 Thanks Steve. First wanted to thank the organizers for inviting me to this great conference and to mention that everything I'm about to say is joint work with my collaborators Kurt McMullen and Alex Wright. So our main theorem is there is a primitive totally geodesic complex surface M13 the modular space of genus one Riemann surfaces with three marked points and I'll spend most of my my talk discussing this theorem before I get to describing F. Let me make a few remarks about what all of these terms mean. Is my font size all right in the back? So M13MGN is the modular space of genus G Riemann surfaces with N marked points. So point in MGN is a pair AP consisting of a Riemann surface A and a subset P of size N. And to talk about geodesics in MGN we need to have a metric on MGN and there's this metric DT called the Teichmiller metric. I say F is totally geodesic I mean for the Teichmiller metric. Every tangent vector modular space generates what's called a complex geodesic by which I mean a map FV from the hyperbolic plane into MGN which is holomorphic complex and locally isometric or if you want the image is totally totally geodesic. So and it's uniquely characterized by the property that the value at I is the base point AP and the first derivative at I is the tangent vector V. And we'll come to this a little bit later but if you were here last week I believe there was a discussion of dynamics over the modular space of translation surfaces and this is closely related to an SL2R action on the modular space of translation surfaces or quadratic differentials. Now if you start with a random vector in tangent vector to MGN you will get a complex geodesic which evenly distributes itself in modular space its closure will be all of MGN. But maybe if you start with a special tangent vector you'll get some complex geodesic with a small image. So we'll say a sub variety and so if you took a sub variety of MGN, an algebraic sub variety of MGN and took a tangent vector to it, typically you'd expect that that tangent vector would not stay within that sub variety but visit all of modular space and uniformly distribute in there. So we'll call a sub variety totally geodesic if the opposite happens. If every tangent vector little V to V has the image of the associated complex geodesic contained not uniformly distributed in MGN but actually contained in V. So our complex surface F, it's too complex dimensional, M13 I'll just remind you is three complex dimensional so this is a hyper surface in M13 and it's totally geodesic. There are two well-known sources of totally geodesic sub varieties. The first arises from covering constructions. We'll come back to these later but these give rise to V commensurable with other modular spaces. MHK for some smaller H and K than GNN. And these are the not primitive examples. So when I threw the word primitive into that theorem that's to say that there's no right for you to expect V to be there. A typical example is if you looked at the locus in MGN where of Riemann surfaces stabilized by a particular element of the mapping class group. You'd get a totally geodesic sub variety coming from a covering structure. And then there's another source of examples. There's the Teichmuller curves which are the one dimensional examples. The first one was discovered by Vich, my colleague at Rice. And now there's a longer list of them and a beautiful literature that's emerged around them. But our example fits into neither of these two categories. It's higher complex dimensional and it doesn't arise from one of these covering constructions. What's that? It means coming from a covering construction. Sorry, primitive means not coming from a covering construction. So the ones that come from a covering construction are not primitive and our surface does not come from such a construction. The typical example, so in M13 there's an involution on every genus one surface and you can ask that the three points be invariant under that involution. If you look at the locus of such configurations you'll get a totally geodesic complex surface in M13 but it's really coming from a lower genus mongeoli space. So when I say it's primitive I mean it doesn't come from that construction. Are there questions? Okay, so also our surface F has a beautiful connection to the theory of plain cubic curves as alluded to by this figure which I'll shortly explain. So let me tell you a bit about that. So I want to tell you about plain cubic curves, what I'll call solar configurations. So for any genus one Riemann surface without market points there is a polynomial F in C-adjourn XYZ homogenous of degree 3 such that A is isomorphic to the zero set of F in P2. The set of points Q or F of Q equals 0. This is a classical fact you can prove using the wire stress P function say. So in the conferences I usually go to in flat surfaces there's usually a fight at the beginning of the conference between people who like to draw Riemann surfaces like this and people who like to draw Riemann surfaces like this. So this picture, I'll allow that picture is the set of real zeros of the polynomial F and it's related to this picture by you take the real points of some complex Riemann surface. Now if I fix a point S in P2 that's not an A, then this gives rise to a degree 3 map on it. So this is the zero of F, the real points of the zeroes of this thing. This point S gives rise to a degree 3 map on A, pi sub S which is just projection. So this is a map from A to P1 where this P1 is intrinsically the projectivization of the tangent space to P2 at S, i.e. the lines through S. And this is a degree 3 map. So if