 Well, thank you for your invitation. And also, I would like to thank Sasha for presenting Maxime's university because I feel a complete intruder here. There would be almost no algebra, well, some geometry. And there would be a little bit of physics, but completely different. I decided so the talk is about aspects related to dynamical systems. I didn't dare to speak about experimental mathematics, which is also part of Maxime's specialization. So I will speak about dynamical systems. And I decided to present the story on a concrete example. So I will formulate first a model problem, and the model problem is quite naive. Then I will have to present some technology, which is useful for solving this problem. Then I will present the solution. So the physics here, well, there is some physics here, which is solid state physics. And a typical problem here is the following one. If you consider the electron transport in lattice, then it's natural to consider so-called inverse lattice. So you consider the space of quasi-momenta. And the electron lives on the surface of constant energy in this quasi-momental space on the one hand. And on the other hand, if you turn on a magnetic field, the electron has to stay in a plane, which is orthogonal to the magnetic field. So the electron trajectories are plane sections of periodic surfaces. My family of parallel planes. And the question is, what is the behavior of these plane sections? It is a classical picture. It is a semi-classical picture. So we can see the momentum as periodic. So our surface of constant energy depends on momentum, but it's considered as a periodic surface. And so that's the problem. And of course, this problem in mathematical formulation can be immediately reformulated in terms of surface filiation, because you quotient everything by Z3. You get a three-dimensional torus, a perfect compact surface embedded in this three-dimensional torus. And the lines which were interested in the leaves of the filiation defined by closed differential one form of this compact surface. So this particular problem is somehow very special from the point of view of surface filiations, because the fact that everything is embedded into a three-dimensional torus is special. So the minimal components of this filiation are usually just toric with holes. So let's consider another problem which leads immediately to surface filiations also related to solid state physics. So this model was suggested at the beginning of the 19th century by Ernfest and Ernfest. And it's formulated in the beginning. I said 19th, sorry, 20th, of course, beginning of the 90s. It's 19, oh, something like this, of course. I apologize. But still, more than 100 years ago. So it is formulated in terms of a billiard. You consider periodic obstacles in your two-dimensional plane. So the periodic obstacles are in the form of a rectangle. And you send a billiard ball. And you are interested how this billiard trajectory expands in the plane while it escapes. Then it comes back. And the diffusion rate, by definition, is either, well, this quantity, or you can consider the same object, well, the same number, can be measured as follows. Just consider your billiard trajectory for some time t, consider it as a set, and consider the diameter of the set. So take the logarithm of diameter and normalize by logarithm of that time. So this is the diffusion rate. And this old theorem, due to Vincent de la Coura, Pascal-Uber and Samuel L'Ilvier, says that the diffusion rate is 2 thirds. So it's not like a random work where the rate is 1 half. In the random work, you have square root of t. Here, you have t power 2 thirds. So it escapes slightly faster than in a random work. And note that I state this theorem regardless. What is there? So this is because of hyperbolic, you said, right? Because it's actually hyperbolic behavior in this case. It is not hyperbolic. No, it's not hyperbolic. So you see, it's not hyperbolic by the following reasons. If you send two trajectories very close to each other, they will stay for a very long time, very close. So it's, yeah. So you know that diffusion rate would not change if I will change the rectangle. So this is also a rectangle, like the previous one. And it's still 2 thirds. And I can replace the previous rectangle by a chocolate plate where we have this very narrow corridors and still t power 2 thirds. So the goal of the talk is to compute this 2 thirds. And perhaps you should give the definition of old. Yes. Old will be explained at the end of the talk. It's old because up to a recent theorem of Esken and Mirza Hani, which will be presented at the end of the talk, this kind of 2 thirds were sort of lucky coincidence. It was in some cases, which were very special, when there was a whole chain of lucky coincidences, it was possible to compute these 2 thirds. Due to a very recent theorem of Esken and Mirza Hani, it became sort of a matter of technology. And this is another goal of the talk to show this technology. So everything which was before Esken and Mirza Hani is old and studying from Esken and Mirza Hani, it's sort of a new age, currently absolutely euphoric. Well, one can change the shape of the obstacle radically, adding corners. And then the escape rate, well, this diffusion rate also changes. It can be computed explicitly. Here double factorial is just product of all even numbers on top and product of all odd numbers in the denominator. This conveys non-convex, which matters. So oops, I can change it by this thing. And it's still the same number. So the only thing which counts is the number of angles pi over 2, which defines the number of angles 3 pi over 2. I can add more complicated angles. If you wish, well, it will be clear in a second. What matters here? Sorry? Discs are not all the angles. Yeah, disks would radically change dynamics because disks will introduce hyperbolic behavior. Ah, OK, because then you stick the convex, because you stick the convex with the help of the hole point and nothing else. Yes, yes. OK, so