 All right, so our last talk, the second talk by George is about the Pico sub-typing. Well, this was, I could also give the subtitle, and the subtitle would actually be my life of Pico sub-typing. More like it, it would be very nice to relate this stuff, big algebra back to Pico sub-typing, and I can imagine that happening. So we will be surprisingly in this story also be working with them. Co-homology of certain varieties, which are not unrelated to the modern space of its bundles. And we will be looking at its intersection homology instead, so very similar roughly what we did for the perverse filtration. We will also have any surprising green structure on this co-homology, which is, which is compatible with the intersection co-homology structures and somehow it feels like we are working on the P side or something. So maybe at the end I will pose this question, what's the value side. So, but so in the whole talk, it will be about the modern space of his bundles, and in fact that mirror symmetry picture, which I was sketching on the in the first talk. So now, for you, it will be less shocking than in my recent talks, I just said one slide about his mirror symmetry. Maybe I push. Okay. So very quickly I recall this motivation from my symmetry picture. If you remember at the very end was concentrated on these heck and wheels and operators each had a nice construction in the group on was explaining the co-homology of them. So we will go back to this space but we will go back to the, like I just think the mirror symmetry origin of these operators where the operators don't directly acting the chronology they actually have higher up, say on co-hrenches on these streets module spaces so let me remind you, that's a notation is going to change from that first of because the letter M, which is way to be overused as you will see usually the letter H because of hodge and coge and that's about here the letter M is over so I have to change the font of the module space because it will turn out to be less important in this story. So as I recall what we were discussing in the first time that somehow we have this classical limit of the mirror symmetry, which can be formulated for the module space of G is one of some occur for a cheap complex feedback group, and it's not as dual. So in a classical limit, we should get an equivalent some sort of equivalence of the right categories, which should be denoted by S, and showing the origin of this from physics the yes to anything for the theory. And also the right categories have to be probably enhanced as we learned from geometric lens. So it's a very schematic idea, but definitely generally, and somehow the focus on the heating vibration, it should just be pretty much a transform along the teaching vibration so let me recall that if we have the leg as your group that it's teaching that we'll have a teaching base which can be identified the teaching base for HD. So we will have these Lagrangian per proper vibrations over the same base and the generic fibers are being varieties which are dual to each other. So it makes sense to think about the relative footing quite transform, and that is going to be our guiding principle of trying to guess or conjecture. What are the mirrors of various objects on the two sides. So the other guiding principle will be this symmetry, which I mentioned already so that just recall it. And today, eventually it will be important to take more complicated tech operators and that once we had seen in the in the food of you are in the first part as over here do you just take simple and what I call fundamental or or elementary hack and modifications but in fact you can do a heck of modification for any type labeled by and an energy representation the lagons in your group, so this is G check at the point C so we fix the point C and the type of the heck of transformation. And then we should have a similar to the previous talk of you know tons we should have a correspondence with space age, which is mapping both to both energies. This is how I can operate there should be a pullback and push forward. Okay, but that's technically not really well defined as the mission because the second consummations don't preserve stability so it's just a schematic thing we should think about again today's talk we will. I mean we will have the module is based on stable ones but somehow the stability issue will be not not really important to not really important. I believe that the categories of current sheets and and downstairs is also about she for this DP. Yeah, that's the first approximation. But downstairs it is for. Oh, this is the classical limit. Okay, so you should somehow generalize on what you know to this much more symmetric situation. I think it's only started from now, although at the very end, we were surprisingly see that the legal algebra also comes from your very different and geometric elements, which has time with what you know that the module stuff but that I don't understand the relationship here in this talk, everything will be current chief on the money on the space of things. So this is kind of hard geometrically non trivial correspondence main difficulty before more complicated use that the space of a cat transformation will be singular. That's really the technical difficulty, but this other operator, the wheels of operator actually just like in on tons of the previous talk is much easier to define the structure of the machine. Now it's important to remember. No, I'm not saying it here. But that means that here is the each check is thought to be the universal G check bundle. So we first stick to this human point on the curse and now I have a principle G channel model on the module is to be the universal G check bundle over the space of G you are G check. So we first stick to this human point on the curse and now I have a principle G channel model on the module is space. And now I can take the associated vector in the representation given by this is a dominant. So that's a very different vector model. It's, it's a beautiful geometry, it has stricter structure so this is exactly the kind of things you would like to see in understand it's mirror in your symmetry. And that's because I'm telling you yesterday that Kapustin and beaten and in fact, I mean, basically, as some feelings on game that story you have this intertwining of these operators with the, with the escalating. So let me show you the simplest test of this and that's basically what we are going to test and see what the mathematical structures we can cook out of it. So let's look at this intertwining equation and apply to this simplest, quite a chief I can imagine on the, the legless you're wanting this space. So, what is this is a structure shift so. The good thing about structure shift is that if you're very nicely feed both to be this intertwining but also the ideas about generally, generally quickly and quite transform. Let me show you how so we will want to compute was the heck a transform of the mirror of the structure she the first thing using the ideas of the way transform, we can convince ourselves that we want the structure she to be mirror. To watch. So then you recall about putting quite transform that the structure she is a freeing quite transform which is the sky paper sheet, the identity element of the dual opinion right. So now we do this relative in a family over the teaching base. So we expect. Your object here that relatively my transform should also be. It turns out the structure shift of this collection of all the identity elements of them. So you have a legless your feet in vibration. And in fact, there is a beautiful section of the