 So thank you very much. It's a great honor to be able to be here and wish Maxima a happy birthday. I wasn't your boyfriend. Yeah. You weren't school or anything. Yeah, so what I'd like to talk about today, well, I'd like to say Maxima, as you all have known and said many times so far, has been at the origin of a number of conceptual revolutions, I think, in a lot of different subjects, in fact. And what I'd like to talk about today is another one of these examples on the subject of basically character varieties or spaces of representations. So the space of representations, so let's let x over c be basically a curve. Maxima Key's asking me, what about higher dimensional case? But for these questions, these questions are already mostly happening for curves. And we don't really know what to say further, I don't think, for higher dimensions yet. Anyway, so let's take a curve. And so we should think of it as basically being either a compact curve or some kind of open curve, but we keep some control over what's happening at the puncture points and so on. And as I think many of you know, an important part of what's going on also happens for irregular connections and stuff like that. But I'm maybe not going to make that much of a distinction for that here. And so the space of representations, let's say, into SLN, let's say R, this is an affine variety, usually called also, the topologists call this the character variety. So in particular, it's open, so it's non-compact. And so I think there's recently been some kind of conceptual revolution in how we can look at the points at infinity. Unfortunately, I don't actually understand this very well. So and also I'd like to thank Tom Bridgeland, because I think in his talk yesterday, he said a lot of things, which one should say here. But let me just, first, before getting started, write down some of the people. So obviously, Konsevich and Soebelman is they have a series of papers, which are really at the origin, I think, of this. Also, Gaioto Morneitzki. So let me apologize especially since I'm on film. Yeah, OK. I'm going to be forgetting lots of people here, so I'd like to apologize to all the people who I might forget here, which are numerous, especially as I said, being on film. Then I'd also like to write Bridgeland Smith. Then there's also people which are basically coming from 3D topology or low-dimensional topology and other similar types of directions. But for example, Richard Wentworth. And there's a whole group of people roughly speaking around François Labouille. Maybe I'll say Lofton and other people working with them. Ian Lee, for example. And their work is based on stuff which we've actually been using a lot, which is a work by Perot. And Perot, I think, was working with Federique Poulin using Kleiner Lieben theory. And this whole thing actually also goes back to Morgan Shailen's compactifications and so on. And I'd also like to mention Thomas Hausel and his co-workers, Rodrigues Villegas, Littelier, and so on. And then there's also a number of recent people which I'm sorry I don't know. For example, Tom mentioned in his talk a certain number of people, Ikeda and some other people. But there's also there's been several different recent preprints on archive. I'm sorry, I didn't collect all this information. And I'd also like to say that what I'll be talking about here aside from the sort of global picture, which is maybe older, but the actual theorems or observation theorems, conjectures, and stuff like that, are joint work with Lune Mill, Alex Knoll, and Ponov Pandit. And I'd also like to say this was on some visits to Vienna. So there's also a number of other people in the group in Vienna, which we were discussing these things with. And I think we should particularly, but Fabian Haydn, who was helping some discussion. OK, and we have a preprint on archive, so I'll be trying to draw some pictures here, but you can also see the much better versions of the pictures in the preprint. OK, so let me just get started. So to get started, I'd just like to draw some pictures these are not the pictures in the preprint. But I'd just like to draw the sort of global picture of the structure that you have on modulite spaces of representations. So let's sort of put that over here. So I'd like to make a box, but instead of making a box, I'll just leave this top open. So this box is representation. The next box is connections, more precisely vector bundles with integrable holomorphic connection. The next box, which I'll put over here, is Higgs bundle. Now I'd like to connect these together a little bit. So this is the Riemann-Hilbert correspondence. And here, well, we have the solutions of the Hitchin equations, which give the correspondence here. But in fact, what I'd like to draw here is actually more the modulite space of lambda connections. Let's draw it this way. So we sort of have a coordinate here, which is lambda. So let me put in quotes that lambda is sort of equal to h bar. But I'm not really sure. I mean, you guys know this much better than I do. Sometimes people put an h bar in here. And I guess the lambda coordinate is often somewhat related to h bar as it occurs in quantum mechanics. I'm not sure whether it's really reasonable to think that this parameter lambda, which is kind of a universal parameter showing up in math, whether this is really equal to h bar or is it just that it plays a similar role in some cases. And so this is lambda equals 0. And this is lambda equal 1. We have a GM action on the space of lambda connections, which scales lambda. And that's why I'm drawing these like this. But the images of the orbits