 So I'm going to talk about, this is joint work with my former student Anya Bankovic. So let's see, so let me start, so S throughout this talk will be a closed, orientable surface genus G2. And so two structures that I want to associate with this. So first of all, gamma of S will be set of homotopy classes of essential closed curves, and let's say maybe unoriented. And then the other thing that I want to think about associated with the surface S, some family of metrics. So omega will be some family of metrics. To simplify things, I'm going to, I'll sort of be slightly vague about this, but points inside of omega will sometimes be metrics and sometimes they'll be equivalence classes of metrics where the equivalence relation is familiar one from Teichmann theory. So family of metrics on S, up to isometry, isotopic to the identity. So I could take some metric on my surface and then I could apply some homomorphism, let's sort of difhumorphism, let's say, that's isotopic to the identity. So I put my hands on my surface and I sort of smear it around and then I could push forward the metric. And I want to think of those two metrics as being the same. And so given an equivalence class of metrics and a homotopy class of curves, I can measure the length of the homotopy class with respect to the metric, where I just take a representative of the metric and then I take the GDSIC representative of the homotopy class, minimal representative. So we'll say that omega satisfies, let's write it, marked length spectral rigidity. So that's the title here, marked length spectral rigidity, almost sort of thinking of the future. As marked length spectral rigidity, if function from omega to r to the gamma, which just takes a metric sigma and spits out all of its lengths, so length with respect to sigma of gamma. In other words, metrics in the family omega are determined by the lengths of their minimizing GDSIC representatives. So this is probably familiar in at least one setting. So hopefully I'll try to leave this up. So example, so the Teichmeler space, so what's the Teichmeler space? That's the situation where omega consists of constant curvature minus one metrics on the surface. And then there's something stronger here, so Teichmeler space of s satisfies marked length spectral rigidity. This goes back, I don't know, to Frick and Klein or something. And you can say more, right? There's, yeah, if, oh, yeah, it is one to one, thanks. Yeah, so if, so I guess I said it out loud, the metric is determined by the length, but yeah, I didn't write it. So no, no, definitely not surjective, yeah, that's right, yeah. So right, so something stronger is true for a Teichmeler space. In fact, I don't need the lengths of all curves to determine the hyperbolic metric, I only need a finite set. So in fact, we can find curves alpha one through alpha k, k linear and the genus, so that the function from Teichmeler space to r, the alpha one through alpha k is one to one. Okay, so this is pretty, pretty easy, let's, let me just sort of say what the point is. So maybe here's my surface. I need to tell you some curves, so maybe I start with picking a pancy composition on my surface, that is a collection of essential curves, in fact I can pick simple curves that cut the surface up into three-hold spheres. So as we know, a hyperbolic metric on the surface will give me a hyperbolic metric on each of these three-hold spheres, which, if I take these to be geodesic representatives, that metric will be determined by the lengths of those boundary components. And then, what should I do? So I've almost determined the metric, but I've got a sort of choice as to how I glue these pairs of pans back together. And I can determine that if I take, say, some curves that are, so I could take a curve that intersects this curve, and then the length of this curve will almost determine how I glue. And then if I, if I want to pin it down so there's, I could take maybe another curve that intersects that curve. And the lengths of those two curves together will tell me how much I've twisted, okay. So from this, it's pretty easy to get 9g minus 9 curves, and in fact you can, you can reduce it down even lower. Okay, but the point is that Teichmeler space is a finite dimensional space. It's dimension 6g minus 6. And so for, for larger families of, of metrics, this becomes a bit trickier. Nonetheless, it's true for, for lots of metrics. So let me just write a sort of big theorem that includes lots of different results. So let me just say a marked length spectral rigidity holds for, and now let's say classes of metrics. So negatively curved metrics, okay. So if I look at all negatively curved metrics, then, so this is due to O'Tal and independently croak. So this is no longer a finite dimensional space of metrics. This space is huge. And it's definitely not the case that I'm going to find finitely many curves. And moreover, I can't even pick finitely many, I can't even pick the set of all simple closed curves. So it's a