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.  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.  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.  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.  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.


Genome, DNA, chromosome, homologous, diploid, haploid, genes, alleles and chromatids, the field of genetics uses a whole bunch of words specific to this area of biology. Are you overwhelmed? Well, let's take a guided tour to understand what they all mean and how they fit together. First, the genome. This is all of the genetic material of an organism. The human genome holds many clues as to how we look and behave, but there is so much information there that we still don't understand most of it. It's a huge area of ongoing research. For most organisms, this genetic information is made up of DNA. DNA, or deoxyribonucleic acid, is the storage structure for our biological information. It is made of smaller units called bases. These are the famous ATC and Gs. Three billion of them strung together on a backbone of sugar and phosphate. The DNA is organized in two strands that fit together in a double helix. The genome of humans is divided into several very long DNA double helices, each of which is called chromosome. The DNA of different humans are incredibly similar but not identical, except for identical twins. And these differences make each of us unique. Each living organism has a characteristic number of chromosomes. Humans, for example, have 22 pairs of autosomal chromosomes, each of them unique in size and sequence, and two types of sex chromosomes, X and Y. Most of our cells, apart from sperm or eggs, have pairs of similar chromosomes. This makes us overall deployed organisms. These pairs of chromosomes, which are called homologous chromosomes, are very similar but not identical, as one is from mom and one is from dad. The human, sperm, and egg, which are called gametes, are different in that they only have one copy of each autosomal chromosome, and carry only one sex chromosome. Having just one copy makes gametes haploid. When a sperm fertilizes an egg, there are now 23 pairs of chromosomes, or 46 in total. The fertilized egg then divides, making the many cells of the human body. Now let's look more closely at this process of cell division, which is also called mitosis. As a cell prepares to divide, each DNA molecule is replicated, resulting in two identical copies of each chromosome. This means that the 46 chromosomes temporarily double to 92 chromosomes in each cell. The two exact copies of each chromosome are called sister chromatids. Initially, the sister chromatids are connected to each other at a central point, which is often depicted as the X shaped drawing of a chromosome. At the end of mitosis, the two sister chromatids detach and are pulled apart in opposite directions. Two daughter cells are created, each with 46 chromosomes in the exact same genome. So let's look closer at the structure of a chromosome. Within the long DNA strand, there are regions called genes that contain the instructions to make proteins. There are around 22,000 protein coding genes in humans. The proteins that genes code for make us who we are. They give us ginger hair or B blood type O. These traits are also called phenotypes in the field of genetics. Zooming back out to the chromosome level once again, we can start to understand where variation in human phenotypes begins. As humans have pairs of homologous chromosomes, it follows that they also have pairs of genes. A pair of genes may be identical to one another, or they may differ subtly in their DNA sequences, in which case they are called alleles. Sometimes different alleles still give rise to the same phenotype. If the two genes on homologous chromosomes produce the same phenotype, for example both causing blue eyes, the person is homozygous for that phenotype. However, sometimes the two alleles produce different phenotypes, for example blue eyes and brown eyes. This makes a person heterozygous. The actual outcome in the person, for example what eye color they end up with, depends on whether one allele is dominant over the other. For example, the allele for brown eyes is dominant over blue, so the person will have brown eyes. The blue allele is recessive, because it has no effect on the phenotype of the heterozygote. Most phenotypes, however, are not as simply explained and understood as eye color. Height, intelligence, and susceptibility to different diseases, for example, are complex blends of numerous genes and their many alleles. Phew! Now that we've covered all of these different terms involving DNA, we can delve into all of the amazing things that DNA does and understand how the study of genetics is revolutionizing medicine and advancing our knowledge of life.


