Scott, and so our second speaker this morning is Hans-Jo Kempain from Imperial, and he's going to be talking about singularities of Taylor-Einstein metrics and Kalabi algorithms. Thanks, so it's a great honor for me to speak here. Like Simon said yesterday, Kalabi's ideas have given us work for the last 60 years, and it can be a little bit more specific. They've certainly paid my bills for the last six years, so I want to thank you for that. Okay, so I want to give some sort of overview lecture about singularities of Einstein metrics. I mentioned some old results, some new results, and some speculation. So looking at Einstein metrics, let's remind in metrics with constant Ricci curvature. So you're saying that Ricci curvature of G is equal to lambda times G for lambda as the Einstein constant, and that's going to be normalized to be either plus one, zero, the Ricci flat case, or minus one. And so more specifically, I'm interested in singularities of Kala Einstein manifolds, and so the Kala case has this beautiful feature that you can sort of split up the problem into a complex part and a metric part, or an algebraic part and a metric part. So you can always look, so forget about the metric data and look at the underlying complex manifold, hopefully. Algebraic manifold, possibly algebraic variety, and then the idea is if you understand the algebraic geometry well enough, singularities of algebraic varieties in some form, then hopefully that's going to tell you something about singularities of the associated Kala Einstein metric. So it's going to be about the correspondence between algebraic singularities and metric singularities. And the talk is going to have two parts, hopefully. So in the first part I want to, so notation, so X is always going to be a Kala manifold. So complex manifold in the first place, or even algebraic manifold, of complex dimension N. You have a complex structure, tensor J, Riemannian metric G, that's Kala, so that J is parallel with respect to G in terms of the associated Kala form, omega. So that's that standing notation. So in the first part of the talk I want to talk about the sort of first case where not everything is known, so that's complex dimension two, the case of complex surfaces, in particular Einstein metrics on real four-dimensional manifolds. And there I'm going to talk about sort of, I'll show you a bestiary of phenomena that are known or are conjectured to appear in all three cases. So I mean the three cases are markedly different. So for all three Einstein constants, lambda equal plus, equal to plus one, zero or minus one. In the second part we're going to go up to the complex dimension at least three and I'm going to restrict to the case that the Einstein constant is zero, so Kala Ricci flat metrics. So in the first part I'm probably going to be mentioning some stuff that I did in my thesis and the second part is parts joint work with Ronan Conlon and Aaron Neighbor. Okay, so the first part is more of an introduction. So we'll just recall that from yesterday that we have a necessary condition at least in the compact case for the existence of Kala Einstein metrics with a necessary topological condition for KE. And so that's the relationship of the first-turn class of your manifold. So let's see one of x be either positive or lambda equals plus one, zero or negative. So in complex dimension two you can say very explicitly what the manifolds with positive first-turn class are. This is so-called del-Petzl surfaces. So what we've got here, so the blow-up of CP2 in k points in sufficiently general position were k's between zero and eight. And then you've got CP1 cross CP1, but that's kind of an easy case because it's obviously Kala Einstein, just a product of two-round metrics on the Riemann sphere. Then here up to, in the Ricci flat case, up to Feynard covers, we have Tori, so C2 mod gamma for some lattice of full-rank gamma, or otherwise famous K3 surface. And then for the C1 negative case, we just attach a label. I mean we would just call them surfaces of general type. They're just way too many, so we don't really know how many there are. So maybe two quick remarks. So the first one is that the subdivision is sort of reminiscent of what we know in the complex dimension one case, the Riemann surfaces. So for complex dimension one, the first on Riemann surface is the first-turn class. So it's just minus the Euler characteristic of your surface. Then you have the sphere, Tori, or surfaces of genus at least two. What's very different from the case N equals one is that there are many more complex surfaces than are exhausted by these three cases. So I mean you could take products like Cp1 across a Riemann surface, sigma g of genus at least two, something like that, because of mixed cases where the turn class doesn't have a definite sign and where there's a priori no hope of finding anything like a Kela-Heinstein metric. So mixed cases or fiber cases. So that's quite different from the Riemann surface case. So that's a necessary topological condition and then we have the corresponding existence theorems that the necessary condition in complex dimension two is almost always sufficient for the existence of a Kela-Heinstein metric. So I'm going to make three cases one, two, three. Just continue here. So case three is in some way, in some sense the easiest case. So that's the Oberleer theorem. There exists a unique Kela-Heinstein metric on, general tribe is really a bit more general even than having the first turn class negative, but in the first turn class it's actually negative. Then there's just a unique Kela-Heinstein metric. So the second case is Yau's theorem that there exists a unique Ricci-Flat Kela metric in each Kela class. And then in case one, that's the hardest case that's been open for the longest time. So if that's been settled in work of Tian, Yau, there exists a unique Kela-Heinstein metric when K is at least three and it had been known for a long time. It's as much a Schema's theorem that was mentioned yesterday that there can't be a Kela-Heinstein metric when you blow up one or two points. So that's what's known about the existence of Kela-Heinstein metrics on complex surfaces. And now I'm going to turn to that question, what happens to the metrics if the underlying algebraic surfaces degenerate? If you have a sequence of algebraic surfaces that acquire singularities, then what do the associated Kela-Heinstein metrics do? How do they behave? All right, so that's the question. What happens if Kela-Heinstein metric when Xj degenerates? Okay, so maybe the basic picture to have in mind is what's going on in complex dimension one. So if you look at hyperbolic Riemann surfaces, then the standard picture out if you start with a genus G surface, you can degenerate that to something that has a node. So it's an ordinary double point from the point of view of algebraic geometry. And then what happens, what happens metrically in this picture is that well, here it kind of looks the same. It's a random hyperbolic Riemann surface.  These hyperbolic cusps really infinitely long. But the question is what's going on in complex dimension two? What can we say in complex dimension two, really? So I'll try to give you one example in each of the three cases, sort of positive Einstein constant, Ricci flat and negative Einstein constant. I'll try to indicate how the cases are different and what kind of phenomena you can see. So if you go to the case of positive Einstein constant, so Ricci is equal to G, then the first observation is Meier's theorem that tells you that the diameter of the manifold is uniformly bounded by the square root of three times pi. So all your Riemannian metrics sit inside some sort of fixed box of fixed diameter. That's some kind of compactness already. And then you sort of have the bane of Riemannian geometry that if you take limits of sequences of Riemannian manifolds, they can drop dimension. It's called collapsing. If the metric sits inside a box of fixed diameter, it could be something like this a priori, sort of shrinking one direction and converting to some lower dimensional space. We don't really want that. Luckily, we don't have that in the positive case. We can use volume comparison to see that that's impossible. So volume comparison with a positive lower Ricci bounds of inequality tells us that the volume of a ball of radius R divided by R to the fourth, so it's volume ratio. This goes down as the radius R increases. So I have a lower bound. Just take the radius R equal to the diameter of the manifold. So it's going to be like root three pi to the fourth. And then up here I have the volume of the manifold. But in the positive K-aligned stand case, it's just proportional to the square of the first churn class. So up to this normalizing factor here, this is basically C1 of X squared. So that's a topological invariant. So it's a topological lower bound on this volume ratio.  