Thank you very much, Nate. So recap. OK, so far, so we have the category of stable maps. Then we showed that there are certain, then the sub-category of holomorphic objects, pseudo-holomorphic objects, is given by a theta, which just associates the weight 1 to such an object and otherwise 0. And the idea is to deform this theta j into a theta so that a certain number of properties hold. And so we discussed that in the first lecture. Then in the second lecture, we showed actually that there is a smooth structure on this category explained what that means. And what we would like to find are smooth things, which do this. And now how do we get them? And that is going to start to happen in this lecture is that we will define another category lying over it, for which the Cauchy-Riemann operator gives a section functor of this. So the fibers actually will be Hilbert spaces over each object as a Hilbert space. And the Cauchy-Riemann can be viewed as a section of this. And then the theta will be obtained in such a way for another kind of functors, which are this time defined here, which you can view as multi sections. And I will explain all this. And this, so here we already know there are some kind of smooth objects, so smooth functors. There will be also some kind of smooth functors here. And if you choose this thing in general position, satisfying some properties like this, but we have to formalize them for these things, then this one will actually have these properties. And it will be of a good type, namely it will be of the smooth weighted category type. So locally it is sort of represented by many folds divided. So it's good enough that you can actually integrate forms over it and so on and actually can define SFT. But there are some issues, there's something that we would have to discuss, namely orientation. And orientations are better done to actually go to a covering, where one actually introduce numberings of the punctures and so on. OK, so that is what we did so far. OK, so this bundle category, so we have our category of stable maps. And we take a functor into Hilbert space. So when I formulate things, I usually give some general formulation, but for the category of stable maps. So you can think of other categories you put in. This scheme works actually for a lot of series. So we associate to an object a Hilbert space. And in this category, the morphisms, so two different objects might get two different Hilbert spaces. But the morphisms are actually lifted to linearize the morphisms. So then you can define a new category. And then we take the object and the vector which lies over that object in the Hilbert space. And what are morphisms? Well, it's a pair of phi E. Phi is a morphism in your category of stable maps. E belongs to the Hilbert space lying over the source of phi. So the source of phi is, say, alpha. And what is the target of this morphism? It's just the vector obtained by applying the lift to the original vector E. So you lift each morphism as a linear map between the fibers. And the objects are the vectors in this fiber. And the target is the image under this linear map. So we have that. So I wonder who did that.  So OK. Ah. Ah. OK. So here we have the object alpha, alpha prime. So here's zero. The other zero. There's a morphism phi between these guys. And we have a vector E. And the lift, mu of phi, is a linear isomorphism which would map this here to mu phi of E. And this thing here, so we can identify then a morphism for this bundle category with the underlying object and the vector E. So morphism is phi. And the source of this is E. And the target is the image which you get here. So in our, what do we do in our case? Well, if you have such, so that's a building of height one, let's say, then what do we do there? Then this, then here, you have an equivalence class of maps up to r-action. So we take a representative. Now you see when you take the tangent space here, then because you have the r-action here, you can just identify this with r cross the tangent space of the underlying v-component of this map. Your twiddle has the r-component and the v-component. And our Hilbert space for this object consists of all maps which are complex antilinear from the underlying agreement surface at the point z into this thing, which is identified. Since I take a representative of here mod r-action, but the first factor is independently defined on which thing I'll take. And this map should have a certain regularity property. Namely, it should be away from nodes of the class H2. And that is because the Koshiriman section will act on H3 stuff when it goes down to H2. And at the punctures, you have to also take, you take exponential decay of this thing. So if you take a puncture and you take cylindrical holomorphic coordinates, you want exponential decay of the partial derivatives up to order 2, which is precisely when you take these stable maps, which are asymptotic to cylinders and have quality 3 delta 0, so 3 times partial derivatives with exponential decay. And then if you apply the Koshiriman operator, they would go to 2 derivatives with the same decay. So that is what you take. So that's the Hilbert space. And more generally, if you have a building, then over each of those, so you have alpha 0 up to alpha n, over each of them, you take such a thing. It's clear then how this acts. Namely, if you have a 0 of 1 form, you just take t. So then these morphisms come from biholographic maps between the buildings. And you map e to t phi inverse or something like this. Yeah, e composed with e composed with t phi inverse, the tangent map of the biholographic map inverse. Last question through this slide. So you said that you wanted a bundle of other spaces, but your definition of going to a bundle has not been given yet. But come. So in my talk, I first did the underlying space just as a category without smooth structure. Then we looked at all the relationships. Then I put a smooth structure on it, could say more about it. Now I put just an algebraic structure on this, discuss that. And then I put a smooth structure on these two things. And at that point, you are on the level where you can just unleash some abstract perturbation results, which brings things in general position. So that's the structure of the talks. So here, we inherit a lot of the structures which comes from this stable map. So we had this input, this evaluation maps, the function E plus minus. Well, we just compose it with this projection down here. And then we get such an evaluation map for E. The grading, the grading we take from the underlying object, then we can decompose this. And that is actually as before, except that this thing now has a little bit more structure over each object. So it has a structure over each object. Over each object, there lies a Hilbert space. And the next thing is to lift the data from S to E. And the data from S to E means, in particular, the covering business which we had. So how does it look like? Actually rather trivial. So in the base, we have this chopping functor. And then you just, over each of these parts, you have the 0, 1 form. You just put it forward. That's it. And it's fiber-wise a linear isomorphism. So I have these objects, which is a building. And over each of these floors, I have a 0, 1 form. And I chop it here. And I just take that forward. And that is an isomorphism on the fibers. So it then satisfies precisely the relationships which we had. So that's basically completely on the nodes. So you don't even have to think about it. And then there's a functor. Maybe if alpha is given, which consists of different buildings, you just, for each map on each building, coming from each building, you just apply the co-chiriman. So that's a functor. And so let's now think of, you might have noticed, it looks really rather like this. The color is better. That is the Joe's color, yeah? It's not purple.  OK, no, let's not elaborate on this any further. So now let's first algebraically discuss what we can do here. So the idea is to perturb theta j. And theta j limits my options, yeah? So if I put a pseudo-holomorphic object into this, then this vanishes. And this thing gives to the 0 vector weight 1, and otherwise 0. So then that's precisely this one. So then we want to perturb this. And what we see here, this gives just the weight 1 for the 0 section. So the idea is now, say, if this is a 0 section locally, I want to sort of have a partition of unity of this and just move. So I've used a 0 section as several 0 sections, but with rational weight. And then I move this individual parts away to achieve transversality. That is what you ultimately want to do. So that's the idea. Of course, then you run it. So this one will be perturbed, let's say. But when you move this away, you want to keep the symmetries. It should stay a functor. For example, locally, you have the action of the automorphism group. This whole thing, which you get, so it might be. So this will be turned into something like this. If this is a 0 section, so here's a base, S, and here's sort of E. It would sort of look like this. And you want to turn it into this. But you want that this whole stuff, these things each have a fractional weight. You want to want that this is invariant. Then, of course, you do this at different places. So it becomes a little bit more messy. So if you go globally, then the things you constructed might be then bifurcated further off, and so on. So that is what you want to do. So you need to develop sort of a machinery to be able to pick such things with sort of this up. So you see, if lambda 0 is replaced by some lambda which consists out of this section, then this is only positive if the Koshiriman operator perturbed by such a section set is 0. What does it mean? If this puts the weight on a graph of different sections, then this here will only, if I perturb this, only become positive if you solve Koshiriman of alpha equals one of the things in the graph. And this you want to achieve transversely. And if this is transversal, then that actually will be a smooth object, a smooth function. So that's sort of the idea. And for this, then you have to develop a little bit of machinery. Is that clear sort of what the aim is? You're the chair. You are not asking questions. OK, you're good. It's like the most basic possible question that I should ask you. What would happen if your perturbations were not on purpose? Well, then you lose some symmetry. And I think there's still a theory, but it's not the theory we want to do. I think you can actually do some brutal, some really brutal stuff and ignore some of the structures that you can do. And that's another way to produce data. And then out of this, presumably, you can produce some invariance. If they're interesting, I don't know. It's like, if you do S1 invariant more theory, then you just forget the fact that it's S1 invariant. And then you have usual more theory, something like that. So it's on that level. But if you work locally, then what the functor is doing is really saying, you choose something locally, and then it's consistent with something else. You choose to show it somewhere else. Yeah, so you have to keep that. But you might not. So in some sense, when you patch it together, you want that it fits. But you might maybe, yeah. It could fit in some slightly complicated way. It could be. Yeah, so you have to think about it. Because just saying, now, this should fit with that is easier to say with a whole lot of structure, which we have, than actually saying, I relax it somewhat locally. Like, I could say it should not be invariant under the action of the isotropic group, and then say it can match it up globally. Because the constructions are local, so you write, as you'll see, a perturbation as a sum of a lot of perturbations.  