 Thank you, and thanks to our organizers for inviting me here. Yeah, so I'm going to talk about counting curves on surfaces and its joint work with Juan Soto. Sigma is going to be a surface with our punctures or boundary components of negative or characteristic. Gamma is going to denote the curve or really a homotopic class of a curve. When I say curve, I always mean close, but not necessarily simple. So it's an immersed curve in general. We want to count curves, and I want to count curves in each mapping class orbit of the curve. So S of Gamma Nod is going to denote the mapping class group orbit of Gamma Nod. So in other words, all the curves of the same topological type as Gamma Nod. So for each mapping class orbit, I want to count the number of curves there are up to some restriction on some complexity of the curve. For example, the hyperbolic length, or length with some metric, so on, but with various things, so complexity as the complexity increases. So as I said, some examples of complexity is the length with different metrics. So with a hyperbolic structure on the surface, we can look at the cardinality of the curves. So as hyperbolic length, of course, I mean what the geodesic representative bounded by L. See what happens to that as L goes to infinity. Or I could put any negatively curved metric on the surface with some variable pinched curvature, for example. And ask the same questions. How many curves are there in this orbit up to a bounded length? Or I could look at some flat metric with singularities. For example, it's translation surfaces and ask the same question. Or I could look at a word metric with respect to some generating set. And here I want to have a puncture boundary so that the group I'm looking at is free. So to be concrete, for example, if I look at genus 2 surface with one puncture, and I look at universal cover, and I pick a standard generating set, so the side pairings. And I look at reduced words in the letters A, B, C, D. And then I can count the number of curves in an orbit that has bounded word length. Or I could do something more combinatorial. So for example, we fix some filling multicurve, say mu. And I can look at all my curves that intersect mu a bounded number of times. We can count. It happens to that as L growth. Or I could do some combination of all of these. For example, if I have two metrics, say two hyperbolic structures, I want to look at all the curves that have bounded length in both. Or basically any combination of these could even do some complexity, could be maybe have some hyperbolic structure x. Take the length with respect to that. I have some filling multicurve mu. I take the intersection with that and maybe divide by the word metric on the curve. And look at all the curves for which this is bounded. So what we will see is that for any up to certain restrictions, so basically how I picked it, I want, if I scale my curve with L, I want my function to also scale by L. But as long as that's satisfied and has some continuity, we're going to see that all whatever complexity I choose, they're all going to be comparable to L to the dimension of the tight miller space. And we're in fact going to see that they're asymptotic to this in any of these examples. So in the case for the hyperbolic metrics, a lot of people have studied this, especially Mercia Connie. So let me say some of those results. So we're looking at the limit number of curves with bounded hyperbolic length. By L compared to L to the 60 minus 6 plus 2. So first, Riven and McShane show that this limit exists if we look in mapping class orbit of a simple curve on the puncture torus. Why not simple? Then Mercia Connie did her famous work show that this is true for on any surface. Not simple. And very recently, a preprint came out this year. She actually extended this for an arbitrary curve. So any immersed curve and the limit exists. And in fact, we can say what the limit is. So this limit exists. Particular, it equals for some constant that depends on the topology of a topological type of the gamma node times a constant that depends on the hyperbolic structure. And for those of you that know, it's the Thurston measure of the unit ball in the space of measure lamination. So from a case of hyperbolic metric, the question is completely solved. I should say something about the word metric too. I'll ask the same question but put the word metric on it for some fixed generating set. Mora Chas has done a lot of work on this. She has a lot of experimental results that indicate that the limit should exist there too. And she also has some results of a distribution occurs with different length, the word metric. She has experimental results. It's something that we very recently started working on together. We can actually show that. So I'm going to talk more in detail what generating set I'm talking about so on. But for the case where we have a boundary, we can also show that the limit exists. But as I said, it turns out that it actually doesn't matter after a certain restriction that what kind of this complexity we consider. This limit with the same polynomial exists. We just stated as a theorem in quotation first. So the limit exists, all the complexity is mentioned. I'm going to say it more precise but in order to do that, I have to talk about currents. So I'm going to let C of sigma denote the space of currents on the surface. So what are currents? So they're really a measure on the space of geodesic in the universal cover. So it's a Pi 1 invariant. But if you don't know about currents, you can think of them just how what simple curves are to measure laminations. So are general curves. So