Hello and welcome to the session. Let us discuss the following question, question says, consider the following distribution of daily wages of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method. This is the given table. First of all let us understand that according to assumed mean method mean is equal to a plus summation fi di upon summation fi. Now in this formula x bar is mean a is assumed mean fi is frequency di is the deviation of a from each of the xi. Or we can say it is deviation of assumed mean from class mark we know xi is class mark. This is the key idea to solve the given question. Let us now start with the solution. First of all we will rewrite the data given in the question. We are given daily wages of workers in rupees and number of workers. Now we know number of workers represent f1 that is frequency. Now we will find out class mark for every class interval. We know xi represents class mark. Now class mark is equal to upper class limit plus lower class limit upon 2. So 120 plus 100 upon 2 is equal to 110. Similarly 140 plus 120 divided by 2 is equal to 130. Now class mark for this interval is equal to 150. Similarly class mark for this interval is equal to 170 and for this interval class mark is equal to 190. Now we will choose one among the xi's as the assumed mean. Let it be 110. So we will denote it by a. So we can write assumed mean that is a is equal to 110. Now we will find out di. We know di is equal to xi minus a. Di represents deviation of assumed mean from xi that is class mark. So we know here for first class interval xi is equal to 110 and a is also equal to 110. So deviation is equal to 0. Similarly here xi is equal to 130 and a is equal to 110. So 130 minus 110 is equal to 20. Similarly 150 minus 110 is equal to 40. 170 minus 110 is equal to 60. 190 minus 110 is equal to 80. Now we will find out the product fi di. We know 12 multiplied by 0 is equal to 0. 14 multiplied by 20 is equal to 280. 8 multiplied by 40 is equal to 320. 6 multiplied by 60 is equal to 360 and 10 multiplied by 80 is equal to 800. Now we will find out summation fi. We know summation fi is equal to sum of all the frequencies. So it is equal to 50. Now we will find summation fi di. We know summation fi di is equal to sum of all these products. So it is equal to 1760. From key idea we know mean is equal to assumed mean plus summation fi di upon summation fi. In this formula x bar is the mean, a is the assumed mean, fi is the frequency, di is the deviation of assumed mean from xi. Now substituting corresponding values of a summation fi di and summation fi in this formula we get x bar is equal to 110 plus 1760 upon 50. Now simplifying we get x bar is equal to 110 plus 35.2. Now adding these two terms we get mean is equal to 145.2. So we get mean daily wages of the workers of the factory is equal to rupees 145.2. Always remember that assumed mean method is used when the numerical values of xi and fi are large. So our required answer is rupees 145.20. This completes the session. Hope you understood the solution. Take care and have a nice day.


No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Okay, let's start our show tonight.  Yeah, the Metro ESP32 S3 is finally in the shop. We had this as it coming soon for a bit. Thanks, everybody, for signing up. We will notify you if you signed up. I want to make sure that we had some for people watching the show live. It's chock full of stuff. We can either go to the overhead or the next day man show. Okay, so you got your ESP32 S3 room module. That's in the middle there. It's pre-certified with FCC and C certifications for the emitter. It's got 16 megabytes of flash memory built in and eight megabytes of octal PS RAM, which means it can drive these cool TFT displays. It can drive a remaker series. It's got so much RAM and so much flash. They'll fit like pretty much every project you can. They do make versions that have even more like they have 16 and 16 or 32 and 16, but I wanted to strike a balance. I thought this was a good amount. Like you can definitely drive TFT displays, but you also aren't paying as much as the higher prices of the modules. I think this is a very roomy module. It's got USB-C. So if that's power and programming and debugging, there's also the debug port. So if you want to connect an external USB to serial converter for debugging the hardware you are, that's available. I also put out the JTAG connector. I'll be honest, I've never really used the JTAG connector on the ESP32 S3, but they do have support for step debugging with open OCD. You can check out their tutorials on doing that. And the connector is there, ready to go. You've got the boot button and the reset button. So you can put it to bootloader mode and reset it. You have the Arduino-compatible pinout. And I did my best to make the pin number match the native GPIO of the ESP while also matching the Arduino numbering. Only thing that didn't make it is 0 and 1 are actually 40 and 41, though the RX and TX because pin 0 is actually not available. It's used for the bootloader. So I figured I'll just name them RX and TX. And then next to the RX and TX is also a battery charging and monitoring circuit. So if you want to plug in a LiPo battery, you can get, especially if you cut the trace for the NeoPixel, you get down to about 100 and 120 microamps. So it can get fairly low power. The NeoPixel does draw like half a milliamps. So you do have to cut that trace if you want to get to the lowest power usage. But it's still a fairly good low power performance. That was good as the feather because it's optimized, but still quite good. Stem IQT pinout for connecting sensors, plug and play, even a spot for microSD card. So great for data logging or reading circuit Python or file storage. And that's connected to the hardware SPI pins on off switch. And you can plug it into a DC jack, 6 to 12 volts DC.  And you just can use it with circuit Python. You can use it with Arduino. It's like super chunky and great for developing with, you know, basically the latest expressive chip, very powerful Wi-Fi and Bluetooth low energy capable Arduino, circuit Python, micro Python, everyone loves this chip. And that's a metaphor.


