3/3 Boundary, corners, strong bundles, and implicit function theorems
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Fish, Joel
In this talk, we generalize the notion of sc-retracts to include cases with boundary and corner structure. In addition, we develop the notion of a strong bundle (of which the Cauchy-Riemann operator is a section) and state an implicit function theorem for transverse Fredholm sections with compact zero-set, which guarantees the zero set of the section is a manifold with boundary and corners, with boundary/corner structure induced from the ambient M-polyfold. [Related literature: Sections 5.2, 5.3, and 6.1 of Polyfolds: A First and Second Look.
 Right, so I guess I want to start by saying that I'm trying to, I think anyway here I'm trying to try to find line a bit. In the sense that my main goal, I think jointly with Catcher in this week is to sort of set up the analytic foundations that we need to make sense of what Helen's going to talk about next week. And the thing is that I could do that completely abstractly because sort of the polyphold theory is sort of completely abstract, but it's also designed to be completely abstract which also contains useful problems. So if I did it completely abstractly, I think everyone would get lost pretty quickly and not have it be founded or you wouldn't have much to connect it to. So I have to try and bring in sort of elements of things that are hopefully familiar and pair them in just the right way I think with certain ideas coming from the polyphold theory to try and illuminate why definitions are the way they are. And so yesterday I think it might have been sort of a good talk for if it was 80 minutes with questions or 60 minutes with no questions. So I think things got a little out of hand at the end. And so I just wanted to kind of review what I had sort of hoped to say yesterday and then build off of that today. So quickly then briefly, what did I try to do yesterday? So the first thing was that well we wanted to try and parametrize a neighborhood of a nodal map and we're not dealing nothing pseudo-holomorphic here at all. We're really trying to build this big ambient space of function. So we're trying to construct some sort of parametrization or be able to write down some sort of a chart for this. And so the first observation that we made was that this pre-gluing map basically yields us the right topology in this really big ambient space in which has this nice property that in fact our compact moduli space. So even if it has, you know, you're dealing with nodal curves or your broken trajectories or whatnot, this big ambient space has a topology in such a way that your compactified moduli space is a subset and that the correct topology on your moduli space, your compactified moduli space is induced from the ambient space, right? And so this pre-gluing map then was something rather useful. We defined it sort of like this. And so we said, okay, well if this gives us our neighborhood in the right space, well maybe we can use it as a parametrization but we can't because it's infinity to one. There's all this information loss, right? So well what do you do? Well I say, okay, HWZ, they sort of, they introduced this minus-gluing map or anti-pre-gluing map, however you might like to refer to it. And so they introduced this and again it was some sort of unpleasant formula but it had this nice property that when I pair it, right, when I pair it with the plus-gluing or the pre-gluing map, that suddenly for each fixed gluing parameter A, this map here became a linear bijection. And so then the idea is to say, well how are we going to make use of that? Well that's useful because what we can do is we can define this set script O sitting inside this subspace here, right? With the property that, well, we'll define it to be all those maps so that when you anti-pre-glu or when you minus-glu you get zero. And that's good because out of, what that means is that basically when we restrict our pre-gluing map, our plus-gluing map, precisely to this set O, it ends up necessarily being a bijection or at least an injection or bijection with its image. Now that kills this problem, right? We no longer have this problem as long as we're happy to sort of restrict our attention just to O. But now O might be some weird set and so now you might say, well, you know, so what? What have we actually gained? And so there was this notion and it's an idea because in the end it's going to work out, but the idea is, well, maybe in some way this set O supports the SC calculus. And maybe there's some way to treat it as if it were something like a Bonoq space or something like a scale Bonoq space. Maybe you can define a notion of smooth maps between them. Maybe you can do that. So how might you do that? Well, you have to do a little, you know, a bit of manipulation here to rewrite the problem. And so we did that by sort of writing down this function R. Again, its domain is the same, but its image isn't in the target space anymore. It's a map from sort of your domain of pre-glued or your domain of unglued maps, basically, together with gluing parameters back to itself again. And we define it in this following way. So we take a gluing parameter and a pair of maps. We apply this box gluing map, which is a bijection. Maybe I should say this here. This was asked yesterday. This is my projection onto the first factor, or sort of zeroing out the second factor at least. And then we take this box inverse gluing. And then we made some observations, or very rapidly I tried to make some observations. I would have spent some more time with if I had the opportunity. Was it this set O that we had a definition for right up here? Well, once we take this definition of R and we see how it's defined, we see that it's precisely the set of fixed points of R. Then we also sort of check really quickly that R composed with itself is just R again. It's a projection, basically. It's a nonlinear projection. That's easy to check here, basically, because this is a bijection. You have its inverse on the other side and a projection in between. And then a quick computation. It's a little one-line proof, basically, shows you that these two properties together guarantee that this set O is, in fact, the image of R. And so then there's this theorem. And this theorem says, after giving this space, C cross E, a suitable scale Bonoche space structure, this map R is SC infinity. So I didn't really give you any justification for that. I'm not going to give you any justification for that. You can look in the literature and you can sort of see, well, why is something like this true? And if I remember correctly, the proof of this essentially boils down to, well, you write out why we have an explicit formula for R here. And when you write everything down, you get some terrible sort of equation, some awful composition of additions and products and blah, blah, blah, you break it down into all the little pieces, though. And at the end of the day, you prove each little piece is SC smooth, essentially by going to that list that is in the lecture notes for day one. And that tells you where to look in the SC calculus book or the SC calculus paper to tell you how to prove each one of these components is SC smooth. And so consequently, you rapidly build back up from this that this is, in fact, SC smooth. So now that by itself doesn't really seem to bias anything. Sure, please do. You say you have a suitable SC Bonoche space structure, so that E itself is an SC Bonoche space. Yes. So what you're going to do is rescale me a zero by some gluing parameter, right? To change the structure me a zero. No. No. So what is this suitable? So I'm thinking of C as, I mean, effectively as R2, which is a nice Bonoche space, a finite dimension, and so I can take a… So it's just a standard product. Absolutely. I mean, when you say it's a suitable SC Bonoche space. Well, well, well. E. E. Well, suitable is really applying to E, not C. C is sort of… It was given a suitable structure, but it says C. Well, E, you gave this example to me. I mean, it was… It was a suitable… …dobbing a paper in Delphina. Yes. Yes. That's not suitable. That's not a suitable… It's not a mysterious thing. No, no, no, sorry. I mean, I think, I mean, it was a… I think it ended up being sort of a slight modification. I don't know if in my lecture notes I actually defined the scale Bonoche space structure on E. I gave you something very close to it, and I told you the base topology on E. And if you understand those two, you can obviously guess what this structure needs to be on E. I don't know if I said it explicitly, which is why I'm saying suitable now. Okay. But thank you. Are there any other questions? Because I don't want to blow through this. This is actually fairly important. Yes. So what was the… What was the purpose of this averaging function that you put in the anti-glue? Right. So, yes. So, let me see. I want to see where this… So here's how I would explain it. So, O is defined to be the set of all points where the minus gluing is zero, right? And we define it precisely this way so that the restriction of the plus gluing ends up being a bijection. So, then what you do is you say, okay, well, if you look at the maps, right, so… Let's go see. So script O then is going to essentially be the set of all points which are going to parameterize your neighborhood of your sort of non-nodal maps plus some nodal maps as well. Okay. So what happens is if you think about having this nodal map when you set up sort of a base problem, you typically want that nodal point to say you want to model it so it goes to say the origin in R2N, say, for instance, right? Okay. But then you start saying, okay, what are all my nearby nodal maps? And your nearby nodal maps, some of them are going to be sort of… Some of them are going to be nodal, some are going to be non-nodal. Those which are nodal, you're going to want to allow that nodal point to move around, right? So, now you allow this… So, now you say, okay, well, I have to allow sort of nodal points to move around. This forces you to add in these sort of… this extra constant C that I had in the definition over here of these corresponding scale Bonach space. Okay. That's all fine. You can do everything with a pre-gluing. There's no problem with the pre-gluing. But in the minus-gluing, right? Well, what happens? If you don't have those… if you're not subtracting off those averaging terms, then what happens is when you look at this set here, it turns out that without those averaging terms, it necessarily must be the case that the only way for this equation to hold is if those asymptotic constants are zero and therefore the node can't move around in the image. Think about it. If you take the nodal value and now you glue and you got this long cylinder, what is the best approximation for the nodal value? It's the integral on the middle loop. The average over the middle loop. That's the approximation for the nodal value. Yeah. I think that shows up. Sure. Any other questions? All right. So this is where we left off. So remember, I guess what we're trying to do is we're trying to investigate maybe script O here supports the SC calculus, right? That's sort of the idea. And so then, I don't know how this was sort of developed in practice, but reading through the material, then you make this sort of observation. You say, well, if it's the case that I have this map R mapping essentially a scale bonox space to itself, which is SC smooth and R compose R equals R, well, then I could do this trick. I could do this trick. I could say that a function F defined from the image of R to the image of some other R in some other space is SCK, if and only if, this is my definition, if F pre-composed R is SCK as a map from this open set in a scale bonox space to this open set in a scale bonox space. See, we already have an SC calculus all defined on SC bonox spaces. We have that defined. But now the idea is, well, how about subsets? Well, if these subsets happen to live as images of these special, what we're going to call them, retractions, then it's the case that we can make this definition. O prime lives inside U prime. O prime is a subset of U prime. So there is sort of one good thing about the terrible HWZ notation is that it's fairly consistent. And so it always ends up being that O is always the image of U and U is supposed to be a subset in E and open until in, I don't know, five or 10 minutes, in which case we add in sort of this additional portion where we put in something like, I think it's called a partial cone. You can think of it as a partial quadrant in between here, in which case U is relatively open in a partial quadrant. But then this always ends up being the domain of R. And R always goes from U to U. It has to go from one set to itself, absolutely necessary as you would want for a retraction. Any further questions? OK. So I make this. Now the idea is we have this, we now have a definition for smooth functions between these weird images, these weird subsets. And then this is sort of the first bit of magic. And the second bit is that actually, that these strange subsets, and they really are, I mean, rather strange. I mean, even when you write down toy examples, I think there's a homework example on this where you can see these things might have finite dimension, but the dimension of the space might jump. And we had locally varying dimensions, locally varying co-dimensions. It might be sort of a full set of sort of, yeah, if you have an SC-Bonach space, for instance, which fibers over some other finite dimensional space, it might be sort of full dimension on one region and then sort of have infinite co-dimension on another. It's a very, it's a a priori, it's quite wild. But nevertheless, despite all the strangeness, it does have a tangent bundle and you can just sort of see what it has to be. And I have a map, R, which defines it, which maps U to U, and R compose R equals R. And so I say, well then, it must be the case of the tangent map of R maps T U to T U, and you apply the chain rule, and the chain rule says, well, T R compose T R is T R, which gives you again a map precisely of this form. And so then you just take as definition the tangent to this subset is equal to the image under this, the tangent map of R of the tangent of U. It's kind of functorial. It's sort of the only thing it could be. And I should say, it doesn't actually depend on how only on the set O. Right. So we're not, I haven't even, I haven't officially defined scale, scale smooth retractions, but this is sort of, this is sort of the first observation. And so then, so then once you've done this, I mean, these are sort of, I mean, almost sort of stupid things. I mean, you just, you're tinkering and you find this sort of stuff, right? Once you see that, you should, then you should have this idea. This is the big conclusion then, should, should be, well, let's try to build manifolds locally modeled on subsets like O. So what are all the characteristics of O that we need in order to actually build some sort of, something like a manifold locally modeled on this? And so this is, this is what I would have liked to have conveyed, I guess, in my lecture last time. And I think it's important to sort of see this story completely laid out like this. To see how, so these things here end up being, so it is the case that this is doable. I'll say this in just a second, but these things are called SC smooth retracts. These are called SC smooth retractions. And these essentially are going to form the, are going to form local models for M polyfolds. And what I want, what I'd like you to be able to see is that, is that if you start with just pre-gluing, which is something that shows up in, you know, whatever framework you want, right? I mean, any framework where you have nodal or broken elements of your modularized space, you start with that in some sort of classical analysis. Then you do this trick. You introduce this minus gluing, repackage it into this, into this weird sort of nonlinear projection and show that it's SC smooth. Then you necessarily are led to this idea of having these SC smooth retracts, which are going to provide local models for our big ambient space, right? So any questions about this outline? Yes, sir. So like this, these retracts that's really, really different from the usual setting, right? Where you have close to pre-glued positions with smooth maps, or the map is always horrible, like in several other spaces. I'm sorry, what was that? I missed. So like this is really different from the normal, like the standard setup, this feature of R. Which, the SC infinity? No, no, no, no, that like post-composition with, or like this. Pre-composition, but. Yeah, a pre-composition with it is exactly preserving the features of F1. But if you think of solar spaces and then pre-composition with smooth maps. Well, I mean, I think, right, if I have a smooth, if I have a classically CK function F, and I pre-compose by a smooth function, I mean, my chain rule should tell me that I just have, that regularity in that sense should be preserved. And so again, that's happening here. So that's sort of not so