So this means that if I take a gram of Hausdorff limit, so I have a sequence of X i, J i that degenerates in some way and have the associated Kepler-Einstein matrix X i, G i. And then I want to take a gram of Hausdorff limit as the sequence goes to infinity. So I'm converging to some metric space, a priori X infinity, D infinity. And I'm definitely going to know that X infinity, D infinity is sort of a metric space of finite diameter and it's four-dimensional because of this non-collapsing notation so X infinity has finite, finite diameter and is four-dimensional. So it's Hausdorff dimension four if you want. That's a good thing. Much more is true actually. So there's a problem I studied and there's actually an ingredient in some sense in the existence proof of Tian in the positive case that you have an a priori result of what the gram of Hausdorff limit could be like so if in the worst possible case... That's... I guess there's Mike Anderson, Bando Kazunakajima, Tian who proved that in the late 80s or early 90s that this gram of Hausdorff limit is actually in orbit fold with isolated singularities. And the singularities are the form C2 mod gamma where gamma is a finite subgroup of U2 acting freely on the unit, on the unit sphere. So what does it mean geometrically? So imagine that's your gram of Hausdorff limit this four-dimensional orbit fold. So the easiest case is when there's only one singularity when gamma is equal to Z2, the smallest possible group, to forming one orbit fold singularity looks like C2 mod Z2 approximately your space. So that's X infinity. And then here you have a member of the sequence sort of X sub i for very large i. You can imagine that there's some sort of little bubble forming. So this is X i, G i for i very large. So there's a little bubble of curvature forming. Symptomology that gets pinched off and disappears into the orbit fold singularity in the limit. And then you say, well, we want to understand what that looks like exactly. It's called the bubble. So what we do is we rescale here to make sort of the topology that's being pinched off have unit size and then what's going to happen is that all the rest of the manifold, if we sort of scale up, if we zoom in with a microscope, all the rest of the manifold is going to be pushed off to infinity and we're going to converge to some complete space here that's Ricci flat because before we rescaled the Ricci curvature uniformly bounded and what we're going to be converging to is then a complete Ricci flat. KLM metric in complex dimension two, that's a hyper-KLM metric. So it's actually KLM with respect to many different parallel complex structures. And that's that if you dilate out, you get a gravitational instanton. So this is actually one I constructed in my thesis. So it has the same basic structure at infinity. So it's a half ray topologically across a three-dimensional nil manifold. Some topology inside. And the volume growth of this thing is... So we had Eguchi Hansen with Euclidean volume growth. We had Taubnacht with pubic volume growth. So the volume growth here is r to the fourth-third. And the sectional curvature decays as slowly as it possibly could in dimension four for an Einstein manifold so precisely quadratically. There's scale invariant decay, so this means the metric size of the plug is roughly one. That's the only thing that fits, but of course you have to prove it. It's a complete type of KLM manifold with a sort of asymptotic behavior. It's the cross-section at h3 or h3 model like this. Yeah, sorry. Let's just say that this means Heisenberg group modulo lattice. Okay, moving on. What is the complex structure of this thing? If you put one here, you get an affine cubic, so if you want it's... What's that? Or you can hypercalabotate it and you get a rational elliptic surface. So complex dimension, at least three, in that case. So here's a simple algebraic singularity model. And we want to understand it from metric point of view. If you take x, let's kd star for a final manifold, d. So d is a final manifold. What this means is you take the total space of the canonical bundle of d and you collapse the zero-section to a point. What does that mean? That's sort of an algebraic cone if you want. So somehow the picture is this. You have d that has complex dimension n-1. x is n-dimensional. And you're taking the complex cone over it. So you have a circle bundle sitting over here. That's sort of the theta direction. And then you just sort of cone it off in the usual real sense. That's the r-direction. So you have r and theta. It's a complex cone. So that's sort of singularity. We'll say d is a final manifold or maybe an orbit fold. So perversely, the most natural examples actually have d in orbit fold rather than a manifold. Or orbit fold. So that's the type of complex singularity that you encounter quite often in limits of sort of Kalabiyao manifolds. Just algebraically if you take an algebraic limit. So what's sort of the metric model for this? Why should there be a metric model in the first place? So there's some sort of theorems in the French school of complex analysis about weak solutions to the Monjom-Pierre equation. Let's say if you have a singular Kalabiyao manifold with sort of singularities, as you break singularities like this, then there's sort of weak or singular Ricci-Flat matrix on these guys. You just can't say anything about what the metrics actually look like. It's sort of an abstract existence theorem on the level of Kepler potentials that tells you nothing actually, but sort of the asymptotic behavior of the metric. So you want to make a good guess for that and try to understand what that could be. There's sort of an easy case where you can make a good guess, at least from the point of view of making a good guess it's easy. That's when D is actually Kepler-Einstein. And then that's construction that I think is implicit in one of Kalabiyao's papers about the Ricci-Flat matrix on total spaces for canonical bundles. So if this complex cross-section of the cone D is Kepler-Einstein, you can lift that to an Einstein metric on the circle bundle. That's what you would call a Sasaki-Einstein metric. And then the cone over that is going to be a Ricci-Flat cone, actually a Ricci-Flat Kepler cone. So if D is Kepler-Einstein, you can sort of lift that metric and cone it off to get a Ricci-Flat cone structure. So there's your Ricci-Flat metric. That's a perfectly good guess for what the model metric should be. So Kepler-Einstein metric on D gives rise to a Ricci-Flat Kepler cone metric on X. Okay? So there's your guess. So now you might want to ask a harder question. Does that globalize? Next question. Suppose you have a compact Kalabiyao X hat. Some Kalabiyao variety in the algebraic sense equipped with one of these weak singular Ricci-Flat metrics that we know is from sort of Pluripoi potential theory. And let's assume that it has an algebraic singularity that's bihologomorphic to X. Just algebraically. Is it then true that the Ricci-Flat metric that's known to exist on the compact guy actually has to converge to that canonical model that you wrote down there? So do the singularities of the metric have to be conical? So is it true that the Ricci-Flat converges to GX always? I think that's that seems to be completely beyond the reach of current technology to say anything much about that problem. So whether you can actually have singular Kalabiyao metrics with isolated conical singularities where you sort of guess the right cone model in advance. There's sort of a theorem approved with Ronin Conlon. You could phrase that as saying that infinitesimally the problem at the linearized level the problem is unobstructed. So if you're trying to solve the Kalabi problem construct a Ricci-Flat metric using a continuity method sort of on a background space that has these conical singularities then you actually do have openness in the continuity method. So if the linear part is okay so you can solve the continuity method for a short while and sort of preserve the conical geometry. It doesn't sort of jump immediately to something else. The hard part is usually this closeness where if you can go all the way from 0 to 1 the continuity method and that I don't know if it's true. But so even the linear case the openness is a sort of non-trivial theorem that's kind of gap theorem if you want. So the statement is if X is a Kepler cone so in a metric sense really a cone with a smooth cross section whose cone metric is Kepler with non-negative Ricci curvature and if you have a harmonic function U on X harmonic so it's a linear theorem so we're passing U is equal to 0 and you assume it's sub quadratic so it's a little o of r squared then it actually has to be plurihomonic some kind of Leovall theorem or cancellation theorem that if you restrict the growth of a harmonic function on your space to be less than quadratic it's actually automatically plurihomonic it's a real part of a holomorphic function and that tells you in some sense the obstruction to openness in the continuity method on those spaces isn't there closeness it's a it's a different story as r goes to infinity so there's this also true on asymptotically conical Kepler manifolds with non-negative Ricci curvature so here's your cone now imagine you have an asymptotically conical space that's asymptotic to that cone at infinity you assume the space is Kepler with non-negative Ricci curvature as well if you have a sub quadratic harmonic function on a space like that it still has has got to be the real part of a holomorphic function and somehow this is false on TELB nut so TELB nut is Ricci flat Kepler but you can make a harmonic function of linear growth on TELB nut that's not actually plurihomonic so somehow this maximum volume growth condition enters into the picture and sort of tells you something about the behavior of these harmonic functions so TELB nut is not asymptotically conical it's cubic volume growth but that's sort of the easy case where we could easily guess the right local model but it's sort of probably still way hard to prove that the model is realized globally in these X hat situations so look at a different case where we can't even guess the right local model a priori so hard case is when D is not when D is not Kepler Einstein so then we have this singularity X is a star that's an algebraic singularity and we don't have an obvious metric model I should give you sort of hands on the example for that the example is you take D so that this has complex dimension N so this has complex dimension N minus 1 so D is actually an orbifold in this case that's Cp N minus 1 the orbifold multiplicity is 1 minus 1 over K along a divisor and the divisor is a hyperquadrex QN minus 2 it's a hyperquadrex and then this space X KD star if I do this complex cone construction I get a very simple singularity I get an ak minus 1 singularity so X is just the ak minus 1 singularity in Cn plus 1 so it's a hyperstrict for singularity that's the locus of all points in Cn plus 1 where C0 to the K plus C1 squared plus plus plus plus Cn squared is equal to 0 so in particular K K equals 2 gives you the quadric for the associated stencil cone metric which in complex dimension 2 is the same as a Gucci Hansen so when K is equal to 2 then on the quadric cone we do have a Calabi-Yau cone metric so we do have a good local metric model so the difficult question posits itself but how about other values of K so that's maybe the most natural kind of singularity that you could possibly look at