 thanks for inviting me. I've spent hundreds of hours with Kashiwara's back catalog and his range is cosmic. It's infinite. I admire him a lot. I don't know if it's appropriate to mention that I have the same birthday as he does. Okay, so in the talk P will be a prime number and K will be the algebraic closure of the field with P elements. And the talk is about the following definition. So an F field on a manifold is a locally constant chief of rings whose fiber is isomorphic to K. So the best example is on a circle. You can put an F field whose fiber at some base point is K that I'll call K with an underline. So his fiber at a base point is K and whose monodromy around the circle is the Frobenius. So it carries an element of a fiber to its p-th power. And every other good example is pulled back from this one along a map from M to the circle. Let's see what when I think of the circle equipped with this chief of rings I'll denote it by S1 with a subscript F sub P. So you might as well define an F field to be a map from a manifold to this reference circle. Well then it's not a good example. Yeah, so if I'd like the monodromy around every loop in M to be to raise to some p to the n-th power. Alright, so this is a chief of rings. So we can study sheaves of modules. So I'll start by discussing locally constant sheaves of modules. So for example the category of locally constant sheaves of underlined K modules on this particular circle is equivalent to the category of finite dimensional FP vector spaces. Okay, the equivalence sends a sheaf to its abelian group of global sections, which is a module for the global sections of the chief of rings, which is just the field with p elements. So incidentally we lost a homological dimension by passing to this twisted category. This is a semi-simple category. Okay, here's another example. So if I take the complement of a knot and I put an F field on that knot complement, who is, let's see, that wraps a meridian of the knot around this circle n times. Okay, then if I look at one-dimensional local systems, isomorphism, that's a finite abelian group whose order is the Alexander polynomial of K evaluated at p to the n. Well, if p is sufficiently large, you have to throw out primes that divide the leading coefficient of the Alexander polynomial. Well, so I don't think that that illuminates the Alexander polynomial very much, but it at least indicates that maybe that F fields aren't boring. Here's another example. So if M is the surface of genus G, let's say genus 3. I want to study rank n local systems on M twisted by some non-zero F field. Well, I can describe it as usual as some space of framed local systems, decide on some way to frame a local system and then divide by the group that changes that frame. And one way to, one way to set up that framing, you wind up with five, maybe a little more space, five matrices, A1, B2, A1, B1, A2, B2, A3, and B3 is missing, or rather B3 is the Frobenius. So subject to these in GLN of K, five times. Subject to this version of the surface group equation, A1, B1, commutator times A2, B2 commutator is equal to the Lang map of A2. This is just the operation that raises every matrix entry to the piece power. A3 at the end. Thanks. Okay, then when we divide by this gauge group, that's just a finite group acting by conjugation. Well, this is some variant of character varieties that have shown up yesterday and today. It's not symplectic in any sense. So the intersection theory on the surface when you have a F field, is there's something weirder about it. For basically any F that is not trivial. Then it would, right, maybe p to some power in that case. For a primitive value of F, maybe it's correct as I wrote it. Anyway, it's not a symplectic manifold, even in characteristic p. But it is at least kind of a manifold in characteristic p. So it has a natural structure of a, not quite of a scheme, but of a perfect scheme. So any automorphism of anything, it can be repurposed as a local system on a circle. For example, on S1 sub Fp. So some examples that are, that seem relevant. So if I have a ring, well, if I have a k algebra, and it's a perfect k algebra. So that means that the pth power map on R is a bijection. We can make some variant of this sheaf of rings on the circle that I'll call R underlined. Okay, that's a sheaf of rings on S1 Fp. And just studying this, this functor, this sets up a moduli problem. And it's represented by a perfect deline Mumford stack. What it means concretely when you talk about perfect schemes, you have like the usual collection of regular functions. And then you throw in all their pth roots, which has some downsides. Like you don't have infinitesimals anymore and you can't do calculus. But, okay, so this thing is a perfect scheme most naturally. What's another thing you could do? You could replace Fp bar with its Frobenius automorphism by the maximal unrammified extension of the piatic numbers with its kind of, well, I don't know what it's called, it's called sigma. It has a field automorphism that lifts this Frobenius map. And, okay, so that defines a sheaf of rings that I'll call Qp infinity underlined. A locally constant sheaf of rings on this reference circle. And the category of local systems for this sheaf of rings is equivalent to, I think, what are called deudenae isocrystals. So it's again a semi-simple category. This is not an algebraically closed field. So you don't get a, you don't just get vector spaces over the fixed field, you get something else. And that something has a name. And, okay, the last variant, you could, let's see, if I have an algebraic group over this algebraically closed field K and it has a rational structure, okay, that means it also has some isogeny whose fixed points are the Fp rational points of this group. And this isogeny is a bijection on K points. Okay, so you can study this sheaf of groups, this local system of groups on the circle. And the category of torsors for that sheaf of groups is naturally identified with the category