 Okay, so thank you very much for the invitation. I'm sorry that I'm giving the stock online as well. At least I'm not alone here. So I have to apologize that there won't be, I mean, in the talk itself, there won't be, there will be a lot of sort of representation theory, geometric representation theory, and actually just geometry, but there won't be any kind of of enumerative geometry, although what I'm going to talk about is supposedly very much related to at least some other aspects which are discussed in this school. So for instance, it's all motivated by some work of physicists and in particular, for instance, some paper of Whitton, paper of Mihailov and Whitton, which discusses Havana homology a lot. So there should be some connection with Havana homology, but I absolutely don't understand what it did. So okay, so let me briefly explain what the plan of the talk is. So for about half of the talk I'll be reviewing some known results. I'm sorry. So, I'm going to start with review of some very basic thing in geometric representation theory review of geometric satake equivalence. So then I'm going to discuss another equivalence which is kind of similar to geometric satake, but which is again, I think pretty well known to for, well, to sort of people working in geometric representation theory, but maybe less well known to other people. This is what's called capital E, which is an abbreviation for fundamental local equivalence. This is a terminology that Dennis Gates was using. And I'll maybe comment on the terminology when we get to it and then we'll discuss Gaiota conjectures, which should be thought of as some analogues of fundamental local equivalence. And then, well, if there is time left, which, you know, I'm not sure about that then I'll talk about idea of proofs. This is a recent joint work with Finkelberg and Travkin, which has just been posted on archive yesterday. So I should also say that I have given a very, in some sense very similar talk also in IHS I think about two and a half years ago, but at that time at that point everything was just 100% conjectural. And now we actually have a lot of theorems so I'll try to sort of kind of emphasize that but again I'm going to begin with review some known results. And one thing that I want to, another thing I want to emphasize before I proceed is that, well, I mean in the end of the day I'm going to get this to get this Gaiota conjectures and then maybe discuss the proofs of some special cases of them but the point is that for people working in geometric representation theory, these conjectures themselves sort of quite strange and I would say unexpected. And I think that mathematicians would never be able to guess those conjectures and thesis somehow can derive them from some kind of string theory calculations. And I'm absolutely unable to follow those string theory calculations, but somehow it's pretty remarkable that by using this kind of very mysterious string theory calculations, thesis can produce conjectures which are kind of absolutely mathematical and also somehow very reminiscent of some other things in geometric representation theory. But again, mathematicians somehow never did anything like that. Alright, so this is this was kind of a preview. So, now, let me start this plan. Implementing this plan so first of all, what is geometric satirical equivalence well. So we work over algebraic writer so we see so everything over. See, and so we fix. Which in the end of the day is going to be g land but for now it's going to be just any connected productive algebraic group. Well, overseas. And so the basic object that somehow which will appear on a geometric side for us is the affine grass manning of G and so the fine grass manning of G is, well, I guess, has been discussed and Joels talks, but let me still just fix notation so I'm going to join by K. I'm going to draw a circle around power series and inside inside there is a ring or which is the ring of the series and then we can see that they find grass manning of G, which is the coalition G of K. And this, well, again, I guess, Joel discussed it. And so this is so confusing dimensional object but it's a pretty nice and logical object so it's, it's union of projective find dimensional projective varieties. And in particular, the group G of K, still acts there on the left and particularly of all acts, and the orbits of geophiles and dimensions in particular we can see that the category of perverse G of all equivalent. She's on the fine grass man. Well, so this is some category this some senior a billion category. And, well, and this turns out to be a natural way it turns out to be a tensor category or symmetric monoidal category. So, this is kind of, you can write it as perverse sheaves on the coalition on the double coalition G of K module for module for and this is, again, this kind of think that you're more or less discussing very close to that so somehow you can use convolution some kind of convolution product here to the tensor the tensor structure, and it turns out that it's going to be symmetric and you know this duality and so on. And the basic basic theorem here is that is the following so in theory which is what is called geometrics of that equivalence is that this category as a tensor category is equivalent to the category five dimensional representations of of the group G check where G check is is the language dual group so this is the group whose root datum is dual to that of G. But pretty soon we're going to just switch to the case when G is GLM so let me just remind that if G is the group GLM, then the dual group is also GLM a self tool. Okay, so let me first say why I mean let me first mention that this is a very good equivalent so it's so it's a starting point for a lot of things in well in geometric representation theory or in algebraic geometry so again, again, I