teaching foundation, which is also called teaching section, which we will recall in the, in the example case you will be studying later. This is the teaching section, which picks up exactly the identity element in the legless your teaching foundation. And that's the input from for him quite trust when we expect the mirror of the structure she this will be the notation for the original G. That's why we don't have check here. So this is the teaching section energy. The structure shift. This should be the mirror of this. The structure shift of the whole space. So that's one side so one side we are going to study the hack and transformations of the section. And on the other side, if we compute this by using the interlining property. And we would first have to use the user operator on the structure sheet but that is just the back of itself. So we know that this should be just this vector. The only representation. And it's me. So when you put this together, what you get is in the mirror of the university of London at that point in the in the, in the, in the, that is the representation should have a mirror, which is the heck a transformation of the teaching section. So that's the right hand side is a nice. We understand the, the quality of the community coming from the university of London. Now we want to understand it's me over. And now we are used to understand how hack and transformations move the teaching section and basically, this is a big part of what we are going to study on to understand the hack and transform teaching section. And the first I just say it to motivate what you will see on the next slide. One nice thing about the heck a transfer of the section. Two nice things is that the heck at operators are running operators or the teaching section as you will see some of them. And so it is always meant to be taken to Lagrangian. And then second is that because the teaching section will be invariant on their energy overseas direction. So we'll be the heck a transform once and it turns out that those will be the heck a transforms will be quite a cheese, but they will have a support. So I'm going to now tell you about these large enough what flows, which are very natural source, very natural Lagrangians on that module this space of hex bundles and one of them, if you already mentioned is the he's a section which is crucial for many applications. And there are many, many more. In fact, we will be paving those of the we were partitioning the issue will be space into these into all these Lagrangian apply flows. So how does that go. So we will first specially specialized in this case, so that it's a center and we can be more concrete so we'll take the PG and X money with space. So we're looking at the point in the region needs money or is based a pigeon and he's fun. Is that care of a brain can back to a bundle and a trace the field, which is, you should think it's a one for value and the morphism of the vector bundle of trace zero. And then the reason that kind of situation, like there's a lot of midline models. So that's represents a point of the future and what is based now we are in the stable, which is basically should have some stability condition, etc. And then we have the teaching map which in this case states a very simple form. I told you in the general story we just stayed there. The finalization of that. Actually, it's accurate. Which in this case, you can just take by computing the characteristic polynomial of the heat fields and the heat field at any point of the curve is just the morphism of the vector space it's a matrix. It's in the chemical bundle, and then when we evaluate it's characteristic polynomial then reveal the coefficients, which will leave me will be sections of these powers of technical. And then we can have the first one because the trace was zero, but you will have all the other coefficients. And so at the end of the day, the base is going to be this vector space it's really just a C star. I find a space but it turns out to be a vector space but you can't really see skills. And it also has a good show the half of the dimensions that will be a space of heat samples, and the total space, and then with respect to a natural, seem like the structure we do have. And this is the only integral system. And so the structures in the constructions of the agent that will be no more geological structure to think about the natural C star action. Which will just be scaling the heat field by what was your complex number, and then it's clear in your preserve. And this action has a very nice property, which is that it is some project, which actually you can do basically talking to the purpose of the teaching that it has so this is like variance of that. You can do that, not to be on the basis they to get such an end on those factors. But it will have a unique response on the base. Because of that, they just with the fixed point set of this is direction, you have to be included in the zero fiber of the heating back. And therefore it has to be projecting itself because I'm at this project. The second property between it's any project is that it's a, it's attracting in the sense that if you have any point on your space and the system or we will have a limit point at zero. Not necessarily at infinity that would be the case for a project the variety but this direction you can always close down your system. And then you can do that, then you have a natural way to partition the space because now we can do any fixed point of this direction, we can collect all those things funnels which flow into that. And then you can do that by collecting those as such that the community is, is a given me, and we will say that this collection of his one goes is the upward flow from E. So these are number close to math that down to me. Do you think of business on the upward flow. This is a beautiful theory, you to be on it's keeping a long time ago in 1973 that shows that two major important things for us one is that this subset is actually a locally close to variety. And second, this is a isomorphic to a vector space in a sea star. The vector space is easy to describe it's just a dungeon space to the fixed point on the system action in where we become the positive bits of the action that will be the zero weights on the fixed point component and negative base so we describe and we keep on the positive base, and therefore we see that as a sea star space it will be like a vector space with a positive action, positive direction of C star. And then one more property which is extra to our case because we have the same like the structure, which we say is homogeneity one, the system action that contains the system acting for my way to one. And from that you can deduce that all these upward flows on the branching. So this way with a beautiful partition of them on the space into a bunch of clothes. And so here is that this inform about because of semi projectivity. Evian it will be one of these upward flow so you bet this partition. And now comes the crucial notion for the first part of the talk where I will be studying only certain type of upwards which are somehow higher highest up in a natural partial ordering of these upward flows, you can say that one will always be another one. If the lower one is not close, but it will have a limit point at infinity. That's striking point of the next one. So that will give us a partial ordering and the maximum amount of respect to this partial ordering. You'll be the ones where the approach is close. So those play a very special role. They will be you will see. That's all in related to minus two representations on the other side. So, then you can