 OK, so today is going to be a little bit less like a mini course talk and a little bit more like a colloquium talk in that I'm going to try to survey some recent developments for you. But I'll start off by reminding you where we were last time. So last time, we discussed the theorem of Eskin-Mozikani-Mohammadi that says that gl2r orbit closures of translation surfaces manifolds linear equations in local coordinates. OK, and just a few remarks on this that I didn't have time to make last time, so remarks. 1, the coefficients must be real and the constant term 0. So I guess it's a consequence of this theorem, but this is sort of trivial. There's an exercise to do, which is take any subset of c to the n defined by linear equations that's invariant under gl2r acting diagonally on each copy. Well, then all of the linear equations have real coefficients or can be assumed to have real coefficients. And they're homogeneous. They have zero constant term. And then that's equivalent to just a sort of trivial statement about gl2r acting on c to the n. Another remark, just a trivial one. So these have a complex structure. They're complex manifolds or complex orbitals because the solution is to real linear equations in these complex local coordinates. And the complex dimension is always at least 2. And that's pretty easy. That's because gl2r has real dimension 4. So gl2r orbits on their own locally already have complex dimension 2. So you can't get any smaller than that. The case of dimension exactly equal to 2 is the case of closed gl2r orbits, which we discussed a little bit. And so the next statement is examples come in what might be called commensurability classes related by branch covering constructions, which is to say if you have one gl2r orbit closure and you look at all surfaces that you obtain as whatever degree 11 covers of translation surfaces in that orbit closure, you'll get a new gl2r orbit closure. But in each commensurability class, there's sort of a unique primitive representative in sort of the smallest genus. And so we're still at the stage where we don't understand these things super well. So generally, I'll consider things that are commensurable to be the same, from my current point of view, far away. There is, of course, interesting combinatorics in terms of how you set up these covers. And we did an example of that last time. Recall, we thought about 4 to 1 covers of the torus. So in fact, I would say this orbit closure is commensurable to the whole modular space of genus 1 surfaces. So there are some trivial orbit closures, which are strata, so whole modulite spaces, and they're commensurability classes. So a whole stratum is obviously a gl2r invariant manifold. It's an orbit closure, too, because the action is ergodic. And here you can also take strata of quadratic differentials. And then you look at sort of the double covers of them that are abelian differentials. And I would say that also would give sort of a trivial orbit closure. Although, again, there can be interesting combinatorics. OK, so a final remark. The coefficients of these linear equations can be taken in q bar. They can be real, so q bar intersect r. So they have to have algebraic coefficients. I mean, you can always take a linear equation and multiply it by pi, or i times pi. But the point is, you can find an equivalent set of real linear equations with coefficients that are algebraic numbers. And in particular, that implies only countably orbit closures, because there are only countably many algebraic numbers, and so there are only countably many linear equations defined by algebraic numbers. So there's some countable set of orbit closures. So that follows from my result that the coefficients can be taken to be in q bar. But it was actually, there was a more complicated proof of that that was part of this. So because there are only countably many, you can hope to make a list. So that's what I wanted to say about Eskin-Mirza-Khani Muhammadi. So as I said today, mostly I'm going to just tell you about results, but I want to explain one more thing in detail. I want to give you a little primer on the dynamics of the one-parameter subgroup, ut01, to give you a little bit of a feel for it before we get to the more advanced results. So sometimes this is called typemolar horizontal flow in the same way that e to the t00, e to the minus t, was called typemolar geodesic flow. So there are a few things I can tell you about this. One is if x omega is not the union of horizontal cylinders, ut x omega is unbounded. The ut orbit is unbounded in the stratum. First of all, this condition about cylinders, so let's go back to one of our favorite examples, the regular octagon with opposite sides identified. So here in the middle, I see a rectangle with two opposite sides identified. That's a horizontal cylinder. OK? So all of these horizontal trajectories, all of these horizontal straight lines close up. There's actually another one, although it's a little bit less obvious. It's helpful if you think about maybe I'll start here and I'll flow in this direction. So that's identified down there. So I sort of flow along here. And that's identified there. And when I flow, I close up. So this whole