Maksha, Motiza, Fubol Kenya Fede, World Team of Tони, Gormaya, and so on and so on. And of course he's here to share his agenda as far as his joint presidential bide alongside Bounfa Sosano in FKF elections. Good to see you, Banna, you've been long time. It's been a long time. I'm good. Question? Quesala ma kabisa. What informed your candidature? First of all, when I ran for the K4 chairmanship, definitely I was ambitious. I was going to be there for the rest of my life. I was looking at the big seat there. An opportunity came in earlier in the year, which someone actually told me to run for the top seat of the president. And I was saying I cannot rival the incumbents because of the resources, how flexibility he has in matters monitoring. And they said, don't worry, we will support you. So after that we didn't talk for a while and I was still thinking. But then I received a phone call from Bonfa Sosano and he told me, Mutiso we have to meet. I want to talk to you about something. And I didn't know what it was all about. And we met actually here in town and he explained to me what he wanted as everyone else knows. He had declared his candidacy last year, mid last year, that he wanted big seat of the health. And he gave me his agenda and it was very impressive actually. And the confidence that he showed on me, a former footballer, he told me that he's been following me. He knows about how vocal I used to be when I was at Madari United at Gorma here. And my input when I was care for president, the way I used to speak out for the players. And actually that confidence in me, I was very impressed. And I could just not say no. And I accepted the bid and since then we were actually going through one or two, three things. So that we unveil ourselves and that happened last week on Tuesday. So it's good that Mutiso is here so that he clears the air because there have been rumors being pedalled around. That these two are being used by somebody to sanitize and justify an illegality. I don't know. Have you had the same as well? We actually talked about that. We met when he told me about what he meant to be his running mate. And that came up because I told him. People know as me especially to be a support of the federation. When we endorsed Nick Mendoa, the incoming federation. During team change. During team change, we endorsed them. And what people also don't want to say is that vote was not mine to give. It was not my decision to make. And we had a meeting with the KFOR board and actually Nick came and we had that meeting. And everyone was impressed by the agenda. We were 14 officials, 13 came, only one missed. So after that, you know very well we could not also still make that decision because it wasn't ours to make. So we sent an SMS to the agenda to all club captains in the KPL, NSL, Women's Premier League and also in Div1 and Div2. We sent the messages one week earlier. So they had time and we also told them the agenda of the meeting. So they had time to confirm with the fellow team colleagues and to come up with something tangible to present on the D-day. So we had that meeting at Panaprikotel and it was a unanimous decision. And the current federation president, he came to the meeting and he gave out his agenda for Kenyan football, for women football and everyone was impressed. So after he left, we had our own small meeting and we asked them, who do you think we can give our votes to? Because all the other aspirants wanted to meet with me alone. And as I said earlier, it wasn't my decision to make, it wasn't my vote to give. So everyone was impressed with the agenda for team change. And it was a unanimous decision because by a show of hands, everyone raised their hands. So it was a unanimous decision and that is what people are not talking about. I did not go with this alone. I was representing the footballers, the club captains and the Kefa board. We have a lot of propaganda. We cannot deny that when it comes to football politics in Kenya. A lot of propaganda is being pedaled here, in here, out and you never know where you stand and who to answer and who is going to expect the truth from. But away from that... No, we stick on that because I haven't answered the full anatomy of it. And the whole thing, it was supposed to come. This is politics and just like football, it's dynamic. One day you're here, the next day you're there. And I had a sitting at the federation at the player status committee. So I continued my mandate for fighting for players rights, advocating for their rights. Since I resigned from Kefa, I continued with it at the federation. And it started unwell, but since last year we had only two meetings and I started asking questions. Like there's a backlog players calling me actually as an individual and I was trying to help them with these cases because I sought out some legal redress. And there's a backlog and I started asking questions like why are we not having this meeting so that we do away with this backlog of cases. And I guess I was set aside from them. From the inside now. Yes, I don't know if there were meetings called behind my back or not, but I have never attended a meeting since last year. What is the name of the committee? Players status committee. You had been seconded by Kefa then? It was just an appointment by the federation. Exactly, because of my role at Kefa. And after I resigned from my role at Kefa, I continued with at the federation. So people have to know that I'm an ambitious guy. And Kefa was not my last stand.  