have several activities of being very stable. One of them is that the map. So the hitching that restricted to disclose some of the results of proper, but it's actually a girl and so this will be close even only the rest of the map is proper. And now these are very special maps so you have two vector spaces of the same dimension with C star acting on both with positive rates. And the map is proper. So from this you can use beautiful results for example that the map has to be even finite. And flat. All the fibers scheme theory can be speaking we'll have the same length. And so, over zero, it's only zero which goes to zero so the pre image of zero is just the point zero. But because of what I said about flatness we will have a non real thickness. So the fiber over zero, we should have the same length as the genetic fiber. And it turns out that there is an exciting and really amazing structure to be seen with the national schemes and generally speaking what we are thinking about. And then there's something that could have thought about 20 years ago when I go to my PhD, I was looking at the support for you have the upper floor. And these guys will intersect the zero fiber in a single point mainly in the that's what it means to be very stable and not recommended definition. And, and then the scheme theory intersection will be very interesting to study. We will do same a few birds one is. For now, let me just formulate this is one side or problem, but in the larger he changed which we can be started this kind of. The idea is not to use this. One aspect if you're interested in trying to understand the teaching on these large yes, but explicitly so if you think about this looks like a very simple map to put it on the map between vector spaces of the same dimension. The coordinates, which are not given to be able to this is not given the political coordinates can be find coordinates where this map we can describe explicitly and that's somehow interesting point of the system theory, because that way you would find half way of the problem of describing the explicitly. Let's do some simplest examples of very stable kicks bundles in the first one will be of course what I mentioned already the teaching section teaching section with them out to be the most important. And it's very stable up our floor. And then start with a section. So the underlying model in the PGM case of the region section is going to be always the same fits the vector. And then we are first describing the teaching section so for any point in the base, which is this characteristic polynomial. And then we will be constructing a Higgs bundle Higgs field of this sector model, which will have this characteristic polynomial. And then if you have a new remember your new customer is a chemical way to do that. You can just stay with the computer matrix, which has precisely the property that if you have the characteristic polynomial as we want it so this guy this is field. We have a characteristic polynomial precisely a so this is the container matrix. And that is zero five a together for every a will give us this so for each section. So this will be nothing else but the upward flow from a particular is one of the one over zero over the origin of the region base that corresponds to milk and Higgs fields are characteristic polynomial zero. So when the last column is completely zero. And that's a very important Higgs bundle in this, you know, in order to work, it's so called chemical uniformize and keeps on going. And from which the upward flow will be nothing else that nature section so the upward flow from this will contain all these Higgs bundles and then you can show that it is the full of what flow. The upward flow from is zero, which will be meaning he from here so that's fine the notation really is just over throwing one notation that you see the plus is the upward flow from the front. It's always the section, which then is being a section of the agent that is clearly closed. So it's very stable. It's somehow it's on the top of everything else so that you know it is very, very interesting. It's really important call this is the top point. And now we do want to make this heck of transformation that I want this time the first part of that okay we'll talk about fundamental elementary hack and modifications. So we will just take a point C on the curve. And first we just take the heck of modification of this case on the heck of modification of the other line bundle, which will just be twisting the last and minus K line bundles with the, with this. And that will be okay, we will denote this okay. And then we will slightly modify these fields. They're mainly the case position that one will replace with a section of this line bundle and maybe the defined section one which is precisely at C. Introduce one CEO at that point, see, so that's the new big bundle which we will call a K. We can check that it is so stable. And then you can think about it's up flat flow. And then, then the first time in the head with Nigel and some results which in this type one one case and the line bundle. We have some of them. We have some of the line bundles characterizes all the very stable ones. And these are some of the most important that's very stable. This type. So the statement is that these are always very stable. Okay, and the way you can prove this or maybe feel so quickly it was running the food was a bit more complicated of course we had a complete classification in this type one one case. So if you just try to think about heck and modifications we will do at the point C, and then you want to take a K dimensional subspace and do a heck and modification there, then you can prove that you can access precisely the points of these more general right for so WK plus you know the output from here, those points are precisely the possible packet transformations of the elements of the section. In general, this map will be. Well, and, and choose K possibilities we will get to generically for the choice of the key dimensional subspace, where we can do that kind of location. But anyway, so and then the idea was to prove that this was very stable is that the Asian section was very stable and somehow the heck transformation should be the closest. But again, if you're not treating to the general story so you take the case of the character of the person in the corresponding heck of transformation will be this, and then you can say that we can think that the transformation of the section is precisely this single part will say the very stable case. We actually just have a single part. The thing which corresponds to being very stable is the fact that the fundamental representation of the SLM or the first case that was the case. And this actually means that they are minimal with respect to some national partial learning of them, of them is dominant. And that's actually basically how you should think about this. We will be able to label all these are type one monopartial somehow we practice, and then then this partially limited partially ordering we already flag the partial ordering of dominant weights. So therefore means you have to be able to correspond to be very stable. And so then the first surprise was when we were so we naturally were understanding these are part was and first we understood the system, a career character of them. We understood that the related quantity for the equity and multiplicity, and we got actually the primary polynomial of the grass mania, but that it was a, I couldn't be in there to imagine that the community ring of the cross money and we show up, but it doesn't, it actually is the community ring is the equity and