 Thank you for the presentation. Thank you. And thank you for the invitation. I have sort of extra sentimental reasons to be happy speaking here. First, because that's an occasion to see old friends like Stefano Lozato, Franca e Cardi, who live in Trieste. And also, I spent here quite a while, more than 20 years ago. And at this time, the story I am speaking about was sort of between conjectures, hopes. And basically, nobody believed that it might be true. And I was very happy to be accepted here at CISSE. And 10 years after, I was in 2003, I was giving a mini course here. And at this time, some conjectures, some first conjectures were already proved. And there was sort of a more general hope that many things might be true. And by now, 10 years after, some major conjectures are already recently proved and the field is still developing. So I would like to illustrate, basically to illustrate the field on one concrete example, not that this example is the most important, but it is sort of the most visible and the easiest one to present in 45 million lectures. So I will speak about diffusion in a periodic billiard as a model problem. So the problem is the iron fast wind tree model. You place rectangular particles in the plane in a periodic way. It is periodic with respect to z plus z. And the rectangles are aligned with respect to the lattice. This is important. Now, you send a billiard ball and you want to trace how the trajectory propagates in the plane. The ball can go far away, come back, go further, come back. So as a measure of the speed with which the trajectory spreads in the plane, I suggest the following sort of the most simple minded estimate. Let's take the diameter of the trajectory or length t. It is some complicated curve in the plane and I consider it as a set in the plane. So I measure its diameter. I take the logarithm I normalize by logarithm of t. So the first theorem due to Delacroix, Huber and Lelièvre is that for all parameters of the rectangle and here it's really for all parameters of the rectangle and for almost all initial directions for any starting point, the billiard trajectory spreads in the plane with the speed which is roughly t power two thirds where this roughly t power two thirds means that this quantity, this limit is equal to two thirds. And this diffusion, well this measure of the speed with which trajectory spreads in the plane is called the diffusion rate. So this diffusion rate two thirds is given by Delacroix's point of certain renormalizing dynamical system associated to the initial one. It is really the pleasure to give this talk at this conference. So please forgive me a part of the talk, probably 35 minutes would be school and 10 minutes would be conference and it's a pleasure because basically everything was already presented. Renalization was presented, homogenous dynamics was presented, modular spaces were presented, even Zygielowicz constants were presented. So all the ingredients of the talk were already mentioned. So before going on I would like to make several comments on the statement of the theorem. First, well what is surprising here? Surprising is two thirds and it is surprising by the following reason. So there are several classes of billiards with convex cutters starting with the works of probably Bunimovich Sina and Chernoff and with the work. So in one context and in the other context, which is closer to this one, by the work of Delacroix and Thomas Vario. So when the obstacle is convex, then the diffusion rate is one half. So the particle roughly moves in the plane as a random walk. So it propagates with the speed square root of t and here it is faster than square root of t. So the person who is moving by the random walk, he is not quite sober, he doesn't go straight, but he is not as drunk as random work when square root of two. It's something in between. So, and this is funny. Another thing which is, however, I should insist that here I measure the diffusion rate really roughly. I pass to logarithms so I cannot control what is in front of this t power two thirds. So I don't know whether it's logarithm, constant, log, log, no idea. So it is really sort of rough estimate. So this is one more remark. And one more funny thing is that it is really cute that these two thirds do not depend at all on the parameters of the obstacle. You can take the rectangle very narrow and very long, leaving just tiny corridors, or you can even take a chocolate plate and leave narrow corridors between these pieces of chocolate plate. So roughly it is still one of the same two thirds and also you can make your rectangle really tiny little bit still two thirds, roughly. I repeat, I sacrifice the factor which is sort of sub-pollinomial. I cannot control it, but roughly it's still t power two thirds. So a natural question is what would be, what would happen if we changed the shape of our wind tree? So now we plant our trees of different species and it was a question, actually it was a question of Jean-Christophe Yorkosso, the thesis defense of Vincent de Lacroix where the one can make the diffusion rate arbitrary small, changing the shape of the obstacle. And the