the space of all curves to the space of currents. So in other words, if I take positive multiples of curves, that space is dense in the space of currents. Just like simple one, sorry. Space of measure laminations. So in particular, the space of measure laminations can be viewed as sitting inside. Space of currents. One more thing about currents. So there's an intersection form. So maybe I should say this. Any immersed curve or multi-curve can be viewed as geodesic by looking at all the lifts in the universal cover. And then the measure is if you have a set, the measure of that set is how many of these intersects. And then the intersection is just the geometric intersection. But in general, it's just a unique continuous extension to the space of currents. So now I can state the theorem. So I suppose we have two continuous real-valued positive functions on the space of currents that's scaling the right way we want with l. So this might look very abstract. But if you want to give things, for example, the length function, the length function, it actually extends continuously to the space of currents and it is satisfied this. But this is more general functions. And then what we really want is at the limit of number of curves. There's a mapping class. Oops, so gamma naught. Touch that. So f is going to represent the complexity. It's bounded by some l. What we want to say is that the cardinality of this over l to the 6d minus 6 plus 2r exists. We can't quite prove that. I mean, Mr. Cotton improved it, but we can prove that the ratio of 2, so if I change this f to g, that limit exists. I should write down what the limit is. So the limit is independent of gamma naught. It depends on my f and g. So it equals the ratio of the corresponding first involved of radius 1 with respect to f and g. So in particular, this means that the existence of the limit that we want, one of these divided by l to the dimension of a type model space, doesn't depend on the complexity. So if it exists for some complexity, it exists for all complexities. So meaning the choice of this continuous map that scales this way. So in particular, since Mr. Cotton showed that if I take f to be the length function of our hyperbolic metric, the limit does exist. That now implies that it exists for any of these complexities. Take any continuous positive function and the corresponding limit exists. And again, it equals what it should. Some constant that depends on gamma naught times the first involved of radius 1 with respect to f. Yeah. Yeah, that's true. So actually, everything that I say works if I consider currents instead of curves. So I count orbits of currents. Okay. So let's see why this result implies the things I stated before, these different complexities. So in other words, why can I view those complexities as functions, continuous functions of a space of currents? Well, if we have a hyperbolic surface and there exists special current associated to it, renewable current. So I said there's this intersection form on it. So if I take the intersection form of this current with a curve, it gives me exactly the length with respect to that hyperbolic metric. Yes. And in fact, you can do this for any negatively curved metric. So these things I've said about currents are really tall and bone-on. And in fact, also, if I look at a flat structure with singularities, there's also such a current. And it also exists for singular, flat, sliding arm. So this is the reason why those complexities I had to do with the length with different metrics is why it's true for those. And it's actually also true for the word metric that you can have such a current. Again, if I look at the example before, the DINUS 2 with one puncture. This is when you reversal cover and we had the side pairings. If I look at the following arcs on the surface, arc alpha, gamma, and this white one is still. If I let mu be the union of these four arcs, that's a current. I mean, any collection of arcs or curves is a current. But the intersection of a curve with this current gives you exactly the word length. So this is with this fixed generating set, of course. And so again, this is something I'm working with more at trust. And we don't know what happens if I change the generating set or if we think this should also be true for closed surfaces. But, of course, the difficulty is that it's not a free group, but I think it should work anyways. Yeah, and as you, Egel, pointed out that all of this works also. If instead of counting curves, we count currents. So in particular, if I look at the Louisville currents for hyperbolic metrics, we're talking about orbits of points in Teichmuller space and we can count. Yeah, sorry. Okay, these are cyclically reduced words. Okay, so let me talk a bit about the methods we used to prove these things. To translate the existence of this limit to a convergence of a measure on a space occurrence. Each L, later we're following measures on the space occurrence. So we're looking at some fixed curve gamma node. So if we each L, let it be 1 over L to 60 minus 6 plus 2R. And then we count number of curves in the orbit we want. And then the counting measure, the Dirac measure is centered at the scale version 1 over L of the curve gamma. So just to see what this means. So suppose this did converge, which we actually can't show quite. But suppose if there existed some limit measure new, then the limit we want is number of curves such that f of the curve is less than L over L to a 60 minus 6 plus 2R. That's the same as taking that limit measure of the set of measure lamination such that f of it is less than 1. So in particular if this, if we can show