thing, that's a second cylinder. It's also horizontal. And when you horicycle flow, so you're shearing this, what that is the effect of is essentially twisting the cylinders. And you see as you twist the cylinders, you're never really going to develop a short saddle connection. Because there's never going to be any short vertical saddle connection. Because the lengths of the verticals are not changing. And any other saddle connection would have to cross a cylinder. And the height is not changing as you shear it. So it's sort of very easy that if you're built out of horizontal cylinders, then your orbit is actually bounded. So let's just think about the other direction. Assume x omega is not horizontally periodic. And this will just be a sketch. So not horizontally periodic, sorry, I should stick to the same technology. I mean not the union of horizontal cylinders. So not all horizontal straight lines are periodic. You can get things of every genus that way, right? Yeah, you can get things in every stratum that are horizontally periodic. And it's also worth mentioning, not every surface has a horizontal cylinder at all. Indeed, this won't if I just rotated the right amount. But every surface has tons of cylinders, infinitely many. They get long and thin, so they're more like these thin ribbons that sort of go around and eventually close up. But it's a non-trivial theorem that every surface has infinitely many cylinders. So assume it's not horizontally periodic. So you can show that that's equivalent to there being some singularity of the metric where I can take a little horizontal line and paths from that don't close up. So what I'm going to do is I'm actually going to take a little rectangle, very thin, and I'm going to flow it until eventually it hits some other singularity of the metric. It might come back to the same one. It has to eventually do something because as I make this rectangle longer and longer, it eats up more area. But the surface has finite area. And the only thing I'm going to use is I'm going to use it's not horizontally periodic, so I can set this up so it doesn't come back exactly there. So OK if it comes back sort of overlapping, but not exactly. So you end up with a situation where you have this sort of long, thin rectangle, one singularity there, and one singularity there. And then in that rectangle, I can draw a saddle connection, which is just a straight line segment joining two zeros. So what I've shown is for all epsilon, there exists a saddle connection with sort of complex length x plus i, y, and y is less than epsilon, but y is not 0. And the fact that y is not 0 is the fact that this doesn't come back exactly to itself. If this 0 was right there, then that would be y is 0. So now I'm going to pick t so that 1t01xy is 0y. So then ut of x omega, so that just shares the saddle connection, and then it'll have a very tiny short saddle connection. So ut of x omega has a saddle connection, and I can do this for any epsilon. So that means the ut orbit is unbounded. An orbit is unbounded. A set is unbounded. If somewhere in that set, you can find a saddle connection that's arbitrarily small. That's the whole question. So it turns out, although it's not obvious, every geotool orbit closure will contain a surface like this. It's not related to primitivity. The generic thing in a geotool orbit somehow won't have any horizontal cylinders, even if you found one, you could rotate it, and then it would have cylinders in some other direction. Other questions? The reason I told you this is, well, a, it's relatively easy, and b, it connects the dynamics of ut, just the fact these orbits are unbounded, to cylinders on a surface. And so it'll make what I tell you next a little bit more plausible. So omega, if c is the collection of all cylinders in some direction, utc is the result of shearing those cylinders, then utc of x omega is in the gl to our orbit closure. OK, so I think this will become much more clear with an example. So let's build an example with, no, no, no, let me do an example. So this cylinder deformation is not just the gl to our action. If you'd like, it's somehow part of the gl to our action applied to just these cylinders, which sort of don't have to be the whole surface. So a good example is take the torus and pick an irrational direction and make a slit and glue in a cylinder. So here's the torus. I make a slit. If you'd like, I can sort of identify the two edges of that slit. And then it sort of looks like this. And then I glue in a cylinder into that. This cylinder doesn't have to be horizontal. It could be in any direction. And since this is an irrational direction, there's no other parallel cylinder here. So what this theorem is saying is it says I can shear this cylinder and it'll stay in the orbit closure. So shearing the cylinder looks like doing something like that. It's actually somehow very concrete. A cylinder somehow looks like this. And you should think of it. You have a cylinder. I'm going to build a set of two pieces of paper. So here's my cylinder. It lives in the surface. So there are some chunks of this that are glued into the rest of the surface. There might be more than one saddle connection. They're