 First, a quick recap of the first two days. So the first day, we defined translation services and gave examples. And the two main definitions are as a Riemann surface with a holomorphic one form, a.k.a. an abelian differential, or a collection of polygons with edge identifications up to cut and paste and translate equivalence. And then yesterday, we talked about the modular spaces, which are called strata. And we use this notation h kappa, where kappa is a partition of 2g minus 2, indicating the cone angles of the surface. And we also talked about the gl2r action, which is obtained by linearly acting on the polygons defining the surface. And we discussed things such as that a surface can have big stabilizer. So you can apply a big matrix, get distorted polygons, and then cut them up and get back to the same surface. And also, given that, you should not be surprised that you can cut them up and get very close to the same surface, but not exactly equal to it. And we also discussed stabilizers a bit more in general. We saw that the stabilizer was always a discrete subgroup of sl2r, but never co-compact. And the reason it wasn't co-compact is the sl2r action can't ever be compact. Strata are unbounded. Even the unit area locus is unbounded. OK. So finally, I'm going to start talking a little bit about why we should care. So you know what I'm talking about. The purpose of this lecture series is to tell you about the gl2r action on Strata. But why should you care? And roughly, I'll tell you three reasons. So the first is the connection to Teichmuller dynamics, or the Teichmuller metric on MG, which I'll remind you. MG is the modulized base of genus G-winged surfaces. It's also a sort of beautiful complex orbifold. There's also a connection to billiards, and all this right, et cetera. There are a number of connections to other elementary problems. And then maybe I'll close by saying new developments, connecting it to even more things. And I'll comment on that later. So first, I want to talk about the Teichmuller metric on MG, because I imagine that a number of you don't know what the Teichmuller metric on MG is. Scott is surprised. I guess, you know what the Teichmuller metric is? OK. So the Teichmuller metric answers a very simple question. What does it mean for two Riemann surfaces, X, NY, and MG, to be close to each other, to be almost equal? OK. Well, as does any metric, it tells you how close points are. But this one, in particular, is framing this question from the point of view of classical complex analysis. So you could remind yourself, what does it mean to be equal? So if X equals Y, that means that there is a conformal map from X to Y. So MG is the modular space of genus G Riemann surfaces. Yeah, it's different from flat surfaces. So I'll just remind you of the relationship. You have the Hodge bundle, which is a vector bundle over MG. And it consists of all pairs X, omega, such that X is in MG, and omega an abelian differential on X. And it maps to MG just by forgetting the abelian differential. And this space is stratified according to the number of zeros of omega. And the strata are our basic interest. So all of our strata live above MG. Other questions? OK, so two Riemann surfaces are equal if there's a conformal map. They do. In a stratum, they have the same genus. When you have the same, so the stratum specifies the orders of the cone points. And by Gauss-Bonne, you can compute the genus for that. You think about one stratum at a time. So the genus has always been fixed. And this whole business is about how do you can with how does the complex. So strata are in fixed genus. You could study the whole Hodge bundle. At least if you remove the zero section, there is a geotour action. But it's strata, for example, that have an atlas of charts to CN with integer matrices as transition functions. And if you look at the whole Hodge bundle, then there's no such atlas of charts. Yeah, you can view it as acting on the complement of the zero section to the Hodge bundle. It does preserve strata. It's more or less obvious. Yeah, if you act, then the number of cone points is preserved at the angle that each cone point is preserved. Which strata you're in is given just by the common. Yeah. Yeah, like for example, you can think like a zero of order one locally looks like this picture. So this is a four pi singularity. And then so you just sort of, this is just a little neighborhood. So you can imagine acting on this picture by a two by two matrix. And it'll sort of move the lines and stretch out the circles, but you'll still see that around this point you have four pi angle. So the strata are preserved by the geotour action. You mean a matrix with zero determinant? You can't act. You could take the polygons defining it and say act by a matrix like 1, 0, 0, 0, which would sort of project the polygons to the x-axis. What Riemann surface is that interval on the x-axis? It sort of has no meaning. OK, so I want to emphasize the, so I said if x equals y, there's a conformal map between them. I want to emphasize the perspective I'm taking, which is a conformal map is one that preserves angles. I'm going to think of it that way instead of as a