mean, so this is very, for example, important for for dual scores, but also I mean this is this is really the starting point for what's called a geometric language correspondence and in fact this is a kind of categorification of something of classical set I guess a more personal which is starting point for usual language correspondence. So this is a kind of very good equivalence, but instead of talking why does good let me talk why this bad. So, so disadvantages of this. So I'm going to name two disadvantages there's a kind of sort of categorical disadvantage and and and this is kind of representation threat. So, the first disadvantage is that it does not work it well doesn't work as stated on the derived level derived kind of what I mean when. In other words, I want to say that you can can see that the it Vivares and derived category defined as money. So this is whenever how a group act on a variety and everything here is kind of essentially find a dimensional. You can talk about corresponding current drive category and that thing is absolutely not equivalent to the derived category for present of fine. Well, actually, when I say here representation of G check I mean, representation of G check is, is an algebraic group so just find dimensional algebraic representations and G check is a reductive group so it's categorical representations is, is semi simple. And so it's derived categories pretty pre well, but the derived category of of geofolk weren't she doesn't have a restaurant is quite known trivial. So. So, so this thing's only close to the fact that it is known what this thing is equivalent to and this is going to call derisive influence, which I actually almost won't use well I won't formally use it but let me just for the sake of completeness let me mention what this is. So this thing is equivalent to the following thing to I should take. Well, okay, what I'm going to say is going to be a slight lie, but I mean it's. Well, let me put maybe bounded derived category here. And, and then this will, and, well, I need to find those conditions that when I'm going to say will actually be true. So she here I should put, I should take the lead algebra of the one goes to a group. I should put it in homological degree negative to regard this as a DG algebra with trivial differentials, trivial differential, then should consider derived category of modules over this DG algebra. So whenever have DG differential graded algebra can consider differential graded modules and we can take it because one drive category and I want them to be sort of let me notation well and also I maybe want to consider here. Sorry. Finally generated modules. Here, I want to consider G check a current once which is, which is just mean stuff so I'll just do know that like this but what it really means is that I can see the modules on which also the group G check acts and the action is compatible with a joint action. So, so this thing is, you know, it's not, it's not a derived category of any a billion category essentially and it's definitely not the right category of SMIC. So the derived set I keep on just kind of much more complicated, although, although it's also extremely important for many purposes and in fact, for this story of cool and branches it's also extremely important, but that's not what I want to talk about says you know, again, let me know if you this is a drawback and the sounds that somehow I have the sequence between a billion categories and I would like to have the. I mean, I would like to have a sort of slightly different set up where, which would extend to draft categories as well. So that's kind of maybe a minor drawback and more important drawback for me is that this dramatic something that does not extend literally to quantum. Well, okay, so this is not a mathematical statement. Well, let me write it like this, let me write to wrap to you to check that means that I can see the representations of the first one. Again, does not extend means that it's, you know, well known natural way to extend. So, now I'm passing to number number two in my plan so I'm talking to this fundamental local equivalence. And the fundamental local influence is it is it is it is a dip is a different to go on some similar nature. But it will be in a course where both of these problems will be cured. And so jumping over to how they should say that is going to other conjectures will be a sort of, it's a set of conjectures which will also extend, which will make which will extend this fundamental local equivalence so it's fundamental local equivalence will be special case with the group GLM. Okay, so. No. Any questions so far. Okay, so now. Sorry. What was Q, what does Q stand for that. What is Q. I mean, okay, so this is this is this is quantum group representations of quantum group. Well Q is typically for quantum groups it's a number although actually, if you want to do things canonically it's not really a number it's, it's some C star valued invariant form on the on the weight on the co weight letters but but you know if Jesus say simple group then q is just a non zero number. So this is just representation of the corresponding quantum group. So quantum sort of this category of representational quantum would say it's a deformation of the category of presentation of G check it and the deformation in the world of what's called braided monoidal categories so it's no longer symmetric so it's it's a dance but it's kind of dance the category but it's not, it's not. It's not symmetric so so it's a well it's braided so if you can see the some V times the W is naturalized amorphous to W tends to be but the square of this of the sequence is not one. So, and so so this this kind of natural thing and and so we'd like to. So here we're going to have some kind of geometric realization of the terms of category of G check representations want to extend