so it actually follows that's known from work of Russ and Thomas also from work of physicists Gauntlet, Metrelli, Sparks and Yau who study Sasaki Einstein Manifolds so using stability type obstructions you can show that when K is greater than 2 times n minus 1 over n minus 2 then this orbit fold there so with this cone angle along the hyperquadric there's no K lines in that metric so with algebraic singularity and ak singularity that occurs very naturally as a singularity of compact Calabi-Yau but there should have no reasonable guess for a Calabi-Yau cone metric in a certain sense there can't be one so then what's the right local metric model for this on the singularity so I'm going to make a table here tic-tac-toe table let's say K and we have n so K could be 2, 3, 4 5 in this dimension 3 up 4, 5 maybe so K equals 2 again is the quadric cone where we have a stencil cone metric so this is perfectly well understood in any dimension this is okay now here you're obstructed in some sense so what's that region so here we don't know what's going on so for the little more word you can sort of extend this argument to include the equality case so there's some boundary cases that are slightly more difficult but you're still sort of ruled out and one mystery case down here there have been papers claiming that there can't be a Calabi-Yau cone metric in this case here it's really the only admissible cases of the quadric cones that were already understood before so then here's the next theorem that I'm writing up with our neighbor so for K strictly greater than 2 times n-1 over n-2 there exists a Calabi-Yau metric so reach if not counter on a neighborhood of the origin on the ak-1 singularity so it's a local metric and the metric has an isolated singularity at the origin but if you zoom out and if you pass to the tangent cone you see something that has a plane of singularities from an isolated singularity at the origin so metric singularity at the origin but whose tangent cone has a plane of singularities let's draw a picture so picture like is this let's say this box here is ak-1 that's singular exactly at the origin so again it's z0 to the K plus c1 squared plus cn squared equal to 0 and now you project that thing on to sort of the special coordinate so this is the z0 axis and you see that the singularity is fibred by quadric surfaces so if z0 is equal to 0 you have a singular quadric cone so the central fibre that passes through the actual singular point and here you've got smooth quadrics so these are all tangent bundles of the sphere so you've got these vanishing cycles here these topological spheres and the metric picture is that the radius of sort of when K is sufficiently large when this power is sufficiently large and this is exactly what sufficiently large means so it's actually sharp it's right at the boundary where these spheres shrink so fast as you move in towards the origin that if you take the tangent cone so if you sort of dilate out by larger and larger factors and pass through a gram of Halstorff limit the cycles get pinched off and the tangent cone that you see is sort of a plane, a flat plane across the singular quadric cone one dimension less but that's true on the level of the Ricci flat matrix that's how the Ricci flat matrix behave an isolated singularity is sort of the tangent cone so it leaves the borderline the borderline case when you have equality here so it's these two guys so we're not writing that up just yet but the only thing that can be true in those cases is that you again have a local Ricci flat metric for the isolated singularity and for the tangent cone which is not simple so whose cross section is not smooth but it doesn't split off a line so if it's more complicated it has singularities but it's just not the product of a plane and a simple cone so the way this is proved actually is so if you guess a good local metric model you make an ansatz for what the metric could possibly look like and then you compute sort of its Ricci potential and you see that the Ricci potential goes to zero so it's approximately Ricci flat in a suitable sense and then you try to perturb it into a genuine Ricci flat metric sort of maintaining the geometry of the derivative problem so you want to solve the Monjampere equation the linearization of the Monjampere is the Laplacian so what you need is good estimates for the inverse of the Laplacian on a space with roughly this kind of geometry you need charter estimates for the Laplacian on a space with this kind of geometry and we do that using blow-up arguments sort of Leon Simon style blow-up arguments you need Liouville theorems for that and the possible blow-ups that you can get and that's sort of the flavor of the theorem that if you don't go too fast actually have to be special real parts of holomorphic functions for instance, things like that if you're ready to have five minutes to say something about the mystery case so what's going on there so again the proof of this this theorem is based on writing down guessing the right ansatz, a right approximate local model but the formula for that local model that we have works for every k that makes sense for every k so in particular for the cases sort of below below the threshold but in those cases the ansatz works at infinity, it tells you something at infinity and not at the singularity so if when k is strictly less than 2 times n-1 over n-2 you can say that our ansatz works at infinity instead of at the origin so you get following geometric picture so again k is strictly less than 2n-1 over n-2 basically only leaves two cases sort of the vertical the column here, we have the stencil cone that's perfectly understood and sort of this one mystery case so let's look at the case first that we pretend we understand completely, so this one so what's the picture there k equal to 2 and n greater or equal than 3 arbitrary so here's this box picture of the A1 singularity I'm saying this, this ansatz that we made suddenly works at infinity in this case so you get an approximately Ricci flat metric at infinity that looks like c cross sort of an n-1 dimensional stencil cone but at the origin we're sure in the good case, at the origin we have the n-dimensional stencil cone metric so we can put that there put the origin further down so here we've got sort of the n-dimensional stencil cone so if you have faith in the solution theory of the complex monogram pair equation what this suggests to you is that there exists sort of a Ricci flat interpolation metric a Ricci flat metric on the total space that interpolates between an n-dimensional stencil cone at the origin and c times an n-1 dimensional stencil cone at infinity so it should be possible to interpolate in a Ricci flat way in between here so here's one obvious sanity check you can make compute the densities so the monotonicity formula for Ricci curvature tells you that the density ratio the volume ratio volume of a ball of radius rd divided by r to the real dimension goes down as the radius increases so you should compute the density of the n-stansel cone and the density of this cone and check that they satisfy the right inequality otherwise this is hopeless so if you compute the density ratio so that's the same thing that Brian White was talking about yesterday it's 2 times 1 minus 1 over n to the n here it's naturally 2 times 1 minus 1 to the n minus 1 that's a popular calculus exercise to show that goes up this way so when the dimension is 3 this is half, the density is 1 half and it keeps increasing and when I let n go to infinity it goes to 2 over e so that seems to make some good sense you know what does it tell us about the mystery case this is k equals 3 n equals 3 so schematically we have the same picture so this is now the a2 singularity in complex dimension 3 we have this ansatz metric this sort of fibrous ansatz metric that's sort of suddenly approximately reach a flat at infinity so here we have sort of c cross the two-dimensional stansel cone but we have nothing to put at the origin we have these claims that there is no nice simple cone structure on the singularity if this works out so nicely then how can this be true can the universe really be so cruel that this space doesn't exist but this one seems to exist you can do even more so the density at infinity here is 1 half and you can show that if this cone if the simple cone existed the density would have to be fader equal to 125 over 243 which is just barely bigger than 1 half so it still doesn't violate the monotonicity it's barely bigger than 1 half and so I was in China last summer sharing an office with Song Sun and I mentioned this problem to him and he was just finishing up this paper with Chi Li about Kale Einstein metrics with conical singularities so sure we can show that the cone exists so question is that cp n minus 1 with an orbital singularity of cone angle beta along the hyper quadric so beta doesn't just have to be like 1 over an integer could be an arbitrary real number so this orbital here according to Li and Song is Kale Einstein again using stability type arguments if and only if beta so it's always a final orbital for any value of the cone angle beta but it's Kale Einstein precisely for of course you can go all the way up to 1 because if beta is 1 then you don't have any orbital points for being a studi metric on cpn minus 1 so you want the lower bound and the lower bound is I hope I get this right so it's 1 over 1 over this critical constant that's appearing in this in this story so it's a saying that in particular for the mystery case so these claims that were in the literature that this cone doesn't exist they were actually wrong and the cone has to exist and sort of the picture is consistent again so you would suspect that there actually exists sort of a Ricci flat space interpolating between these two cones again to show the existence of these spaces that's sort of solving the monjamper equation on a manifold with isolated singularities where the curvature blows up and that seems to be something that sort of they have to exist they just have to but that's something that's sort of beyond reach of what we can currently do okay I'll stop there thanks any questions? that's not known