 Well, thank you very much for inviting me to this conference. The conference, of course, is called Nonlinear Waves. I have an apology. My talk's about linear things. But it's about estimates that have proved to be useful for proving various types of nonlinear theorems, existence, and so on. For instance, the types of estimates that I'll be talking about in the case of domains were used by Burke, LeBeau, and Plancheon to prove existence for the critical wave equation in domains in R3, and so on. And also, we've been hearing about how good estimates for the torus lead to good existence theorems in that setting. There are also theorems about good existence for NLS for the sphere. For spheres, the frustrating thing is that there's really not good theorems for manifolds of negative curvature. So I'm going to talk about improving LP estimates and Cacay and Nicodemus estimates, whatever that is in that setting. And maybe someday there will be some progress on NLS in that setting. OK, let's see here. So the setting is that I'll be talking about manifolds without boundary, compact. Of course, they have a metric g. And the dimension will be 2 or more. My little plot should be negative. And lambda j is the frequency of the eigenvalue. Geometers don't put the square there, usually. So lambda j is the eigenvalue corresponding to the first-order operator, which is the square root of minus the little plotian. All the eigenfunctions are L2 normalized. dv is, of course, the volume element. The little plotian is the little plotian coming from the metric. And so a vague question is, how can you detect and measure various types of concentration of these guys? And you can expand your horizons a little bit and not just concern yourself with eigenfunctions, but near eigenfunctions, quasi-modes, whatever that is. And you expect that you might get extreme concentration at certain types of points, which would lead to bad soup norms or bad LP norms or large P for certain types of points. And you also might expect to get concentration near periodic geodastics. If you're going to get concentration on a different sort of set, you'd expect the set to be invariant under the flow. So the natural thing is periodic geodastics. And I'm going to concentrate on the ladder. So geodastic, of course, is bigger than a point, so you might expect to detect concentration along periodic geodastics in terms of LP norms for relatively small p. And that turns out to be the case. And you can measure concentration or dispersion in various ways. Well, I'm on a compact manifold, so there's nowhere for the eigenfunctions to disperse. But there is a famous problem, which says, for instance, when you have negative curvature, if you take these probability measures, then they should converge weakly to the uniform measure. So that's called the quantum-unique ergodicity conjecture. And that would be an ideal thing. It's very difficult to prove. It's conjectured in the case of negative curvature by Rudnick and Sardak. And it was proved in some special cases, for instance, by Lyndon Strauss. And that was one of the things that they cited when he won the Fields Medal. OK, so if this is your goal to try to show that something like this happens, it fails miserably. This should be the volume in the manifold. It fails miserably on the sphere. The sphere is the worst case for everything I'm talking about today. So let's go over that to sort of set up the better things. So of course, the sphere is up there, the n-sphere. It's a subset of our n plus 1 given by that equation. And the eigenvalues of the square root of the Laplacian on the sphere are just given by this formula, basically k. Square root of k times k plus n minus 1. And they repeat with a very high multiplicity. That's the highest multiplicity that's allowed because of the vial formula, the sharp vial formula. And what's going on there is there's a lot of periodicity. All the geodesics that are periodic with period 2 pi, they're all the great circles. And additionally, if you take p, instead of taking the wave groups that we've been hearing about, but you complete the square by doing that, then these half-wave groups are periodic. They're either periodic with period 2 pi, in the case of n odd, or 4 pi, in the case of n even. And so the periodicity of these half-wave operators also accounts for the bad behavior of the eigenfunctions. The eigenfunctions are nothing but the spherical harmonics, which are the restriction of homogeneous harmonic polynomials in Rn plus 1 to the n sphere. It is hot here. All right, so let's talk about the worst type of spherical harmonics, the worst type of eigenfunction if you don't like concentration near periodic geodesics. So here's a guy which is highly concentrated on the set where the modulus of x1 plus i, x2, is equal to 1. So in other words, the place on the sphere where the other coordinates are 0, it's equator. So it's highly concentrated on the equator. I put that factor k to the n minus 1 over 4 to l2, normalize it. So you can evaluate the modulus of this. It's exactly that, of course, by calculus. You tailor expand the natural log about 1, and you'll find that it's like this. These are called Gaussian beams. See, they live essentially in the set where this has the argument here is of size 1. And so that's a k to the minus 1 half tube about the equator. I'll soon be calling my eigenvalues lambda. And so this is living in a tube about the equator where the tube has width lambda to the minus 1 half. For all practical purposes, because of that Gaussian factor, you could just pretend that this