bihologomorphism. And another way of thinking of that is it takes infinitesimal circles to infinitesimal circles. df maps circles to circles. So df is a map from, you know, I take the tangent space at a point of x and map by df at that point to the tangent space at f of p of y, and it sends a circle to some other circle, possibly of a different size. Conformal maps that are linear are just scaling and rotation. OK, so this gives us a good idea of how to define close before thinking about it from this perspective. So there's no conformal map, but maybe you want a nearly conformal map. So there's no conformal map, but between every two Riemann surfaces, there are loads of diffeomorphisms. And if I have a differentiable map to y, takes infinitesimal circles to ellipses. So here the picture is I have the tangent space at p of x, and I map via df at p to the tangent space at f of p of y. And I look at a circle. And so now the derivative is not conformal, but it's just some linear map, and linear maps map circles to ellipses. So I get something like that. And the sort of fundamental invariance of ellipse are you have the major and minor axes. And I'll denote the sizes of them like that. And how do you tell if an ellipse is a circle? Well, you should look at the ratio of those. It's easy to measure the failure of an ellipse to be a circle. It's just capital R over little r. So you define the dilatation. So I'll denote it kf. And it'll be the max over p of capital R over little r. So for every point, I get a capital R and a little r. And this measures the failure of this map to be conformal in sort of a supernorm type way. And now I can define the Teichmeler metric essentially just by asking I'll look at all possible differential maps and how close can I make it to conformal in this measure. So the Teichmeler distance from x to y. So I'm going to look at an inf over f from x to y. So I might want to do this. What's wrong with this definition? So if I did this, what's the distance from x to itself? One. Because a conformal map, this dilatation is 1. So there's that problem. There's also the problem that this won't satisfy the triangle inequality. Because when you can pose maps, you can check that this dilatation is submultiplicative. But so there's a problem. If you have something that's submultiplicative and you want to make it sub-additive and you also want 1 to become 0, you just take log. So that's the Teichmeler metric. And it's not obvious it's a metric. It's very easy to see it's a pseudo metric. The triangle inequality is just linear algebra. You just have to check that if you have two linear maps, the dilatation is submultiplicative. And you can give that to your linear algebra students as an exercise. But no, it's not obvious that if x does not equal to y that this couldn't be 0. You don't have a conformal map, but maybe you have a sequence of maps that are closer and closer to conformal. So that's one of the basic things you have to prove when you develop the theory. OK, so is the definition clear? OK, so basic things I can tell you sort of facts about this. So one fact is that it's a Finzler metric. What a Finzler metric is, it's very much like a Ramonian metric. The distance between two points is you take the info over all paths of the integral of the norm of the derivative, but the norm doesn't come from an inner product. So it's Finzler, but not Ramonian. But it's still a beautiful metric. It's Ramonian if and only if g equals 1. Other than that, you can show that the unit ball isn't even smooth. So it's geodesically complete. So more than that, there's a unique geodesic between every pair of points. And you can extend the geodesics for all time. And the reason I'm talking about it is because it's very related to the action of e to the t 0 0 e to the minus t. Sorry, I'm not sure you can read that. Maybe I'll move that over here. gt plus e to the t 0 0 e to the minus t orbits in strata project to Teichner geodesics. So the gt orbits in the stratum project to geodesics. And actually, you get all geodesics in this way up to the issue that you have to use quadratic differentials as well as abelian differentials. Sorry? Oh, I mean, I'm just saying the geotour action acts on strata or on the hard bundle. So the way of getting all geodesics through x is you take all ways of presenting x using polygons, and then you look at the gt orbits of those polygons. And those determine a one parameter family of remand surfaces. Yeah, yeah. Even just using abelian differentials, the fibers of the hudge bundle are dimension g. But actually, you have to use quadratic differentials, which includes squares of abelian differentials. Do you think any one of those are geodesics? Yep. Polygons? Yep. And that's how you get all the geodesics. And this is quite amazing. It doesn't happen often in nature that you'd find a metric and you actually know the geodesics in any meaningful sense. Even in remandian geometry, geodesics are solutions to ODE, just forget about it. But here, I mean, here it's not even remandian. It's Finsler, and yet still we know geodesics. So that's a pretty special property. And so this is one of the reasons we care about the geotour