But I think it's possible, at least in a general framework, to if you have a criterion to forget some of the structure. For example, you could definitely forget the structure that you want to, when you perturb, that if you add trivial cylinders to it, that they should just appear as pseudo-holomorphic cylinders. You could use them in your perturbation. So that would stay consistent as long as it's invariant under the morphisms. You could also disregard the fact that things are disjoint unions. I mean, if you have disjoint unions, if this is a solution to this, you could also go away from this. However, I don't think for the latter one, you would get a new theory. Because since you can do it, there's at least a cobertism from not doing it to the one doing it. And then I think, when you arrange the data, you might actually get just a lot of cancellations. But I haven't carried this out. So there are a lot of things you can think about. OK, so what are the requirements on lambda guaranteeing the desired properties for Sita? And here they are. Actually, they are even better to formalize than for Sita. So first of all, we have to define something on the Fibre product. So if this on the Fibre product thing, so I ignore, I don't write down that E zero lies over alpha zero. So then you just take the product of the things. You don't want any zero, do you? The zero? Oh, yeah, so I guess I want everything. So this was my keyboard, I guess. Tap the wrong button. So this is a zero, sorry. So this is a restriction. Then we have this covering function. So here's the algebraic version of this, which corresponds to the version in the first lecture. But these things can be also lined up according to the underlying phases you have, which was lecture part two with a lot of confusion, which hopefully decreased in the discussion session, which would be the same formula. So this is the algebraic version. So is the right-hand side not equal to just taking this maximal breaking of any one object? Right, but yes, if you have this property, yeah. So what this says is basic. So maybe I should never write this formula. But so what it says is. I guess mainly I'm asking, is there some interesting sign that I need to be aware of? No, it's just a lot of consolation. So ultimately it's one term. If you have this. Let's say lambda is always plus 1 0 minus 1. The result here is always going to be plus 1 0. So now here that is now important. So if I have an object alpha, then it has associated remnant surface. And then the remnant surface can be decomposed as the component. So let's first think a building of height 1. The parts of the surface which carry the things which are not trivial cylinders and the things which carry trivial cylinders. And if you have a building, a trivial cylinder building is just a line of those guys. And so when you look at this thing, you can see the trivial cylinder buildings and the rest of the components. So that's a natural decomposition. And you have a forgetful functor. Namely, it forgets the trivial cylinder buildings. OK, we have that already. So now there is, first of all, a Whitney-type decomposition of E. So if I look at my Hilbert space and have a 0 1 form over it, I can put this thing 0 on the trivial cylinder building over the trivial cylinder building. Or I can put it 0 on the complement. So this here is the part which is defined on the original building, but it is 0 over the trivial cylinder component. And this one is perhaps non-zero over the trivial cylinder component, but it's 0 on its complement. So you have this decomposition here. Here it's written. And I have a question. So if you have a two-level building and it has no trivial buildings in it, so it's non-trivial on every level. And suppose on the bottom level, you have a trivial cylinder and something non-trivial. Then what's the corresponding splitting? Is ETC just restricting the trivial cylinder buildings or restriction of trivial cylinder buildings? So if I have something, and then what you said, I have some non-trivial cylinder here and have a trivial cylinder there. And then I have this, which is trivial cylinder. So I have a 0 1 form over this. So the ENTC would just put the value here 0, not here. But it turns out when you do inductive steps for actually constructing lumber, then this bit already appeared earlier. And then the thing was already having required properties here to begin with. And the trivial cylinder is something which is inner. It's sort of 0. It's a homotopic to something. To a geomorphic cylinder, yeah. So here is a picture. So here is a lift of the little c. It just forgets the underlying trivial cylinder building and just restricts and gets this new object here in E. So before, you just forgot about part of the stable map. Now you throw away part of your object in E. So now you have two functors. So one is pi is just the projection here of this bit later decomposition on E on this. So in particular, it preserves the underlying. It covers the identity on objects. But this one, c, does not cover the identity. It covers the forgetful functor below where you actually throw away trivial cylinder components. Why are we having such a careful discussion of the trivial cylinder components? Because of if you want to SFT or want to two, OK. So if you think of this here as a preparation of producing ultimately data by integrating forms of a components and so on, then the next step is what can I do with the data? Can I represent it as a chain complex or something like this? Then if you want this thing to have certain properties, and in this case, an algebra property, you have to discuss the cylinders. They look, of course, completely trivial, but since you make concatenation, then you add something into it and so on.  So if you would disregard them in some way, let's presume you're also some theory, but it would be different or possibly different. So now this is, of course, a projection here. And here, if the underlying object doesn't have trivial cylinder components, then actually you have this identity, obviously. Because there's nothing to put zero. OK, all this functor, so this is also a retraction. It's linear on the fibers. It covers the other retraction. So that's sort of the structure which we have. And these things commute in this way. And then what is important is that if you restrict C to the non-trivial cylinder part, you get actually fiber bias and isomorphism. That's actually important when I, that allows me to pull back perturbations by this because I have the linearity in the fiber, the isomorphism in the fiber. So if you go through this list of things here, you find that it's actually rather trivial in the concrete example, but this kind of things, how I write it, is actually in all the problems, like flow theory and so on. So it's just always this kind of structure. OK, requirements. So there's this pullback operation. And here, so what does that say? So lambda E should satisfy this. So if lambda E is positive, that has to be this. So if this is positive, this has to be 1. And what does that mean? This is 1. This means that over the trivial cylinder part, the component of E is 0. But that means I actually don't perturb over trivial cylinders. So if you don't perturb over trivial cylinders, that means then d bar over trivial cylinder is 0, which means it's actually the pseudo-holomorphic cylinder. I don't have to perturb there. Is that clear? So if lambda E is positive, here I have lambda E lambda E here, then this one has to be 1. So this here is a part of E over a trivial cylinder building. And lambda 0 is our original thing, which puts weight 1 on the 0 section and otherwise 0. So if this is 1, this means this is a 0 vector. And that means the E over a trivial cylinder component is 0. So then the d bar part on this, if d bar is equals that E over a trivial cylinder, it's a pseudo-holomorphic cylinder. So that you see how it already produces one of the properties of our theta. So we're going back to that picture. For the cylinder that's in the middle of the bottom level, can that be perturbed? No, because of the inductive nature of things. So on some level, like a level 1 building, of course, that is something which would satisfy this property. So whatever you construct in the perturbation because of this algorithm, it will actually not perturb over trivial cylinders. So you would always get pseudo-holomorphic cylinders out after the perturbation. So then, of course, there's sort of what we had before. If I have two stable buildings and I put them together, I can move them against each other, then you want that property here. If I take one of the representatives. So what does that now mean if this is positive? Let's just discuss what does that mean if this is positive on an object. So first of all, it means there exists a rational number sigma positive, a vector in the fiber over that object. So that lambda of E is E. And alpha satisfies this equation here with a weight sigma. Sigma is the number associated to this object alpha. So now, if alpha is actually a building of height k plus 1, so top floor is k, then the sigma can be written as a product of positive rational numbers. And each of the alpha i's, and this E then, of course, is a sequence from E0 to Ek. And each of those satisfies this equation with a weight sigma i. Yeah, so if you look at, so what's the interpretation of lambda composed with? So it means, since in the fiber of different vectors, that it satisfies one of the equations coming from the vectors with non-zero weight. And they have a weight coming from the underlying alpha. And so I get a sequence of equations. And each of them carries a weight. And if I add the weights all up, it's 1. That's precisely the splitting of the 0 section. But the equation itself doesn't depend on the weight. No, no, no, no, no, no, yes. So ultimately, in some sense, we count solutions. But we don't count them 0, 1. We just count them with a rational weight. And of course, there might be a sign also. But this contributes, so if I have two solutions, each topologically counting 1, and the equation has weight 1 half, then the total thing I see is 1. 