Am I not allowed to spread my wings and fly? Everybody is allowed to do that. So I have every right to contest for the seat at the helm. Being a former footballer, playing for the national team, winning titles. I have suffered in this Kenyan football. I have cried. I have cheered with the titles I have won. Is there something you left football due to the injury you had suffered? Just before making a late comeback? Exactly. That's why I'm saying I have every right to go at the helm to try and change the way things are being done. And I'm proud of the things I changed while I was at Kefa. The structures that we put there, right now they are enjoying the fruits. And at the federation also right now, there have been structures which have been laid out that were not there before. But sorry. It reached a point where they could not go on from the structures. They did not fulfill the full potential of the structures. And that's why we need fresh faces, we need new faces at the helm. So that we take football to the next level. That was my question. Of the people that are vying for the football Kenya presidency at the moment. You and Bon Fesosano are the people we can outright say are outsiders. Bon Fesosano is a journalist, football writer and all that. He's a complete outsider from the main set people that we have seen there. My question is, what is this agenda that convinced you? When you started Bon Fesosano it convinced you and now you are in the same ticket together. What is this agenda that you have for Kenyan football? Well, that day we met, we took a look at the other aspirants. And these are regular faces that were used to year in, year out. Who have been there at the helm did nothing but still want that seat. And we were underdogs definitely. But you see, when we go to a football pitch, you don't expect to lose. You're going there to win. But to me I have a mantra that even if you don't win, you still win because you take the lessons. You take to the next election. So we are not in this election alone. We are also looking at the next election. But if we win here, we win. But if we lose, we still win because we are looking at the next one. We are planning for this and the next one. So we are not in this to just buckle up and step down just before the elections. We are here until the end. Why do you call yourself a dynamic duo? Well, exactly, that's part of it. But we make the perfect blend because he's been a sports journalist. He is a sports journalist. And you've seen even his social media has been vocal the way he follows African football. And he has travelled to some African countries. And he knows the know-how of what other federations are doing. You get me? And I, for myself, I know what needs to be done in Kenyan football. So that's how we make the dynamic duo because he is bringing his blend of experience. I'm bringing my blend of experience from Kenyan football. You haven't answered my question yet. You are gender for Kenyan football. Well, there are some things that we want to change. And we also want to continue implementing the structures that the current regime have set up. And one thing we really want to concentrate on is club licensing to bring professionalism to Kenyan football. And I feel this is one of the paths that the federation really had a nil rating because they could have implemented this club licensing bit by bit. It's not something that can be done overnight. Where did they go wrong? Where did the federation go wrong on club licenses? They just wanted to implement it in their first year. All of the, exactly. All the teams had to meet the threshold, the criteria. And yet, if that happened, they found out that only four teams could qualify to play in the Kenyan Premier League. And that was very wrong. They could have started with some bits, some threshold like the basics one. The basic one. Having a junior team, which should be mandatory, who should be also in the league that the FKF have formulated or the KPL. I don't remember the last time there was a 20 KPL tournament of which we don't want it to be a tournament. We want it to be a league, an ongoing league together with the Premier League. And also having a secretariat. Many teams here in Kenya do not have a secretariat. Where the chairman is, is where the office is. And that's a very sad state of affairs. We also want the medical cover for the players so that we will see players giving their all in football matches. Because they will know that their medical expenses will be taken care of. Players when going to 50-50 challenges, they will maybe jump up so that they don't get that 50-50 clash. Because they know very well what happens when you get an injury and you don't have any medical cover. And also we want to have a minimum wage for the clubs. It's very sad that most of the clubs in Kenya cannot sustain themselves. They have to depend on the grants from the KPL and the FKF Premier League, from the FKF Federation. So this is for some of the things we want to change in the first 100 days. This is what we will be concentrating on. We have not already laid out our full manifesto. We will do it after we become candidates now, after handing over the nominations. But the things we want to concentrate on is first the basics of the club licensing. Well actually the first one I will talk about club licensing here, of the presidents we have had at the moment. And I have had a keen interest of club licensing. And many clubs like the way you said it before, never met that threshold. And you have said in the first 100 days you are going to do that. The question usually is mausoleum with our clubs is you get into the office, you have been duly elected. But after the election politics follows you again. In that you get onto the Federation, clubs don't forget that politics. They still follow that politics to the office that you are going to. How are you going to deal with that? Because I understand that our education level mausoleum football is very low. We still have that mentality everywhere we go. That mentality is there. How are you going to deal with these people so that you can bring them to a level playing field where they can understand that this is what we need to do for us to move to the next level? Well the kind of politics that we are bringing in the Federation as I said we have fresh ideas, we have new faces there. So we bring a different blend of politics. And we are not in it like we want to bring to have all inclusive stakeholders. We want the politics of inclusivity because that is what has been lacking with the Federation. Like when you are not with us then you are against us. So we won't work with you. So we