here, the images of the points here, their orbits all go to the sort of ground level state in Higgs bundles, which is the nil putton cone. And so I left a little bit of room here for the Hitchin map. The Hitchin map maps the modulite space of Higgs bundles to vector space. So let's just call this A, AM. I guess you can call it B as the base of the Hitchin vibration. B is going to show up a little later for a different thing. And this map is proper. And for example, these guys go to 0. The nil putton cone here is just the fiber over 0 in the Hitchin map. Whereas this space is affine. So let me just say with respect to what Richard was saying in the previous talk. I gather that this is a little bit of an analogy which is considered by you guys, maybe. Which is the fact that all through these are this is a complex analytic isomorphism. So if you have an affine variety, it can't actually have any compact sub-variety. And that's preserved over to here because it's just the same complex analytic sub-variety. So up until here, there's no compact sub-varities. So I gather that there's a little bit of an analogy with this transversal section to the algebraic, to the nodar left-hitch locus that Richard was talking about in the sense that here we have something which has a lot of compact sub-varities because in fact, it has a proper map. So all the fibers are compact sub-varities. But it sort of deforms to a guy which has no compact sub-variety. The nearby fiber here is just the same as this space. So I'm not sure what to do with that further. But I think that's an analogy which you need to take into consideration here. So the fact that this is an affine variety means that it's open. That's why I left this open up here. And so we can think of trying to compactify. Now, what's the compactification going to look like? So I'll draw this. This is actually an example which you can actually calculate. The only one which I know how to actually calculate. In an example, the compactification looks something like this, for example. So the compactification, as I say with a normal crossings divisor, is going to have a lot of different components, and they're going to meet in sub-components and so on. Whereas on this side, we can also compactify better in a different way. So in the modular space of Higgs bundles, we have a C star action. This is the trace on the fiber lambda equals 0 of the C star action, which trivializes the rest of the space of lambda connections. And so we can just take throughout this piece and divide everybody else by C star. So we can just add an infinity here, a guy which is just the quotient of everything except the bottom here, by the action of C star. If you think about it a little bit, that's actually not a, it's a smooth, I mean, if this space is smooth, which it happens in many cases, then that's basically going to be a smooth divisor. It's not actually exactly smooth, because it has some orbital fold points. The orbital fold points are actually interesting too, because they correspond to Higgs bundles which are preserved, which are invariant by some finite subgroup of C star. And those include things called cyclic Higgs bundles, which are being used by some people now. But if you think of it as an orbital compactification, then the divisor here is actually smooth. And this just goes to the sort of Pn minus 1, an infinity of this affine space. Maybe that's cheating a little bit. This affine space actually has the action of C star is actually weighted here. So this is sort of a weighted, maybe, projective space. Now it turns out that you can use this picture here to get the same kind of compactification here. The compactification has exactly the same divisor. So these guys are the same, which is to say the points here are just, again, points in the space here, up to C star invariance. And we can actually see how that works as follows. So if we choose a point out here, which is not on the delbutton cone, then we take a section going out to this point. Then we translate this section back by the C star action. So you can see what's going to happen here from this picture. When you start translating back this section to here, you're going to get a curve which goes out to infinity. OK? Maybe y'all, do we have some color? Let's sort of shade it as it gets closer and closer to infinity. That corresponds to the shading here by going out to here. And so not only do we get a curve here, but we actually get a parameter, too. Let me call this parameter t, which is basically 1 over lambda. So if we're calling lambda h bar, it's 1 over h bar. Well, that's in quotes. So a point at infinity, if we can think of a point at infinity here on this divisor as being a point in the moduli space of Higgs bundles and going towards that point inside here corresponds to just going towards this point sort of horizontally in the moduli space of lambda connection. So from that, I mean, of course, you have to do the geometrical argument. But from that, you can see that the divisor is going to be just the same here and here. So from this point of view, we'd like to think of the behavior at infinity in these two different spaces as being roughly speaking the same. That's obviously a vast oversimplification. But we can think of it as being roughly speaking the same. Now, I'd like to draw furthermore a sort of a cylinder around this guy. So let's think of taking a neighborhood of infinity. Now, the cylinder over this guy is going to go to a neighborhood of infinity in the Hitchin base. So let me just say my picture is actually drawing the picture in the case where this is a1. It's just the affinal 