 Okay, can you hear me? Yes, I think so. So, I would like to thank Sarsapu Fetov for the opportunity to speak here. And it's my first time in Italy. I've never been here before and I'm thoroughly enjoying this country. It's really nice. But it's too hot for me at the moment so maybe I'll come again in the spring or something like this. So this is a joint project with Etienne Lemasson from Crystal and it's something about essentially about the Laplace operator. So I know that in the audience, I mean it's mostly consisting of people working on dynamics but I mean I want to talk a little bit about the Laplace and what it is. So it's a quite central difference of operator in analysis and PDEs and then he just briefly mentioned what it is like intuitive or heuristically speaking. So, you take a combative Riemannian manifold, okay? And then you define the Laplacian of a function F, define on the manifold to be roughly the rate at which these spherical averages deviate from the value of the function of the point X. So here you take a point X in the manifold and it takes just the raw sphere in the usometric in the manifold and then just study the average value of the function. Okay, you get some number and then you just measure how much does deviate from this value. So for example, harmonic functions are those where the Laplacian is zero so then here this average is equal to this thing so there's no deviation. Okay, so in the R2 it's as this usual. You've probably seen this definition before. You can define it using secondary values. So that's the Laplacian. Okay, and so there are these specific numbers if you have a compact surface, these so-called eigenvalues of the Laplacian. So they are just those numbers that satisfy this equation. There are these functions that satisfy this equation. That okay, so take the Laplacian, the point X. So this deviation rate and then you just ask that okay, it's roughly the same as a constant times the function itself. And that's an eigenfunction. And okay, so there are several ways to choose those eigenfunctions that there's no mean, like it's a unique way. In some surfaces, like in the ball, there's a lot of multiplicities. So I'll give an eigenvalue, you may have several different eigenfunctions, but then you can just choose one. And they are really like building blocks for harmonic analysis. That's one of the fundamental things you can use them for. Like if you want to do Fourier analysis or other things, you can define an orthonoma basis using, you can choose them so that they form a basis like if you go through the eigenvalues. And that's a very standard thing that you usually do in the analysis. The other thing is that there are these like more applied things that you can use them to like somehow reconstruct your surface. Like in some cases, there are these questions about whether if you know the eigenvalues, can you reconstruct it? Let's say in here, you have this other, this surface is just actually the disk. And these are some of the solutions to the, I think it's the wave equation in this case. And you can ask if you can solve the, solve the, like finding out those numbers if you can then say something about the surface and also if you know the eigenfunctions, can you reconstruct some of the manifold efficiently if you can do something with it. So the final thing is that these eigenfunctions, there's a natural dynamics in quantum mechanics. I won't talk about that much, but there's, it's all called stationary quantum states. So if you think about just take an eigenfunction to Laplacian here and you take the eigenfunction, modulus of that function squared, then that's a probability density. It's a, it's a probability density function. And it describes you somehow the position, the probability position of a particle in the system. So here these are pictures, these are billiard actually. So corresponding some domain and there are some boundary conditions for the, for this equation here, some directly boundary conditions. And then you're solving the eigenvalue equation and you get these functions and there are some distribution of these things. So they look sometimes like this and sometimes they look like this if you change the energy level for different eigenvalues. Okay, so here are the eigenfunctions and I would like to now use dynamical systems to study these things. So that's the basis of this whole talk. So, so one of the standard dynamical systems on a Riemannian manifold is the geodetic flow. So you put it on the cotangent bundle here. So that's just because you have directions and then you have the points in the definition of the geodetic flow. So what do you do here? So you pick a point and you pick a direction and then you're looking around you to that direction. You want to go to the direction where you minimize the distance. You don't want to go like where the distance is going to be large and then you like locally minimize the distance. So that's the geodetic flow and you get the new point and new direction at that point. And then you just follow around the flow around the space and you get some new points. So that's the geodetic flow on this, this, this place space. So the typical geodesics of interest that you usually study here are closed geodesics. So those geodesics that you return to the same point, you have the same direction. So it's pointing the same place. So it's always looping around in this space. So