result of Berman and series that if I look at the set of simple geodesics, let's say, on a hyperbolic metric and take the closure of all of those, then that misses some open set. So the complement is, is open. And so I can perturb the metric inside of there without changing the length of any of those simple closed curves. So simple curves aren't enough and you really need lots and lots of curves. Okay, so this was then generalized to non-positively curved metrics. And I'll say something about the way that these proofs are related to each other in a minute. So this is due to croak, Fatih and Feldman. So as I'll say in a minute, there's an underlying theme for all of these results. They sort of, there's an initial reduction to a different problem. I'm going to spend some time talking about that different problem. And then from there, the techniques diverge. So the techniques here are sort of similar to the reduction. And then after that point, things become very different. So okay, well we can go a little further. So maybe it's worth pointing out, right, that the negatively curved metrics sit inside the non-positively curved metrics. A larger space of metrics than the negatively curved metrics are the negatively curved cone metrics. And this is due to Hershansky and Feldman. And here the techniques are sort of actually similar to the negatively curved case. So again, this is sitting inside of here. So what are negatively curved cone metrics? So away from a finite set of points, it's a negatively curved Riemannian metric. And at a finite set of points, you have concentrated negative curvature. So you have some comb point singularity for the metric, and the cone angle has to be greater than 2 pi. So in some sense, the proof here is really quite similar to the proof here. But you have to sort of work harder because of these singularities. And then, so the theorem that I want to talk about today is kind of in the direction of this non-positive curvature. So these are, I'll call them flat metrics, non-positively curved Euclidean cone metrics. So this is what we show. So you may be familiar with a large class of flat metrics. They were mentioned this morning, translation structures on a surface. So I start with something like this, and then I glue the sides. You take a Euclidean polygon and glue opposite parallel sides. You get a translation structure. Well actually, this is a singular translation structure. So you have a cone point here, which corresponds to the zero of the corresponding and billion differential. But away from that cone point, the metric is Euclidean. And then this cone point has curvature as a cone angle, 6 pi. So this non-positively curved condition just means that all cone angles greater than 2 pi. So this is the Gromov-Link condition for non-positive curvature. So you have a metric which is Euclidean away from a finite set of points. And at that finite set of points, you have a cone angle which is greater than 2 pi. So let me say something about that in a minute. This class of metrics, these translation structures, or semi-translation structures more generally, are just a very small part of this space of flat metrics. So being a translation structure, or semi-translation structure, means that not only are you non-positively curved, but the polynomial and the complement of the singular points has to be almost trivial. So either the identity or minus the identity. So I'm sorry. Well it's Euclidean, so it'll be, yeah. So just away from the cone points it should be smooth. So at the cone points you have some singularity, but away from there it should be a smoother money. Oh, that's a good point. I think that these, yeah, I'm not sure what the smooth is. Are they C-infinity? C2 at least? Okay. Yeah, so in my situation they're Euclidean, so they're C-infinity. Yeah, so in my situation they're, I mean they're Euclidean, so they're C-infinity. Away from the cone points. The metric just be continuous. That's a good question. I don't know. This is the curvature in the sense of Alexander of Spaces. So negative curvature here we just take, yeah. It's also, because it's assumed to be Riemannian, it's also the sectional curvature of Gaussian curvature. So yeah, probably C2, like I said, this, actually we'll see that here the sort of smoothness is not, well it's not an issue. Yeah, so this, I should say, this result is slightly more general. If you, so you can state it in terms of pairs of metrics. If you start with one metric which is non-positively curved and another that has no conjugate points, then if they're, if they have the same marked length spectrum, same functions here, then they're isometric by an isometry isotopic to the identity. But the, but as far as if you just, if you allow positive curvature that's easy to construct examples, take your favorite, let's say, negatively curved metric and then perturb it in a silly way, build a little bubble here. You can change them, there's no geodesics that are going to go up here. Okay, so