guy, in terms of his size, is the indicator function of that tube times the normalizing factor. You're not really making too many mistakes. And so therefore, you can compute its LP norm. So you have this factor out front. And then if you compute the LP norm, you'll get the volume of this tube to the 1 over p, which is k to the n minus 1 over 2 with a times minus 1 over p. So the norms of these guys are on the nose comparable to this. And this works for all p bigger than or equal to 2. So they have bad LP norms. And so here's a picture of some spherical harmonics. So the ones that I was describing, well, this never vanishes. But if you took real parts of this, you get a real eigenfunction. You usually study real eigenfunctions. So if you take the real part of this guy, it's called the highest weight spherical harmonics. And it's these guys. As k increases, because of the Gaussian beam behavior, like some speakers, almost all the mass is on the equator. Zonal functions. I'm in the audience, too. Zonal functions, which I won't be talking about, but that's a separate talk, are functions which are peaking at the north and the south pole through the middle guys. The first guy is pretty boring. It's just the constant function. And then you have spherical harmonics of higher and higher order. And these pictures also denote where it's zero, which is a very interesting problem, the nodal sets. And already, what is this? The spherical harmonics of degree 1, 2, 3. The spherical harmonics of degree 3, you could see this very high concentration of the highest weight spherical harmonics. OK? All right, so let's talk about saturation of norms. So many years ago, I showed certain LP estimates for eigenfunctions. And there are two ranges. So this is critical exponent, which is 2 times n plus 1 over n minus 1. This should be a familiar exponent to lots of you guys. It's the exponent that pops up in Stryckart's estimates, for instance. So I showed that if you're below that critical exponent, you have exactly the same norms that we were talking about. k is now lambda. And it's k raised to that power, n minus 1 over 2 times the gap between 1 over 2 and 1 over p. And because of the calculations that we did in the previous slide, we see that this estimate up here from the 1980s is sharp. It can't be improved. This universal estimate holds on all compact reamounting manifolds. It can't be improved because of these bad guys, the highest weight spherical harmonics. I won't be talking about this, but I also proved estimates for larger exponents were the power of lambda turns out to be something different. It turns out to be this thing, n times the gap, 1 half minus 1 over p, minus 1 half. Way back when I was doing this, that was the case that I cared about. Because that number pops up in harmonic analysis. It's what's called the critical index for bachnery summation. And what I wanted to do back then was to extend some results about harmonic analysis in Euclidean space to harmonic analysis on manifolds. And I was able to do it because of that estimate. And this estimate in red, I put in my paper because you might as well put everything you can do in the paper, but I thought it was a boring estimate. And now this estimate, this red estimate, is much more interesting. It's linked to several things, actually including this. So the idea is that it's uniform for many, and then it's equal to any many? Yeah, the constants involved, of course, depend on the manifold, but the growth rate landed to whatever power always holds. It's what I would call local estimate. You prove it using local techniques. Soon we'll see the wave equation. I forgot to say that one of the reasons that I don't feel so guilty is that I'll show you how you can use wave equation techniques to study harmonic analysis and especially eigenfunction theory. You prove this estimate, all you need to do is to understand the wave equation for a unit period of time. You don't need to watch a long movie. And that makes sense because on the sphere, these sorts of wave operators that arrive are periodic. So if you watch the movie for time 2 pi or 4 pi, you know everything. That doesn't happen in the case of manifolds of negative curvature. In that case, in some sense, the target manifold won't turn out to be a hyperbolic space, but the ammunition manifold, as we'll see at the end, will be hyperbolic space, whatever that means. All right, let's see. You mind me asking a simple-minded question? Sure. This definitely points out the periodic orbits, but at least some metrics have invariant tori or more complicated higher-dimensional tori or other things. And those also are recurrent. And those also play a role somehow in spectral mass and products. Yeah, like not exactly talking about what you were saying, but a zole manifold will also have these bad properties. A zole manifold, all the geodesics are periodic to the same period. And if you're willing to broaden your horizon just a tiny bit and consider what are called quasi-modes, they're disastrous for the same reasons for estimates like this. OK? OK, let's see. All right, so I was going to say this. Knowing when you can improve these estimates for small exponents is something that's only been recently understood. I've worked with Steve Zeldic quite a bit. And moreover, the work of Bayard from the 1970s says that if you're willing to consider big exponents, in particular p equals infinity, then you get a big improvement. Bayard's work, which is actually very important for what I'm talking about now, is from the 1970s. The pseudonormy estimate, if you plug in