1. So for the picture, this is actually a disk. So let's think of the projective line as being sort of in the middle of the disk. In any case, the boundary of this neighborhood at infinity is actually going to be a sphere. It's a sphere with a kind of hot vibration to Pn minus 1 in the general case. It's just a circle in the case of a weather base has dimension 1. And again, this picture is also a picture. This is actually a relatively accurate picture in this case of where the Hitchin base has dimension 1, where the spaces have dimension 2, complex dimension 2. And it turns out that this is a, I learned this in a paper from Goldman and Toledo, this is a triangle in a cubic surface. So that's a very classic old thing, Frick and Klein or something like that. But if we think of taking the neighborhood of this guy here, it doesn't look a lot like the neighborhood here. But actually, the point is that if you think about it a little bit, it really does look a lot like this neighborhood in the sense that if we look at this configuration of divisors at infinity, we see a circle. If we look over here, we also see a circle. So just from this picture, you can see that what's probably happening here is that that circle there is going to go to this circle here. So when you go once around a little point and really a little disk in this picture, you're going around an entire divisor of stuff over here. And um. Carlos, for SO2, there's a certain boundary of the thickness space. Is that what you're talking about there? I'm not sure, but I'm going to be getting to that. I mean, getting to say something more somewhat closer to that. Before getting there, let me just say a couple of things. So one is that we can conjecture that we can conjecture that the incidence complex of the divisor at infinity in the character, the divisor at infinity at the character variety, is a sphere. And that the correspondence, this maps by some kind of homotopy equivalence to the sphere at infinity in the Hidgen base. So Maxime keeps telling me that this is actually a theorem. So I'm not quite sure what the hypotheses of the theorem version of this statement are. But in any case, I think, and it's stated in a phrase in, Jan pointed out, it's stated in a phrase in their paper, at least something like this. There's also a theorem which says something a little bit like that in a paper by Richard Wentworth, I think. And I think that what you said about the Thurston boundary and so on is fairly close to this. This conjecture is only just motivated really by this picture, in fact. In the case of you. Yeah, so we can think about this a little bit more clearly, but how are you doing for time? You can make this correspondence a little bit more precise in this case, just in this example, which is that if you think about. That one's a triangle, what's the boundary of the middle one? The one that goes up, the connections. Well, as I said, this is pretty much the same as this guy, I mean, roughly speaking. What's the P1, or what? How connections are different? Well, OK, the point is, in the Hitchin modulite space, the fibers of this map are elliptic curves in the two-dimensional case. So we're going to have these, but in general, the general fiber is in an abelian variety in any case. So the fiber is basically going to be some, let's call it abelian variety, some kind of Jacobian. In fact, it's more of a prim or something. But it's a torus, OK? It's a complex torus. But in this one-dimensional case, there's no problems with discriminant loci. And in fact, it's always the same elliptic curve. And indeed, when you go once around, that does a minus 1 on that elliptic curve. I was telling you that yesterday, actually. When you go around here once, that does a minus 1 on this elliptic curve. And now, if you look over here, think of squishing the neighborhood of these three lines to the lines themselves. The neighborhood of a point here is, it's a delta star across delta star, basically. So homotopically, the punctured corner and neighborhood of the polydisk that goes around this point is s1 cross s1. And that's going to correspond to the elliptic curve, in fact. And what about pieces inside here? Pieces inside here, you have to remember that this line is actually a p1. So between these two points, we have a gm. So there's already a one-dimensional circle direction there. And then there's the other dimensional circle direction of the neighborhood of the disk. So those combine together to, again, make an elliptic curve. And you can see in this example that this whole thing follows around. And if you follow along these coordinates, you exactly get a minus 1, also, on the torus. You identify tori, tori, tori, tori, tori, tori, tori, tori. You get minus 1. I mean, something like that. I forget why I didn't do this calculation. You have to do this calculation. There's a matrix with two 1s, a minus 1, and a 0 in it, whose cube is minus 1. That's going around this guy in that example. I didn't want to do too much detail on this example. For several reasons. One of which is that this example is, I think, totally subsumed as a trivial baby case of what Yana and Maxim do. But OK, so maybe just to continue with the picture a little bit. So we can see here there are some interesting regions which are the neighborhoods of crossing points. And some other interesting regions which are the neighborhoods of things which are not the crossing points. Now, you can actually calculate, again, in this example, you can actually just explicitly calculate with the equation for certain values. This example is the case of p1 minus 4 points with a rank 