What's up everybody? Jason here from Jaspys case breaks calm We just sold out a little filler here for the $700 basketball mixer where we're giving away three spots in this 2021 Hoops blaster one box break random teams number one And again the way the work this work start is first we do the break round my customer name the teams This should be the hits out and at the end of the break We'll take all 30 customer names randomize them and the top name will get a spot each top three names get a spot each in The break itself. So let's do the break first guys. Here's the dice really is a customer names roll its Five and a four nine times one two three four five six seven eight nine Robert uncle down to Joshua and five and four nine times one two three four five six seven eight Thunder done of the Indiana Pacers All right, nine nine Again stick to what you got guys quick little blaster 2021 hoops All right, so there you go Let's do the break. All right. So after this filler guys, we have to do a Quick little pack filler for anti random team break and then we'll do anti random teams number five Then we'll do a quick a quick little pack filler as well for the full case R&B 11 teamer then the randomizer and then the full case or Cassie Stanley Zion blue that's pretty nice I'll be topping Tyrese maxi Killian blue Xavier Tillman Co-Anthony a little slam so dig bay Jersey swap quickly Killian Hayes Risenstein Moran Lamello Lonnie with the Hornets Tunis Tyrese Halliburton There you go, and that tree young Hawks Patrick Grizzlies Dylan will do a little randomizer So you get that Jersey swap All right guys the switch scenes And we use the same dash roller to do the randomizer for the Hawks and Grizzlies Little Jersey swap the dice and it is a six and a four ten times. Good luck guys one Two three four five six seven Eight nine ten Hawks on the top after ten times yes ten times you remember top Three get a spot in the seven hundred dollar mixer after ten times one two three four five six seven eight nine and 10th and final time Boom top three Jay Roy and Lewis there you go. Congratulations top three get in Another way to win will be posted up soon guys number two That should be one of the last fillers we need to do to help try to unlock that mixer So appreciate you guys Jatsby's case breaks calm


Next question is from Drew Mosier. What are your fitness trend predictions for 2021? I got one. Oh, go for it.  I have a positive one that I think is gonna be a good fitness trend this year.  I think that's gonna slowly go away and more and more people will find more creative ways to get in shape, you know, and start training at home. And I think more, I think it's obvious that the home gym surge of like equipment is like through the roof, right? It's gonna keep going. I think so too. And I think more and more people are actually having, and this is including myself. So this is the longest I've ever been and not been in a gym in almost 20 years. I have never gone this long. Yeah, you were not a home gym person. No, I was out of the three of us. I was the one who was like, I'm not a fan of it. And obviously because of the time that we're in, I was forced to make it work and figure it out and have now learned to love some of the benefits of that as much as I didn't like it before and said, oh, I love, and I'm not saying that I still wouldn't enjoy getting a lift where there's a bunch of other, you know, people pushing weight and doing things in the gym atmosphere. But I've actually learned to love the at-home workout so much more that I can't be alone in that camp where you probably love the gym setting before, you were forced out of it, you've now been forced into figuring out what at-home workouts look like for you and have now adjusted and have learned to like it.  And I'm gonna continue doing this at home. I think that that's gonna be a big trend this year. Yeah, I agree with you 100%. I think home gym sales, I think digital fitness products and programs, I think those are gonna continue to do well. I think also that the demand for fitness is going to go up. We saw it dip, right?  I think that we're, this is my own, just, you know, belief, I think that we're gonna start to reach a breaking point where a lot of people are like, okay, I need to do something to take care of myself. And I think you're gonna see that more this year than you have in previous years. How can we not when you connect all the COVID stuff? I mean, when you look at the number one cause of fatality from COVID, it's people that are obese, heart conditions, all the things that are preventative if we are dieting and training. So you gotta think that there's a big portion of people that are probably waking up. I know there's a large portion of people that are like locking themselves in their home and not going anywhere and not getting sunlight and not socializing and not doing that in fear of COVID. Then I think there's another big portion of people that realize like, oh shit, like one of the best things that I can do to prevent this or at least mitigate it if it does come my way is to be healthy. I think that's human behavior because look at it this way. I have friends that have served time in prison and the time that they were most consistent with the workout was in prison. Why? Yes, some sketchy friends. It's, yeah, a few, a few. But it's because, why? Because they were in a situation, there was a stress relief. They didn't have a lot of options available to them and they became consistent with their workout. I feel like we're hitting a breaking point. I really do. I think a lot of people are, the demand for fitness is gonna increase because a lot of people are like, all right, that's it. I need to do something. I need to get on a schedule. Things are not open like they normally were. I think it's top of mind, I think it's top of mind because people, I think it's both, like it's sort of the other end of the sword of that is that it's bringing people back to wanting to get into healthier habits and protect themselves and build a more resilient body. And I don't know if it's been announced or not. I know Apple's working on ways to compete with Peloton and all these kind of digital platforms and things, but I really see, we're gonna see some of these big companies jump into the ring here and create experiences like we haven't seen before where it's gonna connect to the watch and you're gonna get points for all this. And all these things like they've speculated about forever in the digital health space, I think are like, and they're gonna double, triple down on them to where you could earn points and then that goes to your insurance and things like that can lower costs for things. Like there's gonna be incentives, I think, within workplaces and stuff. I think it's like finally time where that makes sense. Excellent.