glued in somehow. So I'm going to shear it. That looks like this. It's still a cylinder, but somehow the relative position of these intervals to these intervals have changed. So if I did enough shearing, I'd sort of go around and come back to where I started. But in between, this is something. So I can sort of twist and then just glue it back in the same way to my surface as before. I have to find the notes for my talk. I'm going to use different paper. So you can see that unless the cylinders cover the whole surface, these deformations are not actually in the gl to our orbit. If I sheared the whole surface, so if I did this thing to the whole surface, that would be in the gl to our orbit. But I'm just doing it to part of it. Now if there are two cylinders, so maybe I have another cylinder here that's parallel. Sorry, it doesn't look exactly parallel. I have to do the same shearing to both of them. Because we have to shear all of the cylinders in the same direction. If the cylinders cover the surface in the given direction, then I'm shearing all of them. So I'm just shearing the whole surface. Then it's trivial. So utcx omega is in the gl to our orbit rather than the orbit closure, if and only if c covers, so the union of the cylinders is the whole thing. So if I did it here, I'd have to shear both horizontal cylinders the same amount. I'd just be shearing the whole surface. It wouldn't tell me anything I didn't know before. And by the way, it's easy to show that if you shear, you can also stretch. So I can make the cylinders longer. Although I'm going to have to do it by the same factor to every cylinder. So I can make all the cylinders twice as long in a given direction. So these are somehow geometric deformations. They stay in the orbit closure, but not in the orbit. Yeah, this theorem is proven using the dynamics of the horocycle flow, using something called quantitative recurrence. And it's inspired by what's now a theorem of Simeon Philippe, which is the fact that orbit closures are varieties. So somehow the fact that the structure at infinity is tractable leads you to suspect that this would be true. And somehow this theorem gives you a good hope of being able to compute some orbit closures. They are, but we don't know what manifolds there are. So the space is not bounded. So Gega does not tell you automatically that manifolds are varieties. So if I was inside of Pn, inside of projective space, then I would know by general principles that a complex submanifold had to be a variety. They're cut out by linear equations. In periods, periods are transcendental functions. I mean, periods are literally integrals of omega. Integrals in math are like the prototypical transcendental functions. Yeah, a non-algebraic change of coordinates to access the variety structure of Mg or the Hodge bundle. Now that being said, you strongly believe that orbit closures should be varieties because they have some volume form and they're finite volume manifolds. And if you know about Gega, you know that complex manifolds are varieties unless they have some sort of essential singularity at infinity, some sort of really crazy behavior. But finite volume things shouldn't have some essential singularity. That should somehow use up infinite volume. Real linear equations on complex coordinates. So it's as simple as, so maybe I take like C2 and I look at the equation Z1 plus 3Z2 equals 0. That defines a complex submanifold. Other questions? OK, so I was saying the cylinder deformation theorem sort of gives you hope that you can compute orbit closures. For example, it's actually a pretty easy exercise to show that this has full orbit closure. So its orbit closure is dense using the cylinder deformation theorem. You just have to find enough cylinder deformations. Like there's another cylinder, say, here. There's a cylinder. And maybe it's parallel to some other cylinder, but if you do some little deformation here, then it won't be parallel to another cylinder. So you can just the deformations in this, the deformations in this, and gl2r already give you a neighborhood of that point. So you can somehow hope to actually compute some orbit closures. I'm not exactly sure I understand the question. Yeah? Yeah? Yeah, I mean already this has proven using dynamics. And I'll give examples later of more things that are proven using dynamics. Yeah, it tells you a lot about the geometry of MG and about dynamics on Riemann surfaces. Sort of depends on what sort of question you're asking. If you're asking the right sort of question, it's equivalent to things on Riemann surfaces tend to be studied using the space of Riemann surfaces. And this tells you all sorts of things about the space of Riemann surfaces. So I want to give you a little bit of a sense of orbit closures from the point of view of four years ago, or about four years ago. So I'm being a little bit ahistorical because most of what I just talked about happened in the last four years. But now I'll fill in some of the history. Well, they're very different. If you just apply it to the cylinders, you stay in a bounded set in the same way that I explained that the UT orbits are bounded, if your surface is made out of cylinders. But if you apply