these things are important for algebraic reasons and other things if you want to describe topology and other things. And the other things which is more, also quite important there, but here is more useful is this equity distributed geodesics. So those geodesics that somehow equidistribute the whole surface. So you begin from some area in the surface. So here actually the areas in the cotangent bundle and then you somehow visited uniformly around the whole time. So if you take the statistics, then you're seeing roughly the volume. So this is the Louisville measure. So that's the most uniformly distributed measure on the surface. And you're seeing roughly how much mass you have there in this frequency. And so the Louisville measure just to remind you, it's just the projection of the Louisville measure is uniformly distributed measure on the surface. So if you project it onto this surface, then you just get the so-called Lebeck measure of the volume on the surface. So that's just the natural, absolutely continuously varying measure on the cotangent bundle here that you use. Okay, so an ergodic geodesic flow is a flow where almost all geodesics are equidistributed. Okay, so I'm lying here because I defined the equidistribution to be just for the open sets, you're testing for the open sets. But if you just test it for all the sets of positive measure, then that's real definition or equivalent definition of ergodicity by the Birch of ergodic theorem. So that just means that if you take a typical initial point and a typical direction and it works out the flow, the most of geodesics just look like the blue geodesics. So the red ones are quite rare in this sense. Okay, so there are several examples where you can have this like hyperbolic surfaces where the curvature is let's say constant negative minus one. That's where you have like actually quite a heavily mixing geodesic flow. So these are all playing from the hyperbolic plane by for example, this group actions you can define using them and everything. The other thing is some domains, some billiards. So you can define the billiard flow and define suitable ribbon and metric here and then you have an ergodic flow in this case. So if the boundary has certain shapes then you can have this ergodicity. So why am I talking about the ergodicity and then I was talking about the eigenfunctions of the Laplacian? Well the point is that if you have an ergodic geodesic flow on the surface it's a condition of the surface. Not all the surface is satisfied that the geodesic flow is ergodic but if they satisfy that you can say actually quite a many things about the eigenfunctions of the Laplacian. And there's a numerical evidence that's showing that if the geodesic flow is ergodic then this probability density if you remember for the eigenfunction of the Laplacian. So here we have the eigenfunction of the Laplacian so you're choosing a basis. So it's the orthonormal basis. They should equidispute. So this is just more like numerical evidence if you're just doing, just pick your domain and do some computer simulations. You start to see that they somehow spread around quite evenly. So here for example in this domain you would have this billiard flow and then with some boundary conditions here and then on here you have some energy levels. So some eigenfunctions here and you choose some eigenvalue. You see something like this, something like this. But if you start to increase the energy you should start to see something quite uniformly distributed. And the theorem here is this quantum ergodicity theorem. That if you have an ergodic characteristic flow then these things converge to the function one weekly. So what does this mean? So you take the, remember that except for a density of one subsequence this is very important here. That there is like a, you throw away a few of the eigenfunctions you'll hate and then you don't like them and then you just have the rest and they converge to one. So there is this specific subsequence which is non-constructive in general but how do you find it? And this just means that you have this probability density so this is just a function square and then you multiply it by the volume. So the volume of the surface and this measure converges weekly to the volume measure. That's what this means, this weekly here. So d wall converges to weekly to the d wall. So that's an equity distribution. That's quantum ergodicity, that's what it means. And then you have this density one subsequence which you have here. And this is necessary. So in this recent, like maybe eight years ago something like this Hassell constructed like ergodic examples where you have this ergodic flow but you don't have quantum unique ergodicity. So quantum unique ergodicity would mean that there's no density one subsequence here. And so the question is that this, for example, hyperbolic surfaces and other surfaces are more chaotic in some sense than just ergodic. So for example, hyperbolic surfaces, there's this like very strong mixing properties for the flow. So now the question is can you use that to get rid of these problems that you have here like this density one subsequence? Because in this Hassell's example you don't have this like exponential mixing in things going on. So the conjecture here is that you should be able to do that and but this is still open. So you cannot, I mean it's still, I mean there are some partial things that