 Last figure today is François Raboury from the United States. Hey, though. I would like first to thank the organizers for the invitation. And I would like also to thank you for being here. It's very late. And since you are so good, maybe I are going to give you a treat at the end of my talk. So if you are good. So the plan of my talk is going to be the following. So first I'm going to recall stuff about geodesic currents. And I've been very lucky because it's bright young people like Vidéka on the Tegrin gave a very good introduction already to geodesic currents. And then I'm going to explain what is the UV current of a heat chain representation. And then I'm going to mix that with a complex structures. And I'm going to end up with questions. Because all this is essentially a work in progress with Andres Sambarino. And there seems to be more to be done. But so far we have not done much. OK, so geodesic currents. So as usual, I'm going to consider a surface S, which is going to be closed, connected, G is greater than 2, and mine will be also oriented. And I will denote by one of S, a fundamental group of S. And you know that associated to this group, you have a circle, which is a boundary at infinity of this group. And this is homomorphic to S1. And we consider space of geodesics, which is just the space of pair of distant points in the boundary at infinity. So it's a Cartesian square of that minus a diagonal. It's just a pair of space of pairs of distant point S1. So why is it called the space of geodesics? Because if you take a, so if a G has negative curvature, non-positive curvature on S, then this boundary at infinity of S identified with the boundary, visual boundary at infinity of the universal cover on two points correspond to a geodesic. So what is now a geodesic current? So definition, a geodesic current is a locally finite measure, a pi 1 of S invariant, locally finite measure. So examples, so one example is, well known. So you take now, let gamma, an element in the pi 1 of S. So it has two fixed points in point of gamma in the boundary at infinity. And you consider the following geodesic current. This is noted by delta gamma, which is a sum. So the idea is that you want to take the Dirac measure supported, so it's Dirac measure supported on spares gamma plus minus 1. But of course, you want this to be invariant by the group. So you take the sum for element of pi 1 of S divided by the group generated by gamma of that. So that's the first example. The second example is the UV measure for hyperbolic matrix. This is a well known example, but I'm going to give the construction as a warm-up for the next construction. So second example, UV measure for hyperbolic metric. So this means that in this situation you have pi 1 of S, which is realized as a subgroup of PSL2R. And you can also consider PSL2R itself, so which is identified with the unit tangent bundle of the hyperbolic plane. And you have the action of the geodesic flow, which corresponds to describing that as a R bundle. And the quotient is, well, what it is, PSL2R divided by R. And this is identified with G by space of geodesics. And this carries a one form. So you have a one form theta on S2, which is a UV form. So this form is given by a view, which is the killing product of U with X. That's a generator of the R action. And it turns out that this is actually the connection of a line bundle. And d theta is the pullback of a certain form, which is, of course, the curvature form of this line bundle. But this is going to be the UV measure, the UV current. And of course, what is pi? Pi is this projection, UV current of the hyperbolic metric. So there are quite many things you can do with currents. And I wanted to say. So another thing you can do is to consider the Bonn-Arm intersection. That's a beautiful construction. So if you have two currents, you consider the following quantity. So that's the integral over G cross G divided by gamma of a certain quantity. And this is just the integral of U, cancer, U. You have a measure on that, you have a measure on that. And it turns out that this quantity is finite. And in some cases, we can actually compute it geometrically. The intersection of delta, gamma, delta, eta is just equal to the cardinal of the intersection of gamma on eta in surface S. So this object is indeed the geometric intersection for current supported on geodesics. And it extends continuously to all currents. So the other thing you can do is you have actually a pairing with something which may call parametrization form. So what is this pairing? So remember, you have this line bundle over G. So I describe it in this context of hyperbolic metric. But it turns out that this line bundle is well defined topologically. And you have an action of gamma here. So you have an action of gamma here, which leads to an action of gamma here, so that L divided by gamma is compact. Now you may consider on this object the form. So on L, let's define, so let omega 1 of L is a space of one forms on L, a long orbit of the R action. Or the R orbits. So what does it mean that locally there's an object which