answer is yes, one can take more corners where all the corners are rectangular and keeping the shape of the tree doubly symmetric with respect to vertical, with respect to horizontal axis. We get the diffusion rate which is like this. So for m equal one we get two thirds. Oh, I forgot to tell that double factorial here is product of all even numbers from two to two m in numerator and product of odd numbers from one to two m plus one in denominator. So when m is equal to one we get two thirds, when m is larger we get other rational numbers and when m is large enough we get this diffusion rate arbitrary small. Here the statement of the theorem is already not for all parameters of the obstacle but for almost all in particular for rational parameters of the size, sizes of the obstacle. The number might be different but still for almost all it's one of the same and you can, if you fix the number of angles and you change the shape in this way it would be one of the same number. What is important is the number of angles, that's it. So it's also cute. Now my goal is to tell how can one obtain these values, very explicit rational values of the diffusion rate and also I cannot help mentioning saying that I am especially happy to give this talk in the Institute for Theoretical Physics because once that's mathematicians who realized that the diffusion rate is not one half as a lot of experimental physicists believed but two thirds which is really different. So there were plenty of experimental works trying to prove that the diffusion rate is one half. You have to trap, no it would change the diffusion rate. So here it is important for me that this is a connected polygon. So basically for me it is, so this thing is connected. No, this is part of the theorem, just a second. You can have periodic trajectories. You can have periodic trajectories both in the initial wind tree and any of these kinds. Whether you can have a periodic trap trajectory, I don't think so. I have to think a little bit, I don't think so. Yes, so I have not a slightest idea and I have not a slightest idea and now I have to make a confession if we are speaking about this. So I was so proudly saying that for any parameters of the rectangle and so on we have these two thirds and so on, if you turn all these rectangles by some angle so if you have periodic obstacles but not aligned with respect to the lattice but turned by some irrational angle there is no approach to attack this problem currently and this is extremely humiliating. There are some sort of hoops but this is a sort of very long and complicated program how to even approach this question. Currently everything is purely two-dimensional and it would be clear in a second why. So I would reduce this problem to a problem, in a minute I would reduce this problem to a problem of surface foliage. So once again this is part of this 35-minute school part of the talk. I apologize there are plenty of experts for whom it would be extremely annoying but I have to remind how to pass from a billiard to a surface foliage. So let's forget about the initial billiard, let's take the simplest possible billiard, the billiard and a square or in a rectangle. And it comes actually from mathematical circles for the middle school that it is much easier instead of following trajectory in this billiard it is much easier to unfold the billiard. So let's take a mirror copy of our table and then in the mirror copy the billiard trajectory would continue just a straight line. And if we have a complicated trajectory, every time trajectory hits the border of the rectangular billiard we take a mirror copy of the billiard and we unfold the billiard table along a trajectory. So what we gain is that now trajectory is a straight line. What we lose is that now we have this unfolded billiard. But know that for a rectangular billiard the trajectory at any point goes in one of the four directions. So red, green, yellow and blue here. And I suggest to identify parts of, well, partons of our unfolded billiard tables which correspond to the same color. This is the same as, for example, mark the corners of the billiard and identify those partons which can be identified by parallel translations. So there are four different colors or four different partons if you wish. So we'll get four copies of the billiard. And after identification these four copies are transformed to a torus because we have to also identify the opposite sides. And our billiard trajectory in the initial rectangular billiard is unfolded to a straight line on this flat torus. And if you take these four copies and fold them back you would get the initial billiard trajectory. So basically what I'm saying is that this is two absolutely equivalent problems. The problem of billiard initial rectangle and tracing a geodesic and a flat metric on the resulting torus. This is basically the same problem. And in particular if we want to understand how far this trajectory would go inside the plane in terms of the torus I understand that the question is stupid. I'm suggesting it by purely sort of pedagogical reasons because I would need to reformulate it in a second in a more