 But I will speak about today's loosely intertwined with the Hadamard lectures I'm currently giving here. So I want to talk about the geometric-Satake equivalence in a certain setting. And I don't want to claim that I can prove everything, but I want to say how one can get the handle on some key geometric step in the proof. Okay, but let me first of all recall the classical setup. And so let's say as little k is an algebraically closed field for simplicity. g over k, some reductive group. So then you have several different functions. You have the loop group of g. This is this function which takes any k-algebra to all two groups. I think rg of the wrong series over r. And in there you have the positive loop group, l plus g, which is also a function of two groups. It takes a power series value of points. So these are both, say, fpqc sheaves. And the l plus g is actually in, now we lost everybody. Okay, maybe just a very brief, not so much acceptance. So I want to talk about geometric-Satake and I started by recalling the classical setup. Where I work, say, over an algebraically closed field and have my reductive group g. And then I have the loop group of g, which takes any k-algebra r to the group which is a little wrong series value of points of g. And inside there you have the positive loop group, which takes us to the power series value of points of the field. And then l plus g is actually an affine scheme, but of infinite type. So it's the inverse limit of the r, the joint T mod T to the n-value points, which are some restriction of scalars along this. And l plus lg was some kind of in-scheme. There's an increasing union where you bound the poles of this long series. And then the affine-grasmanian is the, let's say it's the fpqc sheafification. It's not really necessary to do fpqc, you could just do it all. Obviously the loop group was a positive loop group. And actually there's a direct way to say what this is as a functor. It's just classifying the following data. It classifies g-torses. What was my name for the g-torser? My name, let's call it E. Over the spectrum of the long series, of the power series field. Plus the trivialization of E restricted to, okay, so if you trivialize the torsers and giving such a thing as just giving the trivialization over the long series field, which is an element of the loop group, and then you mod out by the ambiguity of taking a trivialization over our power series T. And so, maybe also if GSGLN there's something pretty explicit. It's just a set lambda, or lambda, which are some kind of project. And contain a, ah, lattices, okay. Okay, and so the basic theorem about the geometry of these guys is that, it's what's called a strict int projective scheme. So strict refers to the fact that the transition maps are closed immersions. And, ah, maybe I should have said that you also have an action of the positive, well actually of the whole loop group, but in particular of the positive loop group on the grass mania, which is clear from the description of the quotient there. And the L plus G orbits are in projection with four characters of a maximal torus, and then they take dominant guys. So they fix the maximal torus in the Borel and G via mu maps to mu of t. So mu of t is then an element in the loop group of t. And in particular reduces to the point in the Fn grass mania. And for, so you have the open Schubert cells with the g mu, which are this L plus G times this point mu t. And there, and usually one takes a reduced closures of these guys, called the g mu bar. Well actually now, I mean, now it's the open Schubert cells by a node and then the non-open guys just this way. So there are some projective varieties. Usually singular projective varieties. And okay, and so this whole thing is the union of these guys along close to emergence. And then there is a geometric sataki equivalence. So what's over the complex numbers. Then by Merckwitsch and Villonen, and then there's some other papers, maybe by Timo Wischer, and Shinwen Zhu, which discuss the case of other fields. So maybe use an export to say this. So you can look at the category of purpose chiefs on this Fn grass mania and you look at the L plus G equivalent guys actually. Switch those words, purpose of G equivalent chiefs. It's better if you exchange those words on the Fn grass mania. And actually what Merckwitsch and Villonen would say, well, which hasn't been taken up by Wischer since you so far, I believe, is that if you do this actually with the L coefficients, so integrally, Wischer and Zhu, they work with the QL coefficients here. It should also be true with the L coefficients. And this is equivalent to the category of representations of the dual group. Representations unfoundly generated. So in particular it's true with FL coefficients? It's also true with FL coefficients, yeah. Where G dual is a Lengen's dual group. How does this work? So this works in such a way that if you take the highest rate representation V mu here, which can, which doesn't make sense integrally, then this corresponds to an intersection complex mu here, where mu is a dominant core character for T, but that's the same thing as a core character of the dual torus. And so such core character, say, parameterizes, well, character 6-0 would be the