do not want the visit politics. We are not in it for the long haul. Football brings people together. So we do not want the visit politics. We want politics which will include all stakeholders and especially in negotiating for sponsorships. That is one of the main things where I can say KPL and FKF have been failing because we need all stakeholders to be involved in negotiations for sponsorship. We need the footballers welfare that is now cared for or if there will be a union. We need the referees association there. We need the coaches association there and all the clubs. These are the main stakeholders in football. So we have to include them in the negotiations. How will the players benefit? How will the referees benefit? How will the coaches benefit? Because right now you will find that in some leagues in the Federation you will find that clubs are actually paying referees which is bad for the integrity of football. So these are some of the things that we will want to change as the dynamic duo. A big one. And now let's go ahead and discuss what you said earlier. Grab Lensesing is a big one for you and I'll give you that. But you have also said that there are some programs that the Federation has started. The Centre of Excellence, I think it was a great idea but wrongly implemented because it's not only in Nairobi. Nairobi maybe is just supposed to be the headquarters but the Centre of Excellence is only in Nairobi and we know all over Kenya there is massive talent. I have travelled, I'm a mentor with Chapadimba and Safarekom so I have travelled all over Kenya. I've been to Garisa, I've been to Tukitui, I've been to Coast to Kericho and I could see the talent that is out there but they don't have the exposure. So the under 13 Harambistas, the under 15 Harambistas, the under 17 and the under 20 and the under 23 they have been having regular friendlies, regular matches. This was nonexistent before. Yes, they were never there. But you'd find that the same coach was the coach of under 13, the same coach was in under 15, you'd find still the same faces in the under 17, in the under 20 and also in the under 23. So we do not lack youth coaches here in Kenya. There are so many youth coaches. Why can there be a spread of those football coaches? The Centre of Excellence all over Kenya now. All over Kenya. We want the 47 counties to benefit from this centre of excellence. We want talent to be tapped from every corner of Kenya. That is why we are bringing the politics of inclusivity. We want to include all the stakeholders and you can see the current regime do not have a conducive environment for maybe co-operates all the counties and you can see the last two months we've seen the rise of other federations to rival this federation. Mombasa County have started their own football association and also Makweni County. Is it good or bad? It's very bad, it's very bad for the federation. We could see the extreme super eight league, they broke away from the federation. These are people who are supposed to be members of the federation and that is why we will bring the spirit of inclusivity. We want to negotiate with all stakeholders to take food to the next level. Let me cut you short. Mombasa starting their own football federation Garisa doing all that. And Makweni. And Makweni. Is it believed that they might have believed that we are not well represented in the main federation in Nairobi. That's why they want to do that. Exactly. And as I told you once you are against you are not with the federation, you are against them. So we want to bring everyone together. If we make that conducive environment these stakeholders will come to us and say we want to start a league at our county level and that is how we will try and incorporate the clubs which play in the FKF league to play at the county level because you will find that they will be funding from the counties they will be funding from the constituency level and that is what things we want to change. Here in Nairobi it has been divided into two branches. Nairobi East and Nairobi West of which I think it's a good idea but not very well implemented. We could have had at least eight zones here in Nairobi like eight constituencies that represent Nairobi want to have leagues in all those constituencies and there is money. There are corporates in estates my team I am chairman of Imara Daima football club which is in Imara Daima and there are companies just around us when we go to them to give us uniform they give us, when we go to them to give us both they give us our MCA and our MP who we are very grateful for they always pay our affiliation fees so there is money at the constituency level and there is money at the county level but it has not been tapped so when we come into the federation we will add these small teams to at least work with the county government and also the constituencies because there is money, there is CDF money for sports and there is also the minister of sports in the counties. I think we have moved now to your link for government I think that's where the conversation has led us at the moment you are linked to government because now we have got the national government and then we have got the county government is that your link to government how are you going to work with this government because I understand the county government with devolution at the moment many counties have come up with sports ministries at the county level but it's like there is no impact for them it's like a ministry in Kisi counties like a ministry in Mombasa county but for you how is that going to be inclusive to your government now that is how there have been so many missed opportunities by the federation these are some of the niti gritties that the federation has to work on and that is what we are bringing in as the dynamic duo we will have to tap on these resources there is money as I said in CDF in the county government which usually goes back to treasure exactly it goes back because it has not been claimed now we will try and claim it for the teams which are in the county league and sub branch