Dr. Laura Monk | Counting geodesics on random Weil-Petersson surfaces

Ajouté : 2026-05-12

[00:00:02]Laura Monk from the University of Bristol who will talk to us about counting closed geodessics on random valet and surfaces.

[00:00:12]>> Thank you very much and thank you for the invitation. It's very exciting to have a workshop on this theme and to see so many people working on this topic.

[00:00:19]When I started as a PhD student, it didn't really exist so much. So it's really lovely to see all this happening.

[00:00:27]Uh because it's very much a mandate. I will try to carefully introduce the balpitis model because it's the first time we hear of it this week. So I'm thinking it's a good idea to have one kind introduction. Uh and my aim today is going to be to highlight some technical tools that we have developed with my collaborator Ninian and Paraman in our recent project where we prove that the spectral gap of random surfaces is optimal. You'll hear about this result later in the week and about the spectrum and all these things, but I won't talk about spectrum at all today.

[00:01:02]I'm only going to talk about geometry and counting geoysics.

[00:01:06]But you'll see in the talk how these ideas interact with questions about the spectrum.

[00:01:14]Okay. So the first slide today is going to talk about uh random valid surfaces.

[00:01:29]All right. So um for this talk I'm going to fix some integers G and N such that 2 G minus 2 + N is positive. is so that there's hyperbolic surfaces with this genus and this number of boundary components and I'm also going to talk to fix a vector x with n components which are uh non- negative and these are going to be I'm going to think of them as boundary lengths for my surface.

[00:02:05]So in this uh probabilistic model the idea is to take random elements of this sample set. So the sample set is going to be what we call the moduliz space.

[00:02:25]So it's denoted as m gn of x and it's the set of hyperbolic surfaces of genus g with uh boundary components of length x1 xn. So the boundary is labeled and the length of the respective boundary components are x1 until xn and I take the quotion by isometry.

[00:03:02]So if I draw an element of my set it's going to look something like this.

[00:03:10]So here in this example I have x1 x2 that corresponds to two boundary geodysics and x3= 0. So we take the convention that x equals 0 corresponds to a cusp which is a boundary of length zero really it goes to infinity here and here g equ= 1 and n equ= 3. So that's an that's an example of a point in this space for these parameters. Um the quotient by isometry is just if it says if you look at the space of triangles if two triangles have the same side length you will not you will identify them. So this is the same thing.

[00:03:52]There's very important special cases of this which are going to come up quite a lot.

[00:03:58]So the first example is the thing we actually going to do probability in in this talk. You'll see examples during this week of where we do probabilities in this general space. But for me today, I'm looking only at n equals zero. So there's no boundary components. And my surface look rather like something like this.

[00:04:20]Um and in this case every time because there's no n and there's also no x we're going to not mg because to lightener notations.

[00:04:30]Okay. So that's the important setting. I I will think about there's another really uh good example which is when g and n are zero and free that means that the surface is a pair of pump actually um so it's a surface with no genus and free boundary components the fundamental property that pairs of pumps satisfy is that if I fix x1 x2 and X3 there is one and only one pair of pants which has these boundary lengths. So in particular M03 of X1 X2 X3 is a singleton for any X in R + 3. So there's only one pair of B in each of these spaces.

[00:05:28]In all other cases, uh, cases of like bigger G and bigger N, uh, there's going to be a lot of elements. We'll talk about what this case looks like in general, but in this very specific case, there's only one point in the space for any fixed values of X.

[00:05:45]Okay. Uh, this is a bit of a not so convenient quotient to talk about. And when you introduce something which has a quotient, it's often nice to have the universal cover of the space to be able to study it. When you think about the Taurus, you like to see it as a quotient of R of out n. It's often the case when you have a quotient space that you want to see it as a quotient of something that's simply connected.

[00:06:12]Here the do people see on this.

[00:06:18]So here the universal cover is what we call the t space.

[00:06:33]Um so there's a nice way to construct it which so it's denoted as TGN of X and it's the space of marked elements of the modul space.

[00:06:57]So by mark what I mean is that I'm going to fix a base surface. So this surface is fixed for once and for all SGN uh of genius G with 10 punches and the space of uh the mark surface.

