 about the generalization of the DHT MIRROR construction and MIRROR P equal W conjectures. Thank you very much. First, thank you to the organizers for inviting me to give a talk. Today, I'm going to describe a generalization of the Cologne-Hartmann-Thompson MIRROR construction, which was opened yesterday and also talk about the two different MIRROR P equal W conjectures. This is different from the original DHT with the conjecture in context of the MIRROR construction. The first one is proposed by Harvard-Hartmann-Thompson MIRROR construction speech in case for the MIRROR symmetry for the local avian practice. The second one, you can know recent, in case in the context of the degeneration and vibration. So we view all the things and try to relate to them. So let me briefly recall some backgrounds. Yes, like the most of the audience is familiar with this. Great stories of the MIRROR symmetry. The MIRROR symmetry is for Kallavi-Al, and the other Kallavi-Al many words. There is a relation between the conjecture of one side and then the conjecture of the other side, both inside and inside. And there are several formulations in mathematics. There are several formulations of different condoms. For example, the most basic form using the odd signings. For the more general form, also, probably the category for the MIRROR symmetry in 1994. On the B side, we consider the R category for in sheet and then on the other side, this is called like a tag headache, but we're not going into detail about that further. The more interesting story happens when we try to look at your metrics and interpret the relationship between Kallavi-Al many words. This is called a fidelity proposed by the family of the other. What it says is that they form a special dual lagrangian vibration and they are dual. This idea allows to explore the MIRROR symmetry beyond the Kallavi-Al case. In particular, our main stress is that when we have like the final pair of the final pair, we're actually following and then we can take them and divide them. And then S-Y-D construction tells us that the MIRROR objects are not just a single Kallavi-Al many words, but also to be the certain polynomial function. The idea is that we basically like to construct a polynomial with potential potential inputs from like a lot of accounts of the disk, which touches the MIRROR. In this story, there are like interesting three kinds of MIRROR. The first one is a final self and then the second one is the MIRROR of the Kallavi-Al word. And then the last one is a complement of the unites and then the full capacity. So the interesting question is how these three are related. One way to present this, like to understand the plump reality between them. For example, the MIRROR symmetry connector could be stated as not only just from the equivalent of the category, but we also want to construct something to conquer and then get the equivalence somehow like incredibly. I didn't write down how you and why MIRROR symmetry for you and why is related given by the category localization. But the more interesting thing is that this equivalent is expected to be compatible with a certain old polynomial in this category. On the P side, we have some monitoring of the potential around these things, getting used to a certain conquer and then you would expect that it would be compatible with the old polynomial. So this is, it plays quite a critical role in the way you would say that the line bundle and then the line bundle correspond to the line bundle, and then the line bundle correspond to the line bundle, and then the line bundle correspond to the line bundle and then the line bundle correspond to the line bundle. The main question that I want to understand is this, like how the MIRROR symmetry of Calabria and the MIRROR symmetry of the final and related. In case where you have a Calabria generation of Calabria went to, that was two pieces of the final puzzle. So we're on how the Thompson answers this question, in the case of Korean generations, in many insightful ideas and backgrounds. Then we call a little bit more about the generation story. We're getting to how to generalize this. We're going to think about the generation of Calabria. If you know the generation, the central fiber is similar to the other fiber. If you say that the generation is called semi-stable, let me set the full-touch page as soon, and then the generation of fiber is simple no more important. A semi-stable degeneration, we can talk about the type, or the type M platform, the dual boundary complex of the central fiber is of dimension. So let's access the central fiber to some union of M plus one new variety. The original question is about the freedom on it. How do we know about the what the set is? The smooth ability of this, when we think of this. It's in the degeneration of the generation fiber of semi-stable. And then one of the key necessary conditions is called the de-semi-stable. Here, it's basically saying that the normal bundle of the double loafers, section three, and then you think about, think about it. Then what we have is that we have a this degeneration fiber, and we consider it as a divider in the typical space. And then you can take the one of the components of the double loafers, then we write in the left-hand side that there's a normal bundle of the components and other parts of the original panel. It's sort of like this, right-hand side is called the normal class, or the components of the double loafers. And then collection of all these normal class, also I think, and then this damages can only affect the sensor. So I want to focus on like this condition, because there's triviality of this one. That said, like this is just some necessary condition, not position. You're not going to call it math and math and so on. That when you have a de-semi-stable, and this is a flatial, and with some extra conditions, with the homology, then you can actually smoothen this normal class invariant into the width of semi-stable. So the next definition is not pushing, but the object that I want to study is simple normal class invariant, which is called the semi-stable type n problem. It means that I can put this into the degeneration fiber of the