action. But so this is just gt. What about the rest of geotour? So geotour, really I should be putting a plus for positive determinant. Geotour orbits project to what are called complex geodesics. These are also known as teichner disks. So with a complex geodesic, it's a map from the upper half plane to mg that's holomorphic and isometric. And these complex geodesics are one of the main ways that we study the geometry of mg and the universal cover of mg teichner space. And finally, since we have complex geometers in the room, I should tell you that the teichner metric is the Kobayashi metric, which means it's the largest metric so that maps from the hyperbolic plane into mg are distance not increasing. And that's actually sort of amazing, because the definition very much uses this moduli interpretation of mg. On mg, and nor will I, but I can after the talk, but there is a complex structure on mg. There's obviously a complex structure on strata because of the charts to CN with transition functions in GLNZ, the much stronger structure present there. So it's sort of amazing. You can take mg. You define this metric using the Riemann surfaces and parametrizes. Then you can forget the moduli problem, forget all about Riemann surfaces. Say, oh, I just have this complex orbifold mg. And miraculously, you can recover the metric just using complex geometry. So it's somehow a very important metric from different points of view. So this is what I had to say about the Teichmeler metric. It's one of the two or three most important metrics on mg. The other one's being Baye-Petersen. And now people are also studying a lot that thirst an asymmetric metric, which is not a metric. So maybe I don't have to include that. And it's sort of really one of the fundamental ways we study mg. And it's like the same thing as the GL2R action. So the GT action is Teichmeler-G as it flow. Yep? Mm-hmm. Nope. You'll get different ones, right? Because if you take it in different directions, you get different Geodesics. But I mean truly distinct. If the polygons differ by cut and paste, I would say that's the same. It's the same differential on the surface. But if you take two actually different quadratic differentials or veiling differentials and then you flow, then you really go in different directions. Are there questions? Yeah, so maybe just to complete the discussion, I should be honest. So as I said, really, this discussion is most natural if you talk not about the veiling differentials but the quadratic differentials, which are things that locally look like f of z dz squared in the holomorphic coordinates instead of f of z dz. Or algebraic geometry, they're global sections of the square of the canonical bundle. And so GL2R acts on this. And so does GT. And this is the cotangent bundle is to put this in context. Extension of the Geodesic flow for the description of. Yeah, so usually you talk about Geodesic flow in the unit tangent bundle or on the tangent bundle to a manifold. And here it's more natural to talk about it on the cotangent bundle, which frankly confuses everybody when they learn the subject. That's one of the subtleties. I sort of advise you just to ignore that subtlety when you're first learning about it right now. You can sort of think of it as up to some weird twist in the subject. This is basically the tangent bundle. And so this is Geodesic flow on the tangent. Well, this isn't Riemannian. You talk more about Hamiltonian flows for the vape-Pederson metric, which is a Kaler metric. This metric not even being Riemannian is definitely not Kaler. Other questions? OK. So now I want to move on to the next motivation, which is billiards. And really, you shouldn't think of this as one motivation. You should think I've sort of picked one from a basket of similar motivations, where there are some elementary problem that it turns out you can study. So I mean billiards and polygons. People also study billiards in shapes with C1 boundary. That's like a very different subject. And so a ball moves around in a room. It bounces off the wall, so angle of reflection equals angle of incidence. If it ever hits a corner, just stop it. And you're interested in dynamics, which means the long-time behavior. If you start the ball rolling and you leave for a long time, what sort of thing do you expect when you come back? And there's a trick that people use. So first of all, I should say the polygonal billiards is almost intractable in general. Although I have to preface that being at the University of Maryland, which has one of the few people brave enough to study general polygonal billiards, which is Giovanni Forney, who's not here this week. But with very few exceptions, I mean, general polygonal billiards is sort of terrifying. It's very hard to figure out. For example, we don't know if every triangle has a periodic billiard trajectory. That's on the list of the five most resilient problems in dynamics. So you can have an annals paper. If you sketch a quick proof that every triangle just has a path, where angle of incidence equals angle of reflection, that repeats itself. Although if you just do it in right-angle triangles, that's just a high school exercise. So