first I want to replace billiard by a foliage on the surface and forget about billiards once and forever. And the way one replaces billiard by a foliage on the surface can be, well, the best way to express it is to consider billiard in a rectangle. And there is a classical Katok-Zemlikov construction, which actually comes from school Olympiads. Instead of, well, reflecting the billiard trajectory, one can reflect the table. So in this way, the billiard trajectory is transformed to a straight line. And we can follow the billiard trajectory and unfold the billiard all along the trajectory. And the trajectory will become a straight line. And if we fold back this picture, we'll get. So folding back this thing, we get trajectory on the billiard. And I colored the trajectory in four colors because at every moment, the trajectory goes in one of the four directions. So these four colors correspond to four directions. I suggest to identify now, well, our unfolding is too huge. Let's identify, by parallel translations, the patterns which correspond to the same color. Then we'll have only four patterns. And then everything will be identified to a torus. And our torus is glued from four pieces of our initial billiard. And if we fold everything from the torus, we get a billiard trajectory. So we unfolded a billiard trajectory to a straight line on the torus. Exactly in the same way, we can apply exactly the same procedure to our billiard with this wind tree billiard, except that I will take four copies of the initial billiard table, which is infinite. But I have an action of z plus z on this table. Everything is periodic. I quotient over this action. And I get a straight line foliation on the surface of genus five. So we have four torus with holes. Their identifications between the holes, the parts of the holes, are expressed by colors. So we get the surface of genus five. Straight line foliation on the surface. And our initial problem of diffusion of the billiard trajectory can be reformulated in terms of the surface foliation and the following way. We take our leaf of foliation. We take a very long piece of leaf. We close it up. We get the cycle. And we calculate the intersection of the cycle with the cycle H, which is written down here in cycle V. And it tells you how far in the plane went our trajectory and how far top or bottom or how far right or left. So it's exactly the same thing. I just reformulated it in a slightly fancier way. And from now on, I suggest to concentrate on this problem. We have a foliation defined by closed one form on the surface. We want to study the behavior of leaves of this foliation. Or more precisely, we take a very long piece of leaf. We close it up. We get the cycle and homology of the surface. And the question is, what is the behavior of the cycle as the leaf, as we take longer and longer piece of leaf? Know that actually, there is a very nice structure hidden in this picture. So our surface of genus five is endowed with a flat metric. Well, surface of genus five with flat metric. Of course, this flat metric has singularities. It has conical singularities, which are coming from vertices of the polygon. So moreover, so we have flat metric. We have conformal structure. So we have a Riemann surface underlying this picture. And the natural complex coordinate on this Riemann surface, just coordinate z. If we can see that this thing embedded, if we can see that this plane is complex plane, take coordinate z in the plane, this is the natural coordinate on the Riemann surface. And there is one more structure. Let's take the form dz. All our identifications are just parallel translations. So this form dz is well-defined on the Riemann surface. So actually, this picture encodes Riemann surface and the holomorphic one form. And this holomorphic one form, distinguished holomorphic one form, I forgot to say that I consider I have to choose some distinguished direction. And then I have this holomorphic one form. And conical singularities of the flat metric exactly the zeros of this holomorphic one form. So the answer is the discrete parameter, which is responsible for these two thirds, blah, blah, and so on, is, well, in the first term of approximation is the genus of the surface plus the orders of this holomorphic one form. And one more remark, which would be important at the very end in the solution of this problem, note that our Riemann surface has interesting group of symmetries. So one symmetry is just place everything to the right. Another, everything up. Well, and then up, bottom of two. And actually, there is also central symmetry. So we have a group of symmetries of order eight just directly from this picture. We'll use some of them at the end of the story. Now, let me introduce you some, or let me present you some technology in several words. This technology is technical dynamics. Let's consider just general flat surfaces with, whereby flat surface, I mean the following structure. So we have a flat surface with several isolated conical singularities, and with completely trivial holonyms. So if I take a vector and I make a parallel translation, it comes back exactly to itself. One can glue flat surfaces like this from polygons where all the sides of the polygon are distributed into pairs of where in each pair, the sides have the same length and are parallel. I identify them by parallel translation. I get a flat surface like this. Or if you prefer the language of pairs, Riemann surface plus holomorphic one form, or this is equivalent object, I forgot to say that I have for this flat surface, I also have to introduce to, as part of the structure, I use some preferred direction. As soon as I choose a direction at some point, I can, since holonyms trivial, I can transport it everywhere without any contradictions. Now, well, there are plenty of important structures on this, on spaces or flat surfaces, or if you wish, on modellized spaces of pairs, Riemann surface plus holomorphic one form. In particular, there is a group action. In