 Okay, so I am going to tell you about partial marketing spectrum rigidity. If you have ever seen me give a talk before, like marketing spectrum rigidity is kind of the math where I live now. So I hope you still get something out of this, but I will explain what all of these words mean as we go along. So I'm going to get started and tell you what my big question is. And my big question here for this topology conference is a dynamics question. It is, what can the periodic orbits of a dynamical system X and T tell us about X itself? So a dynamical system, if you're not super familiar, like you've got a space, and then you've got some transformation T that tells you how stuff moves around. And if you're like, why is this hair, I was promised topology, I will convince you in the next couple of slides that sometimes this is still a topology and geometry question. So I'm going to reframe it and ask this in a specific dynamical system, where now my dynamical system is a compact negatively curved surface. And my transformation is geodesic flow. So geodesic flow tells you how things move around naturally on your surface, if that's the thing you're unfamiliar with. It's just the thing that tells you where the geodesics are. So I want to know what the lengths of these periodic orbits can tell me about my surface itself. If you just have the lengths, you don't know enough at all. But if you add a little bit of topology to it, aka you add in like, hey, I want to know that length, where did it come from? And by where I mean, which homotopy classic was attached to, that's enough to tell you exactly what metric you must be looking at in certain cases. So I'll get to what those cases are, but this is what we're going to be talking about. So that like dynamics question suddenly becomes a geometry question here. So homotopy classes, they must be important if adding like that little bit of locational data is enough. So let's talk about them for a minute or two. So if you want to know what's special about them, here I'm going to tell you like why that negatively curved surface part of the qualification is important. So in particular, when you have a compact surface with a negatively curved metric. And if you're like, what does this mean at all? This will mean we are looking at, oh, I meant to draw something, not put an arrow. This will mean we're looking at surfaces of genus two or greater, generally. And we won't have any like weird spherical bubbles or something popping up on them. So when we're in this kind of setting, every free homotopy class has exactly one closed geodesic in it. So if you wanted to pay attention to some class gamma, right, and I'm going to put it in the set of all of the homotopy classes, let's say this is one of the curves in that class, like, we know all of these curves in here are things that you can homotope, I mean, even something that's not great, right? But in this class, there's going to be like one king curve that's going to be special. And this gamma g that I'm highlighting, this is the only geodesic, it's also the shortest closed curve. So this, like, asking about the lengths of the closed geodesics and which homotopy class they're in, it's kind of powerful information because there's only one closed geodesic for homotopy class in the setting. And a quick couple of non-examples, if you're like, oh, this makes sense here, just as a reminder, like, once we escape negative curvature, this is no longer guaranteed at all, like, let's just think about a torus and assume for my drawing abilities that I drawn this super evenly and great, you could have a bunch of geodesics in the same class that have the same length, right? So like, this is no good for us here. But I promised non-negative curvature. If you draw a surface of genus 2, but it's got like a cylinder connecting it, so you've got this, like, flat behavior, excuse my handwriting, you've got this flat behavior here, then what you'll end up with is maybe this strip that's behaving really flatly here. So just like a cylinder in real life, you can have a bunch of geodesics wrapping around, they're all the same length, they're in the same class. So we lose this, like, specialness of closed geodesics. So this negative curvature thing I'm talking about is actually really important to make the homotopy classes be important for what I want to talk about. So just to reiterate what I was saying before, like what's special about these homotopy classes, every time you go in, everyone is going to have a unique closed geodesic. So a point I feel like I should make here is I think of these two things I have drawn here as the same surface, but I've put different metrics on them. So I know the topology of what I'm looking at, it's S, it's a surface of genus 2, but the metric tells me things about, like, the exact shape, for lack of a better term. I know some people will, like, talk about these as different surfaces and I didn't want anyone to be confused about that. I also want to point out that I'm talking about negatively curved metrics here and I don't, I don't just mean constant negative curvature, like, we can have variable curvature here, which is what makes this, like, even more interesting to me. So as we go on, and I'll remind you, like, whenever I put in, whenever I have, like, a curved gamma and I put a little metric on it, it's that special one, it's that special shortest one that's the closest geodesic. So that dynamical question I asked earlier is now a geodesic question. So just to, like, keep us in the frame of mind of where I was going with that, like, these two collections of information I'm talking about, like, they are exactly the same. When we've