you can do in this setting. So there's this, if I'm a hyperbolic surface, actually the conjecture and hyperbolic manifold so it's high dimensions as well. But then if you can have this without the dense subsequence and this is the quite famous result where you, if you assume that your basis of eigen functions so-called Hecke basis, so you choose a specific basis and then you choose the surface such that it has, okay some congruence subgroups of the SL2Z I think or something like this and then from those surfaces you can do something. You can actually prove that there's no exceptions on those specific examples of the or the normal basis. So you can use the arithmetic information about manifold or other information to do overcome this information but still this, if you just have the general thing is the open. So my talk today is not going to be about solving the problem, sorry it's still open but then I'm going to talk about something which is kind of intentioned in developing with this quantum ergodicity theorem. So it's going to be something called the quantum ergodicity in the level aspect. So now you have some idea of what is quantum ergodicity. It's a equidistribution theory of the eigenfunctions of the Laplacian in the high energy limit. So you want to understand what they look like they look like the function one typically but here what we want to do is that instead of taking large frequency we take some fixed frequencies. So we will take some, so here's the spectrum of the Laplacian have the lambda one, lambda two, lambda two, so on and you just fix some interval and you want to study the eigenfunctions of the Laplacian so that the eigenvalue hits this interval just throw away all these others. Okay, well what's the point of this? I mean for example this picture it's just two eigenvalues here so there's no equidistribution for two eigenfunctions. There's no point of this thing. So you have to do something, you have to change something instead of, so you don't want to go to the high energy limit so that would be going to a large eigenvalues but you want to somehow change that there are more eigenvalues here. So okay there's this one four which I will maybe return later it's just for specific reasons you cannot choose the interval to be less than one four. So instead of doing large eigenvalues you just study those, you just study some kind of a geometric chains in the manifold. So what I want to do is that I want to somehow change the geometry in a way that you get more eigenvalues here. So there are some examples like for example volume, genus and injectivity radius which you can use as a parameter to change the geometry. So what I want to do is now study this properties with this eigenfunctions squared Laplacian if you've arrived at geometry. Okay, so let's take this injectivity radius as the main example. So this is somehow also motivated by several other papers where other people have been doing this injectivity radius, the key thing, I will come to it soon. So there's a motivation why do you want to study this as well. And so if you take the hyperbolic surface so you have the constant negative of the current minus one you can define the point wise injectivity radius as follows. So you take a point x here and then you draw the largest possible ball you can have here with the property that if you stand at the point x you're looking around you to every direction you just see the hyperbolic plane. Like you look to the horizon you see nothing but the hyperbolic plane. You just take the last possible ball that looks like around you like a hyperbolic plane. So the surface is not a hyperbolic plane but locally it can look like a hyperbolic plane. So the injectivity radius is giving you the largest possible ball you can have over here. And okay so that's at the point x you can study different other points so you take the infimal point or that's the smallest place so you can have several different points and then you can have the injectivity radius at different points. Okay so that's your geometry parameter and we would like to now study what happens if you change the injectivity radius. So if you change the injectivity radius of the surface what happens is that the number of eigenvalues starts to increase on a given interval which is not at the one four. If it's any interval here between one four and infinity bounded interval the number of eigenvalues starts to increase in this interval. And if you go to these pictures are probably not what actually happens when you increase injectivity radius. I don't know what happens. I cannot visualize it in my head. This is very completely wrong but I think the genus increases by the infinity of the radius. But yeah it's very looks something like this that it just gets bigger and bigger. This is the hyperbolic plane. Okay so yeah so you get more eigenvalues so now you can ask do you have equity distribution? So you have this interval of eigenvalues and then you can ask when you have equity distribution so that would mean that okay you're increasing eigenvalues you get more and more eigenfunctions here and you can ask if there's an equity distribution for those eigenfunctions. So that's the level aspect quantum record is the question or conjecture. And this was asked by Colin de Vertier that okay this is precisely the form so you take this is once again the brody density. So this is an eigenfunction of the Laplacian