difficult context. So in topological terms we can do the following thing. We take a piece of our irrational winding line on a torus. We wait for time t. We join up the ends. We get a cycle on the torus. We consider this cycle, well the closed loop. We consider this closed loop as a cycle in the first homology and to understand how far the corresponding line would go on the universal cover on the plane to the right or to the top we can compute the intersection number of this closed cycle with the cycle which corresponds to the meridian or to the parallel of the torus and this number would give us the distance of displacement of this initial straight line in the plane. So why am I saying all this? Because now I would apply exactly just literally the same construction to our wind tree billiard. First instead of our initial wind tree billiard I will take four copies of it exactly the same way as it was done for the rectangular billiard. Then I will use the fact and there would be some identifications between the sides of these copies. I would quotient over z plus z because everything is z plus z periodic and what I get is this surface which is four rectangles where the opposite sides of each rectangles are identified so that's basically for Toray but this rectangle corresponds to fundamental domain of the lattice and the initial rectangle corresponds to the obstacle so you can consider this topologically as for Toray with holes where the sides of the holes are identified by parallel translations and now we consider we send a straight line filiation in this flat surface which in this particular case has genus five and what is responsible for their diffusion rate is the following thing. We repeat exactly the same thing which I pronounced several seconds ago. We take a long leaf of filiation, we close up the ends to get a closed loop. We consider the corresponding cycle in the first homology of this surface. There is much more room now because this is surface of genus five so it's ten-dimensional space and we take this cycle in this ten-dimensional space and intersect it with this one or with this one where these cycles are indicated on the picture. The corresponding intersection number would tell us exactly what is the horizontal or vertical displacement. It is not obvious from this picture. I just pronounced in a very fast way what is the prescription but it is sort of an absolutely doable exercise. You just apply exactly the same technique as with the billiard in a rectangle versus irrational winding line on a torus and you generalize it to this situation you would obtain immediately this description of this story. From now on I forget once and forever billiards. I'm interested in the following problem. I have a surface, a Riemann surface. I have an analog of straight line filiation on this Riemann surface which would be discussed in a second. I'm interested in the following question. I take longer and longer pieces of this filiation, I close up the ends and I want to understand what is the behavior of sequence of cycles in the first homology which one obtain following longer and longer pieces of leaf of an irrational filiation. For this I need to continue with my school part of the talk. I have to speak a little bit about the asymptotic cycles and so on but I want to start with a few morphisms of surfaces and pseudonauts of automorphisms. Forgive me for a too pedagogical talk. So first remark is let's start with the tori. I want to convince you that tori have a lot of automorphisms. Here is one of them. You glue the torus along the circle and then progressively twist it and when you arrive to the end you make a full twist and you glue it back. So this is sort of topological description. If you prefer algebraic description, more algebraic descriptions like this. If you consider the torus as R2 quotient Z2, then any integer map of R2 to itself factors to a map of a torus to itself. So if you consider a map corresponding to this integer matrix, you get a dent twist. And one more way to see it sort of more geometrical is as follows. You can consider torus as a square with two opposite pairs of opposite sides identified. This map maps this fundamental domain of the torus to the parallelogram like this for which you have to identify the opposite sides. But you can replace this fundamental domain by the following one. Just take pair of scissors, chop off this triangle and since B is glued to B, just place it here. What you get is a square, is a unit square, meaning that it's the initial torus. So this is one more proof that we get a map of a torus, automorphism of a torus to itself. Now one more remark. It would be very convenient for me to consider this transformation in the following way. So the space of lattices was considered in the mini-course of Jens and Andreas many times. But let's recall that any lattice defines a torus. Just quotient R2 over this lattice you obtain a torus. So the space of lattices is the space of flat tori. I suggest to consider this transformation progressively. Instead of taking this matrix, let's put a parameter t here and let make t vary from 0 to 1. What we get is a closed horizontal