got a compact surface with negatively curved metric asking for the lengths of each periodic orbit of geodesic flow and the homotopy class for each orbit, that's the same as saying, hey, I want the length of the closest geodesic in each homotopy class. And when I've got my geometry hat on, like, the second one looks way more appealing to me, even though these are actually exactly the same collections of information. So I should ask if there are any questions before I go on. Okay, I'm going to take that silence as good. I'm going to formalize this notion of asking about length of periodic orbits in each homotopy classes. And that means we've gotten to one of the, like, big things in the title. What is this marked length spectrum? So the marked length spectrum for a negatively curved Romanian metric, oh, sorry, for a negatively curved Romanian metric. And I'll get to what this word means in a second. It's a function. It eats homotopy classes. So it says, hey, I care about curbs in the class gamma. And it spits back out the length of that special closed geodesic in every class. So this is just like it's a machine and it's like, I want to know the special number and I want to know what class it came from. This word, Romanian, I won't define formally, but if you are familiar with metrics at all, if you're unfamiliar and you're unconcerned about the word, you can replace it with nice enough metric in your head. If you really care about this sort of thing, Romanian means your metric has an inner product that varies smoothly, which really means like your angles are varying smoothly. And there aren't any like weird surprises that happen with the geometry. That's what I think about this. This is the fail-safe to make sure no wild things are happening. So this is the marked length spectrum. It encodes this like homotopy, lengths of homotopy classes notion. But I often will actually think about this as like the marked length spectrum of the whole surface in the metric. And it's this thing I have written down here, which says I want all of the lengths, but I want them to remember which homotopy class they came from. So that's why it's marked. You mark the lengths with the classes. So it's not like an unmarked, an unmarked length spectrum where you could count lengths with multiplicity. So that is what the marked length spectrum is. And a little bit, I'll get to telling you about what marked length spectrum rigidity is. But first I want to give you a result that like starts giving us the feeling of how powerful the marked length spectrum can be. So I want us all to remember at this point that at best right now, the marked length spectrum is a lot of numbers with like vague locational data, right? Like just the homotopy classes. That is not a lot at all. So this next result by Krokin-Darbakov back in 2004. This is saying, if you've got two negatively curved romanian metrics, so this is the setting we've been in the whole time on a compact surface S. So just for those of us who are still unfamiliar with the language, right? We've got two surfaces of genus two. I should close it up so we don't think anything weird's happening. So let's put the first metric here. And let's put the second metric here. Oh, man, my pen's acting weird. I'm sorry. OK, if for every homotopy class in your first metric, you go in and check the length. And it's shorter than the length of the corresponding curve in your second metric. I'm saying corresponding curve. I should say corresponding geodesic. Then the area under your first metric has to be less than the area under your second metric. This is weird information to hold in your hands. But really, in every homotopy class, you throw down a rubber band, right? And you let it naturally fall to the tightest spot. Every rubber band in your first metric is going to be smaller than the one in your second metric. That's sort of how I think about it. So in a weird way, I'd be like, OK, it's cool. You're sort of like strapping the first one in tighter. That should mean it's taking up less space. But this is sort of a big deal. It's saying this vague locational data is enough to give you this actual area comparison. This is where it goes from vague to very real. And I promise that this is a marked length spectrum area resolved. It's hiding right here. This is a comparison between the lengths in every homotopy class. So this is the same as me saying this underlined part. If I had said the marked length spectrum under G1 is less than or equal to the marked length spectrum under G2, just enough space. So this is a marked length spectrum result. Sorry, this slide got a little crowded. I'm going to pause again right here and see if anyone has any questions about this in particular. Hi, I have a question, which is sort of for the previous slide in some sense, a closed geodesic can intersect itself. Yes, it can. OK, cool. OK, great. I'm actually going to bring that up a little bit later about geodesics that don't intersect themselves. So great timing. Are there any other questions? OK. All right, this is our very first marked length spectrum result for this talk. I'm going to give you one second one that looks almost the same, except this one's got my name on it. So this is the same setup as before. We've got two negatively curved Bremonian metrics on a compact surface. And now I'm saying there's some subset of the homotopy classes. And maybe I did not formally define the C before, and I'm sorry. This C is just the collection of all the homotopy classes. So there's some set. I'm saying it's got sub-exponential growth rate by lane. I haven't explained what that is yet, but I will. So now we go in. We make this length