[00:07:34]It's going to be not only a surface X like here before.

[00:07:40]I'm going to call the elements of this set big X, but I'm also going to have an identification between S X and my base surface.

[00:07:56]So I'm just adding extra information.

[00:07:58]I'm identifying the points of X to a fixed object once and four, which is going to be convenient when I want to compare different surfaces that they will all be comparable to my base surface I started with.

[00:08:11]So here there's a quotient by isotopy because otherwise this would be infinite dimension space a bit messy but when you when you take this up to omotopy basically you will get a nice space >> uh preserving orientation.

[00:08:41]Okay, so that's my tim space. And so now the question is how do I go from a point here to a point there? Because it's a portion. So I'm and to take to say how we go down how we project this onto this and it's very simple uh simply mgx is obtained by forgetting the marking.

[00:09:14]So, I'm I gave myself a reference surface and ways to compare all my surfaces, hyperbolic surfaces to this reference surface. Now, I'm going to I'm going to forget this map here. And I'm just going to look at this surface there. There's a better way to write it down a bit more formally, which is that MGM is TGN of X quotient by what we call the mapping class group.

[00:09:42]And this mapping class group here is the set of positiveomorphisms of SGN divided by the positive ommorphisms of SGN which are isotopic to the identity. So here what this thing is doing is just saying that in order to forget this marking I just need to apply all the possible homorphisms here and then because I will have turn my surface all possible ways I forgot what the marking was. So now I just have I forgot this map.

[00:10:24]Okay. So this I'm just mentioning this because I just want to say that this is a discrete group.

[00:10:31]It's a very complicated group, but it's a discrete group.

[00:10:36]>> I hate to ask this. Are you is is the stabilizer fixing pointwise the boundary or just >> uh so it fixes each boundary component pointwise >> pointwise >> but each of them. So it it stabilizes the labeling that you picked up the boundary >> right is also the identity on each boundary. Well, here the binary fun are just a it's just a puncture because it's acting on sgn, right?

[00:11:03]>> Okay, then put the x.

[00:11:06]>> Yeah, >> it's a good question.

[00:11:11]>> Oh, have all right. So basically the image if you don't like quotients like I personally don't uh the image would be that this is RM and this is a bit more complicated than ZN but something a bit of that nature. It's a discrete group that acts on Rn and so this is a bit like a Taurus. I'm lying to you a little bit because the action is not as nice as the action of dead end by translation on RN but it's a bit like the image you can keep in terms of keeping a bit of a vocabulary going if you get puzzled >> and it's not obedient right the mapping class >> oh yeah it's not obedient and the problem I was alluding to is that some elements have fixed points so there's problems also there all right so that's my space and when I have a space and I want to so now I want to take a probability measure on this space. It's really nice in order to do so uh to understand it a little bit more. So the first step to understanding it was to introduce this universal cover and it's going to become very clear now because we're going to introduce some coordinates on it which are called pen and neielson coordinates.

[00:12:40]Okay. So because of my friends the pairs of pants and the fact that I can pick three numbers and describe entirely the pair of pants it's really a natural building block to construct bigger surfaces.

[00:12:54]And this is what felon Nilson coordinates are. So any element well okay so if I take any surface so I'm going to take a surface here of genus 2 with two boundary components and what I want to do is to pick some numbers to describe its geometry. So here I have three numbers x1 23 and I know the geometry of this object. Now I'm going to take this and I want to give you a list of numbers so that you can describe the entire geometry of this. This is what is going to be a coordinate for this surface and a very natural way to do so is to cut the surface into pair of PS. So I'm picking a family of curve of simple curves which so that when I take my scissors and I cut it on all of these uh I only get pairs of pumps.

[00:13:52]You can do a bit of a counting games if you get bored about how many there should be and how many such curves there is. Um and so the in order to describe each individual pair of front I need to know what the boundaries are. So here this is x1 and x2 but now I need to to fix a a length for each boundary component.

[00:14:20]Uh, but it's not enough because if I have two pairs of jeans that I brought from the shop and I'm a bit silly and I want to attach them together along the belt, I need to decide where I start sewing. I need to decide, do I align the buttons or not? Uh, once I made this choice, when once I stitch one point, I'll just be able to continue stitching.