the translation structures and the semi-translation structures give you some finite dimensional subspace here of flat S. And so your question about whether simple closed curves are enough, the answer is yes. So this was with Moon-Duchen. So in some work, some older work with Moon-Duchen and Kazaroffi, we showed a stronger statement here. So you have marked length spectral rigidity for translation structures or semi-translation structures with just simple closed curves. And although it's a finite dimensional space, there's no finite set that suffices. In fact, as said, so there's a refinement of this question you could ask, you know, if you take some set of curves and a metric, are you spectrally rigid with respect to that set of curves and that family of metrics? And for simple closed curves, in some sense, you really need all simple closed curves. So if you look inside of PML, if your set of simple closed curves is in dense, then you can find a family of flat metrics coming from semi-translation structures where the length functions of those curves are constant. So for the arbitrary case, you need simple curves that are not sufficient. And the point is that, so this space of metrics seems like a fairly special class, but actually you could approximate any of these metrics by metrics in here. So at least C0 approximate. So you could take your favorite hyperbolic surface, say, and then take a triangulation of that surface with tiny little triangles, and then if you just replace each of these hyperbolic triangles by the comparison Euclidean triangle. So Euclidean triangle with the same side lengths, you'll build a Euclidean cone metric, and at each vertex, you'll have an excess of 2 pi angle. So you can approximate any negatively curved metric or any non-positively curved metric with these sort of Euclidean cone metrics. And in some sense, these are sort of maybe combinatorial metrics. If you just want to build negatively curved or models of negative curvature, these are the kind of metrics that you would use. Okay. Other questions? So in our paper, we asked whether, because looking at the non-positive curve proof and our proof, there are a lot of similarities. So we asked whether or not can you actually do non-positively curved cone metrics. And that was after we did this, David Constantine proved that that's also marked spectrary rigid. So just non-positively curved cone metrics. Okay. So let me take about maybe 20 minutes. So these are Euclidean cone metrics. So he's allowing variable curvature. So in some sense, there are two bad guys when it comes to this sort of problem. One is the places where you have zero curvature and the other place is where you have cone points. They cause problems with sort of the sort of standard approach. And so what I want to talk about is sort of the issue that arises with the cone points. Okay. So there's a reduction of all of these things to, I'm sorry, hyperbolic, yes, hyperbolic orbifolds also, yeah, this is more general than, you could also allow hyperbolic orbifolds. You have, again, finite set of simple orbifold fundamental group elements that orbifolds for, yeah, so that's a good question. I don't know if you allow negatively curved orbifolds, that's, yeah, I'm not sure. Any other questions? Okay. So the key tool in all of these was introduced by Bonahan and then really sort of utilized by Atal. And this is, so there's a reduction of the problem of Mark-Ling spectral rigidity to a problem about injectivity of a different map of the space of metrics somewhere else, namely the space of GDS occurrence. So there's a reduction of the GDS occurrence. So let me remind you what GDS occurrence are or briefly tell you what they are if you didn't know. So we have our hyperbolic surface. So here's our surface S. We look at its universal cover, I should say we have our closed surface. I'm going to fix a reference hyperbolic metric, call it sigma not say, and as soon as I do that then the universal cover will be isometric to the hyperbolic plane and I can take the compactification of the hyperbolic plane and that gives me a compactification of the universal cover by a circle, call it S1 infinity. And then the basic fact is that if I change the metric sigma not to some other, let's say hyperbolic metric, I end up with the same identification of a sort of ideal boundary and more generally actually if I take any metric, so if sigma is any metric then S1 infinity is the space of equivalence classes, weak asymptote classes of geodesics, geodesic rays, geodesic. So if I have any, I could pick a point in my space and with respect to any metric I draw a geodesic ray and that will land at a unique point. If I draw another geodesic that lands at that point these two stay a bounded distance from each other, every point is the end point of a ray, blah blah blah, you don't know this. So what's a geodesic current? So let me say if this is the set of end points of geodesic rays then S1 infinity cross S1 infinity, so these are pairs