Okay, so maybe I will start. So in the last lecture, Jens introduced the hyperbolic plane and described some of its basic geometry. And then he showed how the unit tangent bundle of the hyperbolic plane can be identified with the group PSL2R and how this is very useful if you want to give simple algebraic formulas for the geodesic flow and for the horocycle flow. And at the very end, he briefly mentioned that all of this transfers down to any hyperbolic surface where now the hyperbolic plane will be the universal covering surface of that. But the formulas for the geodesic flow and the horocycle flow transfer down to the hyperbolic surface. And today I will start doing that a bit more carefully. So GE today will be the lead group PSL2R. Recall that this is the special linear group of order two, so the group of two by two matrices, real matrices with determinant one, but you mod out with sign changes. So you identify any two matrices if they are just the negates of each other. So this is modulo plus minus identity matrix. And then today we will also fix some discrete subgroup, let gamma be some discrete subgroup of G. So this is simply some subgroup which as a set is discrete. The one example which will be very important for us is to take gamma equals PSL2Z, the group of all integer matrices with determinant one. And then we will be interested in the homogenous space X equals G modulo gamma. So this is the, as you saw in the introductory lecture, this is the, as a set, this is just the set of all left cosets gamma G where G goes through the group G. And locally since gamma is discrete locally this looks just like the group G but we have to identify various points of G. G modulo gamma, sorry, okay. And the point now for this special group G equals PSL2R which can be identified with the unit tangent bundle of the hyperbolic plane. It follows that this homogenous space can also be viewed as the unit tangent bundle of the hyperbolic plane modulo the action of gamma. I will say a bit more about what this means very soon. So this is using the identification that G can be identified with the unit tangent bundle of H. This is the map, I think Jens called it PSI in his lecture. And then this is almost the same as the unit tangent bundle of the hyperbolic surface H mod gamma. It is the same at least if there is no torsion in gamma. So this is if there is no torsion in gamma. Otherwise they are almost the same but if there is some elliptic element in gamma that fixes some point in the hyperbolic plane then this hyperbolic surface will have a conical point and it's a bit maybe not well defined to speak about its unit tangent bundle. But I want to say what is this H mod gamma. So that is a hyperbolic surface and it is by definition it's what we get if we take H but we identify points which are equivalent under the action of gamma. So here this sim is the equivalence relation saying that any two points in the upper half plane are equivalent if there is some element in gamma that brings one point to the other. So this is the hyperbolic plane but we have identified certain points. But locally unless we are at such an elliptic point that is fixed by elements of gamma this locally this looks exactly like the hyperbolic plane. So all the structure of the hyperbolic plane transfers to this hyperbolic surface. We have the hyperbolic metric giving the hyperbolic area measure and so on. So let me just show an example of what this looks like in the example which will be most important for us when gamma is equal to P cell to Z. If gamma is equal to P cell to Z then a fundamental domain for the action of gamma on the hyperbolic plane is given by the following set. So I will say what the fundamental domain is. A fundamental domain for gamma acting on H and this is the standard choice of a fundamental domain that is always done in the literature in the last century or so. So the following set, consider the following set in the upper half plane. The set of all points in H such that the real part of Z is less than or equal to one half and the absolute value of Z is larger than or equal to one. So here's a picture of that. So we consider all points which lie outside this circle and they should also have real part between one half and minus one half. So it's this domain and I've taken it to be a closed set. So it includes all the boundary. This is F and what fundamental domain means that basically it contains exactly one representative for each equivalence class here but I've included some boundary so there is some over representation but only at the boundary so to state that in precise sense. So H if I consider all the translates of this F by elements of the group gamma then I will get a perfect covering of H. So it's almost a disjoint union except I have some overlap in the boundary so this is a union of all the images of F by elements of gamma and there is only overlap in the boundary. So if I take any two images of F gamma one F and gamma two F then the interiors of these are disjoint for any gamma one gamma two in gamma. So this is what it means to be a fundamental domain. I mean depending on the context sometimes one considers open fundamental domain sometimes closed ones sometimes abstract ones and then one would insist that they are truly disjoint and exactly covering H. Okay so now to see what the hyperbolic surface actually looks like we can also point out what are the identifications. Well there is one map in gamma which identifies which is just translation. This map is in gamma and it acts on the hyperbolic plane by just taking Z to Z plus one and that clearly identifies these two sides and then there is also another map which is the rotation of angle pi around the point I and that is the map zero minus one one in gamma which gives this identification. And then it turns out that so this shows that the point I is actually an elliptic point of order two and it turns out if you compose this in the right way actually this point here is also an elliptic point of order three. But now you can if you do this folding you see that this is a surface of genus zero and so topologically it's just a sphere but we have these two special elliptic points and we have a casp also so it's not a compact surface. And the picture in general for will be kind of similar. So always if gamma is a lattice meaning that this hyperbolic surface is finite area then we can always find a very similar domain some convex domain convex with respect to the hyperbolic geometry with a finite number of sides such that the hyperbolic surface you get it by just identifying the sides in pairs and folding. So now we have this understanding of the homogenous space X. So this by this I mean just I take the unit tangent bundle of H and identify points according to the action of gamma so similar definition as here. And then as I said if gamma doesn't have any torsion in it there are no elliptic elements in gamma then we can identify it completely with the unit tangent bundle of H. But even in the remaining cases we can say that it's we understand what it looks like in the elliptic points and we can see that they are similar. So now we have a very good picture of this homogenous space which I said is a really important homogenous space for us. So in this special case when gamma is equal to PSL2Z then X equals G mod gamma is up to some well up to this minor problem with the elliptic points it's the same thing as the unit tangent bundle of this hyperbolic surface here. But let me also point out that this is the same as SL2Z without the P. So SL2R modulo SL2Z without the P and the reason for this is that the minus the identity element is in here so you get exactly the same quotient because minus identity element is in SL2Z these are the same. So as I said in the first lecture this homogenous space and also for higher order so SLDR modulo SLDZ is going to be a very important space for us. In that lecture I said that this space can be identified with the space of all lattices in R2 in this case so that's a completely different way of viewing this space that we will come back to in a later lecture. But the point here is that when D is equal to 2 we also have this picture of the space it's the unit tangent bundle of this hyperbolic surface. But now when I continue talking gamma will be a general discrete group not necessarily the PSL2Z. So now let's remind ourselves about the flows which Jens introduced in last lecture. How do they look like? Are you want to see a picture of this tessellation? Yeah, so obviously if you translate you get just this but I guess somebody can Google and you will see nice pictures of these tessellations it's very. And then another copy if you rotate it by this you get something like that or you get this. But then is it, doesn't it look like, well I shouldn't embarrass myself by, I suggest you Google and you get this tessellation or we can do it, sorry for. But obviously it will look, when you look down here it will, since the geometry of the hyperbolic plane is as it is it will look really fractal like or something. Okay, so now let's remind ourselves about the flows which Jens introduced. However let's immediately define these as flows on the homogenous space g mod gamma and recall or that means the unit tangent bundle of our hyperbolic surface. So we had the geodesic flow. So in Jens lecture this was a flow on the group g and it was defined by taking g to the point g of at where at is this diagonal matrix. However, now we can define this also on the homogenous space by just saying that the point, the coset gamma g is mapped to gamma g at and you can very easily check that this is well defined. It doesn't depend on