league because you find that a team from Huruma they will play a league match maybe in the sub branch or the county branches you will find that they will go all the way to Kibira and this is a team which do not have money they do not have a source of income so when we incorporate them and we divide the zones they will only play their matches in Starray around Starray you will find others play only around Mbakasi East or Mbakasi West or Mbakasi South or North so it will help it is the burden of these teams and when we find these teams that their affiliation fees which is for sub branch leagues for sub branch teams their affiliation fees is 15,000 and for the county league teams is 30,000 this is a lot of money it is paid per season and you know in every match there has to be security so the home team has to hire a cop and also medical services so these are the things that we want to tap which is in the constituency level and the county level a big one there for you everybody is talking about grassroots football and all that and youth and everything but we've got this stakeholder which you have been helping for a very long time the welfare of the footballers and all that and your generation are getting out of the game most of you are getting out of the game you are now getting onto management and you are very young people getting onto football management how have they received you out there because even if you are a former or senior Nikola Muyoti is now hiring a seat and is a coach and all that but now how have they received your candidature outside there well obviously there have been mixed reactions and in politics that's a given there has to be a split opinions on who is this who wants to this is someone who has worked with the federation and that is something that people have to take it in a positive way because I know what has been going on so I am best placed to change the things that have been going on I have seen what the federation has done I have seen what the federation has not done so I am best placed together with Bonfa Sosano to try and change the niti griti where the federation has failed we have a chance to correct them we have a chance to reach their full potential to continue the vision that they had but did not fully implement and as we know I think there has been no incumbent who has defended his seat ever since I have known football politics there has no one who has come to it a second time and actually it has been a fast for every football president so we have a very good chance we have a very good chance to take this thing and some of those things I have told you we want to change we really want to change how the funds there is also one point I have missed is the funds inclusion there is a funds association but it has been greatly neglected and we want to incorporate them in the federation they have to be members of the federation we have the Kenya Secondary Schools Sports Association and the primary schools sports association you know when these upcoming footballers the teachers at school level their teachers or their teachers in general they are tasked with grooming these upcoming footballers so we want to try and have a balance between education and football and bring them on board they will have to be members of the members of the federation so we want a spirit of inclusivity for all the stakeholders of football with that thought of inclusivity and you mentioned the Super 8 which printed away from the football Kenya federation and these are the stakeholders that we want in the game how will you bring these people on board because now those guys are cold as in the game and they have helped much I think you are one of the beneficiary of Super 8 Yes, I am one of the success stories of Super 8 and actually when the president of the league Husein Muhammad when he was unveiling his agenda for Kenyan football when he was vying in 2011 actually made a speech at KCC when he was launching so as one of the success stories that is I was not really much into politics just giving my story but now in politics fully deep end of politics and we have to bring them on board because you know even corporates shy away because when they look at the federation when they look at the league NSL or KPL they see the same faces and they shy away from working with them so when they are new faces when they have fresh ideas they will come will bring them on board a project like the Extreme Super 8 league it's a wonderful project why would you not want them to to be taken care of the youth here in Kenya and especially here in Nairobi because you'll find that in extreme Super 8 matches there were so many fans better than KPL when you go for Chapadimba tournament you will find so many fans people say Machakos do not like football but I was there with Chapadimba and on Friday there was a match between KCB and Karubangi Shacks you could find you can count the fans who were there they were less than 20 80% of them were journalists true so the next day on Saturday and Sunday the stadium was packed they say Machakos people do not like football but the stadium was packed because of the packaging because of the conducive environment that the corporates are bringing in how they have packaged their whole agenda how they have packaged their whole tournament it brings fans to the stadium so it's not a steep hill to put they just look at the faces which are there and they shy away from working with them from what I'm getting from you is you will work to untrust the corporate trust from corporates because if you realise that Chapadimba coming on board they use their own resources without giving the federation any money you look at super 8 was the same thing just give us the lines and we run without giving the federation any money that trust is what you want to bring that trust is what we want to bring you see with the super 8 league or any league which wants to come on board we can give them 20 leagues all over Kenya run it because you have the resources you have the goodwill from corporates run the county leagues if we have the association the