reducible representation of highest rate mu. And you have this relation. L is not equal to the characteristic of K. So the G dual is split. Yeah, I mean, I'm working on one algebraic closed field, so anyway, yeah. So the topology of this, when you work overseas, and you change, so if you, it's about the topology relative to constructively this shouldn't change when you pass to characteristic P different from L. Is this the... So you wonder why the statement is independent of the choice of the field K. Is that the question? Is it easy to say that... No, it's not. I don't think it's a topology, that if you vary the characteristics, so if you put a family of afengrass manians over spec 0 or something like this, that you get the same category in all characteristics is upper or a clear. I think it's not clear that you get the same answer. Upper. Yeah, there are cell decompositions and resolution, but I mean, there is some finer structure like these. Okay, I mean, this is not just in the terms of categories. There's some extra structure, there's symmetric monoidal structure, and getting all of this equivalently, there's some work to do. But what led this to this highest rate representation? Well, I think there's a unique one which has the highest rate integrally, the highest rate generates the whole representation. If someone takes the highest rate, there's up to scaling, a unique one, just the scaling doesn't matter, so you pick this one and then he takes this letter generated by this. Highest rate. I mean, also, if I work integrally, I have to be careful what I mean by perverse, because there are two dual perversities, and I need to specify which one I take, and, well, you can look it up and work which one you don't, I might get it wrong if I try to say it. All right, I think that's what I wanted to say. Yeah, right, so this is an equivalent of symmetric monoidal categories. And so it's clear what the symmetric monoidal structure is on the representations of the dual group. It's not as clear what you do here, so here it's just in the product. So what you have to do here, you have to, there's a convolution product. So if you do want the monoidal structure, you can define this as a convolution product. So essentially this uses that if you take this LG mod L plus G, well, if you take some kind of L plus G equivalent thing, then this has a natural map to LG mod L plus G. And so if you have two such perverse chiefs, which are also a plus G equivalent, you can make one on this big guy because locally this is more like a product of two copies of this and then you push it forward under this map. But this is not clear that this is symmetric, it's not clear that it's commutative. And so to get a symmetric monoidal structure, the usual argument actually proceeds by giving a different construction of this convolution product, namely it uses a so-called fusion product. And I want to talk about this later in the setting I'm in, but let me only note for now that this, for this you have to work over several copies of the base curve. The base curve in this case is something like k power series T. So in this case you have to work over the power series being in two variables and then specialize to the diagonal. So I have to stick to the middle. Okay, great. So for ZL I think the tensile product has also total one, so you expect to have... Right, I mean so you have to be a bit careful. So if you restrict to some kind of free guys, torsion free guys on both sides, then this works nicely. If you don't, then of course on the category of representations, there are tor one terms. And so you wouldn't expect that the convolution of two perverse things is again perverse, but there might be some extra kind of tor one term appearing which you would have to neglect again. Okay, so I want to talk about a version in periodic geometry. So which entails something like replacing k power series T by maybe the width vectors of k. And so this can be done. And that's the theorem of sigma and u. So let's say g is over the width vectors of k where k, let's say again, is an algebraically closed field of characteristic p, a reductive group. And then you consider the following variant of CF and Grass-Monion. So which you only define on perfect k algebras. Which takes r to the set of all g torsion of the spectrum of the width vectors of r, which are now, in the case of a perfect algebras, some nice ring e plus the trivialization of e restricted to punctured guy, which now means takes the width vectors and invert p. And again you have an action of the loop group, in particular of the positive loop group, which is the same which takes r to the g of the width vectors of r. And let me just abbreviate this theorem by saying that the similar results hold true. Where the small part of this is actually a theorem of Bogov-Batten myself, namely that these guys are, that these closed tuba cells are perfections of projective varieties. So Jue only proved that they are algebraic spaces enough for him to go on. And so the much more difficult part of this theorem is the geometric-Sartake equivalence. And, okay, so Jue actually had to use a trick for the geometric-Sartake equivalence because he didn't have an analog. So to get the symmetric, to get symmetric structure, you