comparison, but we only check it on the classes that are not in the set B. Well, that's cool. We don't know what's happening on the set B, but everything else is still enough to let us make the area comparison. I think about the set B. It's literally named B because B for bad. Maybe the length characteristics don't match up quite well. Like maybe the inequality is flipped. I don't know what's happening, but it's fine. That's that can't stop our shine, right? Like apparently the sub-exponential growth rate by length is the right kind of idea to say, hey, it's cool the rest of the Mark Lane spectrum can do the heavy pulley. So I am going to explain on the next slide what sub-exponential growth rate means later in the talk. Hopefully I'll get to telling you why that's the right sort of idea here. But I do want to point out here is the first time I'm using this partial Mark Lane spectrum idea. And the partial is we're looking at the Mark Lane spectrum, but not for all of the homotopy classes. So we're looking at it on a, we're throwing out some set, which here is B. And at no point in time will B be anything other than a set where concerned is bad. So let's talk about sub-exponential growth. Why is it here? What does it mean? So you've got a subset. I said it has sub-exponential growth rate. The thing to pay attention to here is what I'm really saying is this set B of T, B sub T, where that's all of the homotopy classes in our set, where that special geodesic has length less than T. The growth of that set and T being sub-exponential is what I mean when I say B has a sub-exponential growth rate. And so that means that this limit I have here, like that this is equal to 0. And really what this means is that B does not have any long-term exponential behavior with all the terrible stuff that's going on outside. I see a raised hand. Is that you can ask a question out loud if you'd like? Yeah, right. Yeah, I just wanted to ask, so does this depend on the metric, or is this notion independent of the metric you use? So when it's 0 and we're in negative curvature, being 0 is independent of the metric. But once you get over into positive exponential growth, that depends on the metric. Well, thanks. No problem. Yeah, so that was a great question. Yeah, in general, that's what I said, like by length, because the length does depend on the metric. But luckily, 0 is a real sweet spot. So it's not a real worry here. OK, so what it means, the growth of the set doesn't have long-term exponential behavior. That's cool. Are there any interesting sets? Like, did I say anything interesting on the last slide? But people might care about. Well, if a set's finite, that is going to have sub-exponential growth, because it literally stops growing. It can't have long-term exponential behavior. Someone asked about geodesics being able to intersect. When you're in negative curvature, the way the dynamics work is it's really difficult for things not to intersect on a surface. Like, it's really, really hard. So if you look at the set of homotopy classes where the representative is simple, meaning it doesn't intersect itself, it's an infinite set, but it grows really slowly. So it matches the sub-exponential growth rate here. And the same is true if you just pick your favorite number and say, I want to look at all of the classes where the representative has fewer than that number of self-intersections. That also grows really slowly and up to match this. So these are still interesting infinite sets. So adding the sub-exponential growth rate isn't quite a cop-out when I could have just said finite set. I should also mention exponential growth. So also, that other question was super well-timed. We say it has a subset grows exponentially if this limit is greater than zero. And this is gonna change depending on our metric, but I don't really focus on things with exponential growth in this top. So this won't really come up. But examples are this curly C, which is literally all of the homotopy classes. Hopefully they would have all grown exponentially quickly or I'd be claiming to throw all of them away and still get a marketing spectrum result. And then I have the second bullet that I'm not fully defining, but dense geodesics are really nice because they go everywhere in your space. Like it's really awesome. If you want to see how something behaves, you can follow your dense geodesic and eventually get there. The issue is with the eventually, like dense geodesics are not periodic and they are sort of terrible to study individually, sometimes in certain settings. These epsilon dense geodesics I'm claiming are geodesics that are closed but come within epsilon of every base point and direction. So like almost dense geodesics. So no matter which epsilon you pick, the set of almost dense geodesics, this grows exponentially quickly. So that's second also interesting that the people wanted to know. So now I'm actually going to tell you about marketing spectrum rigidity. Hopefully we're still interested in about this point. So there's this theorem by Kroek and Otau in 1990. I maybe could have written this as Kroek 1990 and Otau 1990 because they proved it independently in the same year. So if you've got two negatively curved permonium metrics on your compact surface, so we're still thinking about surfaces of genus two or greater, then if your Mycelene spectra are equal, so class for class, you go in and you check the geodesic representatives. If you go in and they're all equal, then your metrics have to be the same up to isometry. Isometry is a distance preserving diffeomorphism. So like they're in practice the same for almost everything I wanna know. So what's happening