that the converge, then the limit exists. Does that make sense? In fact, we show that it's equivalent, but okay. So unfortunately we cannot show that that limit, or that those measures converge. If we did then we would have a different proof of mirror's occurrence result. But we can prove the following. If I take any sequence of these measures, you always have a convergent subsequence. And moreover it has to converge to some multiple of a Thurston measure. The problem being that the multiple might depend on the subsequence. And notice that theorem 1, the one that's saying that the limit of a ratio exists, follows from this. We can prove this, we prove that, and from that follows that corollary that the limit exists for all complexities. So I'm going to talk about how we prove this. So notice that these are measures that have support on space occurrence, which is like things that have intersections. While the limit things have support on measure lamination, on simple things. So it's this relationship that we're going to investigate between curves with intersections and simple curves. So basically this is in some way saying that statistically as, so this is of course as, and I go to infinity. So as the length gets very long, your curves start looking more and more like simple curves. So I think that the curves will be looking at two simple ones, to measure lamination. And in fact not all of them, but some generic sets. What I mean by that is the curves I leave out compared to L to the 60 minus 6, it goes to 0. So the limit I don't, I can ignore those otherwise. Okay, so how do we do that? If I'm on a closed hyperbolic surface, it's easier to see how to define this. I'm going to define the angle of a curve with intersection just as being, if I look at all its self-intersection angles, take the biggest one of them. I just want to say if that thing is small, all the self-intersection angles are small. And if I look at the number of curves of length bounded by L, whose angle is bounded from below by some delta, divided by L to the 60 minus 6, this goes to 0 for any delta I choose. So in a way the expected angle of intersection goes to 0 as the length increases. So if we're on a closed surface then, so what this generic set is going, of course going to depend on how small I make this delta. And then I'm just going to do sort of to resolve the intersection since it's a very small angle, very long curves, the resulting curve is very close to. And you also say so if I do have punctures, it means that all large angles have to be around the punctures. And then we develop some new tools, some generalization of train tracks to get things that carry things with self-intersection. And then these things that have intersections around the boundary, we can kind of bound on how you can do surgery and get the bound on the resulting simple curve how long it is. If I have time I might talk a bit about that in a minute. But so it's easier to define this map in the closed case, but we can do it for any surface. So once we have this map, so the goal is to show that these measures, the new converge to multiples or sub-sequences converge to multiple of a Thurston measure. So suppose one of the limit measures, what we will show is the following, that they are invariant under a mapping class group and that they're absolutely continuous with respect to the Thurston measure. So this thing here is relatively easy. So the only problem is that this generic set is not invariant, but it's not too bad to get around it. So this is really the key step to prove this. And why this implies that the limit measure has to be multiple of Thurston measures because of a theorem due to Meister assessed that a Thurston measure is ergodic with respect to the mapping class group. And if you have an invariant measure that's absolutely continuous with respect to an ergodic measure, it has to be multiple. So this implies that the absolute continuity follows from the following. So this is kind of a technical part of the thing to show, but it's really just a combinatorial thing. There exists some constant that only depends on the topology of the surface, such that if I look at this map, then maps things with curves with intersection to simple curves. There could, of course, be many different curves with intersection that maps to the same simple curve. But this lemma says that the number of fibers for each simple one is uniformly bounded by this C. And notice if this was an equality, say, C equals 5, I would say that for each curve, for each simple curve, there's five of them with self-intersections. And then we know that the limit exists since we know the limit for simple ones. So this is not quite as good. And actually, we do show equality for a curve with one and two self-intersections, but it doesn't generalize to higher self-intersection number. But it's always uniformly bounded. So how does this help us? Well, I erased the measure, but we had this thing. So if I push this measure forward to the space measure elimination through the map pi, I get measure mu with support on space measure elimination, which is... And I know that this thing is bounded, so this is... So here I'm just summing over simple curves. And this thing here we know converges to the Thurston measure. Okay, I should have said something. So this is this measure that's slightly different, but we also show that these two are very close to each other. So I have the same limit. So I get closer. They converge to the same thing. And here we see that the mu is always bounded from above by the Thurston measures. In particular, it has to be absolutely continuous with respect. So let