have no analog of such a two-variable algebra. So instead he used this, I think, I think due to Gelfand. So the Gelfand has this trick of proving commutativity of these spherical Hecker algebras by using some involution of the group, some anti-involution, which gave you an anti-self isomorphism of the Hecker algebra, which then showed it must be commutative. And there was a way to do this geometrically and that's how Jue could pull this off. So if you have a two- and an automorphism of G, it nearly acts on this, on all the categories. Right. Okay. And so is there usually the action, let's say for in the automorphism action is trivial or canonical trivial or... Yeah, I mean, there was some subtlety in getting this to work because he had to check some coherences between the monorail convolution structure and the symmetry isomorphism and checking those actually came down to some combinatorial identities, which he could only prove because they were the same as an eco-characteristic and eco-characteristic they follow from geometric sataki. So it's actually this kind of convoluted argument. The group is not constant? It's in the... Well, I mean, in this case it's somehow a constant. I mean, this is a split group in this case, right? Not automatically, but maybe it seems like... Why? It could be ramified. No, it's a reductive group integrally, so it's like, like, un-ramified. It's un-ramified in the work of an algebraic closed field, so it's actually a split group, so I don't have any issues here. Okay, so... Right. But actually, that's not the theorem I need in the context of these lectures I'm currently giving. So for Fox conjecture, we need actually a version for some B-to-round plus cross-magnet. So B-to-round plus is a ring defined by Fontaine and Piede-Cochiery. It's, again, a complete discrete-valuation ring, as k-power series T or the width vectors of k, but now the residue field is Cp. And so, of course, it's abstractly asymptotic to Cp-power series T, but not canonically so. And once I make some functor on some test category, it actually is genuinely different from a power series ring. And so, say if G over B-to-round plus is, again, a reductive group, which, again, must be split automatically, you can do the similar thing. So, again, you have an f-magnet for G. When now the test category is kind of similar as an induced setup, we have to restrict to some perfect algebras to get a well-defined analog of the width vectors. So you have to know perfecto at Cp-algebras, which, by the way, I will maybe in a moment conflate with algebras over the tilt by the tilting equivalence to sets which, well, does a similar thing, so it takes it to the set of G-torsors over, and now there's a construction which takes any such perfecto at Cp-algebra and produces an analog of this ring B-to-round plus, let's somehow know for this family, R. So there's a B-to-round plus algebra which, if I turn that to the residue field Cp, I get back the algebra R. So it's a flat deformation to B-to-round plus plus a trivialization over what's called the fraction field of this G-to-round. Okay, so you can consider this similar object and again, you have an action of an L plus G on it. And now in, so maybe actually make a remark how is this related to the previous guy? There's a relation between these two. If G is defined over with respect to K already, which it's implicitly always is because it's always split and then you can take a split form over with respect to S. Then we can define a family. I find grass manians, which is some kind of Baylent-Drinfeld-type family. Which lives over something I would call the diamond associated with the vectors of K. So these are, technically these are all functors on perfectoid spaces and characteristic P. Whereas the fiber, the special fiber is a bit vector of an gross manian. WK is not there. It's not analytic, but you can still define this as a V-sheave in the context of my course. So the special fiber is this thing and it's essentially a scheme. And the generic fiber, meaning over Cp is a speedy run plus gross manian. And it's, there's a more classical situation if you have a smooth projective curve say and it need a bit of projective, just some smooth curve and you have a reductive group over it. Then you can also build a family of effer and gross manians over the curve whose fibers at each point are effer and gross manian corresponding to the complete local ring at this point. So as usual you have, one can describe the L plus G orbits and define two varieties. Well, let me put varieties and quotes because they are now quite far from actual varieties. And a theorem, the main theorem of my Berkeley course was that these Grigimus, let me put the speed around plus here to make clear what I'm talking about. This Grigimus belongs to this world of diamonds that I'm considering. So technically it's a proper and spatial diamond, proper separated spatial diamond over. And so the course so far implies that you can do a telecomology. We have a six-functionalism of a telecomology for these objects. And so in particular, using some abstract results of Offer-Gerber, one can make sense of perverse sheaves on these guys and then also L plus G invariant, perverse sheaves. All right, I should get somewhere soon. We can