the formula up there, that's the power right here. And if you plug in p equals infinity, it would say the universal bound is this. So in any compact manifold, the pseudonorms grow at worst like that. That's a theorem that's often attributed to Hormander. He wrote a famous paper in 1968 called Spectrophunction of Elliptic Operator, which was actually approved in the 1950s by Avokromovich in Levitin. So that's the universal bound. And Bayard wrote a very beautiful paper in which, implicitly, you get this improvement. If mg has non-positive curvature, we'll use his techniques quite a bit. It's a global theorem. You have to deal with the wave equation for a large period of time, up to basically log lambda. And then Steve and I have written several papers. And we actually have necessary and sufficient conditions, at least in the real analytic case, of beating this. It's a generic condition. This is not a generic condition. So let's talk about some related problems. Well, I did quite a bit of work. There's actually been a lot of activity on studying the size of nodal onsets. Nodal sets are zeros of real eigenfunctions. There's a conjecture of yaw that the size of the nodal set is n minus 1 dimensional house-dorfer measures should be like lambda. And the lower bounds have been studied quite a bit recently. And before, basically, 2010, they were pretty bad. They were exponential in lambda. And then we were quite happy we got polynomial lambda, lambda to some power, negative power, and high dimensions. And there were two competing techniques. Steve Zeldic and I were using wave equation techniques. And another competitor of ours was Calding and Mendenkassi, use harmonic function theory. And in the work that Steve and I did, this was sort of the central thing, to get lower bounds for L1 norms. And when this first came up, I thought this was weird. Because as a harmonic analysis analyst, when you're dealing with L1, it's usually trivial. But when you're dealing with L1 as a harmonic analyst, you're usually proving upper bounds, not lower bounds. Big difference. So Steve and I proved this thing right here, that this is a universal bound, that the L1 norm is always bounded from below by this. It's easy to prove using Holder's inequality. So the L2 norm is 1. You Holder it, 2 is between a 1 and a p. You take p to be in that range I talked about, between 2 and that Sturckart's number, 2 times n plus 1 brand minus 1. Holder it. And then you get the bound that we were talking about for this, figure out what this powers, and do your arithmetic and get that. By this Holder argument, you see that if you can beat the estimate that this is controlled by this, you get a better L1 norm, lower bound. And if you feed that better L1 lower bound into the things we were proving, you get better estimates for the size of nodal sets. And I was pretty happy about that because of the work I'll be talking about. We were able to beat the world record for lower bounds of Koligan-Minnikazi by logs. So in particular, in 3D, the Koligan-Minnikazi, Steve and I came up with a different proof later on about the same time. The lower bound that they obtained was not that the size of the nodal set grows like lambda, but it's bounded for three-dimensional manifolds. Not good, but at least it's not exponential. So this proves the superiority of the wave equation. Well, just wait, Sergio. So we held the world record for a little while for a few months. We could show they blow up logarithmically. But there's been an amazing breakthrough by a postdoc of Sodin and Israel Luganov in 2016. And he completely solved the thing for the lower bound. And darn it, he used very local techniques and harmonic function theory. You get the correct lower bound and upper bound. It's a very famous series of papers with Donnelly and Pfeffermann, circa 1990. And there was all this just for real analytic manifolds, not for c-infinity manifolds. But he proved the correct bound, pretty amazing thing, for the lower bound. What is it called, a bump? Lambda. Should be comparable to lambda. Just like if you take cosine of lambda x and you count the number of zeros. OK, it's going to be like lambda. OK, so that's pretty amazing. So hopefully wave equation techniques will work. We're still trying. I mean, there's still plenty of open problems about nodal domains and nodal sets. But that's a pretty amazing thing. So this works only in the analytic case, you say? No. So this used to be another slide. And I could brag about this and so on. So I should have still made this another slide. Donnelly and Pfeffermann are only the real analytic case. That's 25 years. There's been really no significant progress. But he, the young mathematician, proved the correct lower bound for the c-infinity case. Very nice. OK. All right. So now we come to the cacaonicodon part of the title. So we've seen that these q-landers have almost all of their L2 mass in these shrinking tubes about the equator, this periodic geodesic. OK, and let's see. I guess I didn't say so. But if you can't beat this lower bound for L1 norms on any manifold, then you could show, if you have a sequence of eigenfunctions that saturate this norm, then you could show that for each eigenfunction there's a geodesic and there's a tube where he has the profile of the high suede spherical harmonic. And a large proportion of the tube, he'll have exactly the same size as the high suede spherical harmonic. I was hoping to be able to use this for nodal lower bounds, but it's obsolete. OK, so at any rate, so the high suede spherical harmonic has its mass in these