it to quantum groups. And somehow, this way we can't. Let me say how we can do it and this is also very important for a dramatic langos although I'm not I'm not going to say anything but dramatic langos in this though. So, the story is the following so it's it's going to be it's going to look kind of slightly differently so so let you inside G be a maximal important. Some group. So say if G is GLM we can just take up a triangle or up a triangle of matrices with one on the deck. And so let also cry from you to the additive group be a generic character generic additive character. So example. If G is GLM. You can take you to be. These matrices. With one diagonal zero below the diagonal and anything about the diagonal, and then typical choice of Chi is the summation of all a I plus one. So it's a homomorphism and. Well, you can also take any linear combination of those but with non zero coefficients so it's important that all coefficients for every I is non zero then characters generic. So choose a generic character and then. So we can consider you off. Okay, and they can consider characters and called Chi hat from you okay also to the additive group. And this is just by given by the formula that Chi of you of T is equal to the residue at equal to zero. Chi of you T. Something like this. And so now. If. Now we can consider the following guy so first before you but then we're going to introduce you. So you can see that the wittaker category of the fine grass mind and this is by definition the cataclysm say let me put away the derived category of you off K, comma, I had a covariant she's on a fine grass month. Let me know that here, even to define this have to work a little bit, because I mean the difference between this and. What we did before with geometric said I guess that here the orders of the group you have Chi are you have case or a not a not find dimensional actually infant dimensional so somehow. So this, the, the, the setup is kind of more international than before but sort of modern science knows how to handle such situations very efficiently so some of these actually rigorous definition. What is actually much less of this is that this category is also turns again. And here, I mean here the problem is that there's no I mean, but the tensor is not going to be given by convolution that there's no kind of convolution here. So, so this is a young tensor. But here, you will actually have to believe me how to define, let me just say that for people who know this word the tensor structure comes from fusion, rather than convolution. I will also say that when I say derived category so to hear now. Actually, when you have an important group with an edge of character acting on some space, then when you want to consider she's a current respect this character, then either actually, instead of she's you have to use d modules, or, or you have to work with. You have to work over fine field, because the point is that usually, I mean, the point is that this additive character defines some, you want, you want this kind of character to define some one dimensional local system on your group, and with some applicability properties. And so, in the world of demodules, they can actually do this, if you work over complex numbers. Let me do it for you itself. So if you have time for me to G a, then you can consider the module which is fullback aspect archive sort of exponential demodule on on their fine line on the on the active group and that's, that's going to be one dimensional local system on you which sort of, which has some factorization property. And so therefore you can talk about she's a current respect to or demodules rather than respect to this thing or another sort of equivalent way is to work over fine field instead of complex numbers and work with a lattice shoes. And then you can use the art and try a sheaf, the pullback sorry of the art and try a sheaf respect to kind. And then, so this, so this is about what, what is meant by this, by this category of sheaves equivalent or demodules equivalent respect to this group so so I mean, I mean most people are used to notion of sheaves equivalent to the notion of a group so there's a. So the claim is that there is a kind of enhancement of that so if you have either additive or multiplicative character of the group you can actually talk about she was a current respect action of this group with a character and if this is an additive character, then either you can use the language of demodules or should work with a lattice shoe so fine. So let me not worry about this. So, and the dance structure said, the dance actually is defined in such a way so we actually work define this model is related to formal disk, and in order to define this structuring to use an actual up spray curve, formal disk and let some points. And let them collide and so on so this is kind of typical sort of fusion which I don't have time to explain but again you will have to just believe me that there's a, there is some kind of structure here, but it's kind of defined, I can actually define it for the aggressive for geophonic variants as well. And for, for the obedient category actually get the same structure but derive category actually get a different structure. Okay, let me not go into this. All right, and so, kind of first theorem is that this with a category of the grass modern is now equivalent again, well, I mean I'm using the drive category already so it's just the drive category of again representations of future. So in some sense this way to get categories much simpler than on the derived level is much simpler than than the satirical category because even on the derived level. It's the same as representation to check. So now, where's the quantum group here. So the claim is that you can actually upgrade is the quantum. So let me talk about this. All right, so our way to quantum