looks like f of x, y, t. So this x and y correspond to the part in G. So somehow you can think of this as a family of measures on the line depending on the base point. So you have a natural pairing between a current. So that's a current. And that's an element of omega 1 of L. And this object is just the integral of L of gamma of this same measure tensor. So you have a measure along the line, a measure along the transverse. So this defines a measure on L. So this now is a measure. And you integrate this measure on this compact set. Well, I should have said, of course, it's valiant on the gamma under pi 1 of s, which is the same thing as gamma. OK, so what am I here? Now each in representations. And Graham explained very nicely how it works. So you have the irreducible representation of PSL and R into PSL and R. So that's the irreducible representation. And now imagine that you take a hyperbolic structure. So you have here a discrete space rule representation. And you can compose these two representations. So these representations are representation in PSL and R, whose the ice key closure will actually end up into some PSL to R. And then by definition, we call those functional representations. So then what is a Hitching representation? So a Hitching representation is a representation row of pi 1 of s into PSL and R. I try to keep the fact that the dimension is d, not n, which can be deformed, a function representation. And here deforming means that you deform generator by generator. So then the standard notation is to say that let H of n be the space of Hitching representation. And of course, I would like this to be equivalent up to conjugacy. So the story started by a beautiful theorem by Hitching, which identified H of n with R to the dimension chi minus euler characteristic of s times the dimension of SLDR. So he asked the question, what is the meaning of this Hitching component? And of course, you notice that H2 is the same thing as Tichmer space. And then later on, Trojan Goldman interpreted what is H3. So it turned out that these days we know a lot about these Hitching representations as a Hitching component. And we have an interesting description of them as dynamical object. So they share many features of Tichmer space in the sense, for instance, that associated to such a representation there is a geodesic flow, exactly like hyperbolic structure. But there is something that we know very little about this. Are they related in any ways to complex geometry? We have a nice kind of Thurstonian Sullivan picture, but we don't have the analogous of that. So let's make a beautiful drawing. So that's a Hitching component. And that's a inside of Hitching component. We have this object, which is a function locus. This corresponds to functional representation. And this is, of course, identified with the Tichmer space. So what I'm going to describe is a way to project that, this quotient, in a mapping class group invariant way. For that, I need to, I want to introduce this uville current for the Hitching representation. So let's state a theorem. So if rho is Hitching, there exists map xi on xi star that goes from the boundary at infinity of pi 1 of s to, respectively, the project space and the dual project space. And this satisfies some nice property, for instance, they are continuous. And so you have this transverse property that is if x is different from y, then xi of x plus xi of y is equal to rd. And the other property which I will need is that the image of xi on the image of xi star are c1 manifold. Thank you, thank you. So the images of this curve are c1 manifolds. So it gives a c1 structure on the boundary at infinity. But unless you are in the function case, the c1 structure actually are different. So in other words, you have a map xi xi star from my space g, my space of geodesics, into an object o, which is a subset of rpd cross rpd star. This is a set of those line and co-lines, a line on the epa planes, I'm sorry, so that l plus p is equal to rd. There is a fact that o is a symplectic manifold, of course, in an SLD invariant way. And for those of you who know about symplectic reduction, it is actually the symplectic reduction of the action of r on rd cross rd star. So if you want to describe explicitly what is the symplectic structure, you consider the space l, which is a space of u alpha, such that alpha of u is equal to 1. So that's a subset of rd cross rd star. Oh, I'm so sorry, so sorry. So they are a row on row star invariant. Yeah, I'm sorry, it's very late. So this is a symplectic manifold. So you have this line, this object, and actually this naturally project, of course, by an r action of over o. And you have then actually a one form theta on l. It's given by theta of at the point u alpha of v beta, which is alpha against u minus beta against v. So that's a one form on this line bundle. And you see that d theta actually is going to be the pi star, the preimage, of a certain form, symplectic form, symplectic form. That's very similar that. And now this theorem tells you a little more than the fact that this map is actually a c1. So this map