tubes. If you can't beat the L1 lower bound, then you have a guy that looks a lot like the high suede spherical harmonics. So maybe something to consider is the L2 norms over these types of tubes. So I call the cacaonicodon norm of the function just the supremum of all the L2 norms over these shrinking tubes. So you consider tubes centered anywhere and their orientation could be arbitrary. And you're taking the L2 norms over these shrinking tubes. I call it cacaonicodon because these sorts of averages arose in the work of Cordoba. That would be Antonio Cordoba when he studied Bacna Ries. In his case, he was considering the maximal operator which involved the sup over averages of your function overall tubes of width delta and length 1 about a point x. He called that the cacaomaximal function. And then in 1991, Borgand had a very, very seminal important paper that broke open a whole chapter of harmonic analysis and he switched to terminology. He called it the nicotomaximal function. And then he also considered another problem which is related to the structure of the cacaostat, which is the maximal function where you fix the orientation of your tubes, but you sup overall the centers. So I do both. And so I call it the cacaonicodon norms. OK, so clearly because I L2 normalize my functions, clearly these guys are always less than or equal to 1. I'm integrating here over a subset of the manifold. So that's a trivial bound. It cannot be beat by these highest weights of spherical harmonics because of what I said before. So the problem is when can you beat it? When can you show that these cacaonicodon norms, you're souping over the averages of overall geodesic tubes, when are these a little low of 1 as the eigenvalue goes to infinity? And this came up, I guess, about five years ago in my work and then shortly before that in work of Borgand, which anticipated this. So Borgand was interested in linking improved LP estimates with improved what are called restriction estimates for eigenfunctions. For restriction estimates, what you do is you restrict your eigenfunctions to a unit-length geodesic and you take the L2 norm. Berkshire and Setkoff in 2007 proved this universal bound that the L2 norm squared of the eigenfunctions covered these geodesics are a big O of lambda to the 1 half. That's saturated by the highest weight, that's a 2D. That's saturated by the highest weight spherical harmonic because you had that normalizing factor which was lambda to the 1 quarter in 2D. So this is always true. And people for a variety of reasons, including problems involving nodal domains and so on, are interested in trying to improve this. Borgand showed that if you have better L4 norms, the numerology is that in 2D, the L4 norms always are big O of lambda to the 1 eighth. If you can beat that estimate, then you could beat Berkshire and Setkoff. So Borgand showed that this implies this. It's easy to see that if you can beat this estimate, you can beat the Cauchy and Nicodem estimates. Nothing's going on. Cauchy and Nicodem says you're trying to have small L2 norms over a tube. If you get good L2 norms over all slices, you just use Fubini's theorem. So let's see. 2 implies 3. And then in my paper, I showed that 3 implies 1. Borgand came close to showing that either 2 or 3 imply 1. He was using techniques that really go back to the work of Cordoba. When you study these eigenfunctions, as you'll see in a second, you study reproducing operators for the eigenfunctions. And they're oscillatory integrals. They're the types of oscillatory integrals that arose in the study of Bach-Nerise. And in 2D, these things are completely known. They're sharp estimates. And there are two ways of proving them. One is through the work of Cordoba and Pfeffermann. It's kind of a geometric approach using these Cauchy and Nicodem maximal operators that I was talking about. Borgand only used that method. And he was actually just off by a lambda to the epsilon, showing that this implies this. There's a competing approach, which is due to Carlson, Julian, and Hormander, which is based on bilinear techniques. And what I realize is that you could just take the two things. You just split things up into two cases, splice them together. And that allows you to show that this implies this. And therefore, they're all equivalent. And I'll show you why these are all equivalent in a second in more detail. So the central thing is, as I said, taking the biggest possible L2 norms over tubes. The tubes are very narrow. And you allow their orientation to be anything, and the center to be anything. OK. And let's see. So Blair and I, in a paper that appeared in 2015, but we did the work a few years before that, backlog, we showed that the same thing works in higher dimensions. In higher dimensions, especially for more dimensions, you really are stuck with using these K and Nikonin norms. This is too singular, restricting your eigenfunctions to geodesics for technical reasons. So the K and Nikonin things are natural to use, especially in higher dimensions. And so Blair and I showed that the result extends in higher dimensions. If you have small K and Nikonin norms, then that exactly occurs when you have small LP norms. They're the same. And in a more recent paper, we showed that you can dominate the LP norm in terms of lambda to the correct power times the L2 norm over the entire manifold raised to a certain power. And then this K and Nikonin thing raised to the theta, instead of one minus theta. This is a better estimate than just having L2 norms on the right, because this quantity right here, as we talked