semi-stable degeneration of type n. So I'm basically thinking about this, you know about the complex and standard So the HG construction, let's take n equals one, type 2 degeneration. It means that we have like a lobby out, which is generated of the system problem. Their intersection is n. And then the de-semi-stable condition is given by following the relation. It's a normal bundle in one part, and then we have the center of the normal in the other parts. We already saw in the product's condition. And now, suppose that we have a two-landed-gain stock model, we have pair, and then that's the large enough piece. The mirror of each component of this bundle, in this picture, so we have mirrors of this component of the degeneration fiber, and then we want to impose some psychological condition in the sense that around the boundary, we have a lot of the same type. It reflects the fact that we can actually really need to make sure it's normal for design. So we can just rewrite the de-semi-stable condition under the canonical line bundle of the normal de-semi-stable model. And what this tells you is that the following relation, where P0 is the normal de-mirage big model, but we have all things along the way that don't create a value. And we have the following. So to me instead, it's like we have the normal de-semi-stable model, you know, like the inverse direction, and then it can be identical, P0, P1. So what we can do is, such as to philosophically glue these two random models, which produces a proportional vibration. And then the first observation is that this is a fluid object, which can be politically mirror solution of the normal de-semi-stable model. We are basically saying it's starting from the mirror of the de-semi-stable model, so that's why it's important to glue them to give you the tang, the logical tang of the de-semi-stable model. Volatile mirror means that we only have to have the order-tacture relation. Of course, they did more than this, the period of the number, and you can do it, and then three k's, and then provide like a bunch of evidence. So the direction that I want to go is the following question, which is, like how the general life is going to fall younger? Yes, yes, yes. Sorry, like your x01 equals x10, if you call this smooth D, they contracture the fibers of y are mirror to this D. I just wanted to... Yeah, yeah, when I talk about like the mirrors of here and the fibers of there. Okay, well, thanks. For example, we think about like the degeneration into three components, and each component has an anti-canonical device coming from the intersection with the other two components. So now we have a rumor closing anti-canonical device with each component. And then we have three laws, for example, just write them. I believe we're looking at three lambda-game components. We're doing three disks. It's not quite doable. The idea is that, if we're just looking at the lambda-game component, then this is an anti-canonical device. So we know that the corresponding model will, but whenever you have a rumor-closing anti-canonical device with this, it will only give you one of the components. We should know like more than just telling the device and how you can follow up what we're telling the device and what funds to the certain data. But we need like more than just... The idea is that, like, from the SYD perspective, lambda-game components can actually account for some bit which such as the anti-canonical device. So now we have like the more than one unit of the component in the anti-canonical device. So we think that three counts will be just touching each unit of the component separately. So this idea, I should suggest that we should think of, like, each account separately. So instead of a single potential, we need to look at the multi-potential component. And then take the sum to recover the original lambda-game. So when he has a... He has more than one potential. So say, like, he has two components. So let's be one, two, and then what we would like to consider is the... So this is all the expectation. So the mere expectation shows that X first points to the original lambda-game model. Then B, the anti-canonical device, the fiber of the lambda-game. And then complement the quantity of the lambda-game. But the advantage of thinking about its potential is the problem. We can actually know the mere information of the one of the components. We can treat its one of the components as a problem. It turns out that you take the B image of the genetic fiber of the function H, which comes with the vibration property from the remaining function. And that's also low of the lambda-game. So it's the same for the potential. The DPSing perception in the genetic fiber of the function H, right? So in the picture, we have a base and then this red line corresponds to the mere of the one of the components. So we should think it as a vibration. Much of the state itself always comes with that remaining function. And since the anti-canonical line, that's always why we depend on the fiber of original lambda-game. So this is all from the mere symmetric expectation. So if we find the good multi-potential analog of the lambda-game model, we need to import some conditions. First of all, this blue line corresponds to the red line corresponds to the mere of the one of the two. What's related to the anti-canonical line? Because that's what we want. And that's the first thing. So that means that we should be allowed somehow to move the purple vibration into this black. Moreover, I want to treat this as a lambda-game format, which means I want to talk about the monogram, the monogram, which is pretty good. So I want to, my geometry, nice enough to say that I have the blue monogram near the green thing. These are two conditions that I want to think. The definition of the multi-potential analog of the lambda-game. Much, much better. So if I'm a hybrid or either rank lambda-game model, it's not connected to the notion of not being settled in. Or rank two, rank means that the number of the functions, the number of the functions, the people where y is expected, and this is h. So by the proper subjective, which that we have like a good local trivialization