is c1. And of course, gamma-equivariant. But this map is actually a, the image of that is actually a symplectic sub-manifold. Oh, the image of, wait a minute, I'm sorry, is a symplectic sub-manifold. May I see? I'm not being good. No, I'm really sorry. Thank you for the important interruptions. So the image of that is a symplectic-manifold. So you can produce something which we want to define as a Louisville current, current of rho. So what is it? So let's do it by lambda rho. This is just a pullback by psi, psi star of the symplectic form omega, which you have here. So this Louisville current. So we are kind of pretty sure that's a very bad terminology and it's going to fall back on us as all the previous terminology we have chosen before. So why do you want to call that the Louisville current? And why does it play a special role? So here is a little propos, let's say a theorem. So I'm going to explain the theorem. So lambda rho is either 1D intersection current. And lambda rho is the 1, 2 deep current. I should say rather it's proportional to that. So what does this first sentence means? It means that the intersection of rho of lambda rho with a delta gamma, stuff that correspond to a closed curve, this is exactly equal to the log of lambda 1 over lambda d of rho of gamma. So meaning that rho of gamma is a matrix which is diagonalizable. And I take the larger second value divided by the smaller second value, I take the log of that and that's exactly the intersection of that. So that's the definition of being the 1D intersection current. So what does it mean to be the 1, 2 deep current? Mean that nu is actually proportional to the limit when l goes to infinity. So in short words, it means that you want to consider closed jadezik. And you measure them with their 1, 2 length. And they become equi- there is an equi-repartition theorem. And this arrives to the deep current. So 1 over the cardinal of something which I'm going to call n of l, sum over n of l of gamma in n of l. I take then the log of lambda 1 over lambda 2 of rho of gamma times delta gamma. And I'm totally forgetting something. That's not the definition. You have to divide by, and then still not good. No, no, I'm sorry. There is no log here. So what is n of l? It's a set of gamma such that the log of lambda 1 over lambda 2 of rho of gamma is less than n. Correct this time? It's being recorded. Where is? That's a L'Uville current. Very hard when you collaborate if you sleep during talk. OK. So the fact that this L'Uville current as a 2 nature is going to be helpful in the alpha is a coefficient. The number, which is an invariant of the representation by itself. Maybe you want to call the volume the representation. But it's clearly going to blow up at some point. So we don't want to give it a name. So now let's mix in. So that's the definition of the L'Uville current. And that is a kind of very weird property that it is constructed with different roots. So now let's mix in 1, 2, 3, 1, 2, 3. 1, 2, 3. With a complex structure. So let's assume now that let J be a complex structure on S. So then you can associate to this. It follows that then the boundary at infinity of pi 1 of gamma then becomes identified with the boundary at infinity of H2. And here, let's say I take the hyperbolic, the Poincare half-plane model. Of course now, two distinct points in the boundary of H2, this is question 2 geodesic. But you can associate to say to two points a nice object, which is an abelian differential. So let x, y belonging to G. Then you can associate to x, y, an abelian differential, which I'm going to call theta of x, y, which is dz times 1 minus x minus 1 minus 1, y minus c. So this is an abelian differential, abelian on H2. And which is characterized up to constant by the fact it's a unique which is invariant under the diagonal group, the stabilizer in PSL2R of my pair of points. So now you have something which depends on two points. So of course, what you want to do is to average that with a current. So let's define the definition. So the 8th Poirier coefficient of current mu with respect to G is equal to the integral over G of theta x, y to the power k, d mu of x, y. And let's call that u hat k of j. So that's a very natural object to consider. And actually, it has been considered before. So if you take mu is equal to delta gamma, so if mu is equal to delta gamma, then you exactly obtain the so-called relative point k series for j to the gamma. And we have a little proposition now that first mu kj is well defined. If k is greater than 2, I doubt it's defined for when you have reached a billion differentials. And second property is what is it? Oh, mu kj is actually gamma invariant. It's pi 1 of s invariant. And it's actually a holomorphic differential of degree k with respect to the corresponding complex structures. So another property that if you take, so all this depends on normalization. So forget about, I mean, there should be constants which are factorial of n or stuff or you don't know what. u kj against a holomorphic differential. So if you take the val Peterson product, this is the same thing as