in the space of lattices, meaning that we deform our torus progressively declining the sides. And this picture shows that the corresponding path is actually closed that arriving to t equal to 1, we get back exactly to the same torus we have started with. So we get a closed cycle in the space of tori. Now slightly more complicated diffeomorphism of a torus. Let's apply a composition of horizontal then twist and vertical then twist. Then in algebraic terms, it corresponds to a product of two matrices like this. So it's this matrix. And again, if we want to use the language of fundamental domains, we send this matrix sends unit square to a parallelogram like this. And exactly in the same way as before, we can rearrange this fundamental domain to bring it back to a unit square, meaning that this lattice is the same. The lattice defined by this fundamental domain is the same as the lattice defined by this fundamental domain. Or in other words, this torus is exactly the same as this one. So we get a map of a torus to itself. And once again, it would be very convenient for me to consider this map in the following way. Let's deform our torus progressively. And this time will deform it progressively in the following way. So if you compute the eigenvalues of this matrix, one eigenvalue, well, the determinant is equal to one. One eigenvalue is greater than one. Other is smaller than one. They're expanding and contracting directions. And I will mark this expanding and contraction directions once and forever. And then I will progressively shrink in contracting direction, progressively expanding direction. So at the beginning, I will just move from my initial torus to some other ones in the space of tori. But this picture shows, and all this calculation shows that at some point, when I move along this one-dimensional family of deformations, I come back to the initial torus. So I get a closed loop. Actually, this closed loop is a closed geodesic in the space of tori, which was discussed many times. So it's a closed loop. It's a closed geodesic on the modular surface. So this is just the calculation of eigenvalues and a definition which I would need. These kind of automorphisms of a surface exist not only for the torus, but for surface of any genus. So they are automorphisms which preserve a pair of transversal fallations and which, in appropriate coordinates, are uniformly expanding, contracting in one direction and uniformly expanding in the other direction. And they're called pseudonosses because for high genera there are several critical points. But otherwise, everything is exactly the same as for the torus. So, and this is just the remark which I've pronounced that if you have an onosoph, a pseudonosopholiation, it corresponds actually to a closed geodesic in the corresponding modular space of surfaces. For example, modular space of tori. So this is the promise, the statement. Now, just a word about this family, the space of all tori. It's space of lattices, which we have seen many times. And if, so one of the ways to parametrize lattices up to scaling and rotation is to say that the shortest vector is horizontal and has length one, then the next shortest is, and choose the next shortest in the upper half plane, then it would be located outside of the unit disk and it would be located between two parallel lines. So this is the fundamental domain of the space of lattices or for me the space of lattices is the space of flat tori. And what is important for me for the space of flat tori is, well, what are the geometric properties of the space of flat tori or other flat surfaces which appear in a second. It is never compact because, just because, for example, in the case of tori, we can always have tori which are very narrow and very long. So the torus can be narrow and long and it can be arbitrary narrow and arbitrary long, nothing can be done with this. So this space is non-compact. Also, it is not quite a manifold, it's an orbifold because there are some tori which are more symmetric than others. So this, if you glue opposite sides over regular hexagon, you get the torus and this torus has extra symmetry which is untypical for other flat tori. And if you glue a torus from a square, it also has extra symmetry which does not exist for other tori. So we get two orbifolding points. In a second we would arrive to flat surfaces. The whole picture would be, in a sense, very similar to that except that we'll get a modular space which would be multi-dimensional. Okay. So let me glue something more interesting than a flat torus. Let me glue a flat surface of genus 2. I start from a regular octagon and I identify a pair of vertical sides and pair of horizontal sides. What I get is a torus with a hole. And then I continue my identification and identifying these pair of sides and then the remaining pair of sides. So after the next identification I get torus with two holes and final identification is surface of genus 2. So we get a perfectly flat surface of genus 2. There is no contradiction with the Gauss-Bernet theorem because to be honest, all missing curvature is hidden in the point which comes from the vertices of