So, thank you very much for this invitation and true to what has gone on before, I felt like a young post-doc during the introduction. Okay. So, let's see. So, my title is Limits of Limit Sets. I think the groundwork was set by Caroline during her talk. So, let's sort of recapitulate that. So, what's the setup? You have S. So, for the purposes of this talk, S is going to be a closed surface of genus greater than one. Rho is a representation from pi 1S into PSL2C equal to isom H3. And rho of pi 1S, we denote as G. H3 mod G is a hyperbolic free manifold M. Okay. Then there's an inclusion from the surface S into M, which is a homotopy equivalence. I tilde from the universal cover to the universal cover is a lift of I. And this is a theorem that was referred to during Caroline's talk. Theorem I tilde from S tilde to M tilde extends to a continuous map I hat from D2, which is the same as S tilde, which is the same as H2. Union the circle at infinity to D3, which is the same as M tilde union the sphere at infinity. I hat is called the Canon Thurston map. Why? Because the most famous examples of this are due to Canon and Thurston. Okay. So, let's try parameterizing the problem. So, here we had G. Now you have Gn, converging to G. This is a convergence of clangent groups. I'll say a little more about that later on. Does CTN equal to I hat in previous notation? Let's call this G infinity. Does CTN converge to CT infinity? Okay. Yeah. I is this inclusion of S into M. M is a three-manifold, which is a quotient of hyperbolic free space by this representation, discrete faithful representation. So, M, that's right. No, no. I'm not thinking of this. This is a closed surface. This guy is just a closed surface. This is a topological inclusion. It's just a homotopy equivalence. There's no geometry here. M is, okay. Here's a fact. Here's a fact. No, no, no. Just a moment. M is H3 mod G. It's homomorphic to the surface cross R. So, you take the surface S and include it as surface cross zero, where zero is chosen at random. Zero is any number, okay? So, that's the inclusion that we're talking about. And then the inclusion lifts to the universal curve. Okay. So, yeah. Maybe I should sort of say this explicitly. There's a fair bit of work. I mean, this is not trivial, okay? But to prove that it's a homotopy equivalence is easy. It's a manifold, right? Whose fundamental group is pi 1S, and the universal curve is contractable. So, it's a k pi 1S1. So, there is an inclusion of S into that manifold, which is a homotopy equivalence, yeah? That's all. That's all. That's all we're talking about. Yeah? So, it's just a homotopy equivalence. There's no geometry here, okay? And in a certain sense, what I'll be talking about is largely topology today, okay? Okay. So, does this sequence of Kanan-Thurston maps, I hat, converge, well, CTN, let's say, I hat is not called the Kanan-Thurston map. Boundary I is called the Kanan-Thurston map. Okay. There's some notation, yeah? That's right. This is, so you have an inclusion of S into M that gives rise to a continuous extension. You call the boundary map the Kanan-Thurston map. Oh, oh, I see. Boundary I sub n. Is that what you meant? Okay, yeah, thanks, yeah. So, I've changed that notation here. Okay, fine. So, does this sequence, if you have a sequence of convergent planning groups, does the sequence of corresponding Kanan-Thurston maps converge? This is the question, okay? We have to make explicit the meaning of these arrows. Okay? Because convergence could be one of many things. The way, I think this is the sort of vague formulation of the problem that is sort of second part of Thurston, Bulletin AMS 82, question 14, okay? Yeah, this paper has this whole bunch of problems which is guided research in hand. There's just one problem which is still left, which I think came up in terms of dialogarithms. Okay, so this is the vague sense in which this question was formulated by Thurston and we sort of try to make this question precise. Pardon me? If it does converge, it will converge to CT infinity. There's no choice. So, sub questions, so question one, GN converges to G strongly, which means algebraic plus geometric. Let's say a little more about this in a moment. It implies, question mark, CTN converges to CT infinity uniformly. Okay? So, if you have strong convergence, does it imply, I mean, see, these are now maps from a circle to a sphere. This is sort of, I mean, you can ask very basic questions about them. I mean, these are sequence of maps from a circle to a sphere. You can ask whether they converge uniformly, whether they converge point-wise, yeah? So, uniform is the strongest sense of convergence. So, we sort of formulate it in this strong incarnation. If the convergence of groups is strong, which means that the convergence is both algebraic and geometric. So, what's algebraic convergence? I'm going to say this, not write this. It means you take generators for your group and look at row N acting on those generators. That sequence of elements in PSL2C converges to row infinity of that element, okay? That's algebraic convergence. What's geometric convergence? Look at the entire image of the representation as a close subgroup of PSL2C. That sequence of subgroups in the compact open topology converges to G infinity. That's geometric convergence, okay? So, there are different ways of saying that. So, algebraic plus geometric is what's strong convergence? Question two, GN converges to, this is the weak incarnation of the question. GN converges to G algebraically. I suppose once upon a time, this used to be called weak convergence. I have no idea. Does this imply that CTN converges to CT infinity point-wise? Okay? So, if this arrow is strong, is this arrow strong? If this arrow is weak, is this arrow weak? These are the two questions we are asking. All right. And theorem, this is not a theorem I'll be talking about. Can everybody see here? Okay, let's try it out, yeah. Theorem, joint work with Caroline series is that answer to question one is yes. I'll have very little to say about the proof of this, the positive answer to question one, but I shall use this particular thing in some context. Yes, is there a question? No? Okay. So, the strong version actually, there's a reason why I'm not going to dwell on this because the techniques for proving question one has an affirmative answer, essentially boil down to the techniques that went into this theorem.  Then it boils down to this question here. You have a fixed group. Does there exist a canister map, yeah? And essentially the techniques that went into proving this can be souped up a little bit to give you a positive answer to this strong implies strong question. Yeah, okay, okay, okay. So, maybe I'll write this down. So, what's geometric convergence? Convergence is rho n pi one s. This is contained in PSL2C, converges to rho infinity of pi one s contained in PSL2C in the compact open topology on PSL2C, okay? So, I think, so you just look at this as a set, okay? This is a set, this is a closed set, this is a discrete set, this is a discrete set, yeah? The convergence of these discrete sets is in the compact open topology. So, you take any large ball inside h3 of PSL2C or h3. So, you look at the orbit, look at the orbit of fixed point, yeah? And then the orbits converge, yeah? So, the orbit of the first fellow converges to the orbit of the last one, okay? So, that's right. Yeah, so for instance, yeah? So, for instance, here's one example to bear in mind. Suppose you have a loxodromic, okay? And then you start, so you have a sequence of representations of Z into your PSL2C that has an attracting fixed point and a repelling fixed point. You start bringing your attracting and repelling fixed points together, yeah? You're going to converge to a parabolic, right? But if there's a loxodromic, there's a twist around that geodesic and this collection of rotations becomes dense in the circle dual to this guy, then you're going to pick up another Z corresponding to the linking guy, yeah? Yeah, so you're going to, so the algebraic limit is going to be a parabolic, a single parabolic, the geometric limit is going to be a Z plus Z, okay? All right. So that's not strong convergence. That's an example where the convergence is not strong, yeah? However, this point-wise convergence does hold in that particular case, yeah? All right. So now we'll talk about two. So there are two examples which I'll focus on in the context of question two. A, these are due to Kirchhoff and Thurston from the mid-80s. GN, well, I'll write an equal to here. It's what Caroline called X, let's see, yeah? X tau power N of X. What's X? X is the surface here. This is genus. This is X. There's this curve, sigma here. Tau equal to drain twist along sigma, okay? Single drain twist along sigma.  