[00:14:39]So, there's really one degree of freedom here.

[00:14:42]And this is what is called a twist angle.

[00:14:49]So these numbers here are twist angles and so this gives you a family of numbers uh which are in r plus because the length are positive time r I'll comment on this on one in one second to the power 3 g minus 3 + n if I don't mess this up right now. Uh please tell me if I mess it up. Um and uh these numbers entirely define the metric on this surface.

[00:15:23]>> So it doesn't matter exactly where you place the cuts.

[00:15:26]>> No. So um okay. So I'm just going to write this. So this is a a global coordinate system for the tim space. So the tular space is reidentified with this space through these coordinates. The cuts here are made so that they're closed geodysics.

[00:15:46]So, uh that defines them like uniquely up to moving around. If I did another coordinate system, I would have if I because I could have cut in a different way.

[00:16:00]I could have decided to cut like this for instance and then continued. That would give me a different coordinate system on the space. But both of them are called coordinates.

[00:16:12]All right. So this space is identified to this. I didn't write a circle here.

[00:16:16]It feels like this should be these angles here should be on a circle. I didn't because TG is the universal copper. So all the circles have it opened. I want something simply connected here. So I the quotient go from R to the circle happens down there in these quotients here. They get identified there.

[00:16:40]Okay. Okay, so that's really pretty set of coordinates on this space.

[00:16:46]Um, now I'm able to talk about an actual probability measure here.

[00:17:01]>> This allows cusps >> uh inside. Yes. So I don't have a cost inside. But this can be zero. It doesn't matter. So your parameterization there doesn't include the x's.

[00:17:11]>> Uh no yet the x's affects here won't be later but here. Yeah.

[00:17:17]>> Okay. So now we can talk about the biotitis form.

[00:17:29]Uh so really historically what happened in that bile in uh 58 uh found a beautiful uh simplectic structure on the TM space invariant by the mapping class So at this point in this in this uh in his mind is not talking about any of the things I've just talked about for him a point of tim space is a complex structure and there it's really a completely different viewpoint all of this um but for him with this viewpoint there's a very natural beautiful uh simplexic structure on this space and it's invariant by the portions so in particular It depends on something on this space.

[00:18:31]When you have a simplectic structure, it also gives you a volume.

[00:18:39]So that gives you a volume on tular and modul spaces.

[00:18:54]So that's really nice.

[00:18:56]Uh but here the way it's defined it's really hard to see how it interacts with this and it took a very long time for both birds to real to prove what we call the magic formula which is that for any listen coordinates So see here there was a choice. I could have cut my surface in different ways and I would have had different sets of coordinates.

[00:19:34]This is DL1 DL 3G - 3 + N D1 D2.

[00:19:49]So it's just your leg.

[00:19:51]So it's not at all obvious if even in a very simple example if I locked you in a room and told you that you're not able to to leave until you prove that you can change coordinates from one to the other. I think you would be very miserable and you'd stay there a long time. It's not at all obvious that you can change coordinates between these different things and get the same thing.

[00:20:14]But there's this other thing which happens to have the same formula in all these choices of coordinates.

[00:20:21]What's great is that this means that the modul space has these really pretty local coordinates and there's something that's the league measure that lives on it. So that's really nice setting for probabilities and that's why people have been studying this model.

[00:20:39]Now the following observation is that this space here for legg has infinite volume like Rn for instance has infinite volume but once you take the quotient like the Taurus you can have something of finite volume and it is indeed the case.

[00:20:58]So if I if I write the total volume of the modul space is finite. There's several way to ways to see this. I'm very happy to answer this as a question later. Won't have time to now. Um but the main thing is that because we have finite volume then we can normalize and we have a probability or more simply when n equals zero in g.

[00:21:35]So that gives us a very natural probability measure on the modul space.

[00:21:41]Okay. So that's uh that's the kind of little brown work here.

[00:21:49]Uh now I want to talk about the question of how do I study actually these surfaces sampled this way? How do I prove properties on them?