here and then you take the function one you study the weak distance here. So in the weak topology how far they are and you take the number of eigenvalues in the interval and you sum over the eigenvalues in the interval so take an average. So if this goes to zero when the injectivity radius goes to infinity this means that on average these things are very small. So that means that there's like a density one subsequence in some sense that these things are quite small along those sequences. So let me just go back to the usual quantum record theory. You can reformulate it in a very similar manner. So remember it was formulated like this that you have the this converges function one weekly along a density one subsequence that means exactly the same as this that if you take the number of eigenvalues less than lambda and you sum and you take some kind of weak distance over here then that goes to zero as lambda goes to infinity. So there's a that means you can extract the density one subsequence as a this thing gets very small. So the in the weak topology they get very close to each other these two things. So in this theorem what you have is that instead of you have the fixed interval and you're changing the injectivity radius. Okay, so that's the that's the theorem. So we go to the next slide. So this is some previous work. So this is something that motivated these questions. For example, in so there's this all this field of something called a holomorphic forms and there's like you can prove analysis results of something called a quantum record density for these things. So this is something I almost know almost nothing about. It's just something that I'll be reading because it's kind of motivating on this but then this is like the techniques are completely different here. So you're using this holomorphic structure. So you don't study eigenfunctions of the Laplacian but instead you study more of our forms and those things have some properties that you can have you can ask similar questions about there's something which is similar to eigenvalue and something like this and you can study these things and they have proved this level aspect results for these things already. And this is the other sequences of the other surfaces and the injectivity radius is changing and then they can prove that okay there is equity distribution in the limit. And but there's also like the other motivation is that there's like, there's all this, I think that it was Rutnik and then also Smilanski and a few other people who like have been like advertising this thing that if you cannot solve quantum unique record is it for example other these very important problems for the surfaces what you do is that you discretize the problem somehow and then you try to prove in this toy model of discretized version some similar results whether you can have actually. So there are these papers I think it's most of this paper which initiated this whole thing that well this which has been done before as well but this like quantum record easily part of this thing was redone in this paper where they prove some analog of quantum record easily for discrete graphs. So in that case you can actually prove other things and there's also been this level aspect results have been proven for this. But for the eigen function of the Laplacian I think as far as I know nothing has been done for them so that's kind of been the motivation. So then I at some point I went to do a postdoc in in Jerusalem. So this is an office 82 in the Hebrew University of Jerusalem. I mean the Ellen Linnensraus was in the university where he was postdoc of him and then okay I was not doing this stuff at all at that point I was just doing dynamics and other things but okay you are sharing an office and then at some point you start to discuss what are you doing and you just join common ideas and then kind of like we started to discuss about various things. And then with Etienne we now prove this result. So if you can have this level aspect theorem if you assume that they are compact hyperbolic surfaces. So in general like okay you may be able to ask maybe in general like for compact surfaces with the ergodic geodesic flow but at least for compact hyperbolic surfaces where you can define all these injectivity radius and these things then you can have this problem. And we prove some quantitative bounds with respect to the injectivity radius going to infinity but that's just I mean it's just something some bounds that you can get. But okay so in the title of my talk I was talking about something called Benjaminish-Ram convergence. So Benjaminish-Ram convergence is actually when the injectivity radius goes to infinity at most points that's what it means. That's the definition of Benjaminish-Ram convergence. So there is like you can use the volume measure in the surface and saying that okay at most points there may be some exceptional points where the injectivity radius is not increasing but if at most points it somehow increases then it means that this surface is converging to hyperbolic plane in the sense of Benjaminish-Ram. I didn't want to go into the definition because I just realized that there was just half an hour to give the talk. So before I had like more stuff but I forgot the time so I just had to remove everything. So I just talk about the injectivity radius it's easier. So okay but some examples where you have this like for example this