Take the uniformization, this Q here means simultaneous uniformization of X and drain twist power N of X, okay? Yeah, good, is there a question? It's, you take a compact, large compact ball, you look at the orbit intersect that ball, that collection of compact sets converges, okay? For every ball, okay? And then they take larger and larger balls, yeah, that's it. I mean, I think it's, well, okay. So you're from Grenoble. So it's also called the Shaboti topology. Okay, so now you have this drain twist along sigma and you look at GN, that's the simultaneous uniformization of these two guys. And then geometric limit, okay? So let's call, so this was sort of the first sort of non-trivial example where you have algebraic limit and geometric limit differing. So what's the algebraic limit of GN? What's this? It's the surface here and this guy becomes a cusp. So this is not a very nice picture. What's happening is that this guy is becoming thinner and thinner by drain twisting. So it's the neck is becoming longer. When it reaches the limit, this becomes a rank one cusp, okay? Geometric limit is different. It's still geometrically finite. What it has is a surface at the bottom and a surface at the top and so this is surface cross i and you have this curve here, this curve here which corresponds to sigma, but here inside there's this rank two cusp, okay? So it's a souped up, it's a version of this last thing where you have a loxodromic element converging to a parabolic element and the geometric guy. So what's happening here locally at this place is precisely that. So the drain twist gives you the loxodromic component. It's a twist around this geodesic. So the geometric limit, the algebraic limit is this parabolic here, the geometric limit is this rank two cusp here, okay? So you've taken the surface cross i, take sigma, small neighborhood of that, that's an annulus, take an annulus cross i, that's a solid torus, inside that you remove the core, okay? The core geodesic is removed, okay? So there's a rank two cusp here. So this is a rank one cusp, which means a rank one parabolic and this is a rank two cusp corresponding to a z plus z. This corresponds to just a z, okay? So that's the first example. The second example, this is due to Jeff Brock and his thesis and we were grad students together, I also sort of had a ringside view of his battling it out with these examples. So what are these examples? This is Brock's thesis actually. So what's this? You have now alpha, this was sort of, I mean, it was said, yeah, I think these examples came up in Caroline's talk. So this is what's called a partial pseudo annus half. What's that? So you have again, favorite surface and you have, this is your left half and this is your right half. Alpha restricted to the left side is the identity, okay? Alpha restricted to the right half is a pseudo annus half, pseudo annus half, diffio or homeo half, just this part, fixing, what's that? Fixing sigma. So this is sigma, this guy is fixed. So you can think of this sort of archetypal pseudo annus half, this is a punctured torus, the map at the level of homology or fundamental group is 2111, okay? Just look at, think of that, just that one matrix, yeah? So that's the map on the right side, on the left side, it's the identity, okay? What's GN? GN is again, well, it's the simultaneous uniformization with the group corresponding. That's why there's this quote. The simultaneous uniformization gives rise to either something on the boundary or a quasi-Fuxian group. So I'm going to call this group, the simultaneous uniformization of X, which is this guy, this whole guy is, this is X and alpha power n of X, okay? Yeah, what can happen? Yeah? If you look at it locally, then it's this example that I just illustrated, you have this, you have hyperbolic, yeah? Converging where the endpoints converge, right? So then you're going to converge to a parabolic along that, yes? But if there's a twist, it does not happen in dimension to it, has to happen in higher dimensions, has to happen in dimension three or above, okay? So look at dimension three, there's a loxodromic. So what is the twist along this parabolic? So there's a rotation along the parabolic, yes? So now if you take power n, yeah? And you translate a very small distance along the parabolic, but you've gone round many times, yeah? That is going to give rise to a parabolic in the transverse direction, okay? That's sort of vaguely what's going on, yeah? All right, so now what's the algebraic limit here? Sort of similar to this Kirchhoff-Thurston case. So the algebraic limit on this side, it's sort of fixed, it's SL, but on this side, so this becomes a rank one cusp, and so there's this, yeah? This is a simply degenerate end. What do you mean by a simply degenerate end? You look at the pseudo-anosoph monodromy acting on this, yeah? So then this cross-eye glued by the pseudo-anosoph monodromy is going to give rise to a hyperbolic free manifold of fibering over the circle, yes? So that's well, yes means that's a double limit theorem of Thurston, it's one third of his sort of major work prior to 83, but yeah, I mean it's sort of now part of the culture, so I'm just sort of stating that. It's by no means trivial, but so if you have this convergence, what you have is a sequence of these punctured tori, I mean this is a puncture. So this guy does become a rank one cusp, okay? And you have the sequence of punctured tori exiting the end, okay? That's the algebraic limit. What about the geometric limit? That's harder to draw. Oh boy, the first time I need the best. Okay, both space has been exceptionally badly managed because I have the algebraic limit on the right extreme and I have the geometric limit on the left extreme. I don't know a way of correcting this. Okay, that's right, that's right. So imagine an endless board. Okay, so geometric limit in rock examples, what is this? Well, again on the left side, nothing really is happening, yeah? You have a surface here and you have a cusp here, okay? So this is simply degenerate end. Now, let me use some colors, okay? All right, let's not draw this, yeah? So this is our simply degenerate end. And there's another piece at the top, this is our sigma. So as a first approximate, think there's this surface here, there's a surface at the bottom, puncture torus, there's a third surface at the top, yeah? All these three, there's a book of, I mean, okay, so there are these three pieces. One corresponds, I mean, each of them, say, each of them corresponding to a puncture torus glued along a common meridian, yeah? So this is not a manifold, yeah?  You have a genus to surface, cross it with i on the annulus corresponding to sigma, you glue in another half surface cross i, okay? So that's what it is topologically. So this is, yeah? And this side was degenerating towards the bottom, towards the top, and this top surface is also simply degenerate, but it degenerates downwards, okay? So you have half surface cross zero infinity, half surface cross zero infinity, all of them glued along this common single rank one parabolic sigma to the surface where nothing is happening, okay? So I'm saying that this is the geometric limit.  