that's sort of that's a sub question that's an easier step to the question I wrote down that's not known I mean you would think if you can make a good guess then you should be able to prove that that guess is always realized but you know we don't even know whether you know different compact collabials with exactly the same local singularity types have to have the same local singularity type that's not known but the other singularity types have to have the same local metric models of the singularities can you give us an idea where the 1.5 over 2.3 I mean it's it's a density ratio of the cone so that's the same thing as the volume of the cross section of the cone divided by the volume of the unit sphere we know the volume of the unit sphere now the cross section of the cone that's a circle bundle over this Kepler-Einstein orbifold so you want to compute the volume of the circle bundle over this Kepler-Einstein orbifold the volume of the Kepler-Einstein orbifold itself that's a degree computation you probably know that better than I do how you stick in the beta in the right place to make the degree computation come out right and then sort of you have to lift that to the circle bundle to the metric on the circle bundle and that's sort of the the collabial that's because the length of the I mean it's basically Kepler-Einstein orbifold times the length of the circle because you have a Riemannian submersion so you only have to know the length of the circle and that's something like the index of the final orbifold divided by the dimension so you have to compute the index the divisibility of the canonical class that's a collabial that's a degree computation plus collabial that's the statement of the theorem you say that you can always find some collabial metric even in these X cases where there's an X right so what the X means is that so you have 8k-1 topologically that's a cone topologically this or even holomorphically this is really sort of some compact cross section and then cross for the real line and you put a what you don't put a metric on there just topologically it's a cone so what these X cases mean is that there is no collabial cone metric on these guys so there's no sort of what metric of the form the R squared plus R squared G so L is the length L is the cross section which would be collabial whereas in the good cases there is so in the bad cases I mean it's topologically a cone so you would hope that it's actually medically a cone as well so in most cases it can't be nice cone metrically and what it is instead is the sort of more complicated thing where the cross section isn't smooth but actually singular we have sort of the cone sure we do so we make an ansatz for the metric and then we find that the metric is the ansatz is approximately rich flat but it's not quite rich flat we don't make it rich flat by bare hands we don't write down an explicit solution but so we have to solve the modern pair by perturbation so you can use the implicit function theorem to solve the modern pair so it comes down to inverting the Laplacian and then all the analytic work is solved in getting good enough estimates for the Laplacian on a space like this okay well if there are no other questions let's thank you and I'll be right back


This is stock number 7364A, a 2014 Ram 1500 crew cab, 5 foot 7 inch short box. This truck has the 5.7 liter V8 Hemi motor. From this HD video you'll be able to tell that this truck is extremely clean all the way around. You can see it has the front bumper sonar, factory fog lights, bumpers in really nice shape. It has 20 inch Goodyear Wrangler SRA 27560R20 tires, and the 20 inch factory alloy rims with the wheel covers. These tires look like they have right around half the tread left, maybe just a little bit more, 60% I'd say. And we got the color black over bright silver, didn't see any dents or dings on that front fender. Once again front bumper very very clean as well as the front grille, didn't see any dents or dings on the hood. We take all of our videos in 1080p HD, so if you have HD capabilities on your computer tablet or smartphone, turn them on right now, and it's like you're standing right here next to the truck with me going over it. As you go down this side of the truck you can see just how clean the body is, how glass like and reflective the paint is. We take these HD videos so if you are far away, or even if you're close by and you just cannot make the trip down, you can still see the truck here, the truck, and have confidence in the vehicle that you're looking at before you even get here. You can see that that cab is in really nice shape. And down this side of the truck I did not see any dents or dings, very very nice. It has some chromed step bars. The back rims are in excellent condition as well, no scuffs or scrapes. And these back tires have just as much tread as the front tires. You can see that it does have the wheel well guards in here, which keeps your frame and under body looking really clean on this truck. Just like that. All the lower rockers are in excellent shape. Has the bed rail covers. Rear bumper is in excellent shape as well, no dents or dings on that. You can see it does have the dual rear exhaust chrome tips. The tailgate is in excellent condition as well, no dents or dings on that. Has a backup camera and locking tailgate. The bed is in really nice shape, just a couple light duty scratches, but nothing major. Very nice shape for a truck with this kind of mileage on it. And as we go down this side of the truck, just as clean as the passenger side, once again I didn't see any dents or dings down this side of the vehicle. We pride ourselves in the quality of our trucks, so that's why we do the videos we want to show off. That's exactly how clean they are, so that you can make a smart buying decision before you even get here.  There are no rips, there are no tears in these seats. Very very clean. Both of these seats are heated and cooled and power. You get memory driver's seat with the lumbar, factory floor mats throughout, carpeting is very clean underneath, and it has power windows, power locks, and power heated mirrors. And here we can see that this truck indeed does have 42,870 miles. You get these 7-inch configurable TFT display, so that means that you can change all the stuff in the corners as far as compass, temp, trailer brake control, and then your range. You can do miles per gallon, whatever you want to do in those corners, and then you also have a digital speedometer on there as well as all the information center. You get the heated, leather-wrapped steering wheel, cruise controls, Bluetooth audio controls. It has the 8.4-inch touchscreen radio with the factory navigation.  If you don't like doing it on the screen, don't worry, there's buttons down here for that same stuff. You have the factory brake controller, push-button four-wheel drive, the turn dial, eight-speed automatic transmission. We have dual climate control and all of your other climate controls there, and if you didn't like doing the buttons down here, well, you can do the climate up on the screen as well. This is also where the backup camera shows up, so a really nice, high-definition backup camera. This truck does have the Alpine sound system and has a 115-volt, 150-watt plug-in. We also have remote start on this truck as well as enter-and-go system, so you just have to have the key on you, and the doors will unlock. Pass and receipt is very clean as well, no rips or tears on that. Smells very clean in this truck, I don't believe it's ever been smoked in. Power sunroof, power sliding rear window, you've got your home link buttons up here, and then you have an insist and 911 buttons right there. Those are always active with the vehicle, you never have to pay a subscription on those, so nice feature of the vehicle. Back seats, very clean as well, no rips or tears back here. It does have the latch child safety system for your car seats. These back seats are in excellent condition, there's your power sliding rear window.  If you've got muddy boots or pets, you can put your pets on a nice, flat surface because it goes all the way across here, or if you've got muddy boots, you can put them on there, or tools, keep your seats nice and clean, and then when you're not using it, you can put it back in there. That's storage under there as well, and then there's even more storage in the floor with two container bins. These bins do remove, so you can put ice and drinks, and then take them out for easy cleaning. And then finally we've got little LED lights on the bottom of the seats, so you can see what you're doing when it's dark out. All of these back seats are heated, they're heated seat buttons, and very, very clean inside and out. Take a quick look under the hood, the 5.7 liter V8 Hemi motor, has the MDS system with multiple displacement system, that means that it turns off for the cylinders on the highway. This truck has been fully safety and inspected by our service shop, and has a fresh oil and filter change. The truck has been gone through mechanically 100%, and is 100% ready to go. To see more pictures of this truck or one of our other 400 new and used cars, trucks, SUVs, minivans, Wranglers, you name it, we got it, go to our website, www.summitauto.com. Full pictures and descriptions of every single vehicle on our lot. Videos of every single used vehicle that we have, all at summitauto.com. Thanks for checking out my video. If you like it, want to check out more, go to our YouTube channel, which is youtube.com slash summitauto, like, subscribe, and share. If you want to make this truck yours, give us a call right now,<PHONE_NUMBER>, ask for one of our sales associates to make this super clean, 14 RAM 1500 Crew Cab Short Box here today. Once again, that number is<PHONE_NUMBER>, thanks for checking out my video. In a second, if you want to go to our YouTube channel, there's a link to subscribe to our YouTube channel on your left, a link to more half ton truck videos in our inventory on your right, and if you haven't been to our website, go there, it's www.summitauto.com, see what we're all about, and we look forward helping with this super clean 2014 RAM 1500 Crew Cab Short Box. Thanks again.