of torsors for this finite Chevrolet group. So I want to apply this, these F fields. I want to use these F fields in symplectic geometry. So if I have a symplectic manifold and a net field on that symplectic manifold, you can try to define a Foucaille category with care that's been changed, twisted maybe, by influenced by this F field. I'd like the objects of this category to be Lagrangian sub-manifolds of M together with a local system of modules for the sheaf of rings restricted to the Lagrangian. So I want to set up a simple problem, maybe a sample problem along those lines and then apply Cauchy-Waare and Shapira's theory to analyze it. So my symplectic manifold will be this reference circle times R3 or in other words the cotangent bundle of this reference circle times R. And I want to study always sort of non-compact Lagrangians in this manifold. So I'd like to talk about their boundary conditions, which I'll draw in this way. So here's the surface whose cotangent bundle is M, some surface. So this is the circle down here, it runs from zero to two pi. Maybe it might be better to think of this circle as having volume log P instead of two pi. I don't know, it doesn't make any difference. Okay, so here's a cylinder unwrapped and pressed against the blackboard. And then the boundary conditions, I'll draw two curves, two core-oriented curves on this annulus, an orange one and the green one. So these lift, they lift to Legendrians, Legendrian curves in kind of the boundary of the cotangent bundle of S1 times R. So that's S1 times R times another S1. And I want to study Lagrangians, okay, so first of all they should be exact. Second of all, they should add infinity, at contact infinity, they should wrap around this orange curve exactly once and they wrap around the green curve exactly twice. I mean by that, I mean the embedding from this Lagrangian surface into T star of S1 times R, it carries like one boundary component of the surface, it extends to a map to the compactification, that carries one boundary component around that orange curve one time and the other one around that green curve two times. Yes, although the fun part is that the category that you build this way is a target for invariance of Lagrangians. And those Lagrangians live in a real symplectic manifold and not just in some skeleton formalism. But yeah, yes you could. So there's this story, it's getting older, of Nadler and Zaslo, and they give away to turn such Lagrangians, well local systems on Lagrangians like that, into sheaves on that cylinder and the story works just as well. If you look at local systems of modules over this sheaf of rings, they turn into sheaves of modules over this sheaf of rings on the zero section. And the constructible sheaves that you get, well the sheaves that you get, they're all constructible and they have singular support in the cone over those two curves in that picture. And what they look like, so they'll vanish down here and up here and there will be some two-dimensional vector space here, let's call it k squared, it's just a vector space over k. And the monodrome around this loop in the annulus, this region of the annulus, it sends the vector AB to A to the P, B to the P. And then in this region you have a different fiber, you have a one- dimensional fiber, and it's naturally identified with a line in this two-dimensional vector space, so there's a restriction map from this line to here. So it's naturally identified with maybe the k span of a vector called x comma y. And then when I move that vector along the local system to the other side, it becomes the line span by x to the P comma y to the P. And the singular support condition at the crossing, at this crossing here, it says that those two lines should be transverse or just not equal to each other. So those make an interesting moduli space. Maybe you should talk about exactly how you want to frame that moduli space before I'll describe it. So let me first say that there's a cashier and jupyra that give you a micro-local restriction to the boundary. How should I put this? Let me say I'm going to consider the space of all constructable sheaves that look like this. Let me call that space fancy M. So it's just the set of constructable sheaves, I guess, of this form. And one of the things that you get from the theory of, the micro-local theory of sheaves is a restriction map from M to, in this case, well, there are two components of the so-called, well, of the boundary of this figure. One of them is this circle. And the condition, I guess now erased, that the Lagrangian should wrap around that circle exactly once. It says that this will be a local system, a one-dimensional local system on that circle. And then a true-dimensional local system on the other circle. So those two circles, they have slightly different nature. One of them just is a copy. It projects homeomorphically onto this standard reference circle. The other one is a double cover of that standard reference circle. And it's natural to call it s1fp squared. It has the same relationship to the field fp, to the field fp squared that s1fp had to the field fp. So I want to describe some version of this moduli space where these two maps are framed. So I just, I have my Lagrangian, I could do that by starting with a Lagrangian that had a framing along each boundary component. And the result, I'll just tell you what the result is. The result is naturally identified with, okay, basically there's just that one parameter x, y. So the set of pairs x, y in k squared such that x, y to the p minus x to the p, y is equal to x to the p, y to the p squared minus x to the p squared, y to the p. Maybe there's another sign in there. So that's a famous variety, that's the famous. They also obey x, y is not equal to x to the p, y to the p. So not just non-zero, but, right, this is a famous drain-filled curve or actually something with several connected components, one of which is the drain-filled curve. And that's the first example, historically first example of