I mean, that's just thesis. So anyway, so I won't go into saying why this is the case, but yeah, just, yeah, I think it's appeared in 2001, thereabouts, all right. So these are the two examples that I'll be dwelling on, hope not to erase them till they serve the purpose, okay? Good, and let me tell you the answer in case I don't have time towards the end of the talk. Here in this Kirchhoff-Thurston case, this limit is geometrically finite, okay? So answer to question two is yes for the Kirchhoff-Thurston examples, okay? This is again joint work with Caroline. And answer to question two is no, all right? So what's question two? It's the weak incarnation of the question. GN converges to G weakly or algebraically. Does it imply that CTN converges to CT infinity, yeah? And the answer in this case is yes, and the answer in the second case is no, okay? But I should say that the answer in the second case is barely no, okay? Just about, I mean, the only place where it could possibly break down, it does break down. So there's Mr. Murphy at work there, okay? All right, so now let's try to say how we are going to attack these questions, okay? So these are the examples, I've given you the answers, now let's say something about the proof, all right? So digression, electric geometry. Sturminology is a little unfortunate. Sturminology is due to Thurston, I mean, it's due to Farb really, but he attributes it to Thurston, okay? So this is again Benson-Farb's thesis that we are talking about. So capital X is a geodesic metric space, H is a collection of subsets. E X H, this is equal to X union, H belongs to this collection, H cross closed interval zero one. This is given the product metric, let's call this and each H cross one is given the zero metric, which means any path lying on H cross one has length zero. So this is not quite a metric, it's really a pseudo metric, okay? It's really the geometric counterpart of doing quotient spaces, okay? No, no, no, no. So what you should think about is, I mean, the way to think about this and I think this is how this terminology came up, you have the space capital X, inside that you have this collection of subsets, yes? And you make those subsets infinitely conducting, okay? So it's sort of the way lightning travels. So this is called, yeah, it's given the, and this, so this is called the electric space and so this metric D electric is called the electric metric. It's really a pseudo metric, okay? Alternately, so what you're doing really is on the subsets corresponding to elements of H, you're just retaining the topology, yeah? So it's just making a product with zero one, yeah? Topology means that the notion of open sets and closed sets makes sense, no distance. But on the rest, you have the distance, okay? So I mean, just as topology takes sort of us, topological space forgets everything up, I mean forgets the topology on subsets, it just makes it just, it gives it just a set structure, right? I mean the stuff that you quotient out in the equivalence relation, it's just a set after the quotient out operation, right? There's no topology there. So here, what you're doing is sort of categorically shifting that construction, you're retaining the geometry of the rest of the space on this part, you're just retaining the topology, yeah? Yeah, that's right, that's right. So each H cross one is given and H cross zero is identified with H contained in X. That's right, that's exactly right. That's exactly right. So you could think of, I mean that's, I think the way Benson sort of thought of it first, you look at H and you construct a core, yeah? But that destroys the topology, somehow I want to retain the topology, okay? So it's just take the product metric and the top part you just give, okay? All right, so what's this good for? And this is really what we'll be needing. Proposition, X is now going to be H three for us, H is a collection of convex subsets such that for all H one not equal to H two in H, product, the projection of H two onto H one has uniformly bounded diameter, okay? So what's the hypothesis? You have H three, you have a ball here, you have a ball here, it's easy to see that the projection of this guy is some small set, okay? These guys are disjoint and so if they are balls, then the projection is disjoint. So there's a generalization, the hypothesis is just now, the collection of convex subsets such that the projection of one onto the other has bounded diameter, uniformly bounded diameter. Then electric, so what's going to be an electric geodesic? So now I think we have a chance to draw a picture. So suppose you have, and you have a very small ball somewhere very far away, yeah? So now you want to go, you have another ball here. You want to go from say a point outside this ball to a point here, okay? You can travel for free inside these balls, yeah? So it's like it's going from here and then it hits this convex set perpendicularly. It travels from one convex set to the next by traveling along a mutual perpendicular, then along this, yeah? Then along this and then here. I'm drawing the electric geodesic. So the electric geodesic when it travels from one, so think of these are sort of huge conducting spheres, yeah? How does electricity travel between infinitely conducting spheres? It travels perpendicular to them, right? So this guy, this guy, this guy, and so on, yeah? So it sort of zips across all these spheres and goes from one point to the other. So that's the electric geodesic, yeah? So then electric geodesics and hyperbolic geodesics track each other outside H, okay? So what's the saying? Is that if you look at the electric geodesic and look at this, look at, oh, I wish, yeah? No, this is all white.  You look at the hyperbolic geodesic joining this point to this point and what this proposition says is that outside these infinitely conducting spheres, the red line and the white line stay close, okay? All right. So this is what we'll be using. I won't prove this, this is, so this is really all that we'll be using from electric geometry. Now let's get to criteria for criteria for convergence. So what do you have? GN converges to G algebraically, yeah? Which means what? Which means that GN is equal to rho N pi 1S, G is equal to okay. G infinity is equal to rho infinity pi 1S. Gamma, this is the, so this is the notation I'll use. This is the Cayley graph of pi 1S and this is quasi-isometric to the universal cover of S with the hyperbolic metric, okay? Good. Sufficient criteria, convergence of CTN to CT infinity. These turn out to be also necessary. One, ah, it's not allowed to go back there, I'll erase this. Uniform UEP, uniform embedding of points. What does this say? It says that rho N, there exists F from natural numbers to natural numbers, a proper function, which means going off to infinity as N goes to infinity, such that the distance in the Cayley graph of one from gamma is greater than equal to N implies the distance in H3 from, so let's fix a base point O and rho, rho, let's call this something else, let's call this M, zero, rho N, G is greater than equal to F of M for all N equal to one to infinity, okay? Infinity is included, all right? So which means that all these Cayley graphs are uniformly embedded in your hyperbolic space. That's necessary, that's not good enough. Let's just erase this. Criterion two is UEPP, Uniform Embedding of Pairs of Points, which says the following, for all G, H belonging to gamma, belonging to gamma, such that distance in gamma from one to G, H, look at the geodesic, pairs of points in a hyperbolic space determine uniquely a geodesic, sort of uniquely, okay? So you look at the geodesic joining G and H and gamma, if this is greater than equal to M, this implies that the distance in H3 between the base point and you look at rho NGO, I should have said rho NG acting on O. O is a base point, right? So rho NG acting on O, rho NH acting on O, and now you look at the, oops, look at the geodesic in H3 joining these, yeah? The geodesic in H3 joining these two points need have nothing whatsoever to do with this geodesic, okay? That's important, yeah? That's the whole point of proving the existence of Canon Thurston maps. It says that pairs of points embed properly, okay? If you do this for one group, if you can prove this for one group, then you would have proved the existence of a Canon Thurston map, okay? So it seems sort of a naive thing. I mean, you have proper embedding automatically. If you have a group acting properly discontinuously on a space, the embedding of the group is proper. So all this work is to upgrade something that's trivial, in the case of a single point, to something that's a little less trivial for pairs of points, okay? Took about 30 years to do that, all right. So, pardon me? I didn't finish that statement, that's right. So this is greater than equal to F of N. So you have, okay, here's your gamma, which I, with my bad eyesight, I identify gamma with H2. I have G here, I have H here. I join these by a geodesic, yeah? This lies outside a little n-ball. Map it over to H3, okay? So, yeah. So now you have, this is rho NgO, okay? So let's just call this G.O. And you have some point H.O here. Join these by a hyperbolic geodesic, okay? All right, this is a geodesic in H3, this is a geodesic in H2, yeah? And this is your O. This ball here has radius greater than equal to F of N. So this, if the, if this pair of points defines the geodesic lying outside a large ball, you map the end points over, then join. After joining, the geodesic also lies outside a large ball, yeah? I should