about before, is dominated by the L2 norm of RAM. So this is a stronger estimate than my old theorem, where I just had the L2 norm of RAM to the first power. And of course, this says that if you can improve this, then you can improve that. Just by holders inequality and the numerology here, if you can improve this, then you can improve these L2 norms over these tubes. This gives you the hard half. And let's see. In 2014, Zeldic and I showed that in 2D, you can get these improved K and Nikonin norms if your manifold has non-positive curvature. And in the same paper in 2015, Blair and I showed that for higher dimensions. So armed with these facts and this inequality, you can see that if the manifold has non-positive sectional curvature, you get these improved LPS estimates. And that was a problem that I was interested in for many, many years, motivated by this result. So I was happy when we could prove this. But still, in the theorems that I'm describing on this page, it doesn't give any indication how the norms improve with lambda. There's not a rate, like in Bayer-Ardstam, involving the log. OK? Chris, one quick question on the previous slide. So when you define the K-Kan code in norm, you define it so that it scales somehow in the same way as the L2 norm? It's related to weight packets. You take all tubes of width, lambda to the minus 1 half, about geodestic. And I don't turn it. I guess I think I didn't know what you mean. I don't turn it into an average. I just soup out over the L2 norms. OK? OK. All right, so here's the theorem that we proved using wave equation techniques and actually using dispersive properties of the wave equation and so on. We showed that these K-Kan-Nickelton norms have a certain decay. And the decay is basically one of our lambda. It's worse than 2D because of the worst dispersive properties for the wave equation in 2D. And it's just a little worse than 3D. And we also show that if you have non-positive curvature, you can beat the results of Berkshire and Speckkopf by the same amount. OK, this is when you allow the curvature to be 0, but of course, it can't be positive. If you're willing to assume that the curvature is strictly negative, then you get the same decay rate in all dimensions. And there's a simple reason for that. That's because if you have negative curvature, if you're in Rn as we soon will be with negative curvature, then instead of having dispersion of 1 over t to the n minus 1 over 2, you have exponential dispersion, 1 over sine hyperbolic of t to that power. And so that helps you out. OK. All right, so as a corollary, just feeding it into this estimate right here, you get the log improvements of this. Of course, that's going to give you log improvements of the LP norms. OK. And so finally, we're getting something like Bayerard. Maybe not the optimal power, but some power. n minus 1 over 4? This is n minus 1 over 2. Remember, in my case, my eigenfunctions have eigenvalue lambda, or lambda squared instead of lambda for the Laplacian. OK. All right, so you get this as a corollary. This is not as relevant as it seemed to be before, but you do get an improvement for the L1 norm of eigenfunctions of the lower bound. And that would lead you to an improvement of the size of the neutral sets, but it's way blown away by the recent breakthrough that I told you about. OK. All right. All right, so also, and this may be very happy, too, because this also did not seem to be within reach. As it turns out, if you take these improved LP estimates over here, and you take Bayerard's estimate, and you throw it into a recipe cooked up by Bourgan in 1991, which he used to give a nice proofs. It's the same paper I was talking about, the breakthrough paper that he had involving the Kakea and Nikon and Maxwell functions. If you take an argument that he had in this paper, which gave a very simple proof of weak type estimates for the Stein-Thomas restriction theorem, if you take his recipe, take Bayerard's estimate, and take the improved LP estimates that Blair and I had, it turns out that by using an argument of Bourgan, you can prove weak type estimates, improved weak type estimates for the critical exponent, PC, which is 2 times n plus 1 over n minus 1. I don't feel so guilty about talking about this, because Rowan talked about weak type spaces. And then if you use some more harmonic analysis, something that goes back to Bach and Seeger, but actually it follows from an interpolation argument of Bourgan again, you can upgrade those weak type estimates to get LP estimates. And so I'm happy about this, because nobody had attained improved LP estimates for the critical power. If you can improve improved LP estimates for the critical power, then just by interpolation, you get it for all exponents. You get log logs because of this argument that I'm talking about, but still it's an improvement. It'd be interesting to turn these into logs, but don't know how to do that. OK. All right, so I want to tell you, since this is a wave equation conference, how the wave equation arises. All right, so I'm going to tell you about how you set up these arguments. You want to prove these improved L2 estimates over these types of tubes for eigenfunctions. OK, a problem is that, except for very special cases like the torus or the sphere, there's no way you can write down a formula for eigenfunctions. So you have to attack them through operators that reproduce them. And so here's a typical way of doing this. You choose a Schwarz-class function. You want it to be 1 at the origin. This will allow it to reproduce eigenfunctions. And