general billiards is hard. There's a class of billiards which we know a lot about, and that's rational billiards, which I'll remind you from the first day when we talked about unfolding. And rational billiards are the ones where all of the angles are rational multiples of pi. And the reason for that is this unfolding construction, which I'll now explain what motivates the unfolding construction. And the basic trick is instead of bouncing the ball, you instead reflect the polygon and allow the ball to continue straight into the reflected copy. When you do this, it's sort of a fundamental observation that the new path is the reflection of the old path. So you've reflected the polygon, and you've reflected sort of the continuation of the path. And in this way, billiard paths unfold. So unfold being this procedure of letting it continue straight instead of bouncing to straight lines on the unfolding of the polygon. So I'll remind you, the unfolding of the polygon was what we got when we sort of reflected in every edge, and we kept going until we had everything up to translation, and we could close it up. So when you do this trick, which is very old, you end up studying sort of straight lines on translation surfaces instead of billiard paths and polygons. And if I want, I can rotate the surface to assume the straight line is vertical. So just for clarity, I'll say the vertical straight lines. OK, so maybe this looks like progress because maybe straight lines seem easier than lines with many corners that bounce around. However, the real reason that this is progress is because there's a renormalization operator for the dynamics in this context. And that's the following. So I take my translation surface. This is the unfolding of some polygon. And now I have sort of a very long vertical straight line on it. I mean, maybe it's infinite, but let's just imagine I sort of truncated. So I have a very long segment of a vertical straight line. There's something I can do to make that shorter and hence more manageable. A long line is confusing. A short line is very clear what's going on. And that's that I can apply the same matrix gt, just e to the t 0 0 e to the minus t, which contracts the vertical direction by e to the t. So if I start with something, a vertical line of length l, well, so what I get after this procedure is I get a new surface with a vertical line of length e to the minus t times l, so on a new surface gtx. And this is called renormalization. It's a big theme in dynamics and physics. I start with a long orbit of one dynamical system, the straight line flow on this translation surface. And I get a shorter orbit on a different dynamical system. And the idea is you do this over and over again to understand infinite vertical lines. So this is a big theme. And when you have renormalization, generally, you're very, very happy. And you can prove a lot of things. So just as a sample, what can you prove about this? Well, let's just think intuitively. At least to me, it seems intuitive that this should be most useful as long as somehow this surface isn't degenerating. If the new surface is becoming very weird, there's sort of like a trade-off. Like, oh, it's short, but the surface is very degenerated and hence maybe harder to understand. But if the surface isn't degenerating, then it just seems like a total win. The new surface is just as good as the old surface, but the segment is much shorter. You've lost nothing and you've gained everything. And that intuition is correct. This is Meyser's criterion. So before we had Meyser's criterion for compactness, this is Meyser's criterion for unique or goodicity. And it says the following. If gt x omega, so let's say for t bigger than t, intersects a fixed compact set k in the stratum for arbitrarily large t, then all vertical lines in x omega are equidistributed. Or if your dynamic is the vertical straight line flow is uniquely ergodic. What this means is if you look at a vertical line, not only is it dense in the surface, but if you pick any chunk of the surface of Meyser a half, that vertical line will spend a half of its time. So it's sort of a Meyser theoretic upgrading of density. And this is something that very, very rarely happens, that you can understand every orbit, every vertical straight line, not just almost every, but every. This is for any translation surface. So in particular, the translation surfaces you get from unfolding rational polygons. So this is from the 80s. And it was used to great effect to show that the typical translation surface has this property. But it's always been a thorn in the side. The set of translation surfaces coming from unfoldings of polygons are Meyser 0, because they're so symmetric. They're not at all typical. And so this on its own doesn't actually tell you anything whatsoever about rational billiards. Yeah, so it's difficult to check. But it's been done. So you can show, so for example, so by the way, you might just say the gt orbit is recurrent. It doesn't go off to infinity. That's what this is saying. It's sort of coming back to some bounded set infinitely many times. Later we'll show that the gt, or I'll at least vaguely discuss, that the gt