also define a convolution product. And for this you have to check by hand that the convolution morphism is semi-small, which you can do. So you can also define. But to get to symmetric structure, well, you could cheat again and use trick which probably works. But actually, it's even necessary to have this picture with these higher dimensional bases for Fox conjecture. So I really want diffusion product. Do you want some constructability notion here to define that? Well, I mean, if the L plus G equivalents must automatically be constant on all the open Schubert cells, which gives you some constructability with respect to that, and then of course you want that on each stratum with local systems, I mean for finite rank, which actually must be automatically constant by L plus G equivalents again. And of course you have to check the operations preserved this constructability. You have to check the operations preserved this constructability. I mean, somebody's gone. It's okay. I mean, the way to check this is that you can define some kind of dimmer zero solutions for all these guys here, which are nice morphisms over the open L plus G orbit. And then you first check that everything behaves nicely on this resolution, where everything has some smooth locally, the same thing as some standard guys you know from algebraic, well, which analytifications of algebraic objects. So you can control it there, and then you just push forwards on a proper map and so on. You can control those things. Okay, so for the fusion product let me actually assume that G is defined over, let me fix a model over the width vectors in word P. Start with, and let me denote this guy here actually by L. Then in the world of diamonds, the issue is having these two variable family actually go away. So we have, can take the edict spectrum of L pass to the diamond, and then you have such a two variable guy. Let's call it two for two copies. And you can also define some Baylian's and Greenfield's grass mania over it, which now in the case of two copies of the curve looks slightly different. It's not at each fiber enough on grass mania instead. So what does it parameterize? It parameterizes G torsors over the R-valued points, or actually to be very precise, the R-plus-valued points, parameterized G torsors over the space I denote by this bar of R-R-plus, and then it in some sense takes a fiber product over a bar K with the edict spectrum of L. This can be defined in terms of the width vectors. So it's an open subspace of the thing you get from the width vectors, E over this guy. In the classical picture of the width vectors you described in the beginning, what would be the analysis of this? For the width vectors it doesn't... The two values are just the width vectors. No, already there is no such thing for just the width vectors. So if you want to have these two variable guys, implicitly already for one variable would have to do the spin in Greenfield grass mania which lives over a small part of your curve. So implicitly already for one variable you get the generation from the width vector from grass mania to the B to R plus grass mania. You don't really see this over a curve because it's basically constant X. What did you say? This is another approach instead of the... You said that you can define... I want a fusion product and so for this I need... So yes, I don't want to use trick the skeleton trick. Instead I want to mimic more closely what's done classically with a fusion product and for this I need families of this balanced Greenfield grass mania over two copies of your curve and my small punctured... This is like a little punctured disc and I take two copies of it which now some... So what you order with SPA, W R plus W R plus, that's on the right hand corner. Yes. W R plus is width vector of R plus so it has... It is an attic ring. Was it P comma Pi... Erect topology where Pi is a uniform... So it is still not... It's not analytic but the open subspace will be analytic. So essentially I look at the open subspace where P is not zero and the technical of our Pi is not zero. So I look at these guys but with a trivialization again. X1, it's a graph of X2 which I map into here where X1 and X2 are the maps from SPA plus to SPA L diamond which give you the map here and it turns out that giving such a map to SPA L diamond is the same thing as giving an embedding of this attic spectrum into here with some properties which is in some sense a graph of this map here. So away from the diagonal go the other way. If you have a graph of something then it is a model to the source and you are... Sorry, yes and that's what I mean. Sorry. And that doesn't make sense. Sorry. Thank you. Actually giving such a map is a problem to giving an un-tilt which I call R I sharp. So this corresponds to sharp R I is one and two. And this un-tilt automatically embeds into here. Thank you. Away from the diagonal the fibers are as a morphic to two copies of the speedrun plus cross-monium. So that's because giving a... Like if it's already trivialized away from these two points then until I understand the extension to a G-torque over everything you just have to understand what happens infinitesimally near those two points. And that's given by some usual Bovellastel lemma which has