UT to the whole surface, it'll typically be unbounded. Yes. That's essentially the proof involves thinking about the UT orbit of the whole surface, and seeing that accumulate on horizontally periodic surfaces, and then making some arguments near the horizontally periodic surfaces. So essentially, you produce this deformation at a point in the UT orbit closure of that surface, and then you parallel translate the deformation back. But I don't want to give the proof of this now. That would occupy the whole lecture or more. So I want to say what we know and what orbit closures we knew four years ago. So first of all, we knew the trivial ones. So everything commensurable to a whole stratum of abelian or quadratic differentials. Two, we knew the Veatch. Well, so maybe I'll do this one name at a time. So Veatch showed that the regular n-gon, or I should say 2n-gon, has closed orbit. And he did this via direct computation. He computed the stabilizer in SL2R and showed that it was a lattice. But this was a big surprise. It has miraculous consequences. His student Ward found a few more examples like this. And then decades later, Bau and Mohler, using a pretty sophisticated argument in algebraic geometry, extended this to a slightly bigger family. So this whole family gives finitely many closed orbits in each genus. So it's an infinite family, but it's only finitely many in each genus. I mean, it was an annals paper for Bau and Mohler to sort of extend the family from a one-parameter family to a two-parameter family. So three are what we now call the eigenformulose, which were discovered by McMullen. And some of them were discovered in a different guys by Calta. And again, this was pretty exciting at the time. These were two jams papers when they were discovered. So these are closed orbits and some bigger orbit closures, genus less than or equal to 5. The closed orbits are in genus less than or equal to 4. And then there are some bigger ones also. So these days, we understand these very well. And in fact, you can prove they exist with a page of linear algebra. But they're sort of weird and unexpected. Essentially, they exploit a genus two phenomenon. There's some miracle of linear algebra in genus two. You can try to make the same construction in higher genera, and it just doesn't work for real multiplication. So these are almost but not quite parametrized by Hilbert modular surfaces. And to be emphasized, all of these examples give rise to really interesting examples of MG, some of which that have already been studied and some of which that are studied as soon as you've found them for their own sake, even if they weren't yield to our invariant. These orbit closures tend to be independently interesting things, even if you don't care about dynamics, although I hope you do. And then there were two more closed orbits in genus 3 and 4. One each. So that was the whole list of everything known. Nothing non-trivial known in genus greater than 5, except for things commensurable to lower genus. Sorry? No, these ones aren't closed, but they're all actually fairly close to being closed. So now I'll jump back to the future. So all of these were discovered before Eskimer's Akane-Mohammadi, before the cylinder deformation theorem over there. But now jumping back to the present, for all of these, let me write it, the cylinder deformation theorem. So I'm going to call this the cylinder deformation theorem is trivial. And what I mean in particular is in any direction with a cylinder, the surface is covered entirely by parallel cylinders. So like here, I find this one horizontal cylinder, and oh, there's another horizontal cylinder, and the union is the whole thing. You'll find that there are infinitely many directions where there's a cylinder, and you'll find that in every single direction. It's pretty amazing. It's like an infinitely over-determined system. It's like a whatever-dimensional moduli space. So you expect to be able to come up with this in finally many directions. Although I mean I should clarify, the way you prove this for some of these orbit closures is using the cylinder deformation theorem. So maybe it's a little bit not accurate to say that the cylinder deformation theorem is trivial. But ultimately it is. No, that doesn't mean the orbit is closed. Let me give you a good example. So take this square, and now I want to think about a degree to cover of this square. But now I'm going to branch it over two points. So I can do this with the slit-torus construction if I do the same torus for both. Then that'll cover, it'll be a two-to-one branch cover, branched over two points, whereas before we had covers branched over one points when we were talking about this. So what does the orbit closure look like? I'll tell you, and maybe you'll believe me, but maybe not. I think you should believe me. If you sort of pick everything, if you pick the slit generically, the orbit closure looks like, first of all, you can take these two torus to be anything. And second of all, you take the slit to be anything. Because if you picked the two points to be, say, rational points, that wouldn't be the case. Then you would get a closed orbit. The points could come together in the orbit