which representative of the coset I choose. The reason being that I have gamma on the left but I'm acting on the right by 80. And similarly Jens, we had the horocycle flow and now on the surface or on the homogenous space this will be given by gamma g n t where n is this unipotent matrix 1 t 1 0. And we have the opposite horocycle flow given by gamma g n t minus where n t minus is 1 t 1 0. Okay. And Jens showed you yesterday that this is really the stable horocycle flow for this geodesic flow. So it's a flow along the stable manifold for phi of t. Now I will draw a slightly different picture or I will just point out the following fact. Note that if I take, well, this is a formula and this formula captures very easily, captures in a neat way the fact that this is the stable horocycle flow for this geodesic flow. Yeah, it's just a formula. Phi t composed with H s is equal to H of s e to the minus t composed with phi t for any t and any s. So this will be an exercise to check it. Clearly it's some commutativity relation between the matrices a t and n s. And what it means is the following. If I take any point in my homogenous space or in my unit tangent model gamma g and I flow with the horocycle flow for time s, I get the point H s gamma g. On the other hand, I can flow with the geodesic flow, which is really chaotic, so I want to have some red alarming color. But yeah, I can flow with the geodesic flow for time t. I can do that to both these points and then I get phi t gamma g. And then I could flow for a certain time with the horocycle flow. And the point is that I get back to a certain point here in very short time. I have to flow with the horocycle flow only for time s times e to the minus t here. And then I get the same point as I get by taking phi t flowing time t with this point. So the point is that this horocycle, which may be fairly long, has shrunk exponentially here after. So it's just, maybe I'll write it out. That's H s e to the minus t phi t gamma g. And this is true for any point gamma g in the unit tangent bundle. So I won't do it now, but now you could also consider the derivatives along the flow directions. And this will give you at any point gamma g, it gives you a splitting of the tangent bundle of the unit tangent bundle of the hyperbolic surface into a direct sum of three one-dimensional subspaces. And this gives a splitting of the bundle that is invariant by the flows and such that if you take a tangent vector in the flow direction of H s, that is contracted exponentially by the geodesic flow and so on. So this splitting gives you the fact that this is an unosso flow. I just say that without doing it. So here we have the stable manifolds for that flow. Okay, so now I will give the first theorem. Simply saying that phi t, this geodesic flow, but also the horocycle flows are ergodic and are also mixing. This was proved first by Hopf, 1939, using this geometry involving stable and unstable manifolds and then Birkhoff's ergodicity theorem and it's a very nice proof and it's fairly short. But I will not give it, I will just state it and then I will, I hope to say something stronger. So theorem one, all these flows. Phi t and ht, both of the, both of the horocycle flows. They are mixing and hence ergodic. So I won't talk about what ergodicity means, but I'm sure that many or most of you are well aware of it. They are mixing as flows on, or sorry, I should have introduced the volume measure also. So I have to introduce Mu before stating this. They are mixing on X provided with its Liouville measure Mu. So, okay, before stating the theorem, let Mu be hr measure on G and hence it discussed hr measure yesterday. You have explicit formulas for it on the, on PSL2R. And we will assume that gamma is a lattice. So, we assume that the homogenous space, if I take a fundamental domain for it in G, it has finite measure with respect to Mu. And this is equivalent to assuming that the corresponding hyperbolic surface has finite area, easy to see. And when we have done this assumption, we can just as well renormalize the hr measure. So it's not going to be given by exactly the formula, which Jens had before. But you change it by a constant. So we actually assume that Mu is a probability measure on the homogenous space. And if I didn't do it, let me stress that I write X also for this homogenous space. So now we have X provided with a probability measure. Liouville measure for, okay. And so, here's the statement. These flows are mixing. And let me remind you what mixing means. It means that for any functions, any test functions, I can take them to be L2 functions on X. If I consider F1 composed with Phi t and take it in a product with F2 over X. Then as t goes to plus or minus infinity, this tends to just the average of F1 times the average of F2. Okay, so the reason why do we call this mixing? Well, think of the case when F1 and F2 are characteristic functions of some sets. Maybe take some nice sets. And F1 and F2 are the characteristic functions of these sets. Then what this says is that if I let one of the sets flow by the deodesic flow, it will spread out evenly all over the surface because it's integral against any other nice set tends to become proportional to the area of that other set, so to say. I mean, so if you think about F1 and F2 being characteristic functions, it's clear why this result should be called a mixing of the flow, okay? And I have a couple of remarks, so I hope for this using these properties of expanding and contracting submanifolds and so on. But one can obtain this with a very good exponential rate. In fact, one can have a really explicit understanding of the rate in this particular case when we have just a homogenous space. So let me give one formulation of this. So if you want the rate in this result, then you have to take F1 and F2 to be fairly nice functions. There is no possibility of having a rate for arbitrary L2 functions. It is to show constructions that you can't have any explicit rate for this in this space of functions. But so here, let me take just to have a simple formulation. Assume that F1 and F2 are compactly supported and smooth functions. Then the error, so I mean the difference between this and this side as will be bounded by some constant times. So we will have exponential mixing, the decay is exponential. And we can even get a very precise understanding of the constant here. The constant is typically just minus one half, unless we have small eigenvalues of the Laplace operator on the surface. So it's one half plus epsilon. And then if the, so I just write something out. If there is some small eigenvalue, so lambda one. Let lambda one be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on the surface. So where lambda one is the smallest eigenvalue. So smallest non-zero eigenvalue of the Laplace-Beltrami operator on the hyperbolic surface. So if there are no small eigenvalues, and small eigenvalue here means an eigenvalue less than a quarter. If there are no small eigenvalues, then this is just minus one half plus epsilon. But if there is some small eigenvalue, then the rate is worse. And if the spectral gap, so lambda captures the spectral gap. If lambda is close to zero, then this constant will be small. We have minus one half plus something near one half. So the constant is small. But it's an exponential rate, all the same.  So one can prove it using representation theory. And this is a result of a decay of matrix coefficients. This is really a matrix coefficient for a certain representation. And also, another remark, so if you have mixing for the geodesic flow, then it is easy to get mixing also for the horizontal flows. And what you do is you note that this matrix NT here, you carton decompose this matrix. So carton decomposition is a general decomposition of any element in the group G. And it says that any element can be decomposed as some element in the compact group SO2 times an element in the diagonal group times some other element in SO2. It's more or less unique. And so these are in SO2. And I think Jens introduced a notation k for SO2. This is in SO2. And this is a diagonal matrix. And it turns out if you compute this, you find that when t is large, the value of t prime here will be approximately 2 times logarithm of absolute value of t as absolute value of t tends to infinity. So this shows, and these are in some compact groups. So there's no problem dealing with this. You can have a result which is uniform with respect to both of these. So it's not hard from that to show that if you want a result with a rate, you will get a polynomial rate for the horizontal flow because you have this logarithm. But you can get hold of exactly the polynomial rate of decay for the mixing for the horizontal flow using this relation. OK. So as I said, I hope to say a few words about how to prove mixing with a really precise rate. Or I should mention that, of course, you can get exponential decay for mixing in much more general situations. For example, Dolgo Piat has proved that you have such an exponential mixing result for any unosso flow with some conditions. But that is much more difficult than. So it's a very different method for this result. On the homogenous space, we have access to representation theory, which is really a powerful tool where you can get a really precise understanding of everything involved here. OK. So I wanted to turn to an application of this mixing result to prove an equidistribution result, namely equidistribution of expanding translates of horror cycles. And this is something that is used over and over again in homogenous dynamics, the fact that you can get such equidistribution fairly easily. And it