terms of these polygons, the group action is as naive as possible. You just take your flat surface, you cut it, say, by straight lines to unfold it to a polygon, put your polygon on the plane. Since you have a preferred direction, the way you put it in the plane is defined up to a parallel translation. And then just act on the plane by linear transformation. You get another polygon with the same property. You glue back the sides. You get a new flat surface. So we have the action of the group on all spaces. So the group together ranks will be shaped. Yes, but the fact that they still, yeah. And so, of course, the direction and the length is changed. But the fact that these two sides are still parallel and have the same, yeah. Why did you need to prefer a diagonal? Sorry? Because if I turn. If you remove these lines, you also can consider polygons. It's popular. So if you just consider polygons, they are two unit SL2R actinette spectra. But then the result of the action will depend how you place your thing in the plane. You would lose there. On canonical, canonical action. I don't know. So how do you have the surface? The surface, it's one. It's given by the polygon in the plane. If you put it in a different surface. Then we need the action, action up to conjugation on it. You would want action. No, there's an action, actually. I'm looking at actual polygons, or an actual polygons, in the R2. No, no, no, no, no, no, no, no, no, no, no, no, no. But you get linear transformation in the R2. Of course it does. SL2R acts on R2. No, no, no, no, no, no, no, no, no, no, no, no, no. For the polygons in R2, you should choose the R2. For the plane. Choose the direction. And you love that. You love what we're doing. No, but if it's an R2, if you wish, there's also a, there's the verdict. R2 already has the effect. Yes, yes, yes, yes. Oh, okay, okay. R2 is not the two dimensional vector space. It's R2. It's the coordinate. It's the vector. In this one. SL2R acts on R2. I'm not taking the abstract two dimensional vector. No, you take the surfaces, it's the sort of direction everywhere. It's kind of vector field. And why do you deny it? SL2R acts on R2. No, because R2 has a coordinate. Yeah, but to place, to unfold the flat surface and to place it in R2, I can do it if it's only flat metric. I can do it after rotation. Of course, in space after. And I want to kill this arbitrariness. If they plane after some of the group does that. You need two dimensions. An auto-automated, which is probably. Yes, there is flat metric, so. It's already at the motion. Last limitation. So here's their theorem, which actually is the background of the whole story. Yeah, I forgot to say that by this notation says that I can see the flat surfaces with, well, this encodes the cone angles. All cone angles are integer multiples of two pi. So they're encoded by integer numbers. Or another way, if you consider if you prefer the language with holomorphic one-forms, that's just the degrees of holomorphic one-forms. So some of these degrees is two G minus two. And one means that I consider area one. So the first theorem says that the action of this group and even of their diagonal subgroup, you just contract everything horizontally and expand everything vertically. So this action is ergodic, meaning that if you start with almost, well, you take a surface by random, you apply this flow, the corresponding trajectory visit every tiny domain in the space, and also it visited with a frequency which is proportional to the volume of this domain. So another way to see this action of the group is, so. Yeah, but I'm sorry, I know the torah's not quite rewarding for this stage. One point, space of torah. In space of torah, the space of torah is modulus surface. And my flow, yeah, this is a very good point. In genus one, I consider the space of all flat surfaces. This is a space of torah, and a space of flat torah with the chosen of area one and with chosen direction. So this is an analog of a unit tangent bundle to the modulus surface. And the flow in this case, yes, it is. Yeah, it is it. And the flow is the geodesic flow. And another way to see their action of the group is to note that locally our space of flat surfaces or this moduli space of pairs, Riemann surface plus holomorphic one form is modeled on this coromalgy space. So it's a relative coromalgy of the topological surface underlying topological surface relative with respect to finite collection of points which are just zeros of the form. So we consider not only absolute periods but also relative periods of this form with complex coefficients. And, well, this thing can be represented in this way and then you start acting on this R2 just by linear group and you get the action. And by the way, this representation also explains where does the canonical volume form comes from. So in these coordinates, we have linear vectors, we have vector space. So we have one parameter family of volume elements but as soon as we have a lattice and we do have a lattice considering coromalgy, oops, I had to put I here, Z plus IZ. So we have, we consider integer coromalgy. This gives a lattice just choose the linear volume element which assigns volume one to a fundamental domain of the lattice and you get a distinguished volume element. And when I'm considering surfaces of area one, this corresponds to sort of unit hyperboloid in this space because area is represented by quadratic expression in terms of periods plus periods conjugates. So we can find this volume element of this hyperboloid and part of the theorem which was just formulated says that the total volume of this or hyper volume of this space is finite. Well, this is not quite trivial because the space is not compact. Okay, now why I'm so excited about this theorem because one of the ways to see to express this theorem is as follows. Take by random an octagon like this, well, with this property that sides are distributed in the pairs. Then the theorem claims that if you choose appropriate sequence of times