compact hyperbolic surface you can get these like congruence coverings of compact arithmetic surfaces. So there are these taking like compact subgroup of and no no take a subgroup of the SL2Z which produces a compact hyperbolic surface like choose the subgroup such that and take this something called congruence coverings and those things when you change the parameter of the congruencing you get a sequence where the incitivity radius not just the index there is as the Benjaminish-Ram convergence happens. And then there are some other examples given by surfaces arising from normal co-compact lattices. So there are some examples I can produce of these things. So they're on the paper. So I have at the moment five minutes left I think and I really thought there was slightly more time. So I had something on the proof but there's I don't know what can I say about the proof. So one thing about the proof is that you have this remember that there was this interval you decided the number of eigenvalues in the interval and I said to you that okay the number of eigenvalues will increase but there's actually you need more than that you need actually use some real properties. There's something called the Y law. But the Y law is usually meant for large eigenvalues. You study the asymptotica rate of the rate of growth of eigenvalues number of eigenvalues that are less than lambda. But now we need to study the growth rate in an interval and there are some results where you can use the Benjaminish-Ram convergence. So the incitivity radius at most points increases implies something of the volume growing and then you have some rate. There's this paper of these seven guys where they prove that you can have like these things coming out. This is very important to have the estimating this part of the proof. And the other part of the proof where you have the sum of the eigenvalues and then you have this distance weak distance of the eigenfunction square to one then you need all this like there's like all this kind of machinery because I have only three minutes left. I just say one word, so one sentence that typically in quantum recordicity how do you prove these things? Is that you use something called micro-local analysis or semi-classical analysis or pseudo-differential operator so this kind of machinery. And that's the typical way how you approach this in the large eigenvalue limit. But for a long time like people have been trying to prove something like this and some results like what we did but they weren't able to do it with pseudo-differential methods. I don't know if you can prove them with this so we had to use this like more like a hand-stone method without using pseudo-differential analysis we used this like some sort of disc averages. And there were some very strong results in ergodic theory for those things we were able to employ in this setting and that's the key here. So these things don't appear if you use just the pseudo-differential methods I think but at least in our case you can do like apply this method. And there's also one thing that if you want to do this in a other setting than just hyperbolic surfaces you really need this spectra theory of the radio intercloberators which is very specific for hyperbolic surfaces. So constant negative curvature minus one. So if you go to variable curvature or anything like that then everything collapses here. So I don't know if you can do this part at all. So there was something that we were thinking if you can do variable curvature but then this part really doesn't really work. But this part doesn't, this part doesn't. I think this also you can do something. I don't know. So the next step is my, at the end of the talk I'm asking questions from myself. So can you do non-concrete hyperbolic surface? Yes, there, I think there's many of these things can be, like there are some parts which can be done. These is also the celibate part can be done for non-concrete surfaces. So I think there's some hope but it's the problem is that they can be continuous spectrum and all these things appear and then I don't know what to do exactly there. What's the formulation of the problem? And okay, high dimensions. In the high dimensions I think you might be able to do something but then I'm not sure. I mean, you could use this very similar methods there. I think you can just use the HN instead of H. But then, I don't know. And the variable curvature, as I said, there was also already the problem about the celibate part. So you use the celibate trace formula and this kind of theorem goes quite well. And there's the quantum unique ergo density conjecture which you could ask in this setting as well if you have, instead of having like this density one subsequence averaging, you remove the averaging and take some kind of maximum for example. And there was this one final thing that I shouldn't say usually. I think I've seen some pictures that these things have fractal features. So remember that you have this weak limits of these things and they converge to one in the quantum ergo density theorem but then sometimes they don't and then they may exhibit this kind of like a more non-uniform feature. So this is like they would be converging to one but here it would be converging something slightly different and they can actually look like fractal measures. So that would be interesting to actually understand what they really are. So that's my talk. Thanks.