action is ergodic, which implies almost every gt orbit is equidistributed and in particular satisfies this. So this is useful. But you need some sort of machinery to try to show that you can ever apply this. OK. So again, this is sort of motivating gt. But again, there is also motivation for studying gl2r. And the difference is when you care about more than one direction on a translation surface. So when interested in more than one direction, need to study gl2r. So you need to be able to sort of rotate to make a direction vertical and apply gt. And what are some examples of this? So one of the big successes was for every x omega, if you pick a direction at random, it satisfies this property that all lines in that direction are equidistributed. So if you take any rational polygon and you start at a random direction and you bounce the ball, it'll equidistribute in the polygon. So that's something where it's not just about a single direction, you need to know all directions. So it's somehow about the gl2r orbit. Another problem that gets studied a lot is how many periodic trajectories are there. So just trajectories that close up. So as I said, in the irrational case, we don't know if there are any. In the rational case, we know that grows quadratically. And it's proved by studying gl2r orbits. And the constant in front of the quadratic term depends on the gl2r orbit closure in the length. So if I look at the number of periodic trajectories in a polygon or in a rational polygon or in a translation surface of length less than or equal to l, this should grow like cl squared. It's just the length. I mean, if I have a periodic trajectory, it's like a bunch of, yeah, it's just a length of the trajectory. This is sort of like counting closed geodesics. You've always got to lift upstairs. It's been one of the major challenges of the field to say anything non-obvious about rational billiards because their unfoldings are very special points in the modular space. And so the only thing we know about them is things we know about every single translation surface with some notable exceptions. OK. So as I said, this constant depends on the gl2r orbit closure. So you take the gl2r orbit, and then it's closure in the stratum. It turns out that's what it depends on. So I should say this is an ongoing story. We actually still don't know exact asymptotics for every surface. That's sort of the holy grail. But we know it for almost every surface, and we know it for every surface with some extras as our averaging. You could sort of give a whole talk just about that sort of question. It's related to the volume of the gl2r orbit closure compared to the volume of sort of the part of the gl2r orbit closure that lies near infinity. It's related, in some way, to some intersection theory on MG, which was some big surprise, one of Konsevich's insights. OK. So I want to give you one other sort of theorem from the 80s that's also sort of motivational. So this is called the Veatch dichotomy. And it says, if gl2r times x omega, so if the gl2r orbit is closed. So in a minute, I'll tell you typically gl2 orbits are dense. But if it's closed, then in every direction on the surface, so depending on the direction, one of two things happens. Either every line in that direction is equidistributed, or every line in that direction is periodic. And this is what happens for the flat torus. Example, C mod z o joint i dz, which is just this flat torus. So for the flat torus, the actual, the whole modulite space is gl2r mod gl2z. So the whole modulite space is a closed orbit. So it certainly satisfies that. But it's actually quite easy to see this on its own. If you look in a rational direction, every line will close up. And if you look in an irrational direction, it's one of the sort of first exercises in dynamics or harmonic analysis to show that every line is equidistributed. So any questions about the motivation? Interval exchange transformations is a big one. That's where you have an interval. You cut it up into pieces, and then you reorder the pieces to obtain a discontinuous map from the interval to itself. So it sounds very easy, but this is one of the basic things you study in dynamics of a sort of low, not very chaotic dynamical system. And it's basically the same thing as studying just the GT part of the gl2r action, and not the gl2r. And there are sort of more and more examples that are known that I may be not going to talk about. There's something called the Ehrenfest wind tree model, where you have an infinite grid of trees and a single particle of wind bouncing around. And you want to know, what happens to the wind? Like, does it sort of go off to infinity? If so, how quickly? There, the big surprise was that it doesn't at all behave like a random walk. And it exhibits so-called anomalous diffusion. It sort of wanders away at not the same speed at all that the random walk would. So the next thing I want to talk about is the dynamics of just the GT action, which is e to the t0, 0, e to the minus t. OK. So given x omega, remember we have these coordinates, which are come from the edges of the polygons or integrals of omega over relative periods. So write coordinates