of contracting and expanding, you will approach with this. This thing will approach arbitrary closed, your preferred octagon say regular octagon. Looking at this picture, it seems absurd but the theorem is stated not about polygons but about flat surfaces. So you are able at any point, you are right to take a pairs of scissors and cut and paste, repaste, re-glue your polygon as you wish, preserving of course the relations between identifications of sides. So this theorem can be formulated as follows. Combining these two operations which commute, so you just squeeze and push in vertical direction and use scissors in a smart way. So if you choose in a smart way the sequence of times, you can approach whatever you wish, for example, regular octagon. Arbitrary closed. So the first modification of course changes the flat structure and the second just changes the pattern. It does not change the point of the modeling space. Okay, now let's recall that we are trying to solve very concrete problem. We have a fullation of straight lines on such a flat surface, for example, flat torus. We take longer and longer pieces of leaves and we close them up, we construct the sequence of cycles in first homology of the surface and we want to describe the behavior of the sequence of cycles. So let's start with the model case which is torus, nothing can be easier and the way when we stop to close up a piece of trajectory can be chosen as follows. Let's take a short transversal to the fullation and each time we cross the transversal we close up the cycle and draw a vector in this homology space. Well, there is one of the consequences of ergodic theorem says that, well, which can be applied to arbitrary surface, says that if we normalize the cycles by n, this thing will tend to a limit which is called the asymptotic cycle and the fact that the same construction would work for arbitrary flat, well, for almost any flat surface, well, no, sorry, for any flat surface in almost any direction on any flat surface is guaranteed by this theorem of Kerkhoff, Maser and Smiley saying that for any flat surface, directional flow in almost any direction is uniquely ergodic. So it really mimics the situation with the, well, for torus, yeah, mimics the situation with the torus. And also I suggest sort of the way to an interpretation of this asymptotic cycle. When you have this irrational fullation on a torus, so if I take off the glasses and I look at the picture you cannot distinguish whether it's a rational fullation or a rational fullation with an angle which has large denominator. So morally you can consider an irrational fullation as if it's rational along a cycle which is very, very long and which approximates this irrational fullation very well, except that to be honest, the cycle is not integer, it's real now. So, and this is something, well, it's not only for surfaces or torus, we'll see exactly the same story in much more complicated situation where we'll consider this Daimler flow on this modular space of piers and it's ergodic so we'll pretend that actually it follows very long cycle in this modular space in a second. Okay, now suppose that we are in a periodic situation and suppose that our fullation corresponds to an unosso map on the torus. And suppose that the fullation is taken in the direction of expanding eigenvector of this unosso map and that the transverse segment is taken in direction of the contracting eigenvector of this unosso map. Well, then it's clear that this direction is just the direction of expanding vector because if I take just any pair of integer cycles, for example, I take meridian of the torus, parallel of the torus, they correspond to these two integer vectors, I'm studying and playing the matrix and very fast the images of these vectors and actually the images of the curves, physical curves on the surface, they will get aligned along the fullation. The same is valid for these first return cycles because if I will consider the fundamental domain of the torus corresponding to this first return map to their transverse segment, our fundamental domains wouldn't be squares, the torus will be tiled with these guys and so each rectangle here represents the first return cycle and if I apply my unosso map, the torus will be mapped to itself but this picture, so this segment will get constructed, these guys will get expanded and we see that these first return cycles to a longer interval are mapped by our map to the first return cycles to a shorter sub-interval and we know how the map acts, so it's like powers over a fixed matrix. Now, two remarks, first of course, there is nothing special about torus in this story. I can construct a transversal and first return cycles for flat surface of arbitrary genus and the second thing, pretend that, well, not pretend, so in the simplest situation when the exfoliation is the exfoliation, this expanding exfoliation of some pseudo-unosso map, I can apply exactly the same argument and we see immediately that our sequence of cycles or first return cycles will be stretched along the direction of the dominating eigenvector and what is less trivial is that in generic situation when our exfoliation is just any exfoliation, I will use this theorem of Maser and Witch and we'll pretend that there is some virtual pseudo-unosso map responsible for this foliation and this can be really, so the formalized. So I can, so this theorem says that if I stretch the surface and then cut and paste, then for appropriate times this operation really mimics as if we are following a periodic trajectory of the geodesic flow or if as if we're applying the matrix of one and the same transformation many times and so their quantitative empirical description of behavior over the sequence of cycles. Yeah, I pushed, yeah. These as follows. So they stretch along some distinguished direction which mimics the direction of the top eigenvector but there is also some extra direction which mimics the direction of the second eigenvector. So in the second term of