Hello everybody, welcome back to another AI video and this one going to show you how to create amazing text to video videos and they're going to be free. They're now a minute long. You can use your own music or you can use some prebuilt generated music and they look really good. Here's an example of one I just made a few minutes ago and it's got that cyberpunk futuristic look here. And of course, it has that little bit of wobble in it. That's common in deformed, but you can see here that this is a 60 second video. When it's got zooms and a lot of the footage is really, really usable. It's really good. So let me show you how to do it. First step, go to plasma punk dot com link is in the description below. Step two is you want to just click on this try it out now button. It's going to redirect you. There's a chance now that it's going to ask you to log in again. It is completely free and you can log in with Google and whatever else. And I think Facebook is on its way. The next step, this isn't quite as intuitive as you'd like. But you can actually click on the MP3 button here and you can upload your own music or you can click on one of their styles. They've got four here and then you can listen to it. For example, let's listen to Klein Siler. All right, yeah, that's a bit loud. So let's say I like that one. And again, the point of this is, yes, you can make an AI music video, but you can also just make a video and then take the music off and put your own music in later. So I'm going to go with this one for now. I'm going to go ahead and click on next. And here's where it gets good is this is the part you got to be careful about. You can actually extend this here. So you'll see here that you can select a part of the song, if that's what you're looking for, or you can just move it to the beginning and then look carefully here. It says start time zero and time one minute. That means you get 60 seconds of footage here. So that's why it's so cool right now. And again, it is free. So we're going to go ahead and click on next. But just keep in mind, you can move that around. And then finally, let's take a look here. Here are just some of the styles. This is the last one that I used in the video that I showed you. But if you like something here, you can go ahead and just click on it and select it. Now there's a lot more styles. So you can click this button here, click the arrow and take a look at a few other ones. I like the painting styles. Always very nice architecture, painting, psychedelic, depending on what you're looking for, flip through them, cartoon, photo realistic. Yeah, there's a lot of different styles. Maybe you like this whole Alice in Wonderland painting style. Either way, I'm going to go ahead and click on this and I'll just select this one. Again, you can select whatever works for you. Click on next. And then here's the very here's here's the last part. This, based on what I clicked on, this is a highly detailed matte painting by Jean-Baptiste Meng. But if I wanted that to be anything else, I can put that in there. So if I wanted to be a superhero painting, superhero comic style by, I don't know, Neil Adams, famous comic book illustrator, I could just go ahead and type that in and then presto, there we go. And then the last part is the content description. What do we want to happen generally in the video?  So I'll have a, I don't know, a French bulldog reading a book on the beach, whatever. It'll do its thing and then whatever it is you want it to do. The last step here is click generate video. And this is where it gets kind of good. You can actually now see where you are in the queue. So there's two people ahead of me. And then once, you know, those people have been their videos have been processed and actually it's going very quickly. You can actually see the percentage that the video is done. And you just leave this browser open and then it'll be done. It also has an email where it can send you a link. But in a few minutes here, you're going to have a 60 second, very cool, AI generated video. Thanks for watching.


I'm modeling the transition period, early diesel, late steam period. The layout I started back in the late 80s started out as an article out of an MR. And it met a certain amount of criteria that I wanted in my next layout because this is actually my second layout that I built. I wanted a switchback logging type railroad. And so I started with a basic plan. I expanded it a couple feet each way. If I look at the major construction over the years, I started out with phase one that was basically the plan out of the MR. And then I looked at the basement and said, well, I can bring something around the corner here. And I always wanted, well, I had the roundhouse and turntable for my old layout. And I wanted to incorporate that in some way. So I brought the layout up and I raised it a little bit in elevation and did that section. And then I got with a switchback, really, it's fine in an idea, but to operate it, I like to watch trains running through scenes. So I incorporated the tails of two of the switchback legs and made a reverse loop, or not a reverse loop, but a continuous loop out of those so I could run that. And that was basically phase three. On the first few phases, first three phases, basically I used Code 100 Atlas. I was very happy with that. I did incorporate a few Pico turnouts to fit situations. The last phase, the engine facility, I used Code 83 Atlas trackage on that. If I was to do it all over again, I think that would be the track that I would use on everything. And then phase four was after accumulating locomotives and over the years, I wanted to have a larger, expanded engine facility that I had, but I really didn't have room in this room. So I punched through the wall and designed the layout on the other side of the basement. I have dreams of expanding more on the other side, but technically the layout of the other side, I don't think it's very conducive to layout. I got doors that I'd have to get by and windows and whatnot. But I'm happy with what I have. I use a NCE DCC system. Well, when DCC first came out, it was sort of in this area of Grand Rapids. The club had it, Bruce Chubb, which I did a lot of work on, his layout had it. And it was just, to me, was a natural step to go into something that I was familiar with already. I have quite a few solid locomotives, both steam and diesel. I have a few of the structures are lit. I have some street lighting on the layout, Boulevard lighting. I guess my favorite part of the layout would be the branch line engine facility. I built a scratch, built an engine house. And I was able to incorporate the shea that I had built when they first came out years ago, the MDC shea. And it's just a place where I think if I were to be, that's where I would like to be.


they'll be done. Oh wow that's excellent thank you. Brynn. Yeah great and then did you see that note? I did thank you. Don't embarrass me though. Hey okay. And then um then Mike can you please get Nader the witnesses witnesses right? I don't think that I don't think that's the bear sound but this is the time of night so I'm going to actually I'm going to walk with you. Thank you guys with me um get out of the sun. All right so I just want to make sure that's not really any better I'm sorry. There you go. Okay so I just want to make sure I'm covering all the bases so and thank you Mike for jumping in here uh so Nader you'll have your witness list you'll have a section by section um rest of committee members you can think about speaking to the bill um coach I wrote down what you said which was really wonderful um really appreciated that so anything else? Nope. Thank you. Let me just speak. Thank you. Thank you everybody we'll see you tomorrow. For sure. Are we on tomorrow at 8 30? Oh thank you. No. No you didn't say that. Thank you. All right we can go off live and uh and say our good night.


