 Next, he's going to be in Sheffield. So this project is, at some point, the paper will come out on this, but it's not finished yet. Oh, and he's watching the line as well. OK, so what's the sort of aim of this whole story? The aim is to develop a lattice theory for, let's say, type 2 degenerations. And I'll just tell you all these things are of K3s. That's, shall we say, natural for doing mirror symmetry. And I'm not going to say an awful lot about mirror symmetry, partly for reasons of time and partly because that's part of the in progress. But I'm going to say a lot about the lattice theory stuff. So the first part is some stuff on pseudo-lacknities. And this is mostly background, but they're not sort of common objects. If you're after some references, this was really sort of introduced by Kuznetsov based on work of a bunch of earlier people in a paper of 2017. That paper is called Exceptional Collections for Something, Something, Something. The 2017 Exceptional Collections is enough. And a paper that I wrote with Andrew Harder in 2020, which is called, I don't remember, probably has pseudo-lattice in the title. The unique paper written by Andrew and I in 2020. OK, so what is a pseudo-lattice? A pseudo-lattice is, and I'm going to sort of abbreviate it to PL, that pseudo-lattice is not piecewise linear. G, it's a finitely generated Abelian group, front and generally free Abelian group. And I'm going to equip it with a pairing, a non-degenerate bilinear form. It's an integral bilinear form mapping from G cross G into Z. And the important thing with this, what's the difference between this and a lattice? The important thing with this is we do not assume symmetric. OK, so this thing could be a more complicated bilinear form, not necessarily a symmetric bilinear form. OK, so I'm not going to spend an enormous amount of time going over sort of background theory. I'm mostly just going to give you a sort of example and say, point out some of the basic definitions in this. So here is the running example for the first part of the talk. Let's say that X is a non-singular rational surface. I only work over C. I'm not smart enough to do other fields. So X is a non-singular rational surface. I'm going to assume that the anti-canonical linear system of X contains a smooth member. So this contains all of the delpeto surfaces, but it's also much bigger. I don't assume any positivity. I just assume that there's some smoothness. So what we can do, we can build a pseudo-lattice G, and this is the numerical growth and deep group of the bounded derived category. Well, I've got to get my notation right. Bounded derived category of coherein sheaves on X. And we can equip this with the Euler pairing. The Euler pairing is given by if I've got two classes of sheaves, F1 and F2. This is the sum of minus one to the i, HOM from F1 to F2 twisted by i, or shifted by i. So the shift by i means that this thing is not going to be a symmetric bilinear form. So this is going to be a pseudo-lattice. Now, these things are quite general in general. So we're going to have, I'm going to let P be the class of the structure sheaf of a point. This is what's called. So if I'm looking at pseudo-lattices in general, this is what's called a point-like vector. And the reason it's called a point-like vector is because it looks like a structure sheaf of a point. There's a set of conditions. You can define these things purely algebraically. There's a set of conditions that tell you what point-like vectors are. I'm not going to go through all of that. You can read in Kuznetsov, but these are abstractly, these are things living inside a pseudo-lattice that behave like the structure sheaf of a point behaves if you're living in this universe of work. There's some more things here. So if I take your orthogonal complement of P quotient by P, if I'm working in here, what does that give me? That gives me the narrants of every group of X. And so I call that the narrants of every group of G. Again, if I've got just an abstract pseudo-lattice along with a point-like vector, I can do P perp divided by P. Part of the condition, part of the definition of point-like vectors says that this thing is going to be well-defined. This thing is actually a lattice. So the restriction of the bilinear form to this neuron is very, part of the definition of the point-like vector is that the restriction of the bilinear form to this thing becomes symmetric. So this is actually like a proper lattice and not a pseudo-lattice. I can also define KG. Again, this can be defined purely algebraically in terms of the pseudo-lattices, but we can think of it as the class of the canonical bundle of X. And this is called the canonical class of G. So these are things we do. We can also, we can define a SAIR operator. So the SAIR operator on this category is given by tensor by O of X and then shift by N. This is the SAIR operator, it's got the following property. The following property is if I take a pair of things, V and U, I can sort of commute them if I use the SAIR operator. All fairly standard stuff so far, hopefully. But all of this can be done purely algebraically without resorting to talking about the, without resorting to a specific example. Okay, right. So what can I do? Let's say if I've got C in the anti-canonical linear system, let's say this is a smooth curve. So then I can take I would be