surprising. I think the surprising thing is that, the surprising thing is that, and this was commented yesterday, the surprising thing is that if you reduce this from SC infinity, or rather increase it from SC infinity to classically C infinity, then the image can't have these weird properties. It sort of necessarily must be a Bonoc manifold. And so you then reduce to sort of the standard calculus in that case. Yeah? Am I supposed to think there's a difference between locally modeled on O and locally modeled on the pair is E comma R? OK, so that's a great question. I want to get to that right now, actually. Can you repeat the question? So there's this question of, when I say locally modeled, what actually constitutes a local model? Right? And I keep saying O. And the first time you see this, the natural question is to say, you don't mean O, you mean R in there somewhere, like R and E. So let's see. So OK, so I can say definition really quick. So E is an SC Monox space. Let's say U contained in E is open. And we recall my ambiguity. I really mean open in the Bayes topology. R is a map from U to itself, satisfying R compose R equals R, and R is SC infinity. And all of this says is the definition for an SC smooth retraction. And I'll say, here's another definition, which is that E is an SC Monox space. Which O subset of E is an SC infinity retract provided there exists an R, which is an SC retraction such that O is equal to the image of U under R. So my handwriting is a little sloppy, but everything here is basically just me writing this stuff down in, I don't know, some formalized version. There's no essential change. I'm not sure I entirely understand the second definition. So an SC Monox space comes with a whole lot of structure. How much of that structure is O supposed to remember? Right, so what do I want to say here? OK, so I'll point out that that's your second question. So I want to answer your first question first. And your first question is, well, what do we mean by sort of local? What is the local model? And so that was the previous question. So here's the answer to that question. The local model, our sets are of the following form. And if I want to be, yeah, yeah, not E, not this, and not say R U or U comma E, but we don't want R in here, which is strange. And it's also the case, this is also just in terms of notation. I can say that even recently I've been highly irritated by the fact that you can't just talk really honestly. If you want to really be as honest as possible, you can never just tell me O is the local model. If you're really being honest, you have to give me both O and the ambient space. But it turns out that you don't have to tell me R. And the reason for that is that if you look at this definition, if you look at this definition, you sort of say, OK, suppose I have one retraction, which makes this function here SC smooth, and I choose a different retraction which defines the same set, then again, it's going to have the same regularity. So regularity is independent of this retraction, and the retraction doesn't really tell you any other information. So all of the structure that O has, all of the differentiable structure, I mean, is essentially, I mean, all the scale structure is induced from E, and any differentiable structure is induced from the fact that there exists an R whose image is O. As long as there's one, you're fine. And so in the literature, then, this then becomes your local model. This is replacing, say, open sets in Rn or open sets in, you know, Banoch spaces. And I can make that even more precise now, unless there are further questions by anyone. Can we talk about O also as having a scale structure? Yes. It has a scale. It's a, I mean, the image of the appropriate levels of E. Yes. So O, remember, has to be a subset of E, and so anytime you have a subset, the natural thing to do is to say that the kth level here is just equal to O intersect the appropriate scale structure there. And you know, and... The subset would be zero. Well, remember, my ambiguity at the beginning of yesterday sort of says that, says that when anytime I have a set-wise statement like this, this is like, I always mean the base level. Right. Explain what it means to be an isomorphism of such objects. An isomorphism of such objects. Well... Like for instance, whether I have O, E included inside... Is that an isomorphism, or is that... E cross C. I think that it is O comma zero inside. Oh, I see. Well, I'll be honest and say that I don't quite know what you mean by isomorphism in this case. I mean, you know, it doesn't necessarily have... I don't think it necessarily has a linear... In general, it doesn't have a linear structure to it. So yeah, so in general, this doesn't have a linear structure to it, and so consequently, isomorphism would mean diffeomorphism, and that is a concept I'll define. But in fact, I mean, from just what's on the board, you should be able to conjecture what it would have to be. You would say, well, you know, I've got two such retracts, and we say they're SC diffeomorphism. The SC diffeomorphic provided there exists a function which is a bijection between the two, which is SC smooth, or SC one, whatever. I guess we'll stick to the smooth category, I think, for simplicity. Right? So in other words, the notion of isomorphism doesn't care about you. Well, I mean, E is sort of running the background because I can't talk about a local model unless I have both of them. I have to have the ambient space, and I have to have the subset. Because the definition of SC smooth depends on... This is of course... It passes through you. It passes through you. This of course is actually related to a Facebook post I had about whether or not it's acceptable to write statements like this, typographically. Most people hated statements like this, but writing down what these are subsets of is a useful thing. Vertically it's OK. Vertically it's slanted. It's terrible. OK. Right. I was going to online... I'm surprised he hasn't already. Yeah. I... So what you just said is that the obvious candidate for isomorphism is an F, like one definition which is bijected to SC infinity, but don't you need to require an inverse to SC infinity? Right. I mean, I said bijection and then I want function, and if I didn't say, and it's inverse, to suppose to be SC smooth, I wouldn't want just one direction for sure. Sorry if I missed that. I don't understand why you need to keep E in the local model. I mean, can't you just take some of the knowledge of theta or O plus the knowledge of what all the SC functions from it are? That's out of my pay grade. I don't... Yeah, I'm not sure. How can you define the SC smooth for something not between O's via U, which is an E? Yes, I mean, I can stick another sort of containment in here. I'm making my board work. I'm just saying like that is where E comes into the picture, right? Because you are always just composing all of these maps. Yes. Yeah. Why don't anyone say yes to it? I think it's right. Yes. I'm not entirely sure what the question is. You're just going to define smoothness unless you know E because it contains... Ah, yes. Yes. Thank you. I'm sorry. That was my fault. Use the audience doesn't make remarks. Okay. So now I'm going to do something which is sort of obvious. I mean, I think it's obvious, but I also think sometimes doing obvious things on the board solidifies how obvious they should be. So what I want to do is basically give you the definition of an M-polyfold minus some lying. So let X be topological space. X in X, a chart around X is a tuple, the phi O so that this here is open in X. This is an SC smooth retract. I mean, in this sense here, or it's a local model if you like, O is a retract sitting inside E and then phi is a map from V to O which is a homeomorphism. Because at this point we have no further structure on X. So definition and SC infinity atlas on a topological space X consists of a set of charts of the following form that we've just seen such that they are pairwise compatible. Compatible is the word I missed to define and this collection of V cover X. Two atlases are equivalent if their union is an atlas. No, no, no, no, no. I'm sorry? E is fixed or is it in the definition or? No, E need not be fixed. So any chart of this form where E is some scale bond arc space, O is some SC retraction sitting inside that E, V is any open set and phi is any map. So they could a priori be different. This actually sort of, yeah, I mean things like that are actually important because when you want to consider maps from say your domain is a Riemann surface you could ask questions like well what do we really mean by a Riemann surface, right? Someone might want S2 to be sort of the set of all points of unit distance away from zero in R3 but someone else might have some sort of slightly different, they want it to be sitting inside R4 or something stupid, right? So your domains are different so strictly speaking those are different spaces but you would want to allow that in such a definition, right? So two charts are compatible if associated transition maps are SC smooth. So here's an exercise that the domain of definition is also retry but it's more. SC infinity, atlas. No, no. I've got good. Definition importance. And M polyfold is a para compact Housdorff topological space equipped with an equivalence class of SC smooth atlases. Now I claim this definition should be sort of obvious but also sort of necessary for me to write down. There's a question over here I think. No, I got it. Okay, good. Those are the easy questions to answer. Is there any questions about this? Why is it reasonable to ask a para compact Housdorff? Well because I want to, I would like it to be the case that if I restrict a finite dimensions I recover the usual notion of a differential manifold. But it turns out to be the case that when you build these M polyfolds for instance for a Grumov-Witton SFT and stuff they're para compact Housdorff. All the standard bar manifolds are too. When you say that they're compatible you're asking to be a system and you need to have that, if I remember well that the kth level goes to kth level. So are you assuming that a priori or, I mean, you know you're just having a home or something you can lead to. Right, so you have, well right, so what happens? That's a good point actually, right? So what's happening here is that you're starting with something that's nothing more than a para compact Housdorff topological space. It seems to have no additional other structure. It has no level structure. It doesn't have, it has nothing else, right? And so what it turns out is well this is the structure that you need so that once you equip it with an equivalence class of atlases or even one atlas in particular because it's the case that all of your associated transition maps have to be SC smooth, they have to preserve levels, they have this sort of differentiable structure. Any information that you see in the local model that you would like to see in the M-polyfold that's also preserved in the transition maps, which is essentially everything we've discussed, we can then claim is just induced on M via the local models. So yes, it is the case that after you have an atlas for it, this thing has a nice, you know, what I would call scale topology and variety of other structures. But the transition maps will be defined from a subset of theta to a subset of theta prime, right? Not a, yes. So what does it mean that that maps, that's not a retractable format? This was Helmut's exercise that he stated. So it turns out that it turns out that the restriction of two overlapping, the restriction of two retracts is also again a retract. An open subset of a retract is a retract. An open subset of a retract is a retract. Yes, that's a good point though. Any other questions? Is this all open? I mean, it seems like it's probably been accepted. It's divided by choice condition. Well, it can't be, I mean, in general, it's certainly not going to be open because to be open, it's going to sort of fill up an open set inside E. It's going to be much wilder. But, you know, it is the image of an SE retraction. Yeah, it's the image of an open set in a scale-Bonach space by a scale retraction, scales with retraction. And that's sort of enough to make this definition work. Yeah? And the topology on O is the substaste topology, right? Well, I mean, O, strictly speaking, has, you know, scaled topologies, but the base one, yes, is induced from the base topology. Yes. Yeah. Okay. So, now I can tell you that I'm also lying. In the following sense, so, you know, if you're studying, you know, if you're not studying Gromov-Witton, then pretty much any other sort of moduli problem that you're likely to come across is going to have some more rich algebraic structure, which relies on having met, relies on your moduli space having boundaries and corners. And currently, the way we have this definition of M polyfolds, nothing has boundary and corners. So we need to fix that. Uh-oh. Okay. I'm not lying on the ground. That's triple, triple vowel looks terrible to me. I don't know. Oh, snap. The predictionary.com says the Joel slobbing correctly. Oh, snap, all right. That's why it's free. Ha ha ha ha ha ha ha. Linear SC isomorphism is a linear SC zero map T. This is Helmut's notation. E to F, which is an ISO on all levels. Partial. All right, Helmut. Tell me which word goes here. Quadrant or cone is a closed set, closed convex set. Let me give it a name. C such that C equals T applied to, and T is a linear SC isomorphism. So this, I'm sure, is a partial quadrant. Are we calling these partial quadrants as well? You'll know. Fantastic. We'll call this quadrant. And this is our model, partial quadrant. And you might say, well, why am I doing this? I'm sorry. Yes, absolutely, just a finite number. And you can say, well, why am I doing this? Well, what I want to do is what I really want is to have a notion of boundaries and corners sitting inside a scale-bonox space. And so this is sort of your model for that, the same way you'd write down your obvious candidates sort of sitting even in finite dimensions where you're going to sort of have this shape where this is going to be R to the N, say, for instance. So you could replace this with a bonox space. That's fine. And then we allow this sort of isomorphism so we can move things around. So in particular, this region here is now partial quadrant. I'm sorry? W is just some other scale-bonox space. So your C law is an E. Pardon? C is an E and X is still having a use and T is something. Yes, yeah. So yeah. So C, like I said, over here somewhere, this notation is sort of always standard, it seems. C is going to be some partial cone inside some scale-bonox space. And we want it to be the image of this sort of model one under some linear isomorphism. So like in boxes allowed, but corners. So you wouldn't. OK, so first of all, not the box, but just the corner of the box is allowed. You can't have all four corners because it has to be. No, no, I understand. Which things are I'm asking? He's the following thing, a finite dimensional M polyfoil. So a box, yes, octahedron. Probably not. So for instance, if I have a pyramid with four sides, not allowed, right? You could, but the current theory doesn't allow this. Is there any reason for not allowing that, actually? I would sort of say that that's non-generic. If I sort of think of my faces and I sort of move them around sort of generically, then that's a non-generic position. Actually, you can take any convex set with this interior, and you do every single suspect with that that would cover your case. So you couldn't have spaces that are boundaries just modeled on the boundary of the convex set. So you've seen that in the study? Yeah, you need an interior because. You need a cone condition, right? You have directions to say that the linear map is predetermined. But I'm very curious, and there'd be trivial extensions to polyphones allowing for the opportunity. Yeah. Yeah, I might believe that. OK, so observation. This, the calculus extends to partial quadrants the same way differential calculus extends to manifolds with boundary in corners. It's the same way. This is not a surprising statement. Just look at the definitions. And then now I can make a slight generalization so I can be less dishonest. SC infinity retract, this will sort of replace our previous definition, is really a triple OCE. Previously, we didn't have the C here so that this is a scale, Bonoq space. This is a partial quadrant. And this here is, once again, the image of a retract. But now U is a subset of C, which is relatively open, meaning it's open in the subspace topology induced on C from E. SC sitting inside E here? Yes, sorry. And R is an SC smooth retraction. OK. In the definition of linear SC isomorphism, is it important that you wrote SC0 instead of SC infinity? It's linear, so it should be the case that SC infinity is induced from SC0. So why bother with this T at all and not just take the standard C all in and expect it? Take the standard what? Why not just take the standard C all the time? The standard quadrant, you mean? Yeah, that's a good question for a helmet. I just think