approximation all these vectors live in a two-dimensional plane but they are deviating from this plane already with some fact, as well as the normed power lambda two which is small to one and then if we. What would you mean? That is the orbit of the formation and orbit of the nautical map. So you speak about which one now? I'm speaking, so I consider one individual foliation. For this foliation I consider first return cycles and these first return cycles are represented by these points and homology and I'm claiming that these vectors behave as if these vectors are obtained by acting with some fixed metrics on sort of randomly chosen. I mean, acting by metrics. Sorry, powers of, so I take as if there exists some metrics which mimics a metrics of induced by an automorphism of the surface in homology and as if I'm taking power two, three, four and central of this metrics and. No, no, so it's the transformation space of metrics. You just taught the metric by some transformation. I'm saying that, so I'm saying that this is the empirical that this is the. No, it's the use of the metric in the context to apply metric. You can apply metrics. Yes, it does, it uses it on the same. I will return to this slide. Yes, this slide. So here we had a fallation in direction of expanding eigenvector of this transformation. And it was not surprising that actually all integer cycles are stretched in direction of their top eigenvector corresponding to this transformation. What I'm claiming is that you take now arbitrary surface and you take fallation by random. And this fallation by random. So at random, sorry. You take almost any straight line fallation on almost any flat surface. And I'm saying that if you consider these first return cycles, they actually mimic this story except that now the matrix is, well, the induced. So our homology space is 2G dimensional. So matrix induced in the first homology by an automorphism of a surface of genus G would be, well, in fixed coordinates is 2G times 2G syntactic matrix. So it has more room. And I'm claiming that when, say, if you construct these vectors by a computer, the whole story is as if there is some hidden symplectic matrix which constructs these vectors as before. Well, the powers of which construct this? The weak power of a matrix. The weak power of a matrix. I understand this. Matrix. Sorry, matrix. Yeah, I'm in such a rush. If he says matrix, he means matrix. OK, again, you know, I need a particle problem. OK, so this would be a solution of the problem. Of course. So this is just a formalization of what was drawn on the previous picture. And actually, so everything was already said except one thing. These numbers lambda 1, lambda 2, and so on, which appear, which are responsible for this deviation spectrum, they have a meaning. They are called Lepunov exponents of the Hodge bundle along the technical geodesic flow. And now I have to say what a Lepunov exponents. And this would be done on the next slide. And the only important comment here is that the fact that there is a whole flag up to a g-dimensional Lagrangian subspace in homology, which is responsible for this deviation cycle, the fact that this is a complete flag, it's two complicated theorems proved later. One of them is by Giovanni Forni and the other by Artur Aivella and Marcelo Viano. And they are really non-trivial. And your theorem depends on data theorems. It was sort of conditional. So as whatever sequence of inequalities we have here, there is a corresponding flag of subspaces which mimics these inequalities. So what are the Lepunov exponents? You can forget about everything which was pronounced up to now. Consider the following situation. Suppose you have a vector bundle endowed with a flat connection and a flow on the base of the bundle. Suppose that we are lucky and the flow is, say, ergodic. Then we can play the following game. We take a fiber of the bundle and we flow it along the trajectory of the flow on the base. From time to time, since the flow is ergodic, we come close to the initial point. We can close up the trajectory and measure the monogamy of the vector. But you never said providing a plane. Why did you give this connection? Because I don't want to be dependent on the way I close up trajectories, basically for simplicity. Yeah, you're right. You're right. But in this situation, I don't need to bother about explaining how do I close up the trajectories. Well, then this doesn't have a classical case of tension bundle, which is OK. It's my fault. But here, here we are lucky. So because, yeah, sorry, I forgot. So now I can formulate the multiplicative ergodic theorem and define Lyapunov exponents. So let's consider many returns close to the initial point. And let's produce the following matrix. So matrix A is the monogamy after n returns. So I multiply it by transpose to get a symmetric matrix. Then I can evaluate this root. And the multiplicative ergodic theorem claims that there is a well-defined limit, which does not depend on the starting point. And this matrix mimics sort of mean monogamy. Morally, it pretends that instead of a flow, we have one very long periodic trajectory. And we measure sort of monogamy along this one periodic trajectory, in a sense. And the logarithms of eigenvalues of this matrix, matrix, sorry, of this matrix are called Lyapunov exponents. Now, in our particular situation, the base is this moduli space of piers, holomorphic one forms plus a billion differential, or what is the same, the moduli space of flat surfaces. And the vector bundle is the real hodge bundle. So we associate to every surface just its homology. And the flat connection is Gauss-Mannian connection. In the homology of the surface, we have intergelatis. And there is the only way to transport the fibers in such way that the intergelatis is preserved, is respected. OK, so we're exactly in the same situation. We have this flow, which was just defined on the moduli space. We have a vector bundle, and we have a connection. One can measure Lyapunov