say that just last year, or maybe the year before that, Baker and Riley, Tim Riley and Owen Baker, have given a counter-example to show that if this guy is a hyperbolic group and this guy is a hyperbolic group, not H3, then you do have a counter-example. So can the first and maps do not always exist, right? So it's sort of counter-intuitive, but I mean, how could it go wrong? Well, it does go wrong, okay? So if, I mean, so there's something very specific about H3 going on here. It's not enough space to do crazy stuff. All right, so this is our criterion. And in the last 10 minutes, I will try to say why these two criteria, these are sufficient criteria for convergence, and then we'll, okay, so that's fine. So now I can remove this and try applying this criterion to our examples. So now the last 10 minutes, apply convergence criteria examples. All right, that is the criterion for uniformness. That is the criterion for uniformness, and I'm just going to say what is the criterion for point-wise, yeah? Yeah, so that is the criterion for uniform convergence. From uniform convergence to point-wise convergence, we use this profound observation that uniform convergence on a point is the same as point-wise convergence, okay? So you restrict uniform convergence to a single point. That's all you need, and so what's the, so how are you going to do this? How are you going to get point-wise convergence? You take gamma, again, let's draw this picture. This is gamma, this is boundary gamma, and you have a point, okay? A point defines a geodesic ray, yeah? A geodesic ray is just R plus. It's an honest to God hyperbolic space, right? And you apply this uniform convergence to the space, and then you have point-wise convergence, that's all, okay? So uniform convergence, uniform embedding of points on a ray, uniform embedding of pairs of points on a ray is the same as point-wise convergence, all right? Okay, so basically all that you're doing is you're taking your boundary, and you're reducing it to a single point, and then you have convergence point-wise criterion. All right, so now we'll apply this convergence criteria to our examples. Step one, all right. So now it's time to draw this example a little better, and you have this guy, and you have, so you have this is x, this is alpha power n of x. So which means what? There's one copy of this is SR cross zero, SR cross one by, well, let's call this one, and this is SR cross n, okay? So there are these n blocks from zero to n, pasted on the right. So we zoom in here, this becomes a Margolis tube, which means what? It becomes a picture like this. It's a piece here, piece here, piece here, and there are n pieces, and then there's this one guy, which has length one. So this Margolis tube here blow up, and you have a picture like this. So what's the section? The section is sort of a disk in the hyperbolic space. This corresponds to the left, and then there are sort of n pieces, and then there are, there's stuff hanging off from there. So there's blocks all around this, and there's this Margolis tube. It has length n along the right and one along the left. This is our model for GN, okay? Good. So now how are we going to apply the criteria? Step one, large pieces electrocuted. So now we are going to use our electric geometry. Large pieces electrocuted implies convergence. So this is a mnemonic. What this says is that you have this stuff on the right side, on the left side you have nothing. When you lift to the universal cover, you have this hyperbolic two-plane.  What you do is you put, these are all going to be convex sets, which will satisfy this criterion in this electric thing, that the projection of each of these pieces onto each of the rest is a bounded diameter. So electrify or electrocute, depending on your love for violence. Electrocute lifts of S R tilde cross zero n, all right? And the electric geodesics for rho n and rho infinity are the same. Why? Because the only change is happening on top of this SR, yeah? So if you have a guy, if you have a geodesic ray, you have this point, right? Remember this profound observation of ours, yeah? So there's a point on the boundary and you have this ray on the bottom sheet, yeah? If it cuts infinitely many of these blocks, then these blocks are convex guys, so stuff that's outside the end will remain outside the end. They can't backtrack, okay? So step one is saying that if you electrocute these large pieces and the ray that you started off with has infinite length with respect to this electrocution, then you have point-wise convergence at the end of such rays, yeah? Step two, as usual with such talks, I'll have to give the entire proof in about two minutes. And this comprises about two-thirds of the paper. All right. Step two, so what does this reduces? So this construction reduces the problem to looking at geodesic rays which end on the boundary of a copy of SR2. So infinite length is done. We are left to deal with things which end here, okay? So on the boundary of SR2. So enough to look within an SR2. So you can sort of shift your base point. It's finite length, so you shift your base point and you think of looking at just one SR2. But then we can use this theorem that we've said before. Uniform, so on SR tilde, rho n converges to rho infinity strongly, okay? So this implies that CTn restricted to SR tilde converges to CT infinity, restricted to SR tilde uniformly. This was our theorem which I have said nothing about, yeah? Strong convergence implies uniform convergence. So what does that mean? Are we really done? I mean, I said that there was going to be a negative answer. We seem to have arrived at a positive answer, yeah? Not quite. I have a couple of minutes left. That's fine. So the crucial thing is, here you look at rho n restricted to S, phi 1S. Here you're looking at rho n restricted to just the right side, okay? So what happens is the following. If you have, now you have this canter SR with its canter set boundary. And you have a geodesic, you have your base point somewhere. It enters into this, yeah? And ends somewhere here. So this is X infinity, yeah? So step two furnishes a positive answer to CTn converges to CT infinity, provided SR tilde with cusps corresponding to sigma tilde provided in SR tilde with cusps corresponding to sigma tilde X, OX has infinite length, infinite length in the electric metric corresponding to cusps, okay? See what's happening here, in order to go from step one to step two, you have to sort of ignore information at the places where it cuts across the cusp, yeah? Ignoring that information is really tantamount to saying that the cusps have electric length, yeah? So you electrocute the cusps here, so this is, you can think of this as a step one as large electricity implies convergence, step two as small electricity implies convergence, okay?  And if it has infinite length there, then this strong convergence theorem does apply and we can conclude convergence, yeah? What's left? So, and this is in the last minus one minutes, I'll just say what's left, step three, left are points X belonging to boundary SR tilde which are identified by C T infinity with boundary of sigma tilde, okay? So you had these guys, the sigmas here, these were becoming cusps, yeah? And if you had a ray which got identified with this cusps, you had a ray and if you had a ray which got identified with this cusp here, then you would not be able to apply the small electricity, right? And such points, again there's a theorem here are exactly the end points of crown domains or what are crown domains? You look at SR, you have, this is Phi pseudoanus of acting on this, so there's a stable lamination on the right side. So if you open it up in the universal cover, then there's this cusp here, this is sigma tilde at infinity and then there's this crown domain and then there's this crown, this inverted crown corresponding to leaf of a lam, of a domain of a lamination which contains the cusp, okay? And I think Caroline said something about saying that it just so turns out that exactly at these points we have proved almost everything that it converges everywhere and the only place where it could break down are these crown tips, okay? This is an inverted crown, yeah? And it turns out that CTN converging to CT infinity does break down at these crown tips, okay? So what we have, I mean the convergence of planning groups is fiercely democratic or anti-royalty depending on your point of view, it disrespects the crown tips and respects everybody else, okay?