closure. So that's an example where somehow it covers a closed orbit. But because there are two branch points, it's not actually a closed orbit. So this one isn't primitive. These eigenform loci are mostly primitive. So it's not obvious they have this behavior, but still they have sort of a similar behavior to this example. So it's a good exercise to show if you take any surface with this property and look at any branch cover, it'll still have that property. So that consists of two parts. One, you should show if you have a cylinder and you map it here, then it'll map to a cylinder. This is, after all, a covering map away from the branch points. So it might wrap around a few times, but a cylinder maps to a cylinder. And then you have to show in every direction where this has a cylinder, this is covered by cylinders. Oh, not through these two points. But you only have to be covered by cylinders in directions that have a cylinder. Yeah, you can change the direction. So if I put this at a rational slope, then this would have a cylinder, and those cylinders would lift here. And when I had one cylinder, it would be covered by cylinders. But then I could move the slid and there'd be no cylinders here, and there'd also be no cylinders upstairs. That's consistent. The special property is that whenever you find a cylinder, there are parallel cylinders covering the surface. But there are plenty of directions. The generic direction has no cylinders. You can easily show there are only countably many directions with cylinders. Other questions? So even before I talk about this, people saw this. And a lot of people thought, OK, this is it. Most of them were not brave enough to put it in writing. But several of the leading people in the field told me, they thought maybe this was just the complete list of orbit closures. They're so hard to build, they don't seem like they exist. All of the phenomena that we knew were very somehow special leading to orbit closures. And they said, OK, well, maybe this is just everything. Which, in a sense, would be a very big success, because then it would be very easy to compute the orbit closure of every surface. You have a surface. You'd see, does it cover a lower genus surface? Nope. OK, well then, first of all, if it's genus bigger than 5, then it has dense orbit. And then you could, even in genus 2, you'd see, is it on one of these agoniform loci? And then, no, OK, dense orbit. So it would be very easy to compute orbit closures. Which is somehow the goal, and somehow what you, Eskin Mirza-Kanimo-Hamadi gives you some results on its own. But to fully exploit it, you need to know the orbit closures. The answer depends on the orbit closure. Surfaces with different orbit closures have different properties. So if you ask a general question, you'll get different answers. Well, so do you care about foliations on the Riemann surface? Measured foliations? Yeah, do you? I mean, most people who care about Riemann surfaces care about foliations on Riemann surfaces. Yeah, yeah, the singular measured foliations. This is like a big part of the theory of Riemann surfaces. So say you care about foliations. So you can make any foliation be the horizontal foliation of a translation surface. And then you want to know the dynamics of the foliation. Well, it depends on the geotooler of a closure. You need somehow to understand dynamics on individual surfaces, you understand the whole monolid space. That's somehow a big thing. And you've given a Riemann surface. You want to deform it in a certain way and make it as simple as possible so you could understand it. So in that point of view, you'd always be just especially, because the limiting things are often like this. But also, I think most people are not just interested in individual Riemann surfaces. Individual Riemann surfaces are important, but even more important is the modulized space of Riemann surfaces. It's the central object. I mean, it's string theory and algebraic geometry and symplectic geometry, and everything meets the modulized space of Riemann surfaces. I don't think individual Riemann surfaces are as important as the modulized space. And this tells you directly all sorts of things about the modulized space. So anyways, I was saying, some people conjectured this was it. Mirza Khani was braver. She wrote a conjecture down. Her conjecture wasn't quite so broad as that. So this property, I'm going to call rank 1. Because there is a number you can assign to an orbit closure called rank, and there's a theorem that I proved saying rank 1 is equivalent to this. So Mirza Khani's conjecture from about five years ago, all orbit closures, rank bigger than 1, are trivial. It was certainly supported by the evidence, because in its known orbit closures at the time, everything was either trivial or rank 1. And remember, whatever rank 1 is, it's something that holds for all closed orbits and then all things with similar properties. It's all orbit closures where the cylinder deformation theorem doesn't tell you anything you didn't know already. OK, so this was not an idle conjecture. She had a whole approach to thinking about this, extremely intricate. And I spent a year or two working with her fairly intensively