exponents. And the numbers which were presented just a second ago are exactly at this Lyapunov exponents. OK. Now, I promise to arrive to numbers, so I have to evaluate them. Well, Lyapunov exponents are defined very generally. Very nice object, very important in dynamics. The main trouble with this is that usually it's impossible to calculate them. They're just disastrous. Well, sometimes when one is lucky, one can estimate them. But here, where all our structures are so reaching, geometry that will arrive to true evaluation. I need to define one more ingredient to formulate the statement about the Lyapunov exponents. So let's temporarily forget about this moduli space of flat surface and so on. Let's consider one individual flat surface, and let's count geodesics on this flat surface. So one can count geodesics on a flat torus. It's allowed to go to the secretion. No, I want to count regular geodesics. So if one counts regular geodesics on a flat torus, the number of closed geodesics of bounded length grows quadratically with length. So if you count how many geodesics of length at most one milliard, you can find it. No, not the same. Yeah, yeah, sorry. Yeah, of course. Classes of geodesics, absolutely. So we get the approximate, well, the number is gross. Well, the number is the same as the number of primitive points, primitive integer points in a disk of radius 1 million, which is the area of the disk times pi squared over 6. Yeah, yeah, yeah, of course, the LS, yes. So one can do the very same thing on any flat surface. So again, of course, closed geodesics appear not individually, but appear in families. And let's count them, but this time, so on a torus, a family fills the entire torus. Now, if we have a closed regular geodesic, we can move a little bit to the right or a little bit to the left. There would be another closed regular geodesic like this. But at some point, it will hit a conical point. So at this time, we stop. So we have a band, we have this flat cylinder filled with closed regular geodesics. Let's count them with a weight, which is the thickness of this cylinder in the sense that let's count them with a weight, which is the area of the cylinder. And I recall that we normalize the area of the entire surface by 1. So we count the closed geodesics with weight, which is the area of the cylinder. And here's the theorem of which, well, which was adopted to this particular weighted count by Vorobets. It says that if we count the number of closed regular geodesic, the weighted number of closed regular geodesic on a flat surface, Ls bounded by length L. And then we average with respect to, say, entire stratum. And then normalized by this L squared, the result does not depend on L. L disappears. Well, this, no, no, I average S, sorry. This is a family of flat surfaces. Well, more formal theorem is for any SL2R invariant family of flat surfaces. You measure for each flat surface of this number and then average with respect to all flat surfaces in this family. Then the only parameter which, since we averaged with respect to S, doesn't depend on the first L. Yeah, yeah, they're two different Ls, sorry. Yeah, that was very, yeah, this L is a family of flat surfaces in, well, say, like this one. Calligraphic letters are always brothers and sisters of this thing, who are SL2R invariant. So this constant is called Siegelwitz constant. And by the way, this quadratic asymptotics, the same constant appears in quadratic asymptotics for almost all individual flat surfaces in this family. But what I need is this theorem, because this theorem suggests how to compute this number. You see, since it does not depend on the bound for the length of trajectory, initially this theorem was designed for bound, which is large. But nobody prevents us from taking a bound, which is very small. Let's replace this L by a tiny little epsilon. The theorem is still valid. Now, it's bound for the length of trajectory. We're counting how many closed geodesics of length. What is L again? This regular L is the bound for the length of geodesic. We're counting how? It's going to go back to L. Yes, at most L, at most L. Most L, but that can be epsilon. Yeah, you are right. For most of flat surfaces, if I put a bound epsilon, I just get 0. There are no short geodesics. So there are some surfaces for which we will be lucky. And there is sort of a bottleneck where we'll find one short geodesic, and this is rare. And there will be even less flat surfaces for which we can find two short independent geodesics. But remember, we normalize here by 1 over p epsilon squared now. So I'm claiming that one can forget about surfaces where there are two or more independent short-closed geodesics. And basically, this integral is reduced to the volume of the part of the space of those surfaces where one can find at least one closed geodesic. Now, the structure of this part of the space of flat surfaces is like this. We have several sort of reasonable flat surfaces joined by narrow long cylinders. And this enables to evaluate the volume of the space. Because from this picture, it's already clear that everything would be reduced to products of volumes of moduli spaces corresponding to this sort of thick pieces with some combinatorics involved. So these combinatorics can be done. And one can evaluate this Siegelwitz constant. And it's really an expression in terms of volumes of this adjacent strato-flat surfaces normalized by the volume of the initial strato with some explicit combinatorial factor, which is a rational number, which can be calculated. It's some combinatorics, but it's doable. Also, so combinatorics is explicitly described. Volumes are fortunately calculated by Eskin and Akunkov. So for holomorphic one-forms, this constant can be computed in full small genera. There is an analogous story for slightly more general flat metric with almost trivial holonomies. They correspond to quadratic differentials. There is a problem. The volumes are not computed as numbers. But let's stay for a while