groups is, well, again, we're going to have an equivalence of well braided monolid categories. And then, while it's easy to say what we're going to have on the right on the right we're going to have derived this representation so let's fix we fix you and see star. So, we can see the right kind of representation for quantum rule. Yeah, I mean maybe before, before I go to quantum rule, let me say that, you know, this is in terms of the rival but it but it also induces the corresponding equivalence and the sense that this equivalence between derived categories is compatible with natural t structures on both sides so you can see the perverse sheaves on the left and just representations on the right to get in front between this ability. So, and same is going to be true in this Q kids. And so here, let me write this thing like this. And let me just explain what I mean by this so what this thing. And what with Q is so this is, this is the category of you of Chi comma Chi had a quiverion. She's on some determinant line bundle L over the grass money and so the grass money and actually, well, okay, so here we we gain some words which I don't want to discuss but it's very simple than all the just once I mean questions what is Q I mean so if I think about Q is a number, then, I mean, it's, it's, I can really think about Q is a number which is simple otherwise I may have several parameters Q in fact try to think what Q is canonical and then there's actually some choice of somebody in your form on the on the lead algebra and such a termed line bundles also parent threads by this. But if G is a simple group, then you can actually think about the number and some determine line bundle. Well, I mean, actually I have to remove with the zero section, the research removed. I mean, when I say online bundle means on some, you know, some total space of this line bundle, which have monodromy Q along the fibers. We have this L minus the zero section over the grass money and there's some kind of well for simple group there's some canonical line bundle otherwise some choices. And so here, well, it's a line bundle zero section removed so every fiber here, every fiber here C star. And so we can see that she's we can send some kind of twisted she's so we can see that she's in the or well not I mean, you can do it for any, for any variety of the line bundle you can consider Q two is that she's without that means that we can see the she's not on the variety but on the total space of this line bundle without the zero section, which have monodromy Q. I mean, the fundamental group of the I mean, I mean, they're going to be on every fiber it this thing is going to be a local system and the monodromy is going to. So this again, this is like extremely general procedure in which is used in geometric observation theory over time for many, for many, for many purposes. So, and so, and then let me go back so then we have this equivalence, which is again, this is a braided manual. Now, if you haven't seen this before, and if you're following what I'm talking about you might ask why couldn't I do this for the original set up equivalence I mean, I could have tried to take. So, Geofoic variant perverse Geofoic and see perverse she was again on the total space to this line bundle with monodromy Q. And the question is that and the claim is that that that would not work. And so, for instance, if Q is not root of unity, it turns out that if you can see the geofoic variant she is then there will be essentially no geofoic branch she is with monodromy Q. Almost all of them, I mean, they'll be just, well, it is simple, they will be, they'll be just one, I mean, this category will just get over that. So, so somehow it happens that if you can see the, if you do it with geofoic ventures if you if you can see this Q twisting then the category becomes much smaller I mean for it for if user with a few and it will become just smaller if Q is generic, it will just essentially collapse. But for this with a category that's not the case and we get this quote. Okay. Any questions so this is a. This was the review of this fundamental local. And now I should say that maybe I should. Because this FL E FL, he was a conjecture of Jacob. Okay. And then, and then was proved by gaze gray for generic Q, and then by gates gray. And independently. Okay, so I'm exactly more or less at halftime so. So this was, this was a review some known results. So now, let me go unless there are any questions let me go to the other conjectures. I asked a question. We go back to the statement of the categories of the last one that you had. Which one, this one. Then you have just she's coherence or which she perversion. No, no, no, no coherence is anywhere. No, no, here in Korea and she's do not appear here anywhere. She's she's misconstructible. She's or actually, I mean I said that, I mean, when you work with this with a category, I wish to work with constructible she's over fine field or I should work with corresponding demon. Because the ones that they're kind of this she's so fine field, they're going to be not same so in particular or because when demon is will not have regular similarities. So, but again, this is, this is a mild, this is not very important point. And again, I mean, I mean, even, I mean, before that so, so, okay, so here. So here, for instance, in this equivalence. I mean, in this definition, same thing happens and then you know if you can see this, this theorem then here all she's a constructible actually if you go back to this set back equivalence then again here on the left you can see the construct. I mean, when you work with perverse shoes, I mean, means that you construct, you, you work with constructible shoes. So it's rather, you know, in some kind of more general setups related to dramatic when you have usually kind of a typical situation that you have a