community ring of the grass mania, which will be describing these apart flows. So for that, let me quickly recall the notion of equity and community in this edge of the geometric situation so we will have a complex deductive group we will have the corresponding classifying the principle, the universal principle, but all over the classifying space PG, where G acts on the contract in the space, he G up to one of the pizza that defined the institute or gene bundle. So the coefficient ring will be the common of the ring of the classifying space, which is, of course, as important in the characteristic last theory. So it's a beautiful way to compute it, it will be important both ways that it's both the G invariant polynomial of the algebra, but also the line invariant polynomials on the cartons of algebra. So this was described to be this ring, which is a great thing that to agrees. And what's the geometric important for us to know this by showing this theorem. This is a polynomial ring. And yes, you will see in a moment that all or academic offerings will be over this polynomial. And one of the topics that we have now a smooth one of some graduate variety with the action of a complex deductive group. And then we can form this home of the quotient or water portion by the diagonal action of G. We combine the action on x and the inverse of principle. And we will just define the equivalent common logic is the common logic is the water portion. One important thing is that we can project to the e gene or G in the second factor you can project. And if you'll be, then the portion will be a fiber bundle over BG in the fibers you can identify with the space axis itself. So that's what's important then and pull back homology classes from the base. It is actually naturally receives a map on the common logic of the classifying space. And that will be the reason actually we will see more structure to a very important knowledge because it's an algebra over a much more interesting geologically much more interesting logic rather than just a point. And the field C, you'll get this point on the algebra positive dimension. Okay, so we will be in the situation always that or can continue to be accurately formal. And that means several things. One is that you can recover the ordinary homology of x by by specializing over the over zero in a study that the augmentation ideal of a strategy and divide, or you can tensor, your, your agenda and that's the specialization over zero. And there's something that can be formally the ordinary homology is over zero. Or another way to say this is the, if the ring is a free module over the coefficient. It's going to be a finite free module. And this finite free module will be the analog of finite flat not from the, from the hitching story. And the basement we don't have even homology, the odd, homology as will be the case for us, all the examples. Basically, then it's a cure to say that the map, the issues not on this FI spectrum. So now you have these two FI varieties, the base of the map is there. And the quotient GIP quotient of T by W which is now an FI space because of the political ring property of the invariant ring. So we're this way that this finite flat morphism, the spectrum of the economy of the variety. Yeah, this is a general story. We assume X is a key form of so that these properties for other we could be crazy. You have to assume this otherwise you don't. But this map in general, but in the form of case, for example, and that when it doesn't have what come over G, then it will be automatically for more. So let's assume the varieties. And then we will get this nice geometrical picture. And the surprise is that you can use this type of abstract link is similar to the hitching on a very stable off our floor. It's actually can be used to model the hitching that on the upper foods we have good that on this first time new case, miniscule of what flows. You can use the ecology of the grass mania of Caitlyn's in CN and to to model the teaching that so what do I mean. So I have and the complex grass mania of Caitlyn's which is a homogeneous space for PGLM, so you have the security ecology. It turns out to be the spectrum is irreducible. And in the base is also vector space and, and this has no homology so it is accurately formal so this is a finite that morphism. For example, or hitching maps on this particular of what flows are also can be modeled by this in the sense that you can just pull back this map by a natural evaluation method that won't see you will end up there, you can just pull back that diagram. Yeah, it's a full day diagram of this. In that way, you get in some nice sense, an explicit or conceptual description of this map as the pool that they're going to get the grass mania. Okay, so you might, I guess you might see why the grass mania. It is here and the reason that the grass mania parameterizes the possible hacker transformations that we can do at that point so at the point C. And this is the key to measure subspaces of the fiber of the vector bundle, where we can attempt to do a transformation. This was the space of hacker transformations will be called the label gr for my God, okay, that's the so called FI sugar variety. And this is in our case is the whole grass mania. The reason for this result is to, to understand, you start to start from the section, and we look at the point C and we will start to get this one or the key planes and see which one of them is preserved by the and then you'll be on the each section everything is regular. So then you'll be fine. I think many such subspace this pleaser so you're going to find many many choices. And then when you do that in this family, you will compute some fixed point scheme inside the grass mania time stage in this in the order that base and you will turn out to be completely precise in the span of technology and you will be soon a paper with a PhD student of mine, where we generalize this to more genetical situations. So this now is a more established fact that fixed point schemes, which in this case described the region is actually iso in respect to the field of technology. And now we come back to this original question over zero here by a formality, we have the technology ring of the grass mania means that the recording of that fact point. And that that was our original observation with Nigel, which was a very nice explanation of just this numeric quantity to be computed in the paper, which I mentioned may have mentioned it was the binomial proficient entries key, the q binomial proficient, which is we know that we know the grass mania, but this is a beautiful geometric realization of that because this actually is the conogering of the grass mania at that big point. And so that's why we saw this form of the multiplicity. Okay, so that was the first observation. And now we will be thinking about what should we do on the mirror side what should this accurate and respect to the factory and homology map on the mirror side you remember that we want the heck a transfer teaching section to become the universal bundle in the corresponding representation which in this case is the exterior case exterior power of the universal bundle. And now we will be doing something on that side inside the representation, we will be and now in the universe and bundled with the structure of bundles of algebra structures and Kirill of algebra. So, let me remind you, and this construction of Kirill over from 2000 and he attaches to any. Now we go, it should be G check but for simplicity on this I mean at G, but in more of the area or now on the legacy was signed the SNL side in the previous point. So we