2 system and some conjugacy classes at the four points. For some special values of the conjugacy classes, you can actually calculate the map and see what happens. What actually happens is that these regions here, which are small regions here, go to entire sectors over here. More precisely, there's three maps, like there's three walls. So these are really the walls. Then you let's look at the pullback of those walls in the cylinder. And it's the regions which are not on the wall here, which correspond to regions like this here. So basically, this map from one side to the other has this property that it's exchanging big regions in small regions. I think that's what I would say, at least as a first epsilon approximation, is really the conceptual, new conceptual thing which appears in Conceivage-Sableman. So they basically have a picture like this in general. And these are the walls of which the wall crossing formula is talking about. So let me just mention here, as far as this is concerned, which is that this conjecture can also be viewed as a geometric version of the P equals W, of the P equals W conjecture of Thomas Hausswell and his co-workers. Now, why do we say that? Because basically, the map from the neighborhood of infinity onto just the real incidence graph on this side. So take the incidence writing, just take a real simplicial complex. That's the thing which we're conjecturing is a sphere here. So that's supposed to go to the map here, which just maps you to the sphere at infinity in the Hitchin base. So it's some kind of vague statement that's saying, wait a step, because this map to this real incidence complex sort of corresponds to some top or bottom piece of the weight filtration. The map that induces on cohomology is going to be the highest or lowest piece of the weight filtration in the cohomology of maybe a neighborhood of infinity of the character of Reine. Whereas this piece, if you're looking at the sphere at infinity, but in the Hitchin base, it's clearly the cohomology class there pulls back to something which is clearly somehow related to the LaRae structure for this vibration. And Heusel's conjecture is a more precise conjecture saying relating some perverse LaRae filtration on this side with the weight filtration on this side. So I don't know whether you can, this is only a tiny piece of both sides, if at all. So I don't know, one could ask can we really geometrically say that the rest of this conjecture corresponds to some geometric picture? I think that's probably actually feasible once we really understand the wall crossing picture here. I mean, you might say that maybe wall crossing should actually imply the speed equals W conjecture or something like that. Let's see, so what did I want to say next? So this is kind of the global picture here. Now, the next thing I'd like to talk about, maybe I'll leave that up here since the tradition here seems to be to use the sideboards. So let me just leave that up here. Now, what I'd like to talk about now is, well, we were just in Vienna. We were just trying to sort of understand this picture and understand the relationship with things like stability conditions and stuff like that. Oh yeah, maybe I should say that before. So as I said, I don't really understand this. And I think Tom actually gave a good, a better discussion than I could here. But what's the kind of zero-th order output of the papers by a Konsevich Soybleman? It says basically that the hitch and base should be considered as corresponding to a space of stability conditions. This is really more of a philosophical. Yeah, I don't know. Well, I mean, we've been working on trying to figure out how to get the category and everything from this picture, which we haven't found yet. But maybe you guys know how to do it. But maybe we didn't see how to prove that. I mean, in any case, I think I'm not sure about it. OK, maybe considered as corresponding to or is equal to the point I'd like to make here is that, in fact, the idea is that we have these structures. So on some kind of stab of D, we have a structure of walls. Also, maybe one thing to say here, which Tom said yesterday, is that we have to integrate this whole thing over the family, over the modular space of curves. We're not supposed to just fix a single curve, but we're supposed to put those in together in a family. And so in some cases, this is the result in the paper of Bridgeland-Smith. It's not necessary. It's a few points for the ledge movements. You have small percentage of parameters in the sense that it's a large curve. It's a dimension for which this is much larger. You never get the curve on the space of the gate. You can go to the left and the center. OK. So you mean there will be other stability conditions which are not covered by this. Yeah, OK. But in any case, in the space of stability conditions, we have some walls. So if this guy is sitting inside the space of stability conditions, we'll in any case have walls, which are the intersection of the walls into here. So we're going to have some wall. And these walls are the things which I drew in the example. And the idea is that these should be, maybe let me say the chambers, in fact. The chambers should correspond to points in the character variety, in the compactification of the character variety. So points over here. So chambers over there should go to points over here. And the walls are supposed to correspond to p1, basically. So in general, I don't think anybody knows how to write down equations for this in any good way, in general. Maybe there I'm talking about the compact case of compact drainage surfaces. In the open