[00:22:04]So with respect to this simple form is is it the L's and the toss that are sort of >> simpic coordinates exactly >> you get a probability measure of what space what's pg on mg on >> so that's the case n equals Nice.

[00:22:41]Okay.

[00:22:43]So, I'm going to sample a random surface. Now, from now on, it's going to have no boundary, but surfaces with a boundary will reappear, which is why I went through the bother of talking about that. So I have my fixed my fixed surface that I sampled according to this thing and my question is what are the closed geodisics? So there's countably many closed geodisics.

[00:23:06]Some of them are simple. I'm drawing here a few example of simple closed geodysics. They have no self intersection.

[00:23:14]Some of them are not. You can have for instance this is an eight which has exactly one self intersection. Or you can have much more complicated ones which do crazy things and you have a countable family of these geodeics.

[00:23:30]Each of these godesic I'm going to call them for the most part gamma and they have a length l gamma.

[00:23:39]A very good way to compute things about these questions about this geodysics is to study this kind of average.

[00:23:51]So I'm going to consider f a function from r plus to r which is nice. You can take it smooth and completely supported or positive. It needs to be measurable.

[00:24:05]It's a nice function. It's a function you like to integrate.

[00:24:10]And I'm going to define this to be the expectation of the sum over all gamas all the closed geodysics like this of f applied to the length of gamma.

[00:24:22]So a nice example of that is if f is an indicator function this is the number of closed geodisics in this interval.

[00:24:30]So I I want to study these kinds of averages here. This is the expectation associated to this probability.

[00:24:41]So the question for the rest of the talk is how do I compute such a thing?

[00:24:46]Oh, it's a very big sub. I think I made hopefully made it clear by showing a few examples. There's going to be some very simple geodysics and there's going to be some slightly more complicated ones and there's going to be some crazy ones which go all over the place. And in order to study this this big average, a natural idea is to separate it in different kinds of geodysics. I'm going to look first at some simple ones and then some more and more complicated ones.

[00:25:22]And this is the first way it's been studied is to really separate it in simple versus non-simple.

[00:25:33]So simple being that there's no self intersection.

[00:25:37]So I write the full average as an average over just the simple ones.

[00:25:52]So these are defined by taking the same sum but I just restrict the sum to simple geodysics or non-simple physics respectively.

[00:26:02]Here Mahali developed in 2013 a very beautiful formula to compute exactly this quantity and also it's asotics for large genus.

[00:26:20]It's a very beautiful and simple idea uh which I'm going to try and present.

[00:26:27]So she wrote uh this average as an integral of f of l. So my test function if you want to you can think here again about f being an indicator function of a segment.

[00:26:48]And here I have a function here which I'm going to call Fiji simple of LDS.

[00:26:58]So it's just simply an integral showing the existence of such a function and with such a rewriting is just measure theory. It's not very deep. But what happens here is that she wrote an expression for PG simple in terms of volumes of moduli spaces with boundary.

[00:27:32]You don't have to write the formula.

[00:27:37]because it's the reason I'm writing it by heart is that there's a very pretty picture to replace the formula with which is that basically this number counts how many how many surfaces of genus G contain a geodic of exactly L and the answer to that is how it were constructed such a surface. How do I construct a surface which I'm sure has a jodic of length exactly L and the answer is I take my jodisic of length exactly L. And now I wonder what can I glue to this to make a hyperbolic surface of genus G.

[00:28:25]One possibility is that I glue a surface of genus G minus 2 G minus one sorry with two boundary components of length exactly L. That's one way to construct a surface of genus G which will have exactly which will contain my geodysic of length L.

[00:28:50]When I do that, I need to glue this guy to this guy. And I have a free parameter, the twist that I was talking about earlier.

[00:28:59]This is what this is counting. And this is the number of possible surface here.

[00:29:05]It's the volume of the modulized space of such surfaces.

[00:29:09]Another possibility, which is why there's a plus here because we're doing some probabilities. to be like adding the different possibilities is to take a joy L and to also pick an integer I and to glue. So here for instance I equ= 2 and two bordered surfaces of genus I and G minus I on each side and this is what this second term is counting. So really what this thing is saying is I'm counting all possible ways all all possible surfaces I can construct this way >> and so I divide by VG because there's an expectation down there. So it's a proportion of count.