of points and let's take pairs of distinct points, so this is the diagonal, okay, so there's x, x, so we throw away the diagonal, now we have distinct pairs of points and I don't really care about the orientation for my geodesic so I'm not going to care about the order of this, so let me mod out by x, y is equivalent to y, x, so if I take two points then at least with respect to my hyperbolic metric there's a unique geodesic between those points and with respect to any metric, any sort of reasonable metric, maybe I should say these are geodesic metrics, I'm going to restrict to non-positively curved geodesic metric soon. So with respect to any geodesic metric I take a pair of points and there's a geodesic connecting those by infinite geodesic and any two by infinite geodesics will stay a bounded distance apart, okay. So let me call this g of S tilde, so these are unordered pairs of distinct points on the circle at infinity and if I let, yeah, that's exactly right, it's a mobius thing. So another identification, if you want you can think about the hyperbolic plane sitting in RP2 and it's the complement of the hyperbolic plane, the compactified hyperbolic plane. So okay, so given a hyperbolic metric or a negatively curved metric, sigma, I can also look at the set of geodesics, so this is sigma on S but then let's lift it to S tilde. I'm not going to give it another name, let's just continue to call it sigma, so now I have my hyperbolic or negatively curved metric on S, I lift it to the universal cover and I look at the geodesics, so these are by infinite geodesics with respect to this lifted negatively curved metric. Well what I was just explaining is that, oh and I should say these are unoriented and unparameterized, so I'm just thinking about the subset of the universal cover. So there's a map from this to this space and this is a homeomorphism, not only is it a homeomorphism but it's a Pi 1 of S invariant homeomorphism, equivariant. The fundamental group is acting on the universal cover, by infinite geodesics, yeah, G sigma is the set of by infinite geodesics, yeah, oriented and unparameterized. So the fundamental group acts on the universal cover by isometry, so it acts on the set of geodesics, it also acts by extension on the circle infinity, and hence it acts then by the diagonal action on there, so I have a Pi 1 of S, equivariant homeomorphism. This is for negatively curved metrics, but for non-positively curved metrics, I'm just going to say NPC, so non-positively curved metrics, we can also define this G sigma, and it's a little exercise that this is, well it's still Pi 1 of S, equivariant, what I was sort of saying over there is that it's surjective closed map, that's not too hard to work out actually what the fibers of this map are, okay, so, but in general it's not a homeomorphism and this causes some problems, okay, so let's tell you what a geodesic current is, so far I haven't done that, yep, say it again, yes, yes, that's right, so because we have non-positive curvature, the fibers here are exactly sort of strips of geodesics, so if anywhere this map is not one to one, so I have a pair of points and I look at all the geodesics that connect those points, there's a bi-infinite flat strip in the universal cover, fully aided by geodesics that are the fibers, I'm sorry, that's right, yep, yes, so the, yeah the inverse images are either points or intervals, yeah, okay, so what's geodesic current, so geodesic current is a locally finite Pi 1 of S invariant borrel measure on this space G of S tilde, so I'm gonna stop making the distinction maybe between G of S tilde and let's say G of Sigma naught which is my fixed hyperbolic metric, okay, so I'll refer to the points here as geodesics, okay, so well if you haven't seen this before then this may look like a funny thing to do but here's, so here's an example, so let's start by taking a homotopy class of curves, this is good now we're bringing back in the sort of objects that we're supposed to be interested in, so I take a homotopy class of curves, I look at its geodesic representative gamma, I'll just call the geodesic representative gamma also, so I look at the geodesic representative with respect to my fixed hyperbolic metric Sigma naught and then I look at the preimage, so here's P, gamma tilde, this is P inverse of gamma, you get a bunch of geodesics, there may be some intersects, some are disjoint, the intersection pattern here is the intersection pattern down here and now this is a collection of geodesics, so this is the subset of this space of geodesics and so and not only that but it's locally finite, so I can just take the counting measure, so notice that, so counting measure on gamma tilde, so gamma tilde is pi one of s invariant, it's the lift of something down here, so this measure that I get is also pi one of s invariant, so let me not even call it something, let's just still call it gamma, so counting measure on gamma tilde will just be called gamma again, so this gives me