will take a dominant character of a highest rate presentation of of G, if you act on this act of space to be new. And to this field of attached to what he called the family classical family algebra, I just simplified the Kirill of algebra. This algebra, the tensor algebra, the cynical algebra on G with the endomorphisms of the presentation, which is G invariance of G acts on both sides. Okay, so that's some magically definition. So here whether I like to first identify GBG star because then the symmetrical algebra can be thought of as polynomial functions on G itself and you can do this in what case G will be here and there to be a simple group so we can identify GBG star. And we just thought of as polynomial maps on the algebra to endomorphisms of the representation matrices, really so we should think of an element in the Kirill of algebra is a matrix acting on the representation that the entries are polynomials on the algebra. But now everything is G invariant. So G acts on the representation GS or X on G, and they have to have a G invariant polynomial that from G to N. Okay, so that's a more explicit what these elements in the Kirill of algebra are. So first thing to notice that if you have just a. Okay, again, I already changed to G star. So if you have a invariant polynomial on the algebra, then you can just multiply such a polynomial function point wise. So here, and it will of course be difficult to change the variant as it uses the reserve G equip variants. So this algebra which is nothing else but that will coefficient on the ground. The classifying space is. It's going to act over this it maps there and this is such an algebra associative HG algebra. In general, it will, it's not commutative and this will be crucial later the case is that it is not commutative but for us in the first part where we will look at the very stable case, it will be commutative. In particular, the most important example for us is when the representation is minus give them the algebra will be commutative. And in that case, one can use in the case is here the way we will do the most few case before G equals SNN is just a fundamental representations. We can use this kilo algebra structure to build a bundle of algebra structure on or universal bundle along the HG section. So how do we do this? So first, we can see that at the point C of the universe of all. So we are going to have a vector above over the agent section. At every point I just have this vector space on which that point I have. So if you have a point in the base of the kilo algebra, I will have a morphism from G to there, I will have a matrix acting on this so I can act with this kilo algebra on that vector space and that over that point. That way I can embed the kilo algebra into the endomorphism algebra of this bundle. And now what's known trivial is that this action is an important, it's very, this algebra is very nice. It's what I call cyclic algebra. This means that applying this to the identity vector or some vector here. This can actually generate our vector space. It can actually identify the kilo algebra with the underlying vector space itself. And using that identification, you can build a multiplication on the on all these vector spaces on this vector and you will get a bundle of algebra structure on the universe of all over the teaching section. The relative spectrum will be the same as the spectrum of the key in large. Again, you know what this is coming to think I can take it spectrum, and it will be the over the spectrum of the base. And it turns out that this is the right middle picture to the, to the one we had seen from the other side on the other the previous slide. So what's just a few words, how you construct this one with larger structure, we are using in this case of the combative case of the support and operators. And one just applies those operators to the Higgs field. So the Higgs field you should think about at every point the Higgs field is just an element of the algebra. And you can use the key a little bit or two that to an endomorphism of the corresponding representation. And that's how you can build and morphisms there. And then you use the cyclist property of the algebra machine is going to be crucial. This is proved by function in the case of the key of logic. And I will just aside remark that the Higgs one this construction of the bundle of algebra structure appears in the BNR correspondence when you hook up the spectral curve, you actually use this simplest bundle of the original E, and when you take that out the spectrum. You, you can, without the C depends as you can actually hook up the spectrum this way. But that's not going to that what's important by the way that you can generalize the BNR correspondence. Well, at this point any means to a representation, but later to any visible representation. Okay, and now let me conclude the first part by showing you how the two sides fit together I want to tell me it's you that this observation of the hitching that will be a new skill awkward to model model by the property that the middle of the universe about this representation should be the hacker transfer teaching section. And now we are in a situation which is PG and then you use a fundamental representation. In that case that gets on so teaching section is actually just that the structure she offers some money so in itself it has extra structure. You can multiply sections because it's a structure sheet of something is the sheet of algebras. And then now you can think about what happens to a sheet of algebras. When you do really want to transform for that I mean you can multiply sections. But of course, if you're going to take putting questions for me to not give you multiplication to relate the conversion of the of the three questions one thing itself. But over zero. And this very compulsion is actually making zero zero. You will get a sheet of structure and that's because somehow they have we stick to says that the sheet of algebra should have a mirror, which restricts to the recent teaching section to the identity elements in the field of transport as a sheet of algebra. And in the short case it will be a vector bundle, the mean or say to be a bundle of logic. So that's the motivation to seek and this bundle function of structure universal. He's one of them, you know, section. So I just write this argument out here. And somehow, again, you use this heuristic we have from annoying that we have some sort of frame quite chance for them. And then you get to be completely identified as to these two sides of the mirror symmetry. So, here you see for construction of the bundle of our structure here. And also on the killing of algebra. But the other side you see more modeling of the art floor with the, with the equity of the cost money. And we need one more team put which was also observed by a new show, although it's an important distinction that one you should prove that the end. The killing of algebra for me was to the representation is the equity of the community of the community. variety, or of the millions to a flag variety, but we actually will go to the language you will sign and they will be taking the community flag right in the year, the case. Actually, the same it's always the cross money and so it's more blood there but basically, or you should know it's on this side. And actually, but this theory shows us that we should actually go on the other side and instead of taking the, the communo skew variety on this side you should take that means to the other side, which in Taipei. It is the same, but you have some interesting examples there, which are used to buy some of the other