 Okay, welcome to part two. Thomas, please go ahead. Yeah, okay, so, right. So to recap, both in the Grisimani in case and in the Toric case, there is a positive part of these spaces. The positive part is a real semi-algebraic thing as a topological space. This topological space has a phase post set and the phase post set matches in the Grisimani in case the positive or a stratification and in the Toric case, it matches the Torus orbit closure stratification. And this analogy is pretty potent. So there's a lot of things that are in parallel. I wrote down what the top form is on the Grisimani in, but I think I won't explain this formula. Let me go on to talk about coordinate rings. I'll come back to this analogy between the Toric case and the positive Toric case. Maybe I should have done this first is, since this is more an analogy with Schubert geometry than with Toric geometry. So if you look at the homogeneous coordinate ring of a Schubert variety, which you can define to be this ring, which is you take global sections of these various line bundles, various twists on the Schubert variety, then you get this homogeneous coordinate ring of the Schubert variety. And these vector spaces, the global sections of a line bundle on a Schubert variety, they are things called Demezurial modules. There's a similar description of the coordinate ring of a Positroid variety. It's again, what you do is you take this line bundle of D and you look at the global sections along the Positroid variety. And I call these things Ciclic Demezurial modules. And up to a dual, they're basically defined in this way. Actually, this is not a hard theorem at all. It's that you can take a bunch of Demezurial modules and then appropriately rotate them. So this chi here, this chi here, your Ciclic rotation. So it's whatever the map is on these global sections induced by the action on the underlying vector space, C to the N. On the underlying vector space, there's a Ciclic rotation that takes the basis vector EI to AI plus one. And that is an element of GLN. And that will act on these global sections. And if you sort of intersect N of these Demezurial modules and you get something called the Ciclic Demezurial module, and this thing is quite interesting because it has a crystal basis like the, like, well, like sort of irreducible representations of GLN and also like Demezurial modules have canonical bases and crystal bases. So does this intersection. You can show that this rotation is compatible with these constructions. So this was also studied by Lakshmi Bhai and Littleman and studied from a different perspective in this recent work of Moosa, Gao and Huang from a Grubbner basis perspective. I think something that I've been wanting for a while but have sort of made no traction on is the question of what is the character of this Ciclic Demezurial module. For example, what is the dimension of the space of global sections of a line bundle on a positroid variety? Can I find a formula for that? And we know there are nice formula in the case if instead of positroid variety, if I put Grismanian then there's this wild character formula and wild dimension formula. In the case of Demezurial module there's Demezurial character formula for this a recursive formula that computes what the character of this thing is as a T module. Same thing, I would like to know what the character of this is as a T module. As far as I know, nothing is known about this question. It doesn't seem to be an easy, at least as far as I know there's no easy way to get it from something that's known. Now there's another coordinate ring to look at. So this was coordinate ring of a closed positroid variety. You can also look at a coordinate ring of an open positroid variety. And a couple of years ago with Galaxian we proved that the coordinate ring of an open positroid variety is a cluster algebra. So the open positroid variety, so the closed positroid variety is irreducible normal CNF. Open positroid variety, it's not projective, it's smooth and affine. So the properties are a little bit different. So you don't need to take homogeneous coordinate rings you can just take its coordinate ring and this open positroid variety is just spec of a cluster algebra. This cluster algebra has been around since the work of Posnikov but it took a while and there's here a bunch of references. Now, before we eventually proved that it was a cluster algebra. And actually this is a little, it's a weak analogy but it is a sort of an analogy of the toroid case. In the toroid case, the strata are tori and tori are very, very simple cluster varieties. And something that I thought would, I really underestimated when I was working on this problem was I thought, oh, surely this result that the coordinate ring of the open positroid variety is a cluster algebra means the coordinate ring of the closed positroid variety is also a cluster variety by somehow sort of de-projectivizing in some particular way. In the way that Scott presented the homogeneous coordinate ring of the Grismanian as a cluster algebra, it also showed that the top open positroid variety was a cluster algebra as well. So there were two things being shown there. A little bit surprising is I think it is still not known whether the homogeneous coordinate ring of a positroid variety, which is this thing that I'm talking about here. The thing who is a graded ring whose graded pieces are these cyclic demazurial modules. I don't know whether that is a cluster variety and has something to do with obviously these two guys are related by, I mean, these two rings are related by some kind of localizing. Here we have to invert a bunch of Pruca