on trying to prove this. And something came out of that. I was always trying to write down the simplest translation surface with some sort of linear equations that might be a picture of an orbit closure of higher rank, but somehow I couldn't rule it out. And eventually I wrote down something that looks like this, opposite edges identified. And there were linear equations that went along with this. This guy, so let's start writing things, v0, v1. Or let me say maybe v1, w1. So w1 is the golden ratio times v1. And then v2, w2, v3, w3, wI equals golden ratio. So eventually I wrote down this picture of some genus 4 translation surface with some linear equations defined over q adjoint squared with 5 that I couldn't show wasn't an orbit closure. And indeed, fast forward to the present. So theorem, S-Gandm, McMullen, write, there are at least eight primitive rank 2 orbit closures that are non-trivial. And this is one of them. So this actually took us a very long time to figure out how to prove this. And this wasn't the first example that was figured out. There was a second example I found that had some special features. And so just with Kurt McMullen and Ronette Mukamel, not with Alex Eskin, I wrote a paper on that which just appeared in the annals that showed that that was an orbit closure. And part of the difficulty is at least what we were doing is we were starting with some picture like this. And the picture is totally unhelpful. I mean, OK, here's a surface with some linear equations. The problem is as you continue these linear equations, you might get some subset that's dense in the stratum. And it's just really hard to show that doesn't happen. We had very strong computer evidence in addition to very strong theoretical evidence. You have to have strong theoretical evidence to find this, because how do you pick out equations? I mean, there are infinitely many equations you can write down. So you need some good understanding to write this down. Essentially, the reason I could write it down is because Mariam and I were fairly close to proving her conjecture, which it's a good thing we didn't because it's not true. But if you get close to showing that something doesn't exist, you can sometimes just write down an example of sometimes what it takes. But anyways, ultimately, the proofs that these are orbit closures are purely algebra geometric. You take the point of view of algebraic curves, and you build some low side of algebraic curves with differentials that you can verify, that you can compute exactly, like polynomial equations for everything, and you verify that there are linear equations. And you don't have this picture anywhere in sight. And at the end, you do some truly annoying computation to see that this is in the orbit closure you just constructed. So it's two orthogonal perspectives. You have flat geometry and dynamics, which helps you figure out what's there. And once you have it, you try to forget most of what you know about it, except for some general properties. And then just start trying to think about algebra geometric constructions that could build such a locus. And then having built it, you try to show it's the locus you thought existed to start. Well, so we can actually give equations, which I don't want to write for you. The true insight lies in the fact, so these are low side of x omega. And in all of these, the Riemann surface has interesting maps to p1. Now, every Riemann surface has maps to p1. But these have low degree maps with dihedral monodromy. And these aren't normal, these aren't Galois covers. So here there are generally no deck transformations, sometimes as an involution. But you can look at the normalization so that this is a Galois cover. So here you have the deck group acting. And here you pick the omega to be some sort of eigen form for something in the deck group. Actually, it's an eigen form for something in the group algebra. And the linear equations you get here on the omega, so you pick the omega here to be some sort of eigen form for something in the group algebra. And then when you project it down here, it still satisfies those linear equations, even though there's no more automorphisms, no more deck group. And what made this construction really hard to find is that we don't get the low side of all covers. So there's some sort of special relations that have to hold among the branch points for this to work. So actually, a construction like this is one of the first things that was tried. And it didn't seem to work. And the tricky thing is that the branch points are constrained to make it work. Yeah, you switched to some other coordinates. For example, you can specify the branch points here. Those are algebraic. Or you can just write down families of algebraic curves. And then you have to check, do they all give different Riemann surfaces? We don't actually use coordinates on the Hodge bundle. We just produce algebraic families. So the most basic example of an algebraic family is something like this. As T varies, this gives you the family of genus 1 Riemann surfaces, one of the most famous families in math. So we do something you can give much more complicated, really quite complicated, parameterizations like