with holomorphic one-forms. So here's the formula for the sum of this G Lyapunov exponents. It's this simple-minded expression in terms of degrees of 0s of holomorphic one-form plus the Siegelwitz constant normalized with pi squared over 3. So this thing, well, it's already there. This thing can be computed. So this Lyapunov exponents could be computed. Now, just a word about the proof. It's based on the initial maximum observation that, on average, with respect to a circle in the Tehmel space, which corresponds to orbit of the Cecil 2R, of the variation of a Lagrangian subspace does not depend on the Lagrangian subspace. And this is an enormous simplification. It reduces the whole computation to some integral over the moduli space. And then computation of some integral which involves the curvature of this time holomorphic hodge bundle. And then one have to struggle with analytic Riemann-Roch theorem, compare asymptotics of determinant of flat metric and underlying hyperbolic metric when the surface degenerates. Apply some cutoff and get this formula. Yeah, 130 pages and 15 seconds. That's my personal record. OK, now we arrive to nowadays. A very recent theorem of Eskin and Mirza Hani says that, well, everything which I told are applicable to this SL 2R invariant submanifolds, they claim that any SL 2R orbit in this space of flat surfaces, the closure of any SL 2R orbit, is a nice orbit fold. This is a miracle in dynamical systems. When you consider closures of orbits in dynamical systems, you get just arbitrary, complicated objects of irrational Hausdorff dimension and so on. And here, this theorem mimics a theorem of Ratner for the corbitlosures of unipotent flow. We know that the modelized space is not homogeneous. No, not the equal and generated but unipotent. Yeah, sorry. Yeah, yeah, yeah, I was too fast. Yes. OK, so the development of this story and I did, sorry, I forgot to say about this. Moreover, in the homological coordinates which I represented, these invariant suborbive folds are just affine subspaces. And not arbitrary affine spaces. I just don't have time to describe them. But there are plenty of constraints what affine spaces can, in principle, appear as orbitlosures and SL 2R invariant manifolds. So this theorem before something like this was already known in genus 2, because in genus 2, there is a classification of all SL 2R invariant manifolds due to Kurt McMullen. But genus 2 is very particular. And nobody knew before this theorem whether it was something specific about genus 2 or not. No, it's not. So this is a general theorem. Another development is that they're really efficient methods of constructing orbitlosures. Another recent development says that for any given flat surface, it's really. So this theorem generalizes this theorem of Kirchhoff-Mezus Smiley saying that for any flat surface, almost any direction is uniquely ergodic. This theorem says that for any flat surface, almost any direction is Lyapunov generic. And this shows what should be the technology of trying to understand the topology of any flat surface. So you take any flat surface, and you want to understand the behavior of a directional fallation in almost any direction. So the technology is as follows. Apply this theorem. So push the button and find the orbit closure. It is really with all this recent development that became a technological question. So you just have to work hard. As soon as you found the orbit closure, you have to calculate the Lyapunov exibands for this orbit closure. And this is the answer to all the questions. Now, in the wind tree problem, there was extra lucky coincidences. One can notice that our flat surface is a cover of a surface in the hyperliptic locus in genus 1. And that the cycles H and V, which we're interested in, are coming from this genus 1. Now, you apply this technology and you prove that the orbit closure of almost any wind tree surface is the entire hyperliptic locus. And then, since it's in genus 1, the sum of Lyapunov exibands is just one Lyapunov exiband. You compute it. You get to thirds. If you consider this wind tree flat surfaces with more complicated shape, there is another hyperliptic locus. You compute all the same. You find Lyapunov exibands. You get this number, which was there. Now, I still have several minutes. I want to present one application. I will skip the other one. I want to compute the volumes in genus 0 for quadratic differential. So I suggest to consider slightly more general flat surfaces in the sense that, for a flat surface like this, the holonomy is already in the group Z over to Z. But otherwise, it's the same. But I want to restrict myself to genus 0. And we have seen that computing volumes is actually equivalent to computing, well, the number of the volume of the modulite space. I want to compute the volume of the modulite space of flat surfaces like this. So genus 0 and, say, three singularities, conical singularities with an angle like this. What kind of volume? You say it's a stratum. Yes. So for each stratum, I have these cohomological coordinates, which are slightly more complicated in this case. Is there a special coordinate for volemics? Yes. And basically, I have to count the number of integer points on a huge ball in these cohomological coordinates, which is equivalent to counting how many flat surfaces styled, let's say, 1 zillion squares. And with this singularity data, I can find connected. And as a matter of fact, for genus 0, this is counting who its number with the ramification data like this. This guy is, so yeah, that's another several words. So here's the guy. And it's not me who did this guy. It's Maxim, actually. I cannot help telling the story. We were desperately counting these volumes, using some combinatorics, relating some terrible sums to multiple zeta values, then using Don's wonderful code for relations between different multiple zeta values to find the final answer. And at some point, it wasn't born more than 10 years ago. At some point, Maxim comes to my office