the octagon. All the vertices of the octagon are identified to one point so we get a subtle, a conical singularity there. So our surface, our flat metric has conical singularity. But otherwise all the geometric properties of the surface are, well, they mimic the properties of flat torus in particular the holonymy of this metric is trivial. One can generalize this considering any polygon of the same kind with pairs of sides distributed into pairs, with sides distributed into pairs of sides of which are parallel and have equal length. So this is the generalization of the previous picture and you can easily imagine further generalization. And what is important is that there is a group action on the space of flat surfaces like this which can be defined already on the level of these fundamental domains on these polygons. So I forgot to say that I consider the choice of the vertical direction as part of my geometric structure. So for me, from now on and for the rest of the talk, this surface and the surface obtained from this polygon by rotation say by 30 degrees are completely different. I do distinguish them. And then my polygonal pattern is defined inside the plane up to a parallel translation. So when I apply the linear transformation, the result important is also defined up to a linear translation so it would define one of the same surface. My action is well defined. And now the first key theorem of the story is the theorem of Mazur and Vich which says that the action of the group SL2R and of diagonal group are ergodic with respect to some natural finite measure on each connected component on every space of flat surfaces where this notation stands for flat surfaces where I fix the cone angles of the singularities. So recall that our flat surfaces have some cone angles but all the cone angles by construction are integer multiples of 2 pi. So for fixed genus, there is a finite set of combinations of conical angles which are integer multiples of 2 pi and here I denote what are these multiplicities of the angle 2 pi. So here I consider flat surfaces with n conical singularities with angles d1 plus 1 times 2 pi, etc. dn plus 1 times 2 pi. This one means that I normalize the area of the surface to 1 because otherwise to avoid sort of stupid reason of non-compactness. Excellent. Now why I find this theorem completely magic? Because it claims that you start with, for example, basically any octagon like this and applying this shrink horizontally, expand vertically transformation you can approach at some point any other flat surface in this family. For example, the flat surface obtained from a regular octagon and there is no paradox here, clearly the fundamental domain would never approach but I'm speaking not about fundamental domains, I'm not about polygons, I'm speaking about flat surfaces. So for one of the same flat surfaces there exist many fundamental domains and the true statement of the theorem in these terms is that if you apply this shrink and expand and for an appropriate amount of time and also cut and paste in a smart way, then combining these two operations you can approach or approximate arbitrary well, for example, regular octagon or any other surface in the same family. Okay, now I have to come back to my problem I want to study straight line foliations on these flat surfaces and let's start once again from the torus. For example, one can study the sequences of cycles which correspond to irrational foliation and the following way, choose a transversal, the particular transversal doesn't really, is not important, any. And every time the leaf of the foliation crosses the transversal just join the ends by a segment along transversal, get a closed loop and consider a cycle. We obtain a sequence of cycles in the first homology on the surface and let's normalize this cycle just by, so this is a cycle in the first homology let's normalize this length by the number of returns to the transversal The statement, well, the corresponding limit, this is one of the theorems that almost, that usually it exists, that this limit is called asymptotic cycle and the most general theorem which says that the limit exists is the theorem of Kirchhoff, Maser and Smiley saying that for any flat surface and for almost any direction so the directional flow is uniquely ergodic meaning that there exists well defined asymptotic cycle, one of the same for all starting points. Excellent, now suppose that I suggest to find this asymptotic cycle in the situation when the foliation is not arbitrary but pseudonosoph. This is a very special situation and in this very special situation it is really easy to find the asymptotic cycle just like this. So by the following reason, for example, we can start with the torus and with our favorite Anosov map on this torus, take any closed curve and any cycle corresponding, well, the cycle representing it. For example, one like this or one like this can apply our deformorphism to the surface several times. So in terms of homology, we are applying this matrix to some integer vector and you know what happens when you apply hyperbolic matrix, powers of hyperbolic matrix to a vector. So it is