what is called the period of q. So this is something that I want to call qat. On the unit tangent bundle of s, I'm sorry. Actually, I introduced a definition for that. At this pairing of some object which I called, so there is a current against this one form which I called qat, qch. And qch is equal to q of x, x, x, x, x dt. And this is a generator of the geodesic flow. And so why do I call them Fourier coefficients? That you can actually interpret them, indeed, as Fourier coefficients. I'm sorry, I wanted to give you a treat. So now, that's a holomorphic differential of certain degree, let's say degree k. And this object is, so in other words, you take the integral of a US of a function which is the real part of q of x, x. So x is the point of US. There is no real part, I'm sorry. And then you take dt. So you take the form along the geodesic flow, and you take the tensor with the current that you have. So for instance, if you take the UV current itself of the hyperbolic metric, all this for a coefficient of 0. So maybe I should write that as a remark, that if mu is equal to nuj, the UV current of that, then all these Fourier coefficients are actually 0. So what's a theorem that we like with Andres? Just wanted to check if I've done my homework so far. Right, a theorem that given a teaching representation, there exists a unique complex structure up to the form of course on S. Such that, J on S, such that, the quadratic Fourier coefficient, to turn out the proof of that is relatively easy. Once you know enough math, so what is the idea? So what is the idea? So we consider the function, so let i rho be the function which is defined on Taichman's face, 2r, actually 2r plus, which is defined by 2 a complex structure J, associate the intersection of the UV current of rho with the UV current of the complex structure. So now you prove the following thing. So you first prove that i rho is proper. And that's, I mean, it's not very difficult, but it's even easier when you use a Hengren polymer. But it's not as deep as that. Then you prove that i rho is convex along earthquakes. So actually I should make here that the fact that you prove that i rho is proper using the fact that you have this interpretation as a intersection current. And here you prove that this is convex along earthquakes. So that's a follow from Steve's works. And here you use the fact that i rho is an average of current supported on geodesic. So now you know that you already know that you have a minimum because this is convex. So then you use the earthquake theorem. So it tells you that given two points, there is a unique earthquake flow and unique earthquake. And you want to know, now you want to show that there is a unique minimum. So you know that your function is convex on that. So it has a minimum, but this minimum could be not unique. So here to guarantee uniqueness, that's the word that trump used. I can spell it, I hope. Guarantee uniqueness. So you use the analysis of i rho. And for instance, this is done by Bridgeman and Arino. So the conclusion of all this piling up of here I mean that i rho has a unique minimum. And then you finally use this formula here. So then you realize that the minimum, you have this formula, this d i rho of a quadratic differential. So you know that the tangent space of the stationary space identify with quadratic differential. And you see that the variation of i rho in the direction of q2 of the quadratic differentials is exactly by this formula here. That against the quadratic differential here. So you see that's something that's not very hard to obtain. But that's still a nice result. So we have defined this way a map. So now let's go to the questions. So we have defined a sort of a projection from the Hitching component to the stationary space. So this is indeed a projection in a sense that you start from a point in the function locus. You end up in the function locus itself. But apart from that, we do nothing. For instance, it's pi a submersion. So we don't even know this object as this is a submersion. And what about the fibers? So we hope that the fibers are the fibers contractable. And if it were true, that would imply that Hitching component dividing by the mapping class group is actually topologically at least a nice vibration of a Riemann-Moduli space. And so now we are very greedy. So we have other questions. So we're just so far used only the quadratic and the second Fourier coefficient. But of course, we hope that the other one makes sense. And we have this nice map, psi. So it depends on the complex structure. That goes from the Hitching component to the sum for k equal to dh of d of the holomorphic differential of degree kd. So these two space have the same dimension. So by the inverse invariance of the domain, this should definitely be a homomorphism. So it's psi a homomorphism. So that sounds very greedy. And again, if this is true, then that would imply a nice result. That would imply, again, the description of the Hitching component as a vibration of a Tajmer space. So that's a yes. So that's it. So I could say more, but I promised a treat.