here? This side of the arrow, like this implication up, this isn't as interesting because it's like saying, hey, these two things I have look exactly the same. Check the lengths of the geodesics. Are they the same? I mean, they should be. Like the metrics are exactly the same. Like this is information we had, but this arrow here, this is saying, hey, check those lengths in the homotopy classes. I won't tell you where the geodesic is, just check to see if they're the same lengths in both metrics. That's enough to tell you that we have to be exactly the same. And that is wild to me. Like that, like how, right? This is my link spectrum rigidity because the marketing spectrum, which is like an infinite amount of data, but doesn't feel like enough is enough to tell you exactly what metric you must have. And so that's how the phrase rigidity gets used a lot. Like sort of floppy data is enough to give you all of the information. So that's what's happening here. So this was like a big result. I'm still, I'm not surprised every time I read the proof, but like the overall proof strategy is like, wow, how did you think of that at all? But this is sort of, this is the sort of thing that I think about now I'm gonna show you like my improvement on this result, which is maybe what you would expect at this point. I wanna throw more stuff out. So I still want the setup of negatively curve for money and metrics on a compact surface. I still wanna do something with the marketing spectrum, but now I wanna restrict the marketing spectrum. That is I only wanna look at it on the conjugacy classes that are not in this set B. That might be that. I don't know what's going on with it, but if off of this set, you check all of the homotopy classes and the lengths are equal, that's still enough to tell you that you have to be looking at the same metric. So not only is like sort of floppy data enough to tell you exactly what you must be looking at, even less floppy data is enough to tell you what you must be looking at. So this is exciting to me. Sorry guys, like my thesis to Francis this week, so you're getting like my thesis work. This partial marketing spectrum rigidity like is a real improvement because I could have written this instead of like partial marketing spectrum rigidity as maybe like marketing spectrum super rigidity. Like it's so good. It doesn't even need all of it to tell you exactly what you're looking at. Another way you could think about this and I wrote it as a corollary because this has made it at home for me sometimes. I have talked about B as maybe being a bad set like maybe a quality doesn't hold or something else weird that's going on, but you can really think of this as saying, hey, you checked the partial marketing spectrum as long as that set you didn't check is growing sub-exponentially that it means it couldn't have been a bad set at all. Like they actually had to have been the same length the whole time. Like the marketing spectrum so good that it makes the rest of the marketing spectrum falls into lines like that's, maybe that's the end didn't make sense. But this is partial marketing spectrum rigidity which was the part of my fight. And hey, we're gonna talk about this for the rest of the time. I'm not gonna try and like give you a full proof of this, right? Like, because one, we don't have enough time. And also I'm really just gonna give you the parts that have lots of pictures in them because they are a little bit better to follow. But I think they give you the best feeling for what's happening. But I'm gonna do that by spending some time talking about why all of these assumptions and the theorem are important. So this negatively curved part, we've touched on it before but I'm just gonna remind us why this is important. I'm gonna talk about why a surface is the right sort of thought to have here. Like why two dimensions? Like why aren't we doing this in higher dimensions if we could? And then if there's some time at the end, I'll tell you where the sub exponential growth rate comes in. So first I'm going to explain to you like or just bring up again, like what we talked about with negative curvature at the beginning. The reason we care about negative curvature, right? Like you've got this one periodic orbit for Homo Tofi class. Like it really carries us through a lot. Like we really need it to make like the Mark Lane spectrum at least in the way that I'm going to use it. Like we really need it to carry us through. And also that point I made about the closed sheet Essex that are almost dense like that's growing exponentially quickly. That's also a hallmark of negative curvature or like some hyperbolic behavior. So that's why negative curvature is coming in. It might come up again a little bit later but these are the two points you've already come through. So like, look at that. We could already track something off, right? This second thing about two-dimensionality I am going to have to explain a part of the proof because but there's going to be a picture at the end. And when we see it it's going to be like, okay there's no way to like nicely transfer this to our nicely adapt this to higher dimensions. So right now we're going to work under the assumption like, we've got a mark my inspection in our pocket like we've got all of the information and we just want to see if we can figure out what the metric is. But our goal while it's going to be a sketch of a part of the proof like the goal is to really get you to understand why two dimensions is the right setting for this. And so to motivate like where we're about to go Otal proves this thing in his paper about something called a geodesic current which I will at