has numerous applications, as you will see in later lectures. This fact extended to higher dimensional settings. So here's a theorem. So I will consider a piece of a horror cycle. And the starting point will be, I call the starting point P. P is a point in my unit tangent bundle of the hyperbolic surface. And then I fix the length of this initial horror cycle piece. And then let curly H be the horror cycle starting at P and having length u. And the claim is that if I take this piece of a horror cycle and I transport it by the geodesic flow, as in the picture which I just removed, then as T goes to minus infinity, this tends to become more and more equidistributed in X. Clearly, T has to go to minus infinity because when T goes to infinity, it becomes more and more just like a point as this picture shows. Now we are going the other way. So phi Th equidistributes in X mu as T goes to minus infinity. And what I mean by that is that if I take an arbitrary test function, a bounded continuous test function or equivalent, it is a compactly supported continuous test function. Then if I integrate my test function along this horror cycle, so I take the average of the test function along this transported horror cycle, which is again a horror cycle. So phi Th up du. And I take an average. So I take 1 over u here. As T goes to minus infinity, this tends to the volume average of F. OK, so a picture. I had a picture over there. The point is we are now, let me do a picture on the, here's my three-manifold, the unit-ended bundle. And I have some horror cycle H. And I'm transporting it with a geodesic flow, but backwards. So if I would go this way, I would shrink the horror cycle exponentially. But now I'm going the other way. So the horror cycle becomes a longer and longer curve. And if I go sufficiently long, now in the picture, I don't go very far, but it is phi Th. But if I go very long, then this curve will spread out all over the surface in a more and more evenly distributed way, as this shows. I'm writing too small, maybe. So it's F phi Th U of P. Maybe I rub it out and write it in larger signs. F phi Th U P. Sorry? No, no, U is, I integrate always. U goes only between 0 and my fixed number, capital U. D U. D U is the parametrization. P is a fixed point. Yeah, yeah. So I don't know. Maybe it's better to write it something like this. So this is the average along the horror cycle. It's H T P. This is my horror cycle. And then I want to plug this. If I write this and I take some test function and plug it into, then it is a one-dimensional measure along the horror cycle. Now I actually plug it into F composed with phi T or something like that, then I average. Say it again. T goes to minus infinity, very importantly. And do I repeat the same stupidity? No, I didn't. So it's U here, of course. In this, in the picture, thanks, thanks. So P is here, just a starting point of a horror cycle. P is the starting point. And then this is the horror cycle I get by following the horror cycle flow for a fixed time U. This is H capital U of P. So it looks the same, I guess, more or less. So I just told you that this is a three-dimensional picture if you want to believe it. But if you want to see this as just the hyperbolic surface, I would draw the same picture. OK, OK, but with the unit. OK, so I see what you mean, maybe, as Jens wrote it. Here's the horror cycle. And then here's the geodesic flow. And OK, so this is really some, I'm not sure if this is a good, this would be one. If it happens to be, if the point P happens to be a unit tangent vector pointing upwards, then the horror cycle would look like this. And I would have all these unit tangent vectors. And then I follow the geodesic flow but backwards so that this is expanding. And I will get a really long horror cycle curve down here. Yeah, I was actually going to mention, yeah, OK, say again. OK, I didn't hear. But I'm considering the orbit, the horror cycle orbit of the point P, but just for a finite fixed time. So here is phi t of P. And here is phi t of H u of P. Yeah, clearly I must learn to present this in a better way until next lecture. I mean, next time I give the course. I will come to that. Here I stated just for some interval. But OK, I will point out the corollary. OK, so yeah, maybe we can, I will be happy to try to explain it more in the tutorials. Yeah, I wanted to point out that one application of this, one special case would be if H is a piece of a closed horror cycle. For instance, in the modular group as I had before, then perhaps I have my fundamental domain of the modular group sitting here. And then since I have a casp in this case, there is an associated family of closed horror cycles. If I have a completely horizontal horror cycle here, then it's clearly a closed curve on the surface. I come back to the same point after having flowed a certain time. And so there is a one parameter family of closed horror cycles. They can be parameterized by the length, for instance. And when I'm flowing in this direction with the geodesic flow, the length of the horror cycle is increasing exponentially. And I have a, well, so this says that a really long closed horror cycle sitting down here. And if I want to see it on the surface, I have to pull back. I have to take each point on this horror cycle and pull it back by the appropriate element in PSL2Z. I will get some curve, which I can't draw. But I will get some curve inside this fundamental domain. And the point is that this curve, when the horror cycle is really long, it will look almost equidistributed according to the hyperbolic area measure. And even if we lift it to the unit tangent bundle, we will have equidistribution also direction wise. Date, ah, OK, I think Selberg has it in unpublished work in this case, at least in some setting. But Sarnak, the paper by Sarnak, which we have in our list of references, 1981, well, he proved it. OK, no, no, no, he didn't prove it. He proved it for closed horror cycles. So who should one? OK, Eskin and McMullen, they do it more generally. So maybe Eskin and McMullen, 96, I think. But I'm not sure if there is a bet. Again, Margolis, ah, in Margolis thesis, it is, ah, yes, of course. I'm going to explain the proof of this. And the proof, you can prove it using a certain thickening technique, or it's also called Margolis trick. And he actually pointed this out for more general and also flows in his PhD thesis. So sorry, this is way, yeah, it was much earlier. Sometimes late 60s maybe, or early 70s. Margolis thesis, so this is rubbish. I mean, no, no, no, the papers are not rubbish. Sorry. So I want to outline the proof of this. But I first want to point out that from this, you can actually generalize it more or less directly by noticing that we get the same limit for any piece of a horror cycle. Then this means you can, using that just, you can, one can see that instead of having Lebesgue measure along a piece of a horror cycle, we can have just any absolutely continuous measure on the real line. Any absolutely continuous probability measure, say. And we will get the same limit. And we are also free to let the test function f depend on this parameter u. This is technically very useful in many applications. So I will leave it as an exercise to prove it. And I'm happy to discuss it and so on. But I want to state it in a precise way. Corollary for any absolutely continuous probability measure new on the real line. So absolutely continuous with respect to Lebesgue measure. And again, say for any compactly supported test function on x. We have that if I instead, I take the same average as here, but with respect to the measure new instead. So f phi t, h, u, p, d, nu, u. And this again tends to the same, the volume average of f as t goes to minus infinity. And even, as I said, for any test function, and now it's a compactly supported function on the space r times x. And I do the same integral, but I now have two parameters and I let f depend also. I have u, phi, t, h, u, p, d, nu, u. And this again tends to the same thing. So f depends also on the parameter u. And the proof of this is quite simple. Basically, you decompose. OK, so we will discuss it as an exercise. But by approximation, clearly, we can replace the then, because f is a bounded function. So I can approximate the density in L1 sense. So I can easily reduce to nice, compactly supported, continuous density. All right, I don't get the same limits. Thank you. Thank you. Clearly, you have to integrate now. So r times x. And then f, u, p, du, no, no, no, no, d nu, no, d, and then the other order, d mu, p, and d nu, u. The same average here. So that is no good notation. So thank you. Q, maybe? Sorry if it's not legible. So this is an exercise to deduce this corollary from the theorem. But now, in the last six or seven minutes, I hope to just outline the proof. One proof of this theorem, this equidistribution theorem, using a so-called Margulis trick or Margulis thickening technique, which he introduced in his PhD thesis. And it will use the fact that the geodesic flow is mixing, outline of proof. So we would like to apply the mixing theorem with F1, OK, I just erased the mixing theorem, but you have two arbitrary test functions, F1 and F2, in the mixing result. You would like to take F1 to be equal to the test function F. And then you would like F2 to be a one-dimensional measure along the horizontal. If you were allowed to do that in the mixing result, you would get this directly. So I just write this out. Want to apply theorem 1, the fact that the geodesic flow is mixing, with F1 as the given test function F. And F2 as one-dimensional Lebesgue measure along the horizontal piece H, the fixed