coordinates. But it seems a little bit surprising to me and maybe I'm wrong and it's really easy that there's some obstruction to easily proving that you can get rid of the things that you invert and it's still cluster. Okay, yeah, so I won't say much about this. Let me move on to, I want to try to state one of the mirror symmetry statements for the Grismanian and the sense of mirror symmetry that I'll be talking about is the sense due to Given-Tau. I mean, there's a lot of work on mirror symmetry and Given-Tau popularized a point of view where you're allowed to pair a funnel variety, not a kalabiyau as initially in mirror symmetry but a funnel variety with something which is supposed to be mirror to something else and that something else is not a funnel variety. So it's not mirror to the same kind of thing. It's mirror to something called a Landau-Ginsberg model. So in the mirror symmetry statement for the Grismanian that I want to talk about is that the Grismanian is mirror due to the open positroid variety and a potential. So this thing just to start out, so this is the dense open positroid. So there's one positroid variety which is dense in the Grismanian. So that's the top one. So that's what this notation is. And so it's mirror not just to the open positroid variety but the open positroid variety equipped with a potential. This potential is going to be a holomorphic function on the open positroid variety and it will be a rational function on the Grismanian. And the formula for it, so there's a number of different ways to write it. The formula, the way I've written is due to Martian reach. This potential looks like this in terms of Pucca coordinates. So in the denominator are the cyclic miners. So the miners, the Pucca coordinates labeled by I up to I plus K minus one. And then you change one of the, you swap out one of the indices to make one of the things in the numerator. It won't be important for us, but there is a queue here if you, there's a quantum parameter somewhere in this potential. Okay, so what is the statement of mirror symmetry? So when you say the sentence that the Grismanian is mirror due to this, what is the statement? So I want you to find a version of mirror symmetry where the statement is sort of in the spirit of all the things I've talked about so far. So here is the statement. The statement is that on one side, we just take the cohomology of the Grismanian. And on the other side, we take the twisted cohomology of the open-positroid variety. So what is this thing here? Yes, first you take algebraic forms on the open-positroid variety, which means the algebraic DRAM complex of the open-positroid variety. So remember this open-positroid variety, it's a special case of one of these things. So its coordinate ring is just this cluster algebra. So we're just taking the module of scalar differentials of that ring. But instead of using the usual derivative, we use this twisted differential. And so the usual algebraic DRAM complex would just be this part of the definition. So you just take the forms on the open-positroid variety, algebraic forms, and just take the usual exterior derivative. Now, the extra information of this potential, what we're asked to do is to twist it. So instead of taking the derivative to be D, we twist it by also doing DF wedge. So this DF wedge, so DF is going to be a one form, and DF wedge will do the same thing as D. So D will take a zero form into a one form, a one form into the two form, and so on. DF wedge will take a zero form into a one form, one form into a two form, and so on. So it goes from the right pieces in the DRAM complex. And so the mirror symmetry statement is that if you take the cohomology of the algebraic DRAM complex twisted, twisting the differential in this way, then it's isomorphic to the cohomology of the Christanian. Let me say that clarify this even a little bit more. So the right hand side should be somewhat familiar. So the left hand side is the sort of foreign object. This left hand side is a complex, and it turns out the complex, so without taking cohomology yet, this complex only has cohomology in one degree. It only has degree, it only has cohomology in the top degree. And in that degree, the cohomology group has dimension and choose K, which is the dimension of the cohomology of the Christanian. So that's one of the statements of the, a consequence of the mirror theorem for the Christanian is that this weird thing that you define from the open positron variety, it has cohomology just in one degree and the dimension is equal to this number and choose K. Yeah, so let me emphasize again that, I mean, this right hand side is familiar and this left hand side is not, that this left hand side is a purely algebraic construction. So now that we know that the coordinate ring of the open positron variety is a cluster algebra, that's a ring and if we think we understand that ring, then we can apply just algebraic constructions to build the algebraic Dirac complex from that ring and compute this. That's not how the proof goes, but I'm saying, in principle, this left hand side is some purely ring theoretic digits. The open problem is, and people have looked at this for a while now and I think still we're not very far here, is that what if instead of doing the top piece, so I said, Christmanian is mirror due to the top positron variety with this particular potential. The conjecture is that the closed positron variety should be mirror due to, I think the open positron variety with some potential and therefore there should be some, what this mirror duality statement should say is something analogous to this. So it should say that the cohomology of