[00:29:58]>> Sorry. So yeah, do we have any any coefficient >> uh in this form if the joistic is oriented? No. But uh oh at least I don't think so. There's a bit of a problem in the literature with constants. But this one I checked many many times. If you take the dis oriented should be okay.

[00:30:18]Mahani only writes it until integer divided by two and there one have somewhere and it's unoriented for her.

[00:30:25]Okay. So this is really beautiful and what's amazing and the reason why there was after Mahal's article and in her article a big uh explosion in the field is that actually this term is the leading order term. If you look at G going to infinity, it's kind of used in in some ways in his work, but really the best way to do this is to use some estimates by Wuen uh from a spectral gap article uh where they prove that um oh I forgot to say one thing. So the other the thing I forgot to say is that this I can compute to the leading order.

[00:31:16]I can say that if I want to consider big surfaces now, so G becomes big. This is going to be 4 L cinch squares of L / two plus an error decaying in one / G. And there's a good bound of the error in terms of L just don't want to state.

[00:31:37]So I have an explicit leading order term as d goes to infinity and some decay in uh terms of the genus.

[00:31:45]Now what happens is that I can prove that win proof that the non-simple terms has uh a decay of one / g uh with a very well controlled error. I'm saying that because actually their paper is really not just about the one over G but really precisely how this behaves and this is a key part of the spectral gap result is to control very well this error. But for the most part what you can remember is that at the leading order only simple geodysics contribute and uh we have a formula for this term and then if we want it not simple they're all in one / g terms. So now the question becomes what happens if I want more precision if one / g is not enough in my expansions if I need to compute things more precisely.

[00:32:47]Well, this is why this is the results I'm going to talk about today.

[00:32:53]Um um okay there's no title here. Uh okay.

[00:33:08]So I'm going to talk about the question of like now what happens if I want to understand these things more precisely?

[00:33:15]If I want to go forward and not just say simple non-simple but more precisely enumerate all the possible fates.

[00:33:23]So this is results by uh Anant Man uh so presented in three articles uh and uh yes so I I won't cite this every time but everything from here is new uh except if I specifically say so okay so the first step in order to describe describe this average in a more precise way than just say simple non-simple is to define another notion of like class types of geodysics. We are going to put geodysics in boxes of different types.

[00:34:17]Okay. So, oh I should have what slightly informal definition gamma 1 gamma 2 on SGN or SG I guess. So this is my base surface. I'm going to draw everything on my base surface because my mounting brings things on my hyperbolic surfaces. So we only look at curves here.

[00:34:46]So I'm looking at two curves gamma 2 and gamma 1 and gamma 2 and I'm saying that they are of the same type if there exist uh embedded surfaces uh sigma 1 sigma 2 in SG contain containing gamma 1, gamma 2 respectively and a homorphism from sigma 1 to sigma 2 such that the image of gamma 1 is freely homotopic to gamma 2.

[00:35:42]I'm going to put the picture now.

[00:35:45]So let's look at the genus3 surface I'm going to write to to show you two different simple geoloysics maybe I have gamma 1 and gamma 2 here and what this definition so these are going to have the same type they're both simple so I want to put them together. And the reason why they have the same type is that there's a little cylinder because they're simple.

[00:36:20]There's a little cylinder around each of them. So that would be sigma 1 and sigma 2.

[00:36:28]And there's homorphism from this cylinder onto this cylinder sending gamma 1 onto gamma 2. So I'm not asking for homorphism of the whole genus free surface. But I'm just asking for a local thing sending gamma 1 a neighborhood of gamma 1 to a neighborhood of gamma 2.

[00:36:48]Similarly, I can look at figure eight. So that's physics with one self intersection.

[00:37:02]Here are two examples.

[00:37:05]In both cases, there is a small pair of fonts around gamma 1 and gamma 2 and there's uh homorphism from this pair of pants onto this pair of pants sending this curve onto that curve. So all the pairs of pants are going to be put to all the figure eights are going to be put together in one type. So that's a very nice way to group things together in different boxes depending on the shape of the jig that I'm looking at.

[00:37:52]Now what I want to do is to write an integration formula like mahalis.