1 half for that equation plus 1 half from the other. So it's a system of equations where if the equation is true and you have a solution, you just, it contributes according to its weight. It's like the S and P index. How big is the capitalization of a company or something like this? I was also interested in buying stocks. I mean, it's a sort of this kind of thing. But you're also going to say if you cap lambda of E equals sigma. That's what that means. Cap lambda of E, yes. So this is, of course, what happens here, which I said somewhere here, lambda of E is sigma. OK, there it is. Here it is. So that means if I solve the equation d bar equals E, this equation counts. And it's taken into the general bookkeeping with a weight sigma. So then this one here, if there's a length, it's decomposed in a certain number of EIs. And we have this product structure. And so each equation is this here, has this weight, then I can put this together to this one. And the sigma comes from this individual weights of this parts. So then because of this property here, each EI vanishes on the trivial cylinder components. Actually, on every I, because this one was already perturbed. And this EI, so this is a building of height 1. And this EI already satisfies a property that over a trivial cylinder component is 0. So all the trivial cylinder bits, for example, this one here, if you look at the blackboard, all over this, if there are solutions, would actually be real J-holomorphic cylinders. So then if the alpha I has different components, so I'm looking on a floor, then this components, you can decompose it according to the different components. And some of them have trivial cylinders in it. And each of them actually has a weight. And the sigma I would be a product of the weights for the individual components. So this is what the perturbation all does. So individual components are perturbed separately, pseudo-holomorphic, trivial cylinders turn out to be pseudo-holomorphic, and so on and so on. So now we come to the smooth. So algebraically, I think now it's all clear. And now we have to put a smooth structure on this thing and see that we can define what is a smooth lambda and so on. And then we are ready for the perturbations here. Can I ask a question? So what's going on? So at the very end, do you end up with something which is a Q linear combination of manifolds or something which is only locally Q linear combination of manifolds, whatever that means? No, OK, so model of the following is actually one of the things you said, which I'm explaining now. So if I have too many faults, weight of each is 1 half, I could view them as four manifolds by taking a copy of each of them with weight 1 quarter. That would be considered equivalent. So then if you have overlaps, so you have here some manifolds and here some manifolds with weight, what does it mean they fit together? Basically, it means that if you take a certain number of copies here and a certain number of copies here, you can match them up so that the weights are the same. So they're smoothly fit together. So that's one of the things which I said. Well, I don't have the four. I think the second one, for example, it's not a. It's in the middle. You can't break it up into manifolds and then give each of them a weight because they may be. There's no natural identification. You just can identify after. Locally you can do that. But not globally. No. I mean, you can put some artificial structure on it to say what you have to do at any given moment in the overlaps. I mean, you can say, you put the structure on top and says, you have to take so many copies and that you have to identify with this one. Which is actually, you have to do when you do extension properties like for sections. I mean, the same thing for sections. Because when you look at how, if I have a section defined over the boundary, how do I extend it to the interior? Well, the only method is you make a local extension, take a partition of unity. But if I don't know what to identify with, what do I add actually up then? So you need that structure of this identification extend and then according to this identification, you glue the things together to get an extension of the boundaries. Thank you. Good question. Any more? OK. So now the theorem is there exists a natural star means up to some fixing some discrete set of data. A strong bundle structure for this thing here. So it is basically like the polyfoil structure for this, except that we here have a strong bundle line over O. We have the associated translation group part. So the objects on top are vectors in a strong bundle over E. Then we have an action by the isotope group on this. And here it covers this thing. This is one of the size from the polyfoil structure on S. And we have this in a coherent way. And then if you have two of those guys, then we get the transition set. And that transition set is actually a bundle over the transition set here. And this has a strong bundle structure, as it was defined in one of the lectures last week. So this would be the generalization of the structure which we had here. And at this point, you can start talking about smooth multi-section functors. Is that sort of clear? OK. So now it takes a Cauchy Riemann functor. And it goes from here to here. And actually, well, just look at this composition here. Then it lies in the image of this one. And you get a local representative. And it turns out it's S.C. Freton, which was defined by Cartrain and Droll. And the modeler category, where this is 0, has a property that its orbit space intersected with each connected component in the underlying space. The orbit space of this is compact. That's Gormov compactness. So now we are in a smooth setting. So Cauchy Riemann section is an S.C. smooth section. That is what that property means, that this is Freton. Then if this thing vanishes, this means we have pseudo-holomorphic objects, which is sort of this associate modeler category to this one. If I take its isomorphism classes and intersect it with the connected component of the orbit space, it's compact. That's Gormov compactness. So that's by definition what it means, as you have a Freton functor. I forgot what O and S. O and S. So S stable maps. Good, that is what we're talking about for a while now. Good, good. Then O, this functor defines the polyfoil structure. So these are the things which are injective on objects. If you pass to orbit space, you get... So there's a point here, an object here, which is mapped to the original given object, alpha, and so on. But O is retracted. Well, it's just taken an M polyfoil. It's an M polyfoil. So then this is a strong bundle of an M polyfoil that was also introduced. So that is the model for... So is the statement that if I take... So E over S is some bundle that we already said that way. It's a statement that whenever you pull back to a polyfoil bundle, to a polyfoil trunk, I get a strong polyfoil bundle. Is that what it means? So that means that given any object here, whether there's a selection of a set of those guys, and if you put that in, then that's sort of the local structure near the object alpha. Let me try that again. So is del y bar defined by pullback? Yes, yes, oh yes, right. So in this case then, for this construction which you have, if you take any of those guys, it is a Fretom operator in the sense as we have discussed. So the pictures here, when I described before, you have the smooth functor theta, and I put my hand in, then I see sort of many faults. Now I have a bundle over this, lying here, and when I see the Cauchy Riemann functor, the trace, what it does here, and where it's mapped to, that's actually a real Fretom operator. And then I can put anywhere, and this structure gives, if I know something here, I can always transport it to a neighborhood of any isomorphic object. So that is what it... That means, so now here's a polyfoil packaging of the SFT data. So we have a strong bundle structure over polyfoil. The Cauchy Riemann section functor is S.T. Smooth and Fretom. We have S.T. Smooth covering functors with compatibility and some additional stuff, where we have for each phase, so I haven't defined this, but self-clear, we defined it for on the level for S. So if I face here, then this here is just the stuff of E lying over S.T.T.A. Then this was a covering functors. Then there were a certain number of compatibility conditions, which we discussed in the last lecture and also yesterday in the discussion. So you have this diagrams of these things and you have that. So I suppressed here the moving of components against each other. Then out of this data, then one can write down which we did before a requirement for the perturbation you want to do. But that is basically sort of the smooth packaging of the data, which you need to do, to produce the data, which you need for S.T. The last diagram also is commutative, if you replace P by D bar, right? All these functors commute. No, I mean, in the first two, which you talk about this or this or this. So there's three diagrams, right? And then there's two equities below that, which I read as if you change the arrows down and label the P to arrows up and label them D bar. Yeah, yeah, yeah, here. So it's compatible. So if you put the sections, if you put the Cauchy-Riemann section here, the local representative. So here this is, so this is a restriction of the Cauchy-Riemann and here the other one. I'm gonna have to ask you, why are there not three equalities at the bottom of the three diagrams? Yeah, okay, so yeah, so that's good because I forgot to write them. So if you apply, if you apply, yeah, so what do I want to say? Yeah, well, this more controls this one. So if I, you see here, here's the identity. So there's not too much happening with respect to D bar. So it's just controlled by the C, but the C has certain properties with respect to the P. I mean, that's what they want to say here if you have. But it is true that if P composed with D bar is simply zero, probably. No, no, no. If you have a solution, on the solution set of what you're interested in is identity minus pi composed with D bar would be zero. Which means on the trivial cylinders, you would be pseudomorphic, yeah? Identity minus pi composed with D bar equals zero means on the trivial cylinders you're pseudomorphic. Right, so that's exactly the equation here, right? So pi composed with D bar is in fact equal to D bar. It would be.   And then the other part of that would not be zero. No, then it would not be zero. No, I just said only on the solution set, ultimately. So I think you can write certain things under the sum. So identity minus pi D bar equals zero provided actually, what does that actually mean? Equals zero means, well, that is precisely what you can say. Identity minus pi composed with D bar equals zero precisely means the trivial cylinders which you see are pseudoholomomorphic. Right, but that is exactly your right hand diagram today. Ah, OK, good. Now, OK, so something in that direction. So let's put a weight on this 0.1. And so there's some truthiness to it. So now, constructions of SC plus multi-section functors which are a particular class of those guys here. So this is an important class. It's sort of, these are multi-section functors which you can sort of view as multi-sections of, these are sort of compact perturbations of this. So multi-section functor is of that particular kind, provided as the following properties. So if you take sort of this uniformizers and the underlying thing, the q0 would be the object where you're looking at, then this composition here is a count of the number of indices for having, it's a number of indices where this age satisfies this. So you go into the base, pH, so age lies in, so age lies in k, pH lies in O. These things are defined on O. And if Si of the underlying base point is equal to the vector you put in, you count the number of indices, this is that, and divide by the number of indices you had. And these things here should be locally SC plus sections. And let me remind you what that was. This strong bundle comes with a double filtration. Namely, it made sense to talk about mk, and here m, where k is less or equal to 0, less or equal to m plus 1. So in particular, you have a k01 lying over an O0.  