it's useful to have this definition at your disposal when you need it, because otherwise you might come across a collection of functions so that the way you've defined it, so the easiest thing you have defined, is such that I think it's convenience. So I didn't get the question. The question is, why have we bothered to define partial quadrant this way? Why don't we just always work with the model partial quadrant? Because if you have a tangent, if you look at the tangent space, there's a natural notion of a partial quadrant, the tangent space at a point. And then there's no zero infinity. So maybe I didn't get it. Why have we defined C instead of working with the model partial quadrant? Yes. So for example, when I have an endpoint for it, and it takes a tangent space at a corner point, then there's a naturally defined partial quadrant with a tangent space. Yes, it is naturally defined. Is there something, and then what can you say? You can only say it's either more of a true standard. I'm not convinced either way yet, but I have some more to talk about, I guess. So how do we know that taking finite k will be enough for applications? Because conveniently, our moduli spaces should be of finite dimension, and therefore, the dimension of their boundaries. I mean, the highest order corner you could possibly have is finite. So the redundant dimensions will always be actual full dimensions, not that's what we expect. No, no, no, not the full dimension. But the corner structure, we usually define as. Just in how many ways can the stuff break? And then usually the energy is fixed in the final model. Yeah, usually. I mean, in order to have an infinite dimensional corner, you need to have an infinite level building. And I mean, I don't know of any sort of standard analysis which covers it. Maybe the applications generalize it. I'm sure it's not difficult to generalize. For some. Right, so then what I would like to do is. But that's the case for when we make them holomorphic. When we make them holomorphic, there is energy lower bound on energy. But if it's not since nothing is holomorphic now, we don't have any lower bound on energy. Potentially there could be. Right, but what you're saying is, I mean, but what ends up happening is you're looking at some big union of spaces, B. So these are sort of non-nodal curves. And then I guess what I want here is some sort of fiber product here. So in other words, you're looking at B, union, B fiber product with itself, union, B fiber product three times. And so you take this big long union, and no point in here is ever a point in an infinite number of. I mean, you could have mentioned that if you look at arbitrary maps, if you can break them infinitely many, often even if you have a finite energy window. But no one map has infinite number of levels. I mean, you could think of. Oh, like 1 over x squared. No, no, look at the sum of 1 over x squared. But you're trying to connect it. But the thing is, you need an ambient space. It just contains this stuff. So in the ambient space, there's no need for it than having more corners built in than for your model aspects. But you could, of course, pursue in certain cases. OK, OK. So since this works, eventually we will pass the whole multiplication, this will work. Can I add a voice from the back? Please. So usually, your space B is all things that are in one homogeneity class. So if you define the energy as actually the sublactic area, you actually have finite energy on all of B. I mean, you could imagine, if you look at the function, R n, you put a lot of critical points in the vector infinity. You couldn't have it in such a way that the flow cannot escape. But you have infinitely many critical points to infinity, which sort of the difference of the energy goes close to 0. Then you could actually break things as often as you want. So that's possible. But for each space that you fix some points, it wouldn't happen on the level of the gradient flow. So we have a polyphol discussion this afternoon. So I think Joel should. Right, which Nate has generously let me take about 10 minutes of, I think. So let's see what would I like to do. So my notes are more detailed than what I'm going to write on the board here. But I'll draw some pictures just to present the idea. So what you want is to say, OK, if x is an m polyfold, let's see if this is what I say this. So OK, so once I have, OK, as I should say this. So once you have an SC retracts modeled more generally, then you run through the same collection of definition in terms of charts and atlases to have a new definition of m polyfold, which allows these as your local models. That's the honest definition of an m polyfold. So now we have finite boundary and corners. But yes, but it will be the case that in that m polyfold, you have sort of corners of whatever dimension you would like, not infinite, but finite whatever you like. OK, so then suppose we have an m polyfold. What we'd like to have is sort of a measure of cornerness. I'm sure that's correct. So what do I mean by that? So what do I mean? So what we really want to do is define something and how many will probably use this a lot called the degeneracy index. Degeneracy index, which really what you want to be is a map defined, well, I think we'll just say dx is a map from your m polyfold into the natural numbers, but can sort of be 0. And so what I would like to do is say, well, how do you define this? It's not too difficult. Since m polyfolds are effectively defined through transition maps and local charts, let's just define it for sort of the model case. And in the model case, here's a nice SE-Bonox space. And if you want to, you could cross it or plus with another scale, Bonox space. And these points in here that are interior points, this is where the degeneracy index should be 0. And these points here are degeneracy index 1. And these are points so that your degeneracy index is 2. And so once you have it in this sort of local model and the standard, and you see how this sort of generalizes more, this is to see how this generalizes, you can then do it for partial quadrants. You make that definition. You then you can then define it in a chart. So in the notes, for instance, you'll see that you have a degeneracy index defined like this. This is some natural number. And then from the charts, you want to define it for an m polyfolding. There's just this little trick here because presumably you can write down local models where the degeneracy index is different depending upon the local model. So you take the minimum over all your possible charts. And this allows you to essentially define this degeneracy index, measuring what stratum of the corner are you in. And then after one defines this, it's important to state the following proposition. If x and y are in polyfolds and u is contained in x, some open set, v is contained in y, open set, and f maps from u to v is an sc. Does it have to be smooth? Let's say it's smooth, diffio. Then the degeneracy index on x of a point x is equal to the degeneracy index in y of f of x. Here we're assuming that x is in u, f of x is in v. So the conclusion here is that sc smooth diffimorphisms on m polyfolds preserve the sort of corner strata, which is exactly what you'd expect in finite dimensional manifolds with boundaries and corners. And also, no one could say that dx is equal to the case of c, which is the trivial thing which you define for c. So the partial quadrant, you can see u as an m polyfold. And then when you look at the corner, as you can just look which way described in this picture, just some definition. It's not necessary. It's because you took an infinite number of possible charts. It's a good thing. You're saying in that picture, every chart you recognize that corner where d is 2, if you always need to. 2, whatever. But do you really have to take a minimum? Yeah. Because you could have a retrack which is this line going like this. Then the naive definition with the degeneracy of that corner is 2, which was taken as a chart. But the line is better for the retrack in general position. Oh, I see. You can have a bad retrack here. Yeah. The sort of mistakes. Because you take just big space and put that same somewhere and then retrack it. Yeah, this would retract onto that. So you look for the sort of. So the point is you might have some set like this, and you're going to have a retrack sort of sitting inside here. And yeah, so you have a retrack sitting inside here. And then the image of a retract in this even this two dimensional case might be this sort of line. And then if you have this line, you would say, well, what's the degeneracy index here? Well, d should be 0, of course. What's the degeneracy index here? Well, is d equal to 1 or is d equal to 2? In this model, it's in an ambient space where it should be 2. But if you take a different local model here, you can find one where it's 1. That's why you need this minimum. Any other questions? How do you know that you actually have the. So like I said, well, you told me that I can make all kinds of weird spaces as these SD manifolds. Maybe I was able to also make the corner of this quadrant. The quadrant itself. How do you know that's not it? But those are the line definition. How do you know that's not possible? How do you know that I can't make something like an M polyfold with boundary in corners making use only of this definition which doesn't allow for the boundary in corner structure? I mean, I. I mean, presumably you have to know that otherwise the degeneracy index breaks. No, you cannot make it better for the real corner. You can take the over corner. You cannot make the corner structure better. That's the zero. Yeah. Can you say anything about the truth of that theorem? Well, I ask later. I think, yeah, don't worry. I started late, so I'm only two minutes over based on when I started. But yes, I can go ahead and stop here. Well, I mean, if you have a final statement, but I was saying the audience should maybe allow you to make a statement. Yeah, so I guess the last sort of thing I wanted to state about this is that finite dimensional manifolds with boundary and corners have this nice property that if you look, so here's a toy example. Here's a toy example. They have this nice property that along the boundary of your finite dimensional manifold with boundary and corners, you can do the following thing. You can look at the set of all points of degeneracy index 1, and then you can look at the closure of this. And the closure of this is, again, a manifold with boundary and corner structure. And so you can see that from this picture. That's just this segment here. And that's a property that finite dimensional manifolds of boundary and corner have. You'd like to lift that up to M polyfolds as well. And that can be done, but there's an additional condition that one needs to place on your definition of retraction that define your local models. So this is called the taming condition. It's in the lecture notes. I'm not going to present it here. And the key thing to know is that the conclusion guarantees for you this nice property that you look at the closure of your faces, or the degeneracy index 1 portions. And these are, again, M polyfolds with boundary and corner. That's one. And the other thing to remember is that essentially all retractions that occur in practice are of this special type, they're sort of splicing type retractions. And in these cases, the additional taming condition is automatically satisfied. So it really ends up being something you don't have to check too much. I mean, it should be completely straightforward. And then you have this additional property. OK, and that's where I'll finish. Thank you for letting me run over. Thank you. Are there any more questions? When should people read the lecture notes by Mr. Bees? I assume it's done already, right? The lecture notes are online. Read them at your leisure. And, yeah, and of course, if you have any questions, you can ask. And I will be speaking briefly in Nate's discussion. I'm sorry for that. But I have to make sure enough stuff is done for Katrin to start tomorrow. Says Katrin. Any other questions? If I take the, I suppose I have some SC manifold. Sorry, just I hate to interrupt. SC manifold or do you want M polyfold? Because there is a, OK, thank you. And suppose I take the algebra of SC smooth function. OK, continue. Does that determine the M polyfold? I think logically that I'm going to work on one of these. No, I mean, is this GELF-on doing it? But what is the GELF-on doing, saying? OK, so I think there are better spaces. We don't even have smooth functions. I think this is a good talk for T-Tag. Let's thank Joel again. So Nate was very generous. And let me talk for a few additional minutes just to finish up the stuff that I wanted to make sure you guys. I had presented before Katrin tomorrow. And as my voice is just about to give out, I think that's the universe saying it's really time for me to stop talking. So I'll try to be as quick as possible. So the main thing, so we'll start over here. Strangely, that's just sort of how things got laid out. So it goes one, two, three, four, five. So the main thing that I want to tell you about is the statement of this theorem. So what I've spoken about mostly so far is M polyfolds, which I said this is like a generalization of a manifold. And it's a generalization in such a way that it's going to have boundary in corners. And it has both topologically and differentially, it has enough structure so that it will contain the compactified, modulized space as a subset with the Gromov topology just induced from the ambient scale topology, or base topology even. Great, so now you have manifolds. But of course, if we want to study pseudo-Huller-Worfer curves, say, for instance, for other moduli problems, we want these to arise as zero sets of a nonlinear fret-home section. And so that means we need some notion of a bundle. And so the main theorem that I wanted to point out here is this implicit function theorem. This is sort of the first step, which basically says, well, what would you expect a finite-dimensional version to say? The finite-dimensional versions say, without manifolds with boundary in corners would say, I have a finite-dimensional manifold, I've got a nice smooth bundle over the top of it, I have a section of that bundle. And assuming that the zero set is the linearization of the principle part of that section is subjective, then it should be the case that you have an implicit function theorem, which guarantees that the zero set is in fact a manifold. And then if the base of your bundle is a manifold with boundary in corner, then you'd like it to be the case that the corresponding solution set or zero set also has a boundary in corner. And so that's essentially what this theorem says. So top of this board down to this line here. That's just essentially the M-polyphold version. Yes? There's x equal y. No, so y is the bundle and x is the little x and little y. Little x and little y. The solution set equals y. This is cap x. I mean, I could rename this. Well, x, if you like. Is that the issue? But x was a specific term for x, x, 0. What was x in the way of the equation set? It's OK. It's at every point x in the solution set. At every point x you want? I'm fine. OK. Thank you. Right, so that finite dimensional version of what you would sort of expect for this sort of an implicit function theorem is exactly what this says. But then I've underlined all the words that at this point in the course of lectures we don't know yet. So I wanted to kind of go over those somewhat quickly. And some of them I can go over quickly and some of them I can't. So for instance, a big one here is an S.C. Fredholm section. So this requires an entire lecture, but that's the point of tomorrow's lecture by Katrin. She'll be introducing S.C. Fredholm sections. And then there's a tame strong bundle, which also takes a minute, and that's essentially what takes up the bottom two boards here. So I want to pause on that for just a second. And the remaining two pieces are good position and sub-M polyfold. So let me go ahead and drag this top board down just so I can point a little bit easier. So good position has a precise meaning. It is in the lecture notes, as are essentially these pictures here. The point is that, I mean, I think the point is that in order to guarantee that you're sort of the sub-manifold that you get, this sort of cutout has a nice boundary and corner structure, then the behavior of that sub-manifold, before you've shown it, has a nice boundary corner structure, needs to interact nicely with the boundary of sort of the ambient space. The boundary and corners of the ambient space. And so general position would be sort of somehow it intersects transversely. And that's sort of ideal. That's what you would expect. I should probably mention this about HWZ sort of terminology. So when you read, when they say general, I think generic. The general case, in my opinion, covers all possible cases, but the generic one is the one that happens sort of most often, but not necessarily always. And then when HWZ say good, what they really mean is good enough. Right? Because