and says, oh, I found this thing in Puppin's Koenig, I guess, close to their train station. One can really create, one can construct them. And another story which I want to tell is that, well, we did this thing. It was in my office. And at some point, we brought the family of my colleague to Max Black Institute to show how mathematicians work. So there was the three-year-old son of this colleague and a friend of his wife. And the son said, oh, I know now what mathematicians do. And the friend of the wife of my colleague who was an artist said, oh, I had it in my childhood, but I never had an idea that one can construct volumes. I always did flowers which were flat and so on. And I'm telling this to store, well, to things because to my mind, this really characterizes the way Maxim does mathematics. It's really the way he does mathematics. He plays with it. Like, no, well, you started it. Yeah, I just finished. Yeah, you started it. And so he really plays like a kid. And he sees things which other people do not see, which are just behind them. And, well, he creates, well, so this sort of things out of nothing. And working with him, it's like broad, well. What's supposed to represent? It's, yeah, it's square tile surface. It's square tile surface. It's an integer point. Well, OK, rectangle type. Yeah, it's integer point in this space. So basically, counting volume is equivalent to counting how many, so you are given 1 milliard squares like this, how many connected surfaces you can construct using at most 1 milliard squares, having exactly these conical singularities. OK, we'll show you the zero. Yeah. So. And you have a specific focus. Maybe go on, yeah. So let's, for quadratic differentials, there is a similar formula for the sum of Lyapunov exponents. Let's apply it to genus zero. In genus zero, there are no first homology. So on the left-hand side, we just have zero. The sum is equal to zero trivially. So we get a relation between the Siegelwitz constant and this expression just for free. We have this expression for Siegelwitz constant. On the other hand, one can use the expression for the, yeah, sorry. Now I will tell the answer and then the solution. So let's define the following function of an integer number n. It's double factorial of n divided by double factorial of shifted n, corrected by pi power n, and corrected by this last factor. And let's define, by definition, double factorial of minus 1 and of zero as 1. Then here's the formula for the volume of strato of miromorphic quadratic differentials with at most simple poles in genus zero. Just the product of these guys evaluated at d i's. So Maxime conjectured this formula about 10 years ago using approximate values of Lyapunov exponents computed by computer and looking in this particular case and then making a guess for a general case. And yes, this guess is true. And to prove this guess, you use this formula for the Siegelwitz constant and the expression of the Siegelwitz constant in terms of volumes. So you get series of relations for the volumes. And to prove that the formula for the volumes is sufficient to plug in the guessed formula in this thing and prove that the relations are really true. This is some combinatorics which can be done. So here are the relations. And one have to write down, well, generating function for this thing. This is the generating function for this sum and for this sum is the same. And this is one does not obtain this relation on the nose, but using some functional relations between these functions, one can prove this. This I will skip whether it, yeah. So this is a description how the computation sort of fails trying to make computation honestly. The Lyapunov exponents were not designed to compute volumes. Everything goes vice versa. The volumes were supposed to be used to compute Lyapunov exponents. And it was just in a protocol how we tried to generalize Maxim's combinatorial results well, to compute volumes honestly. And this is just a list of problems with which I want to finish. So there are plenty of, well, there is this brand new theorem of Eskin and Mirzaghani and all the results of it. But now the question is whether it's possible to make a classification of a Seltor invariant submanifolds. And it's really very interesting. And there are recent indications that there are nontrivial interesting Seltor invariant manifolds, the origin of which we do not understand currently. And another thing is, well, there are plenty of problems. Well, another thing is, say, to compute volumes when genus tends to infinity. There is when genus is, well, finite genus, even for a billion differential, the formula of Eskin and Nukunikov is rather an algorithm how to compute. It's not a compact formula. For quadratic differentials, it's even more complicated. But it looks like when genus tends to infinity, there is enormous asymptotic simplification. And there is a confirmation for the principal stratoments confirmed in recent work of Dawi Chen, Martin Merler, and Don Zagir. Well, and also it would be interesting to do everything not analytically, but algebraically. It would be great to compute this Zigalovich constant, for example, to find an interpretation of it in terms of some intersection theorem, which is probably not constructed yet, say, like ELSP formula, or something like this. And finally, one more problem is to study dynamics of the Hodge bundle of other families of complex varieties and some experimental results for families of Calabiyaus are already obtained by Maxime. And another thing which would be extremely interesting would be to go backwards. So here we started with billions, and we found an interpretation in terms of modular space of curves. It would be funny to start with some modular spaces of complex varieties and to go back and to find an interpretation with some renormalization of simple-minded dynamical systems like billiards. So thank you for your attention, and congratulations from the entire billiard community to Maxime. Thanks, Speaker Ging.