So the Si's go from, so they're defined on O0, but they go into k, m, m plus 1. And then, of course, what is important, that is why I said it's compact. If you go, this is a fiber regularity. And if you view them with respect to the different norm, that is a compact inclusion. That is why I call this compact perturbation. So these are some kind of sections, but they are constrained by having this property. And I think you talked about that, or it was mentioned maybe last week. I don't know what the definition is saying. It's saying that there exists Si's. Yes, so there exists finally many Si's, so indexed by the set i. And you look at the coincidences. So basically, the picture here is, if this is O here, and this is a fiber, then locally, so you have a certain number Si's, i and i. And each of them carries the weight 1 over i, 1 over the number of elements in this thing. And you just look at this vector here, how many graphs are there in which it lies? So you have this vector, so Si, this is an i here. I just don't understand. We have a definition of multi-section function before that. The only difference now is that we require the things that we locally represented by the s c plus. So locally in a chart or a uniformizer. So first of all, the multi-section functions were in each fiber, there were a finite number of vectors with which having weights adding up to 1. So now, if I put the chart in, then I have this, of course, on the image, but these different things should lie on graphs of an s c plus section. Is that clear? So if you put your hand in and you see in the fiber the different points, they line up as lying on a graph of s c plus sections. So now we want to, and this section should be sort of compatible with the group action, and that's sort of the compatibility. So there is an action of our automorphism group on the set i. And you have the orbits under the conjugation by this thing lying in there. So let me first say certain properties what it can do with this guy. So you can build this sum here, which is sort of a convolution. And this is smooth. So if each of those guys is an s c smooth, or s c plus, so I forgot the plus here. So if this is s c plus, then this is s c plus. Because what is the representation? What is the local section structure of this thing? You just have the section structure s i for 1, and t i for the t j for the other, and you just take all possible additive things and just take as a weight to take 1 over the number of the indices here times the indices of the other. You have even here lambda 1 and lambda 2 sitting on different bundles? No, no, they are there for our bundle e. Why do you call this a sum in that order? Well, it's a convolution. That's better. Because on the section structure, you take basically all possible things how you can add up things. So if locally, the first one is given by s i, and the other by s i prime prime, and this in index set i, index i prime, then the sum is given locally by taking all these combinations here. But where the index set is actually i cross i prime. So plus because of that. So then this one here just replaces the sections locally by t times the section. So of course, it's 1 over t on the other side. Otherwise, so this is a smooth family. So this is OK. So if t is 0, you just get lambda 0. So I put the lambda 0 up here. Otherwise this. What is lambda 0? So 0 should be up here. It's a section which just is a weight 1 on the 0 section. I mean, this is a smooth procedure. So this one here, what does that mean? The indicator function here, it just means that the local structure is t times s i. And then if t goes to 0, then you get the 0 section. So lambda 0 of e is 0, unless e is 0, in which case it's 1. Yeah. And all that works exactly because the total suburb weights over any fiber. It's 1. It's 1. So that is actually a smooth family if you change t. So then if you have an s c smooth functor into r, so it's clear what that means. That means if you compose it with a uniform as s c smooth, then you can put that in front of it here. So you can use partitions of unity to cut off such multi-sections smoothly. Then this makes sense as long as locally, near a point in a uniformizer, you just have that the family is locally finite. So if you take a point and then you have only finally many non-zero vectors there, and you just add them all up in this way. So then this is also, again, a good section. So these are important facts for actually constructing perturbations. This allows you to construct things locally and then just add things up. And then a good fact is if I give you any, so it should be a smooth object and a smooth vector, then there exists actually such a lambda of s c plus multi-section functor where lambda of e is positive. So I'm going to show you this, how to prove this. So now, for example, locally, remember when Katrin was describing the transversality result and perturbation result. So if you have a freterm, or even in finite dimensions, if you have a section of a vector bundle and you want to make it transverse by small perturbation, what you take is you add to it some ti times the perturbation to fill up the core kernel. Then you solve this with respect to the additional parameter. You get a manifold. And then you project onto the parameters you added and take a regular value. And for every regular value, that is a good perturbation. So now, what do we have locally? Locally, what we want to achieve locally is we want to break the symmetry. That is generally what we have to do to achieve transversality. Of course, sometimes we can avoid this. Then we take the orbit of this perturbation, which is also transversal. And then we have maybe some more perturbations. But for each of these problems, it's precisely that argument. So when you look at this, you just have to make sure that one of the local problems is transversal. You get a set of full measure for the perturbation. You take intersection. And then you take some of the values there. So that's the only additional complication. But otherwise, you use precisely this thing. So what that means is that actually rather than taking local sections, you construct local multisections and take that sum. And each local multisection depends on a few real parameters, t. You take the right sum so that it fills up the core kernel. And then you get sort of this branched manifold. And then you have a projection on t. And then for each piece of manifold, you require that that projection is regular. Now, these are countable conditions. So you find regular things. So that's the only difference. So it's a straightforward thing coming from there. Then you can even go further.  You can actually tilt the kernel into the manifold and make it transversal. But that is also a little bit. So for this, you only have to construct a section which takes enough values to fill up the core kernel. Here, you have to think about that it has a specific, it might have, say, value 0 there, but it should have a particular derivative which, together with the linearized coefficient operator, has a certain thing. But that's the same problem, like in finite dimensions. So there's nothing new. I mean, it's, of course, not so surprising because Fritam's theory is locally a finite dimensional problem times something you don't have to care about. And for that finite dimension thing, these perturbations are as rich as in the finite dimension theory. OK. So let me just explain you how I construct a section. So I want to construct at alpha in the neighborhood. So I want to construct a section which has a certain property at an object alpha. Didn't you just explain to us how you construct a section? So now I do it. So no. Let me, on some level, now I give you on a precise level and still I have 10 minutes. OK. Five. OK, good. So you just have to put something on the table and then you get a good answer. So I want to construct something at the smooth object alpha with a given smooth vector over it. So what do I do? So I take such a uniformizer. So here's a picture. The underlying thing, so this is an orbit space that would be psi, the image of this one, if I pass to orbit space, would be this red stuff. So there's a point somewhere here which corresponds to the object. I take a neighborhood u there. So now, what do I need? So in Hilbert spaces, you always have smooth bump functions, but on certain Banach manifolds as well. But unfortunately, on certain Banach manifolds or Banach spaces where you don't. So I think C alpha, I think, does not have smooth bump functions. But so there was a study 30 years ago. People were interested in it. So there's a lot of literature on which Banach spaces have smooth bump functions. But SC smooth bump functions are a little bit more there because it's a vehicle requirement. But in any case, on Hilbert spaces where we're all set up, you don't have to worry. So here is my Z set O. And here is sort of a neighborhood. So now just here, say this is a point which goes to the object alpha. Now you just construct with a bump function this object here. So you just take a bump function, which at this object corresponding to alpha takes this value E0, which corresponds to the given thing in your fiber. You take the support in the small set. And then you rotate it around by the action. So now you have a local thing. So this is now on the image of psi bar of k. So then we define it by this formula, which is precisely the definition. And now we extend it to the whole category. Namely, if you have any vector, then if there is no more fism, which brings the underlying base point into the image of O, you just say it's a multi-section which has one on the zero section. And if you actually can reach this patch here, then you just define it by this, by what you reach. And that is a smooth function. Because if I go from one to the other, I have this smooth transition. So if I have a local section structure here, I just can move it over there. So that is a local construction. So now you can take a finite number of those parameterized by p to fill up the core kernel at alpha. Then the Frayton property actually guarantees that it nearby is also the case. You do this at different spots, covering the compact solution space. And then you have enough things to do precisely what Katrin says. Said some time ago. OK? So now I generously got five minutes. And I only use three of them. So I stop here. Otherwise this, huh. Are there any questions for our speaker? Can you go back just to the last slide, please? My computer is very