[00:37:57]there's going to be some new uh things to introduce in order to do that to to do that. Uh mainly two things.

[00:38:09]The first thing is that earlier when I was picking my clos of length L really there's only one such jodisic. There's only one of simple close jodic of length L. I don't really have a choice. To the contrary, if I think about pairs of pants, imagine I fix a certain length, what's going to happen is that there's going to be a lot of different pairs of fonts which have different geometries, but for which the length here is exactly the same. So, if I want to do the same thing as here, I will have to kind of pick a pick an eight and then pick a thing to put around.

[00:38:50]So, There's many pair of pants for which eight is of length L. So here if you think about this specific case uh what we would want to do is to have the length of the pair of pants my x is from earlier.

[00:39:29]We would like these to become three variables and I'm going to vary those three numbers x1 x2 and x3 on the level set where this guy has length exactly L.

[00:39:41]So cuz now I want to look at all the possible eights of length of a length L.

[00:39:47]So that will require an extra integration on the space. So more precisely um if uh I am looking at a type see so my type is going to always be a surface and a curve on it. So here for instance the surface is a pair of PS and the curve is the eight. But more generally, I have my type is going to be given by a little surface on which my curve is drawn and my curve.

[00:40:31]I'm going to need to do an integration on the space of matrix on S which is the space of S here.

[00:41:04]So as opposed to earlier, I'm now considering that the length of the boundary of S is now a free variable. So I'm I'm deciding to pick some length for the boundary of S, but also a metric for my surface.

[00:41:26]So an example of that is that t of the pair of pants is r + three.

[00:41:33]So earlier with the pair of pants I had only one element but now my boundary is a free variable. So have more.

[00:41:41]Okay. So that's my space and I can equip it with the volume by just taking uh the lebe measure on the x component and the valpit measure on the second.

[00:42:05]This is really just the valition on the bigger space where I let the boundary be a free variable.

[00:42:13]So that is one element which is completely absent in the simple case.

[00:42:19]Then the second element is actually was actually already there. It's just a generalization.

[00:42:24]So what we do is we introduce a concept of realization of S in SG.

[00:42:35]So it's exactly the enumeration I just erased of I was enumerating all the possible ways to put the cylinder in the genus G surface. So now for instance on the case of a pair of pants if I want to include a pair of pants in a genus G surface I can do it this way where the rest of the surface is connected or I can glue something to one component or three different surfaces. to three different components.

[00:43:22]So there's all these different ways to view S as a as an embedded surface in a genus G surface and all of these are going to be called realizations.

[00:43:34]Uh R is a name of a realization. Uh is there a specific reason why like non uh like positive reels only and not non- negative ones like what happens when you what happens if you have cusps?

[00:43:50]>> Um well we want the surface to be connected. So here we want to imagine S to be like this and if I put a length zero here it's not going to be it's going to have a cut and not be connected.

[00:44:01]>> Okay. Thank you.

[00:44:03]>> Uh okay. So that's realizations. And in each of these cases I can I can create a function vr of x which is the if I give if I write x1 x2 x3 or more generally with more x's the length of the boundary this function is the valpeten volume or the product of the valites and volumes of all these modul spaces. So it's the number of participle metrics I can put here.

[00:44:37]It's my uh freedom for what happens outside of X.

[00:44:47]So it's an associated bip volume.

[00:44:52]So it's just the product of the bips of all these pieces outside of S.

[00:45:05]So now we know everything.

[00:45:07]Uh so if I want to now compute which is the expectation of now I'm I'm not restricting to simple but to type t.

[00:45:31]So I want to take the average of a random surfaces of a sum but only the sum of eights for instance.

[00:45:39]Then I have the formula for the average.

[00:46:13]I didn't write the formula for five.

[00:46:40]Okay, so this is a formula for the average and it's an integral. Earlier there was an integral. It was a very simple integral on the just the length of the curve. Here it's more complicated because the length of the curve is going to depend on the metric. It's go I need to take a metric on my neighborhood here.

[00:47:03]So it becomes an integral on a bit more.

[00:47:05]It becomes an integral on the space of all possible matrix on S. You can think a second and the space of matrix on a on a on just one jic is just R plus which is what this integral was earlier.