for every homotopy class of curves an example of a geodesic current, so in fact we get an injection, so we have a one-to-one map of this into the space of currents, so let's call that c of s, so geodesic currents, okay? I'm counting, so gamma tilde, this is a subset of this space and I'm going to count, so I need to define a measure, so given a measurable set, I'm going to intersect it with gamma tilde and count how many points are in there, so put a direct measure on each point, gamma tilde is a, yeah, that's, yeah, so, right, any, right, okay? Make sense? Okay, now in fact there's more here, so this, this, this definition, this particular definition of geodesic currents I believe is due to Bonohan, and he proves more, so he actually proves not only is this one-to-one into here, but it's also if I take positive real multiples of this, right, so now I, every curve gives me a measure, I can take multiples of that measure, positive multiples, and he actually proves that, that's still injective and in fact dense, the image is dense, so every geodesic current can be approximated by real multiples, positive real multiples of these guys, okay, so what do we do with, with this, so now we've got lots of examples and we know how to approximate everything, so the key theorem, the Bonohan, is that there exists a symmetric, bilinear, continuous function called I from currents on S, cross currents on S to R, and what does it do? So the intersection, so this is called the intersection function, if I take two currents that come from, so for all gamma and delta, homotopy classes of curves, well if I have two homotopy classes of curves, there's a natural number associated to it, namely the geometric intersection number, namely, so I, I try to minimize the number of times that these curves intersect in double points and under all, because these are homotopy classes, so I'm allowed to homotopes, I take the minimum of those, that's the geometric intersection number, and this is, there's this nice, really nice function that extends the geometric intersection number, okay, and the key point is that geodesics minimize intersections, so if I take geodesic representatives, they're gonna minimize the number of times that anything in the homotopy class can intersect, so I guess I don't have time to tell you, I mean this is not a super mysterious construction either, I mean the, the, if I have two geodesic currents, so these are measures on the space of geodesics, and then I can look at the set of all transversely intersecting pairs of geodesics, and then I can take the product of those two measures on that subset, it's an open subset, and basically I take the volume of, of that measure once I pass to the quotient, so take a fundamental domain and take the measure, okay, so, so far we've related now geodesic currents tell us something about homotopy classes of curves, actually give us a lot of information, what about metrics, so this is the other, so Bonahan was trying to give another interpretation of the Thurston compactification of teichner space, and, and he, he used geodesic currents to do that in a very beautiful way, so let me just state this as a theorem, and this is, this is really due to Bonahan, in the case of, you know, of, of hyperbolic metrics, and then Otaal for, for negatively curved, and Krog, Fati, and Feldman, okay, and, okay, and, and, you know, you can sort of push this more, Hussanski and Paulin, and, and what's the point, so given sigma a non-positively curved metric, maybe even cone metric, so I'm not going to qualify here, saying Ramanian let's allow also singularities, cone singularities, there exists, so what's the, what's the name of the metric, let's call it, let's call it sigma, there exists a current called the Louisville current in this space of currents, such that nobody can see, move it over here, such that if I look at the intersection number of any one of these curves, this homotopy classes of curves, I take the homotopy class of curves, and then I associate to it this geosic current, this counting current, and then I'm going to take the intersection number with this Louisville current for the metric, so, and what you get is the length, okay, so associated to every metric, there's a current, an intersection number with that current is the metric, is the length with respect to the metric, okay, so this is, this may seem sort of mysterious, I will tell you just very briefly where this comes from, at least in the case of like a Ramanian metric, so the Ramanian metric on the unit tangent bundle, I can look at the geosic flow, and there's a natural invariant volume form, and okay, well if I contract that volume, so it's invariant under the geosic flow, if I contract that volume form with the geosic vector field, in this case I get a two form, I lift that to the tangent, unit tangent bundle, the universal cover, the flow lines in the universal, in the tangent bundle, unit tangent bundle, the universal cover, are exactly the space of