entertaining. OK. Can you go through this again and tell me again which spaces are which in this picture? OK. So in this argument, actually, it wasn't so apparent. It's actually important that the underlying space is actually at least power compact. So I take a look at psi of O. It takes the associated isomorphism class, which is sort of this red stuff. Then that is here, yeah? So then in this isomorphism class is the class of the original object alpha given. And I take a neighborhood around this. And then this thing is that, so this is a metrizable space so it's normal. So I can actually find a small neighborhood around it that the closure in the whole space is still contained in it. That's important because otherwise that thing will actually not become even continuous. So then you take the preimage of U in O, which is this blue thing. So this red stuff is O. And this is a preimage of this U. So now in this one, you take a bump function which has support in this. And here somewhere is a point which corresponds to the object alpha which lies here. It's the object alpha, this is a category. There's the object alpha somewhere here. It comes from a point which lies in the blue region. So over the blue region, there's this point representing alpha, say Q0. And over this alpha, there was this fiber. There was a vector E, which corresponds to some vector lying over this Q0 in the bundle K. So now you just take a bump function, which is one in the neighborhood or at this point here, times this vector and extend it. So I haven't talked about extension results, but they are on the polyfoil level quite easy. So you can ask me maybe on Friday and I can show you how to construct them there. So anyway, so there is a section with support in the blue thing. And now you want to construct a functor. So what I do is I transport this section around by conjugation. And then I define, then I get as many, of course, some of, it could be if you have a symmetric section, then some of them are the same. But that doesn't matter. This is your index set. And you give each of them the weight one over the order of the group. One over the other. The definition of FG, suppose this Q0 that you had isotropy, how many you can still. Yeah, yeah, yeah. So this is actually coming from the isotropy group of this element. So if it has a large isotropy, then in general, I would, for example, construct something like this F, which is achieved some transversality nearby. Then since the bar is a functor, then if I conjugate the functor, it doesn't change. But this then, so then the perturbation by this one is a conjugation of the perturbed thing by G. So it's also transversal, at least in the region by 1. So moving this around doesn't destroy transversality. And then, of course, in general, you might see some other sections coming from the overlaps or so. But for the construction, that is sort of the minimalistic thing you have to do. And then you give each of them the weight one over the number of elements in the group. So that's precisely the requirement. So that means now on that slice, when you look at what happens, so the section is now defined on that same slice here. So now I have to extend it. So then I get an object here, here with some vector. If there is no morphism from this one, which reaches a point which lies in here. So if I cannot, so either I can reach this slice or not. If I can reach this slice by a morphism, if I cannot reach this slice, I put the weight on the 0 section, 1. And if I can reach this slice by a morphism, then I define it like this. So I look at what point is there. And I give it the same value. And that is an SC plus section now defined globally. So it's not very difficult. It's just really always the local constructions. And the language is sort of so high level that you basically always see everything on the nose. You don't have to go in complicated coordinates and say what it actually means, what you're doing. So that makes it, of course, it could have been equivalent theory. But the language level is much easier if you stay on that high level. In particular, since on that high level, every information is there. And there are abstract results who produce whatever you want. OK, so good? So I will ask a question that maybe I kind of know yes or no, I'm not sure. So what is the reason that we need to go to M polyvolts instead of working with retracts as a local model for this kind of category? Well, you could put retracts there if you want. You can also put M right now. I mean, this is a little bit larger. So this M polyvolts is locally modeled on retract. So rather than taking just something which has one chart by retract, then it could replace this one chart by the actual retract. I do this. It also has some advantage when I do cover, when I discuss the coverings. So at some point, of course, in the whole thing which has been suppressed, I have to give a definition what is actually a covering functor in the whole thing. And then I have to give a local model for this. Then on the top, generally, I have more points. So it's actually more union of retracts going down to the other thing. So the things are easier. But one could, but I think it would be unnecessarily restrictive to say that. I mean, it's a fair question. I mean, it's like, I mean, the equivalent would be in many folds. And I define a manifold as something which locally has charged isomorphic to an open set in Rn. And I just define this is a manifold because it's locally homomorphic to some manifold, to some smooth manifold with a transition smooth manifold. Yeah, so the manifold is like locally a manifold. No, no, no, no, no, no. Right, that's what this is. Also, one side is a category, and the other is actually some well-defined, smooth kind of object.


Hey there friends. Thanks for checking in at the range now and today. We're looking at my star BM 9 Very similar to a Browning high power. I consider this to a degree a High-power subcompact now it does have a four-inch barrel all steel frame a single-action trigger with an eight round magazine that does have a magazine disconnect But it mimics many of the features of The high-power what makes this ultra attractive is how much it was selling for it was around 230 bucks and I'm impressed with it, and I've been impressed with it now. It does have a five pound trigger Obviously single-action only does have a thumb safety Right there that is engaged because I have a round in the chamber But it also has a magazine disconnect which does not allow me to fire that round the thumb safety Already engaged, but it's meant to be carried cocked in lock so a total of nine rounds Blacked out sights all across, but I have found it to be accurate and Reliable for an incredible price that many people were picking this up for and I've I've seen it even around $200 now. This is a military surplus gun. It was in the hands of Law enforcement and military and it's a maiden Spain So in Spain no civilians use it and this was made in 79 So let me stop yapping here. We'll take a few shots and see how I do with this star BM 9 Yeah, just as I remember it impressive gun For the money and even if it was more money, it would still be an impressive handgun Now when I first got it I didn't know the year that it was produced But it it's stamped right there on the trigger guard right on the top side there. It says 79 So this was actually made in 79. This one is an excellent condition It really is you look at some of them. There's a lot of slide where You know chips and scratches throughout this one not so much. So I'm really happy to have it Another thing that makes it similar to the high power is the disassemble it disassembles the exact same way And it does have an all steel guide rod and a captured Recoil spring along with the magazine disconnected. I don't find too attractive, but that does mimic The high power So let me go ahead and we'll try some one-handed shooting here And that's it these were typically coming with One mag But being that it's a surplus gun It is out there It's a good looking gun or I shall say I should say mags are out there From what I could tell single stack nine millimeter handgun The star bm9 has a barrel bushing there that needs to be disassembled. I'm not going to take the gun down entirely But I will show you once you dis or once you engage the thumb safety The pin just drops out and then you Take it off and you can see what you're looking at Does have a link like a 1911 and the four inch barrel Let me take a few shots And then I will turn the camera around. I'm using cheap reloads that I picked up Probably a year ago So we'll see how It looks like I have a double feed So I'll lock the slide back And I will Let that round or case drop out And I'll go ahead and turn the camera around Picked up a explorer recently 2015 Explorer I'm not big on Buying new vehicles I Disengaged the thumb safety. Let's see how And I have another double feed We're going to try this again with some factory ammo that I that I have Get out of there I have I have one round left But now I have fjoki 115 grain ball ammo Whoa boy, so maybe it wasn't the rounds that I had There goes one out of The mag that's okay Thumb safety, here we go All right, so I have a little issue here I was talking about how much I enjoy the star and I do But I need to clean it up because right now it is not ejecting Properly on we'll try it one more time See what's going on here so There it is my star bm 9 made in 1979 I do enjoy it. I have to say Probably needs a good bath And I'll have this out again and see if it can perform a little better But it has not experienced or I have not experienced any issues until today Go figure you turn the camera on and this is what happened, but It's real And I still love the gun If you like videos like this, please subscribe and share I always appreciate thumbs up button. Thanks for watching and you guys be safe


This paper proposes a novel myoelectric control scheme that supports gesture recognition and muscle force estimation. It uses a multi-task learning approach to reduce the influence of force variation on the accuracy of gesture recognition. The proposed method achieves a 27% reduction in gesture classification error and a 0.1479 RMSD force estimation accuracy. Additionally, the proposed framework is applicable in different numbers of electrodes and has a low response time delay of about 28.22 milliseconds on average. This makes it suitable for use in myoelectric prostheses and exoskeletons. This article is authored by Ruanchen Hu, Xiang Chen, Hao Yanzhang, and others.


On the mats, a class A matchup between Black Duck CLB and Fauston Bagley wrestling at 126 Christian Praetor against Christian Schmidt. Praetor with a take down, he wins the battle of the Christians by decision four to three. For Fauston Bagley, Carter Sorensen at 145 able to control the match gets the pin for the win. It was a tight one bonus points to play a big part in deciding this one. Landing Gross secures the pin here for Black Duck Castley Bina and they get the win by just three points. If you enjoyed this segment of Lakeland news, please consider making a tax deductible contribution to Lakeland PBS.