[00:47:19]So here I put the length of my curve in terms of a metric and there's this function here which is enumerating all possible way to embed my surface in the uh surface. So this function was the same in the case simple it was the function associated to the cylinder here it's a bit it's it's very similar in the situation it's just that I need to embed S and not a cylinder.

[00:47:47]Okay. So there's one thing to notice here is that the only dependency in the genus G is in PGS.

[00:48:05]So in this formula there's this quite complicated integration on space but it's really happening here. the complication of like and so we'll see an example of that soon but there's this kind of mysterious integration here in here there's no genus we're just looking at the shape of the geobysic on a little scale where where we zoomed in and the place where the genus of the overall surface appears is only in this function here which lists all possible ways to put my little pattern in my big surface.

[00:48:43]So this is the only dependency. And here this guy only depends on s. It's only a list of the ways to put s in my big surface. Um it doesn't have uh any it doesn't see gamma. It's just a way to list all the possible embedments.

[00:48:59]I forgot a little symmetric constant which appears if if gamma has some symmetries.

[00:49:14]Uh so that kind of splits the problem.

[00:49:18]The problem now is to describe um the integration on uh t of s and the length function because these elements didn't appear in the previous.

[00:49:51]So actually uh with Nini uh we prove that this function here has a full asic expansion computed in the first two orders and uh the general form of the coefficients the first the leading order is from meahan tree.

[00:50:15]Okay. So really the problem in this formula is to understand the dependency on the on the h space and uh the length of the geodetic.

[00:50:40]And in order to understand and compute these very precisely, we introduced a new family of coordinates on tim space.

[00:50:49]So so far you had your felson coordinates in your backpack. What we did is we constructed a lot more than those felson coordinates.

[00:50:58]Um so they're obtained this way.

[00:51:03]So the first step is to open the intersections of gamma.

[00:51:10]So when you have a curve gamma and an intersection, so you can zoom in so that you can only see something like this. You zoom in very close to the intersection. You assume that there's no triple points or anything like that. You zoom in very well. And there's two ways to open this intersection.

[00:51:43]So that's uh so that's a way to kind of transform the intersection on something which no longer has an intersection and but we want to record that there was an intersection. So we put a bar here.

[00:51:58]And then if I want to still follow my curve, I can do this by following my bar.

[00:52:06]So actually I'm going to define two bars because actually I'm going to have to go through the bar twice. There's always going to be a bar B+ and a bar B minus.

[00:52:24]depending on the situation. We don't define them the same way. But what's important is that um is that when we want to follow the trajectory of my curve, it's going to correspond to following pieces of these and the bars.

[00:52:41]And I want it to I want the two directions to be actually the directions that go through and the shape to be the same for both. Well, that doesn't We won't have time to get into that anymore.

[00:52:57]Okay.

[00:52:59]So, when I do this on a simple example, let's do the pair of parts.

[00:53:06]What's going to happen?

[00:53:08]Sorry, the eggs is I'm going to get a picture like this.

[00:53:14]So, that's in a pair of pumps or a picture like that.

[00:53:23]So this is my bar the line here.

[00:53:30]And so you can you can have fun and do the intersection here. And the rest of my curve was actually this one and this one. Oopsie.

[00:53:45]Or this one respectively.

[00:53:50]So in yellow is the bar and in green is the curve after I opened it.

[00:53:56]So the curve is going to be beta 1 beta n. It now doesn't have any intersection anymore. I opened all the intersections and I'm going to have some bars here.

[00:54:10]So what I want is to use this to have some coordinates here. It's a pair of pants. So there's three numbers I need.

[00:54:17]It's going to be the length of those three elements here. More generally, uh the bars have lengths L I uh here R is the number of self intersections and they separate the curve. So if I look at the points here, they're going to cut the curve the beta in some portions and the length uh there's going to be twice as many. Sorry, I'm rushing a bit to give you one formula.

[00:55:09]So here there's two R of them.

[00:55:16]So what we assume is that uh if gamma is a generalized state which is a condition that guarantees that beta is a multicur so I can detail on that later if you ask me outside this talk Then the L's and TAS are coordinates on tular space and I have a formula for the Jacobian of the change of coordinates.