the embedding of C into X. Then what I can do is I can, I can take the pullback, the pullback by I, this is a map from G to the numerical growth in the big group of the different, all this stuff. And the derived category of coherent sheets and C. So C is going to be a smooth elliptic curve just by a junction. So this is what we call a spherical homomorphism. It's an example of, again, this is something you can define completely abstractly, it's unilatses, but I'm not going to do so. Part of the definition of that is that it has to have what's called a right adjoint. So the right adjoint in this case is by lower star and the defining property of that is the, if I do I up a star of U with V, that gives me the same, this is just gives me an integer, right? That's the same as doing C. So we can sort of pull classes up and push classes down and that respects the bilinear forms. So everything works in this nice example, but we can also, you know, the sort of the thing I want to get across somehow is that we can also do all of this purely algebraically in terms of abstract structures, unilatses, bilinear forms, this kind of thing. Okay. So I'm going to go back to the last one, I'm going to go back to the last one, I'm going to go back to the last one, okay. And I will say I'm going to get even more vague. We're going to get back to being specific in a minute. A spherical homomorphism, let's say F from G to E. E is isomorphic to this thing of an elliptic curve, brackets. So a spherical homomorphism from a pseudo lattice to the boundary, the numerical growth in the group of the boundary derived category of coherent sheaves on elliptic curve. This thing is called quasi-delpetso, quasi-delpetso, if it looks like this thing, I have a star in this example. So again, this is something that, you know, one can make a purely algebraic definition of in terms of the pseudo lattice, but you know, to give a talk where I give all these definitions, I would spend an hour and not get through the definition. So I'm just kind of saying, I'm giving you an example and saying all of this can be generalized purely in terms of, okay, we're all happy. We're given value of happy. Okay, so right now we're going to get into the, well, we're getting closer to the meat. So let's suppose I've got F from G to E and G from G to E. These are two spherical homomorphisms. I can use this, I can put a pseudo lattice structure on the direct sum, G direct sum H, and how do I do it? I do sort of my bilinear form is given by, make myself to say, block diagonal stuff. So I've got, if I've got two things, U and V, what do I mean here? I've got two elements, U and V that are both in G. I take the intersection in G. If I've got two elements, U and V in H, I take the intersection in H. I take zero down there. And if I'm up here, I take F of U with G of V inside E. So this is what to do if I get something where one of them is in G and one is in H, then this is extended bilinear. So we call this F semi-direct sum G mapping from the definition right, G semi-direct sum H to E. Okay, so this is, you can prove this is also a, this is also a spherical homomorphism. You can prove that certain properties of this can be deduced from certain properties of the factors. So in particular, things like, you know, this curve C being anti-canonical is something that, you know, gives you certain properties. Again, I don't know if they're too far into these, of course. Okay, now I have to figure out how to clean the board. Watch the least professional example of this you're ever gonna see. If I do it without getting myself covered in filthy chalk water, I'm gonna be happy. That's the bare minimum. Oh, look, I missed a bit in the middle. This is not too disgusting. Don't shake it. That's how you get it all over. Okay, right. So now, you know, what's the main kind of construction here? Let's say I've got FI from G, I to E. These are Pozzidale-Petzso spherical homs on pseudulaxis, my line down the middle. I'm gonna make an assumption. I'm gonna assume, remember, if I had these canonical classes, I'm gonna assume that the K of G1 intersected with itself is the same as the negative of K of G2 intersected with itself. And we'll sort of, I'll talk about where this condition comes from in a moment when I give you an example of where this stuff comes up. And then what I can define is I can define F to be the semi-direct sum of F1 with minus F2. This thing works out to be a spherical homomorphism. I'm gonna let R be its right adjoint, F. And then we have the following lemma. This is a purely, purely a statement about pseudulax. This does not use any of the geometry here. It's just a purely sort of algebraic statement. So let K be the kernel of F and E bar be the saturation of R of E inside, R of E, R of E, inside G. Then the following is true. The bilinear form is symmetric on K. Bilinear form is symmetric on K. And E bar as a subset of K is totally degenerate. So what do we do if we've got a totally degenerate sub that's of a lattice? We can quotient it. So what I can then do is I can write down an exact sequence. Zero goes to E bar, goes to K, goes to K mod E bar, goes to zero. I can call that thing M. Okay, that's the kind of algebraic background. Now I'm gonna show where this comes up in John. All right, so everything I've said so far, I sort of, you know, I've waved my hands an awful lot. But, you