in general, I would say this position is better than whatever I've drawn right here, which I'll explain in a second. But I mean good enough because it's good enough to get the result that you want, which is that your sort of sub-manifold has nice boundary and corners, a nice boundary corner structure induced from the ambient boundary corner structure on the M-polyfold, say, for instance. Right, so that's terminology. And so here's general position. And so what does good position mean? Well, basically what it means is that you have a tangent. And it's a little trickier because in general, of course, I've drawn everything with your subspaces being like one-dimensional. And so it's a little bit trickier to sort of make these statements, especially since your ambient space is not finite dimensional. But the point is that you want it to be the case that you have your tangent plane. So it's going to pass through the origin. And you want it to be the case that you can sort of wobble it a little bit. It takes sort of an open neighborhood of nearby planes. And then you want to make sure that that open neighborhood stays, or the corresponding rays, if you will, sort of stays on the interior of your ambient partial quadrant. And so that's why what you have this line right here is supposed to represent this finite dimensional subspace. And it's good because I can wobble it, passing everything still through the origin. And all those corresponding planes are going to pass through the interior here. That would be not the case here. This is a not good case where you have, say, this ray sort of travels precisely along the boundary. Because now you wobble a little bit, and suddenly it's gone. Suddenly it's outside that partial quadrant. I see. So your picture, though, is that it's not a two-dimensional cone coming into this two-dimensional. It's a one-dimensional thing. It's a sort of red thing. And you're a two-dimensional triangle, sort of very quick wobble. Yes. Oh, yeah. I think that's right. So it's really that when you say it's not good, it's because that one-dimensional thing is coming along the edge. Absolutely. Absolutely. Right. And then, of course, here's something particularly not good. When you have sort of a line that sort of, apparently this happens in the theory sometimes it has to be dealt with, you have some line which passes like this just through this point, and then you can sort of see even no perturbations of this line sort of bring you into the interior. So this is just sort of a picture definition. The precise definitions in the notes. But I wanted you to make sure that there at least be aware of something like that. Oh, and then a sub-M polyfold. So I can read this. Let X be an M polyfold. So we're thinking this is an ambient space. A is a subset in there. And we want to know when is A a sub-M polyfold? Well, that's going to be the case if, for every point in A, you can find an open neighborhood V around A and an SC infinity retraction R taking that open neighborhood to itself so that the image of that is A intersect with that neighborhood. I mean. How about your retraction hands? Of course. It just retracts right down. It's much easier. So I think if you think about this definition for just a moment, you should hopefully convince yourself that this is the only possible definition that it could be. In some sense, replace X with SC bond-ock space. And you should essentially get, you nearly get the definition of just a retraction in a model case. So all we're really doing is saying now you can also have retractions in M polyfolds. And that retractions in M polyfolds can cut out sub-M polyfolds. And that's good because this theorem tells you that that subset that you get in your M polyfold is a sub-M polyfold. And then the bottom end of this guarantees that that sub-M polyfold is essentially just a finite dimensional manifold with boundary and corners. Any questions about this? So having a retraction on your polyfold means that locally on charts, it is a retraction. Right. So I mean, a sub-M polyfold, because of the definition, it must be the case that you can find a local model in which it's actually an M polyfold as well. No, but I mean, because when you define your charts on this side, I'm just wrong. Your retraction is defined on an open set. Right. But you should see that as long as you have something which supports the SC calculus, as long as you have a domain which supports the SC calculus, then you should be able to have a notion of a scale retraction. So it's just a one-step generalization from that. Good. So the last thing that I haven't explained then that I need to is the notion of a tame-strong bundle. This tame-ness, I'm not really going to talk about too much, although it's discussed in the lecture notes, basically just saying there's these additional conditions on your M polyfolds to make sure they have a nice boundary corner type structure. So what I do want to talk about, though, is this term strong bundle, which the first time you see it might see a little bit strange. And so I wanted to make you aware of something so that when you see the definition, it doesn't seem like what the heck is going on here. So to do that, I can review some stuff that hopefully we all recall just from the classical theory. So the classical theory says a linear, I should have said, linear Fredholm operator, it's a linear map between Bonach spaces with closed image, finite dimensional kernel and co-kernel. So this is fine. And it's the case that Fredholm operators are stable under compact perturbation. So I've said what I mean by this. You can add on compact perturbations and you still stay Fredholm, and it doesn't change the index. So then if you have a classically smooth nonlinear map, then we say this is Fredholm provided it's the case that when you look at the linearization of this map at any given point, the linearization is Fredholm. So again, this is classical. And so then we come down to this point here. We say, well, let's not worry about domains too much. I mean, I don't want to have topology of my domains. I want to try and keep this someone as simple as possible. And in that case, we see, well, the Cauchy-Riemann operator is Fredholm as a map from something which takes maps of regularity k plus 1 to maps of regularity k, maps, sections of pullback bundles, and so forth. But the main thing I want to focus on is the fact that this is of regularity k plus 1 down to regularity k. So now, if we see that, and that's true for any k that we choose, then the natural choice is to try and fit this into the SC structure, like the SC-type calculus. So then even in toy cases, then, for instance, we think of the Cauchy-Riemann operator as acting from one scale space to another, where this is your first scale space, where the kth level has regularity k plus 1, and the kth level in the target has regularity k. That's just this statement here translated into the SC calculus language. So that's not surprising. But then something a little bit strange happens, which is, well, what are our compact perturbations? That's what we had right here in the classical sense. And so now what you say is, well, we remember from scale bond-ock spaces that the higher levels compactly embed into the lower levels. And so as a consequence of that, if we're going to perturb our compact operator, we expect it to be the case that it's going to be a map from one scale bond-ock space to the other scale bond-ock space, except we've shifted by 1. So it moves one level up. And if you write this down, we look back in this case, well, that just means that we're tacking on a perturbation, which maps hk plus 1 to hk plus 1. So you have a differential operator, which drops you down, a compact perturbation, just adding on a lower order term. So that's not so surprising. But then we point out one last complication just in terms of, and this is really in some sense it's a notational complication, and then you just have to push through, carry through with it, is the fact that, well, the reality is that in our applications, the Cauchy-Riemann operator isn't just a map between, say, scale bond-ock spaces. We really have to think of it as a section. And so consequently, we have to think of it as a section from locally, at least, from e into e plus f. So this is our total bundle. And if this has regularity hk plus 1, then this has to have regularity hk, which is essentially what I've written right here. And then you have your perturbation. But your perturbation sort of lifts you one up in the target here. And so I've written that down here. So your sections have to go from ek to ek plus f1 plus k. And this is sort of frustrating. It's frustrating because that means in local coordinates, in a local chart, it's irrelevant. You say, OK, this is easy enough to do. But the problem is that you then have to build a bundle. And the thing is that what that means is you essentially have a bundle with two different structures. You have this structure that you have to keep track of, and you have this structure that you have to keep track of. And so the fact that you essentially have two scale structures that you have to keep track of in order to have a good notion of perturbations together with your differential operator motivates the word strong in strong bundle. So anywhere in the literature, in HWZ literature, any time you see the word strong, that means that you have two scale topologies, two scale structures that you have to pay attention to, essentially because you're dealing with bundles. Does that make sense? Damn. So this is to be able to have a notion of compact perturbations. Yes. But doesn't it automatically have two structures? If you have FK sitting over EK or something in bundle, and then you look at FK is governed by FK plus one in it. But in practice, it's actually always apparent. I mean, once you see one example, you know the more. But on the abstract level, the formula is S, G, or coincide. So if you couldn't hear Helmut, he said, it sounds so just to sort of highlight something about the polyphonic theory is that on one hand, it's built on this totally abstract level, and on the other hand, we make sure it contains all the theories that we have, all the theories that we want it to. And the downside is that in practice, when you actually write things down, you probably could get away with not having to keep track of these two structures because they would be fairly apparent. But in order to abstractify that and then have this abstract theory in which everything fits, you need to have this notion of having two different scale structures essentially on your bundle. Please. So just as, right, isn't she that you can always get a strong bundle structure on a bundle where the strongness comes from this shifting by one, like you've got in this example? I think that would be more for sense. You have to be careful that the transition that's preserved the scale thickness, but that's a difficulty. That's the taming of the tamed sort of the quality of the transition. I mean, taming the sense that it doesn't allow on stuff. It seems that it draws on comparably nice. For instance, I mean, for instance, it doesn't make sense, like you might say, well, maybe you're lucky and you've got a scale structure here and a scale structure here, so maybe instead of taking the usual diagonal filtration, we take a double filtration, keeping track of both simultaneously. But then you write down a transition map, and you see that you get some of that, but not all of that. There's this unfortunate fact that if I consider a map of regularity, classical regularity CK, it doesn't make sense to talk about a vector field along that map of regularity K plus 2 or something. That doesn't make sense. That doesn't hold in transition maps. And so then consequently, in your transition maps, you don't keep track of the double filtration. And if you look at what's the best you can possibly do in applications, well, the best you can possibly do is essentially to keep track. In applications, in general, is to keep track of precisely these two filtrations. And then from this, you can get a partial double filtration, which is the way HWZ do it. So in the end, what ends up happening is you end up with local charts that are written like this, where we use this left triangle here to denote the fact that this, well, HWZ will tell you that you have a double filtration, m, k, where m is, I'm going to forget these. Is this right, Helmut? Damn it. Hey, as best as you can, it does work. Right, yes, yes, of course. Yes, so you get this sort of double filtration, but it doesn't extend all the way up. And again, the reason is essentially because it doesn't make sense to talk about vector fields which have much higher regularity than the map along which they're defined. That's essentially what this problem boils down to. But also simultaneously needing to have a canonical class of compact perturbations in which to do the theory. So yes? So this is implicit function here, right? And this tamed, strong, bundle condition is a condition so that we have compact perturbations. Yes. So how is it related? We're not going to perturb it. It's already perturbed so that it's transverse. So for today, yes. Tomorrow, no. So tomorrow you will need to have a perturbation. And the point of me presenting this is that I have to tell you what the strong bundle is precisely so that you have a space in which you can make perturbations. Yes, precisely you can make perturbations. For this statement, I suppose I could remove, yeah, for this statement, I think I probably could remove this strong condition and work with something sort of less. I mean, you could probably work through the implicit function theorem. It's not clear to me where that would cause a problem. But how do you define phretom? I think it's sort of one project. But phretom actually requires that when you bring it into normal form, you use the strong bundle property. That's on the target. Yeah, that's right. OK, yeah. So inherent into the definition of an SC-Fredholm section is the notion of a strong bundle. It's inherently built into it, which you haven't seen the definition, so I'm just being vague at the moment. But I guess the one thing that I kind of wanted to clarify here, which I doesn't seem I was successful at, is that when you see strong bundles, they're going to have this weird new symbol where you're thinking of sort of this as being a total space, this is the total space over the base E. And everywhere you see strong, it's essentially because you have to keep track of two different SC structures. And the need for having these two different SC structures is the fact that you have both the Fredholm section and you need to have a compact perturbation. Hopefully I can get that message across, if nothing else well. Better luck next. It's a little bit stronger because the Fredholm power, as I said, the Fredholm power involves this and constrains the flexibility of your corner changes. So Fredholm is defined by looking good sufficiently nice in certain corners. And since in the target, you can only use things which preserves a double-fit fraction, you can't really completely wide sense. So it enters there a little bit. And then concerning your question, then having these perturbations comes in when you do the softness. But if you have surjectivity, I mean it seems like you really need it. Then surjectivity, but the Fredholm property involves something of discovery, in its definition. And if you don't have surjectivity, the perturbation that you can achieve this, generate. Generic, surjection, perturbation. I'm a little confused. So if you have a bundle, then you have a total space and you have a base. And if you have one filtration on the total space and one filtration on the base, why isn't that sufficient to recover the other filtration in this amount of farther direction? I think this was essentially your question earlier, right, Duzza? Yeah. And I think the answer is that if you write everything down in terms of the applications that you want to study, the problem becomes much more concrete. Why is it not formally positive? Because you have coordinate changes. So in one local chart, you have EK and FK. And the coordinate changes are, say, CK. And the fiber, it's things of quality, CK minus 1. And therefore, at level K. And so at level K plus 1, that's why you're allowed to go K up to N plus 1 at level K. And CK, the CK coordinate changes, because I have CK things at the bottom. And basically, over CK things, you have coordinate changes to CK. It's K changes. So the attention bundle is not a strong one, to put it this way. OK, so I went longer than I wanted to, and I really wanted to hand things over to Nate, so I'm sorry for running over.