given horizontal piece H. So the one-dimensional Lebesgue along H. However, this is a distribution. It is not an L2 function. And the mixing result doesn't extend to arbitrary distributions or arbitrary measures. It's easy to show by examples. It doesn't. For example, if I would take an average along the unstable horizontal cycle instead and then flow backwards in time, it clearly I would get some result which is false. So I'm not allowed to do this. But still, OK, this is forbidden. But still, one can start asking, what happens if I try to approximate this Lebesgue measure by some characteristic function, some tube around H? And it turns out that since H is an unstable curve for the expanding deodesic flow, I mean I'm flowing in the direction such that it expands, since I'm averaging along an unstable piece of, since I'm expanding along an unstable sub-manifold, this approximation will work. It will work for me. So OK, so the trick is to thicken the horizontal cycle piece H. And OK, maybe I want to leave the statement of the theorem, so I move here and discuss. So OK, we want to thicken H. So we fix some small epsilon and let H sub-epsilon be an epsilon neighborhood of H. Now, this is not completely, this is a bit vague because I've never introduced a metric on G. One could introduce a metric and this would make sense. But I'm outlining the idea of the proofs. Then I hope to say a bit more about how you can make it more precise. So this is an epsilon tube around the fixed horizontal cycle piece H. And let F2 be the characteristic function of this tube. So characteristic function of H epsilon. But let's normalize it so that it has total mass 1. So I divide by the volume of the tube so that F2, the total volume average of F2 is 1. OK. And now we want to see what happens if I flow this tube. I have this initial horizontal cycle H. And now I have fixed some epsilon neighborhood of it in some three-dimensional neighborhood, which is H epsilon. Ah, yeah, of course, thanks. OK, so I have H and I have H epsilon. And we flow with the geodesic flow. And I know that the curve H, well, by definition, that becomes the curve phi th. So t is negative, and the geodesic flow is going in this direction. But the point of this proof, and I'm not going to prove it, I'm just stating it. The point is that actually also phi t of H epsilon is a nice neighborhood of phi th. It remains so when t goes to minus infinity. And the reason is that the horizontal cycle itself is the unstable direction for the flow. And so when I go out, go away from this horizontal piece, I can see it as moving along the geodesic direction and the other horizontal direction, which is stable, so contracting for the geodesic flow. So that approximation, when I transfer it by phi, will remain a small approximation. Of course, some point sitting here, say, one-third between the starting point and the end point, that point can move quite far along this unstable direction. And it can. But if you write down this with coordinates, it turns out that that averaging remains nicely. It spreads out nicely, so that averaging remains good. So that's the point of it. I'm going to write something. It's basically epsilon. OK, so you have to introduce coordinates. And the point is that you don't have to take epsilon neighborhood. You can even take some fixed one neighborhood along the unstable direction, but it will remain epsilon if you have fixed the metric and so on. It will remain. So in this proof, we can fix epsilon and we get the result. And then only at the very end, we will let epsilon go to zero to get a good approximation, a bit vague. And I want to write at least a few lines to say something. So the point here is that phi t of h epsilon is near the transported horocycle piece. And this means that here, now, to conclude the proof for this hand-waving proof, here's the thing that I wanted to compute. I wanted to compute f of phi t of hup, the u, one of the u. And since this, here I'm moving along this transported horocycle, and since the tube phi t h epsilon is always near this, and the averaging is as it should be, this turns out to be approximately equal to f phi t p and f2 p. Remember, f2 is the characteristic function of h epsilon and d mu p, because this is an indicator. This kicks in only when p is in the epsilon neighborhood of hu, and that means that p is in hu, is in the original h. And then I get exactly this. So how good this approximation is depends on epsilon. But now, here I can apply the mixing result. This, as t goes to minus infinity or plus infinity, if you like, this tends to the volume average of f d mu times the volume average of f2 d mu x. And I chose this, so it's equal to 1. Otherwise, this approximation, that would be a constant here. OK, so that's really hand-waving. I was, yeah, hope to give a bit more details, maybe, in the tutorials. Thank you.


the difference, if I may, between martial fighting and martial art. Martial fighting is I must cause someone to fall from myself to advance. So if you want to be a martial artist, you would actually sacrifice something of yourself to help someone else rise above as opposed to you causing them to fall so you can rise above. And if I think if we use that as a template of how to choose to be a martial artist over a martial fighter, that it's not about the, let me put it this way, the lowest common denominator that we share as martial artists is that we can beat somebody up. Is it necessary? Absolutely. So yes, the application, the street application of the arts is absolutely imperative, but it is not the end game. The end game is to enlighten yourself and to raise your own personal vibration so you can help to raise the vibration of other people around you.


Today I announced my retirement from YouTube. This will be my last day as the founder of Freedom, making videos for you because I am 38 years old. And here in the Philippines, if you're 35 or older, you can apply for a special retirement visa, which I've done and I was accepted. So today I am making my last video of The George Show. I hope you will not forget me. I founded some amazing brands on YouTube, like TGN, NETV, and now Freedom. And I really want them to keep growing and be successful and help you grow. Guys, it's been an amazing few years, but I bid you farewell on this, my final episode of The George Show. Did I get ya? April Fools, of course I'm not retiring, of course. This is not the final episode of The George Show. And of course, we are going to keep growing together with Freedom. We have so much to build, so much to do. Music, more game licenses. We're going to announce Nether, a unique game for you this Friday and a lot more coming in the next few episodes of... Secrets, if you really look at yourself you will be speechless. What happened to being doctors and teachers? What happened to being your brother's keeper?


Hello and welcome to another preview show here at Vitality Stadium, a muggy and rather foggy Vitality Stadium you'll see. I'm joined today by matchday commentator Chris Temple. You'll notice I'm not Zoe, my name's Ian and over the next few minutes we'll be going through a fun filled packed festive season and what we've got to look forward to over the next few weeks. Here's what's coming up on today's show. First we talk about the injury to club captain Simon Francis, the cherry's head to Old Trafford looking to cause an upset, Eddie Howell marks his 10 years in charge with the club and we look ahead to the game with Watford in our first game of 2019. Before we get started on the football then Chris, unfortunately on Thursday night we had some really bad news. Simon Francis ruptured ACL and that's the season done with. Yeah, horrible first of all for Frano. We obviously send our very best wishes to him. Again, it's just so innocuous isn't it? We've seen it with Lewis Cook on the training ground and we've seen it with Adam Smith at Newcastle. No contact, just that the way his foot planted in the ground, a horrible one for him. It looked awful at the time and you know he's in considerable pain. So, yeah, not, I guess it capped out a pretty disastrous day all round really in terms of the score line. But for him now I mean he's been, he's the club captain. He's the focal point. He's a hugely respected figure in the dressing room. You know, the players look up to him and he has been a huge support. I know for a lot of players when they've been going through difficult times. So the roles are averse now. He needs his teammates around him to help him through it. The one thing with Frano is that it happened to Callum Wilson when he was what, 23, 24, 25, maybe Lewis Cook, 21, 22. He's 33, which is a whole different ballgame I think in terms of the recovery because you know with the greatest of respect he won't have as much time left in the game as those guys. His body has taken more wear and tear down the years than other guys who are trying to recover from this. So I mean that's the key for him now is to try and get back in the early part of next season and try and be the same player that he's become because he's come so far as a player. You wouldn't have thought when Bournemouth paid 25 grand for him that he would be here, you know, playing the part that he plays in the Premier League these days. So I've no doubt in terms of his character, we see a lot of highs and lows as an individual and here at the club of course as well and that he will be well placed mentally to deal with it. The club obviously have the best possible treatment available now. You know, they'll be going out, I'm sure, to Dubai as Adam Smith has done recently as part of his recovery and of course the other side of