know, the takeaway message is we can do these things called pseudul lattices. They can have various properties if you don't care too much about if they're sort of particularly nice, we can glue them in pairs. When we glue them in pairs, we get this set of properties. So we get, we take the kernel of this, this gluing map, we take the saturation of the white adjoint, then we get this. Okay, so where is this gonna, where is this gonna show up? Let's look at sure in the generations of capers. We're in the generations of capers. So I'm gonna take curly V over a complex disk. This is gonna be some degeneration. In here, I've got a point zero. Over that, I'm gonna have V zero. I'm also gonna have V and that's just gonna be over some sort of generic point T. I'm gonna assume, this is what's called a Turing degeneration. So what's this gonna be? It's gonna be V naught is gonna be a union. If I got my notation right, if I don't get the notation right, it's a C. So this is a degeneration. I'm degenerating my K3 surfaces. I'm taking a very special type of degeneration. I'm degenerating them to unions of two surfaces, glued along a curve C. So my degeneration looks something like this. So X1 and X2 are rational. And C is smooth anti-economically. Yeah, C is smooth anti-economically. So somehow, these things, there's old results of Friedman. Somehow these things can be thought of as primitive type two degenerations of K3 surfaces. You can sort of think of these things as like minimal type two degenerations of K3 surfaces. Other type two degenerations can to some extent be obtained from these by base change. Do I want to say anything else now? Probably, actually, let's put some more on here before I raise things. So as we saw before, I can let GIs be these things. I can let the FIs be the map from GI to, you know, this thing, K naught num VB co of C. And, you know, everything is I'm in the setting that I was in before, right? You know, I've got these things are quasi-delpetsosunilatices. I've got a pair of them. I've got these maps. So what am I going to do? The semi-stability gives me this condition that the intersection of the canonical devices on one has to be negative the intersection of the canonical device on the other. This is condition on such a degeneration for it to be smoothable. So we should, in theory, well, the idea is we will see these suilatices showing up kind of naturally in the situation of these generations, getting better at this thing. So here is the next lemma. So over Q, what do I get if I take this suilatice G tensed with Q? So what was G? G was defined just exactly as, oh, I didn't tell you what G is. This is the gluing of these along F1 direct sum, minus F2. So it's that over Q. This is isomorphic by the churn character map to the even cohomology of X1, direct sum even cohomology of X2, where this is Q cohomology. And moreover, KQ by the same thing is isomorphic to H0 of V0 direct sum second weight graded piece of H2 of V0 direct sum H4 of V0. So somehow this is the middle graded piece. I mean, these ones, the mixed hodg structures are not interesting. I mean, they're pure on the top and the bottom. So this is the middle weight graded piece of the mixed hodg structure on the cohomology of the central fiber. So what have we achieved here? What we've achieved here is we've achieved in a sort of natural way coming from the lattice theory of the two components. We've achieved the lattice theory on the appropriate graded piece of the cohomology of the central fiber of the generation. So we've got that, then we can look, we've got this exact sequence. Can we identify what that is? Those of you who study degenerations and mixed hodg structures probably know where I'm going already. So M, this portion up here is isomorphic to the limiting mixed hodg structure of H0, second weight graded piece, limiting mixed hodg structure and H4 limb. Yes, K mod E bar is this thing. This is K mod E bar. Wait for the next statement. So now I haven't missed a cue here. This is a legitimate isomorphism over Z. There's actually a Z isomorphism between these two things. So over here we've got the cues, but this one we do not need the cues. So what does this give us overall? I can take my exact sequence, tensor it over Q, and this is gonna give me zero goes to, I get two classes here, Q eta and Q xi. I'll tell you what those are in a second. This goes to H0, V0, direct sum. The thing that's K, and this goes to H0, limb, V, direct sum, blah, blah, blah goes to zero. What is this? We all recognize this sequence. This is Clemench-Schmidt. So this is the Clemench-Schmidt exact sequence, and what we've done is we've endowed it with a lattice theory from these, coming from these two relax. Yeah, maybe, maybe I need like just second weight graded piece. It's just second weight piece rather than second weight graded piece. Let me check it off. I've got a set of notes and maybe there's, maybe there's, maybe this is not exactly, it's something very much like. It's some weight graded piece in the Clemench-Schmidt, but I'll need to check my notes that aren't these ones because it's sort of technical. I can tell you what xi and eta are. xi is the class of minus C, C in H2X1, H2X1, direct sum H2X2, and eta is the class of minus a point, a point in H4 of X1, direct sum H4 of X2. So these are sort of very nice identifiable things. Now