sort of constructible side and coherent side and so. You know, here, if you look for instance, this equals then this is, this is the left hand side and, and the right hand side or position to check it should think about it as coherence side so it's this rather, because it's actually here and she's on the stack point more G check. And so, and so for instance, we look at this derived subject equivalence here, then this is also she's on actually coherent shoes and actually some derived stacks so so somehow usually it's a kind of typical situation we have some equivalence on the left and have some constructible shoes on the right and have some here and shoes and stuff. But in this QK somehow, it's slightly different because you don't I mean this this coherence side becomes sort of. Well, it becomes kind of Q twisted so to me it's I mean it's not really here and she's on anything anymore. So, so let's proceed to get the connection so so now the idea is that. So the auto producing so. Defined. So, let me actually. Sorry, let me let me say one. There's one notation so fix. Now two numbers to natural numbers. M and M, and I assume the M is less or equal to them. And just for simplicity. Then the other produce the following the other produced the geometric category by geometric again again mean it will be some category of some kind of she's on. Some kind of a fun cross money. This is a geometric category, such that. Which is equivalent, let me say like this, which is, which is conjectural equivalent equivalent to the following thing to representations. Q. So this is a quantum quantum group but this time it's going to be quantum super. So this is the super world so first of all before Q you can consider the super group kind of algebraic super group plm. And this is automorphisms of the super vector space C of M and so this is a super vector space which has even dimension M and all dimension M. So this is some you can consider super go for the morphisms and there's a, there's a well known. Well, maybe less not super well known, but kind of known at least Q different mention of that as well. Let me just. Actually, I mean, you can actually do it for non generic you as well. But, but you have to be. It will also be true for non generic you if you define this quantum group careful. So. So you have to choose, you have to work with some particular form of the quantum group. And so I'll maybe specialize to the case. I'm equal to n minus one, because that case in that case will be it will be the simple to explain but before I do this let me say that if. Well, we're working here with arbitrary m and m, but so if m is equal to zero. Then all the super part goes away. So then GL man is equal to just g one. So it's supposed to recover the same thing we had before. And so this guy or the category. So this will become just with this with the category of the progress money of g one. But I'm going to look at the at the other extreme maybe not the real assembly I mean that the other extreme will be m equal to m. But it turns out that the easiest thing is the easiest example to explain for me will be n equal to m, and minus one. So it turns out that. So let me explain what happens in this case. And it will actually not look at it will it will look pretty differently from from this with the category but somehow it turns out that this is kind of, well, not a continuous deformation but if you sort of mean. And this guy or the conjecture stuff for arbitrary m and m and if you sort of move on from zero up to minus one then you you're going to move from this with the case story to what I'm going to tell you now. And, and this actually in this case it's extremely simple to explain. Anyway, this guy or the category in this case is the following thing, let me write it and then I'll explain it. Well, is. Well, again, we can work with a billion categories over derived categories that makes you work with, let me formulate the statement for a billion categories. What I'm going to explain is that the same thing will be true for derived categories as well. So I think you can see the perverse shoes on the affine affine Drasmanian of GLM. Well, I need to put q here, because, because I want to do for quantum robot particular I will be able to specialize it to people to one as well but I will have to do it carefully and so what they should put here should put things equity variant with respect to GLM minus one of all. So here I have the group GLM minus one can be naturally embedded into the group GLM just in the most stupid way possible. So name you can see the just matrices like this you can see the matrices which have one here. So this is, this is just GLM minus one sitting inside GLM and you can see the things inside GLM minus, which I correct like the GLM minus one Oh, so it looks like so from so kind of symbolically it looks a lot like the geometric static But it turns out that that it behind is much more like this FLE than the geometric static faults. So. Okay, so now if you want theorem, which is reading our recent so. And again, so I should say that I'm only considered this example of M equal to M minus one but this is maybe I should I will make some comment in the moment. What happens for, for other m's but right now you can see the only example when m is equal to M minus one, but in this case we have a theorem that this category. Well, holds both a billion and derived level category reverse GLM minus one, or a covariance she's with Q on the famous mind of GLM. And this is equivalent to wrap you GLM. M. Well, the only thing is that I need to, I need to say one thing careful so somehow of course when you start, let's forget about Q for a second. So if you started a category of representations of an algebraic supergroup. I mean algebraic supergroup will usually act on super vector space. So you should consider. So, so the meaning of