case, we have the cluster coordinates. So we can do a little better. I'm not sure whether it's very clear what the higher dimensional pieces look like. And here maybe you know. But in any case, what Maxime points out is that we have these p1s, which join together points. So part of the idea, I think, is that on a given p1, there's only going to be two points. Is that accurate? So basically, if we look at the stratification that you get from compactifying the space of representations, you'll have some points, which will be the lowest dimensional strata. Between the points, they'll be connected by some p1s. Those p1s are supposed to correspond to co-dimension 1, real co-dimension 1 walls over here. When we go from here to here, we do half the dimensions. So things which are complex dimension 1 there could well correspond to real co-dimension 1 pieces here. So now, the SL2 case, the case of some kind of character varieties with SL2 coefficients, has been treated by a number of people and in lots of different ways. So it's a, let's say, well-treated, among other things by Bridgeland Smith. And in this case, so this is kind of the start of our discussions that we had in Vienna. In that case, the Hitchin base is equal to a space of quadratic differentials. A quadratic differential, q, just corresponds to the spectral curve y squared equals q in some kind of cotangent bundle of x. And a big role is played, an important role is played by the fullation, say, a real part of q equals 0. And this actually goes back to Thurston theory and so on. Let me just explain maybe. So underlying this situation, there is something called a WKB question. So we can draw the WKB question here, basically. So here I drew a curve. Here I took a horizontal curve going out to a point in the space of the Higgs bundles. If we translate that back into this picture, we get a nice algebraic curve in the modular space of connections, which goes out and has a nice transversal structure to the divisor at infinity. And in fact, we have this canonical parameter. So at the divisor at infinity, there's sort of a good way of measuring how close we are to the divisor at infinity. Now if we take that and look at the monodroming representation, I think this is what corresponds in Tom's talk to the place where he said, OK, let's take a complex projective structure and look at its monodromy. So we look at the monodromy. But here we can just say, let's take a connection and look at its monodromy. Then this is going to give some kind of weird path inside here. So this is going to give some kind of path, let's call this rho of t. So we have a map from basically c or some neighborhood at infinity of c into t mapping to rho of t. And this is actually what's known as the WKB problem. And so it turns out that this function has some very nice properties, which I think we could sum up as saying it's the example of vorus resurgence, maybe without writing this down on the board. I can say more precisely, even though we don't really know the equations, we can write down some coordinates, the Prochese coordinates, which are just the traces of the representation applied to some group element. And so we can think of, I mean, this is an affine variety we can embed it by some coordinates. If we look at those coordinates as functions of t, then we can take the Laplace transform of those functions of t. The Laplace transform of those functions of t has a very nice analytic continuation property. And that's vorus resurgence, basically. So this is a very exponential kind of a curve. But it does have some sort of hidden regularity properties. And what Gaiot and Morneitzky say is that they analyze. And what's the curve in the connections? You've just scaled the connection. Yeah, OK. We're taking a curve here. So I can be more precise about this. I mean, let's say a typical example, let me write that down here. A typical example of the connection is E, say vt is just, say, a trivial bundle. Nabila t is equal to d plus some sort of initial connection, where if we take the trivial bundle here and we just add a diagonal thing here. Well, OK, let's not have that trivial bundle, just some bundle. Let's just choose some connection, Nabila 0, on the bundle E, and then we add a multiple of the Higgs field. So the limit over here is E, E phi. This is one typical example. I mean, there's different ways of, if you've got a point here, there's different ways of getting a horizontal section that goes out to that point. But roughly speaking, we expect that the behavior over there is sort of the same. That's going to give you a number of different ways of having a family here that approach this point at infinity here. One sort of the easiest conceptually easiest way to write one down is to just take a bundle, assume that we have a Higgs bundle whose underlying vector bundle admits a connection, which is not always true, in fact, but assume that it's true, then we can just choose some connection, then add a large multiple of some of the Higgs field. So this is what's called singular perturbation, and this is where this WKB thing comes in. We're taking a connection which has the property that it has a large algebraic term. In terms of differential equations, that's what the same thing you're going to get if you do something like this h bar squared d by dz squared plus the potential. If you turn this guy into a matrix form, to a 2 by 2 matrix form, and then divide by h bar, then you're going to exactly get a 1 over h bar here, basically. And so the WKB problem is just in general what happens when you have a system of ODE's like that that has a large parameter