as, so I have sort of xj plus i, yj, equals 1 to n in CN. So I'm going to write them in terms of the real and imaginary parts. But I don't actually want to write it like this. I want to write it as this, x1, y1, x2, as a 2 by n matrix. Yeah. So x omega is a translation surface. Yeah, x omega is a translation surface here. Yeah, no, x omega is a translation surface here. x omega always denotes a translation surface. Remember, by definition, a translation surface is any x omega where omega is not 0. So there are two ways. Let me remind you about the coordinates. So there are two ways to define the coordinates. So we discussed this in a special case of like an octagon where we had v1, v2, v3, and v4 that gave local coordinates in c4. And what I'm saying is you should write each vj as xj plus i, yj. So in other words, xj is the real part of the jth coordinate, and yj is the imaginary part. And the other way of thinking about these coordinates was I had gamma 1, gamma 2, gamma 3, and gamma 4. And vj was the integral of gamma j. Other questions? So I want to write them like this because this action comes from the action of gl2r on r2. So really, instead of c, I'm essentially writing r2 now. So I have a bunch of vectors in r2. And I know how to act on r2 by a matrix. So if I have just one vector, I have a 2 by 2 matrix. Let's say the 2 by 2 matrix is e to the t0, e to the minus t. So if I have just one vector, it's very clear how to act. I act in the usual way, multiplication. So here, it would increase the x-coordinate by a factor of e to the t, and it would decrease the y-coordinate by a factor of e to the minus t. So in fact, in these coordinates, the action is just given by matrix multiplication. This just acts by multiplying every column by this 2 by 2 matrix. Right. So right now, it seems beautifully, beautifully simple. And based on this picture, we could formulate a number of guesses. So let's sort of make a conjecture. If x omega and x prime omega prime, so these are going to be two surfaces that are nearby each other, have the same real parts of coordinates then gt x omega and gt x prime omega prime get close to each other exponentially fast. OK, so that's the perfect question for this exact moment. So I've forgotten something over here, which is eventually I'll leave the coordinate chart. And then I'll go into a new coordinate chart. Since I'm discussing this in coordinates rather than more abstract language, we have all this trouble of, well, which coordinate charts are we picking? But let's just say we magically picked some coordinate charts. And so really, I needed some change of coordinates. And the change of coordinates acts linearly on the periods. So it sort of acts by recombining the v's. So this is v1, this is v2. So maybe a new coordinate would be v1 plus v2. So the coordinate changes act by taking sort of linear combinations of the columns. So here, in fact, I have something that I'll denote a tx omega. So this is a matrix in gl and z. And it's the change of coordinate. Assuming we've magically come up with some way of picking coordinate charts. As I said, there is a more abstract framework, but I think it's better to see it in a concrete framework first. OK, and this causes a giant mess. So it's also called the Kinsevich CoCycle. It's a co-cycle in the same way that the derivative of a diffeomorphism is a co-cycle. It satisfies the chain rule. So it's pretty obvious this is the hard part. It's pretty easy to understand what this is doing. This is not so easy. This is essentially the cut and paste. Or rather, I would say it's the linearization of the cut and paste. Like you cut and paste your crazy shape into a more reasonable shape, and it has some effect on coordinates, which is given by this Kinsevich CoCycle matrix, and what's going on. And now we're sort of at bad news because it's not really so clear what's going on at all. If there's one thing that's clear, we have no idea how cutting paste works, not in any sort of elementary sense. So the study is not at all combinatorics. We can't keep track very well, except for maybe some very, very special cases of what sequence of cut and paste you should do to keep the shape looking reasonable. So how are we going to study this? So I'll explain that now, or at least start to explain it. But let me first tell you the moral, which this guy loses to e to the t0, e to the minus t. So it's not as strong. So for example, this conjecture is true, although the Kinsevich CoC puts up a fight. So you might expect the exponential rate is e to the minus t. That's obviously how, look, there's the e to the minus t. But if not e to the minus t, it's a slower order of exponential convergence. So this Kinsevich CoC really puts up a fight. But ultimately, it loses to the very strong e to the minus t here. Yep. And in fact, that's what I'm going to sketch a proof for you over the next, I don't think I'll finish it today, but that's the next topic. All right, so how does this work? How can we possibly study this, given that we can't understand the cut and paste required? It's not even clear. That's a super well-defined question. So you guys will be happy, because the answer is you should