it which is less important I guess from the grand scheme of things is no right backs. So if you've ever played there, PC, you could be in line for a game at Old Trafford. Absolutely right and just to sort of draw a close to the Boxing Day game. A very strange game. Eddie House said himself 5-0 perhaps wasn't a clear reflection of what happened but I guess whatever way you dress it up, it wasn't a great day at the office for the cherries. Yeah, I spoke to Azmi Begavic after the game and I sort of said to him, you know, it's 5-0 on paper but actually there were a few positives in there and I think probably biggest players don't want to seem to be saying there were positives after a 5-0. The manager can probably say it but for the players who've been involved, maybe they can't be seen to say there were positives after losing 5-0 but it was a funny old game as they say. You know, everything spurs hit went in with a bit of luck, a couple of defensive mistakes as well which are unfortunate and are just a bit too prevalent in the moment but from Bournemouth's point of view, missed some good chances. You know, the wee man with the header, very unlucky hit, Junior Stanislaus. David Brooks couldn't quite lift it over Larisse. Stanislaus's header, Charlie Daniels' header, obviously the offside goal as well so it just didn't work out for Bournemouth on the day. It wasn't a 5-goal game. I think, you know, spurs by a couple would probably have been fairer because they were comfortable. I don't think they had to play necessarily that well to win but it's the errors and things at the moment and just that just lack of clinical edge, maybe a lack of confidence a little bit in front of goal. Callum Wilson wasn't able to get into it much at Wembley so yeah, that's one to quickly draw a line under and move on fast. So Eddie Howe has said recently that he thinks his side have actually played quite well against teams like Manchester United, teams like Tottenham and of course it's Manchester United coming up next. We know it's not going to be easy but what are the sort of points to be positive about from a cherry's perspective? Well against United recently, they've been competitive games. Let's, for example, as we go into Old Trafford, let's reflect on the last two there. Last season a narrow one-nil defeat. Bournemouth had quite a lot of pressure at the end of the game, could have nicked something and then the season before of course was that the famous 1-1, it's not often, there's not many famous 1-1 draws in the world but having gone down to 10 men in that game with Andrew Sermon being sent off and it was of course a day when Eddie Howe was sick as a dog and barely even made it to the touch line that day as well. So yeah, that was a highlight the first time they'd got something at Old Trafford and then of course here, what, just a few weeks ago done in the last minute by Marcus Rashford in a game that Bournemouth probably not only deserved to get a draw from but possibly could have won as well. So yeah, I know Manchester United have changed manager since then and things have taken a bit of an upward curve for them but in terms of the last couple of visits to Old Trafford, it seems to be a ground now that doesn't overhaul them, it seems to be a place that there, it's a big open pitch, you know, you can go and express yourselves and from that point of view, no fear really but the only problem is that they've just got off the back of a five-nil hiding so the key is to move on very quickly from that. They do come up against the United side who have turned the corner at a bad time as far as Bournemouth are concerned and then Olly Gunnar Solskjaer has gone in there and somebody who obviously gets the club and understands what United fans want to see and the best way for United to play, I mean, it's amazing how players can suddenly start playing two weeks after they couldn't play. Paul Polk where suddenly is, you know, bounce back to be some sort of world beta. So, yeah, they go in there at a bad time in terms of the Solskjaer honeymoon if you like but you know, we've seen Southampton's honeymoon bubble end this week for their new manager. So who's to say that Bournemouth can't go there and puncture Olly Gunnar Solskjaer's dream start. Exactly, right? And they have scored eight goals in the last two games. Bournemouth, as you say, maybe not going into the game with the back of the best form after the box and a defeat. What's the best sort of setup? Bearing in mind, of course, Simon Francis isn't available either. What's the best sort of setup in terms of trying to get a result out of Trappard? I think the best sort of setup is at the moment what everybody has got. To be honest, I mean, I can imagine they're going to go three at the back. They presumably have to. The likes of Diego Rico and even Jordan and I, you can imagine coming in for in contention having with the Chelsea game in mind because actually you wouldn't necessarily want to throw those two in that sort of game as wingbacks without that Chelsea game for a bit of confidence for both of them and for the manager in putting them in. But they, you know, they showed in that game that they can handle not only the big stage, playing away from home against quality players with the home side, having all the expectation and a lot more of the ball. So I wouldn't be surprised to see Diego Rico given that given a go this week. The Ryan Fraser is the one, isn't he? Because we've seen him play right back in the past, but I think I said on the last preview show you don't really want your most creative attacking player playing it right back or even necessarily at right wingback. So that is an option for Eddie Howe. If we're talking about a back three, then it's a sort of, I would toss up between Tyrone Mings and Charlie Daniels. Charlie's had a difficult spell recently. Whether he, you know, it gets taken out the firing line or whether, you know, Eddie Howe's been pretty loyal to him, whether we see him play as a left wingback maybe and Tyrone Mings on the left of a back three. But I think that looks like the only way they can go. I don't think they've got anybody to play as a right back in a flat four. So from that point of view, it's not ideal because you want to go to the places that, you know, like Manchester United with the ideal set up with everybody available and being able to play the best way that you can that you think will nullify them. But it could be a case of this is the team. This is how we've got to play. Do the best you can. But it still could be effective. We've seen the back three work out well against big teams before. And I guess the thing is with a big game like an old tripped old traffic is that you've got to make the most of it. And you've got to almost revel it and enjoy it and see what you can do and see if you can get the best result you possibly can. And Bournemouth second favourites, of course. And that's, you know, again, that might be helpful in terms of going there. No one will expect them to get anything. United expectation from their fans all of a sudden will be through the roof as well against a team like a Bournemouth who they will perceive to be beatable opposition. You know, you look at the games coming up around the corner for the Chorries Watford obviously here in the first home game of 2019 and then Brighton and the FA Cup after that then Everton away looks a bit tricky and then there's obviously midweek games in January as well with Arsenal and Chelsea looming not too far in the distance as well. So yeah, United away, it's a game that you wouldn't bank on needing some points from. But I think that the performance is the key thing. If you get a point there you'd be absolutely delighted. But then we look back at the Carabao Cup what a week last Wednesday lost the game but everybody was feeling quite positive about it. So a similar, if you can't win the game a similar sort of performance at Chelsea.  We've just seen Leicester throw in two results that nobody would have expected them to get. So yeah, let's keep the fingers crossed that this could be a, I guess, a revelation of a moment at the end of 2018. And before we're back here at Vitality Stadium for the next game, it's Watford. It's also a special day for Eddie Howell. 10 years since he was appointed on the 31st of December 2008 that would be around there.  Yeah, of course he's been away to Burnley in the middle of it as well. But yeah, 10 years ago since he first got thrown the reins, if you like, on the way at Darlington. And here he is 10 years on about to go away to Manchester United. So I think that is pretty much about his polar opposite ends of the scale as you get. As you say, what a rise it's been in that time. So many highs and lows for him, mostly, you know, mostly highs, one or two lows that in the early stages of his career that he didn't have a lot of control over. But you know, what he's achieved since then, you know, speaking to him about that, he says, you know, he doesn't feel like he's changed too much as a manager in terms of how he approaches things. I was asking, you know, how the preparation of things differed back then. And he said, he covers a lot of he covered a lot of motorway miles back then doing a lot of the watching himself. There wasn't the analysis and the the scouting network around the country back then. I think the scouting network probably when Eddie House started in 2008 was probably if you knew someone that happened to be at the game, give him a ring and see what they thought, you know, with no disrespect to Eddie and the setup. But I think that's probably how it worked. But yeah, you look at him now, I'm sure he's so invested in the job. He if the one thing that probably has changed, he probably devotes even more time now, probably the maximum amount of time that he could devote to the job. But it is, you know, it is amazing what when you when you look back at those those 10 years and where we all were 10 years ago and what we were doing, I mean, I was working for Radio Solent, so I haven't gone anywhere. But yeah, Darling to the way to Manchester United away. And of course, he lost his first two games in charge.  