because you're going to be dealing with a wave equation, you want this Fourier transform to be supported in, say, the unit interval. Then if you let P be the square root of minus partial plushion, then this function of P, rho of lambda minus P, is defined by the spectral theorem. So if you act on an eigenfunction with eigenvalue mu, then it's just multiplication by this Fourier multiplier, rho of T of lambda minus mu. T will turn out to be, later on, a small constant times log lambda. So you have that formula. They reproduce eigenfunctions. So you're trying to show that the integral of your eigenfunction over these small tubes is small. So if you simply take f to be e lambda, this will be 1. And you get the bounds that you want. So if you can prove this estimate, then simply by taking f to be e lambda, you get what you want. So you want, of course, the constants to be independent of lambda and so on. So this is what you need to prove. OK, so how do you do that? Well, you just take this guy right here and you rewrite it using the Fourier transform. So it's, what, 1 over 2 pi. There's a 1 over T because of the dilation. There's rho hat of T over capital T. There's e to the i lambda T. And then e to the minus i Tp dt. And because rho hat is supported between minus 1 and 1, this integral just involves fairly big times, but times which are at most log in size. OK, so that's a formula for this operator that we need to estimate in this way. And then you just use Euler's formula. You add to this operator rho of lambda plus p. p is a positive operator. Lambda is a large number. So this guy will have a kernel which is rapidly decreasing. So when I add this operator to this, I'm just making a mistake involving an operator which is trivial, which has a kernel which is rapidly decreasing. When I play this game of adding these two guys, I'll be adding e to the i Tp with a plus sign. OK, and then I just use Euler's formula. So after using Euler's formula, I can replace this by cosine of Tp. So I have to show that this guy has small, when acting on a L2 function, F, has small L2 norms over these tubes. I like cosine because cosine by Huygens principle, this kernel will vanish. If I take the kernel, it'll, well, I'm going to lift it up to the inverse of cover. I'm getting it out of myself. So I want to use Huygens principle. That's why I introduce cosine. And so now I use some geometry. So by the Cartan-Hadamard theorem, I can rewrite this kernel. I can consider the universal cover of my manifold because my manifold is non-positive curvature. And so I take the universal cover, which is Rn. And the covering map is just the exponential map. Identify. So the exponential map is, of course, the map from the tangent space at any point to the manifold. I identified the tangent space with Rn. And if I were sensible, I would be playing this game where I take the exponential map over p, which is the center of the geodesic. So if I play this game, I can lift my metric on my manifold using this covering map, this exponential map, to get a metric here. If I'm assuming, as I am, that my manifold has non-positive curvature, then g tilde, the lifted metric, will also have that property. And then you have this formula right here. The wave kernel for the metric downstairs agrees with the sum overall that, OK, there's more here. So it's best to go over the model case. So the model case, of course, is a torus. The torus you identify with minus pi to pi to the n. And then this is a fundamental domain for the torus. And in addition to having this covering map, you have deck transformations. In the case of the torus, the deck transformations would just correspond to translating by elements of zn. I'll have a picture for this in a second. And then you have this formula here, which relates the wave equation on your torus, if we're dealing with the torus, with the wave equation upstairs, which would be an Rn. And so you're summing up over all the translations. You're taking the solution of the wave equations kernel upstairs. You're evaluating at a point x in your fundamental domain. And then you're translating the other point y in the fundamental domain. This formula right here, where this would be the Laplacian on the torus, and this would be the Rn Laplacian, this formula right here is easy to prove. It simply follows from the fact that c infinity functions on the torus are in one-to-one correspondence with smooth periodic functions in Rn. You couple that with uniqueness for the Cauchy problems on the torus, and on Rn, you get this formula. And the very same argument works in this more general setting. I couldn't find very good pictures, but this is supposed to be the picture for fundamental domains for manifolds of negative curvature, compact manifolds of negative curvature. And these are the translates by the deck transformations. I'll tell you why it's a bad picture in a second. But it's a good picture because it depicts the way things, if you have a manifold strictly negative curvature, the geometry from our Euclidean eyes is becoming very, very warped in the angular direction. There's no warpage at all in the radial direction. Why is this a bad picture? It's a bad picture because it's supposed to really depict the fundamental domain for, say, the two-fold torus, the manifold of curvature, the simplest type of manifold of compact manifold of curvature minus one. Now, if you take the standard torus, S1 cross S1, you get its fundamental domain, say the two torus, by chopping it up. You chop it up once, that gives you a cylinder, then you chop it up like this, you