in it. So then there's the whole question of, I mean, vorous resurgence is sort of what happens when you try to do exact WKB approximations. Well, so what happens in this SL2 case is the following, which is that you can actually say what's going on a lot more clearly for this WKB problem. So the WKB problem in the SL2 case, which is, OK, I'll just draw this picture. We have the fulliation defined by the quadratic differential. So in the case of a generic quadratic differential, this fulliation is going to have just triple points here like that. So these are points where the quadratic differential looks like z dz squared. Now, suppose we have a point here and a point here. And we'd like to calculate the transport for our connection from the point p to the point q, OK? Then am I going to succeed here? Let me do this with this color. Then we take a path joining p to q. If we take a path which is transverse to the f, oh, what's going to happen here? This is not good. If we take a path like that, which is transverse to the fulliation, so gamma, if we take a path which is transverse to the fulliation, then the transport for the connection, so t, p, q of t, it's a row of t applied to a group element. But you should actually think of it as you should look at the fundamental group rather than the fundamental group. So we should look at the transport for the connection going from the point p to the point q. But the connection depends on t. Then this is going to look basically like e to the sum constant times t. And the constant is going to be the length of the path transverse to the fulliation. So the exponent of the transport matrix is the path length transverse to the fulliation. This is not true if you take a bad path, which is not transverse to the fulliation. If you took a path like this, then if you try to calculate the length of the path transverse to the fulliation, then this real part of q changes sign as you go past this point. The you can see from differential equations you're supposed to be taking the absolute value of this guy. But this is going to give you the wrong answer because we sort of went too far up and then too far back down again. So this doesn't give you the right answer. It just means that if you try to integrate the ODE along this path, it looks like it would have this exponent, which is the length along this path. But actually, a whole bunch of stuff is canceling out due to moving the path from here to here. Maybe that's what's called quantum tunneling, possibly. OK, so I should be stopping pretty soon. I think I'm not an expert on this by any means at all. I think we can say that what's going on in Konsevich Soilbaum and which is now in this paper of Bridgeland Smith and so on, there's this notion of BPS states. And I purposely drew this fulliation as to be close to a fulliation, which is going to have a BPS state. So if you rotate things a little bit, you can see that there's going to be a stage where the leaf coming out of this singular point equals the leaf going into this singular point. When you go through that stage, then this blue path, you have to sort of change the direction of the blue path as you go across that state. So our idea was to say that somehow or other, the geometry, but we don't know how to make this precise, in fact. But somehow or other, the geometry of the BPS states and so on. So we'd like to say that the geometry of the BPS states corresponds in some way. And this is supposed to be the thing which is governing in some way, which I don't understand, the wall crossing formulas, which fit into this global picture. The geometry of the BPS states should correspond to looking at the tree, which is the quotient by the fulliation, which is another way of saying it's the space of leaves of the fulliation. So let's call that t. And we have a map, just the projection from x to t. So this guy is a tree. And this map, let's call this h, h is what's known as a harmonic map. And in fact, it's a harmonic phi map, where phi, in this case, is the quadratic differential, but in general, is the spectral curve, just phi 1. So it's sort of the square root. In this case, it's just the square root of q, the quadratic differential. If we take the square root, that's a differential, but it's only defined up to sine. And the differential of h is equal to the real part of phi, basically. That's well-defined, because there's no well-defined direction on the tree, just up to plus or minus 1. Now we have a situation where, in this SL2 case, the quadratic differential itself leads to this map from x to t. Let's call this t phi. This tree only depends on the quadratic differential itself. Now. Maybe you do us some cover. Yeah, OK, sorry. And this whole thing fits in, which I don't have time to talk about this too much, but fits into the theory of Perot and so on, which is really going back to the Morgan-Chalen convexification, in fact, which is going to say that if we look at a sequence of connections, then we're going to get a limiting harmonic phi map, x tilde, into some tree, t. So let's call this t omega, because it depends on the choice of ultra-filter or something like that. And this is for those of you who might want to look where that is in our paper or know what it is yourself or something like that. It's sort of the cone. It's cone omega. It's the asymptotic cone. Underlying here, as I mean, again, you'd have to write this down, but as many of you know, the underlying, especially the Hitchin correspondence, there's this notion of harmonic map to a symmetric space. And if you're thinking, oh, can we think of this tree as being the asymptotic tree