geosics, the orientation, so, and this two form just pushes down to a measure on that space, which is invariant under pi one, the two form on the unit tangent bundle of the universal covers invariant, so when I push it down, I get this invariant measure, okay, so there's this construction, and the key thing that I want to maybe just say in just a minute is what L sigma looks like, let me just write it right here, so in the case where sigma is a Riemannian metric, there's a nice form for this L sigma, so let me write down, so here's the universal cover, so here's a nice set of geodesics, let me fix some segment alpha, okay, so here's some geodesic segment alpha, and let's look at all of the geodesics that cross alpha, okay, so all the geodesics that cross alpha are parameterized by the point of intersection together with the angle, so let's say alpha here is, sorry, this is horrible, so let's say alpha here maps zero to L, which is the length into the universal covers, this is my geodesic, I can parameterize all these geodesics by T in zero to L and theta in zero to pi, okay, so these are all the transverse geodesics that are transverse to the segment alpha, and then here's what L sigma is on zero L cross zero pi, it's just one half sine theta d theta dt, whatever, okay, so if I were to integrate over this set, this function, I exactly get L, okay, I'm calculating the length of that segment, and so I was mentioning trying to sort of very briefly say what this intersection function does, sort of integrating over all transverse intersections, so if I take a curve and take its geodesic representative and I integrate this over that, basically over all the points of transverse intersection with that curve, I'm calculating the length of that curve, so that's sort of the idea here, okay, one more fact about, are there questions? Some questions seem to have died down, so one more fact about these geodesic currents due to O'Tull, and this is really what now, you know, this already suggests some connection with the marked length spectral rigidity, so the theorem of O'Tull says that the map from currents on S to R to the gamma of S, so I'm going to take a geodesic current mu and I'm going to take its intersection number with the curve and record all those numbers, so this should look familiar, right, this is sort of like what I'm doing here when I'm calculating lengths, this is injected, okay, so geodesic current is determined by its intersection number with all closed curves, okay, okay, so now we can put all this together and we see what the reduction is, if I have a metric, almost, you need to know, well, yeah, so there's a little bit more than just density, yeah, almost, so yeah, there's some, we don't know sort of non-degeneracy of this intersection, anyway, so there's a, we have our function now from here to the space of currents, so we have our metric sigma and we read off the Louisville current and then we follow it by this, this is already one-to-one by O'Tull, so this map will be one-to-one if and only if the composition is one-to-one, but the composition by this formula is exactly the length, this is sort of length, okay, so we reduce the problem of proving marked length spectral rigidity to proving injectivity into the space of currents and now maybe it seems harder because the space of currents maybe seems more complicated, but it's got more structure than the set of closed geodesics, right, the set of closed curves is just a set, here we've imposed some additional structure and that gives us some leverage, okay, so 10 minutes, let me, let me come back to this picture again of what this Louisville current is, so for this Euclidean cone metrics, one way to take a singular metric and to try to produce this Louisville current is to take a sequence of, say, negatively curved metrics that approximate it and then you could take the corresponding Louisville currents and it turns out the space of projective classes of Louisville currents is compact and so I can always extract a convergent subsequence and so that would be one way to produce the Louisville current associated to sort of singular metrics, but it turns out that you lose too much information and in particular we're going to use a lot of information about what this Louisville current looks like in the case of non-positively curved cone metrics, so the formula, one half sign, so this is this formula, this is still valid for Euclidean cone metrics, so this is still valid for phi, this is formula L phi, so where phi is now one of these flat metrics, so I should, I should clarify, right, because if I take a geodesic in the universal cover, so here's my geodesic segment alpha and if I look at the geodesics that intersect this point and make an angle theta, well if this geodesic hits, hits a cone point then there's infinitely many ways to extend it, okay, so I'm no longer parameterizing all geodesics by that go, you know, that transversely intersect alpha with just the angle and the point of