this we can see the representation of this in super vector spaces. And then let's put also a letter s here and the letter s here means that we can see the first she's we can see the constructible she's with key fusions not in vector spaces but in super vector spaces. That makes perfect sense and then somehow I mean that this is what you need to know for this. So this is again a braided manoid of equivalence. And so here for this to be true is stated queues to be generic generic essentially means not the word of unity. So, this is what we call for derived categories. So, I should say that it's kind of funny that if you if you replace, you know, if you look at this. If you look at this category here. If you replace GLM minus one by the full GLM, then somehow things become very different because first of all, I should say that now on the, if you can see because one derived categories will not be derived category and second, if you if you if you try to put you there if you put generic you there. And as I said before this category will just essentially collapse, which has become the category of a little bit extremely small, but for some reason, if you GLM minus one instead of GLM. Sorry, actually, I should have read my Amazon minus one session right here. So, so if you put GLM minus one instead of GLM, then somehow miraculously, all the problems disappear. And you get this thing. Now there's some kind of also funny combinatorics here because for instance, this representation so the quantum or even non quantum just usual supergroup. The quantum and correspondence was actually a useful presentation of it's even parts to know that the even part of this group is just GLM times GLM minus one. So, and this should have something to do with the orbits of this group GLM minus one all in the front guys mind of GLM. And the claim is that so the claim is that this group has discreetly many orbits and the orbits of rhyme tries by pairs of domain weight domain weight of GLM domain weight of GLM minus one. I mean, such things happen pretty rarely. I mean, usually if you put some some kind of random group here then it will not have discreetly many orbits on the fine grass mine and but this one. Now, let me make some comments about this. Let me make some comments about the shape of the guy or the conjecture in general. Now I should say that I'm only formalizing this get the conjectures in this GL case, although they're kind of more general I mean, I mean, they're kind of. There are other, say classical supergroups for example, some degree school or the symplectic supergroup and there's some version of this get a conjecture for that one. So let me stick to GLM case conjectures for, let's see, arbitrary. Well, let me say less than M, because the case M equal to M is also slightly different. Well, what Gaiota does, Gaiota, so he tells you how to. So Gaiota produces for you some unipotent group UMM inside. So actually, well, it's inside GLM. And, well, yeah, so it's inside GLM but you can also then we're actually going to also then for future purposes embedded into GLM cross GLM so somehow getting into GLM is. Well, it's just the GLM part is three bill here. But so, and this is normalized. This is normalized by GLM inside GLM. And also you can basically character of this UMM into the additive group, which is also normalized by GLM. And this in general is the other category. The category is the following guy, it's, you can see that well, let me just hit a drive category of the category of perverse shoes, well, it's not perverse shoes. So in retrospect to GLM, oh, semi direct product UMM of K, well, comma, chi hat. So similar notation as before I should also put this queue here and I should put the fine grass money in of GLM. Now, it's actually convenient to write in the following ways the same as well. I mean, especially if you're not afraid of various infant dimensional problems. This perverse sheaves and respect to GLM of K, semi direct product. UMM of K, also character kind hat. Also, you and here should put the fine grass money in of GLM times GLM, which is actually the product of the grass money. And this is kind of an exercise to see that this is absolutely that the logical this thing considering she's here grants back this group is the same considered the same as considering she's in the fine grass money with product of this two groups considering. So, so, so in some sense, so the advantage of writing it this way is that here we can see the shoes that grant respect some group of K. So this kind of the way I should think about this. This is this this group GLM semi direct product with UMM so I didn't tell you what UMM is, but there's some particular definition of it. This is, well, it's, you should think about it as sitting inside GLM times GLM where again the embedding of this important part is goes only into the second factor and GLM goes into this can think diagonally. And you should think about the subgroup. It's, you should think about this one as analog of the maximum important huge. So it's analogous in many respects or for instance has the same name of you sort of in for again for this group for you know for you inside GLM cross GLM. This is going to have the same dimension as the maximum important subgroup in here. And so the point is that if I'm as equal to zero, this new man will be just the maximum important and we're just going to get back to the same with your story. So, so the kind of special cases is that if I'm as equal to zero. So if UMM is then this GLM it disappears because it's just GL zero. This is going to be maximum important in GLM. And if I'm as equal to n minus one, then UMM is going to be trivial. And that's that's another case that we consider it and and so this so it means that. So the point of saying that this GLM minus one seating diagonally inside GLM minus one times GLM for many