integer vector so it cannot be collinear to either of two eigenvectors but actually your images of your vector get stretched in the direction or in the expanding direction. So to find the asymptotic cycle, basically you take any and you apply several iterations of your map and what you see is up to a very small arrow, you see the asymptotic cycle. In particular you can apply this to first return cycles because in, so suppose that your cycle is the first return cycle to a transversal. Then if we choose the transversal in direction of contracting filiation and, well, this is by definition the leaf of the expanding filiation, when we apply our Anosov map, what happens is that our transversal follows itself but gets contracted and the piece of filiation gets much longer. So we'll get first return cycle under the map or iterations of this Pseudonosov deferomorphism. What we get is, we get sequence of first return cycles to shorter and shorter transversals. That is, we get exactly the asymptotic cycle. Well, I'm cheating just a little bit but morally everything is absolutely correct here and now I have to make a, I have to state a theorem saying what is the behavior in the most general situation. So take now any directional or, sorry, almost any directional filiation on almost any flat surface and consider the sequence of first return cycles. They behave exactly as iterations of some intergevector, well, sort of, they mimic the behavior or iterations of some intergevector under Pseudonosov map. So now we have many dimensions, not two as before, so we'll get this asymptotic direction which was already discussed but we'll also get further asymptotic direction so if we consider the projection of all these vectors to a screen which is orthogonal to the asymptotic one so if we do not want to see the asymptotic direction but only the deviations we'll see that all these points would get aligned and in the case of Pseudonosov map it is sort of obvious because we have the top eigenvector but we have next eigenvector and the corresponding plane would be spent by the top eigenvector and the second one and then if we consider the screen which is now orthogonal to the plane we'll get similar picture and so on. So it's like having decomposition of an image of a large iterate of a matrix with respect to eigenvectors. So here's the formal statement of the theorem there for almost any surface and there is well in the vertical direction there is a flag of subspaces which was presented on the previous slide with these properties and what is the essential part of this theorem is that the corresponding numbers which are on the previous slide which were responsible for this deviation so here this deviation is already the length power lambda 2 which is smaller than 1 this deviation is length of the cycle power lambda 3 which is smaller than lambda 2 and so on in the case of Pseudonosov map these numbers are just the logarithms of corresponding eigenvalues in most general situation that's the Lyapunov exponents of the Hodge bundle along the Tecmeler geodesic flow and the corresponding connected component of the stratum now Tecmeler geodesic flow I would not introduce Tecmeler metric or whatever we have seen that we have a group action on our families of flat surfaces so by definition for this lecture the Tecmeler flow is the diagonal one e power t e power minus t on the diagonal so this is the definition of the flow now and the Lyapunov yeah and this is a very important remark so the theorem got this form only after the work of Giovanni Forni who proved that the Lyapunov exponent number g for the stratum is positive and later work of Arthur Avela and Marcelo Viana who proved the spectrum is simple so before it was sort of conditional theorem where the inclusions were non strict and to have all the strict inclusions one has to have this property which are two difficult theorems proved later now what are the Lyapunov exponents in this particular situation so we have this families of flat surfaces and in each family of flat surface for each family of flat surface we have group action the action of SL2R and we have a flow which is the action of a diagonal subgroup excellent so we have a base space on which we have dynamics and ergodic flow now when you have a flat surface you can consider its homology and taking for every point of our family of flat surfaces the vector space of homology with for example complex coefficients we get a vector bundle on this space this vector bundle is not trivial it's not a direct product because when you follow some closed line in the family of flat surfaces you come back with some twist as we have seen we have considered closed oocycle or closed geodesic we get a deformorphism of the surface so when we come back our vector our bundle twists and we get a co-cycle so every time when you have a vector bundle endowed with a flat connection and a flow on the base you get a co-cycle which is the natural co-cycle which is responsible for the mean monogamy of the bundle of the connection in direction of the flow so the Lyapunov exponents of this vector bundle excellent now I arrived to I still have two or three