county level where they are setting their own associations we could give them the regional league take the regional league run it in your county so that way we will have these resources being tapped teams will no longer have teams not going to marches because they don't have fare they don't have food they don't have a hotel they can sleep in so once we tap into these resources I think like clubs paying the referees will be a thing of the past that is what we are assuring the football stakeholders that is the fans the coaches, the referees the footballers and also the delegates who are very important but in these elections we will be big one there for you thank you very much the Y254 it is the touchline here on where Maxwell has had just a bit of an emergency we will be talking to him to understand what has happened to him but the show must go on here and if you are joining us we are having a conversation with Innocent Mutiso who is the first chairman and their leader is Bonfuss Osano as post journalist who is actually now running for the presidency of football Kenya federation here in the country he is giving us the agenda and what they plan to do with football many things that we are talking about but now it is the ultimate one professionalism because the league has locked that and that's why we've got all these problems coming in I remember reading one of the first football chairman in Kenya has got to be the late honorable Kenneth Matiba and back in the 90s professionalism was running all over the world and that is what we wanted to introduce in Kenya but many people could not allow him to do that I don't put you on that hot seat for professionalism but can we achieve it Yes we can and as I said once we implement the club licensing once we start implementing it step by step we will reach those goals that is where the federation has failed if they started it bit by bit since they came into the foray in 2016 right now we could be full professionals we are semi professional currently our leagues are semi professional and below KPL they are just armatures so we want to bring that professionalism and it is achievable once we implement the club licensing fully we want clubs no longer operating on zero balance or operating on overdrafts and we will be monitoring these things at least 10 million so that we know this is the minimum balance for this team they will be able to pay this player from January to December we will start a circle where the players can contribute the referees or the coaches because they know very well that at the end of the month or at the first or the fifth of the month there is something that it will get into the accounts it will take something from there and put on the circle this is their money it's not our money but they will have a way of investing I'm very lucky to have grown up in the Madari youth or the Issa setup where when I was in Madari youth we had the Jamibora microfinance and every month because we used to receive money every month salary amount of money and put it in our Jamibora accounts it was mandatory for everyone it was mandatory so if we do that when doing that after 3 months you can take a loan if you want to start a business you start if you want to expand your business you can if you want to buy 3 things and actually that is the money which made me shift from my parents place to go to my place because I took a loan after 3 months to buy things for my house so it will be a very good thing if we incorporate this in the KPL in the NSL and this is where KFOR comes in or if there will be a football union this is where they come in this is where they will they will have huge responsibilities of also taking care of the footballers in the grass roots because also them they will be having their manis from tapping from the constituency level and from the county levels they will also have their money that they can save in the circle so this will bring at least a level of professionalism also in the clubs we no longer want clubs to be depending on the KPL or the FKFOR grants that will be a thing of the past and your final word now as you finish tell us what is your final word and do you see a bright future for Kenyan football that depends largely on the delegates because since the courage regime was brought in it was all rosy and things were good but there's a level that it reached that people got fed up and everyone was complaining so if they want another four years of complaining then they will vote wrong people but if they want a whole new setup if you want football to be taken to the next level then vote for the dynamic duo we do not have money to give the delegates we only have our agenda to sell and they should vote with their minds and their hearts they should know that we have football at heart the people who are actually our rivals do not have football at heart I saw someone one of the aspirants on social media donating football and you can see these are just footballs which are being hooked in the streets there are like 300 Kenyan shillings which when you use it because in the grass root you find they are not good grounds so after two days the ball is done it looks like a duster I have seen the other administrators who have been there and did nothing they want to come back but they are they are the people who have quoted actually live on TV saying that he is hiring a seat in the gobernatorial race in 2022 and the term right now from 2020-2024 it will be the running term so are they using the football elections as a springboard for Kenyan politics so people should try and at least vote for people who have football at heart who have the players at heart and we will take football to the next level Wukwe na tatuja bla? Wukwe na tatuja bla? Wukwe na tatuja bla? Magadana kufuwa rei ni fortuna kufuhu wu kuwaiti, kikiwa na mga jambu wa yaw제 bangu tomatoesu kufuwu wa kufuwa rei. but they are the outsiders in this musical game of chairs. We'll be right back, but for now, enjoy the PSG and Bayern Munich Champions League final. When we come back, it will be the fun zone here on the touch ring.