 Right, so I guess I want to start by saying that I'm trying to, I think anyway here I'm  trying to try to find line a bit.  In the sense that my main goal, I think jointly with Catcher in this week is to sort of set  up the analytic foundations that we need to make sense of what Helen's going to talk about  next week.  And the thing is that I could do that completely abstractly because sort of the polyphold theory  is sort of completely abstract, but it's also designed to be completely abstract which also  contains useful problems.  So if I did it completely abstractly, I think everyone would get lost pretty quickly and  not have it be founded or you wouldn't have much to connect it to.  So I have to try and bring in sort of elements of things that are hopefully familiar and  pair them in just the right way I think with certain ideas coming from the polyphold theory  to try and illuminate why definitions are the way they are.  And so yesterday I think it might have been sort of a good talk for if it was 80 minutes  with questions or 60 minutes with no questions.  So I think things got a little out of hand at the end.  And so I just wanted to kind of review what I had sort of hoped to say yesterday and then  build off of that today.  So quickly then briefly, what did I try to do yesterday?  So the first thing was that well we wanted to try and parametrize a neighborhood of a  nodal map and we're not dealing nothing pseudo-holomorphic here at all.  We're really trying to build this big ambient space of function.  So we're trying to construct some sort of parametrization or be able to write down some  sort of a chart for this.  And so the first observation that we made was that this pre-gluing map basically yields  us the right topology in this really big ambient space in which has this nice property that  in fact our compact moduli space.  So even if it has, you know, you're dealing with nodal curves or your broken trajectories  or whatnot, this big ambient space has a topology in such a way that your compactified moduli  space is a subset and that the correct topology on your moduli space, your compactified moduli  space is induced from the ambient space, right?  And so this pre-gluing map then was something rather useful.  We defined it sort of like this.  And so we said, okay, well if this gives us our neighborhood in the right space, well  maybe we can use it as a parametrization but we can't because it's infinity to one.  There's all this information loss, right?  So well what do you do?  Well I say, okay, HWZ, they sort of, they introduced this minus-gluing map or anti-pre-gluing map,  however you might like to refer to it.  And so they introduced this and again it was some sort of unpleasant formula but it had  this nice property that when I pair it, right, when I pair it with the plus-gluing or the  pre-gluing map, that suddenly for each fixed gluing parameter A, this map here became a  linear bijection.  And so then the idea is to say, well how are we going to make use of that?  Well that's useful because what we can do is we can define this set script O sitting  inside this subspace here, right?  With the property that, well, we'll define it to be all those maps so that when you anti-pre-glu  or when you minus-glu you get zero.  And that's good because out of, what that means is that basically when we restrict our pre-gluing  map, our plus-gluing map, precisely to this set O, it ends up necessarily being a bijection  or at least an injection or bijection with its image.  Now that kills this problem, right?  We no longer have this problem as long as we're happy to sort of restrict our attention  just to O. But now O might be some weird set and so now you might say, well, you know,  so what?  What have we actually gained?  And so there was this notion and it's an idea because in the end it's going to work out,  but the idea is, well, maybe in some way this set O supports the SC calculus.  And maybe there's some way to treat it as if it were something like a Bonoq space or  something like a scale Bonoq space.  Maybe you can define a notion of smooth maps between them.  Maybe you can do that.  So how might you do that?  Well, you have to do a little, you know, a bit of manipulation here to rewrite the problem.  And so we did that by sort of writing down this function R. Again, its domain is the  same, but its image isn't in the target space anymore.  It's a map from sort of your domain of pre-glued or your domain of unglued maps, basically,  together with gluing parameters back to itself again.  And we define it in this following way.  So we take a gluing parameter and a pair of maps.  We apply this box gluing map, which is a bijection.  Maybe I should say this here.  This was asked yesterday.  This is my projection onto the first factor, or sort of zeroing out the second factor at  least.  And then we take this box inverse gluing.  And then we made some observations, or very rapidly I tried to make some observations.  I would have spent some more time with if I had the opportunity.  Was it this set O that we had a definition for right up here?  Well, once we take this definition of R and we see how it's defined, we see that it's  precisely the set of fixed points of R. Then we also sort of check really quickly that  R composed with itself is just R again.  It's a projection, basically.  It's a nonlinear projection.  That's easy to check here, basically, because this is a bijection.  You have its inverse on the other side and a projection in between.  And then a quick computation.  It's a little one-line proof, basically, shows you that these two properties together  guarantee that this set O is, in fact, the image of R.  And so then there's this theorem.  And this theorem says, after giving this space, C cross E, a suitable scale Bonoche  space structure, this map R is SC infinity.  So I didn't really give you any justification for that.  I'm not going to give you any justification for that.  You can look in the literature and you can sort of see, well, why is something like this  true?  And if I remember correctly, the proof of this essentially boils down to, well, you  write out why we have an explicit formula for R here.  And when you write everything down, you get some terrible sort of equation, some awful  composition of additions and products and blah, blah, blah, you break it down into all  the little pieces, though.  And at the end of the day, you prove each little piece is SC smooth, essentially by  going to that list that is in the lecture notes for day one.  And that tells you where to look in the SC calculus book or the SC calculus paper to  tell you how to prove each one of these components is SC smooth.  And so consequently, you rapidly build back up from this that this is, in fact, SC smooth.  So now that by itself doesn't really seem to bias anything.  Sure, please do.  You say you have a suitable SC Bonoche space structure, so that E itself is an SC Bonoche  space.  Yes.  So what you're going to do is rescale me a zero by some gluing parameter, right?  To change the structure me a zero.  No.  No.  So what is this suitable?  So I'm thinking of C as, I mean, effectively as R2, which is a nice Bonoche space, a finite  dimension, and so I can take a…  So it's just a standard product.  Absolutely.  I mean, when you say it's a suitable SC Bonoche space.  Well, well, well.  E.  E.  Well, suitable is really applying to E, not C.  C is sort of…  It was given a suitable structure, but it says C.  Well, E, you gave this example to me.  I mean, it was…  It was a suitable…  …dobbing a paper in Delphina.  Yes.  Yes.  That's not suitable.  That's not a suitable…  It's not a mysterious thing.  No, no, no, sorry.  I mean, I think, I mean, it was a…  I think it ended up being sort of a slight modification.  I don't know if in my lecture notes I actually defined the scale Bonoche space structure  on E. I gave you something very close to it, and I told you the base topology on E. And  if you understand those two, you can obviously guess what this structure needs to be on E.  I don't know if I said it explicitly, which is why I'm saying suitable now.  Okay.  But thank you.  Are there any other questions?  Because I don't want to blow through this.  This is actually fairly important.  Yes.  So what was the…  What was the purpose of this averaging function that you put in the anti-glue?  Right.  So, yes.  So, let me see.  I want to see where this…  So here's how I would explain it.  So, O is defined to be the set of all points where the minus gluing is zero, right?  And we define it precisely this way so that the restriction of the plus gluing ends up  being a bijection.  So, then what you do is you say, okay, well, if you look at the maps, right, so…  Let's go see.  So script O then is going to essentially be the set of all points which are going to parameterize  your neighborhood of your sort of non-nodal maps plus some nodal maps as well.  Okay.  So what happens is if you think about having this nodal map when you set up sort of a base  problem, you typically want that nodal point to say you want to model it so it goes to  say the origin in R2N, say, for instance, right?  Okay.  But then you start saying, okay, what are all my nearby nodal maps?  And your nearby nodal maps, some of them are going to be sort of…  Some of them are going to be nodal, some are going to be non-nodal.  Those which are nodal, you're going to want to allow that nodal point to move around,  right?  So, now you allow this…  So, now you say, okay, well, I have to allow sort of nodal points to move around.  This forces you to add in these sort of… this extra constant C that I had in the definition  over here of these corresponding scale Bonach space.  Okay.  That's all fine.  You can do everything with a pre-gluing.  There's no problem with the pre-gluing.  But in the minus-gluing, right?  Well, what happens?  If you don't have those… if you're not subtracting off those averaging terms, then  what happens is when you look at this set here, it turns out that without those averaging  terms, it necessarily must be the case that the only way for this equation to hold is  if those asymptotic constants are zero and therefore the node can't move around in the  image.  Think about it.  If you take the nodal value and now you glue and you got this long cylinder, what is the  best approximation for the nodal value?  It's the integral on the middle loop.  The average over the middle loop.  That's the approximation for the nodal value.  Yeah.  I think that shows up.  Sure.  Any other questions?  All right.  So this is where we left off.  So remember, I guess what we're trying to do is we're trying to investigate maybe script  O here supports the SC calculus, right?  That's sort of the idea.  And so then, I don't know how this was sort of developed in practice, but reading through  the material, then you make this sort of observation.  You say, well, if it's the case that I have this map R mapping essentially a scale bonox  space to itself, which is SC smooth and R compose R equals R, well, then I could do  this trick.  I could do this trick.  I could say that a function F defined from the image of R to the image of some other R  in some other space is SCK, if and only if, this is my definition, if F pre-composed R  is SCK as a map from this open set in a scale bonox space to this open set in a scale bonox  space.  See, we already have an SC calculus all defined on SC bonox spaces.  We have that defined.  But now the idea is, well, how about subsets?  Well, if these subsets happen to live as images of these special, what we're going to call  them, retractions, then it's the case that we can make this definition.  O prime lives inside U prime.  O prime is a subset of U prime.  So there is sort of one good thing about the terrible HWZ notation is that it's fairly  consistent.  And so it always ends up being that O is always the image of U and U is supposed to  be a subset in E and open until in, I don't know, five or 10 minutes, in which case we  add in sort of this additional portion where we put in something like, I think it's called  a partial cone.  You can think of it as a partial quadrant in between here, in which case U is relatively  open in a partial quadrant.  But then this always ends up being the domain of R.  And R always goes from U to U. It has to go from one set to itself, absolutely necessary  as you would want for a retraction.  Any further questions?  OK.  So I make this.  Now the idea is we have this, we now have a definition for smooth functions between  these weird images, these weird subsets.  And then this is sort of the first bit of magic.  And the second bit is that actually, that these strange subsets, and they really are,  I mean, rather strange.  I mean, even when you write down toy examples, I think there's a homework example on this  where you can see these things might have finite dimension, but the dimension of the  space might jump.  And we had locally varying dimensions, locally varying co-dimensions.  It might be sort of a full set of sort of, yeah, if you have an SC-Bonach space, for  instance, which fibers over some other finite dimensional space, it might be sort of full  dimension on one region and then sort of have infinite co-dimension on another.  It's a very, it's a a priori, it's quite wild.  But nevertheless, despite all the strangeness, it does have a tangent bundle and you can  just sort of see what it has to be.  And I have a map, R, which defines it, which maps U to U, and R compose R equals R.  And so I say, well then, it must be the case of the tangent map of R maps T U to T U, and  you apply the chain rule, and the chain rule says, well, T R compose T R is T R, which  gives you again a map precisely of this form.  And so then you just take as definition the tangent to this subset is equal to the image  under this, the tangent map of R of the tangent of U. It's kind of functorial.  It's sort of the only thing it could be.  And I should say, it doesn't actually depend on how only on the set O.  Right.  So we're not, I haven't even, I haven't officially defined scale, scale smooth retractions, but  this is sort of, this is sort of the first observation.  And so then, so then once you've done this, I mean, these are sort of, I mean, almost  sort of stupid things.  I mean, you just, you're tinkering and you find this sort of stuff, right?  Once you see that, you should, then you should have this idea.  This is the big conclusion then, should, should be, well, let's try to build manifolds locally  modeled on subsets like O. So what are all the characteristics of O that we need in order  to actually build some sort of, something like a manifold locally modeled on this?  And so this is, this is what I would have liked to have conveyed, I guess, in my lecture  last time.  And I think it's important to sort of see this story completely laid out like this.  To see how, so these things here end up being, so it is the case that this is doable.  I'll say this in just a second, but these things are called SC smooth retracts.  These are called SC smooth retractions.  And these essentially are going to form the, are going to form local models for M polyfolds.  And what I want, what I'd like you to be able to see is that, is that if you start with  just pre-gluing, which is something that shows up in, you know, whatever framework you want,  right?  I mean, any framework where you have nodal or broken elements of your modularized space,  you start with that in some sort of classical analysis.  Then you do this trick.  You introduce this minus gluing, repackage it into this, into this weird sort of nonlinear  projection and show that it's SC smooth.  Then you necessarily are led to this idea of having these SC smooth retracts, which  are going to provide local models for our big ambient space, right?  So any questions about this outline?  Yes, sir.  So like this, these retracts that's really, really different from the usual setting, right?  Where you have close to pre-glued positions with smooth maps, or the map is always horrible,  like in several other spaces.  I'm sorry, what was that?  I missed.  So like this is really different from the normal, like the standard setup, this feature  of R.  Which, the SC infinity?  No, no, no, no, that like post-composition with, or like this.  Pre-composition, but.  Yeah, a pre-composition with it is exactly preserving the features of F1.  But if you think of solar spaces and then pre-composition with smooth maps.  Well, I mean, I think, right, if I have a smooth, if I have a classically CK function  F, and I pre-compose by a smooth function, I mean, my chain rule should tell me that  I just have, that regularity in that sense should be preserved.  And so again, that's happening here.  So that's sort of not so surprising.  I think the surprising thing is that, the surprising thing is that, and this was commented  yesterday, the surprising thing is that if you reduce this from SC infinity, or rather  increase it from SC infinity to classically C infinity, then the image can't have these  weird properties.  It sort of necessarily must be a Bonoc manifold.  And so you then reduce to sort of the standard calculus in that case.  Yeah?  Am I supposed to think there's a difference between locally modeled on O and locally  modeled on the pair is E comma R?  OK, so that's a great question.  I want to get to that right now, actually.  Can you repeat the question?  So there's this question of, when I say locally modeled, what actually constitutes a local  model?  Right?  