intersection, but it turns out that this current is actually supported on a much smaller set, so let me, let me write down some facts, the point, so if you like you could think about where is the geodesic flow defined, so if I throw out all of the singular geodesics, all the ones that go through the cone points, then I still have a geodesic flow and that turns out to be defined on essentially, on it, well on a, with respect to let's say theta and t, a set of full measure and so I can just ignore all the geodesics that go through the cone points and the L-P measure of a set of geodesics is equal to the L-P measure of that set of geodesics after I've thrown all the ones that go through cone points out, so the way to say that is that the support of this L-P is equal to the closure of a set of non-singular geodesics, okay, and there, there was actually another point nobody complained because it's like 430, so another issue, right, is I could also have, you know, one of these flat strips and as far as a measure on boundary cross boundary, all of these, all of these geodesics give me the same point, right, these are the things that are in the fiber, so really what I should do is I should get a measure on the space of geodesics in the universal cover and then push it forward to a measure using this closed map and that's, that's what we do, so this is, but let's not worry about that, so this is the closure of the non-singular geodesics, those are the ones that don't go through cone points, so that's useful, let me just state some other properties here, so this is one useful fact, another useful fact is that, well we're taking the closure of that set of geodesics, so that means we are going to see some geodesics that go through cone points and actually that's the sort of whole key to this, to this sort of proof, so we have all geodesics, all geodesics meeting exactly one cone point are in the support, so if I look at all the geodesics that go through a single point I can approximate those by non-singular geodesics, oh I should say it and, and make an angle pi, so let me just draw a picture, here's my singular point and I've got now a geodesic that goes through here, it only meets this one and it makes an angle pi, so the, the excess here is two pi, so on the other side I see something greater than pi, okay those are in there and then there are only countably many geodesics meeting greater than or equal to two cone points, okay, so this is maybe a technical point I shouldn't have even said it, but this basically says the ones, so these geodesics that meet more than one cone point you can ignore them, there are no atoms in this measure, so from the point of view of the, the measure they don't, these are you know irrelevant, okay, so in the last, I don't know, five minutes, let me just say what the, or three, let me just say what the, what this buys you and then I will, I'll stop, I'll leave the rest of it as a mystery, so the point now is that the support, the support of this current determines a lot of information, so it determines all the cone points and in fact their cone angles, so just from the support we get a whole bunch of information, we also get a set of all geodesics between points in the support LV between cone points, so maybe rather than trying to, yeah basically that's, that's right, so somehow you know here, let me just draw a picture, so here's, here's some, here's a cone point, how am I going to identify this cone point knowing the support, well I've got this geodesic in there and then I can go and make an angle pi over here and then I can also make an angle pi over here and so you see some sequence of end points of geodesics that build some kind of a star-like thing and then there's some equivalence relation you can put on these and from that you get, you can, you can sort of identify the cone point with some subset of the support and that identification I should also say is pi one of s invariant, equivariant, so all of this is pi one of s equivariant, so that's the first point you know, that's how we figure out what the cone points are and then how do I figure, so what is this thing about geodesics between cone points, let me just draw another picture, so here I have two cone points zeta one zeta two, I can look at all the geodesics in the support that separate the universal cover into two components with these on either side and why do I want to do that because the L-thee measure of the set of those geodesics is going to be the distance between these two points so from the, once I know this, the support I get a whole bunch of information and then I use the measure to actually tell me the distance between these points so now given one of these flat metrics phi I can reconstruct all the cone points in the universal cover I can tell you what the cone angles are and I can tell you what all the distances between them are and that turns out to be enough to completely determine the metric because it's Euclidean, so that's, so that's it, I'll stop