you take, if you take the line through S, it'll meet the curve A at 3 points and these 3 points are mapping all to the same point under this map. What's that? Absolutely. In fact, so a typical line meets this curve at 3 points, but if you start with a special line it might meet the curve only in 2 points. So the critical values of pi S are the lines through S and tangent to A. The critical points of pi S are just the points of tangency and the critical, and then the co-critical points are the other intersections with these critical lines. So right here is a critical line and this right here is a co-critical point. In the celestial analogy, if you think of S as the sun and A as the earth, then these lines correspond to lines where it's, where it's dawn, some point where the sun is rising on earth. And these points are codon. Sorry, absolutely. And in that case, so, and that occurs exactly when the dawn equals the codon point. Yes, so if the line happens to meet the curve to order 3, then the dawn point and the codon point are the same. It's also easy to compute, so by Riemann Hurwitz there should be 6 of these. And actually it's easy to compute where they are. There's a, there's a conic called the polar. And it's given by the very simple formula is you take the standard inner product on, on C3, on R3 and complexify it to C3 and you take the, the homogeneous coordinates for S and inner product them with the gradient of S. So this is a vector whose coordinates are degree 2 polynomials. And when you inner product them with S, you get a linear combination of such, this is a degree 2 polynomial. Its intersection with this degree 3 polynomial is 6 points and they're the 6 points of tangency. They're the similar formula for the co-critical points, but it's, it's a little more complicated. Now for typical S, this polar is, is smooth. But if you're very careful about how you choose S, if S happens to be in the zero locus of the Hessian of F, which is just the determinant of the matrix of second order partials, then the polar is singular and in fact it's a union of two lines. So for typical S there are 6 of, of these dawn points and you can't really distinguish them. But, but if S happened to be on this Hessian curve, then the polar splits into two groups of three corresponding to, so now we can label the points as dawn and dusk. And the, the co-points for, or the codon points associated to L1 form a configuration, your ration P that has, that has the property that AP lies in F. It's a little hard to draw an exact picture on the board of that so I did an exact picture here. So this is one of these codon configurations. So here's my plain cubic, A, I chose a pretty random example. And, and here's the Hessian of A, this orange curve is the Hessian. So if I take a sun and place it on the Hessian, then there are three, three lines corresponding to critical values of projection from S that happen to be tangent to the, to the curve. And their points of tangency are because S lied on the, on this Hessian they actually form a line. That's the, the dawn line. And then the, that implies actually that the co-critical points are also lie on the, on a line and that's the, the codon, the locus of codon points. So now I've described to you how to take a plain cubic and come up with a curves worth of configurations of three points on it. And as I, as I move A, the, the polynomial defining A and the, the location of the sun I sweep out a complex surface. So the picture you should have in mind is we've got F. It has a map to M1 just by forgetting the configuration of three points. And if I take a particular point in A in M1, then the fiber over it is more or less the Hessian of the associated plain cubic. That's, that's the fiber over right. It's not quite true because there, there was the issue of choosing dawn versus dusk. So it's really a degree to cover of the Hessian on which you can consistently choose the dawn, the dawn line. This is kind of a, a parameterization of the surface F. We can give a more intrinsic definition F consists of the configurations A, P, to the condition that there is a degree three map to P1, subject to the condition that one, P is co-critical for pi. And two, P is linearly equivalent to a fiber of pi. Linearly equivalent just means that they, the points lie on a line in the plain cubic model. And the translation between these two definitions is you just think of pi sub S when, when you think of pi. So the upshot is that F is an irreducible surface, irreducible complex surface in M13. We've called it the flex locus. There was actually vigorous debate over what to call it, but we decided the codon locus was a little too cute. Oh yeah. I'm taking the co-critical points. I take the, I take the green points. Yeah. So I, so to each son I get a collection of green points and I, as I move the sun, the green points are wrong. Yeah. Yeah. So we're taking the codon, the codon points. Yeah. Sorry, I can't hear you. Yeah. I mean, this is, this is, are you not happy with this? What, what was that? Well, how do you take a point in M13 and come up with the embedding? You look at the linear system associated to P. Okay. So really this is a very natural setting in which to study points in M13. Whenever you have three points on a genus one Riemann surface, you get a, you get an automatic embedding into the plane on which, in which they lie on a line. Okay. And if you believe that, then, then asking for them to be arranged in this way becomes a little more intrinsic. This, this is, this, this does not make, this does not make reference to the plane model. It just says there