And I keep saying O. And the first time you see this, the natural question is to say,  you don't mean O, you mean R in there somewhere, like R and E.  So let's see.  So OK, so I can say definition really quick.  So E is an SC Monox space.  Let's say U contained in E is open.  And we recall my ambiguity.  I really mean open in the Bayes topology.  R is a map from U to itself, satisfying R compose R equals R, and R is SC infinity.  And all of this says is the definition for an SC smooth retraction.  And I'll say, here's another definition, which is that E is an SC Monox space.  Which O subset of E is an SC infinity retract provided there exists an R, which is an SC  retraction such that O is equal to the image of U under R. So my handwriting is a little  sloppy, but everything here is basically just me writing this stuff down in, I don't know,  some formalized version.  There's no essential change.  I'm not sure I entirely understand the second definition.  So an SC Monox space comes with a whole lot of structure.  How much of that structure is O supposed to remember?  Right, so what do I want to say here?  OK, so I'll point out that that's your second question.  So I want to answer your first question first.  And your first question is, well, what do we mean by sort of local?  What is the local model?  And so that was the previous question.  So here's the answer to that question.  The local model, our sets are of the following form.  And if I want to be, yeah, yeah, not E, not this, and not say R U or U comma E, but we  don't want R in here, which is strange.  And it's also the case, this is also just in terms of notation.  I can say that even recently I've been highly irritated by the fact that you can't just  talk really honestly.  If you want to really be as honest as possible, you can never just tell me O is the local  model.  If you're really being honest, you have to give me both O and the ambient space.  But it turns out that you don't have to tell me R. And the reason for that is that if you  look at this definition, if you look at this definition, you sort of say, OK, suppose I  have one retraction, which makes this function here SC smooth, and I choose a different retraction  which defines the same set, then again, it's going to have the same regularity.  So regularity is independent of this retraction, and the retraction doesn't really tell you  any other information.  So all of the structure that O has, all of the differentiable structure, I mean, is essentially,  I mean, all the scale structure is induced from E, and any differentiable structure is  induced from the fact that there exists an R whose image is O. As long as there's one,  you're fine.  And so in the literature, then, this then becomes your local model.  This is replacing, say, open sets in Rn or open sets in, you know, Banoch spaces.  And I can make that even more precise now, unless there are further questions by anyone.  Can we talk about O also as having a scale structure?  Yes.  It has a scale.  It's a, I mean, the image of the appropriate levels of E.  Yes.  So O, remember, has to be a subset of E, and so anytime you have a subset, the natural  thing to do is to say that the kth level here is just equal to O intersect the appropriate  scale structure there.  And you know, and...  The subset would be zero.  Well, remember, my ambiguity at the beginning of yesterday sort of says that, says that  when anytime I have a set-wise statement like this, this is like, I always mean the base  level.  Right.  Explain what it means to be an isomorphism of such objects.  An isomorphism of such objects.  Well...  Like for instance, whether I have O, E included inside...  Is that an isomorphism, or is that...  E cross C.  I think that it is O comma zero inside.  Oh, I see.  Well, I'll be honest and say that I don't quite know what you mean by isomorphism in this  case.  I mean, you know, it doesn't necessarily have...  I don't think it necessarily has a linear...  In general, it doesn't have a linear structure to it.  So yeah, so in general, this doesn't have a linear structure to it, and so consequently,  isomorphism would mean diffeomorphism, and that is a concept I'll define.  But in fact, I mean, from just what's on the board, you should be able to conjecture what  it would have to be.  You would say, well, you know, I've got two such retracts, and we say they're SC diffeomorphism.  The SC diffeomorphic provided there exists a function which is a bijection between the  two, which is SC smooth, or SC one, whatever.  I guess we'll stick to the smooth category, I think, for simplicity.  Right?  So in other words, the notion of isomorphism doesn't care about you.  Well, I mean, E is sort of running the background because I can't talk about a local model unless  I have both of them.  I have to have the ambient space, and I have to have the subset.  Because the definition of SC smooth depends on...  This is of course...  It passes through you.  It passes through you.  This of course is actually related to a Facebook post I had about whether or not it's acceptable  to write statements like this, typographically.  Most people hated statements like this, but writing down what these are subsets of is  a useful thing.  Vertically it's OK.  Vertically it's slanted.  It's terrible.  OK.  Right.  I was going to online...  I'm surprised he hasn't already.  Yeah.  I...  So what you just said is that the obvious candidate for isomorphism is an F, like one definition  which is bijected to SC infinity, but don't you need to require an inverse to SC infinity?  Right.  I mean, I said bijection and then I want function, and if I didn't say, and it's inverse, to  suppose to be SC smooth, I wouldn't want just one direction for sure.  Sorry if I missed that.  I don't understand why you need to keep E in the local model.  I mean, can't you just take some of the knowledge of theta or O plus the knowledge of what all  the SC functions from it are?  That's out of my pay grade.  I don't...  Yeah, I'm not sure.  How can you define the SC smooth for something not between O's via U, which is an E?  Yes, I mean, I can stick another sort of containment in here.  I'm making my board work.  I'm just saying like that is where E comes into the picture, right?  Because you are always just composing all of these maps.  Yes.  Yeah.  Why don't anyone say yes to it?  I think it's right.  Yes.  I'm not entirely sure what the question is.  You're just going to define smoothness unless you know E because it contains...  Ah, yes.  Yes.  Thank you.  I'm sorry.  That was my fault.  Use the audience doesn't make remarks.  Okay.  So now I'm going to do something which is sort of obvious.  I mean, I think it's obvious, but I also think sometimes doing obvious things on the board  solidifies how obvious they should be.  So what I want to do is basically give you the definition of an M-polyfold minus some  lying.  So let X be topological space.  X in X, a chart around X is a tuple, the phi O so that this here is open in X.  This is an SC smooth retract.  I mean, in this sense here, or it's a local model if you like, O is a retract sitting inside  E and then phi is a map from V to O which is a homeomorphism.  Because at this point we have no further structure on X.  So definition and SC infinity atlas on a topological space X consists of a set of charts of the  following form that we've just seen such that they are pairwise compatible.  Compatible is the word I missed to define and this collection of V cover X.  Two atlases are equivalent if their union is an atlas.  No, no, no, no, no.  I'm sorry?  E is fixed or is it in the definition or?  No, E need not be fixed.  So any chart of this form where E is some scale bond arc space, O is some SC retraction sitting  inside that E, V is any open set and phi is any map.  So they could a priori be different.  This actually sort of, yeah, I mean things like that are actually important because when  you want to consider maps from say your domain is a Riemann surface you could ask questions  like well what do we really mean by a Riemann surface, right?  Someone might want S2 to be sort of the set of all points of unit distance away from zero  in R3 but someone else might have some sort of slightly different, they want it to be sitting  inside R4 or something stupid, right?  So your domains are different so strictly speaking those are different spaces but you  would want to allow that in such a definition, right?  So two charts are compatible if associated transition maps are SC smooth.  So here's an exercise that the domain of definition is also retry but it's more.  SC infinity, atlas.  No, no.  I've got good.  Definition importance.  And M polyfold is a para compact Housdorff topological space  equipped with an equivalence class of SC smooth atlases.  Now I claim this definition should be sort of obvious but also sort of necessary for  me to write down.  There's a question over here I think.  No, I got it.  Okay, good.  Those are the easy questions to answer.  Is there any questions about this?  Why is it reasonable to ask a para compact Housdorff?  Well because I want to, I would like it to be the case that if I restrict a finite dimensions  I recover the usual notion of a differential manifold.  But it turns out to be the case that when you build these M polyfolds for instance  for a Grumov-Witton SFT and stuff they're para compact Housdorff.  All the standard bar manifolds are too.  When you say that they're compatible you're asking to be a system and you need to have  that, if I remember well that the kth level goes to kth level.  So are you assuming that a priori or, I mean, you know you're just having a home or something  you can lead to.  Right, so you have, well right, so what happens?  That's a good point actually, right?  So what's happening here is that you're starting with something that's nothing more than a  para compact Housdorff topological space.  It seems to have no additional other structure.  It has no level structure.  It doesn't have, it has nothing else, right?  And so what it turns out is well this is the structure that you need so that once you equip  it with an equivalence class of atlases or even one atlas in particular because it's  the case that all of your associated transition maps have to be SC smooth, they have to preserve  levels, they have this sort of differentiable structure.  Any information that you see in the local model that you would like to see in the M-polyfold  that's also preserved in the transition maps, which is essentially everything we've discussed,  we can then claim is just induced on M via the local models.  So yes, it is the case that after you have an atlas for it, this thing has a nice, you  know, what I would call scale topology and variety of other structures.  But the transition maps will be defined from a subset of theta to a subset of theta prime,  right?  Not a, yes.  So what does it mean that that maps, that's not a retractable format?  This was Helmut's exercise that he stated.  So it turns out that it turns out that the restriction of two overlapping, the restriction  of two retracts is also again a retract.  An open subset of a retract is a retract.  An open subset of a retract is a retract.  Yes, that's a good point though.  Any other questions?  Is this all open?  I mean, it seems like it's probably been accepted.  It's divided by choice condition.  Well, it can't be, I mean, in general, it's certainly not going to be open because to  be open, it's going to sort of fill up an open set inside E. It's going to be much  wilder.  But, you know, it is the image of an SE retraction.  Yeah, it's the image of an open set in a scale-Bonach space by a scale retraction, scales with  retraction.  And that's sort of enough to make this definition work.  Yeah?  And the topology on O is the substaste topology, right?  Well, I mean, O, strictly speaking, has, you know, scaled topologies, but the base one,  yes, is induced from the base topology.  Yes.  Yeah.  Okay.  So, now I can tell you that I'm also lying.  In the following sense, so, you know, if you're studying, you know, if you're not studying  Gromov-Witton, then pretty much any other sort of moduli problem that you're likely to  come across is going to have some more rich algebraic structure, which relies on having  met, relies on your moduli space having boundaries and corners.  And currently, the way we have this definition of M polyfolds, nothing has boundary and corners.  So we need to fix that.  Uh-oh.  Okay.  I'm not lying on the ground.  That's triple, triple vowel looks terrible to me.  I don't know.  Oh, snap.  The predictionary.com says the Joel slobbing correctly.  Oh, snap, all right.  That's why it's free.  Ha ha ha ha ha ha ha.  Linear SC isomorphism is a linear SC zero map T.  This is Helmut's notation.  E to F, which is an ISO on all levels.  Partial.  All right, Helmut.  Tell me which word goes here.  Quadrant or cone is a closed set, closed convex set.  Let me give it a name.  C such that C equals T applied to, and T is a linear SC  isomorphism.  So this, I'm sure, is a partial quadrant.  Are we calling these partial quadrants as well?  You'll know.  Fantastic.  We'll call this quadrant.  And this is our model, partial quadrant.  And you might say, well, why am I doing this?  I'm sorry.  Yes, absolutely, just a finite number.  And you can say, well, why am I doing this?  Well, what I want to do is what I really  want is to have a notion of boundaries and corners  sitting inside a scale-bonox space.  And so this is sort of your model for that, the same way  you'd write down your obvious candidates  sort of sitting even in finite dimensions  where you're going to sort of have this shape where this is  going to be R to the N, say, for instance.  So you could replace this with a bonox space.  That's fine.  And then we allow this sort of isomorphism  so we can move things around.  So in particular, this region here is now partial quadrant.  I'm sorry?  W is just some other scale-bonox space.  So your C law is an E.  Pardon?  C is an E and X is still having  a use and T is something.  Yes, yeah.  So yeah.  So C, like I said, over here somewhere,  this notation is sort of always standard, it seems.  C is going to be some partial cone inside some scale-bonox  space.  And we want it to be the image of this sort of model one  under some linear isomorphism.  So like in boxes allowed, but corners.  So you wouldn't.  OK, so first of all, not the box, but just the corner of the box  is allowed.  You can't have all four corners because it has to be.  No, no, I understand.  Which things are I'm asking?  He's the following thing, a finite dimensional M polyfoil.  So a box, yes, octahedron.  Probably not.  So for instance, if I have a pyramid with four sides,  not allowed, right?  You could, but the current theory doesn't allow this.  Is there any reason for not allowing that, actually?  I would sort of say that that's non-generic.  If I sort of think of my faces and I sort of move them around  sort of generically, then that's a non-generic position.  Actually, you can take any convex set with this interior,  and you do every single suspect with that  that would cover your case.  So you couldn't have spaces that are boundaries just modeled  on the boundary of the convex set.  So you've seen that in the study?  Yeah, you need an interior because.  You need a cone condition, right?  You have directions to say that the linear  map is predetermined.  But I'm very curious, and there'd  be trivial extensions to polyphones  allowing for the opportunity.  Yeah.  Yeah, I might believe that.  OK, so observation.  This, the calculus extends to partial quadrants  the same way differential calculus extends to manifolds  with boundary in corners.  It's the same way.  This is not a surprising statement.  Just look at the definitions.  And then now I can make a slight generalization  so I can be less dishonest.  SC infinity retract, this will sort of replace  our previous definition, is really a triple OCE.  Previously, we didn't have the C here  so that this is a scale, Bonoq space.  This is a partial quadrant.  And this here is, once again, the image of a retract.  But now U is a subset of C, which is relatively open,  meaning it's open in the subspace topology induced on C  from E.  SC sitting inside E here?  Yes, sorry.  And R is an SC smooth retraction.  OK.  In the definition of linear SC isomorphism,  is it important that you wrote SC0 instead of SC infinity?  It's linear, so it should be the case  that SC infinity is induced from SC0.  So why bother with this T at all and not just take the standard  C all in and expect it?  Take the standard what?  Why not just take the standard C all the time?  The standard quadrant, you mean?  Yeah, that's a good question for a helmet.  I just think it's useful to have this definition at your disposal  when you need it, because otherwise you  might come across a collection of functions  so that the way you've defined it,  so the easiest thing you have defined, is such that I  think it's convenience.  So I didn't get the question.  The question is, why have we bothered  to define partial quadrant this way?  Why don't we just always work with the model partial quadrant?  