types. Okay, and then you have the isomorphism of the spectrum of. And you can use this to catch what you want to do. That's the upload flow can be understood as the spectrum of the relative spectrum of this budget. So, that's the first part of the talk which already give some surprising appearance of the economy and quality of the Tasmanian that cities. But what I will talk about in the second part is, is how to generalize this to known and very stable monogamy skewer rates more general than just basically any way. And we will see that this picture is not yet completed in the other side, but we will be able to be able to be able to find very interesting arguments here, they will be called the big arguments. And let me just say to finish the first part that you can somehow consider this generalized story that is geometric as some sort of classical limit on the geometry. So that's the first part. No. So then. Okay. And then you'll be short. Short. No, we don't have a point on that. So we have an infinite ism. Yes, we are at that point. So some of the genius, they didn't say doesn't do much. We somehow factor out. Right. There is no genius on there in the models. Of course, there is a genius here is the very large vector space. But this one shows that most of the directions here are the uninteresting there on that here is in the window in the window of the action. Okay, so last bit of these two days. Right. So what I'm going to do will be hopefully much faster than the first part and you just tell you what how can one generalize this picture. It's actually to any dominant co weights, not just the new skills. And that will be given by these big algebras, which I have recently been studying. So let me remind you this, the, these big algebra should be what properties they should have and why we are interested in them so the setup is this we have an irreducible representation now we work on SLM of SLM of not the selling big multiplicity free representation. And, and then, because of killings result is killings of algebra is typically not going to be commutative so it's on non commutative associative algebra. Because in order for the mirror symmetry to match the two sides, you could not you can ask this what is the mirror of the universal bundle in the in this particular representation. And so what you can do by the, this in the tiny property should be the hacker transform of the structure shift of the section. And then if you think about how you construct roughly the that that hacker transform you do this hacker correspondence, and you pull back and push forward. The point is that this bundle of the sheep of algebra structure you have on the section you will be there, you will pull back a sheet of algebras, you will push forward, you should get a sheet of algebra structure on this one and support. And, and that's the reason we expect the sheet of algebra structure bundle of algebra structure on the universal bundle. And that at the end we will want to have as we did before we want to have the commutative sub algebra of the killings of algebra, which is, which is cycling. And then that will be your aim to find the big algebra, you will call it the big algebra and coming to think that cyclic sub algebra of the killing of algebra. And it's just like, it seems in the simplest case, which is not trivial in the case of SL three and the adjoint representation, we understand that you know algebra explicitly. So, you know, had two students working on the skill of algebra. And there you can see that there is a unique such commutative and slightly sub algebra. And interestingly, the students did not study this commutative sub algebra. Sorry, what was it? Okay, this is a first just them on a vector space and matrix committing me to its very sacred secret if there is a vector in the vector space, so that you get the whole vector space. And this way you will be able to have an isomorphism between the algebra to define a dimensional algebra with the vector space itself. And then, yes. And when does it on the project. It's on the representation, right, it's always act on the representation, what you will have issues that identity element there there is a wonder, and we will let the algebra act and we should get back the base times the vector space. But you need to pick an element in the new, but we knew who has won. I mean, the highest they say, but, and then what do you think in SLM, or because you're in the organisms are, so you have maps from SLM to. Yes, but I think that every point you can take is the highest rate. And over every point this will be a psychological. And I think we can just say the highest rate at every point. Over the region is a morphism from the leogical team. So, but every point I have a finite dimensional matrix algebra acting on the tractor space being new at every point of the leogical every point of the leogical. And this, all of them will be cyclic at the same time. And I think I just take the highest rate vector, and they will generate the vector space. Okay, so let me now tell you the hints which I had got before I came up with the, with the construction of the big algebra. So these are two PhDs of students of. So the first one had this explicit computation of the field algebra for the joint representation of SRP. And there you would actually find my inspection, such a bigger algebra. And so you could see that it exists there but more interestingly tie brought down that for any simple, the algebra, he wrote down in the adjoint representation, the killing of algebra, sorry, it should be on the top, but okay. In his description, he has some operators which commute to each other. And he actually writes down the monomial basis and you can see in there that those operators will generate the cycling sub algebra. And that's because the dimension matches the dimension of the algebra but again he does not study the sub algebra. But you can see it in those examples where you can actually have a complete description of them, of the killing of algebra, you can buy it. And I believe that it's a unique one. So it looks like this will be sub algebra. And that was one of the hints of that I had more than half a year ago in these cases I had a sub algebra which should be the one we need. But the second team was that we had some operators in the key of algebra which were explicitly understandable. So we have some operators of Kirill which I will show you on the next slide. Kirill shows that they are in the center, and it turns out that they are actually the center so we have the center of the algebra, any maximum sub algebra contain the center. So these operators should be in whatever, and maximum competitive sub-algebra we want. And do you see how the operators are defined by Kirill. And there, already there is a guess what might be the right algebra but also back then I was, I forgot the precise order of things, but I was already reading about the mission for my co-integral systems, which are now Poisson, and they have everything algebras on the algebra, or the rule of the algebra, and they are also obtained by the success and successive differentiation of invariant polynomials exactly the thing which we will be doing here. And so that these are somehow the hints and now I'll show you my first which was just a guess, but it looked very good from many angles. So this is the construction of the big algebra. In these elementary terms the original way how I worked with them. So this is how Kirill will define his M operators in concrete coordinates, he took some coordinates for SLM and then the dual coordinates are the dual basis with respect to the Kirill form. So you just