Welcome to Toffy TV. I'm joined by Andy. We are previewing a six-pointer. Let's call it what it is, Southampton versus Everton, St Mary's, three o'clock. This Saturday is the place to be or not to be, as someone once said. Most likely. Andy, without getting to talk sport on everything, this is a big game for Everton, isn't it? It is, but I don't want to go, like, because I listened to talk sport yesterday and the same, we could be relegators. I'm not going to be relegators, let's be honest. But we need to start picking up points now, don't we, realistically, because 11 points from 11 games that put us for 38. It's not clever as it is, but this is the type of game that we should be winning now. If we want to get eight, seven, six, these are the type of games. Top half. Yeah, top half forever. We need to win these type of games. Southampton's still, obviously these didn't place you badly against City last weekend, but it's still a bit shell-shock after getting shell-act on that. The manager, supposedly not under pressure, but from the outside he is under pressure, I guess, from the media. So we need to take advantage of that. Our manager is under pressure on both inside and outside. That's what it seems like. But yeah, we need to win this game. But we have got a better side if you turn off, that's the problem. You don't turn off any games. I mean, a way on the south coast, last 11 games, three draws and eight defeats lost a blighting to me, so not a great place for us to go. Southampton last time he won there was 2015 and we beaten 3-0. It's been a bit rubbish playing Southampton away, particularly since we took the manager off them. That seemed to not make them not like us. And they've had good results against us. We should have left them. We should have left him there and got someone good. It's a tough game this one, it really is. For lots of reasons. Southampton have obviously got plays like Nathan Redman do on his day as a handful. Danny Ings loves a goal against Everton. Yeah, they've got teams that were proud to have some set-pieces. So Shane Long is still there. He'll come out for us and run the channels for the last 20 minutes and all that. So, notoriously, not a great place for us to go. But from our perspective, obviously the mid-field is a big issue for the manager. We've lost this, like I've said before, on a final where. This first judge mid-field in his own mind would have been Govaman and Gomez. Maybe Delfin, including us. And we've got no Gomez now till next season. No Govaman till probably January, February. Delfin was clutching his hamstring at the end of the game at the weekend. We don't know whether he's going to be available. He hasn't trained so far this week. It's me and the trained. Excuse me, it's me and the trained. He was clutching his hamstring as well. He was walking out. That was it, that mid-field. He's walking out holding this hobbling round on Sunday where he's still brilliant. And that'll be potentially where it's won and lost, I think, because obviously they've got Rameo in there, who's what he is. James Wall Pradd, who's full of energy in there. Two of his arms strong as another one. So they've got good legs in there. We have to match them up. Obviously Rameo's not quick. He's just going to sit in here. There's two ways you could go with this. They played two ways, Southampton, obviously. I think they'll play a way against us. They can either be very high pressing and start to try and play on the counter. Or they'll sit deep, like they did against Man City, beat themselves hard to break down, then obviously sendings, and maybe mousaging up. I think the last time he won, he was the one who scored the goal. He won 1-0. They need him back because he is a match winner. But apart from that, they haven't gone forward today. They should defend in a part of moments of the Leicester game. They're not overall too bad. But I hope they play a bit more. They're trying to attack us. But then they're under pressure because they haven't won a home for what. I think so. It's all well and good. We're worrying about them sitting deep and us trying to break them down. But I think the fans will want to see them playing the attack. They don't want a reaction. You lose 9-0 at home. Your next home game, there needs to be a reaction, doesn't it? You know, they had two... Unwinnable games. Luckily, or however which way you look at it, they had a city of 8-1 a week after that. They played very well against them. They were on the island or not. If they lost both games, if you can look at it like that because it was the way they expected to get anything out of either game. But they'll be looking at Evan and thinking, Evan, I've won away all season. So this represents an opportunity for us to get three points here and move a level point with Evan. From our perspective, we need to be looking at it and thinking they haven't won a home all season. They've been hammered 4-1 and 9-0 the last two home games. But we know it'll be a tough game, but it's a game we can win because we're better when teams come at us and you'd like to think them being a home team. You look depressed, and that is Ralph Hasson-Hutl's game when he was at England's start and he played a very high-pressing game for 2-2-2. They haven't really done that at Southam, so I think he's realised he hasn't got enough talent to play mostly three at the back. We said before off camera that he did go to a fourth at the start of the season, but they usually play three at the back and the wing backs provide the width. I hope they do press, but if they sit in like that, they shouldn't treat Evan like they treated City, but obviously you don't want to lose that game with Ralph Hasson-Hutl. So would you sit in against Evan and move lost games this season when teams have sat in? I suppose if you were in, possibly you would look for a basis to start. I've heard that on one of the shows when I was in the Premier League, they did their lowest ebb this season, losing 9 million, and it's time for them to bounce back and a point might be a way to start. You might look at it and go, don't lose, that's a starting point. Don't get beat, put a bit of press, put a bit of passion into your playing for once you get your own fans. Now we might look at it and go, Evan are quite weak aren't they at the moment in terms of mentality, but if you get a goal ahead against us, it's not stockings up.  I honestly think if either being a little bit more conservative, away from home this season, we'd have probably been up by up the table. I say he does play to win every game, but there's a disconnect between how we play and the plan on the pitch but that's a different subject. I agree what you're saying is when it hasn't worked away, why hasn't it gone, if you listen to any other manager, if it doesn't work for a long time, you go back to basics. Play on the counter, try to nip some points, a couple of draws on the counter. We had two more draws, we beat in the top ten. Exactly. So get back to basics, maybe not send the full backs as I, maybe sit the defenders, whatever, just try something different, but we play the same way, no matter whether home or away, who were against, maybe not Man City, because we could sit deep against Man City, which is, let's be honest, sensual, you don't want to get annihilated, do you? That's where Marko, the disconnect is the philosophy's principles, but probably away from home, we should have been a bit more pragmatic, play on the counter, go a bit more direct, play Darmwall, Ian Richardson, I can't sat there, I like it when you've got movement against three at the back, because their whole width is going to come from the two full backs, and let's be honest, that Southampton, the manager being the way he is, he's going to send them a bit, he's going to send them higher time. Well Berton's going to be out isn't he still, I think it's the final game of his three match back, so he's still out at the moment. So, I mean, I think he's a good player, Darmwall, that's not a sad. Yeah, he's a good player. He's obviously getting towards the... But what you do, I think what everything needs to do is, we've discussed this before, is not to have orthodox wings where they're playing in the wing channels, let the full backs play big, get them in a bit tighter to the striker, and get Richardson up against Jan Vestergaard, Jan Vestergaard's a big 6 for 5 centre off, he does not like to be moved around, get Richardson, don't have him against Bednarich, don't have him against Jack Stevens, and I think Richardson have some... I play Richardson off the left in this game, and then we don't usually discuss first 11, but I play Richardson off the left in this game and get him cut in on that right foot, cos, to be honest, he's our only consistent goal threat when he's on... I know people have got on his back this week, but that's the type of player he is. Has a heart as the same, everyone loves a heart, so... Oh, I mean, yeah, you're talking obviously about the nonsense article of Paul Joyce's written. Well, we know where Paul Joyce is coming from that week. The team, he supports Shrike, he gets booked for diving. That there's very little... And consistently diving, and that not led the article. Where's the article about how mannau o Shalala, the damage in Liverpool? Well, my question is, he's authorised and says you've got an article within Evan, maybe, and that's a different subject. For Evan, it's poor from Evan.  