an analog here by this B-run plus cross-monium. But over the diagonal itself the fibers are just one copy. Because if you're over the diagonal well then giving us more than away from these two guys is just the same as away from one guy and then you just get one copy. So this picture is actually really close to the usual picture for a curve. And so what you want to do is that if F and G are L plus G equivalent, which maybe for the moment is not critical perverse sheaves on the speedrun plus cross-monium then their exterior tender product the box product defines the sheave on this Binance and Duhinfeld guy away from the diagonal because away from the diagonal this is a morphic to two copies. And so let's denote this open inclusion J into the whole guy. And what you want to do is you want to canonically extend the perverse sheave here to perverse sheave on the whole guy. And it turns out that there is such a canonical extension and for this let's me assume that F and G are 0. If I look at the box product this has a unique extension to what's called a universally locally acyclic ULA and this is a notion I want to discuss in the five minutes or so that remain ULA sheave on this whole Binance and Duhinfeld perse-monium and this extension is perverse it's actually equal to the intermediate extension this is what's called J exclamation mark star and the restriction to the diagonal is equal to the convolution product. Where the convolution product upper depends on the choice of ordering but this picture is obviously invariant under flipping the two factors which gives you the symmetry okay and so this follows the usual arguments so this is how one proceeds classically but one needs a good analog so once one has a good definition the right definition of ULA I can say this universally locally acyclic so in the monoidal structure one has also to verify an hexagonal and pentagonal right so there are these higher axioms these higher compatibilities these you check by three copies of this picture and so on okay so you need analogs well I mean I mean there is this thing that if you try to do the same for the whole derived category there's actually not any infinity monoidal structure on this it's only E2 or E3 or something like this so actually so this nice symmetric structure is actually really restricted to the perverse chiefs so let me recall briefly the usual definition of ULA chiefs let's say f from y to x is a map of schemes let's say finite type separated map of schemes and f over y is some sheaf well it's actually be more precise it's something deep plus constructable sheaf on y some coefficients l is invertible on x as usual is f universally locally acyclic if and so universally refers to the fact that this should be true after any base change the following holds true that for all geometric points y bar of y so this maps to x this maps to some geometric point x bar of x and I have another geometric point eta bar which specializes to x bar and so you want that if you look at the sections of that's a strict generalization of f the same over x bar is over eta bar meaning if I base change this guy over the strict generalization of x at this point to eta bar this is unchanged what's your definition of the drag category is it well let's assume that everything is a finite type over a field I don't think it's in there is there a subtlety then no because it depends if you use your definition you can use the because you're working on this usually it is the let's work with torsion coefficients to this issue okay so it says something that if you have a specialization so it essentially says something like the consequence is the following say if f is proper then actually the homology of the fiber y over x bar is the same then if I take rf then this is actually a local system it's a locally constant and how do you prove this because it's always constructible it's enough to prove that it's under specializations it's unchanged and this invariant under specializations is precisely what's encoded in this kind of condition but it's somewhat encoded locally whereas this is somewhat then a corresponding global statement so it's a statement that some of the homology of y with coefficients in f is somehow invariant as you move on the base x in a sense okay so an analytic geometry let's say for 8x spaces so for analytic at x spaces the analog does not work the obvious analog does not work the problem is that there are no interesting specializations not sufficiently interesting specializations so you mean to make sure that this thing about proper morphism to make sure that the thing about proper morphism say is true so you certainly want a notion which has this implication so for example you might have situations that you have some ball here and the point inside which is somewhat roughly the situation we're in here with this diagonal inside this two variable guy and then you have some y over here and then you have the fiber y0 in here and then the kind of situation we're in is that f over y is maybe locally constant on y minus y0 and we want to we want to characterize uniquely the extension to x to all of y then you would somehow want to look at specializations which specialize into this closed point but from a point in B but there are no