a closed positron variety is isomorphic to the twisted Dirac cohomology of the open positron variety with respect to some potential. This potential should have a shape that's not too different to this. So in the mirror symmetry world, this particular form of the potential, actually each term in here is supposed to correspond to one of the boundary devices. So actually there's a bijection between these terms in the potential and the positron devices in the Christmanian. So if we understand the positron geometry well enough, we should try to guess what this F is for a positron variety. And this is a place where the, I think the analogy with the toric case is very potent. So one of the test cases for given how final variety conjecture is if you take a final toric variety, then it's mirror due to, what is it mirror due to? Is it mirror due to just a Laurent polynomial? So it's mirror due to a super potential on a torus. And one way to think about this torus is that this torus is sitting inside a toric variety. No, maybe a mirror due toric variety, not the same one, but this is a dense torus. This is one of the dense torus orbits. And then on it, the super potential will be some, some Laurent polynomial. There is a question in the chat. Yes, pi F final. Yes, I think I would call it, I would call it final. So the anti-canonical divisor is the class of the union of the positroid divisors inside that positroid variety. And that's effective. Okay, so finally, I want to end by talking a little bit about the topology of these positroid varieties. And so I can compare to Schubert's cell, then it's a little bit boring. So I want to focus on the topology of the open positroid varieties. If I compare to the Schubert cells that Schubert cells are boring, they're contractable, they're just affine spaces. So there's no interesting topology in the Schubert cell, but there is interesting topology in the Schubert variety. However, if I look at the open strata, then I could also compare to the toric case. And there it's not contractable because it's a torus, but it's still a little bit boring. So it turns out that the cohomology of these open positroid varieties turn out to be pretty interesting. So as I said, the smooth affine complex varieties, and it's got a mixed hot structure. And the mixed hot structure of an open positroid variety is not pure, but it satisfies some condition that it's mixed-tate or hot-tate, which makes it a little bit nicer. So the mixed hot structure of Deline endows the cohomology, a particular cohomology group, say the cave cohomology group of a complex variety with some additional filtration. For simplicity, I'm gonna think of it as something called a Deline splitting. So it's just a direct sum over p comma q of some additional pieces. And in the mixed-tate or hot-tate case, these pieces are non-zero only when p is equal to q. So then what you get is basically a bi-grading. So I'm just gonna think of a mixed hot structure as a bi-grading of the cohomology. So an additional grading. So a relatively recent theorem of mine in Galaxian is that we can calculate the mixed-hodge polynomial of this top positroid variety. And it turns out to be a sort of familiar polynomial. So as I said, in general, the mixed-hodge structure gives you sort of the free indices k, p, and q, but in the mixed-tate or hot-tate case, there's only two indices. And I'm not gonna write down the exact definition because there's a lot of indices to remember, but you get a bi-grading. So you're gonna end up get a bi-grade. The mixed-hodge polynomial is gonna have a two-variable polynomial. And it's a little bit more elegant to state the result when you mod out by the free torus action. So there's a torus sitting inside p, g, l, n, that acts on the columns of this k-bi-n matrix, acts on Grismanian. If you mod out by this torus, then this torus acts freely. It turns out on this open positroid variety. If you mod out by it, then the mixed-hodge polynomial is just this qt-catalan number. So I pretty well studied two-variable polynomial that generalize that. So if we forget the two variables, so we just plug in one, then we just get the catalan number back. So a consequence of this theorem is that the Euler characteristic of the space is just the catalan number. So it has this really elegant simple formula, one over n times n choose k. And this is, I mean, this is relatively new stuff. So we've known this for two or three years now with Pasha. And we're still thinking about, we're still thinking about how this works. So in some ways it's still a bit mysterious to us, but it does show that this open positroid variety has topology that's extremely interesting. It's Euler characteristic is a catalan number. So it's a very nice number. One feature of this is this generating function only has positive signs and that reflects, I mean, it's in the conventions of mixed-hodge polynomial, but basically it reflects the fact that this space only has even co-amology. And this is something that I think one of the immediate open problems in this is, why is it that this particular positroid variety, when I'm mottled by Taurus, I get something that has only even co-amology. So it doesn't have a stratification to even cells. It doesn't have, so it's not like the Grasmanian, it's not a projective thing with a bunch of affine, with an affine paving, but its co-amology is even. And the open problem is for which positroids is it true that the co-amology is even after you mottled by the Taurus? And that's something that's been puzzling us for a couple of years now. Okay, I think I'll stop here. Thank you. All right, well, thank you very much for a great talk.