near the bottom. First, the condition says that the function h and h1, h1 and h2, it's going to be a hybrid alone, large complex. And then this condition basically says that the pathology will not be drastically changing if you slightly move it slightly like this, the pathology will not be, of course, and all of this compatibility can be very good. And of course, if you find a higher rank lambda-game model for any of them, then the idea is, for example, so the first proposition says like anti-diagonal hyperplane, it's basically the union of all the general coordinates. So this is come from the mirror symmetric expectation. And you also consider like a monogram operation. Which is confused by taking a loop along the one of the four-game hyperplanes near the infinity. If you have like two components, two n equals two, then you have a dysmonodermy, and then you have this one. And this monodermy uses not only the automorphism of a generative fiber, but also like that. So you can now define like what we mean by like the mirror hyperlambas, the higher rank lambda-game model. And you start with like the unfair n component. And I have a n function, which is mirrored through this. It basically encodes all the mirror symmetric information of the intersection. Then n equals two k's, you have a v1 and v2, and v1. I want to write that out. Yes. No, no, no, no. I haven't got into there, but what I'm going to do is that I want to start with like a labial, in which to generate the three components. And each component has an entanglement to the fiber, but at this time it gives the union of the second in the degeneration further. So that's why we're actually talking about this. What happens if we have a normal processing entanglement. Monodermy or symmetry should be stated as the kind of height and width. The equivalent of the cuticle diagram. So n equals two k's, we have that variable of x, we have that variable of v1, we have that variable of v2, and v1, v2, and then we make the cuticle diagram. The challenge is that we're not going to talk about this. The answer is that the need is that actually we can expect a line-bundle and monodermy. So instead of like talking about the canonical line-bundle here, if we know how the chemical fiber decomposes into a bunch of these components, we can also talk about this. If the line-bundle corresponds to monodermy along the height of the entanglement, and by definition we take the composition, we take the sum, we recover the line-bundle, canonical line-bundles, and then global monodermy. So this is a employee for a mirror-symmetry refining with the choice of normal processing entanglement. Now we can actually talk about the how to generalize EHD construction. Now I consider a semi-stable degeneration of collabia of any kind, but with the three components. Then I want to construct a closed mirror of this kind, and we have a three component, and each XI comes with the canonical choice of anti-canon to device, which is the union of the intersection with the other. Then so from the story that I introduced, we might be able to start the hyperlambda-pinnacle model, not just a single and now we have like more space, which is a mirror of this pair. Also I need to like to put some logical constraints, because also on the degeneration side, we already know that like using the answers, we can rewrite the de-semi-stability solution, which is the vanishing of the normal part with the mirror counter part. There's this with the de-semi-stability solution before YI, and then modeling along the base format. I'm going to treat this as a model of this. On the second components, I have a line, you can model them, it goes along the base of model, but we treat it as a model. Now we just need to figure out what the solution means. To call this like a, because we are the good candidates, which is P2, as you know that like in P2, there are like the three polydisk, three polydisk consists of, you know how the three polydisk consists of, exactly the equation to what they call a five section of the P1. Then you can write down like a relation of the three disks, and that's basically the same with the chart pool. It turns out that this model condition that I wrote down here is the famous case. Then coming from the topological constants, we obtain a topological five dimension. Then the claim is that this topological five dimension is not the key to the topological key, but we claim that it's the topological key. We can do this more generally when we have a simple number of coding varieties with a higher type, you're bound to complete the standard and then what we obtain in the end is just for vibrations, that is the function here, it needs a whole factory. So if I'm not doing the zero program, that will apply under some factory, in that case, we can compare the four dimensions and then the generation components has a zero dimension. Yeah, well, of course, that's a mystery. It's perfect. Well, I don't know how to obtain the vibration from this, but that's the whole story, you know, we'll find it, we might be able to get the vibration, but that's going to be different then. Yes, yeah, it's only just before we can include that in the topological key. I should also mention that there is another direction of the generalization for semi-stable generation to the bottom, and also, as I said, this the function is purely the prologue, not possible to study it from President John, but there's something that works by Annabella in the context categories. So I can do a little bit more, because on the degenerative side, we have like a monitoring system around the central fiber, and this gives you some sort of filtration on the homologous and the major question is what will be the corresponding formations on the mirror, and then it turns out to be all like perverse ratio of patient to the vibration. And the different formulations and then several evidence systems are given in paper. We call like what was the formulations and then what kind of features. I have a picture of you. So no matter if you have like a, you know, with the hemorrhagic varieties, with the homology of the type of piece. From a little further, I have the homology of the push form of the