least partly explain in a little bit. And one of them is called the Leaville current, this lambda G. And what Otal proves is that if you're in the starting of the proof we're in and your marketing spectrum is equal then these two Leaville currents that have metrics attached to them so they've got something to do with the metric they can be equal if and only if your metrics are equal. And why is this exciting? Like why is this helping us at all? That really means that what he's saying is if you can reconstruct whatever this Leaville current is that's enough to uniquely identify the metric G. So if it turns out it's easier to take your marketing spectrum and turn it into and use it to reconstruct whatever a current is you can use it as a sort of middleman to figure out say, hey, I've uniquely identified the Leaville current and that's enough to say I've uniquely identified the metric. And I don't wanna trivialize this theorem at all. Like this is like large part of the paper he spends doing this but I feel like find it's like less enlightening than what I'm about to talk about. So, okay, I wanna tell you sort of what a Leaville current is and then we're gonna reconstruct it. And while that may sound like a big task I think that we can do it in the next not that long. So, all right, what is a current? We've gotta talk about something called the boundary at infinity first just for a little bit because I'm gonna start pointing at this and some people may say, what are we doing? So what I've drawn here, this S this S is just the universal cover of our surface. So we're a negative curvature it's gonna look like sort of like the disk model of hyperbolic space which is why I have this drawn here. And we draw a circle around it because although we don't really talk about the boundaries being there like we like to contain things and we draw pictures but there is a way to discuss like the boundary of this space. So for this part if you'd like to just keep hyperbolic space in mind while you're doing it this like boundary at infinity talk still totally transfers. So what I have written here is that the boundary at infinity is the equivalent class set of equivalence classes of geodesics at infinity. What do I mean by that? If you were to pick a point on the boundary like let's just pick this point here. This can be thought of as the set of geodesics that all would like head toward this point like given infinite time. So all of these geodesics eventually like become bounded distance apart and stay that way. So if you were naturally gonna think about what the boundary would be this is sort of what I would have guessed. I would not have written it down as like explicitly but this is the boundary at infinity. But what's exciting about the boundary at infinity for us to negative curvature? One thing that's really nice is that because we've got those this is tied up in that whole like hyperbolic behavior uniqueness of geodesics and homotopy classes and things like that. If you pick two points on the boundary there's only gonna be one geodesic that could have like started at one like started at negative infinity and ended up at infinity at the other point. So two points on the boundary are enough for us to uniquely identify geodesics which is like, I mean, it doesn't seem like it would be a problem except it does become a problem once you're in non-negative curvature. So this is like sort of a bad example I'm maybe gonna draw. So I guess a non-example of that case. If you remember our friend from earlier our surface of genius too but we've given it a little flat behavior. What happens here is that these geodesics in the middle that are all like these geodesics in the middle that are the same length and that are parallel all of these if we were to look at them in the universal cover which would still sort of look like hyperbolic space the thing that would happen to us is we'd have lifts of these geodesics we'd have these two points on the boundary but because they stay bounded distance from each other like all the time they ended up being identified by the same two points on the boundary and that's like not great for us. We don't really like that. So I'm gonna get rid of this after picture in a moment but the negative curvature is important for this like unique identification. So this is the point I was really trying to make here we've got this nice thing it's a boundary it helps us talk about sets of geodesics. So I have conveniently drawn one here it's all of the geodesics going from one loose that into the other and this is one of the topologies we could put on like the space of geodesics if we wanted to talk about it. Is there another point I wanted to make here? I also really like the boundary because you can sort of forget about the interior of this space if you just want to think about geodesics you can think about all of the geodesics as just being pairs of points on the boundary. Now I've been on this slide for a while and I promised you like two slides ago that I was gonna tell you what geodesic currents were. The reason I talked about the boundary at infinity is because this is sort of where geodesic currents come into play. So geodesic currents they are locally finite depth transformation and variant measures. So just like all around nice measures on the space of geodesics on a universal cover. What this being a measure on the space of geodesics means it means that geodesic currents measure sets like the one I have drawn over here on this picture. So a geodesic current will eat the set and spit a number back out. One of these currents is the Leaville current which I mentioned a little bit earlier and the Leaville current has the