[00:56:26]Okay.

[00:56:28]So, sorry I'm not finding the time to say more about this. uh but basically it shows you that there's a very this is a very nice change of variable which allows you to go from valson on this space so here it's like x x1 x2 x3 to these new coordinates I introduce we introduced and we can compute the change of variable in terms of the geometric data uh that we introduced. Thank you for your attention.

[00:57:05]Thank you very much Laura for lecture questions.

[00:57:12]>> Don't you need several figure eights? I mean there's a lot of coordinates on the original motor space but here there's how many how many coordinates are there?

[00:57:18]What what is going on with this?

[00:57:20]>> Uh well so here for instance this gives you two different choices of coordinates. You could express this in this or in that you have a choice. So there's a lot of fetch on sets of coordinates, but you only ever need one because it's just different coordinate systems. You can pick one. You don't need to look at several at the same time. Here it's the same. You can define from any change any penis coordinates.

[00:57:43]You can go to this and the change of variable will be this one. Sorry, I can see it's not convincing.

[00:57:50]>> Okay, never mind.

[00:57:50]>> But you just have two two different choices. You really can make a choice.

[00:57:54]It's global coordinates and space. But there are some applications where your new choice of coordinates is more advantageous where you >> for instance with asthma we have a uh we computed the like the average number of offics between two boundary components and these coordinates are a lot more easy to use than uh and coordinates. So there they can be used for their own rights. You don't need to be >> get a specific for each topological type choose a different coordinate.

[00:58:35]>> Can you speak a little louder? I can't >> I mean the coordinates that we choose depend on the curve that we want to study.

[00:58:41]>> Yes.

[00:58:41]>> So here this choice was for the figure eight.

[00:58:46]>> If you take a more complicated one you have more >> more bars and more. Each of these choices gives you a different global set of coordinates on the entire Tesla space and you can just pick one for one and four. You don't need to consider several at the same time, but you have many that you can pick from.

[00:59:07]You have a question.

[00:59:08]>> Oh yes, sorry.

[00:59:10]>> How does this look when the genus is large?

[00:59:14]>> So here the genus one.

[00:59:16]>> Yes. Uh so this has a full asic expansion in powers of one / g >> and if you want the expansion of that you can just literally plug it in >> like literally because there's no genus anywhere else.

[00:59:27]>> Yeah. But where does it start?

[00:59:30]>> Uh it starts at it starts at 1 / g to the power the other characteristic of s.

[00:59:35]Okay.

[00:59:35]>> So that's why at g at the leading order you only simple because that's characteristic zero.

[00:59:43]So >> uh well that would be it. So in this one you'd have that the you'd have that eight cinch of x1 over two is equal to so here there's two components if I well this is x1 x2 and x3 um and if l if this is l Or I can also write it here. If this is L, I have alpha and beta here.

[01:00:46]Uh so now that's what.

[01:00:58]So see it really gives you a formula in both the sets of coordinates and you can decide to pick the change of variable.

[01:01:08]It wouldn't be more costly to do a more complicated example.

[01:01:13]>> I mean probably not just in a pair of pants probably. How about you know?

[01:01:17]>> Yeah. So you can so this the prop is really like prove these two cases and also the case in one solorus and then it's a um induction argument where we redo a similar change of variable at each step. It's an induction in the number of intersections.

[01:01:35]way over I think.

[01:01:39]>> Any further questions, comments? Yeah.

[01:01:42]>> So for uh these averages for the simple curves, you say there was a leading order, right?

[01:01:48]>> Yes.

[01:01:49]>> And here for your is the figure eight the next leading order that you have some.

[01:01:53]>> So anything with characteristic minus one will have a contribution to the leading order. So here this is characteristic minus one the pair of So it will this eight or any gamma on the eight sorry on the pair of plants any gamma will have a contribution to the next order. So the next order there's already a lot of things but anything on the s which has bigger ear characteristic will be even lower order.

[01:02:16]So it depends on the characteristic of s here.

[01:02:22]Okay any further questions? If not, um, we'll now have our coffee break and we'll reconvene at 3:30.