piece. We can, on perfect representation, given by some temptation of this context, basically we call the size of the supporting properties of the homologation. It's like definition is not moving. It's not moving. When construction is like a descent, and it's like the pure way of the way. But the, what we do, here, and by the problem with reality, we have pure homologous distribution on this part of the population. This is going to be all good. So consider the pair of general flags of the piece. So flag means that we first like a fixed embedding of the base, and consider linear flag there. All that. And general means that it's somehow intersects the first solution to the first one. So given this data, we consider. The flag of the propriety of the words of this premium. And the first, in case that these are fine. For example, in our case, multiplication, the presentation is given by default. It's the return of the distribution. We're very thankful for it. But in this case, we don't need to play. Parallel. The more generally look at these public perspective, which is vibration. When the presentation is here. I still haven't watched this. Homology quoted on some data. We just take like this a position. And then we are a big question. The state is one more definition. Okay. I'm buying it with a good economic. But any product. We can define the pervert mix. So this is going to be the whole variable. Where you. And then we have a lot of. And where the perfect presentation corresponds to the. And there are people. So the structure. For the low Calabi out there. Identity between. but the connection in regard it as a refined version of the and you can see that the P and the role of the P and W are so that's why we call it and the seats like here we have like both perverts and weights. I want to see the weights on one side and perverts on the other side and with some like the variable changes then we have the following. For this is simply the fear of real weight inflation but we're going to promise what we do together we only have the weight. So my hope is that like those you may actually make this from not just a quality of the dimension but also like the difference between the two. In general for the competition for the weight of creation we really use a competition and like on the specter thing and you can do the same for like it's ethanol agent math and then if this competition is pretty enough for example this is the final term and then this is what we call multi-dimension and we might be able to do something on the level like specter thing using the mere symmetry expectation of this there and then I did a higher rank round up while I was down here at the top stage probably lifts this form of the sequence of the statement as a multi-dimension but the problem only problem we had is that the sequences for the weight of creation because we look like the specter sequence that it's just a flat non-dimension. The specter sequence looks like a bunch of more bits more. So what we need to do is that we need some sort of another description for the weight of creation. And I can write down the typical dive in form of which is just like a copy of the specter sequence on the weight of the weight of creation under the it gives you the one specter sequence for the weight of creation. The idea is that instead of just using the general class I want to consider starting from the bunch of numerous parts in union of the general hyper plane then there's ordinary hyper plane and then it has like a natural class intersections. And then using that we have a kind of check pad complex and the description of the plane. A little bit more but so I'll say that there's the people in self-expecter sequence called the homological theory. The final thing is that coming back to our degeneration story X be like a smoothing of clavium and then this is what we need to think of. It's a combustion here and then simplified again, we have a limiting mix of specter and then here there's a reverberation to this plane is that if we know what happens for the components of the degeneration part of the degeneration part of which component of the degeneration part of the homogenous for near near the public and then it's actually followed if it should follow this if it implies the idea is that you just write down the vector sequence for the weight of nation and consist of the function consists of the components of the degeneration and then we have function between that and then hold on that. We're just able to write it down and then try to do it probably I think the difficulty is that we need to obtain this completely. The one part that we need in the primary logic in the primary logic we need some obfuscification of this data and in the future. Any question? I think like if you interpret your multi-payery when you consider this multi-payer hun2c and in terms of teta functions like because in the original harder Doran Thompson when they have only one divisor so each component with respect to the when you have only one divisor you have only one teta function and this teta function determines your Landau-Gamesburg mirror if you have several devices you will have several teta functions and I think what you want amounts to working with several teta functions to describe your multi-payer and compatibility with the several teta functions I think this would this should help to show also compactification. I think this is some things we are also interested in definitely very nice talk thank you. Yeah this is a general question I guess I asked before but um these multi-payer hybrid Landau-Gamesburg model which have several potentials question what these several potentials means from the physics point of view I mean Landau-Gamesburg models were supposed I mean the potential is part of the world action somehow and actions are usually scaling anyone else has a comment on this I mean it's kind of from the physics perspective it's a bit worrisome to have this many potential I don't know what other words what what this is telling us about I realized but it's bringing it up again I think I want to make further questions or comments right so if there are any no further questions we thanks again for your