throw down the differential operator like this, it's not really a differential operating that it's not doesn't satisfy any sort of lightning school. You sum together the different, the derivative of the original element in the Kirill algebra again you should think of the Kirill of algebra as a matrix with entries from polynomials on the algebra. You can differentiate all the entries in one direction in the algebra and multiply the salt and matrix with the matrix given by the dual element in representation and you sum these things together. So this is just his definition, he did show that, and that this will land in the Kirill of algebra G invariant, and he showed that if you took an invariant polynomial, and you differentiated that. So for example for SLM, we will take the coefficients of the characteristic polynomial. And amazingly you turn out for the general story it matters which basis of invariant polynomials we take. Well for Kirill of business it was not necessary but anyway so we take some basis elements of the invariant polynomials in the algebra and then we will differentiate it. So Kirill of stem operators. Kirill of because a medium operators that's the only use of the letter m starts to be here. And then we regenerate all the this sub algebra of the Kirill of algebra which as I'm saying Kirill of short it's in the center of the algebra. But actually you can show that it's the whole center so you have a nice explicit way to describe the center of this algebra. But this thing from machine for my ecosystem, or otherwise you can just guess maybe differentiating with further decreases the degree of the genesis that is the polynomial system degree and you start to differentiate to get smaller and smaller elements in this great Kirill of algebra, and you have these operators of HK I call them in degree I minus K, these are the successive very that these are the same. These are the big operators, and we just take the sub algebra generated by them over or coefficient. And this was correct in the case of the edge of the representation that we only had one more in the type a case one more operator needed the second day of the of the of the third and second day of the turn class. And so this is the big algebra. And then it was a long process until I could be proved with an intern over the summer that it is actually a coma piece of algebra, you can compute in a computer that's what I did for several months. Suggestion I use magma, and we managed to program this in the computer and we had all these matrices, and amazing community and they also had the right dimension, it was a cyclic sub algebra in a computer experiments up to SL six I think, and in the big representation stuff stuff stuff and the financial matrices. We, it is always commuting and and amazingly, even today I don't have any kind of elementary proof that they commute. And let me first talk about that community property, because at the end of the day, it comes from something quite deep and actually going back to the geometric arguments, construction of the and related to that because it actually predates you. So, it turns out that the big algebra. It comes from a universe of the algebra sitting inside that as a product of this. And it is actually what you can call it some paper later because they call them algebra. And this is the image of that fading Frank's Center in the finder text article at the critical level. And then, and then it turns out to have a geometric language explanation. And it shows up in the paper of a different this big, big algebra. So our algebra should be for all of them is somehow a, it's an image of this big, big on the subject. So that's the reason we know this community. And then how do you do cyclicity and again, here we were lucky, because in a later bird by fighting Frank and reading or they don't really consider the whole golden algebra, but only all the elements and single element. And they show that this golden algebra on ties with the mission for my ecosystem. But I saw they show that it is always for a regular element in the base it's always a cycle project. And then you can they use it's a finite free algebra and therefore maximum. Maybe at the end of the day, although all of us very explicit and you can compute it. It looks like the accurate homogenous or something that's why I like to work with such things because basically that's what I compute all the time like homogenous of things. And then, but then it's quite deep mathematics which explains why it has the properties that you. Okay, and then let me finish with the geometrical picture, which, which you can use this algebra which will be actually even more enlightening because maybe it's too much. So let me, again, list you some properties that you can just prove again by using other people's computations. So it turns out that the big algebra is intimately related to the intersection co-mology of FI sugar varieties. So we take the FI just money and put it in this case of PGNN. And inside this, if you look at the singular, the singular variety, the closure of this orbit of the group of this group on the FI sugar variety. This is often from earlier in the case of the fundamental representation. And this is the gas money. And in general, this should be thought of as the space of you can do like a transformation at the beginning point of this time. And then, then again, now the, both the co-mology of this academic and co-mogering and the intersection co-mogering is a module where the co-mogering is described explicitly by Bessie, and I think that they do see from the geometric certificate. The theorem of, of near co-vision, you know, and that's against work and then already several ideas about exactly these kind of applications. For any new, this is a complete description. And then if you go there and unrevealed, you get that basically what algebras are precisely these algebras. So the medium algebras, the center of the kilo algebras is precisely the ordinary accurate co-mology of the FI sugar variety. You have to just unravel their description, their result. And more exciting, they are just realized recently that actually there, you can interpret in other ways and then you get that the endomorphism algebras, the intersection co-mogering, the way how they describe that module. It gives you precisely the killing of algebras. So you can now think of the killing of algebras instead of abstract algebra or in our case it has to do with mirror symmetry, but it is a very nice topological meaning. The endomorphism algebras of the intersection co-mogering of the FI sugar variety over the co-mology. And the center, of course, is just the ordinary co-mology itself. And now because the big algebras is a cycle, you can identify it with the intersection co-mology itself. And this way, as an MU module, the big algebra will be precise with the intersection co-mology. And this way what we get is a great digital structure on the intersection co-mology. This does not have a canonical product by topology, but in this particular case we have some canonical multiplication on the ring. So I should say that in the ordinary co-mology ring, the same was observed by the paper of Faden, Franklin, and Niko. This is the accurate and extension of that observation. But you will see that the accurate and extension, we see actually much more the structure of the aquarium. The co-mology at the intersection co-mology or the action of the big algebra carries much more information than just the ordinary co-mology. And then the conjecture is that now you can put everything back and use this big algebra to describe the mirror of the universal bundle in