is a degree three map, which you can later extend to. Well, I don't, I mean, I don't know how to phrase this without, I mean, we, we have several other alternate definitions, um, but none of them are very obvious from this. And I, I do view this as kind of an, this is just referring to the complex geometry of A and how the locus of P sits. There's no extrinsic information here, I, I think. Pi also plays a set, a sexual role as will, as will come later when we talk about quadratic differentials. We call it the flex locus because if you think about the zeros of the Hessian, those are, those are the flexes of F as points in C3. So the claim is that this surface, this flex locus is a totally geodesic sub-variety of M13. To convince you of that, I need to tell you a bit about geodesics in moduli space and, of course, quadratic differentials. So now we're going to consider the bundle over M13 whose fiber over AP is the vector space QAP, consisting of integrable column holomorphic or meromorphic quadratic differentials, um, which are holomorphic in the complement of P. I, integrable here just means the poles are all simple. And they're only allowed to be a P because it's got to be holomorphic on the complement of P. The reason this is a relevant space is because QM13 is naturally isomorphic to the cotangent space, to moduli space. And there's this wonderful SL2R action on QM13 whose orbits project to the complex geodesics I referred to earlier. And if you prefer the language of translation surfaces, any particular quadratic differential in here gives AP the structure of a translation surface or more precisely a half translation surface. And it can be presented as a union of polygons and you can just hit all those polygons by an element of SL2R and that gives you the action. Okay, so if I'm going to claim to you that F is totally geodesic, I should give you a locus of quadratic differentials generating differentials, generating SL2R orbits and complex geodesics which happen to lie in F. So the definition is UF consists of triples AP and Q where A and P happen to lie in F and pi satisfies, I erased my definition already. Oh, no, I didn't, satisfies one and two. And the zeroes of Q form a fiber of pi. So the fibers of, there's a one complex dimensional family of fibers of pi. So there's a one complex dimensional family of location, possible locations for the zeroes of Q and combined with a scaling action that means there's a two dimensional fiber over any given AP. So we get a fiber bundle structure. A typical AP has a C2's worth of such quadratic differentials so we get a fiber bundle like this and in particular QF is irreducible of complex dimension four. Now our main theorem follows easily from the theorem QF is SL2R invariant and the implication is just transport the SL2R orbits, complex geodesics. If you believe that this is SL2R invariant then each quadratic differential here generates a complex geodesic in F because it remains in QF which all protects down to F. And at any point in F I've given you a two dimensional space of directions to go which is a, which is a dimension of the space of F. So this, this shows F is totally geodesic. What's that? Primitivity is not obvious although it's not that hard. So in any M13 the primitive, the non-primitive, there are finally many prominent non-primitive examples and you can directly verify that this locus of quadratic differentials can't arise in that way. I'll discuss that at the end of the proof. So strictly speaking this will prove it minus the primitivity assumption but we, but I'll discuss that once we complete the proof that it's totally geodesic. So a very useful thing to do to a locus of quadratic differentials or half translation surfaces is to pass two covers on which they're actually translation surfaces. They're, where they're, where they're associated filiations or half translation structures are actually orientable. And that process, there's a well-known process for doing that and that's by taking square roots. So given any Q and QAP it may or may not be the case that Q is actually the square of a holomorphic one form on A but whether it is or not there is a square root X omega where X is a Riemann surface probably of higher genus and omega is a holomorphic one form and this process basically takes Q and makes it orientable. So with, so this pair has the property that there's a two, two to one map from X to A and the pullback of Q is equal to omega squared over the odd order zeros and the poles. Yeah. Yes, but for simple poles it all, it all works. Yeah. And in fact the, I mean I could write a formula but it probably wouldn't be very enlightening at the moment. It's a beautiful formula but there's, there's a, yeah. It works with simple poles. If you like the simple pole and the flat structure has cone angle pi. When you branch over you get cone angle two pi I, it's no longer a zero. And the key point, the reason this is relevant to this discussion is the square root is SL2R equivariant. So if I want to show QF is SL2R invariant it's enough to show all the square roots are SL2R invariant. So let's define omega G to be the locus of square roots. X omega where X omega is the square root of AQ for AQ, APQ in this QF. And I'm going to impose just a generic condition. There are three zeros of Q. So this, this, this Riemann surface is a degree to cover of A branched over, when there are three zeros of Q there are three points P. It's total of six branch points. This is a genus four Riemann surface. So this is a sub locus of omega M4. The hodge bundle over in genus four, the locus of abelian differential in