Ar eich ddadu gynesai, roedden nhw'n mynd i ddechrau sianfod 1, Ff5, 0,5, 2 yn y mae'r gael Ross i ddweud Mae chi'n mynd i'r ffyrdd yng Nghymru i'r mewn iawn i'n gymryd yn credu'r lleolau i gynhyrchu i'r ddiolchol. Rhywbeth yn leiriadau o'r ddwaith gyffredinol os ygrydigaeth aelodol i'r leirio iherwydd ei tyw'r cyffredinol. Roedd hynny mynd i ddod chargesion teulu Tiwrn i'r ddweud, ac mae'n rhaid i'r ddweud yn cael sicrhau ond ei amser. Roedd angen i'r ddweud i'r跆ונהeth aelodol i'r brosesion i amser Work and study—some were for people to come home to. Although retaining our young people has been difficult in the past, new strategies offering the conditions they need to thrive in the Highlands and Islands are starting to have effect. In November, Highlands and Islands Enterprise published the results of their latest survey, Young People in the Highlands and Islands maximising opportunity. That study had two main aims. From the document itself, it says to provide an overview of the evolving attitudes and aspirations of young people in the Highlands and Islands, how those have changed since 2015, and to identify gaps within the current provision of education, training and employment, and consider how opportunities can be maximised for young people. The population of the Highlands and Islands was approximately 470,000 in 2016, according to the study. However, those aged 15 to 30 only comprise 17 per cent of the population compared to 21 per cent across Scotland as a whole. This deficit is mostly acutely felt in the Outer Hebrides, Lochaber, Skye, Wester Ross, Argyll and our Islands. The first stat that jumps out of the report tells us that 55 per cent of young people are committed to staying in the region. That is up from 43 per cent in 2015. There is also evidence of an increase in potential returners, those with an interest in an attraction to living in the region that are from the Highlands and Islands but living elsewhere. 69 per cent of those who do stay feel that they are lucky to be able to work or study locally. That is up from 62 per cent in 2015. 60 per cent of young people think that the Highlands and Islands has a good educational offering. That is up from 56 per cent in 2015. There is now less of a perception that the young people that stay lack ambition due to the range of further and higher education available. There is no doubt that the University of the Highlands and Islands is having a positive effect on keeping our young people in the area and attracting young people from throughout the UK and the world to study and stay in our area. North Highland College, which is a partner in the UHI, is one of Scotland's top colleges for positive student destinations, boasting rates of 90 per cent, a statistic that I know everyone involved is extremely proud of. To add to that, North Highland College and West Highland College student leavers have the highest progression rates into work at 40 per cent, well above the national average. This is North Highland College's third year of delivering foundation apprenticeships, a very welcome endeavour supported by the European social fund and enjoyed by young people who have taken the opportunity to learn this way throughout the area. This year, the subjects that young people can study have been increased by introducing business skills, IT, hardware and system support and engineering is awaiting approval. These new subjects send a powerful message to young people that the Highlands are not only open for business, we are open for innovation and success too. At this point, I would like to draw the chamber's attention to my register of interest as an adviser to the board of North Highland College. 87 per cent of young people are proud to be associated with their community, and this is up from 78 per cent in 2015. 64 per cent want to work in the region, up from 44 per cent. I mentioned foundation apprenticeships before, and there is no doubt that the developing the young workforce initiative has been pivotal in pulling together schools, colleges and the public and private sector and encouraged them to work together in a way that they never did before. The Cadence and Sutherland group is facilitated by Cadence Chamber of Commerce and has provided a wide range of employment and career development activities and support, which has been given and led to an increased number of work placements, employability workshops and events, employer-led mock interviews and STEM opportunities. The development of STEM has been further boosted through support from the Highlands City region deal. This is £315 million worth of funding, consisting of £135 million from the Scottish Government, £127 million from the Highlands Council and its partners and £53 million from the UK Government. Those are good stats, but, just like real life, not everything goes up and not everything is good news, so for balance. The numbers participating in their communities is down by 9 per cent, and that statistic surprised me because I attended a youth volunteer award ceremony in Wick in November, and there were loads of young people there. All active in the community, all extremely proud to be receiving their awards, and I know that our islands also have a strong cohort of young volunteers having met some of them in the Parliament last year. Another 38 per cent say that a lack of local opportunities is a barrier to achieving employment goals, and while 71 per cent were happy with the choice of subjects they can study, 46 per cent felt that the range of subjects available will limit their post-school options. That rises to 63 per cent in fragile areas. There is no doubt that there can be issues with school subject choices in rural areas, and one of the main barriers to that is teacher recruitment. I raised the subject of probationary teachers being allocated earlier in the school year to allow more flexibility with subject choices and timetables, and the education secretary assured me that that was something that he would look into. Half of young people agree that their community is a place where it is okay to be different, and that is the same as 2015. However, research by LGBT Youth Scotland shows that, in rural areas, 81 per cent of LGBT young people experience at least one form of bullying in education, and that 9 per cent of lesbian, gay or bisexual and 27 per cent of transgender young people left education because of that. That is only one of the reasons that an inclusive approach to education is even more important in the Highlands and Islands. In conclusion, that is a welcome report that shows more and more young people are realising and taking advantage of the opportunities offered to them to enable them to live, study and work in the Highlands and Islands. We have always had an issue with depopulation, but we should never completely halt this outward migration and neither should we seek to. They will always want to broaden their horizons, whether that is in other parts of Scotland or in other parts of the world. What that study shows is that we are empowering young people with more choice, and the days that have been forced away from your community because there are fewer prospects of studying and employment are becoming a thing of the past. We need to build on that work, ensure that those figures continue to rise, especially in our fragile communities, in areas such as provision and access to opportunities, education, training and engagement with arts, leisure and culture activities. We also need to make sure that there is adequate housing for those who choose to stay or want to come back. Two or three new houses could make a huge difference in places such as Loch Carin, Kinloch Berfey or Apple Cross. I will leave you with a quote from the report. The Highlands and Islands is such a wonderful place and I always love telling people that that is where I am from, me too. We move to the open debate. Speeches of four minutes, please. Jamie Halcro Johnston, followed by Rhoda Grant. Thank you very much, Deputy Presiding Officer, and I welcome this debate. I congratulate Gail Ross on bringing it to the chamber. As the Highlands and Islands MSP, I am perhaps biased. I think that our region is one of the most diverse, most beautiful and friendliest in Scotland. The region has drawn many in, charmed by our landscape and sense of community, and it is undoubtedly a spectacular place to live. It is certainly not without its challenges nor are they new. For generations, many people born in the Highlands and Islands have looked further afield for opportunities. They have felt that they cannot continue their education or progress in their chosen career and remain in the land where they were born and raised. There is undoubtedly a sense of this in rural communities across our country. Young people will often move away for university or work. What cannot be denied is the extent to which this is more pronounced than the Highlands and Islands. As high survey indicates, people aged 15 to 30 are 17 per cent of the population of our region, as opposed to 21 per cent across Scotland as a whole. There is a divide here too. In some of the Western Isles and West Highlands, the figure is even lower yet. One of the most pronounced impacts of rural living early on in a person's life is education. The impact of subject choices is significant. Even where we see innovation, such as the extension of foundation apprenticeships, we still see choice narrow in more remote communities. Some may suggest that this is natural. That is a consequence of the choice to live in rural communities. It is simply what you would expect where a lower density of population exists. That will certainly be true in some aspects of life, but there is also a stronger theory that Government exists to expand opportunity to share prosperity and to provide services that are similar across its population. As far back as we can reasonably analyse, Scotland has always had a higher public spending than the UK average. Today we see that delivered through the Barnett formula. The chief justification for that disparity is that we have these geographical and demographic challenges, where we might ask, is that additional expenditure going if not on creating a level playing field in public services within Scotland itself? There will always be a pool to a local community being close to friends and family, the sense of home. For many that pool will not be loosened by the odd difficulty. However, for these communities, the challenges of staying can be overwhelming. As a young person leaves school and looks to their future, they might be able to accept a narrower level of choice in their education, but they might consider other possibilities too. Can they get public transport to college, university or training? Can they be sure that they will be able to pick up a part-time job to supplement their income? Those are areas where Government can take a greater lead. Preferably Government at a local level, Government that is responsive to the particular needs of remote and rural communities. My party has often pushed for localism. Decentralisation will be part of any solution, and yet our established organs of local government are struggling more than ever. I mentioned foundation apprenticeships, but I should also touch on some of the opportunities presented by graduate apprenticeships too. We should also consider how other elements of our education structures, like articulation from college to university, can be valuable in our region. If we can properly adapt those educational routes to our difficult geography, there not only are we increasing life chances but also providing a route for those reluctant leavers to stay. Those are just a few of the areas that could be mentioned. We could look at infrastructure from roads to broadband or the need to encourage entrepreneurship. I do not feel that those approaches are not understood by policy makers. They are. However, choices are made or not made and change can be glacial. The challenges of not making progress are stark. The future of