this representation. It's still outstanding because the varieties where you would have to compute are similar and then it's kind of still on trivial how to complete the picture. Okay, and let me finish this talk just by listing some properties and some interesting observations. The first thing is that we always work with the explicit generators of the algebra, but in general, they don't exist. So it turns out, in the work of Yakimov, who writes down this good algebra. You can do explicit generators only in the classical types and then surprisingly, it matters which they, which generate the set of the immediate polynomials you've started. You can do the same thing, although the ones you would take naturally for a salam, those two are known to be living inside the Feynman-Frenk algebra. But anyway, so it turns out that we don't yet have general, so that in the exception of case types we don't have generators of the algebra, except we have this abstract theory of vertex algebra which gives us this big algebra. So we have to be able to, and they conjecture us of Yakimov and their collaborators as a conjecture that there is always some set of generators of invariant polynomials. I mean, in order to generate the big algebra. Then what's very nice is what I mentioned already that the accurate apology has much more structure because it's not just a fat point, it's a scheme of an FI space. You can compose it into irreducible components, the medium algebra, and then you can actually understand each component of the medium algebra which is the accurate homology of some partial flag variety. You have some Yanis Yibirula, the composition here, and okay, that's some interesting thing only for them. Let me now go to the next one because this would be much more interesting. So I can take now this component of the medium algebra, and over this there will be a component of the big algebra. And then this way, we can define the grade in the ring structure on the intersection homology of the slides to the FI grass mania. And that's very interesting because, for example, this primary polynomial is known to be the custom stick polynomial but now we have a great deep ring. This is the final dimension of a big ring whose associated is the question I'm sick polynomial. And again, you can start to try to compute it in the adjoint case, I, because of those other works, I think I can explicitly compute it but it's like a whole set of examples that you can try to compute and you will get a great ring on this intersection and, for example, the zeros way to actually be can be computed just by as the portion of the big algebra by the augmentation ideal of the medium algebra, this somehow is a very interesting. Right. Then there will be a very nice way to relate it to the real funds for the group, and this is exciting because I believe this will make contact with the legends program for the real groups. And then the carton in motion of the real for the act on the big algebra, and I started to compute the invariant of the fixed point on the spectrum that is the point of algebra. And then it looks like a very long trivial and much more complicated than the original algebra it should describe real needs one of the fixed points of the real evolution acting on the space and it should have to do with another group, not there's PhD on the on the real geometric events. And I expect this one to to to be a very rich story. So I'm just giving one example and then the inform is of what style. And this one, this looks like a very simple evolution by my mind minus one to the power of the degree. The degree gives us the C's directions which is minus on the C's direction. And then you can compute it's invariant when it looks like very interesting the dimension of this is the going to be the signature of the and coached Riemann, by any relation on the, in the intersection college of data visible representation but this is something that will be the formation of that was the library. So this will be a very interesting objective for me. This is the quantum version, which is now very mysterious. I didn't make this point, maybe clearly, but the, this flag in Franklin thing which describes all or big algebra actually come from the actual So this is for them, it's about ring of functions on spaces of opers, but somehow for us, but they test it for them it's the gamified version so you have even sometimes I think they have irregular similarities everything is on P one for them. And I don't see the genetic relation between our Higgs modeling spaces and their spaces, but the funny thing is that here you can further quantize. And then you would get a quantum Kirill of algebra. You would have a big universal quantum Kirill of algebra which again is some sort of golden algebra. And that also should be commutative, but I don't know, we don't know what is the genetic meaning you would think maybe some sort of quantum chronology of this. Schubert varieties, but it's more complicated than that this one to algebra that very hard to understand. Great. Thank you. Yes. Yeah, we shouldn't be trying to do some thinking in the ordinary case but it's more complicated in that case you have multiple small resolutions you could have different structures in the finite. So it's more complicated. But in particular, I find. Because that makes you put it on else. It corresponds to reduce. Sorry. So this intersection from all that is there. As if you see go over there. Then why don't you. And what is it. They kind of simple one or some nice. Well, I don't yet know I mean we are starting to think about this perspective in just a couple of days. So it will be the intersection, come on, it will be a module, well it's an endomorphism algebra over on the intersection causes so it will act on it. By, by this construction. They are not commutative so and then what now what we are thinking of that any present anything you can do the same kind of construction for any singular space you can take the ordinary cosmology of the single right you can take the intersection and you can look at the endomorphism. Why would you do that well for me because it's the end of the key of algebra, but also, if you have a small resolution, then the ring structure there will give you. A maximum cyclic commutative sub algebra for this endomorphism. So you can actually put all these possible ring structures into the same. So to any single space, you can think about the endomorphism algebra of the module of the ordinary coding of intersection. And then you can think that cyclic sub algebras here should somehow be giving the ring structures on your intersectionology but sometimes for example in the finite sugar right the case you can have multiple small resolutions, which have one isomorphic instructions but that means that they should still be inside the same. Then you can, I don't know, you can maybe classify all the cyclic sub algebras in this case I feel or at least in some cases I see there is a unique one so you cannot choose anything else. But they don't have small resolution systems. In this case I understand the joint representation, you have only a semi small resolution a unique semi small resolution or this was some answer on resolution, but we know that that our big algebra is not the sub algebra of the resolution. So it's not cannot be seen there, but there's no small resolution, but we still have some in those cases that can be going straight here on the intersection. Alright, so maybe thank you for a great talk, thank you so much. Thank you for being at the end of our today. Thank you everybody online and we'll be there. We'll put these things up for access later.