So he won his first game here against Wickham. And that he said again, speaking to him, he said that gives him much pleasure as it does winning a game now. So yeah, it's what a decade it's been. I suppose a lot of people would just like to rub the Burnley bit out in the middle, maybe at times Eddie would as well. But yeah, remarkable, remarkable decade and who's to know how much longer he will be here, hopefully for the foreseeable future and beyond. But there's no doubt in what he's done in those 10 years for this club. Yeah, we fast forward on 10 years from that landmark opening, if you like, back here against Watford, that's sort of apt Amisham's ways from. He was on the terraces at Vicarage road before we moving down here as a young lad. So it probably makes sense that we're playing Watford and it's a really good blockbuster sort of game to start 2019 here. Yeah, I like that. Absolutely. And Watford, of course, if we're looking at performances this season with the Watford away game back in his old neck of the woods Eddie was probably right up there with the best performances this season. Four nil winners. A team that again, you've probably gone there happy with a point on the day. But Watford got blown away and they'd started the season really well. Don't forget they'd won the first few games. They're like Bournemouth in that middle portion of the table at the moment in terms of the teams who are capable of winning two or three in a row, but also capable of losing. And at the moment, the gap between sort of seventh and eighth down to about 14th to 15th is pretty tight. We've seen West Ham obviously beating Southampton this week. They've shoved Bournemouth down a place as well. Just on the subject of that, by the way, if we're talking about calendar years, there was an interesting league table I saw this week of the calendar year of 2018 rather than the the season, if you like. And Bournemouth was seventh in that table outside of the top six and had a great second half to last season and a great first half to this season, which for a 12 month calendar year has put them in a great position. Not bad at all. But yeah, all in all, coming back here, good way to start, you know, under the lights. But again, that I would say off the back of Tottenham and obviously we're speaking before the club have been to United. So they may be coming into Watford off the back of a fantastic four nil winner, Old Trafford. But either way, it's a game that you've got to look at and think a bit like Brighton, bit like Huddersfield. That's a game that needs to probably beat what needs to be a point at least. And I think probably that's one that's a bit of weight on to get three, I would say. And it's been well documented the form recently, of course, doesn't help when you've got trips to both Manchester side, you're playing arson, you're playing all sorts of teams from the top echelons of the league. But now I suppose is the time of the season where it's sort of win or bust, isn't it? Now it's the time of the season where we're playing some of the teams around us, maybe even below us in the table. And now's the time we need to win and get our season back to the heady heights we had at the start of the season. Well, that's the thing of the platform they've got from that from that record breaking start, you know, 26 points here, probably three more wins and safety, I would say. I don't think people think that anything more than mid 30s is going to be needed to stay up this year. But yeah, as you say, it's a period where and again, we keep harping back to the same things. But you think of the new cast of game and people look back and go, what a missed opportunity that was. Huddersfield didn't play that well, managed to win. Brighton, it was a bit more like it here and the feel good factor was back in the stadium. So let's hope that by the time they get to the Watford game that, you know, the positivity that that Brighton result exuded in the performance of David Brooks, who didn't get much of a chance to get in the game at Wembley, you know, let's hope that the players revelant in that occasion as well and, you know, be able to capitalize, I guess, on the good start to the season and make it the best ever finished because that's the platform they've set themselves now. There are definitely a number of worst teams that are going to be fighting out the bottom three places. And the question now is can Bournemouth hang on to the likes of Everton and West Ham, who are obviously picked up in a going, well, even Lester, look at Lester now, they've they had a pretty non-spectacular start to the season and they're seventh. So that is that the sort of middle chunk of the table now that Bournemouth are in. No reason at all why they can't, as we now enter the second half of the season, the United Game is the first game of the second half of the season. They've played everybody. They've had a look at everybody. Now's the time to start doing a few doubles over people. And we'll be here, of course, for every step of the action from Watford, Manchester United and beyond. We hope you join us too and we hope you have a wonderful 2018 end of the year and a wonderful new year, Ziv.


The Social Economic Rights Accountability Project has asked Niger to return more than one billion Naira in the nations from Nigeria. Is this request in order? And the English Premier League is back. We'll talk sports this morning with a look also at the ongoing Commonwealth Games in Birmingham, the United Kingdom. And G.D. Johnson joins us later for in-depth analysis of today's newspaper headlines. Good morning to you. We're back with the breakfast of Pasadena Afqal right here on Channel 408 on the STV. And of course, 308 on start times. We're reaching a live from our studios in Victoria Island because my name is Kofi Bartels. It's a beautiful Friday morning and we're back with interesting discussions. Of course, an analysis of the issues that are of importance to you this morning. And we usually would start off with a look at what's trending in the social space. We have, of course, the Internet and the social media networks being a very active space with discussions on very important issues and we call it a top trending segment. Let's go there. The first one we have to look at happens to do with the Economic and Financial Crimes Commission declaring a popular Nigerian Instagram figure. All right. Let's call it that. Mustafa. He's probably known as Monfa but his name is Ismaila Mustafa.  He's always, you know, being invited for questioning and all that. So they have come up with a new one. This is the latest one. They're declaring Monfa wanted. And of course, they put out a tweet yesterday and this is what the tweet said. It says anyone with useful information as to his whereabouts should contact the Economic and Financial Crimes Commission, EFCC, or the nearest police station. This is what EFCC said. It said Monfa is wanted in an alleged case of retention of proceeds of criminal conduct, possession of documents containing false pretenses, false asset declaration, and laundering of proceeds of unlawful conduct is what they are saying. Of course, there were reactions to this on the social space. But let's do a background to this particular latest request by the EFCC now. In June, a Lagos State Special Offences Court sitting in Ikeja had issued a bench warrant. Ikeja is a part of Lagos for those who are tuning in from other parts of Africa. Anyway, the additional bench warrant for the arrest of Monfa and the judge issued the warrant following the continued failure of Monfa to present himself for trial. You can see a picture of the gentleman there being guided into court by an EFCC operative. He was arrested on January 10th by the EFCC and arraigned on January 12th alongside his company, Ismalop Global Investment Limited, on an eight-count charge of alleged money laundering of over six billion Naira. You can see some of his Instagram pictures there. He's sort of a celebrity that I think he was rubbing shoulders with the Hush Poppy terms of flaunting his wealth and celebrity lifestyle, expensive lifestyle. Let's call it that on Instagram. Now, on January 18, this same interview was admitted to bail in the sum of 200 million Naira with two shirtees in Likesum. One of the shirtees the judge ordered was on a property valued at 100 million Naira within the jurisdiction of the court. The court also had ordered that the defendant return to his international passport to the custody of the court while ordering his remand in any correctional centre of his choice in Lagos pending the perfection of the bail conditions. He was able to do that and therefore, think about two months later on March 28, he was released to his lawyer, Boyga or Yehole who is a senior advocate of Nigeria. Shortly after the court varied his bail conditions and of course that was that. But the EFCC have now had to come out to say that this gentleman is wanted for the reasons the earlier put out. And of course it's got a lot of Nigerians talking and you know tongues have been wagging but Monfa has come out in his defence also to resupply the EFCC by saying that he didn't steal or loan