me talk a little bit about how we get this thing, which also has to do with how we get at the... For closed surfaces, the measure or the angles go to zero. So we define something that we call a radala. Think of it as an immersed version of a train track or a diagonal extension of a train track. So we have a train track. So there's a complementary median or maybe puncture. We're allowing some extra edges or allow to cross. With certain conditions, they're not allowed to be homotopic to one or the other edges and so on. Or if you have a puncture, you could wrap around the puncture a number of times and need to go back to the same place or go somewhere else. And then, of course, we do this in a more precise way, but this is the basic idea of it. So we show some of the things that hold for train tracks also works. So for example, if I look at simple curves, there's finite collection of train tracks that carry all of them. The same thing holds here. So finite carry all, say, all the curves in an orbit. Let's just say all curves with bounded self-intersections of all curves. Self-intersection less than or equal to k. Yeah, radala, sorry. And moreover, since, yes. No, so not necessarily. You could have a very small intersection that's still carried here. Yeah. So basically, so we define like an epsilon geodesic radala. So then the smaller epsilon is, and if you have a big crossing, it has to actually happen in one of these crossings instead of here. Yeah. Yeah, so I didn't really give a precise definition, but so you have a train track. And then, so basically, so if a train track is an embedded graph on your surface, this is an immerse graph such that there exists a subgraph of it that is a train track. And then, so, and the extra edges satisfying certain things that they have to start in in this, the casp of a complementary regions, and they're not allowed to be homotopic to the, to the branches of a train track and so on. Does that make sense? We can talk more about it after. But what I want to say is point. So I claim that if I have, right, so if you're given a train track, you know how many curves are carried by the train track. And what we show is that given a radala, there are, there exists some K such as, there are K times as many curves as self-intersection carried by the radala as simple curves for the train track. And the reason to see that is first, you can only go finer than many times over these extra edges. It's each time you pick up an intersection and we bound them in an intersection. So it really is just counting the ones that are on the white part. So, and this is actually how we get this bound C. So if you're somewhere along the train track and you, you put a crossing for two of your curves, you can slide this around and you get the same curve up to homotopy, but eventually you can slide it anymore. So eventually you get to a vertex, you can slide it, you go back, eventually they diverge again and you can slide it. So you can get some combinatorial bound on how many places you can put the crossing on. And that's how we get inequality for K equals one and two. Then it's easy to actually get it, but as soon as you get to three, it gets very complicated and depends on how the train track looks. But you can always get an upper bound. Okay. So now to see the thing with that the angle of self-intersection goes to zero. Well, so basically by this argument I said, we know that for train tracks, when we're counting how many curves are carried by, or how many simple curves are on the surface and we have finally many train tracks, we know we only have to count the maximal ones because the other ones correspond to a lower dimension of curves. So a maximal train track are the ones that all the complementary regions are either triangles. If we're on a closed surface, or if we allow punctures, we could also have punctured one of us. So in particular, if I'm on a closed surface, the train track parts on my radar are all triangles, which means I can't have any other red edges, which means that my curve is carried by an actual train track. And by making this as epsilon geodesic as you want, that means that all the angles, all the crossing angles have to be in a very small neighborhood of these branches, and therefore arbitrary small self-intersection. Maybe I should write some of these six. All the red dollar tau hat instead of tau for a train track. Look at the limb soup of a number of curves carried by it, bounded number of self-intersections, bounded length. This limb soup is always finite. And moreover, it's zero unless I have a maximal red dollar. Get a maximal red dollar. I mean, the train track part of it is maximal, so it only has one of these complementary regions. Again, there's only finally many to consider. And for each one, I only get contribution to my limit if I look at a maximal one. But then it follows a maximal red dollar, maximal train track, segments of a curve, either angles or arbitrary small. And then the last thing I want to say, so this is, now we know how to define it if we're on a closed surface because the angles are arbitrary small. So if we're on a red punctures, things that wrap around come back. So what we do is define this map high first on the red dollar. So just take whatever weights, probably like one you have here, add it to corresponding branches here. So you get map from a red dollar to a train track. So, and then you map the corresponding curves carried on that to the corresponding simple curve. And then we show that this actually makes sense also on the curve. So just do the surgery on the cutoff each loop. And you get the same thing. So let me end here.