There seems these suns, the victim now. Did he say that Evan was going to get relegated? He said Evan and Cuppie were relegated to you. Cuppie. Well, I suppose, at the moment, it's hard to argue with Gannis. I don't think... If we go on to our January, then, all right, fair enough, yeah. But then even so the talent's still lit. There's still enough talent there with a bit of brains to win a couple of games. But the longer you leave, the current situation, whether the Marco can turn it around, then obviously I don't believe so. And then you're going to have to get a new manager? I look and just... I think Evan is still more than capable of finishing in the top eight. Easily. What's your five points off the top... But what I would say to that, which kind of funders my own narrative, is that we don't look capable of putting two wins together and at the moment, until we can put, do you that, you won't? So therefore you would have to say that at this moment in time you wouldn't definitely back against Evan not being relegated. No, I know that. We know, we watch them and we know that there's talent there. But someone like Durham who, hey, he's trying to get a reaction. Of course, that's exactly what he's doing. But if he doesn't watch Evan all the time, and he just sees them with 11 points after 11 games, he will go, well, they're in trouble, because they're where they are for a reason. The fact is, if you look up the league, the one win off 10th or 11th or whatever. All of those teams are in trouble. Well, that's to be honest, because apart from, I think Chelsea will probably get the, I guess, they seem a bit more, people forget about Chelsea. People go, oh, Frank Mampard, he was doing this great job. They got half the team, won a championship two years ago. So it's, there's a perfect blend for Chelsea. But in time, the little rest of them... At the moment, it's off for a flying, Liverpool are the thing, you bought Leicester and Chelsea a flying, but then there's a... The seven, six points then, behind. Everyone else is six points behind. There's not really any good teams in the league. I'm having with a modicum of consistency. I'm not talking about winning every game. I'm talking about winning three, lose one, draw one, three. We're asking for to win 10 on the bounce here. We're asking for just a bit of consistency. I think having to be fine, but... It has to start with a game like this, and if Marco hopes to stay in charge or whatever is going to happen to Marco, he's going to have to win games like this. Because the away form, when the next manager comes in as well, the away form is actually what's going to keep them in the job, it's the away form which has cost-managed the jobs in the last couple of years, because Cumin couldn't be away from home. That's been the difference, isn't it? It'll be the difference with this fella. Everton's not only Liverpool have took more home points than Everton since March, which is bonkers when you look at it, when you consider how good Man City is. But we can't win away games. There's a mentality issue. We're horrific away from home. Any manager that could come in from Everton or Marco Silva, the big thing for him would be getting some away wins. And this game at the weekend representing opportunities to do shop. I mean, just looking at Southampton have won 11 Premier League home games against Everton. They've only won, which is only beating Newcastlemore in the Premier League, which is 12. You've got a long time beating Newcastlemore, having lost down there the two teams. We've lost five of our last six away games at Southampton. 11 of Southampton's 12 Premier League wins against Everton have come at home. 91.7%. Fantastic. Southampton have won just one point in there opening five home games. Yeah, they've been poor away from home this season. You did quite well at home last season to be fair. I like Arsenal, but they're sure to confidence. They're sure to confidence. Something's got to give, really. A draw isn't really good enough for Marco and saying no. I also think there's performance aspects here to this. Again, at the start of the season, I said, one draw around, but you would have took it, wouldn't you? At the start of the season. Not now. So when you look at the team, but then it's the style of performance that the performance was absolutely terrible. There's a disconnect between them and feel me attacked still. Do you still haven't got any blends at the moment on how much you've losing by now? It comes as we bore players in the summer to blend and unfortunately we haven't got that blend because they're all injured or whatever. Yeah, that is on Marco, but there's still no style of rhythm to the team. And this game here, you need to, this is a game where I think Southampton probably will sit deep. So we need to see those ideas about to spread the team around and that for me involves having Alex O'Rubian rather than the Icelandic master there. Well, we've got to have, we have got to play for me. We have got to have legs in midfield. I'm not saying I'll get sick or something like that. No, not for me, because I think Snyddlin will play. Can't have him in the Snyddlin. I think Snyddlin will play. But I think him and Tom Davis will be fine Snyddlin because Tom Davis then has got the licence. All Snyddlin needs to do, and I don't know why, but when he plays with other players, he seems to start drifting up the pitch, he's sitting and screening the back two. That's all he needs to do. But then your full backs go on them. Three drops in between, you make a back three, then that's all you need to do. No, but that's what we need. If we've got that movement in midfield and legs. And that's how you go up and call-gates instead of a key. You've got a call-gates. Everton are windless in the last eight Premier League away games. Three draws, five defeats. That's disgraceful. However, ten of our last 18 wins on the road have been against sides, please 15th or lower, which obviously is what's all done tonight. That's not great though, isn't it? That is why Everton are finishing top six because they don't win away games against teams around us, which is why we're not. Theo Walcott has been involved in six goals in his last four appearances against Southampton, scoring five and a six and one. First goal for that funny bats. It's his old club. He's done well. He wasn't great all weekend, but he's done really well. I thought he took them off too early. I think it's pace at the game, especially when they can't be bothered chasing people. I think the team has to be pace-y. I know probably Towson's going to play because he's turned into Levan Dowsky in people's eyes because of a couple of something being sport. I think he's done like half a loo, and he's not done anything wrong for me. He's not perfect, but he's scoring goals. He should be in the side. Moise Keane's got to wait in the inside of it. Maybe Moise Keane's game will come, nor it's your own. See what frustrates me with Moise Keane just quickly on him is that we've played him with the static midfield.  It's like, well, he's back out now, and I can't believe that all of a sudden people are looking at him and going, he's not really your option.  We know what that is, but that's just idiotic. Anyway, to get the best side of players, same with a Wobie, same with Davieson, you need to have players round them. That's what it's all about combinations. I try to say this on Dom. We don't work because there's a combination. Dom just stands, but they don't work. You can tell me how many times. So you need a combination that works. And a Wobie, and I think Keane would work, and a Wobie and a Charleston seems to work as well, because Charleston makes them run in behind, and a Wobie's got the pace and speed of thought to release them. And a Wobie is, even though I thought the man has to start with the correct team and play him on the wing. But I thought Sunday. It was the right team to play. And I thought a Wobie had to play there just to make sure we could get the others in the field. But now, obviously with losing hands, it's a big opportunity. Now he goes back into the field and he stays central. And if that means that Charleston going out of the left well, then so be it. Well cut, assuming at the moment, he has to be. He's won with pace, he's playing well, why wouldn't he be in there? I think he's going in for his centre forward. He's done well in the last two games.  I don't think the manager rates him, I'm going to be honest with that. Whether rightly he's right or wrong. The manager, I think he, well saying that the manager changes his mind doesn't he? But I think that's what happens when you've got no consistency. You're looking to stumble on something that works. And we've tried different things. You've tried too many things. We're throwing a lot at the wall. Not in terms of formation, because I think we've played one different formation. No, but he's throwing different things. So that's the one thing I don't think people can criticise him for. At least he's not just playing, because there was that period of four defeats. He played the same players basically every game. Now I don't think anyone can have a go. He's dropped Michael Keane, he's tried a Wobie inside, he's tried Tom Davies, he's tried Hullgate in. Moise Keane's probably the only one who can probably count himself unloch i gŵr. I'm not bothered about him bringing Moise Keane along slowly. I've got no problem with that. The only problem I have is done play him as a right winger in a 43-1. It's just not going to work. But whatever. I think I understand why managers are hesitant to bloody players who are not experienced, because they make mistakes. When you're under pressure, you might not be all... I would as an impact, so I haven't made a team, but whether he does or not. It's energy and excitement. Excitement, and it's a fresh face for the players. There's fear as well. It's a fresh face for the players. The players look and go, and it's a fillip for other players.  He does get it, he is getting a chance here. I've just got to get me, whereas if they're never getting made the 18, you start thinking I've got no chance. Let's see what happens. It's a difficult dish, because we don't know what's ever going to turn up. Sadly, and probably sadly for the manager as well, that's been the story of the season. Decent at home, terrible away. Against the team, that's poor at the moment in Southampton. It's a must-win game. Especially for the manager. For me, it's a must-not-lose game face the former. We have to take something, because Southampton below us don't give them the opportunity to get above us. So don't lose the game. But to me, it's a game that I have to win, but should be winning. How many games have we got before? The horrendous run might not be as bad as people think though. It might spark. All the teams who we play in December are going to come on to us and evidently better when teams attack us. I'm not saying we'll win all again. Well, that's not a team we can beat, I guess. Arsenal Chelseaite fan United, I don't worry about any of them three. Good as Chelseaite, they don't bother me in the slightest. Liverpool and Leicester will be very tough away games. But that's football, they're all tough though. This will be a tough one at the weekend. So let's see what happens. Big thanks to you and let us know that this make or break for Marko Silver. That's what we've been at certainly on our live shows this week. Is Saturday make or break for Marko Silver. It feels like this question flips your hand every week. It's actually boring. People are laughing at it. The reason for me, it's too ambiguous to know and knows there can't be this. We've got to draw on that last game so it goes on. There needs to be some clarification either from the club that he's going to be at the minimum or they need to remove me either or because otherwise it's just a question Marko for the whole club forever. You need some clarity really. I suppose the only answer to that question is we'll know by the next preview which is after the international break. Big thanks to you and like I say thanks for watching. Subscribe if you haven't get the video. Thumbs up if you want more videos. Join us on Patreon. See you later.