the money. That's what he's saying. He didn't steal or loan the money. So he's absorbing himself. He went to, I mean I'm sure there are no surprises that he chose his verified Instagram page. That's where he went and he went there to declare that he could not be declared wanted by the EFCC as he has neither stolen money nor gotten involved in money laundering. That's what he's saying. This is what he put up and I quote, After my arrest and arraignment on the first case of the Federal High Court in Lagos on the 10th of December 2021 the court gave me judgment and ordered EFCC to release my wristwatches and other valuables in their custody to me for being unlawfully seized. He went on to write being a bitter agency caused the EFCC a bitter agency. The EFCC in dissatisfaction with the judgment of the court on the release of my wristwatches came up with another trumped up charge against me he says. Moffa continued to write by writing on Instagram on the 10th of January 2022 while I was at their office at a court able to receive or retrieve my wristwatches in compliance with the judgment of the court. Immediately after collecting my wristwatches they held me down at their office and charged me to court the following day on the frivolous charges he says. I was arraigned before a familiar judge to the EFCC and in fact I was given a very difficult bail to meet as a result of which I ended up spending over two months in prison custody while making efforts to meet the difficult bail. After several attempts to meet the bail conditions failed I successfully applied for the bail conditions to be reduced which eventually led to my release he wrote. Upon my release I traveled outside Nigeria to check on the welfare and well-being of my family and returned to Nigeria shortly only for the EFCC's corrupt lawyer he uses the word corrupt lawyer not our words I wrote to me Oyedepo rather than focus on establishing that his criminal case against me along with the EFCC operative Keina Garba he chose to capitalize on the fact that I traveled to Blackmail me into entering plea bargain he says else he would inform the judge I traveled and ensured my bail is revoked therefore making me have no choice but to dance to his tune so he's alleging Blackmail here he says I initially refused but after so much Blackmail and intimidation by Rotimi Oyedepo I succumbed and was told to pay 142 million to the EFCC not that he's admitted that he broke his bail conditions so he said they made him ask him to pay 142 million to the EFCC as compensation and also signed for a one-year imprisonment sentence I wasn't comfortable and they thereafter if you reduced it to 14 million and one year imprisonment when it became obvious to the EFCC or to the EFCC that I wasn't going to sign the plea bargain and to play the game they informed the judge that I traveled so he's saying despite his breaking the bail conditions meaning that you give us your passport you can't travel out he found his way out of the country he said he went to attend to look check on his family that he came back to Nigeria and the EFCC knew what had happened and they found out and they decided to ask him to pay some money so they keep it away from the judge so he says it is Blackmail he said it is Blackmail but if you actually might say it is a plea bargain so he says that the judge without any concrete evidence and disregard of the fact that I was never absent in court revoked my bail but fortunately the judge again released me on temporary basis and I was asked to appear in court a week after so he goes on and on and on it's a long one that he wrote on Instagram if we want to go into it all we would not take the next story but you get a sense of what his response is which is he is saying that he is innocent and the EFCC is merely trying to frame him up it's a case that of course it's sure to garner a lot of attention as Monfa is quite popular on Instagram with his lavish lifestyle I think the difference between him and his fellow Instagram celebrity who shows an expensive lifestyle it's that Monfa has a little bit convicted and also has over the years chosen to return to Nigeria to face any allegations we'll see how this goes another one that caught attention is the Acquavum State High Court sitting in New York ruling on the murder case of the young woman the job seeker whose disappearance went viral on Twitter me and the family were asking for her on Facebook also everyone was worried about her whereabouts because she put out information that she was in trouble and nobody could find her In Yoban Umoran it's her name I think one of the investigative journalists from Nigeria who is very active on social media David Andrei spearheaded some efforts to get information out regarding this girl and indeed he was trending yesterday people giving him credit for the fact that information was put together to enable the authorities to arrest Uruak Akpan who is said to have been behind Yoban Umoran's disappearance all the cases were in court and the Acquavum State High Court sitting in New York state capital yesterday sentenced Uruak Akpan this 21 year old to death by hanging for the murder of Yoban Umoran the job seeker from Acquavum State the court also found the accused guilty of rape accordingly sentenced him to life imprisonment for having carnal knowledge of the deceased whom he lured to his father's house in Uruak local government area of Acquavum State on April 2021 in April 2021 another pretext of giving her a job so this judgment lasted for two hours the delivery of the judgment lasted for two hours yesterday the trial judge Justice Basi in Kanang discharged and acquitted the convict's father Frank Akpan and his sister Anwan Basi the second and third defendant in this case who accused of being accessories to murder after the fact to murder respectively so this is what the court said the judge said that this court has proved beyond reasonable doubt that Uruak Akpan is guilty of murder and you are charged accordingly you are to die by hanging until you rot away those were the words of the judge however you know this is it wasn't without drama and of course they also got people talking there wasn't there was confusion in the court as the gentleman the young man the 21 year old young man after his conviction attempted to escape from court after the sentence was announced by the trial judge attempted to run away the convict came out of the dock and immediately grabbed a lawyer by the neck before the scary operative swooped on him grabbed him and beat him mercilessly causing a stare in the court so that's that a lot of people on social media expressing satisfaction you know that this case has run its course and the justice has been served to this young man you can see him being dragged away from the court let's listen and watch a bit of what happened there we get up from my face we get up from my face we get up from my face we get up from my face it was really a dramatic scene at the judiciary the court premises there in a quibble state and you can see the gentleman the young man being dragged being dragged from court into that waiting vehicle the case was really followed in fact the search for Inil Moran was really really hot and active on Twitter with a lot of people asking for help for whereabouts looking at the last message she shared online and trying to get clues information coming out that she's going for a job and she also put out information a distressed information on Twitter this helped people to know what was going on and of course the investigation and the search yielded some results the arrest of this young man I think this is one of the cases that has received swift judgment it's not always usual it's not always usual let's quickly take one last one social media was also responding to the appointments by the two major political parties in Nigeria the All Progressives Congress announced an appointment Governor Simon Lalong of Platter State as the Director-General of the APC Presidential Campaign Organization appointing Festus Kiamo as a Spokesman a number of appointments were made excuse me now before these appointments you had Commidas Adamso Shimuli former governor of Edo State being referred to as a DG of the Bolame Tinwoo APC Presidential Campaign Organization it seems there's been some reshuffling if you want to call it that why people were talking a lot about Lalong is because he's a Christian from Northern Nigeria and some feel that this may be a way the party is trying to sell themselves to Christians around Nigeria especially in the north following the same faith ticket of the APC now the People's Democratic Party's Presidential Candidate Artiku Abooka also made some appointments appointing two notable vocal Nigerians of vocal and political issues one is a professional politician Dino Mele when I say professional politician I think I mean that he's been in politics for a long time and has held positions in the legislature so Dino Mele who came from the APC to the PDP some years ago it's been appointed as a Spokesman for the Artiku Presidential Campaign also Daniel Boilap is to serve alongside Dino as a Spokesman of Artiku Daniel Boilap is a very vocal on social media and on electronic media as well and he has been known as an APC a staunch APC supporter and so it was surprising to many that Daniel Boilap was appointed to speak on behalf of Artiku during the presidential campaigns but Daniel Boilap last month decamped from membership of the L Progressives Congress citing the same faith ticket of the party all right we have to go I'm up next Jideh Johnson joins us as we flip through the pages of the National Daily analysis of the headlines