We hear a lot about the sacred right to abortion, which is never specifically spelled out in the Constitution or the Bill of Rights, while in reality the most sacred right is free speech. That's why it's the First Amendment. You couldn't even demand a right to abortion without free speech. Yet this right is under heavy assault by the very people that should be protecting it. Listen to Democrat Amy Klobuchar from Sunday on MSNBC's Meet the Press, hosted by Chuck Todd, a lifelong Democrat whose wife makes millions working with the Democrat Party. You can amend it and focus on certain kinds of speech misinformation, disinformation, and all you're saying is we know people are going to put stuff on your alleged town square, which has become really a communications company. Your network, other news organizations have limits in place and standards. Look, free speech is over if Amy Klobuchar, the media, the government, or anyone else gets to police misinformation or disinformation. They can't even define what that is. We've seen over and over again what happens when the government appoints people to police free speech. You end up with people like Nina Jankiewicz and organizations like CSIA, which were tasked with doing the government's otherwise illegal censorship with guys like Chris Krebs leading the way. That's where we need the platform to be more transparent and how their algorithms work, how engagement works, so that outside security experts and researchers can dig in and hold them accountable. So is Donald Trump a domestic threat? There is disinformation that he is spreading. The Hunter Biden laptop, that was remarkably similar to a hack and leak operation from 2016. It was just in a different model. It is well established at this point that the media is corrupt, is partisan, and is likely controlled by the CIA and FBI. They've spread so many fake stories and disinformation, which always conveniently cuts against the Democrats' political opponents and the media's critics. So it's nothing new. They should have no hand whatsoever in policing free speech. And if they do have a hand in it, then you know it's a threat to free speech. But holy shit, if this isn't a great example of exactly why that is the case. Somehow, I'm astounded to hear this on MSNBC, a day after multiple people called for the policing of speech on their shows. Yes, I mean, we can't forget that this is a strain in the American character and American history. And I think it was not only a failure of imagination, it was a failure to reckon with history. And the fact that the military was so slow to react, on the one hand, I commend that. I also agree it's a problem of optics of having armed national guard troops going into the Capitol. Wait, no, sorry. I'm going to have to quickly stop and interject here. The fact that the Capitol wasn't totally blocked off in the first place is suspect, considering that it had been blocked off before during Democrat riots at Trump's inauguration. They knew it was coming and purposely allowed them to come in.  But it's more than a problem of optics when when Capitol officers are being killed and wrong, wrong, wrong. Exactly. Zero officers died on that day. None. One officer died of natural causes the next day, with the autopsy showing no blunt force trauma, which conflicted with earlier media lies that Brian Sicknick had been beaten to death with a fire extinguisher. Making matters worse, The New York Times has quietly retracted its story about the death of Capitol police officer Brian Sicknick on January 8th.  But now medical experts are saying that's not true. And the paper is backpedaling, admitting that it's possible he was never even hit at all. All they do is lie, while at the same time demanding that their party have the power to police disinformation, which they're experts at spreading. It's just a preview of what the future will be like if they get their way. They will be able to say whatever they want to, free of accountability, while your speech will be severely limited if it exists at all. Did you find this video informative?  Thanks a lot, I'll see you all in the next one.


Gweithio i chi. Maen nhw'n hyfforddiad o peeledwch, y romant. A gweld pobl yn y gyllid yn hyfforddiad hwn. Mae'n mynd i gael y Llyfridol. Ond, yn y cyfrifoldaeth sy'n adegyswer i y cyfrifiwr i ni o gan pawb cyffredin y Cixinkwyr, mae fydd y cyfrifiwr yn y chyfo'u cyfrifiwr o'u cyfrifiwr ar yr adegwch. Fyngfaen chi'n gweithio'r cyfrifiwr o bobl yn gyfrifiwr o bobl yn cyfrifiwr, i'r gwleidio cynyddu i gael y gweithreulau a gwheilwyr y nautaryg datblygu sydd yng Nghymru? Mae gwirionedd nifith gweithreulau ac mae gennym, nad oedd gwneud o bobl rereinwyr. Mae nid i gweithreulau i'r gwleidio ond uchydig, yn weithglwyr arweinydd, ac mae'n gydag i'i gweithreulau iddynt i gael yn gweithreulau i gael gweithreulau i gael gyllidau. towaedden eligibility? Tenant grant fund spend data to the end of January 2022, broken down by local authority, was published in March 2022. Since then, all local authorities have been asked to provide up-to-date reports covering the financial year 21-22 quarter four, as well as for the first two quarters of this year. This information is being collated and quality assured, and it will be published by the end of this year. As per the programme for government, the local authorities will be able to use any unspent funds to support people who have built up more recent arrears and guidance will be issued to local authorities shortly. Neil Bibby. I thank the minister for that answer. Scottish Labour welcomes the fact that the government widened eligibility for the fund in the programme for government. Once again, it is something that we call for in our cost of living plan in August that the government adopted. However, the government has not published its updated guidance on the widening of the scheme, and more critically, the renew of the £10 million has not been provided. Given that three of the six councils wholly in the West Scotland region have exhausted the original funding and the other three have just £35,000 between them, this empty gesture will do nothing to help tenants struggling with arrears through the cost of living crisis. Will the minister therefore renew the funding and ensure that the grant fund is fit for purpose? I welcome Neil Bibby's support for this move, even if it's a little bit short-lived, because he seems to have not welcomed it by the end of his question. The fund was set up to support tenants during Covid, and it was announced that we would extend that eligibility in the way that Mr Bibby has explained. Local authorities have to consider outstanding applications in relation to arrears accrued during the Covid pandemic before considering applications in relation to more recent arrears, and that will be made clear in updated communications to local authorities and in the guidance. However, I do not think that any of us have to look far to find areas of the Scottish budget where we would all like to put more money, and I very much hope that Labour colleagues will join us in calling on the UK Government to inflation-proof the Scottish budget to enable us to do that. Will the minister outline how the Scottish Government intend to make the private rented sector remain affordable and sustainable over the winter period? Indeed, we have asked local authorities to ensure that the private rented sector is able to benefit from the tenant grant fund as well as from the other support that we make available. It's the information and data that is currently being collected by local authorities that will be collated and published later this year that will show us whether that emphasis has had the desired impact. I hope that we can all recognise that the Scottish Government is putting substantial funding into supporting tenants in all parts of the rented sector throughout these difficult times. To ask the Scottish Government what impact inflation is having on its affordable housing programme. We are aware of the global issues affecting construction, including the war in Ukraine and rising inflation, which has been exacerbated by Brexit and the current cost crisis. We are working closely with the construction industry and housing partners to mitigate that where possible to achieve our shared goal of delivering more affordable homes for Scotland, including operating a flexible grant system that can take account of increased costs. I am heartened that the affordable housing sector continues to show signs of recovery with completions having risen by 17 per cent compared to the previous year to June 2021. I thank the cabinet secretary for her response. Does she agree that new affordable home construction has been undermined by the UK Tory Government cutting Scotland's capital grant this year from £4,973 million to less than £4,468 million, a fall of more than 10 per cent, £504.1 million? Unless the reverse, this double whammy of cuts to our capital grant, coupled with rising inflation due to Tory economic incompetence, will further reduce new build and that next week's UK budget will be a good place to start reversing these cuts? Yes, inflation and the economic chaos caused by the UK Government means our annual budget today is worth £1.7 billion less than last December. We are facing an enormous strain at the same time as focusing on protecting people from the cost of living crisis and mitigating against many of the UK Government's cuts, particularly those impacting people on low incomes and child poverty. Instead of cutting Scotland's capital grant, we have urged the UK Government to release additional public spending on infrastructure and consider other measures to help ease those pressures and allow our capital programme to continue at the required