minutes and I arrived to the conference part of the talk so I want to report several recent results one of them is that there is a formula for the sums of Lyapunov exponents and it contains two parts so one is just this multiplicities of conical singularities the integer numbers which are responsible for geometry of the flat surface so this is really the sort of a trivial part of the formula and the other ingredient is Siegelwitz constant which is one and the same for every family of flat surfaces and what is great is that one can compute this for example for strata one can compute the Siegelwitz constant in terms of volumes of the strata so this is the two theorems are difficult theorems so this one took us about 15 years of work to prove so I use it as a black box and now the most striking result of last years is the theorem of Eskin-Mirzachane and Muhammadi which is sort of analog of Ratner's theorem in this situation I should say why it is so striking so genus the space of flat torre which appeared in Jens and Andreas series so this for genus one the corresponding modular space is a homogeneous space but starting from genus two the corresponding families of flat surfaces are not homogeneous space it is known that these spaces are not homogeneous so a priori there was no reason to expect that such a miracle as the fact that all orbit closures and all invariant measures are extremely nice exactly in this situation of homogeneous spaces it was absolutely over-optimistic and still it is true and finally it is proved it is very complicated theorem and the proof is really difficult but it immediately produces wonderful results and well there are effective methods of construction of orbit closures which are under consideration and also one of the colors of this theorem is that for any given flat surface almost all vertical directions define a Lyapunov generic point in the orbit closure and this shows how one can sort of approach in a sense at least conceptually all kind of wind trees as soon as all the angles are rational and as soon as my unfolding construction produces a surface of finite genus so first you start, you pass from a billiard to a flat surface then you take a magic wand of Askin, Mirzahari and Muhammadi you touch your flat surface you apply the action of vessel to R something happens and you get a complicated complex manifold of some dimension orbit fold but which is actually extremely nice and this is the miraculous part of the story it is under construction there is no classification of SL2R invariant manifolds yet but in some cases it is possible to find the orbit closure excellent then you compute the corresponding Lyapunov exponents and in a sense you are done in the case of wind tree or in the case of this generalized wind tree which I suggested everything can be reduced to genus once so the formula for the sum of Lyapunov exponents is reduced in a sense to a formula for a single Lyapunov exponents which is computable because we know the volumes of corresponding moduli spaces so all this is sort of quite a technology but still it is at least conceptually it is clear technology how to produce things and I want to finish with several challenges and open directions and now when we have some fantastic theorem new horizons appear so one major question is to study all GL invariant suborbials all these flat surfaces this question is solved only in genus 2 by Kurt McMullen in 12 years ago and there are indications that the life is really interesting there are known examples which are sort of miraculous there is no understanding what is the geometric origin of these examples now one more question is basically differential geometry and it goes sort of backwards this dynamics and Lyapunov exponents some universal properties with respect to some universal estimates for the curvature of holomorphic sub bundles of the Hodge bundle and on the one hand and on the other hand it gives a possibility not to compute exactly but at least to estimate Lyapunov exponents this is also wonderful now one more very interesting direction is to study what happens when genus tends to infinity and there are many indications telling that many quantities sort of stabilize and do not depend on particular shape of the flat surface or particular collection of conical singularities as soon as your surface is sufficiently complicated and not too symmetric it seems like all of them are in a sense the same and maybe this is an approach to Win-Tree Billiard with irrational angle and so on it would be very interesting to relate this Siegelwitz constant to intersection theory and to find the answer for the expression for it in terms of complex algebraic geometry and also here I considered families of Riemann surfaces but one can try to consider families of complex surfaces of complex three manifolds and they are corresponding moduli spaces there is a group action so some things can be done and it would be very interesting to recognize and the corresponding dynamical systems renormalization systems of something which is analogous to billiards so this is to my mind a very cute question thank you very much for your attention