on some kind of Gromov boundary of harmonic maps to symmetric spaces, the answer is yes. And in fact, that's the way you get it, and that's what Perot's theory says. But now, what's the relationship between this guy and this guy? It's just that this guy has an obvious universal property, which is any time you have a harmonic phi map from x tilde into anything, into any tree, it obviously has to factor through the space of leaves, just because phi is constant on the real part of phi is 0 along the leaves, so the harmonic map has to be constant on the leaves. So this has a universal property like that. So now, this is what we wanted to look at in the higher rank case. Why did we want to look at this, basically, for a number of many of these reasons here? Notably, I guess the idea is that the stability condition is supposed to somehow just depend on the choice of point in hitch and base. So whereas this limiting tree here might be, might depend on more than just the point in hitch and base. So we'd like to get a structure which only depends on the point in hitch and base. So now, in our work, we've been looking at how to extend this to the case of just SL3. So SLR, we don't actually have anything. We don't know what to say. No, sorry, I guess some parts of the argument work for anything. So basically, Perot's very nice paper, extended to group voids, extended in a relatively simple way to group voids, yields the following thing, that if we have a phi, which is a, sorry, if we have a WKB question, let's say delta t equals delta nabla 0 plus t times phi, so if we have a Higgs field phi and we look at a modular space for connections then I think this should also work for a more general family where the bundles lie to vary as a function of t and stuff like that. But we just wrote this down in this case. Yields a construction which corresponds to this situation, limiting, and also you have to choose an ultra filter, limiting harmonic phi map. So I'm using this kind of capital phi here for the spectral, for the Higgs field. This smaller phi is the multivalue differential, which is the spectral curve of the Higgs field. I mean, the values of the differential are just all the different eigenforms of capital phi here. And you just get a limiting map to some building, to some kind of R building. And in the SL2, in the SL3 case, this is modeled on the apartment system, it's just R2 with the S3 group of symmetries and so on. But I mean, in general, I think this should work for any kind of group or something like that. In general, you'll get the apartment system is going to be the one which happens for the Buatitz buildings for that group, basically. So you get a limiting harmonic phi map. But that's just kind of a formal thing where it actually depends on the ultra filter to take the limit and so on. Where, roughly speaking, the distance from h of p h of q is some kind of limit from, it's just that it has to take the ultra filter limit. But it's the limit of 1 over t times the log of the absolute value of rho t. So it's basically the scaling factor c here. It's just that we need to take a limit. And you might want to take a lim soup or something like that. If the limit exists, then any ultra filter limit is going to be equal to that limit. If it's just a lim soup but not necessarily limit, which actually happens, in fact, notably at the boundaries along the wall, then the ultra filter limit might be different from the lim soup, for example. But it's roughly speaking some good approximation to the WKB exponent. This is just the distance in this building. This is just kind of a formal thing, which I think basically puts into detail what was always kind of a hope, I think, ever since the theory of harmonic maps to buildings came about. I mean, this is sort of what you think of them as corresponding to in terms of, as related to, say, harmonic maps to symmetric spaces. But then the question is, what about, let me try this operation here. I should stop. So then the statement which we like to show is just that we have the same universal property. But let me put this as a conjecture. So the conjecture is that there exists a universal. In fact, for some reasons, which I can't explain, there exists a universal harmonic Femap to a building from x tilde into some b, some building depending on phi. So this is only supposed to depend on the spectral curve. So it's only supposed to depend on this multivalu differential, but it's supposed to have this universal property, except it's only a versatile property. Which is that if we have an actual map to a building obtained by that type of a condition, then it's supposed to factor. The difficult to hear is that x has a real two-dimension. This guy, for bigger groups, has sort of arbitrarily big dimensions. So this can definitely not surject onto this guy. So on the many points here, which don't come from points here, there's no uniqueness here. And you can actually construct situations where you can see that you shouldn't expect a uniqueness of this map. So that's why it's a versatile guy. And so we could just show that in one particular case in our prepram. Yeah. So we think this may be a pre-theorem in progress, maybe a three-theorem in progress in the SL3 case. And the SL3 case, but that used to kind of cheat anyway. Here we're cheating a lot, you might say, because we're using the fact that the tree is going to be a quotient of x. So the points in the tree, we know what they are. They're leaves inside x. Here we maybe cheat a little bit, which is that in the case of SL3, this has dimension 2 and this also has dimension 2. So this guy is actually