there's results of, so you can ask like, okay, I've got these cues here. The cues are not really ideal. This isomorphism over Z, what goes wrong? There's results of freedom from 1984 that say that actually, this exact sequence at the bottom actually holds over Z. So do we have any hope that this is actually a set of the dead isomorphisms? The answer is no, but it's not very far off. So actually this KQ, there's a failure of isomorphism between the integral part of this and the integral part of that, but that failure is restricted to H4. You can get rid of the H4 by doing the following thing. So if you restrict to Neuron-Severi, what do you get? You get zero goes to, then everything becomes exact over there. Well, I'm gonna write this and it might make Matt angry. Now over Z goes to GRW2, H2-LIM, BZ goes to zero. And this is something which, this limiting host structure is something which you can really identify if you've got a degenerating family of K3s, this is something you can work out lattice theoretically. This is a lattice you can calculate. So here we've actually, here we've actually, yeah, so maybe, yeah, we'll figure it out. So, here we've got an actual sort of proper recognized exact sequence of lattices. Okay, so so far, what have I done? I've done some complicated algebra. Well, not so complicated, linear algebra. I've done some fiddly linear algebra to define these pseudo lattices. And then I've derived using them some results about the generations of K3 surfaces that have been known since before I was born. Big deal. So, what's kind of the point of this and where is it going? The point of this and where it's going is that I can make statements about what the mirrors for all this stuff should be. And that's what's coming. That's sort of the next bit when I have watered the board. So this whole picture shows up in a second way. Elliptic vibrations. So, let's suppose I've got pi from Y to P1. This is a elliptically-fibre K3 surface. I'm gonna let f be a fiber. And I'm not, when I say elliptically-fibre K3 surface, I'm specifically not going to assume there's have to be a section here. I'm just gonna say we've got something like this. I'm gonna have gamma living in P1. This is a loop and using it, I can decompose P1 into a pair of disks or, you know, things topologically equivalent to disks glued along the curve delta, glued along the curve gamma, sorry. And I'm gonna let pi i from Y i to delta i. These are just the restrictions of this vibration to each of these two disks. Let's freshen up the chalk. Now I can define a, so I can look at the second relative homology of these things Y i relative to a fiber. If I've chosen my fiber appropriately to be on the curve gamma, there is a bilinear form on here. So this has a bilinear form. It's called the cipher pairing. And with this, this gives it a pseudo lattice structure. And I can get a spherical homomorphism. What's the spherical homomorphism? I can take the boundary map from this to the first integral homology of an elliptic curve. And this, you can give it symplectic basis. It's got a symplectic intersection form. So this thing is a pseudo lattice. And as a pseudo lattice, it's equivalent to the, it's the same pseudo lattice as the numerical growth and deep group on the right category of boundary coherent sheaves on an elliptic curve. So this thing is isomorphic as a pseudo lattice to what we've been calling E before. It's the same thing. And that's really a manifestation of homological mirror symmetry for elliptic curve. It's just saying that, this is sort of the symplectic focaya category statement. I mean, well, okay. This is numerical growth and deep group of the focaya, appropriate set of adjectives focaya category for elliptic curves. On the other side, you've got the boundary graph categories. They're the same thing. Okay. So you've got that thing. I'm going to make two assumptions on the loop gamma. I'm going to make two assumptions. I'm going to assume there exists a symplectic basis, AB for E such that anticlockwise monodromy around gamma acts as, and when I say anticlockwise monodromy, I mean for either of the discs. I look at each disc and this should be true for each of the two discs. It acts as follows. If this is A and B, A and B, I should get one, one, zero. And the number up here should be E of YI minus 12. What's E of YI? E of YI is the topological oiler number which you can compute for these vibrations over discs by just counting up the contributions from each of the singular fibers living inside each disc. The second condition is if R is the right adjoint, so phi, phi I, think R I, then R I of A is primitive. We think this second condition might not be necessary. We don't know any examples where it fails, but in order to prove the things that come afterwards, we need to know this to be true and it's still a work in progress. So I'm allowed to make assumptions that I might get rid of later. Under these two assumptions, these two maps phi I, these are then quasi-delpezzo homomorphisms of pseudolattices. So I'm in the setup I had before. I've got two quasi-delpezzo homomorphisms of pseudolattices and so using the stuff I was doing before, I can glue them together. So if I glue them together, what do I do? Yeah, I'm doing all right for time actually. I was a bit worried. This was a one hour talk and in the half hour before now, I have cut 20 