unroll it, and you get a square. So if you do that for the two-hole torus surface, you have to make twice as many cuts. So there should be eight sides instead of seven sides. And this picture, there are seven sides. It's not my picture. So when you deal with, there's lots and lots of pictures for fundamental domains, of hyperbolic quotients, where the picture depicts the Poincare disc model or the upper half-space model. This is the model corresponding to using exponential coordinates, geodesic normal coordinates, which is the right sort of coordinates for the way I'm gonna be looking at these problems. Okay, so I couldn't find a good picture. Well, you guys know this, I'm sure. So if you have a manifold and negative curvature, of course, the sum of the angles adds up to less than pi. This isn't so relevant. So now let me tell you about how we're gonna try to use the wave equation, how we're gonna prove these things. Hopefully I have enough time. So this is the guy that I need to estimate. This is the kernel of the operator that I need to estimate. I'm trying to show that if I act on a function f and I restrict this guy to a tube like this, I get something which is small. Okay, so let's erase something. So what I'll do, as I talked about, is I'll use exponential coordinates. So here's my fundamental domain. My geodesic will be something like this. I can always use geodesic normal coordinates about what I call P right here. Okay. And then I can extend, this is a geodesic gamma. Okay, I can think of this as a geodesic on the manifold or a geodesic upstairs. And then since I'm working upstairs, I can extend this. And if I choose geodesic normal coordinates like I've been talking about, this'll just be the line. So this is the fundamental domain. And these are translates of it, like in that picture. Okay, so here's one that hits the extension of the geodesic and then here's, there's a whole bunch of them, like in the picture, actually exponentially many of them. Okay, if the radius of a ball is T, then the number of these fundamental domains is really large. The number of fundamental domains could grow exponentially. If the curvature is negative, it'll look like that. Okay, and so there's sort of two types. So here's alpha of D and here's another alpha of D. So if you think about Hormander's theorem of propagation of singularities and so on, if you have some experience with this, okay, it's a disaster, as I said, because the number of nonzero terms in the sum is actually growing exponentially. You won't get anything because of Huygens principle and the fact that this integral is supported between minus T and T. You won't get anything from fundamental domains that live outside of this ball. So that'll contribute nothing to the sum. So you'll have a whole bunch of terms. It'll be exponentially many terms. It'll turn out that it's fairly easy to show that individually they're all well behaved. They actually have perfectly good norm. You just have to add them up. So it'll turn out that there's two types of fundamental domains. The ones that intersect this geodesic and then everything else. The terms that intersect the geodesics, actually the number of terms just grows linearly. It grows like T. But there's exponential in T terms that aren't there. So you expect the main contributions to come from these sorts of guys. And you'd be kind of happy if you can show that if you consider these guys that intersect this geodesic and are within the ball of radius T, if you could put a small cone through them. Or if you could put a fairly large cone through them. So if you can control this angle theta, it depends on T, if you could put inside of there by choosing this constant C appropriately. If you could put inside of there an angle which is bigger than or equal to lambda to the minus delta where delta is very small, you might be in good shape. And it turns out that you can because it's a pot in the golf. So I wanted to find clip art with the poodle, but I couldn't. So if you have your friend who's a girl in this case walking away from you in Euclidean space, then of course the angle that she makes with the horizon is basically one of her T's. It's just decaying like one of her T. In hyperbolic space, this angle is like one of her the sine hyperbolic of T. It's decaying exponentially. So Tupanegov says that if you assume that your curvature is pinched from below, which it automatically is, I'm dealing with manifolds of non-positive curvature. They're compact. So I have a lower bound on the curvature. I might as well assume the lower bound is minus one. Okay, I could just multiply with the metric by a constant to achieve that. When I lift the metric upstairs, okay, on the universal cover I'll preserve that lower bound. So I can always assume that the curvature is pinched from below by minus one. And then Tupanegov says that this angle we're talking about is bounded from below by the corresponding angle in hyperbolic space. And because that angle is what I just told you about, that allows me to get this lower bound, okay? And that's terrific because I can add up these guys. I have universal bounds for all of them. I'll have this one of a log here. They'll come with nice bounds and they'll also come with the j-th term. They'll also come with something which is of this size. Okay, if that's a j-th term. And so that allowed me to add things up. And I get different bounds if I'm in dimension two, three, or higher, okay? And then finally, because I have one minute, let me tell you about how to handle everything else. So these guys are good. I have to