genus four. And in fact the, by enforcing that there be three zeros of Q, I restrict myself to a particular stratum in omega M4. The locus where the differential omega has three double zeros. Generic one form has six simple zeros but each of these square roots has three double zeros. So this is the stratum thereof. Now a stratum like this has period coordinates locally modeled on on the vector space h1x relative to the zeros of omega with coefficients in C near the point x omega in omega g. And the map is just, if you give me an omega prime that happens to be near omega, I send it to its comology class in this vector space. This is a nice system of coordinates on this stratum of abelian differentials. It's so nice that the transition maps are all R linear or even Q linear. And the SL2R action is the obvious one on rewriting h1x this comology group. So this is is isomorphic to C to the N for some N which is R plus IR to the N because there's a canonical real structure which is just R squared to the N. And then an element in SL2R acts on R2 in the obvious way by a linear transformation and you're just taking the diagonal action of SL2R. So the theorem that QF is SL2R invariant follows very easily from the theorem that omega g is locally defined by R linear equations in period coordinates. So once you write down you're an open set in the stratum locally modeled on R squared to the N, the locus we've defined this omega g is cut out by just R linear equations in this real vector space. And if you believe that the SL2R action is really just the obvious one here, it's clear that it will preserve that linear subspace and the locus becomes SL2R invariant. So this is so omega g is SL2R invariant as an elementary consequence as the elementary you have famous theorem by Eskin, Mersikhani and Philippe. So I mean the what I just said proves to you that a locus defined by real linear equations in period coordinates is SL2R invariant and they've proven the very not obvious converse that every SL2R invariant locus is actually a sub-variety of the Hodge bundle defined by a linear equations in period coordinates. And then once we know that omega g is SL2R invariant we know that QF is SL2R invariant because square root is a covariant and omega g is just obtained from QF by taking all the square roots. Okay so if I'm going to convince you that omega g is defined by really linear equations in period coordinates we need to figure out how to read off period relations and the key to doing this is something a little bit mysterious called the mystery torus. So let's count dimensions for a second so QF in QM13, M13 is three dimensional so QM13 is six dimensional, F is two dimensional, QF was four dimensional so this is co-dimension two. So by my count I owe you two period relations I need to explain two equations on the periods of X omega in omega g which are not explained independent of by the fact that omega g is contained within this locus of square roots to extra conditions. Okay so let me tell you how to see that. So let's fix an APQ in our QF and a solar configuration. A is a zero of a plane cubic, a sun, a pi equals pi s, solar configuration. This picture implies that there are polynomials linear polynomials li, z, cd such that well so the locus p is let's call it p1, p3 so the zeros of li is just the line between pi and s those red lines. The intersection of a with the zeros of z are the zeros of q. The intersection of a with the zeros of c is the p, the codon line for the poles of q and a intersects the zeros of d is the dawn locus. So here's a picture of that. So the only extra information is now we have this line z recording the zeros of q. We've got l1, l2, l3 they're the lines connecting the sun to the to the marked points. We've got the line d which is connecting all the associated critical points and the line c which is connecting all the associated co-critical points. So what picture do we have? We've got our it's got this degree three map pi to p1 which remember is the lines through s and we've got this degree to cover x, x omega is the square root of a q. I can form from this whole picture a torus, the mystery torus by taking the two to one map to p1 branched over four, I need to branch it over four points and I've got four points going through the sun. I have four lines going through the sun there so l, i, m, z. So the claim which will explain the period relations is that this diagram can be completed. That was a definition of qf. So the zeros of q lie on a fiber of pi. Other questions? So the proposition is there is a degree three map from x to b polymorphic making this the square commute and I hope to convince you that you can sort of see this fact this is not such a complicated fact you can kind of see it from this figure. If you think where the map x goes to p1 is branched it's totally branched over the intersections with those four lines and therefore at least it has a good shot of lifting to b. But the proof of this proposition is obtained as follows. So first we're going to de-homogenize p1 using the polynomial c and to show we get this map to b what we want to show is that the product l1 times l2 times l3 times z over c to the four to de-homogenize it has a square root in the field of meromorphic functions. Well let's look at our quadratic differential. The quadratic differential it's supposed to have zeros at z and poles at c. So it's actually z over c times the square of the holomorphic one form. And that implies that to form x the function you get in addition to the normal functions on a is precisely a square root of z over c. So that implies that the