minutes out of it. So you may notice that there are 20 minutes of this talk missing and they've been excised from the middle and that's why none of it makes any bloody sense. If you want the other 20 minutes, buy me a beer in the pub later. Yep, I'm completely agnostic. We're in Germany, there is no bad beer here. And if you think there is, you haven't been somewhere that's got really bad beer. Okay, so I can again take this direct sum. This is now the direct sum of these, direct sum over I of these two things. What do I get for K this time? K works out to be the compactly supported second homology of Y with F taken out over Z quotiented by the class of a fiber. So this is what my thing K works out to be M works out to be the orthogonal complement of F quotiented by ZF. So both of these both, well, M, this orthogonal complement is taken in H two of the K three surface. So this is sort of subset of H two of Y integral second homology of Y. Y homology, yeah, homology of Y. Make sure I'm on the right place and what does our exact sequence become? And this is exact over Z not exact over Q M, the other thing, right, zero. And this is zero goes to the homology of F goes to a compactly supported homology of Y less F over Z quotiented by ZF. I've made a hash of this, haven't I? And this goes to F over ZF, this goes to zero. That's an isomorphism, that's an isomorphism, that's an isomorphism. And moreover, you know, this is not just an isomorphism of Z module, this is an isomorphism of lattices where this thing gets the usual lattice structure from H two of Y, right? So I can induce a lattice structure on this from H two of Y and this thing is getting its lattice structure off of these unilattices. So what have I done? I've sort of identified this picture and what is this sequence? This maybe doesn't look terribly familiar. This is what Chuck and I in a paper last year called the Mira-Klemmenschmitt sequence. When I say last year, I really mean two years ago. So you might ask, you know, okay, but on the other side, really the interesting information was contained in the Neuron-Severi part, right? I mean, you know, if you take the Neuron-Severi part, it cuts off the H naught and the H four and then what you're left with is sort of a standard exact sequence that we recognize for type 2 degenerations of K3 surfaces. If you do the same thing on this side, I mean, you can do it, you can take these Neuron-Severi lattices, but what the picture you get is not, does not seem to have an obvious geometric interpretation, right? You get some sort of, you know, complicated sub lattices of these things, but you know, we don't, there's not some nice clean way that we could think of to write these down as, you know, nice comologies. So there is, you know, this sort of, we take this as kind of an indication that, you know, working on the Neuron-Severi's, if you want to do mirror symmetry, it's a little bit of a red herring, the Neuron-Severi don't work so much, don't work so well over here. Really, if you want to do sort of some lattice theory here, you've got to be working with the pseudo lattices. Okay, so I've said mirror symmetry a lot. So in the last closing minutes, let's say something about mirror symmetry. So what's the sort of motivation behind all of this and where we're going to, this conjecture was never really written down, but you know, sort of comes out of a circle of ideas by Chuck and Andrew and me. So in this sort of situation, it says something along the lines of, if X and X prime are a mirror pair of K3s, then we should have a mirror correspondence between type two degenerations of X being mirror to, let's say, elliptic vibrations on what? Elliptic vibrations on X check. Now, if I've got a type two degeneration, these are not necessarily unique. If I go to a type two degeneration, you know, there might be multiple different models for the same degeneration, multiple ways of completing the family. So we have something like projective Kulikov models should correspond to these splittings of P1. So taking a projective, yes, I don't know. That, yeah, I don't, yeah, I don't know the answer. It's not that's not what you really need to do. But maybe we can check about, maybe you know something right. Okay, projective Kulikov models should correspond to splittings of P1. And moreover, if you take these splittings of P1 and cut them apart, the components of the degeneration here should correspond to LG models. So you can think of an LG model as like an elliptic vibration over a disc or more generally elliptic vibration over an A1. So the components here are rational surfaces. No, in nice situations, there'll be del petso surfaces. Those are mirror to LG models. You can think of a sort of elliptic vibration, you kind of think. But the point is that like, the pseudo-laxis picture that I very hastily presented is the same on both sides. So over here, we've got these pseudo-laxis G1 and G2. On the other side, we've got the same two pseudo-laxis. These correspond in this picture. So what we're hoping to do is we're hoping now, well, okay, all of this is kind of the ambient space to work in, right? So when you're doing mirror symmetry of K3 surfaces, you're talking about lattices and these sort of orthogonal complements of lattices happening inside. Yeah, you talk about lattice polarizations, orthogonal complements, lattice