metric G attached to it because it does super depend on the metric. I have chosen not to explicitly define currents because it would take us not to explicitly define it but we can talk about it later if anybody has questions but it was important to know that to reconstruct it but it's gonna be a measure on the space of on sets of geodesics. And while you may think we definitely need the full definition to do this with a proposition I'll give you in a moment we actually totally won't. So please be patient with me. So this theorem I have on the bottom here it's a shortened version of O'Tal's theorem earlier. So this is saying again once we're in the setting we're in which is the Markling Spectra are equal then this Leaville current is enough to uniquely identify the metric. So if you're wondering where like the partial Markling Spectra part of this went to turns out it still holds with some adjustments to the proofs that he did. The Markling Spectra is if you have the partial Markling Spectra equal then the Leaville current still determines the metric. So that's the kind of place we're at right now. So the goal which I had pointed out before we knew what a Leaville current was was if we can reconstruct what the Leaville current is we can actually tell what the metric had to be. So that's what we're gonna do now. I forgot to think about this picture and I hope people can see it but I'll point out what that smaller word is in a moment. Okay. So I promised you a proposition that would help us understand how to reconstruct this metric. Well, Bono improved in 88. So this is two years before that rigidity result came out. If you've got a homotopy class scammer if you've got a homotopy class scammer and it's got a geodesic representative on your surface right there then the length of that curve on your surface is going to be equal to the how the Leaville current measures a certain set of geodesics. Okay, so right now this is saying without talking about what's in that bracket somehow the Leaville current remembers something about lengths of geodesics and why this is exciting to us is that this thing that's written here like this is Mark Lane's vector info. Like that's totally what that is because we're thinking we've got all of these lengths of homotopy classes, right? And this proposition says, okay we can translate these lengths into something to do with the Leaville current. So what is this thing the Leaville current is measuring? It's saying geodesics intersecting one copy of gamma G. What does that mean? So over in this picture, I've got on the surface that geodesic representative and then here I've got one lift of it. It doesn't matter which one I chose and then I've got a highlighted segment of it that's the same as the length of the curve on the surface. And again, it doesn't matter which one I chose but you can think of this if you want to seem like a fundamental domain of the curve on the surface. When our Leaville current measures all of the geodesics that intersect the special segment, it says, oh cool, I know what that is. It's exactly the length. Like the number it fits out to you is gonna be exactly the length and it's really good at processing like that as geometric information. So right now with the Mark Lane spectrum, if we take any length, if we take any lift of any close geodesic we want, as long as we look at geodesics that are intersecting some segment that would project exactly to one copy of the geodesic on our surface, the Leaville current can tell you what it is because it's the length of the segment. It's really good at that. It's really bad at, right now we don't have any other information like once we wiggle the length like all of a sudden we've lost it but right now this is a ton of information in our pocket just from the Mark Lane spectrum. So we're gonna take this and we're gonna really like press it until we get the information that we want. So right now what I'm gonna do is show that this proposition is enough to say using the Mark Lane spectrum, we can find the how the Leaville current would measure any given set of geodesics. Okay, that's what we're gonna be doing for like the next two slides. And once we've done that because of that theorem from before that's gonna be enough to have you need to be identified the matrix. So we kind of would have found, so we'd be like done with the proof based on that. So let's see how we would do that, right? I've put up some random set. I wanna look at all of the geodesics that go between I and J. So that means I want all the geodesics that are sort of like here, right? Like all of these, not covering those letters on purpose because I wanna reference them. The thing that we can notice here is that all of these geodesics, they're going to intersect this 801 and 802, right? Like it's sort of acting like a marker for them. They'll intersect both of them, but they will not intersect gamma one or gamma two. So once we have that information, we can remember like, hey, we actually know how to talk about sets of geodesics that intersect things. Like we know how to measure sets of geodesics that intersect things. Okay, so how do we do this? We can measure the geodesics that intersect 801, right? Let's measure those, but we know that according to that proposition, that's gonna be the same as the length of the segment. Then we add it to the measure of these because we just wanna make sure we've crossed both of them. Okay, that's that length, but we don't want any of the geodesics to cross these because we only want geodesics that are going between I and J. All right, well, that means take away the length and take away the length because that's the same as the