young people in the Highlands and Islands is the future of our economy, of our public services and of the opportunities for generations to come. Rhoda Grant, followed by Jenny Gorruth. Thank you, Presiding Officer. I would like to congratulate Gail Ross on bringing forward this debate. I would also like to express my support to Highlands and Islands Enterprise for surveying young people. It is the benefit of having an enterprise company that has a social as well as an economic remit. It is heartening to know that so many young people want to live and work in the Highlands and Islands. That said, I am not against people spreading their wings and seeing a bit of the world. The Highlands and Islands are famous for sending people around the globe. Unfortunately, much of the outward migration was and is not choice but due to a lack of education and career opportunities in the region. We need to make sure that the decision to stay or leave is a real choice, enabling people to stay without compromising their life chances and career. Gail Ross said that one of the biggest developments that has stemmed outward migration is the development of the University of the Highlands and Islands. It offers excellence in many of its institutions. She spoke about North Highland College but there are many others as well. They provide normal courses that you would expect in a further education college but there is the excellence that comes from a higher education facility. It is often the combination of both that provides opportunities that are not available in other institutions. However, it is still the case that young people have to move away when their chosen course is not available locally. Otherwise, they are forced to compromise which is shown clearly by the statistics that show lower educational qualifications in those that opt to stay compared with the national figures for qualifications. Those figures also disguise movement in the area with people having to move a distance from home just to access further and higher education. That leads to internal migration by young people to the more urban areas within the Highlands and Islands. The study also showed that young people wanted to stay in the more fragile areas due to the stronger sense of community but, again, were less likely to be able to do that for career and educational reasons. A recent study in the US showed that more young people were either returning or joining the community there. There was an increase in children in primary schools because of that. That trend is due to a number of reasons, but what is crucially important to young people is work, housing and access to services. If we can provide that, we can halt depopulation and bring new life to the communities that would otherwise be dying. Another issue that Gail Ross touched on in her speech is the issue of diversity. The LGBT Youth Scotland 2017 survey showed that young LGBT people living in rural areas were more likely to have poor mental health because of fewer safe spaces for socialising. Gail Ross talked about bullying and that is totally unacceptable. While rural areas can be the best places in the world to live, no one is anonymous, it makes coming out difficult because you cannot do that gradually due to the lack of privacy. That lack of privacy makes it difficult for people with poor mental health to seek help when added to that. There is the perception and stigma. Therefore, those communities can be the most supportive. Everyone knows everyone else so that you are less likely to be looked upon as a one-dimensional person but for all your attributes. However, we all instinctively want to conform and not stand out from the crowd, so anything that appears to make you stand out can be much more difficult to deal with. The Highlands and Islands is a wonderful place to live. While I had to move, for all the reasons highlighted, where I stayed when I was a child, my first home will always be home to me. I never chose to leave and I don't want any other young person to be forced to make that choice. I start by congratulating my colleague and friend, Gail Ross MSP, on securing this evening's members' debate. Today's motion is focused on the Highlands, so I would like to begin by declaring an interest as proud descendant of a Highlander. My granny, Flora McRae, was the daughter of a crofter. My great-granddad, Donald, was also the postman for Muiravord and Muir of Taradil, where the family croft was and still stands today. For my granny and for so many of her generation, leaving the Highlands was almost a necessity. Indeed, the depopulation of that part of the country is remembered as one of the saddest times in our country's history. That is why the Highland Youth Survey is important, because it tells us the story of the next generation's ambitions and aspirations for Highland Scotland. Gail Ross' motion today rightly points to the important part that young people play in supporting thriving communities. Indeed, 70 per cent of participants in the survey considered that those who stayed in the Highlands were lucky to be able to work or study locally. I think that that speaks volumes about the opportunities now present for young people in the Highlands. Members will remember that last year marked the year of young people, which focused on inspiring the country through our young people, celebrating their achievements, valuing their contributions to communities and on creating new opportunities. Although there is no direct Fife equivalent, the Scottish Government commissioned a young people in Scotland survey and asked a number of questions to young people about their ability to make their views heard and on decisions that affect their lives. When asked about adults in general, more than half of young people surveyed and agreed that adults were good at listening to their views and that adults were good at taking their views into account when taking decisions that affected them. Young people need to have their views listened to, but they also need to be part of that decision making process. The Highlands use survey is there for encouraging, as it shows that increasing numbers of young people want to live and work in the Highlands and Islands with the proportion of committed stares increasing from 36 per cent in 2015. Deep Population, however, is not limited to the Highlands, and this week I've been lucky enough to be shadowed by an S6 pupil, Jennifer Smith, from Ock Muti High School in Glenrothes. Ahead of today's debate, I asked if she would stay in Fife when she finishes school this summer. She told me, when I leave school, I want to go to Edinburgh to live and study, and then maybe to London and maybe one day I'll come home to Fife. I completely understand Jennifer's motivations for doing so. I also grew up in Fife, and I left to go to the big smoke in Glasgow. Gail Ross did likewise. We need to balance the needs of our rural communities, like the Highlands and like Fife, with the needs of young people to explore our cities and to experience different places. We should not place a limitation on their ambitions, but rather seek to empower our young people to have a real voice in decision making from the outset. Whether that is through the school council or modern studies, 2019 is no longer the year of young people, and perhaps last year's real tests will be whether or not we continue to engage young people in our work as parliamentarians. The Highlands use survey is certainly an invaluable tool for measuring societal shifts in that part of Scotland. Reflecting the shift in responding to the survey in the press and journal, historian Jim Hunter wrote about his experiences of growing up in the Highlands last month. He said, so prevalent was the conviction that success could only be achieved elsewhere, that someone still at home when in their early twenties was likely to be seen as, by definition, a failure. Contrast his view with the young person that Gail Ross quoted earlier, who said, The Highlands and Islands is such a wonderful place and I always love telling people that is where I'm from. For Jim Hunter, what's made the difference is what he describes as a transformation in the attitudes in young people is the work done by the Highlands and Islands Enterprise through the backing of successive governments. Perhaps today I should really be calling for a Kingdom of Fife Enterprise. On a serious note to the minister, the Highlands use survey provides us with invaluable information to measure active shift. With that in mind, I will now be writing to Fife Council to ask if we can seek to learn from the survey by listening to the views of young people across the kingdom. John Finnie, to be followed by Donald Cameron. Thank you, Presiding Officer. Like others, can I start by thanking Gail Ross for bringing this debate? I think that it's a very interesting debate and congratulating him for bringing it here. An early declaration is a man lured by the bright lights of Fort William. Many years ago I've spent all but a handful of years living, born and bred in the Highlands and living there. I'm delighted that one of the early findings is that there's an increased number of young people who are committed to staying in the Highlands. I'll also absolutely go along with everyone who said, let's not be negative about people who leave. I think that it's very important that we have a rich mix. Part of that rich mix comes about because people do leave. After all, it's many people from the Highlands Islands who have gone throughout the planet doing lots of good things, often returning or often continuing to contribute there. I want to talk about something that, perhaps a small negative of the report, I was disappointed not to find that there was no mention of Brexit and there was no mention of Europe, because part of that rich mix and the big change in my time in the Highlands is the number of people from all sorts of different cultures who are contributing to making it the vibrant place it is now. I think that Brexit looms over everything. The Erasmus programme and Dr Winifred Dwing, which was the MEP that was instrumental in bringing that in and the benefits that there has been from that. The reality is that, as things stand, there is the potential that people are going to be denied some of these opportunities. The comparator for this survey was the period 2015, the previous survey to now. I think that the change is that there are the raw statistics and I've just remembered another very minor negative. Although I'm very happy to go along with my colleague Gail Ross's comment about the Northern Innovation Hub and the Science Skills Academy, it does disappoint me that £119 million, which could be doing something very constructive, is going to construct roads in the Inverness area. Roads are going to take people 12 seconds between locations at peak time. Things are about priorities and the priorities for a lot of people will be to understand the needs in any community. I think that the comments about youth participation are heartening the level of participation that there is. I think that it's very important that there are more parliamentarians that look like Gail Ross and less that look like John Finnie. I think that there are far too many men in suits and that is the very position that puts people off. I think that things are changing and they are changing for the better. I thought that it was significant that the outward migration was particularly concentrated in 19 groups. On the question of education, which has been repeated, and indeed my colleague Gail Ross was with me at Androda Grant as well, when we visited eSchool, it was an important development in the Outer Hebrids, which catered for remote learning. I think that significantly the opportunity to use video technology to allow teachers to deliver classes from their homes. When we were there, I think that it was local authorities in the north-east of Scotland for which some of the services were being provided. We need to embrace the technology. The whole collegiate