Hello, everyone. In this module, you will learn how you can apply validation in lightning or our components. So jumping into developer console. So I'm going to create new lightning component where I will be demonstrating you validation. So naming it as validation, clicking on submit. So here in this component view, I need to create the UI part. So entering the text like enter a number, then UI input text or a colon ID equals to input component. Then placing a BR tag, then putting a lightning button, label will be submit and on click of this, I'm going to call do action method. So what will happen? This enter a number will be displayed as a plain text. Then this UI input text tag will be showing input element where you can type some value and when you will be clicking on this button. So this do action method will be executed and the label of this button will be submit. Right. So I saved this dot CMP file. Now I'm going to click on this controller where I will be implementing the JavaScript function. So we used my action as a function name. Right. If you want to change it so you can do that as well. Now here I'm creating one variable as input component, then component dot find input CMP. So here you will also understand the use of RID. So if you provide RID with any element, so through that RID you can find that element in the JavaScript controller. Right. So creating one more variable as value then input CMP dot get V dot value. Right. So through this component dot find we identified the properties of input CMP element whose ID is this then through this input CMP variable. I'm getting the value of that element and that will be available in this value variable. Now we need to test this test this is input numeric. So I'm applying if is not a number value input dot input CMP dot set. Right. So here in this input CMP variable we have the properties of that element whose RID is input CMP. Right. So we are going to set an error message. So V dot errors, then message colon input not a number and then plus value. So this is part is completed. Now I'm going to implement the else part. So here I'm writing clear error. So this is a comment and input CMP dot set. Then V dot errors. Right. So if number is sorry input values not a number then error will be displayed like this. And if this condition is false then error will be clear like no error will be displayed so error will be added this null value. Right. So this way I implemented this controller part as well. So in the view like we have this input element and the button and when you will be clicking on that button. So these operations will be performed. Now I'm going to open the application where I will be using this. So demo application and here I'm going to use this validation component. Clicking on preview. So later on like in previous module I demonstrated you how you can place or a component on record phase or any application. So if you want to use this component somewhere so you can do that. So I'm putting a number and clicking on submit. Okay, unable to. So there's some mistake. I need to identify that. Here it is do action and we use it as my action. So my bad I updated the code refreshing the page putting 10 clicking on submit so you can see no where it is coming. If I put a click on submit. So it is showing error message input not a number. Right. So this way you can simply test like whether input is number or not. So this was a simple demonstration. So I hope you understood now I'm going to demo you one more validation. So closing these components and I'm going to create that new letting component and naming it as date validation. So in the previous example I showed you how we can apply number validation. Now we will learn how we can apply date validation. Right. So we need to implement the code here as well. Okay, so our column attribute name equals to test date type equals to date. So this this is first attribute that I created of type date. I'm going to create one more attribute. So our column attribute name equals to date validation error and type equals to Boolean. So two attributes I created then I'm going to create a dev class I'm using as wide. So inside dev I'm going to use lightning input class equals to V dot date validation error. Right. So actually I'm using the attribute for this class. So we will be changing a value of this attribute. So accordingly this class will be having the value. Right. Instead of closing it I'm using question mark and SLDS hyphen has error. Right. So if this Boolean is true, then the CSS class will be applied like SLDS has error. And I'm closing single codes then colon blank curly basis double quotes. Right. So this way this class attribute of this lightning input is having some value through this date validation error. Now type equals to date. So actually we are implementing input lightning input element. So it's first properties class second is type then label. So label will be entered date. Then value equals to V dot test date. So whatever date you will be selecting so that will be available in this test date attribute. Name equals to date on change equals to C dot date update and closing this. Right. So whenever you will be selecting any date so this date update method will be executed because of like we are assigning or we are like using it with on change. Right. So now after this, I'm going to use aura if so in this demonstration you will also learn like if you want to apply if else on the UI itself. So how you can do that. So for that we need to use or if then is true equals to V dot date validation error. Inside this aura if I'm going to use a div class equals to SLDS. Text color underscore error space SLDS hyphen P hyphen left underscore X hyphen small. And closing the div inside div I'm writing date must be in present or in future. Right. So if it is true. Then only this will work otherwise not. And after this I'm placing a VR tag. Right. So this is basically lightning input that will take date as input. And this is or if which is used to display the error message. Now, after this we are creating a lightning button. So I created one more element or one more tag lightning colon button. So disabled will walk as per this attribute. The variant will be brand on click on click of this button. The submit function will be called. So we need to define this in the controller and label of this button will be submit. Right. So this is the UI code. Now I'm clicking on controller. So I need to implement the controller part. So here I'm updating this to date update. Then creating a variable named as today. New date. Then variable DD equals to today dot. Get date. So they date part will be available in DD. Then for month I created one more variable mm. So today dot get month plus one. Then variable four times y equals to today dot get foliar. Now, if date is less than 10. Then up and zero before date. So if you're entering one, two, three, four, so zero will be appended automatically. So if DD is less than 10. So DD equals to zero plus DD. Then if month less than 10. Then up and zero before month. So if mm is less than 10. And then equals to zero plus mm. Then I'm creating a variable today. Formatted date. Then equals to then we need to put four times y then plus. Then in single quotes, I'm putting hyphen then again plus mm. Then plus then in single quotes hyphen then plus DD. Now, if component dot get V dot test date. V dot test date is attribute that we created here. So we are fetching its value to component dot get. So if it is not equals to blank and component dot get V dot test date. So here component dot get then V dot test date. Then after this bracket. Okay, so let me shift it to new line. So here what we need to put. So I'm just copying this. So I pasted today formatted date. Right. Then close this packet. So we are checking whether test date is empty or not. And whether test date is less than to this. Today formatted date or not. Right. Now, if these two conditions are true. Then component dot set V dot date validation error. We are setting it true. And in else, we need to set it as false. So else pasting it here. And here we need to put it as false. Right. So this way this first method is implemented putting a comma. And here I need to write one more method. So here we are calling two methods. One is date update and one is submit. So date update we implemented. Now we need to implement the submit method. So I'm copying these from here pasted. Now inside these curly basis, I need to write the code. So variable is date error equals to component dot get. Then inside this I'm putting attribute V dot date validation error. So I'm copying it pasting it here. Then if is date error. Not equals to true. Then alert and data is valid. Right. Okay. So now I'm going to. Use this lightning component on this application so that we can test it. So date validation. Clicking on preview. So let's see this code again. Right. So here basically. Date format is created till here. Then we are checking through this component dot get like test date is not equals to null or blank. And it should be component. So component dot get V dot test date is less than today formative date. Right. So date must be in present or future. So if date is. Enter date is less than this today's formative date. Then that error message will be displayed. Here you can see we can enter the date. So. This org current date is 22. 25 select 21. Okay. So it is showing some error date a bit component. Okay. Component is not. Okay. I just need to refresh because I modified the component spelling. That was a typo. So selecting 21. So here you can see it is showing date must be in present or in future. And here you can see submit button is disabled. Now how it happened. So here. Yeah. For this lighting button, we are using this disabled property. And we are using this attribute. So if it is. True. Like validation is there. In that case, a button will be disabled otherwise not. Now, if I select today date. So here you can see it is working fine. Okay. So basically what is happening, it is not taking the org date. It is taking the system date. Right. This error message on 22 because today is 23. So if I select 23, so here you can see. Summit button is visible. And if I click on this, so it is showing entered data is valid. Right. So this way I hope you understood how we can use different features. Of aura. So. Here a lightning input is used or if is used lightning button is used where disabled property we saw. And in controller we implemented two methods to support them. So this is all about how we can apply validations in our component. Thank you.


Photocatalytic technology has been gaining increasing attention due to its potential applications in industry and environmental protection. It is important to develop an efficient and cost-effective photocatalyst that can be used in industrial processes. This review provides an overview of the current status of photocatalytic technology, focusing on the characteristics of photocatalysts, their preparation and their application in industry. This article was authored by Fu Baozong, Xian Mingwang, Haonan Lu, and others.