Neville measure of geodesics intersecting these sets. We end up double counting because you like count the geodesic when it crosses 801 and again when it crosses 802, but like, I mean, it's cool. Like we've pretty much done it, except like the problem is that this statement I threw up above here, like this length translation only works if like these gamma one and two and 801 and two are like segments of lifts of closed geodesics that are the correct length, right? Like that's the only way it works. And it also only works if these are lifts of geodesics that are not in that bathtub B. So we're like almost there, right? But the good news is is we can always take limits of things that look like this, except what we're gonna do is we won't have them going all the way out, right? So instead of having these pink geodesics like head all the way out to the boundary, like maybe we'll stop them here and here and here and here, right? So you can find like four sets of four geodesics that get longer and longer and longer heading out to the boundary and you can use them to approximate. So as long as we're not using any like bad tomatopi classes, we can take this limit and be totally fine. And this is like the setup of Otal's group. So once we know that we can like reconstruct the measure of this arbitrary set, like this would have worked for any set that we wanted. So we could have, so we can actually reconstruct the Liyaville current based on this information, like with the copy out of those bad sets. And so once we know that, based on that proposition before, we can uniquely identify the metric and we're like totally done guys. Like we've proved Mark Lane's spectrum rigidity in not that much time. The issue still comes with like these bad sets, right? So that's a thing to bring up maybe not actually talk about for that long because there's not a ton of time. But although we may have forgotten at this point, I did wanna tell you why this totally won't work in higher dimensions. Like essentially what we did was trap a geodesic with other geodesics. And you cannot do that once you get up into three dimensions, like you just can't. Like this only works because this picture is flat and you can't just like lift this purple geodesic up and like put it somewhere else, right? Like there's no way to work this in three dimensions not this exact argument. So like that's sort of why, but not sort of. This is one of the places where like being a surface it's like really intrinsic to making this argument work. I'm not as familiar with Croak's approach. So I showed you like what Otal did, but like this is the one that I've paid the most attention to because it's the one that the proof style that I've ripped off of. So we have answered why dimension two and we've still got a little trouble, right? Like what if all of the choices we've made somehow like when we're doing this approximation cause we're awesome, what if every choice we make somehow has one of these bad geodesics show up? Like that's the only way we can get to partial Mark Lane Spectrum rigidity. But right now I would just like to point out we've got another check mark. Like we've done two out of the three things. And so I won't really take a lot of time to talk about sub exponential growth. That'll give you like a little bit of an idea. So just now what we did was we showed Otals sort of we gave a sketch of Otals proof when there is not partial Mark Lane Spectrum rigidity that we've shown if you have a full Mark Lane Spectrum you can uniquely identify the Leeville current which will uniquely identify the metric. The issue is now like what if there are some classes we don't want to use? So this, which I think I'll take to be my final slide. What you do or what I did to talk about slow growth is when I'm trying to approximate an arbitrary set if you see instead of having these like single points like this these end points of I and J on the boundary that I'm trying to approach I need a little bit of wiggle room. Cause like what if in this approximation sequence I always have like bad home to be classes show up like it's not an especially fast growing sequence it could show up. So I needed a little bit of wiggle room. So I introduced like some sort of error and this set on the boundary I'm trying to approach which is going to give me like more spots I can send this like approximation sequence to land and doing that sort of gives me the opportunity to like generate a ton of sequences like instead of maybe one or I well I don't know how many there would have been with Otaal's group but instead of being like I found one all of a sudden I say, okay I found like a bunch of sequences of these like crossing geodesic pictures that could help to approximate. And then once I've done that there's some work to be done to show that okay I've got a ton of sequences they're all approaching the boundary what if somehow like they've all still got like these bad homotopy classes show up. So I've also got to do some work to show that there are a lot of unique homotopy classes showing up so like one bad set can't one bad homotopy class can't keep showing up repeatedly and ruining my life or at least it's showing up only a much smaller number of times than like it would need to ruin what was happening here. So I show that there are many, many unique homotopy classes that are showing up and in fact the number of unique classes showing up is growing exponentially quickly since our bad set B is not growing as quickly like they can't stop me from finding good unique homotopy classes because the number of good classes is growing so fast that they're outpacing the bad classes. So maybe I've got to start this