tell you how to handle these guys that are outside of, that do not intersect, that avoid this cone. So all I do is I take this operator and I compose with a pseudo differential operator that localizes to this cone we're talking about and then everything else. So q theta is localizing to this cone in Fourier transform space. So this is my angle theta. Okay, so these guys will really only see these guys because of this picture. So this piece is very good. So let's ignore it. So therefore I only have to consider this operator composed with this. And then I can undo, I can go back to this formula, okay? So I take this operator and I compose it with this and then I have to prove estimates for this. I can use Euler's formula again to really reduce it to this. So this composed with this, I need good estimates. And this involves an average, okay? And this is a unitary operator. So since I've run out of time, what I really need to do is to show that if I take this sort of thing, I'm rushing through this, I'm acting on an F, DT, then I get good L2 norms over these tubes if F is supported in a tube. And because this involves an average over an interval of length capital T, and since these are unitary operators, I could just reduce to this. That's a trivial reduction. And here's why it works. Here's the wave equation. So at the end of the day, I'm trying to prove good estimates for L2 norms of this expression inside of this small tube. Oh, I forgot to put in this cutoff. That's important. L2 norm over the tube. I've got to the punch line. So I need to estimate this. So having the pseudo differential cutoff means that I'm just only considering waves that are traveling at an angle theta or more from the direction of the central axis. So if I wanna prove this estimate, I just have to know if I have a particle inside of the tube, and it forms this angle, how long will it take till it exits the tube? And the time, the escape time, it's easy to see, is gonna be less than or equal to lambda to the minus one half, and then you're gonna have to lose, of course, if the angle's small. You lose like that. So this will be trivial because F is supported in the tube. I'm taking the L2 norm of the tube if time is bigger than this. So really, I just have to estimate this. I'm rushing through this, and then I just use synergy estimates, right? I just bring the L2 norm out, and I use the fact that this is bounded in L2 on the manifold with norm one. And so when I play this game, I'm gonna get a gain, which is like this. And that's a really terrific gain if my angle is like this. That'll give me a total gain, which is like lambda to the minus one half plus delta. And because I'm just trying to beat logs, that's way more than I need to handle the contributions of this remaining piece. And it looks like I'm over time. Sorry about that. I think you use a wave equation just at the end. Yeah, just at the end. Just at the end. Domain of dependence you use, but that could be. Domain of dependence? No, it's actually, I didn't emphasize this as much as I should because I forgot to say, I really, really use the global parametrics of Bayard. It's pretty subtle to use the wave equation. You run into disaster if you didn't have Huygens principle and things like that. Yes, really, really, really. Yes, and I need very, very, so everything goes wrong logarithmically and that's implicit in his parametrics. So I use quite a bit, we use quite a bit about the wave equation in hyperbolic type manifolds. Very sensitive to that. So this parametrics is only put up to logarithm in time. Yeah, exactly. Did you have a question, Walter? In negative sectional curvature case, there's a huge amount known about the geodesic flow. Yes, the geodesic flow, yeah. You're using like one geodesic, or sort of even a part of the geodes because you're not translating it around by the deck transformation. You're just using one geodesic on a covering case. Is there, if we say strictly negative sectional curvature, does the situation get better? Yes, it'll get better. For instance, it's just a technical theorem that I showed you gets better. And the most interesting case, if you have the most interesting, I consider it all geodesics, but the interesting geodesics are the periodic geodesics. So if you have a periodic geodesic, you can put one of these tubes about it, right? And then you can care about how the estimates depend on the length of the tube. And there are other problems that are also very related to the types of things that you're bringing up. A slide which I skipped. There's something called period integrals. The log improvements that you get, are they expected to be optimal, or is it possible? No, well, it depends. So there are some far out conjectures. So it's conjectured that by, somewhat by Tarnac and by physicists, that if you have strictly negative curvature in 2D, for instance, then the eigenfunctions, as they are in the torus, should be essentially bounded. But people are way, way far away from priming that. This was from the 1970s. And nobody's improved on this. Okay, just like, this conjecture is implicitly, it's from the late 80s, but except in very special cases, nobody's proved them. So there's optimism, but... Your result implies the improvement of the Schrodinger equation on 6.24? That's a good question. It should. It should. I haven't looked into that. Actually, I didn't really think about that till I was here feeling guilty about the fact that there was nothing nonlinear in my talk. Spectral problem's not on your problem. You multiply a lambda by a solution. One more question.