this. OK. These are all the rank 2. Well, actually, we missed two of them until recently, which was sort of embarrassing. Two of them were coming from a variant on our construction. And the computer program we wrote to figure out all parameters worked, actually found it. But it really looked like it was giving trivial orbit closures. And it wasn't until we looked carefully that we noticed it was non-trivial orbit closures. It's actually really amazing. So all of these come from unfolding quadrilaterals. All come from unfolding quadrilaterals. So each of these is the orbit closure of an unfolding of a quadrilateral with some special shape or some special angles. And Alex Eskin is a computer program that will run some tests on an unfolding of a quadrilateral, which doesn't compute the orbit closure, but will provide circumstantial evidence as to what type of orbit closure it has. And Alex's program didn't find any evidence of any additional higher rank orbit closures, even though he was just doing this sort of very elementary and involved sort of searching for cylinders. And you sort of need a lot of theorems to interpret the data, but you can interpret the data. So it didn't find any evidence of any others among unfoldings of quadrilaterals. And yet the proof here is amazingly special. There is no reason that such an orbit closure would have to come from a family that he recovers. There's no theoretical reason whatsoever. So it's really quite curious that we have an extremely special seeming proof that seems like it shouldn't work all the time. And yet it works in every case where we have any evidence, any computer evidence that it should work. So conceivably, this could be all of them. I have lots of conjectures about orbit closures, but I have no conjecture about whether these are the only higher rank ones. Or at least not whether these are the only rank two ones. These are all rank two, it turns out. So can you say what rank two is? Oh, I didn't say. I just said rank one was equivalent to an any direction with a cylinder you're covered. If you want, though, I'll tell you the definition because it's pretty easy. So the tangent space to the orbit closure lives naturally in the relative homology group. And the relative homology group maps to the absolute homology group. So you take the tangent space, you map it to absolute homology, and you take half its dimension. That's the rank. So it's half the dimension, but you exclude some of the dimension, the relative part of the dimension. Yeah, you can interpret it geometrically in terms of how many disjoint independent families of cylinders you can have. OK, so I want to tell you something else in terms of the geometry that we got out of this quite surprisingly. Six of them have dimensions 8, and then one has dimension 9, and one has dimension 10. But they're all, sorry, I got that wrong. Six of them have dimension 4, and one has dimension 5, and one has dimension 6. Two is closed orbits. Yeah, yeah, somehow. You can describe some of the deformations with cylinder. So you have more cylinder deformations. You can't have cone points come together. So actually, in most of them, you can't just have two cone points coming together. You have to degenerate the whole surface. So most of them have sort of more complicated generations. Yeah, like a repair, more than one would get, you'd collapse more than one. But it's sort of hard to have a good, I mean, we have this algebraic picture of an, well, actually it's only birational. So we have an algebraic picture of most of these orbit closures. But even so, it's a little bit hard to really understand the boundary. There are compactifications that are better than Delene Mumford. The ultimate versions are still under construction by people other than me. And that's a fun conversation to have. But I just want to end by saying one other thing. So this theorem with the same authors, again with the same thing that one of them came first just with Kurt McMullen and Ronan Mukamel, totally geodesic complex surfaces in M13, M14, and M21. So here these are moduli spaces, say, of genus one surfaces with three marked points, genus one surfaces with four marked points, et cetera. So these are complex dimension two things. And you should think of these as analogs of what are called Teichmeler curves. So in general, you have these Teichmeler disks that wind around densely in MG. But sometimes they actually close up and form an algebraic curve. These are the projections of the closed yield to our orbits. And those Teichmeler curves are totally geodesic of dimension one. These are the first non-trivial examples we know of things of totally geodesic of dimension bigger than one in the Teichmeler metric. It's the Teichmeler metric that's connected to GL2R because GL2R orbits project a Teichmeler to complex geodesics for the Teichmeler metric. You can also have in mind the Kobyasian metric. So this is really exceptional, the geodesic complex. For example, we don't know if there are three points whose convex hull in the Kobyasian metric is like all of Teichmeler's space. We don't really understand the convexity property very well. So it's sort of surprising to find these totally geodesic things. So