pace. I would agree with Kenny Gibson that the UK budget would be a good place to start. Housing associations across Scotland are reporting that the Scottish Government's rent control scheme is resulting in its chaos to have to rewrite its plans and economic plans for new construction. How many fewer homes does the cabinet secretary think that that will now deliver? The big impact on the social rented sector and local authorities at the moment are the issue of high interest rates, which affects their loans and rampant inflation, which has been exacerbated by the actions of the UK Tory Government. On the rent freeze, as the member knows well, we are working closely with the sector to establish the key considerations for any cap on rents beyond 31 March next year, because there has been no impact on rents in the social sector this year, because they were already set. What we are talking about is from 1 April next year. We have said that we will provide certainty by 14 January at the very latest. Meantime, we operate a flexible grant system, which we expect will allow the continued delivery of affordable homes. Of course, we are working very closely with the sector to help them to address some of the challenges. As I said in my answer to Kenny Gibson, completions have risen by 17 per cent compared to the previous year to June 2021, so we are still seeing projects coming in, and I would encourage RSLs to continue to submit those. I am keen to understand where mid-market rents sit, whether they sit within the affordable sector or the private sector with regard to the rent cap legislation, because I am keen for more mid-market rented properties to be built, but they need certainty for that to happen. I will write to Willie Rennie with further detail, but in essence some mid-market rent sits within the private residential sector and built by the private sector, and some sits within the social rented sector that are built by RSLs, so it would depend on whether they sit, but I am happy to write to the member with more details of that if we would find that helpful. To ask the Scottish Government how it is supporting local authorities to deliver relief for residents at risk of poverty during the cost of living crisis and in light of rising energy costs. We are working with local authority partners to support people facing the cost of living crisis. Through the emergency budget review, we have taken a number of actions, including allocating almost £20 million of additional funding to double the December Scottish child bridging payment to £260, benefiting around 145,000 eligible children. We are also making up to £86.6 million available for discretionary housing payments, mitigating the UK Government's unfair bedroom tax and benefit cap, and giving local authorities more flexibility to support people with energy bills. Of course, we are also providing over £260 million to support council employee pay rises, especially benefiting lower-paid workers. I thank the minister for that answer. In July this year, concerned with the looming energy crisis, I met with local churches to discuss the idea of warm welcome spaces for the winter, and I am glad to see now that it is being rolled out throughout churches in Air, Presswick and Trun. During these difficult times, local authorities are putting measures in place to assist local communities, and SNP-run North Ayrshire Council has launched a massive £450,000 fund to help out residents through the cost of living crisis. Does the minister agree with me that all councils across Scotland should take similar steps? I know that all councils are considering those matters, and many local authorities are working to help people with the cost of living crisis using their resources and powers. That includes exploring the establishment of warm spaces and the fund that North Ayrshire Council has set up. I am also aware, if I recall correctly, that Glasgow City Council has created a £3 million fund, including £1 million for fuel top-up cards. Midlodion Council has put £29,000 into a heat and eat fund for families to help them when they are not eligible for Scottish welfare fund support. Falkirk Council has allocated more than £0.5 million for its household support fund, providing cash for support to 1,000 households since September 2022. There are a number of actions taking place, and that symbolises how we all need to work together in this cost of living crisis to support people. The minister will be aware that one of the best ways to help people is to get the appropriate social security benefits, but Social Security Scotland announced yesterday that, for four days next week, there will be taking no online applications, and people will not be able to apply for benefits during that period. Is this acceptable, and what measures will he take to make sure that my constituents and his constituents are adversely affected by this? Social Security Scotland is providing more benefits to people in Scotland than is available elsewhere in the UK. The reason that Social Security Scotland is having to pause applications electronically through its systems through the period of time that Mr Balfour has stipulated is because there are system upgrades and processes that need to run through in order to deliver the really significant intervention on Monday 14 November of our Scottish child payment, which is only available in Scotland, going up to £25 per week per child for eligible children and being extended to children who are eligible who are under 16. Taking the possible take-up of that benefit from around 100,000 children to an eligibility figure if people apply of 400,000 children. We are focused on running really good systems and doing things correctly, and that involves making sure that the IT systems and the operational systems in Social Security Scotland are all running the way that they need to be for Monday 14 November, and we will help all the people that we will. Scottish Public Services' Ombudsman report found that there was a 36.7 per cent increase in the Scottish welfare fund review applications received from the previous year. Can the minister set out when he will be able to tell us about the review of the Scottish welfare fund and when it will be complete? Is it briefly possible, minister? I thank Pam Duncan-Glancy and other members for their interest in that important area, and the Scottish welfare fund is an important aspect of how we help people every year, especially this year. I will be updating the committee very shortly on that review and look forward to doing that. To ask the Scottish Government for its response to the latest registers of Scotland, UK house price index figures. Housing to 2040-hour long-term housing strategy is designed to be agile and may assess programmes and make adjustments as needed. Recognising wider market conditions in August, we increased the thresholds for our open market shared equity scheme by an average of 9 per cent to support more first-time buyers and priority groups into home ownership. We also operate a flexible grant system that can take account of increased cost to partners when purchasing properties on the open market for affordable use. The economic chaos of recent months caused by the UK Government has, of course, not helped. I thank the cabinet secretary for her reply. The latest report showed an annual increase of nearly 28 per cent for house prices in the western isles. That is part of a trend that has seen local house prices rise by more than 81 per cent since 2015, higher than any other local authority area. Meanwhile, we have areas like Harris where something like a fifth of the housing stock is tied up in second homes and short-term lets. Is the cabinet secretary willing to meet with me and partners such as Corinne Neal and Shear to discuss possible solutions to the serious problem? Of course. I am always open to discussing housing issues raised by members and councils, and I am happy to arrange that. It is worth noting that £43.3 million is being made available in this Parliament through the affordable housing programme in the western isles. I would expect the council to be working closely with relevant partners to ensure delivery of the affordable housing that local communities need. My officials are working closely with the council to achieve that, and we are meeting in the islands next week, as well as progressing the development of a remote rural and island housing action plan to support housing delivery in rural Scotland. However, we will get that meeting established as soon as we can. 5. Donald Cameron To ask the Scottish Government what its response is to the recent Highlands and Islands Enterprise survey in which 76 per cent of people who responded said that there aren't enough affordable houses for rent or to buy locally. While we delivered 3,417 affordable homes over the last Parliament in Highlands and Islands, we recognise that challenges remain. I am pleased that we are making available more than £422 million to support the delivery of affordable homes in the region during the current Parliament. In recognition of the challenges that are facing our more remote communities, we are working with stakeholders to develop a remote rural and island housing action plan that will be published in the spring. It is clear that the lack of affordable housing is one of the main drivers of depopulation of the Highlands and Islands, and there are now over 9,000 households on the waiting list for social homes in the Highland Council area alone. Given that the Scottish Government has continually failed to meet its building targets for affordable homes, what action will the cabinet secretary now take to help people in the Highlands and Islands to get a home? We have made available