Because if you have a tangent, if you look at the tangent space,  there's a natural notion of a partial quadrant,  the tangent space at a point.  And then there's no zero infinity.  So maybe I didn't get it.  Why have we defined C instead of working with the model  partial quadrant?  Yes.  So for example, when I have an endpoint for it,  and it takes a tangent space at a corner point,  then there's a naturally defined partial quadrant  with a tangent space.  Yes, it is naturally defined.  Is there something, and then what can you say?  You can only say it's either more of a true standard.  I'm not convinced either way yet, but I have some more  to talk about, I guess.  So how do we know that taking finite k  will be enough for applications?  Because conveniently, our moduli spaces  should be of finite dimension, and therefore,  the dimension of their boundaries.  I mean, the highest order corner you could possibly  have is finite.  So the redundant dimensions will always  be actual full dimensions, not that's what we expect.  No, no, no, not the full dimension.  But the corner structure, we usually define as.  Just in how many ways can the stuff break?  And then usually the energy is fixed in the final model.  Yeah, usually.  I mean, in order to have an infinite dimensional corner,  you need to have an infinite level building.  And I mean, I don't know of any sort of standard analysis  which covers it.  Maybe the applications generalize it.  I'm sure it's not difficult to generalize.  For some.  Right, so then what I would like to do is.  But that's the case for when we make them holomorphic.  When we make them holomorphic, there  is energy lower bound on energy.  But if it's not since nothing is holomorphic now,  we don't have any lower bound on energy.  Potentially there could be.  Right, but what you're saying is, I mean,  but what ends up happening is you're  looking at some big union of spaces, B.  So these are sort of non-nodal curves.  And then I guess what I want here is some sort  of fiber product here.  So in other words, you're looking at B,  union, B fiber product with itself,  union, B fiber product three times.  And so you take this big long union,  and no point in here is ever a point in an infinite number of.  I mean, you could have mentioned that if you look  at arbitrary maps, if you can break them infinitely many,  often even if you have a finite energy window.  But no one map has infinite number of levels.  I mean, you could think of.  Oh, like 1 over x squared.  No, no, look at the sum of 1 over x squared.  But you're trying to connect it.  But the thing is, you need an ambient space.  It just contains this stuff.  So in the ambient space, there's no need for it  than having more corners built in than for your model  aspects.  But you could, of course, pursue in certain cases.  OK, OK.  So since this works, eventually we  will pass the whole multiplication, this will work.  Can I add a voice from the back?  Please.  So usually, your space B is all things  that are in one homogeneity class.  So if you define the energy as actually the sublactic area,  you actually have finite energy on all of B.  I mean, you could imagine, if you look at the function,  R n, you put a lot of critical points  in the vector infinity.  You couldn't have it in such a way  that the flow cannot escape.  But you have infinitely many critical points  to infinity, which sort of the difference of the energy  goes close to 0.  Then you could actually break things as often as you want.  So that's possible.  But for each space that you fix some points,  it wouldn't happen on the level of the gradient flow.  So we have a polyphol discussion this afternoon.  So I think Joel should.  Right, which Nate has generously let  me take about 10 minutes of, I think.  So let's see what would I like to do.  So my notes are more detailed than what I'm going to write  on the board here.  But I'll draw some pictures just to present the idea.  So what you want is to say, OK, if x is an m polyfold,  let's see if this is what I say this.  So OK, so once I have, OK, as I should say this.  So once you have an SC retracts modeled more generally,  then you run through the same collection of definition  in terms of charts and atlases to have  a new definition of m polyfold, which  allows these as your local models.  That's the honest definition of an m polyfold.  So now we have finite boundary and corners.  But yes, but it will be the case that in that m polyfold,  you have sort of corners of whatever dimension you would  like, not infinite, but finite whatever you like.  OK, so then suppose we have an m polyfold.  What we'd like to have is sort of a measure of cornerness.  I'm sure that's correct.  So what do I mean by that?  So what do I mean?  So what we really want to do is define something  and how many will probably use this a lot called  the degeneracy index.  Degeneracy index, which really what you want to be is a map  defined, well, I think we'll just say dx is a map from your m  polyfold into the natural numbers, but can sort of be 0.  And so what I would like to do is say, well, how do you define  this?  It's not too difficult.  Since m polyfolds are effectively defined through  transition maps and local charts, let's just define it  for sort of the model case.  And in the model case, here's a nice SE-Bonox space.  And if you want to, you could cross it or plus with another  scale, Bonox space.  And these points in here that are interior points, this is  where the degeneracy index should be 0.  And these points here are degeneracy index 1.  And these are points so that your degeneracy index is 2.  And so once you have it in this sort of local model and the  standard, and you see how this sort of generalizes more,  this is to see how this generalizes, you can then do  it for partial quadrants.  You make that definition.  You then you can then define it in a chart.  So in the notes, for instance, you'll see that you have  a degeneracy index defined like this.  This is some natural number.  And then from the charts, you want to define it for an  m polyfolding.  There's just this little trick here because presumably you  can write down local models where the degeneracy index is  different depending upon the local model.  So you take the minimum over all your possible charts.  And this allows you to essentially define this  degeneracy index, measuring what stratum of the corner are  you in.  And then after one defines this, it's important to state  the following proposition.  If x and y are in polyfolds and u is contained in x, some  open set, v is contained in y, open set, and f maps from u  to v is an sc.  Does it have to be smooth?  Let's say it's smooth, diffio.  Then the degeneracy index on x of a point x is equal to the  degeneracy index in y of f of x.  Here we're assuming that x is in u, f of x is in v.  So the conclusion here is that sc smooth diffimorphisms on  m polyfolds preserve the sort of corner strata, which is  exactly what you'd expect in finite dimensional manifolds  with boundaries and corners.  And also, no one could say that dx is equal to the case of  c, which is the trivial thing which you define for c.  So the partial quadrant, you can see u as an m polyfold.  And then when you look at the corner, as you can just look  which way described in this picture, just some  definition.  It's not necessary.  It's because you took an infinite number of possible  charts.  It's a good thing.  You're saying in that picture, every chart you  recognize that corner where d is 2, if you always need  to.  2, whatever.  But do you really have to take a minimum?  Yeah.  Because you could have a retrack which is this line  going like this.  Then the naive definition with the degeneracy of that corner  is 2, which was taken as a chart.  But the line is better for the retrack in general position.  Oh, I see.  You can have a bad retrack here.  Yeah.  The sort of mistakes.  Because you take just big space and put that same  somewhere and then retrack it.  Yeah, this would retract onto that.  So you look for the sort of.  So the point is you might have some set like this, and  you're going to have a retrack sort of sitting inside here.  And yeah, so you have a retrack sitting inside here.  And then the image of a retract in this even this two  dimensional case might be this sort of line.  And then if you have this line, you would say, well, what's  the degeneracy index here?  Well, d should be 0, of course.  What's the degeneracy index here?  Well, is d equal to 1 or is d equal to 2?  In this model, it's in an ambient space where it should  be 2.  But if you take a different local model here, you can  find one where it's 1.  That's why you need this minimum.  Any other questions?  How do you know that you actually have the.  So like I said, well, you told me that I can make all kinds  of weird spaces as these SD manifolds.  Maybe I was able to also make the corner of this quadrant.  The quadrant itself.  How do you know that's not it?  But those are the line definition.  How do you know that's not possible?  How do you know that I can't make something like an M  polyfold with boundary in corners making use only of  this definition which doesn't allow for the boundary in  corner structure?  I mean, I.  I mean, presumably you have to know that otherwise the  degeneracy index breaks.  No, you cannot make it better for the real corner.  You can take the over corner.  You cannot make the corner structure better.  That's the zero.  Yeah.  Can you say anything about the truth of that theorem?  Well, I ask later.  I think, yeah, don't worry.  I started late, so I'm only two minutes over based on when I  started.  But yes, I can go ahead and stop here.  Well, I mean, if you have a final statement, but I was  saying the audience should maybe allow you to make  a statement.  Yeah, so I guess the last sort of thing I wanted to state  about this is that finite dimensional manifolds with  boundary and corners have this nice property that if you  look, so here's a toy example.  Here's a toy example.  They have this nice property that along the boundary of  your finite dimensional manifold with boundary and corners,  you can do the following thing.  You can look at the set of all points of degeneracy index  1, and then you can look at the closure of this.  And the closure of this is, again, a manifold with  boundary and corner structure.  And so you can see that from this picture.  That's just this segment here.  And that's a property that finite dimensional manifolds of  boundary and corner have.  You'd like to lift that up to M polyfolds as well.  And that can be done, but there's an additional  condition that one needs to place on your definition of  retraction that define your local models.  So this is called the taming condition.  It's in the lecture notes.  I'm not going to present it here.  And the key thing to know is that the conclusion guarantees  for you this nice property that you look at the closure of  your faces, or the degeneracy index 1 portions.  And these are, again, M polyfolds with boundary and  corner.  That's one.  And the other thing to remember is that essentially all  retractions that occur in practice are of this special  type, they're sort of splicing type retractions.  And in these cases, the additional taming condition  is automatically satisfied.  So it really ends up being something you don't have to  check too much.  I mean, it should be completely straightforward.  And then you have this additional property.  OK, and that's where I'll finish.  Thank you for letting me run over.  Thank you.  Are there any more questions?  When should people read the lecture notes by Mr. Bees?  I assume it's done already, right?  The lecture notes are online.  Read them at your leisure.  And, yeah, and of course, if you have any questions, you  can ask.  And I will be speaking briefly in Nate's discussion.  I'm sorry for that.  But I have to make sure enough stuff is done for  Katrin to start tomorrow.  Says Katrin.  Any other questions?  If I take the, I suppose I have some SC manifold.  Sorry, just I hate to interrupt.  SC manifold or do you want M polyfold?  Because there is a, OK, thank you.  And suppose I take the algebra of SC smooth function.  OK, continue.  Does that determine the M polyfold?  I think logically that I'm going to work on one of these.  No, I mean, is this GELF-on doing it?  But what is the GELF-on doing, saying?  OK, so I think there are better spaces.  We don't even have smooth functions.  I think this is a good talk for T-Tag.  Let's thank Joel again.  So Nate was very generous.  And let me talk for a few additional minutes just to  finish up the stuff that I wanted to make sure you guys.  I had presented before Katrin tomorrow.  And as my voice is just about to give out, I think that's the  universe saying it's really time for me to stop talking.  So I'll try to be as quick as possible.  So the main thing, so we'll start over here.  Strangely, that's just sort of how things got laid out.  So it goes one, two, three, four, five.  So the main thing that I want to tell you about is the  statement of this theorem.  So what I've spoken about mostly so far is M polyfolds,  which I said this is like a generalization of a manifold.  And it's a generalization in such a way that it's going to  have boundary in corners.  And it has both topologically and  differentially, it has enough structure so that it will  contain the compactified, modulized space as a subset  with the Gromov topology just induced from the ambient  scale topology, or base topology even.  Great, so now you have manifolds.  But of course, if we want to study pseudo-Huller-Worfer  curves, say, for instance, for other moduli problems, we  want these to arise as zero sets of a nonlinear  fret-home section.  And so that means we need some notion of a bundle.  And so the main theorem that I wanted to point out here is  this implicit function theorem.  This is sort of the first step, which basically says, well,  what would you expect a finite-dimensional version to  say?  The finite-dimensional versions say, without manifolds with  boundary in corners would say, I have a finite-dimensional  manifold, I've got a nice smooth bundle over the top of it, I  have a section of that bundle.  And assuming that the zero set is the linearization of the  principle part of that section is subjective, then it  should be the case that you have an implicit function  theorem, which guarantees that the zero set is in fact a  manifold.  And then if the base of your bundle is a manifold with  boundary in corner, then you'd like it to be the case that  the corresponding solution set or zero set also has a  boundary in corner.  And so that's essentially what this theorem says.  So top of this board down to this line here.  That's just essentially the M-polyphold version.  Yes?  There's x equal y.  No, so y is the bundle and x is the  little x and little y.  Little x and little y.  The solution set equals y.  This is cap x.  I mean, I could rename this.  Well, x, if you like.  Is that the issue?  But x was a specific term for x, x, 0.  What was x in the way of the equation set?  It's OK.  It's at every point x in the solution set.  At every point x you want?  I'm fine.  OK.  Thank you.  Right, so that finite dimensional version of what  you would sort of expect for this sort of an implicit  function theorem is exactly what this says.  But then I've underlined all the words that at this point in  the course of lectures we don't know yet.  So I wanted to kind of go over those somewhat quickly.  And some of them I can go over quickly and some of them I  can't.  So for instance, a big one here is an S.C.  Fredholm section.  So this requires an entire lecture, but that's the  point of tomorrow's lecture by Katrin.  She'll be introducing S.C. Fredholm sections.  And then there's a tame strong bundle, which also takes a  minute, and that's essentially what takes up the bottom two  boards here.  So I want to pause on that for just a second.  And the remaining two pieces are good position and sub-M  polyfold.  So let me go ahead and drag this top board down just so I  can point a little bit easier.  So good position has a precise meaning.  It is in the lecture notes, as are essentially these  pictures here.  The point is that, I mean, I think the point is that in  order to guarantee that you're sort of the sub-manifold that  you get, this sort of cutout has a nice boundary and corner  structure, then the behavior of that sub-manifold, before  you've shown it, has a nice boundary corner structure,  needs to interact nicely with the boundary of sort of the  ambient space.  The boundary and corners of the ambient space.  And so general position would be sort of somehow it  intersects transversely.  And that's sort of ideal.  That's what you would expect.  I should probably mention this about HWZ sort of  terminology.  So when you read, when they say general, I think generic.  The general case, in my opinion, covers all possible  cases, but the generic one is the one that happens sort of  most often, but not necessarily always.  And then when HWZ say good, what they really mean is good  enough.  Right?  