polarizations. We're hoping that using this, we can lift that whole theory from lattice polarizations on X and X check to pseudo-lattice polarizations on the type two degenerations or the elliptic vibrations so the whole picture will be compact. So that's where it's going. Okay, thank you very much. Well, thank you. So we have time for questions or comments or discussion. Yes. Yeah, I'm just having general questions just out of curiosity. Do you know kind of the string theory literature in this direction? Because this is something which has been studied quite extensively where these type two degenerations are like stable degeneration limits and then you have a dual heterodic model there and you have to respect it. I know little bits of this, but probably not as much as I do. And I don't know what the necessarily, what the interpretation of this is going to actually be connected to this. I know, I mean, most of the things, so the challenge here, you know, there's a lot of things I can get into and sort of the challenge here is understanding the structure of the pseudolattices. So the challenge of what we've done here so far is understanding the pseudolattice structure on these where these YIs, the fibers are not necessarily all I want. So the fibers don't necessarily satisfy this thing where they're like simple pinch tori. And I haven't, that's something that I haven't really seen much in the literature anywhere. In most cases, people seem to, it's sort of, you know, a lot of the literature, people seem to consider vibrations where they say, okay, you know, we assume that all of our singular fibers are like simple, more singularities. And then we do some protective geometry kind of style stuff. Yeah, yeah, yeah. And I've seen, I mean, actually now, I mean, there is some stuff here because the You want to mention the string junctions motivation that you and I- Yeah, yeah. So we started off with this with, we started off looking at string junctions and there's the junction pairing in the string theory literature. And the junction pairing works out to be precisely the symmetrization or it's either the symmetrization or the anti-symmetrization of the pairing that we've been looking. So this, there is a relationship there, yeah. So, and I'm now trying to, we ran into some problems with that but it was about eight years ago and so I've forgotten what the problem's done. Yeah, I mean, we have, there have been sort of, there have been relationships that we've observed here. So, you know, there's sort of some old work by, what am I saying, old, like 10 year old or so worked by sort of Grassy and Halverson and Shanison, where they study these kinds of vibrations and they get, and also like the Wolframs, why that? And they get these kinds of decompositions and we show that the same decompositions hold on the pseudolattice level, which is sort of, you know, you can deduce their results from ours but the other way, as best I know, doesn't hold. We've got, you know, they've got these sort of symmetrized pairings and ours has got this non-symmetry which makes it more, but I guess ours is our picture is more natural from a homological mirror symmetry standpoint. So this is all kind of, you know, pairings on numerical growth and deep groups of, you know, derived castings of Coherent sheaves or focaya castings depending on the size of it. But if you do want to see a tie-in to the heterotic type two and F3, Mohsen, my postdoc is going to be talking about that in this lecture, that we're turning it around using this to perform the other dualities. Yes. In the situation when you don't have maximally unipotent monodromes, is there any version of SYZ mirrors, symmetric conjecture or anything like that? I'm not sure. I mean, there is some maximally unipotent degeneracy hiding here. Okay. And the place that it's hiding is that, you know, there's some technical assumptions on your type two degeneration has to be, has to occur along a locus, which is incident to a point of type green, like maximum degeneracy. So you can, I mean, the idea is to try and sort of interpret, make some mirror symmetry interpretation statements, which work sort of away from, you know, allow you to move away from smooth lobes, talk about the singular things. There is some SYZ related stuff here because ultimately, I mean, you know, we've got elliptic vibrations on K-P-Zero. So, I mean, these are to some extent an SYZ type object. And I think the predictions of SYZ effectively boil down to kind of, I think there's some statements you can deduce from SYZ, which give you kind of some mirror symmetry between these two exact sequences I've got. But we're hoping to go further and do something really with the lattice polarization and that's sort of where this is heading. But I'm not sure SYZ has a lot to say in that sort of, but yeah, you can, you can to some extent see this as a duality between, I think if you're looking at sort of generic K-free surfaces with a sort of generic factor degeneration, you can see this as some sort of duality between, you know, like a symplectic vibration and now it's a great vibration. I don't see any more questions. Oh, you're fine. Okay, fine, thank you. I was going to check it, but I won't now. Ha, ha, ha. You're done. That was it. Okay, thank you. So let's think, I'll speak to you one more time. Maybe I'll go through to stop the video and start.