specializations from B minus a point so all the specializations they somewhere either happen in this closed stratum which here is just a point anyway or in this open stratum but then you're only talking about what happens on the individual stratum where you know things are well behaved anyway and say anything about the variation as you change the stratum for the kind of stratum we care about here so I'm already over time but I want to give the definition the usual definition of ULA it's with a paper stick on the product it makes sense what is what? the definition with a paper stick on the product of ULA okay there's another definition of ULA in the literature and I don't know so let me give a definition which is which works anyway which is closer to the classical one so we asked the following one is the same condition for the specializations you still put it in but it's well in the case of interest we always satisfied anyway so that's not the critical part same conditions for specializations but then you put a second condition which is the following and for all constructable sheaves G on Y the trig push forward of the tensor product what does it mean constructable constructable means it's locally constant after passage to a constructable locally closed stratification but you have to be a little bit careful about making this precise because the stratum might not be attic spaces themselves only pseudo attic spaces but there's a way to make sense of this condition the point is there are constructable spaces in the sense of spectral spaces in the sense of spectral spaces right so this actually I'm sorry for going over time but if I'm not going over time this whole talk doesn't have any sense so what does constructable mean so again in our example that you have this open this closed immersion of a point into a ball the issue is that i lower star of z mod l to the nz this is not constructable that's because part of the definition of a constructable locally closed stratification is for example if you have an open stratum then the open stratum needs to be quasi-compact and the issue is that if you look at b minus a point and this is not quasi-compact so if you turn things around the consequences are following if you have g over the ball some constructable sheath then there exists an open neighborhood u of the point such that g restricted to you is actually locally constant that's because constructable subsets of these attic spaces are slightly funny so any constructable subset of a if it contains the classical point it automatically contains the small open neighborhood of this classical point and so this means that if you apply this conclusion here that actually in the neighborhood of each classical point there will be a small neighborhood such that the cosmology over this close this one point is the same as the whole neighborhood and in this way you actually get this kind of local constancy around this closed point and so right so if you put this extra condition in which is for example satisfied if f is a local system and f is smooth and has some other nice properties which resembles the known properties of ULA sheaths for schemes then you can actually get the argument running so we will take a few questions so maybe we start by Tokyo so two questions from Tokyo yeah so are there any questions so I suppose that you want to prove this Fargo's conjecture so you so is this construction are you ready to do that prove Fargo's conjecture or what I mean so far I'm just trying to give a proper formulation of Fargo's conjecture so so I mean implicit in Fargo's conjecture is that there is a geometric set up equivalence and then only then you can really make sense of and that there is a nice theory of sheaths on Bungie and some there is this factorization sheath property and you really need this fusion product to make sense of it right I mean so you really need this fusion product to make sense of everything on the other hand once you have a proper formulation of Fargo's conjecture then as I said in my first lecture here you actually automatically get a construction of say my simple air parameters for all irreducible smooth representations and maybe in the case of GLN or so there's also it might be possible but anyway to do something but but right I mean for the moment it's just about getting a proper formulation of all the objects in Fargo's conjecture and then you have a question yeah that's a question from Tokyo, thank you so the gene veterans in your remarks it is possible to construct a nearby nearby cycle banter to build a fusion fusion product argument to construct a symmetric monoid stock around the previous sheath okay I didn't completely understand the question but I think it was whether there is some good theory of nearby cycles in this setup and so that's actually a good question so for usually nearby cycles for schemes there is in particular the theorem that if you have a perverse sheath on the generic fiber and you take nearby cycles it's again perverse and this fails let me give you an example so this means that you have to actually be a little careful so let's say you have the ball times the ball why I'm projecting to the first factor I'm mapping to the ball and then the fiber here is the ball why is inclusion to the