square root of z over c is a meromorphic function on x. In fact the field of meromorphic functions on x is obtained from the field of meromorphic functions on a by just adjoining this one square root. It's exactly what you need to be able to take a square root of this of this quadratic differential. But moreover on a the polynomial l1, l2, l3 it's the union of those three lines the three red lines. What is it doing? It's meeting the curve a at each of the yellow points to points of order two each of the points on d to points of order two and it's meeting the curve along the line c to points of order one to order one. So up to adding functions which are zero on a the product l1 times l2 times l3 is just d squared times c up to adding multiples of the homogenous polynomial little f. This is true. And now what we wanted to show is very easy. So l1, l2, l3, z over c to the four is d squared c over c to the four which is z over c times d squared times d over c squared. Z over c is a square on x and d over c squared is a square on x. So this is a square on x and that shows this function has a square root in the field of meromorphic functions on x. So that proves this proposition and then this implication is pretty easy. So the zeros of q they were equal by hypothesis to a fiber of pi and that implies the zeros of omega equals a fiber of p. The zeros of omega come from the zeros of q and they're all mapping to this one critical value point for the map from beta p1. And so if I take the push forward of omega to b, I get a holomorphic one form on b that has a zero at this fiber. But a holomorphic one form on b with a single zero is actually identically zero. And this is exactly the type of thing that gives you linear period relations. So this says that the image of the induced map on homology obtained by you first take the induced map on homology, the normal induced map on homology which typically goes the other way and then you take its dual via the symplectic form. The whole image of this has zero omega periods. If I take a cycle in the image of this map I get a cycle on x whose period in omega is zero. And since this is a two-dimensional space I get a two-dimensional space of linear conditions on the periods of omega explaining why these x's of magas are locally defined by our linear equations and period coordinates. To address the question which was raised earlier, so this shows that f is totally geodesic in M13. Let me address primitivity. I said covering constructions give rise to totally geodesic in MgN. I said before that v is commensurable with a different modular space. In particular the universal cover or the universal cover of a normalization is isomorphic to a traditional teichmeler space. So this is bi-hologmorphic, equivalently isometric. Hologmorphic and isometric. As I said, you can see by pretty elementary means that this f is not a totally geodesic. It is not primitive. By just enumerating all the possibilities for this and checking it's not one of those. But we showed a much stronger version of this because it's totally possible that even though f is primitive, its universal cover is still some traditional teichmeler space. In fact, the universal cover of the normalization of f is not isomorphic to any traditional teichmeler space. To prove this theorem, we adapted a technique pioneered by Jordan for studying maps between teichmeler spaces. That other theorem. So the complex dimension of f is two. And there's only one two-dimensional teichmeler space. So it's enough to show that f twiddles is not holomorphic or isometric to T05. You might think there's another one, T12, but T12 happens to be isomorphic to T05. It's the one coincidence. So we're going to use the technique of Reuden for studying maps between teichmeler spaces. So Reuden showed that for every A prime, P prime in T05, the teichmeler norm, the norm associated with the Finzler metric, the teichmeler metric on the cotangent space, which is canonically isomorphic to the space of quadratic differentials. He showed it's C infinity, in fact, analytic, in the complement of five lines. And it's not C2 on those five lines. And the lines are pretty simple to describe. A typical differential here has five poles and one zero, but there are five lines in which one of the poles and one of the zeroes collide and you get four poles and no zeroes. So there are five lines in here. And the norm is C infinity on the complement of those five lines, but it's not C2 on those lines. And in this way, this was a keen greeting to proving the bi-holomorphic, automorphism group of teichmeler space is the mapping class group and no larger. So to show that F twiddles is not isomorphic to any traditional teichmeler space, we showed that for typical AP F twiddles, the locus of differentials, the restriction of the teichmeler norm to the cotangent space to F twiddles, which is this locus of quadratic differentials whose zeroes lie at a fiber of pi, is not C2 on the union of six lines. In fact, it's the lines in QAP pi, which intersect the strata of differentials minus one cubed, one two, and the stratum minus one squared zero two. So as you move Z around, there are six special locations for Z where Z has fewer higher-order zeroes. And using the same techniques as Roy's, you can show that the teichmeler norm is not C2 along those lines. And so if you're going to give me a holomorphic isomorphism from F twiddles to T05, it will give me an isometry between these two spaces, but they can't even infinitesimally be isometric because their regularity doesn't really match up. Okay, that looks like a good place to stop. Thanks.