what are called deline-lustic varieties. So I'd like to talk about deline-lustic varieties next. So I'd like to talk about it in language a little bit similar to this. So fix a group and a rational structure on that group. So an algebraic group over fp, so that'll be the histogeny on the algebraic closure. So it induces, like I discussed, a chief of a local system of groups on an annulus. It'll be helpful to me to think of this annulus as having a boundary. So let me just write it as s1 times the extended real line. It also induces, it's in fact almost equivalent to the data of a Dinken diagram automorphism, u3, u4, this one. And then fix, just an example, then fix a sequence of simpler reflections for the vial group, or just a sequence of simpler roots for the Dinken diagram called that capital I. So I is just 2, 3, 2, 4, something like that. Then I want to define a space called the deline-lustic variety, or just called x of I. So it's the variety that parameterizes some structure on this annulus. So here's this annulus. And I'll use the sequence 2, 3, 2, 4 to mark some points on the boundary at the top, 2, 3, 2, 4. So parameterize, and this is an annulus that carries a sheaf of groups whose fiber is some reductive group over the algebraically closed field K. And I want to parameterize a G of K torsor together with a family of B reductions, reductions to the Burrell subgroup of G along the boundary, and not along the whole boundary, but along the spaces in between those marked points on the boundary. And then subject to the condition that the Burrell here and the Burrell here are in the relative position indexed by this simple root, relative position conditions given by the sequence I. So these are the deline-lustic varieties. When you set it up right, x of I is a variety over K, and it depends only on the element of the variable corresponding to... Sorry, I should... These are a little bit more general, because there's no reason that these... Yeah, what is it? Yeah, there's no reason that that's a reduced word, but... It's a positive element of the braid group, and yes, it depends... That part... There's something that's pretty easy, which is, I think it's easy, that it depends only on that element of the braid group, and slightly harder, well much harder, is the fact that it depends only on the conjugacy class of the braid, some twisted conjugacy class, which is what I will talk about in a minute. So you have this variety over K, and it has the right framed version of it. It has an action... So, framed version. It has an action of this finite group and a commuting action of some torus that depends on I, some finite subgroup of a torus that depends on capital I, and that brings me to... So the non-commutative group is just changing the frame of the torus, and somehow that torus is changing the B reductions. Yes, so what are delinuistic varieties for? So they're cohomology, they're allatic cohomology. It has an action of this finite Chevrolet group, which is very interesting, and this finite commutative group, which is easy to understand. So it's a bimodule for these two groups. So it gives a map called delinuistic induction, depending on the sequence I, from the representation theory of this finite group, finite commutative group, to the representation theory of this non-commutative group. So everything that we know about the representation theory of finite groups like this, with some consequence of knowing something about these varieties. And I guess one of their important properties, their first important property, is called disjointness. So if I have a character of a torus, this finite subgroup of a torus, and I delinuistic induce it up to a character of the Chevrolet group, and then I do it again with a different delinuistic variety, and I want to compute their dot product. Their dot product is zero, unless i and j map to the same element, same Frobenius twisted congexie class of the vial group. Oh, that's true if i and j are reduced words. And I come to think of it, I don't know what's true for very general braids. I'm only on the image in the vial group, not in the braid group. Okay, okay. Right, if, well, if these are the only chance they have to be the same, to have a non-zero dot product as if i and j are the same congexie class. And then once they're in the same congexie class, then maybe it's more difficult. But this is a theorem of delinuistic, and how do they prove it? Well, pairing those two characters should involve pairing these two bi-modules. So you should study the variety of i times x of j, and then divide by a single copy of the finite non-commutative group. So I'm going to call this a pairing variety. So it continues to have these actions of commutative groups. But anyway, you need to prove this disjointness property. You need to study this variety. And what's useful to know about it is that it has a stratification by algebraic tori. So there's some way to decompose it into algebraic tori. And then what contributes to this inner product are the zero-dimensional tori. So it's helpful to know whether or not there are any zero-dimensional tori. So all that I really want to say is that you can see these tori in this, well, in symplectic geometry, at least for type A, and in this sort of formal surface description, surface topology in general. So I'll say something about that. So here's the simplest example. So if g is SL2, and both i and j are just the same, the only node of the Dinkin diagram. Then the pairing variety, well, it parameterizes data, is it still up there? Data like that. But now with i going across the top and j going across the bottom of that annulus. So here's i up here, and here's j down here, and that's it. So you have some b reductions on the boundary in between these marked points. So in this case, what does that look like? So we have a SL2, we have basically a two-dimensional vector space over this annulus. And then up here at the top, we have a line of that two dimensional vector space. Might as well be some