purposes is analogous to maximum important subgroup in the same. For instance, you can do a simple exercise and check that has the same dimension. This is analogous is not a mathematical statement but in fact it has the same dimension is a mathematical statement I can check that. So, so somehow what Gaiota does for you he produces this, the sub and then, and then the rest and, and actually, they also come with a character, but again for M equal to n minus one case this character will be trivial, and then we should consider she was in the case corresponding to fine grass man and equity variant was group with the character, and that recovers well if you put also few in the picture that recalls the corresponding category of representation of quantum. Maybe I mean I have something like five minutes left. Yeah, some point my zoom stopped working but if I can people hear me now. Yeah. Okay, let me discuss briefly what happens in q equal to one case. And this is a subject of our previous paper from two years ago. This is joint paper. So, in this case, well, and again for simplicity, let me take my cooking minus one. So then the claim is that if you think the kind of gross a perverse she is on respect to g l n minus one, oh, on the fine grass man of g l n. What you get is, well, you would like to say that you get represented well I mean the principle should also put us here just because for the same reason as before I would like to write that they get represented representation of g o m m, but I said that when you specialize to non generic you have to be careful. And, and, and actually so this particular thing is actually not true you actually have to put gel on the line here and this is going to be some degenerate version of super cool. So, and let me say briefly what this thing is so So, well, if you can see that the lead algebra, because one in the super algebra, this is essentially, I mean, looks like matrices of size m plus m times m plus m. So let me write them as like blocks like this. So it has here. This is the even part this is even part. This is all part this is all right. Well, and then I have various. Well, on the level of the algebras have very super super commutators. And so the you introduce the I mean this is this is the least of your g l m m but then introduce the kind of degenerate version of that, which means that it's same vector space the same super vector space. And the commentators super commentators and define I mean, most of them will be again the same except the super commentator of any two odd elements. But the bracket of any two elements equal to zero, if a and b are odd. And if you if you drag it even with even or even with odd. It's the same, the same thing as before as, but odd with one will be zero. That's that's a kind of degeneration of this usually super algebra, and then we can also do it on the level of groups of algebraic super groups and it turns out that we get this equivalence. Now, maybe, let me just, I'm tempted to make the following comment that we call this star. So star is actually a special case of another set of conjectures, but this time by completely different people by Benz v circular radius and then can touch. And those conjectures are motivated they can have supposed to be categorical version of a lot of known results about special values of automorphical functions. And, but the, well, that's a kind of motivation but the actual form of the conjectures is the following so. So, well, let me tell you what the sort of left hand side is left hand side is roughly the following sorts you can see that some group G is before. And inside you can see the some age which is a spherical sub rule and spherical means that I think some of this microphone is on age has an open orbit on the flag right to G. Maybe, let me say an example that, for example, this GLM minus one inside GLM minus one times GLM is spherical. So, as Benz v circular radius and van can test, they study say the derived a bit of perversions doesn't matter derived category of H of K equivariant sheaves on their fine grass mine and which. And they sort of describe what it's equivalent to but it will take me some I don't want to do it right now but it's described in terms of some kind of dual groups and some representation of some groups. So this is some, well, they have some precise conjecture and this process but this precise conjecture is based on some kind of theory of the spherical subgroups. And so this, and this theory of spherical and so it's based on some theory of spherical subgroups. And the. So, but the real sort of motivation for them is to, well partly proven, partly build up on some results about the morphical functions. So somehow if you look at this example if you go back this example that on the one hand so so let's let's look at this as I said it on the one hand, it's a special case of the school and so. It's easy clarity and van can dash on the other hand, it's the q equal to one specialization of this conjecture so go out, which are motivated by some completely different things and by some again calculation string theory which I'm unable to to reproduce or to order again. Now, unfortunately, this conjecture to be in cease to create some van can dish have no known q version. And so I actually talked to them and, and it is not. And they're really. Well, it's not clear how what to do that. But on the other hand, somehow they can take so much more general because they're not. I mean, many situations when it can actually talk about the spherical subgroups and when there's no super groups. So I don't really know what's going on here. And again, the most surprising thing for me is that I mean, especially if you go back to this theorem that I wrote here, then, again, it's, you know, if you look at the formulation, it's not something that you might expect to be corollary or string theory