to Highland Council a 25 per cent increase on the funding provided over the last parliamentary term over £240 million. We have also, of course, got the rural and island housing fund. We have also delivered more than 1,600 more homes in rural and remote Scotland this Parliament. There is an increase in the number of homes being delivered in remote and rural Scotland, but I think that Donald Cameron and his colleagues on the Tory benches need to be consistent because he mentioned the issue of affordable houses for rent or to buy locally. One of the issues that he will be aware of is the loss of homes to holiday lets and short-term lets. When we brought forward legislation to address that and to avoid the loss of homes in his area and other member's areas to short-term lets and holiday lets in order to address some of the issues that he raises in this Parliament, how many colleagues voted against that legislation? What we need to see from Donald Cameron and others is a bit of consistency because when we develop and deliver the levers to help address some of those problems, I just do not understand why how many colleagues come along here and vote against those levers. There is no consistency in their position whatsoever. We also need slightly shorter answers. Brief supplementary, Gillian Martin. Thank you, Presiding Officer. I am notwithstanding the cost issues that Kenny Gibson has outlined. Can the cabinet secretary comment on the progress of the affordable housing supply programme that has been undertaken to make sure that the target of 10 per cent of the 110,000 affordable housing targets met? Gillian Martin raises a really important issue here. The 110,000 target has been really important and it builds, of course, on the 113,000 affordable homes delivered since 2007 with more than 6,000 of those delivered in rural and island communities, but we recognise that there are particular barriers to delivering affordable housing within rural Scotland, which is why the remote and rural housing action plan, which is going to be published in the spring, is so important because it specifically addresses how we can remove those barriers to speed up the process of delivery of affordable homes within rural Scotland. I am sure that, when she sees it, she will recognise the importance of that plan. To ask the Scottish Government what discussions the social justice cabinet secretary has had with ministerial colleagues on any potential impact on levels of personal debt of the inclusion of individual consumers under the Moveable Transactions Scotland Bill. The bill is a long overdue reform of the law of security relating to movable property, making it a matter for the economy portfolio rather than the social justice portfolio. The main reform in the bill, which may impact on individual consumers, is the introduction of the statutory pledge. I am in no doubt that it should not be possible to grant a statutory pledge over ordinary household goods, and so its impact on individual consumers is expected to be limited. However, I recently met citizens advice Scotland and other debt advice agencies and listened carefully to what they had to say on this, and I can confirm to the men in Parliament that I am very well disposed to strengthening the consumer protections in the bill. I thank you very much for that answer. As you said, the bill that is currently constituted would allow people who are in very difficult financial circumstances to borrow money-based of the value of their items around the value of £1,000. It seems that, as you have said, almost every consumer debt and money advice organisation has highlighted the serious pitfalls that prevent for those struggling in debt. In short, a bill designed to help businesses is suddenly incentivising irresponsible lenders to target individuals in financial distress. I am glad of the cabinet secretary's answer, but can I just clarify that you will seek to speak with colleagues in other portfolios to amend the bill accordingly? Yes, I am very happy to engage directly with the member. My position right now is that I am awaiting the report of the DPLR committee. I gave evidence last week to the committee. I want to carefully consider what their recommendations are, but I recognise the concerns that have been raised. I am considering them very carefully. I will consider the report very carefully and I am happy to meet with any member ahead of the stage 1 debate to discuss these matters further. To ask the Scottish Government what actions are being taken to end the practice of children living in temporary accommodation in light of recent homelessness statistics. Cabinet secretary, we want everyone to have the stability of a settled home that meets their needs and to ensure that the time spent in temporary accommodation is as short as possible. Our strong homelessness legislation means that homeless households, including those with children, have a right to temporary accommodation, which provides an important safety net. However, I have asked an expert group chaired by Shelter Scotland and the Association of Local Authority Chief Housing officers for an action plan to help to reduce the numbers of people in temporary accommodation with a strong focus on households with children. The group will produce the final recommendations in early 2023. I thank the cabinet secretary for that answer, because Shelter Scotland says that our housing system in Scotland is broken. In the last year alone, the number of children stuck in temporary accommodation rose by 17 per cent, the highest since records began and the doubling since 2014. The situation in Edinburgh is now beyond crisis levels, with more than a quarter of all children in Scotland living in temporary accommodation here in the capital. We need to see an emergency response, cabinet secretary. I also ask if she will agree to personally chair and establish an emergency task force for the capital to specifically look at the issues that are facing children living in temporary accommodation here. First of all, we already have an expert group chaired by the very Shelter Scotland that he has just mentioned along with a that are looking at Edinburgh and the rest of Scotland. I think that we should allow them to get on with that work. In terms of Edinburgh specifically, in the summer I met the housing convener for Edinburgh and other local authorities under most pressure. I have recently written to both Edinburgh and Glasgow housing conveners to follow up on those discussions and have reiterated the ask for them to submit proposals that could relieve some of the pressure on temporary accommodation. I have committed to considering all the options that are being brought forward to help with the pressures on temporary accommodation, so they need to come forward and I have very much an open door to responding to that. Finally, although Miles Briggs and those on his benches need to recognise that the links from temporary accommodation and people ending up in temporary accommodation are very much linked to poverty and people ending up in debt and poverty and the cost of living crisis is a major factor in that. While we will do what we need to do, I would urge him to also make representations to the UK Government to support people at this difficult time. To ask the Scottish Government what discussions the local government minister has had with local authorities regarding running effective consultations on the delivery of front-line public services. I regularly speak to representatives of local government on a variety of issues regarding front-line services. Consultations are important to ensure that local people and communities have meaningful decisions on public services. However, councils are independent of the Scottish Government, and as long as they are meeting any consultation statutory requirements, it is entirely a matter for councils how to carry out consultations, and the Scottish Government has no involvement in those processes. If there is a particular issue or service that Jenny Minto is concerned with, then I am happy to consider that and, as appropriate, I ask the relevant portfolio minister to respond separately. I thank the minister for that very helpful response. As the minister will know, Argyllin buta is a hugely diverse area with no two communities the same, and consultations can be on topics as diverse as education change to improving peer infrastructure. When it comes to consulting on local services, one size does not fit all. What guidance would the Scottish Government offer to local authorities to ensure that individual consultations get the best responses to reflect the community's needs? I agree that the diverse and varied needs of Scotland's communities is an important aspect to consider when carrying out consultations. That is why I believe that councils are best placed to determine the needs for each consultation locally. The Scottish Government has its own consultation guidance, and we seek to ensure that we engage with a wide range of stakeholders to take on a broad range of views and experiences to inform policy and decision making. I would be happy to have further correspondence with Jenny Minto on ways in which the Scottish Government can perhaps assist her more in her area. Thank you very much minister. That concludes portfolio questions. Can I just add a reminder that those participating in portfolio questions or indeed any debate are expected to stay in for the duration of portfolio questions, except if they have been given prior consent to leave early? I noticed that a couple of members dashed out during the course of proceedings, so I would appreciate that if that did not happen in the future. There will be a brief pause before we move to the next item of business.