picture out at like a much greater length because the growth rate is by length but it's okay. So once if I can show those things then like once I've tossed out like some subset B that's bad but grows slowly it still leaves most of the sequence I wanted to build or most of many of those sequences I'm seeing I can build but I am actually gonna be done here because I would like the rest of this to run on time. So thank you everyone. All right, thank you Noel. So much, if you guys would like to you can unmute yourselves and applaud or you can use the participants pad or participants break out and try to be clap emoji there. So thank you so much Noel. So I'll go ahead and open it up. Are there any questions for Noel? So does this depend on the fact if you are manifold as a boundary or not? This, I am thinking manifolds without boundaries. So like those surfaces of genus too. Right, so could you like tell us where it would fail suppose if I had a pair of pants or something like that? If it has a boundary. So I just, I don't know if a lot of the like geometric meaning of a lot of the things I'm talking about like hold up once there's a boundary. So even like what is the universal cover of a pair of pants? I don't, is it, do you know? Wait, this is a real question. Like I'm thinking like holds might show up. Well, okay, holds can't be any universal cover. Wow. Okay, so I'm just flipping back for a second. So like these, like this proposition and these things with geodesic currents depend on like manifest places where geodesics are allowed to go on forever. So I'm thinking if you've got something like a pair of pants where like there's a boundary or geodesic could hit and just like not be able to go forward there might be a problem but I would have to think more about that to give you a better answer. All right, thank you. No problem. I have a question. Is there a sense in which there's like the smallest partial length spectrum that you could consider or like a way to say that this is like the largest bad set that I could remove? Now, I mean, that's a great question and people are thinking about it, right? So things I want to think about later are like can I increase the exponential growth and still make this tick? Like can it have positive exponential growth? But some other people I'm thinking about are asking like maybe not what's the largest set you can throw out but like say I just pick how big the set is that I throw out like how good of a bound can I get? Like how close do things have to be together? But I strongly suspect if you let me flip back awhile to that first Mark Lane's spectrum result. Oh, wait, where am I? That here with all of the things drawn on it that this is probably allowed to just be strictly less than the exponential growth rate of your home at OB classes, but I don't have a proof yet. It's just a strong hunch. Thanks. A question. Yeah. Could you sketch a bit of a definition of a little bit current? Yeah, I can. If you, I'm gonna find a place where I have got circle with some space on it. Okay, the Leeville current. Okay, so once the Leeville current is, so I can tell you how you would integrate it and then I'm gonna draw you a picture and show like where the factors come in. Okay. So hopefully I can write this well enough to be legible for us. Okay, so D lambda, D lambda G is equal to, I think it's half sine theta D theta D T and where this theta and this T come in is you have got, so like say I was trying to measure this set, I should have set that. You take some geodesic that's like transversal to it. This is independent of which transversal you pick, but I don't have like the answer in my pocket to give you quickly for that. This is independent of which transversal and where the theta comes in, is it asked what angle your set is making as a process and then the T comes in, like you pick somewhere to be like zero and you are asking like where the geodesics are crossing this transversal. Okay. So this is why it's like really intertwined with the metric because it like really cares about the geometry inside. One question, does it relate to the limit measure? Yes, so this projects down to the, this projects down to the Leoville measure in some sense. So I don't know if this is gonna be a satisfying answer to you, but it's like if you take the Leoville current and you like cross it to also care about like length from how that gives you the Leoville measure. Okay, yeah. Yeah. Yeah. Okay. Thanks. Wait, okay. Does anyone else have any questions for Noelle? I have a question. Yes. So in your, so you want to say that if the Markl and spectra are equal, then the metrics are equal. I'm sort of confused about what he exactly said. Are you saying that the metrics are equal on the nose or is that an isometry, a self-diffuomorphism of the surface that takes an isometric to the other one? This is a, they are isometric, so there's a distance preserving the defuomorphism. Right, and that defuomorphism, so could the defuomorphism be different from the identity? Yes, it can be different, but it preserves like how all of the homotopy classes work. I don't know if this is exactly the correct way to think about it, but like a defuomorphism that like might somehow just like invert everything would like make a change, but like would preserve the lengths. I don't, I could give you probably a better answer in just a little bit. Oh, sorry. Maybe I should say this differently. Do you know if the defuomorphism would be isotopic to the identity? I think so, yes. Okay. I think so. Okay, okay. All right, so it's one o'clock, so let's go ahead and thank Noel again for giving such an awesome talk. Thank you so much, Noel. No problem.