do some analysis. So really what's going on here, the more abstract point of view is I'm just taking some homology class. So this is the integral of gamma 1 omega. And I have this gt orbit, so maybe I'll draw a picture. So here's a gt orbit. And I maybe start here. That's my start point. And I have this basis of homology gamma 1 through gamma n. And the coordinates are the integrals of omega. And now I can parallel translate this basis along here. And so here, I'm sort of in the same maybe simply connected set here. So here I have the parallel of this basis gamma i. So the parallel translate is you have this gt, and you just drag the homology class along. You're sort of changing the flat metric, but the topology isn't changing. So you could just drag the homology class along. And so really, you just need to know the transition function from this basis, from the new basis to the old basis. And you're interested in how large is this matrix. So how large is the change of basis matrix, which is exactly this matrix, at. And the way they'll study this is we'll fix some sort of family of norms on homology. And we'll just ask, how quickly does gt does the parallel translate of some fixed homology class grow as I drag it along here? Is the parallel translate of some fixed gamma? So how quickly does the parallel translate grow along the orbit gtx omega? Oh, it just means so I'm changing the metric, and the homology class comes along for the ride. gt. Yep, it's a different metric on the homology. I mean, the homology is sort of a locally constant. It's a vector bundle. It's, in fact, a flat vector bundle. Because it's locally trivial, right? You change the flat metric, homology's still homology. You haven't changed the topology of your surface. But I'm going to choose some continuously varying family of metrics on homology that changes with the x omega. And then I'm just going to ask, so I have this one homology class, and I have this path in the stratum. And I sort of sort of this flat bundle, and I just sort of parallel translate this vector along in this flat bundle. And I have this family of metrics, and I want to know how big does this section. Yeah, I'm going to put a metric on each fiber of the bundle. And from the dynamical point of view, it doesn't really matter what sort of metric I pick. Because the bundle's locally trivial. It's not trivial. It's really only locally trivial. And that's the problem, because we're going to see growth. If it was the trivial bundle, I'd come back and I'd get what I started with. And this matrix AT would be the identity, but it's not. So you have to be a little bit careful. And in fact, so one thing you want is you want your orbit to come back to itself. Or else it's hard to make sense of the question. But then, no, it doesn't really matter what coordinate you pick for the starting neighborhood of the point, as long as you're coming back. The exponential rate won't depend on which coordinate you're picking. No, so the condition that the x-eyes are all the same does not depend on which coordinate you use. And that's because you use a different coordinate. You're just applying some matrix here. You're taking some in your combination. So it's like if I have v1 and v2, and v1 prime and v2 prime, and vi and vi prime have the same real part, then v1 plus v2 has the same real part as v1 prime plus v2 prime. So it's sort of just linear algebras, as it doesn't matter what coordinate chart you pick. So there are a few bad ones. As I said, you want ones that come back to the compact set a lot. And I'm going to sort of brush that under the rug. But it'll sort of be implicit to the discussion I want. Gdsics that come back a lot to the compact set. You're going to get this estimate for this. I'm going to get A estimate for any homology class. But as I'm about to discuss, there are two very special homology classes, namely to switch to the dual point of co-homology for a second. There are two very special co-homology classes, namely the real part of omega and the imaginary part of omega. So one of those will actually have exponential growth rate e to the t. But the theorem will be that aside from that, everything else has a smaller exponential growth rate as long as you're sort of spending enough time in a compact set. So in particular, if you want, this will apply to almost every x and then any x prime omega prime nearby. And if you're a dynamicist, you know that what I'm proving is that this gt is hyperbolic, non-uniformly hyperbolic. OK, so I think I'm about out of time. But what I'm going to do next time is I'm going to just review this setup. And I'll tell you what the family of norms is that you want. And it's going to be called the hodge norm. It's very basic and important in hodge theory and in the theory of Riemann surfaces. And what's beautiful about that is, although we have a dynamical question, we only care about the long-term behavior. With this metric, you understand what's happening infinitesimally. So I'll state an estimate for the derivative of the hodge norm. And integrating that estimate will give the estimate you need for the long-term growth of the hodge norm. OK, that's all for today. Thanks.