property to identify the connection. I mean, after we construct the map, we have to see that B-model connection goes to A-model connection. And these properties are used in the proof. So yeah, that's the end of the talk. Thank you. Thank you very much.