Okay, for the next part of this series, I want to prep my journal. Now I'm going to again use what I have, and this is a moleskin, moleskin something. I don't know, I wrote the page title page, oh no I didn't. It's a moleskin weekly notebook from 2017-2018 that I never used. And normally I would have these sort of dashboards protecting the pages I'm not working on, like something like this. But for what we're going to do, I'm not sure this is enough protection. And I kind of want to make a clear plastic something I can slip one side of the journal in and then flip up the pages I'm going to work on and flip the other side in. Does that make sense? So I have these sheet protectors I've had for a while. So I thought I would take one or two of these. I'm not sure one is going to be big enough. I'd rather have two I think. That way if the journal gets puffy, there's room. So I'm going to take two of these and I'm going to cut the holes off because I don't need them. Then I'm going to open the sheet protector and I'm going to put one inside the other. I realize this might be hard for you all to see on camera because these are clear. So I'm going to tape that together with some clear packing tape and we're going to flip it over. Now I need this to not overlap on this side. So I'm going to cut away the part where they overlap on each side. It doesn't have to be neat and perfect and pretty because it's just not about that. So then I'm going to tape the top shut and basically make a clear plastic book cover for my art journal. Again it doesn't have to be perfect. I'm sure you could probably go buy something but I don't want to spend any money and I want to just use what I have. So now I've got this thing that has these two pockets in it. So I can take my art journal that I'm going to use and I can put one side over here. I can lift up the two pages I want to work on to start with and take the other side and put it over there. It works very nicely. I can take some blue painter's tape which is somewhere around here. I can rip off a piece of blue painter's tape and you'll have, because my cover is bigger than my journal, there's a gap so it's not going to be perfect and things are still going to leak but we want to minimize the leakage. So I'm going to put a piece of tape there and then I'm going to push the journal up against the tape down here. The other thing that we will do is fold this back on itself and put a clip. Just hold things neat and clean and hopefully sort of flat like that. And again, things may still get through and I might put some masking tape on the spine so that what I'm using doesn't leak through the spine here to the other pages. For that I would just put some masking tape. You don't have to. You could just not worry about what's leaking. I don't know that I'm going to. But that's all for the next episode. Right now we're just making a clear plastic protective cover for our art journal which you could use for any art journaling project or you could cut dashboards out of random plastic or these are clear plastic cutting mats from the dollar store. So use what you have. Make something to protect your journal. The pages you're not working on or semi-protect them. And join me in the next episode.  I'll see you in episode three. Bye guys.


So, let me make a few sharp points as a collective answer. One, no question the world is more nationalistic than in the past and a lot of that nationalism is economic nationalism and cultural nationalism. Second, I would say where India is concerned, we in a way are a standout, we are an exception. Because in this country you could say, yes we are more nationalistic, but at the same time we don't see a tension between being nationalistic and being international in the sense of dealing more with the world and engaging more with the world. So, the nationalism is not a kind of a negative sentiment directed at the world. In fact, people generally feel if you're going up, you should be doing more things with the world, not less things with the world. Third, in a more multipolar, your word, a nationalistic, my word, world, I think we need, we will see diplomacy take different forms where the old ways of working will now not go away, but be tempered by much more, I would say, creative, innovative ad hoc kind of working arrangements often centering around issues rather than across the board. So, the character of international politics will probably change in many ways. Four, you have, I agree with you, I think a lot of multilateral regimes will come under stress. How they survive will depend on how they respond to that. They will come under stress partly because of this nationalism that I spoke about. To some extent, a lot of them are also being critiqued for how well they work or don't work. So, there's a kind of performance audit on multilateral regimes also going on. Sometimes that can take very unfair directions because if you do a performance audit with a very self-centered nationalistic view, then I'm not sure, I'd agree with the conclusions of that audit. But certainly that too is a factor. So, all in all, I would say more complicated world, definitely a more interesting one, possibly a more difficult one. But where India is concerned, you also made a reference to G2. That's something we have never accepted, we've never been comfortable with. I think partly what would also distinguish us from other countries is that we still have a very strong relationship with countries of the South. And we, in many negotiations, which we would have seen as trade ministers as well, we not only stand for our own interests, to a large extent we voice collective interests of the developing world, G77, for example, certainly when it comes to trade or when it comes to climate change. So, I think that's also a constituency for us. Now, when you look at all of this, you put on the one hand say, okay, it's all getting more difficult, probably true. But you could also say that we are in a very unique position, that today being a market economy, a democracy, socially pluralistic, we have comfort with the West. Being Asian, part of the rise of Asia, we have comfort with a lot of this rebalancing in Asian countries. Having a G77 constituency, working with countries of Africa and Asia, we have a much stronger bonding with those countries. So, I think we could be at the right intersection, a lot of it therefore requires active diplomacy to make sure that those constituencies are all sort of brought together. That's very much the intention of the government. So, what you have seen really definitely over the last five years and you saw recently at the UN, is a willingness today to go out, engage countries, visit more countries and therefore, you can see a new energy in foreign affairs.


Migrants break through security barrier at U.S. border. A crowd of migrants has torn down a concertina wire barrier and broken through a line of Texas National Guard troops while trying to cross into the U.S. from Mexico. The standoff occurred amid heightened debate over border security with Republicans accusing President Joe Biden of dereliction of duty. Footage of the incident posted online shows migrants dragging down a section of the barrier and a group of five National Guardsmen forming a line and trying to stop them. Many of the men hold up their hands signalling for the troops not to shoot them, but seconds later migrants push through knocking the heavily outnumbered soldiers out of the way and running to the border gate. The New York Post, which had a reporter on the scene, said the incident occurred at a border crossing in El Paso, Texas. About 600 migrants massed on the Mexico side before a section of the border barrier was torn down. After racing to the border gate, the men shouted at the troops on the other side. An unidentified source told the newspaper that the group was pushed back to the Mexico side of the border. The Texas National Guard was deployed as part of Governor Greg Abbott's $12 billion operation loan star cracked down to stem flows of migrants illegally crossing into the state. Biden's administration has filed lawsuits against Texas, challenging the state's authority to put up barriers and police migrant traffic. The federal government normally has jurisdiction over border security, but Abbott has invoked the state's constitutional right to self-defense, arguing that Biden's dereliction of duty has triggered a migrant invasion. Earlier this week, a federal court that is hearing one of Washington's legal challenges temporarily paused implementation of a new Texas law that would allow state authorities to arrest and deport illegal migrants. Last attack showed us that yes, it's a violent invasion. Conservative commentator Mike Chenevich said in a post on X, the true nature was hidden as the regime left the border open. Now there can be no denial. Germany won't allow Putin dictated Ukraine peace, Olaf Scholz. Germany will not let Russian President Vladimir Putin forcibly alter Ukraine's borders or impose the terms of peace. Chancellor Olaf Scholz has vowed, we will not accept a dictated peace at the expense of Ukraine. Scholz told German lawmakers in Berlin, law is stronger than violence. He added that Putin sought to violate that principle with the launch of Russia's military operation against Ukraine in February 2022. We will not let him get away with this, he said. Scholz insisted that Germany's backing of Ukraine in the conflict with Moscow will not decrease and that expecting otherwise would be a miscalculation. He reiterated his criticism of Putin's re-election last weekend for a fifth term as president saying it showed that Russia is not strong. If the Russian president believes that he just has to wait out this war and that our support will weaken, then he is mistaken. Scholz underlined while addressing the Bundestag. He reiterated that the Weimar Triangle and the entire European Union stand together in their support for Kiev. In his remarks, Scholz criticized the debate in Germany around the potential delivery of tourist cruise missiles to Ukraine, which the Chancellor opposes. The debate in Germany is nothing short of ridiculous, he declared. This is embarrassing for us as a country, he said, arguing that the debate was not understood outside of Germany. Scholz has argued that tourist deliveries could spark further escalation with Russia and cause Germany to be drawn directly into the conflict. He said Ukraine would be able to use the missiles to strike deep into Russian territory and that Ukrainian forces would need guidance from German personnel to use them. He pointed to the fact that Germany is Ukraine's second largest supplier of military aid, saying that this must be recognized. However, as EU foreign policy chief Josep Borrell acknowledged in an interview, European allies will be hard pressed to fill the funding gap if Kiev's biggest backer Washington reduces its support. US President Joe Biden's administration ran out of funding for Ukraine in January and has struggled to secure congressional approval for over $60 billion in additional military and financial aid.