point in the projective line. Let's call it z. And then this one is, I can get whatever is labeling this other side of that marked point by continuing this line, this is an annulus, remember, around to the other side of the annulus. And then when it crosses the boundary of this annulus, it gets raised to the power of p. Down here something similar happens. And then the condition at this node is that z and z to the p are different. So it's a set of pairs, z comma w, such that z is different from z to the p, and w is different from w to the p, so that's some subset of the affine plane. And then I divide that by the action of the finite group SL2fp. So it's some algebraic surface. And how can you see a lot of tori in it? This is this spectral network story, or maybe it's a special case of that story that I heard are called BPS graphs. So draw, but basically you draw a graph on the annulus. Oh, I forget. It won't, I guess I have. It won't, I don't understand exactly what it changes there, but it doesn't change the fact that the thing is covered by algebraic tori. Covered is a little bit too strong. But yeah, I want to draw a graph in the annulus. Well, I want to emphasize that I could draw a really complicated one. So I could draw like this one. I don't know if I'd be able to work that out on the blackboard in real time. So I could also draw a simpler one like this one. So this graph maybe I'll draw it in color. And now these labels on the boundary, well, once I've drawn this graph on the surface, they extend into the interior of the graph. So they give labels for each chamber, each face of this graph. And I want to consider an open subset of an open subset, depending on that graph of this pairing variety, cut out by open conditions. So it's just the set of z comma w that obey the usual conditions that z is not equal to z to the p and w is not equal to w to the p. But then also for each edge of this graph, if two numbers are labeling faces that are that meet at an edge, then they should also be different. So here you'd have z different from w to the p. And then also here you should have z different from w. So this is some open subset of this surface. And I claim that it's an algebraic torus. And in kind of a natural way, up to powers of p. So the map to this torus, so the two coordinate maps, they're cross ratios. So I send z comma w, sorry, there's one cross ratio for every compact edge in this picture. So there's a cross ratio for this blue edge. And there's a cross ratio for this red edge. Oh, maybe this one. So the blue one takes z comma w to the cross ratio of the four numbers that are surrounding that edge taken, say, in clockwise order. So the cross ratio of z, w, w to the p, z to the p. And the other one takes you to, maybe I have to draw, I have to continue this story. I have to draw another fundamental domain for the annulus to get this right. Okay, so here's still z, here's this red edge. And then this is z to the 1 over p. And that's the only, okay, that's the only new guy. So this red, the four numbers surrounding that red edge are z, z to the 1 over p, w, and w to the p. So take the cross ratio of those. Well, so there are a lot of tori like this inside of this surface, each one indexed by a graph that looks like that. There's some kind of, when you can obtain one graph from another by mutation, there's some kind of transformation, which is a little bit different than the usual cluster story, because the thing is not Kalabiao in any sense, I don't think. But I've run out of steam and pages. Okay, I'll stop. Some questions and remarks? So you seem to suggest that you can rotate the points by closing the, closing your lines, rotate you, and then you get the conjugacy environment? Yeah, Frobenius twisted conjugacy, or, or, yeah, yeah, exactly. And the grade relations, or you'd forget, or? Grade relations are easy, but, but things that are just, there should be more to it than that, because when you, sorry, when you continue something around, it might not be a reduced word anymore. So, so now, now there's some, well, it changes the dimension of the delinuistic variety, but it doesn't matter for the character. So the dimension depends only on the scalinality of time, right? But if, so if you have two, if it's not a reduced word, and you want to, maybe that's what I'm saying, to cancel. I have some questions. So, so this story, I like the cluster Torah, yeah, only, yeah, it's a big Torah, but in this theorem, it can be composed in very small Torah, which is not total logos in cluster geometry, the whole story. Yeah, it's not truly. That's true. There is, in, in type A, there's the, this story of, these are equivalent to Manjuli of sheaves on a, yeah. Legendary knot, and there's the story of ruling stratifications, this Henry Rutherford story, which is, is sort of, which gives you these, these closed, these positive co-dimension Torah, so I don't know how they fit in with the cluster story, it's true. Okay, and just one small remark in general for this idea of sheaves of categories, one can put any local system of local fields on symplectic manifolds, and then begin to be focacate categories, but yeah, that seems to be, yeah, I agree. So, so, so it, you started from the FP, but if you start from the, or the Calypso Xero field with a notarism, then what happens? Yeah, I, what? Yeah, for example, take the, polynomials and t2, that, t2, Yeah, one, it's a little bit of a miracle that, Yes. This is defined using some field automorphism, but it's linear, or it's geometric over the, not over the fixed field, but over the, it's, it's, it, you get some variety over FP bar, even though you've invoked this, this field automorphism of FP bar, and that's something really special to Frobenius. More questions? I don't, I don't know. I mean, just, I mean, if you take, for, I said two, for example, if you take a, high with one element, I mean, can you add anything? You don't impose that transversality condition there? Yeah, I haven't thought about it for that, yeah.