these are not foliated by Teichmeler curves. They are foliated by Teichmeler disks. So maybe I'll end just by saying a result that I proved on my own that contrasts these totally geodesic things of higher dimension to totally geodesic things of dimension one. So totally geodesic of dimension one is a Teichmeler disk. So it's a manifold and Teichmeler space, the universal cover of MG. There are tons of them. You can join any two points. And then you project them to MG and typically it stands. But sometimes it covers a Teichmeler curve. And that actually happens infinitely often. Some countable infinite set of Teichmeler disks that cover different Teichmeler curves. So now we say, okay, what happens with higher dimensional totally geodesic things, these higher dimensional versions of Teichmeler disks that we now know examples exist. So I was able to show that the opposite occurs. So first of all, every totally geodesic sub-manifold in Teichmeler space covers a variety. So it's the analog of the totally false statement every Teichmeler disk covers a Teichmeler curve. And there are only finally many. Which is the analog of the totally false statement that there are only finally many Teichmeler curves. And so this result was proven using joint work with Alex Eskin and Simeon Phillip in which we compute something called the algebraic hull of the Consavage-Georges Coast Cycle. Which is somehow a very powerful result. Which let us do something we're really not supposed to be able to do, which is really understand things in the universal cover really well. Where you don't see the dynamics as much. Okay, so I'll stop there. Thank you very much. Any questions? I like questions. Yeah, geol2orbitz. So geol2orbitz. Yeah, so before I said the gtorbitz. So the gt action is ergodic. So if you take even just gt, e to the t00 minus, e to the t00 even the minus t. Even that orbit will be dense in the unit area locus of the stratum. So say take the principal stratum which is the big dimensional one. So then you can project that to mg. Well if you take a dense set and you project it to mg, you again get a dense set. It's one dimensional, just the gt orbit. So already that one dimensional thing projected to mg is dense. And so then if you take the geol2orbit, that's even bigger. So it's still dense when you project it. Yeah, yeah. So the density upstairs is actually stronger than the density in mg. Six, seven or eight dimensional, yeah. Most of them are six, sorry four. I don't know what I'm having numbers today. Four or five or six dimensional. Most of them are four dimensional. You get a three dimensional variety. Ah, yeah, so that's interesting. But those are not the modular spaces where you find those. So these guys in, and notice I said eight there and I only have three examples there. So in three of the examples is an involution. And you quotient by an involution and mark the poles of the resulting quadratic differentials. And you lose another dimension and you get it as geodesic surface. Actually, whether or not an orbit closure gives you a totally geodesic variety, you first need an involution and then you quotient by the involution and then it's just a dimension count. You need the geol2orbit closure to be twice the complex dimension of the resulting variety in mgn that you get. So one thing I didn't have time to talk about or I chose not to because maybe this isn't the audience but when Simeon Phillips showed that orbit closures are varieties, he actually showed that they're cut out by conditions involving the Jacobian. So endomorphism, the Jacobian torsion conditions and the Jacobian twisted torsion conditions. So at the very least it's clear that these are cut out by sort of algebraic conditions. In a sense, you can think that these orbit closures witness unlikely intersections. If you look at a loci in the moduli of PPAV of abelian varieties where the Jacobian has special properties, this is some nice homogeneous space. And then you intersect it with mg and by a naive dimension count you expect the dimension to be wrong, to be small. And so in here it's like the dimension is bigger than sort of you would expect it to be. But even more than that, some of these are very interesting. Like this one was the one that we wrote up with Curtin, that Curtin, Ronette and I wrote up and it's a beautiful story with the classical geometry of cubic curves. Hessians, Caleans, we cite a book from like the 1800s with some of the formulas that we use. There has not as I know been any number theory application, but there is hope that some orbit closures will give rise to interesting modular forms. Well, we already know they give rise to interesting modular forms, but they haven't been used to construct Gaoua representations or anything like that. I don't know that the number theorists really care that much. There are reverse applications, number theory gets applied to this a lot. So for example, Ronette is thinking about teichmeler curves and characteristic P. And obviously anytime modular surfaces come up, you'll have number theory lurking in the background. And some diophantine approximation results has played a role in some of this story. When we can continue their conversation.