calculations. But, but somehow it is and again, you know, mathematicians weren't never able to to produce such a conjecture and even after this connection is formulated somehow. I don't know how to sort of motivated mathematical. That's one thing I want to say that thing I want to say is that that thing I was planning to say but I don't have time for that is, is safety words about the proof. Let me just say that the proof is similar, although slightly more complicated. And then it goes on to the proof of the original FLB by Gays, Gary, and it goes through some kind of, and it's built, it builds up upon upon some work, old work of this recording of Finkelberg and Schachman, who realized the revolutionization of quantum group in terms of certain factorizable sheaves, and later on Luria somehow explain why, why the work is sort of essentially three bill from authorization, or from the two algebras. So somehow something like that is also used to also use our usual quantum group but also for quantum supergroups and so you realize that in terms of some kind of some kind of kind of factorizable sheaves and piece of configuration points on the curve, and then use some kind of similar argument to those of Gays, Gary, to prove this. The last thing I'm going to say is that this should be an analog of this, I mean, again, well, I know how to formulate it, which is don't have a formal proof yet. When instead of the category find dimensional representation of a supergroup, you use, you get a category all for that for a particular choice of a browser, and I should say that in the quantum supergroups there's notional subgroup or subalgebra, but unlike in the usual case, not all of them are quantia. So for a particular choice of a borrel, you can look at the corresponding category all. And that should be geometrically realized in similar terms, but instead of the fine grass money and you should use the fine flag variety. And, well, there's some kind of representation variety complication of that because in this way, you actually, you can actually use that to get some character formulas for simple models and category all for for supergroups, which actually not known. But I should, but again, this is maybe not my main interest, my main, my main this in the second of two fold. So one, or maybe three fold one is to understand how visas are able to actually produce such statements. Second is to understand the kind of better the connection to this work of been three secularism Venkatesh. And third is that something again I don't have time to talk about but it is actually clear that this results are very can be used for various applications to something which called quantum geometric langos program. But again, I don't have time to talk about this. So I think I should better stop here. Thank you very much. Any questions online or offline. Physical calculations you're talking about exactly. Well, you know, I bet I would not even try to start answering this question. Well, I mean, it's just the, you know, it's, for instance, there's this paper by me how it can we turn so there they somehow on the one hand they have to understand a theory for the corresponding super group. They started a three dimensional from some of the startups as dual and somehow it's it has to do with sort of some brains colliding and so on. You know, I mean, I mean, I mean, you know, I made some kind of effort to understand this word but I'm absolutely unable but but it's it's very much connected to the paper to old to some well not very old but like from 12 years old papers by Gaiota and with and about boundary supersymmetric boundary conditions in four dimensional gauge theories and so on. So for instance, even if you look at these papers by get get with and then you see that they produce some relation between supersymmetric boundary conditions and super groups. So it's, it's boundary conditions for gauge theory for usual, even groups, but some kind of interesting boundary conditions for them are related to super. When you started as duals of those boundary conditions and somehow when you can actually compute them and I mean the computer. Well, I mean, the actual statements so I can actually format the kind of physical statement which is essentially almost equivalent to this theorem. And that involves field theory that involves four dimensional field theory so it's really a statement about one boundary condition going to some other boundary condition are there as well. But the real question is how the thesis know that somehow some kind of two boundary conditions are related by astrology. And for that some kind of string theory calculations are used and and that I really, I can't explain. There's something in the chin. Okay. Any other questions. Right. Thank you very much, Sasha. No, but the big question. Yeah. The browser group which appears here is, is kind of mixed for all. So, in order to choose a browser group, essentially, well, I mean, the reason the right, I mean, one way to think about why there are different browse for for for super group is that, I mean, this kind of I mean, okay, so if we're working with just the real thing, you know, browser group and just flags browser group in in CN browser group and GLM and our flags and the super vector space CMN. And so, I mean, complete flags. And, but, I mean, every time you add dimension I mean, when you have a flag that's my have space v one, see which is inside v two which is inside the three and so on. Every time you add one more dimension. But in the super world, you can add either one, even dimension or one odd dimension. And so the barrel which is the one way where when you're alternated. So when you add sort of first one, even dimension than one or dimension then again even then again or the soul. That's kind of. That's that's the barrel which is natural to use. Any other questions. Thanks.