system of UHI, which has been a great boost, was that small groups of people together, who by their very nature are going to be still in the communities, are contributing to the whole. I think that that's an opportunity. Many rich cultures in the highlands, the north culture in the north, there are elements with the Gaelic, particularly in Skye. In a situation in which, in my day, Gaelic was done at Calys by largely old people, and now the innovation in which there are people making very good livelihoods as a result of taking highlands and highlands culture throughout the planet, and that has brought young people to it. I think that there is a lot to be positive about, and I thank you all for being in the motion. The last of the open debate contributions is from Donald Cameron. Thank you, Deputy Presiding Officer, and I'm delighted to follow John Finnie, who of course spent some of his childhood in my local village. As he's one of our younger MSPs, I hope he too may one day return home at some point. I'd like to thank Gael Ross for bringing this issue to debate this evening. I feel that it's been a very worthwhile discussion. In what I think has been a start to this year, which we've seen a lot of pessimism and negativity for all sorts of reasons, it's very pleasing to take part in such a positive debate. As the Highlands and Islands MSP, like others, I'm all too aware of the issue of depopulation and its major impact on rural and remote communities. I'm just taking a few statistics from the HIE report. The report says that in the 15 to 30 age group, 17 per cent of that population age group is a deficit compared to 21 per cent in Scotland as a whole. The report also says that the population in Highlands and Islands is predicted to be stable to 2041, but the 15 to 30 age group is expected, nevertheless, to decrease by 15 per cent, which is a significant number. In our Garland Bute, for example, there's long been a prediction that there'll be a decline in the working age population of more than a third. Likewise in the Western Isles, it's estimated that their working age population will decline by 27 per cent, so there is, I think, a real problem. It's not just a recent phenomenon. The history of the Highlands and Islands in the last 250 years is one that tells of more people leaving than arriving. However, with all that said, it's abundantly clear that there are many signs of improving attitudes to living in the Highlands and Islands and that more young people see a future for themselves in the region, and that is something that we all rightly welcome. I think that one speaker spoke, I think that it was Rhoda Grant who spoke about the Uists, because last year there was a report that highlighted that, unlike other island communities, the Uists appeared to be bucking the trend, and there's a birth rate rise of 67 per cent in the last decade, which is very marked. There was an article in the Herald last year which described that, young returners were helping to reverse the depopulation trend, and there was a new generation of young people who were keen to lay down their roots in these communities. Many of the reasons that were cited were things such as a feeling of greater safety for bringing up a family, the landscape, the untapped market for business enterprises, and it's clear, I think, from the HIE report that there are many more opportunities for young people and even just a few years ago, the report notes that the Inverness and Highland city deal is clearly going to have an impact. It mentions projects such as the Northern Innovation Hub and other rural growth deals, especially. Other speakers, including John Finnie, mentioned eSchool, which I visited in Stornoway last year, using technology to enable, in this instance, school children, but could easily be extended to other areas. There's been investment from STEM industries in the region, which I think also gives hope to more long-term economic regeneration. Long-standing businesses and organisations have committed to a long-term future in the region. We've had a BASF farmer in Kalanish in Lewis and MG Alibar in Stornoway will ensure that there remains a demand for skilled jobs in the area. One thing that struck me particularly from the HIE report is that 87 per cent of young people think that, although life in the Highlands requires making compromises, we accept that to be true, that they still believe that there are growing opportunities for young people. Lastly, I was struck by Gareth's comment about housing, because I think that that is absolutely right to recognise that this is an issue. Likewise, there are issues with ferry connections. All the issues that we know and talk about, such as the issues that represent the Highlands Islands, Paul Roy's broadband, et cetera. All that said, there is so much impetus for young people to stay and work in the region. I am delighted to take part in this debate, because if we do get it right now and ensure that we continue the economic regeneration of the Highlands and Islands, then it will only serve to encourage more young people to live and work and make their life there. I now invite Jamie Hepburn to respond to the debate for around seven minutes. I join with others in thanking Gail Ross for bringing forward her motion for debate this evening. I thank members for their contributions. As Donald Cameron sent out, it is quite welcome to be able to take part in a positive debate. It is fantastic to see progress in supporting more young people to live and work in the Highlands and Islands and to see their changed attitudes in terms of their desire to do so. Jamie Halcro Johnston said that he might be biased in celebrating the area that he represents. Of course, he should not be, because it is an outstanding part of the world. Again, that is demonstrated by the trends that we debate this evening. Of course, it is very welcome that Jenny Gilruth and others place this debate in its proper historic context. Like Jenny Gilruth, most of us in other parts of Scotland have some part of our historic family experience rooted in the depopulation of the Highlands and Islands. It is welcome to see that situation turning around. I will place a lot of my comments this evening in the context of developing young workforce, which Gail Ross's motion mentions. I believe in making a positive difference to the length and breadth of the country, but certainly in the Highlands and Islands. I want to talk about the role of Highlands and Islands Enterprise. Clearly, they have had a role in relation to the very welcome report that they have pulled together. We should thank them for that work, but they are also integral in creating the right conditions for the Highlands and Islands to thrive. They have been involved in supporting the right conditions for Scotland's employers and their progress in implementing the city-region deal, delivering projects such as the Northern Innovation Hub and the Science Skills Academy. We should commend the work of HIE and the role of UHI in creating opportunities for young people in the region and acting as a hub for research and innovation. The Scottish Government is committed to improving outcomes for those who live in the Highlands and Islands. We are investing £135 million over 10 years through the Inverness City region deal. The deal is at our investment of £135 million, further £53 million by the UK Government and £127 million from Highland Council and other regional partners, representing some £300 million of investment in the region over 10 years, delivering a step change in transport, innovation, digital connectivity, housing skills infrastructure and tourism, improving the lives of many living, working and visiting the Highlands and Islands and being able to further the trends that we are debating this evening. Of course, it could have been £82 million more if the UK Government matched the Scottish Government investment. I am very happy that the Parliament voted this evening to reiterate its position that the UK Government should match that level of investment with the exception of the Conservative and Labour party, but it will not linger on that too much more. As the minister was responsible for developing the young workforce, I want to say a little bit about our progress in supporting schools, colleges and employers to widen choice and improve outcomes for young people. Of course, the headline achievement of that initiative, that programme, has been achieving the reduction in youth unemployment by 40 per cent and 14 levels by 2021. We achieved that ambition four years early, but we want to build on that progress. In terms of the Highlands and Islands, it is critical that we continue our long-term plan to strengthen education skills partnerships between schools, colleges and training providers and employers based on the local circumstances. I am pleased to note the positive shift in perceptions of young people in the area. I think that that is undoubtedly down to a variety of factors, of course the fact that it is an outstanding part of the world to live in, but also the efforts that schools and others are taking to redesign and refocus their curriculum offer to better meet employer and young people's needs. We spoke about the issue of young people leaving the Highlands and Islands, of course, as Gail Ross, Rodi Grant and John Finnie and others made the point that there is absolutely nothing wrong with young people leaving any part of the country that they live in if that is what they aspire to do. That is an opportunity that they want to take up, but all too often when I visit areas, any part of the country, but particularly rural communities, it is often because of a misperception that there are limited opportunities available on the young person's own doorstep. We have to break down those misperceptions because that creates the idea that a young person has to leave the area that they have grown up in. That is not always the case. I have seen that happen on more than one occasion. I have been able to, through the developing the young workforce initiative, engage with young people who have actually taken up an opportunity that is on their doorstep. Quite often it will be with an employer that has been there for generations based on their own community. That is why it is so important to have employers involved in the delivery of developing the young workforce. Of course, we have 21 industry-led regional groups, six of which are in the Highlands and Islands, all working towards the same aim, but being locally responsive to their own local economic and skills needs. I have been very happy to visit across the country and to those groups in the Highlands and Islands. I look forward to being able to return again in the future. I know that there are particular challenges in supporting education and training opportunities in Scotland's rural communities in the Highlands and Islands. Of course, there have been improvements. The University of the Highlands and Islands has made a system change in the delivery of higher education opportunities in the Highlands and Islands e-school, which was mentioned by John Finnie and Donald Cameron. It is also an organisation. I have had the privilege of visiting as well. It is a fantastic model for the delivery of education. We know that there are barriers and additional costs to the delivery of employment and training. That is one reason why, for example, we have adapted our modern apprenticeship system by the creation of a rural supplement, something of importance to the Highlands and Islands. We have funded support training providers and employers' overcome barriers that are traditionally faced in rural areas in the delivery of modern apprenticeship opportunities. Those are just some of the ways through our DIY agenda, our skills system and our strategic economic investment. We are committed to supporting young people in the Highlands and Islands and Gail Ross and other members can be assured of that on-going commitment. That concludes the debate and this meeting is closed.