Because in general, I would say this position is better  than whatever I've drawn right here, which I'll explain in  a second.  But I mean good enough because it's good enough to get the  result that you want, which is that your sort of sub-manifold  has nice boundary and corners, a nice boundary corner  structure induced from the ambient boundary corner  structure on the M-polyfold, say, for instance.  Right, so that's terminology.  And so here's general position.  And so what does good position mean?  Well, basically what it means is that you have a tangent.  And it's a little trickier because in general, of  course, I've drawn everything with your subspaces being  like one-dimensional.  And so it's a little bit trickier to sort of make these  statements, especially since your ambient space is not  finite dimensional.  But the point is that you want it to be the case that you  have your tangent plane.  So it's going to pass through the origin.  And you want it to be the case that you can sort of wobble  it a little bit.  It takes sort of an open neighborhood of nearby planes.  And then you want to make sure that that open neighborhood  stays, or the corresponding rays, if you will, sort of  stays on the interior of your ambient partial quadrant.  And so that's why what you have this line right here is  supposed to represent this finite dimensional subspace.  And it's good because I can wobble it, passing everything  still through the origin.  And all those corresponding planes are going to pass through  the interior here.  That would be not the case here.  This is a not good case where you have, say, this ray sort  of travels precisely along the boundary.  Because now you wobble a little bit, and suddenly it's  gone.  Suddenly it's outside that partial quadrant.  I see.  So your picture, though, is that it's not a two-dimensional  cone coming into this two-dimensional.  It's a one-dimensional thing.  It's a sort of red thing.  And you're a two-dimensional triangle, sort of very  quick wobble.  Yes.  Oh, yeah.  I think that's right.  So it's really that when you say it's not good, it's because  that one-dimensional thing is coming along the edge.  Absolutely.  Absolutely.  Right.  And then, of course, here's something particularly not  good.  When you have sort of a line that sort of, apparently this  happens in the theory sometimes it has to be dealt with,  you have some line which passes like this just through this  point, and then you can sort of see even no perturbations of  this line sort of bring you into the interior.  So this is just sort of a picture definition.  The precise definitions in the notes.  But I wanted you to make sure that there at least be aware  of something like that.  Oh, and then a sub-M polyfold.  So I can read this.  Let X be an M polyfold.  So we're thinking this is an ambient space.  A is a subset in there.  And we want to know when is A a sub-M polyfold?  Well, that's going to be the case if, for every point in A,  you can find an open neighborhood V around A and an  SC infinity retraction R taking that open neighborhood to  itself so that the image of that is A intersect with that  neighborhood.  I mean.  How about your retraction hands?  Of course.  It just retracts right down.  It's much easier.  So I think if you think about this definition for just a  moment, you should hopefully convince yourself that this is  the only possible definition that it could be.  In some sense, replace X with SC bond-ock space.  And you should essentially get, you nearly get the  definition of just a retraction in a model case.  So all we're really doing is saying now you can also have  retractions in M polyfolds.  And that retractions in M polyfolds can cut out sub-M  polyfolds.  And that's good because this theorem tells you that that  subset that you get in your M polyfold is a sub-M  polyfold.  And then the bottom end of this guarantees that that sub-M  polyfold is essentially just a finite dimensional manifold  with boundary and corners.  Any questions about this?  So having a retraction on your polyfold means that locally  on charts, it is a retraction.  Right.  So I mean, a sub-M polyfold, because of the definition, it  must be the case that you can find a local model in which  it's actually an M polyfold as well.  No, but I mean, because when you define your charts on this  side, I'm just wrong.  Your retraction is defined on an open set.  Right.  But you should see that as long as you have something which  supports the SC calculus, as long as you have a domain which  supports the SC calculus, then you should be able to have a  notion of a scale retraction.  So it's just a one-step generalization from that.  Good.  So the last thing that I haven't explained then that I need  to is the notion of a tame-strong bundle.  This tame-ness, I'm not really going to talk about too much,  although it's discussed in the lecture notes, basically just  saying there's these additional conditions on your M  polyfolds to make sure they have a nice boundary corner type  structure.  So what I do want to talk about, though, is this term  strong bundle, which the first time you see it might see a  little bit strange.  And so I wanted to make you aware of something so that  when you see the definition, it doesn't seem like what the  heck is going on here.  So to do that, I can review some stuff that hopefully  we all recall just from the classical theory.  So the classical theory says a linear, I should have said,  linear Fredholm operator, it's a linear map between  Bonach spaces with closed image, finite dimensional  kernel and co-kernel.  So this is fine.  And it's the case that Fredholm operators are stable under  compact perturbation.  So I've said what I mean by this.  You can add on compact perturbations and you still  stay Fredholm, and it doesn't change the index.  So then if you have a classically smooth nonlinear map,  then we say this is Fredholm provided it's the case that  when you look at the linearization of this map at  any given point, the linearization is Fredholm.  So again, this is classical.  And so then we come down to this point here.  We say, well, let's not worry about domains too much.  I mean, I don't want to have topology of my domains.  I want to try and keep this someone as simple as possible.  And in that case, we see, well, the Cauchy-Riemann operator  is Fredholm as a map from something which takes maps  of regularity k plus 1 to maps of regularity k, maps, sections  of pullback bundles, and so forth.  But the main thing I want to focus on is the fact that  this is of regularity k plus 1 down to regularity k.  So now, if we see that, and that's true for any k that  we choose, then the natural choice is to try and fit this  into the SC structure, like the SC-type calculus.  So then even in toy cases, then, for instance, we think of  the Cauchy-Riemann operator as acting from one scale space  to another, where this is your first scale space, where the  kth level has regularity k plus 1, and the kth level in the  target has regularity k.  That's just this statement here translated into the SC  calculus language.  So that's not surprising.  But then something a little bit strange happens, which is,  well, what are our compact perturbations?  That's what we had right here in the classical sense.  And so now what you say is, well, we remember from scale  bond-ock spaces that the higher levels compactly embed into  the lower levels.  And so as a consequence of that, if we're going to perturb  our compact operator, we expect it to be the case that  it's going to be a map from one scale bond-ock space to the  other scale bond-ock space, except we've shifted by 1.  So it moves one level up.  And if you write this down, we look back in this case, well,  that just means that we're tacking on a perturbation,  which maps hk plus 1 to hk plus 1.  So you have a differential operator, which drops you  down, a compact perturbation, just adding  on a lower order term.  So that's not so surprising.  But then we point out one last complication just in terms of,  and this is really in some sense it's a notational  complication, and then you just have to push through, carry  through with it, is the fact that, well, the reality is that  in our applications, the Cauchy-Riemann operator isn't  just a map between, say, scale bond-ock spaces.  We really have to think of it as a section.  And so consequently, we have to think of it as a section  from locally, at least, from e into e plus f.  So this is our total bundle.  And if this has regularity hk plus 1, then this has to have  regularity hk, which is essentially what I've written  right here.  And then you have your perturbation.  But your perturbation sort of lifts you one up in the  target here.  And so I've written that down here.  So your sections have to go from ek to ek plus f1 plus k.  And this is sort of frustrating.  It's frustrating because that means in local coordinates,  in a local chart, it's irrelevant.  You say, OK, this is easy enough to do.  But the problem is that you then have to build a bundle.  And the thing is that what that means is you essentially have  a bundle with two different structures.  You have this structure that you have to keep track of, and  you have this structure that you have to keep track of.  And so the fact that you essentially have two scale  structures that you have to keep track of in order to have a  good notion of perturbations together with your  differential operator motivates the word strong in  strong bundle.  So anywhere in the literature, in HWZ literature, any time  you see the word strong, that means that you have two scale  topologies, two scale structures that you have to pay  attention to, essentially because you're  dealing with bundles.  Does that make sense?  Damn.  So this is to be able to have a notion of compact perturbations.  Yes.  But doesn't it automatically have two structures?  If you have FK sitting over EK or something in bundle, and  then you look at FK is governed by FK plus one in it.  But in practice, it's actually always apparent.  I mean, once you see one example, you know the more.  But on the abstract level, the formula is  S, G, or coincide.  So if you couldn't hear Helmut, he said, it sounds so just to  sort of highlight something about the polyphonic theory is  that on one hand, it's built on this totally abstract level,  and on the other hand, we make sure it contains all the  theories that we have, all the theories that we want it to.  And the downside is that in practice, when you actually  write things down, you probably could get away with not  having to keep track of these two structures because they  would be fairly apparent.  But in order to abstractify that and then have this abstract  theory in which everything fits, you need to have this  notion of having two different scale structures  essentially on your bundle.  Please.  So just as, right, isn't she that you can always get a  strong bundle structure on a bundle where the strongness  comes from this shifting by one, like you've got in this  example?  I think that would be more for sense.  You have to be careful that the transition that's preserved  the scale thickness, but that's a difficulty.  That's the taming of the tamed sort of the quality of the  transition.  I mean, taming the sense that it doesn't allow on stuff.  It seems that it draws on comparably nice.  For instance, I mean, for instance, it doesn't make  sense, like you might say, well, maybe you're lucky and  you've got a scale structure here and a scale structure  here, so maybe instead of taking the usual diagonal  filtration, we take a double filtration, keeping track of  both simultaneously.  But then you write down a transition map, and you see  that you get some of that, but not all of that.  There's this unfortunate fact that if I consider a map of  regularity, classical regularity CK, it doesn't make  sense to talk about a vector field along that map of  regularity K plus 2 or something.  That doesn't make sense.  That doesn't hold in transition maps.  And so then consequently, in your transition maps, you  don't keep track of the double filtration.  And if you look at what's the best you can possibly do in  applications, well, the best you can possibly do is  essentially to keep track.  In applications, in general, is to keep track of precisely  these two filtrations.  And then from this, you can get a partial double filtration,  which is the way HWZ do it.  So in the end, what ends up happening is you end up with  local charts that are written like this, where we use this  left triangle here to denote the fact that this, well, HWZ  will tell you that you have a double filtration, m, k,  where m is, I'm going to forget these.  Is this right, Helmut?  Damn it.  Hey, as best as you can, it does work.  Right, yes, yes, of course.  Yes, so you get this sort of double filtration, but it  doesn't extend all the way up.  And again, the reason is essentially because it doesn't  make sense to talk about vector fields which have much  higher regularity than the map along which they're defined.  That's essentially what this problem boils down to.  But also simultaneously needing to have a canonical class of  compact perturbations in which to do the theory.  So yes?  So this is implicit function here, right?  And this tamed, strong, bundle condition is a condition so  that we have compact perturbations.  Yes.  So how is it related?  We're not going to perturb it.  It's already perturbed so that it's transverse.  So for today, yes.  Tomorrow, no.  So tomorrow you will need to have a perturbation.  And the point of me presenting this is that I have to tell  you what the strong bundle is precisely so that you have a  space in which you can make perturbations.  Yes, precisely you can make perturbations.  For this statement, I suppose I could remove, yeah, for this  statement, I think I probably could remove this strong condition  and work with something sort of less.  I mean, you could probably work through the implicit function  theorem.  It's not clear to me where that would cause a problem.  But how do you define phretom?  I think it's sort of one project.  But phretom actually requires that when you bring it into  normal form, you use the strong bundle property.  That's on the target.  Yeah, that's right.  OK, yeah.  So inherent into the definition of an SC-Fredholm  section is the notion of a strong bundle.  It's inherently built into it, which you haven't seen the  definition, so I'm just being vague at the moment.  But I guess the one thing that I kind of wanted to clarify  here, which I doesn't seem I was successful at, is that when  you see strong bundles, they're going to have this weird new  symbol where you're thinking of sort of this as being a total  space, this is the total space over the base E. And  everywhere you see strong, it's essentially because you have  to keep track of two different SC structures.  And the need for having these two different SC structures is  the fact that you have both the Fredholm section and you need  to have a compact perturbation.  Hopefully I can get that message across, if nothing else  well.  Better luck next.  It's a little bit stronger because the Fredholm power, as  I said, the Fredholm power involves this and constrains  the flexibility of your corner changes.  So Fredholm is defined by looking good sufficiently nice  in certain corners.  And since in the target, you can only use things which  preserves a double-fit fraction, you can't really  completely wide sense.  So it enters there a little bit.  And then concerning your question, then having these  perturbations comes in when you do the softness.  But if you have surjectivity, I mean it seems like you  really need it.  Then surjectivity, but the Fredholm property involves  something of discovery, in its definition.  And if you don't have surjectivity, the perturbation  that you can achieve this, generate.  Generic, surjection, perturbation.  I'm a little confused.  So if you have a bundle, then you have a total space and  you have a base.  And if you have one filtration on the total space and one  filtration on the base, why isn't that sufficient to  recover the other filtration in this amount of  farther direction?  I think this was essentially your question earlier, right,  Duzza?  Yeah.  And I think the answer is that if you write everything down  in terms of the applications that you want to study, the  problem becomes much more concrete.  Why is it not formally positive?  Because you have coordinate changes.  So in one local chart, you have EK and FK.  And the coordinate changes are, say, CK.  And the fiber, it's things of quality, CK minus 1.  And therefore, at level K. And so at level K plus 1, that's  why you're allowed to go K up to N plus 1 at level K.  And CK, the CK coordinate changes,  because I have CK things at the bottom.  And basically, over CK things, you have coordinate changes  to CK.  It's K changes.  So the attention bundle is not a strong one, to  put it this way.  OK, so I went longer than I wanted to, and I really wanted  to hand things over to Nate, so I'm sorry for running over.
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