going to cover some percentage of the points here, you might say. I mean, it's going to cover some open sectors inside here. Let me just finish by maybe the drawing of what this actually looks like. So this is a theorem in the BNR in the example. So CR pre-print. Let me just say what the harmonic map looks like in this example. In this example, we have the spectral curve has two different ramification points and there's what's called a caustic line joining these two guys. This region inside here, and these are the Gaiot-Mornitzky spectral network curves. This region here bounded by the Gaiot-Mornitzky spectral curves maps into here. The rest of the picture actually maps down here. And this whole thing maps into an initial piece of a building. Then you can fill it out to make an actual building. But the initial piece of the building is actually pretty easy to think of. You just take two sheets of paper and glue them together along this region. Let's just take two sheets of paper and glue them together along a region that looks like that. So just put some glue here and glue your two pieces of paper together so they're not glued together down here. And this region here, it sort of folds over along this caustic line. And the stuff above here goes to the back sheet and the stuff below here goes to the front sheet. So that's the example we know how to look at. And then based on what's going on here, doing a seven trying to redo the proof a zillion times, we think we may know how to see what to do in the general case for SL3. Is this compactification compatible with mirror symmetry in the sense that before compactification, we can do kick bundles for a group of length and stools of mirror pairs? And what happens after compactification? Maxim. I think mirror would just change to be the correct way to do it. But it doesn't change the base. Yeah. I don't understand this at all, obviously. But my vague understanding is that we should actually somehow or other think, but I don't know how this is supposed to work. But somehow or other, more think of this guy as being mirror-dual to this guy. I think mirror-dual is a representation of 5.1 to mirror-dual to that set. OK. Oh, so you think that if you do mirror-dual here, then you should get a mirror-dual here? I don't do mirror-dual, mirror symmetry at all. And he says something like that. Just on the painting. But then what happens over here? I mean? The mirror symmetry situation, it is clear here. You look at the whole piece carefully, and you put the painting with zero. That's the case on the services compact. If you've got punctures, the story is a little more complicated. No, it doesn't matter. Punctures, you don't play your role. It's easy. It does, you have to touch it. Well, the question is if mirror symmetry is spoiled by adding divisors and finishing it. Oh, that's it. What happened? I mean, I'm not sure. I think at least in some way, you can think of going from a symplectic guy to a Poisson guy, right? Is that true? When you add the divisor and infinity? Maybe at least in this picture. Yeah, I remember to suggest you some time ago just because you have, for the last thing, you have problems that curve it's too low dimensional. Is it possible to extend to some kind of high-dimensional right in your kind of connections? Oh. If you try to replace x by a higher-dimensional right? Yeah, or if you're able to curve inside a high-dimensional right in your null equation, you get kind of connections. I mean, try to look at just an analytic neighbor out of the curve. I don't know about that. I mean, I see. I think I understand your remark a little better. I'm not sure. I mean, certainly the real problem with this whole thing is that we don't have any way of saying what the points in the building, which are not touched by points of curves, should actually correspond to. That's the real kind of a question here, which in the tree case, you can say the points of the tree correspond to leaves. So there's some geometrical thing. So that's a good question. I don't know. I don't know. And I think it's also the other part of this remark. We try to understand what is how to calculate the DT invariance from this character of a right? It's actually a very neat question. We should start off with algebraic maps of C star to the character of a right. And this has the same back. It will get some work. But you have a right, which has the same dimensions. Original character of a right, and you get some kind of algebraic change of form. So you think that counting C stars inside here? Oh. The space of C stars inside here? Do you group the numbers? Learn the numbers. So it has the same dimension as the character of a right. It makes some kind of two different ways to get some great change of coordinates. And it will be DT invariance inside of the tree. You mean the two different ways are the two end points, some rather? Yeah, kind of. We have two end points for all two devices to DT. Yeah, so you're looking at C stars, so there's going to be two different points here. So you were thinking of this? No, no, C star will connect two different points. But the interior of C star will be exactly inside the character of a right. Yeah, yeah, OK. But that's going to give you a correspondence between the divisor and itself somehow there. Like maybe a heck of a type of correspondence or something. Yeah, I thought that just it's a neat question here, too. Yeah, yeah. OK. Depending on the situation, it's a lot of elements. OK, good. So now we have a correspondence between this divisor and itself, which is the modular space of C stars mapping in two different ways to DT.