second factor and then I claim that I can find a a perverse sheath on here whose nearby cycles are not perverse and so okay let me try to draw a picture for this so let's say this is the ball which is a special fiber and then I have the generic fiber here the ball times the ball and what might happen and then you have somewhere a fixed point X in this ball and it has the pre-image and the specialization there is some subset here some open subset well okay so let me talk about the open if I take it in the attic world it's closed but then I can take remove one rank 2 point and it's open and what I can do is I can put a sheath which is concentrated on some small part here so actually oh I mean you can take J. Loiswig from a small ball in size so this is a small ball small quasi-compact ball last one it's a quasi-compact one and if this inclusion take J. Loiswig of my coefficient rank and then of course the nearby cycles so this can be checked to be perverse then the nearby cycles are 0 everywhere except at X but at X you get the compactly supported cohomology of this open subset so if I take the nearby cycles then this is concentrated at X and given by the compactly supported cohomology of this U which is a lambda degree minus 2 and that's too far off to be perverse so a related phenomenon is that the art in vanishing theorem fails for rigid spaces again because I mean the ball if you just take a close ball it's an affinoid space and so you would expect that the cohomology of all constructable sheaves vanishes above degree 1 but then this J. Loiswig of lambda for a smaller open ball inside gives you a sheave where the cohomology is a compactly supported cohomology of this U but this goes up until degree 2 and so this art in vanishing fails. So here you are considering because I don't understand what situation you are considering or considering the formal model of a ball of a spec of the variation ring or you consider a ball cross a ball well ok so right I'm implicitly passing here to maybe the completion so that so sorry I mean it doesn't really make sense let's talk about the inverse here well imagine that inside the ball times the ball you have some kind of cone which contracts very sinly to X that's the kind of situation I'm considering so you have this maybe there are several different fibers here and you have this thing which very sinly contracts down to X and if you have a sheave there then the nearby cycles will be concentrated at this point but the cohomology there will be the compact support and so you actually you have to make sure that these kinds of examples don't occur in this situation this condition of constructable I mean is it enough to check this for the skyscraper G or something like this skyscrapers are never constructable but I mean in this way this is a very difficult to check if you have to check it for right but I mean you can check it in practice because it's a theorem that it's true if F is a local system and little I mean script F is a local system and this small F this morphism is smooth comodologically smooth and further compact and and it's preserved under push RF lower shrink in general and so we're using some form of procedures I mean for example on the DeMazur resolution you have some local systems they have this property and push them forward to the stratum and then they're still locally universally locally can be done so do we still have other questions from Beijing? no thank you that would be all for Beijing okay so other questions in there okay yeah the category progress is equivalent to the category of representations or two that means this setup of D the run plus representation category is supposed to be well so the same so the same and so this is if you work with Cp but actually this F and grass manual is defined over Qp and then if you encode the descent data to L group is it possible to define near by second from Pia Gershwin on the B-derarm of FGM so the question was whether you had to space between FGM generic FGM plus FGM especially FGM and so whether you can have some nearby cycles there and yes I think you do and it might be possible to deduce this result by nearby cycles to the result of the I think if you have group as you find over Zp you can also directly define all the experience in Winfeld-Grassmine it's not just over spa L times spa L but over spa OL times spa OL and then have somehow this split vector of grassmine always in the picture still other questions in your abstract I think you mentioned something like collapsing two points of space there well that's basically what you do right I mean spa L times spa L looks like spa spa Z times spa Z essentially locally at P like L is essentially QP and then you have something like spa QP times spa QP over some more absolute base and then I'm restricting to the diagonal and so this essentially means that it's locally the local picture is an open subset of spa Z times spa Z and then I restrict to the diagonal so I'm taking two points and collapsing them what do you mean the same well but I mean I'm infinitesimally away from P already right so I'm deformed a little away from P now I have two points which are close to P but different and then I can collapse okay so there are no other questions let's try to speak again