1/4 Polyfolds and the construction of Symplectic Field Theory
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Hofer, Helmut
Topics: Short overview. The category of stable maps. I/O structures. Covering structures. Additional structural constraints.
 So if you want to take notes, I would only write down what you don't see here. So, okay. Yeah. So, this SFT business, actually, maybe most people are not aware, started in December 1992. So, it was in Basel, and I gave a talk, you remember this? So, I gave a talk about the Weinstein Conjection, my first talk, and then Yasha and I were running through Basel. I think at this point, we already had an idea about contact homology, but it took them some time to go through business. So, now, yeah. So, of course, there was a certain number of issues. So, for example, in 1992, nobody knew even how to deal with the transversality issues in Gromov Witten and so on. So then, we knew that we should perhaps do some version of cylindrical, but it was always uns, not particularly satisfying. So, then in 96, and there were certain solutions proposed for dealing with transversality issues. And then, I think at that point, it became clear that in one way or the other, you could do things, but it wasn't so clear if the methods would actually be so great. So, I think in 2000, I was convinced that I could do SFT in a thousand-something pages in full detail. But right from the beginning, I thought of the feeling it would be sort of the wrong methods to use. And if you have to write a lot of things and you're never convinced about the project, it's really hard. So, then, with Sotzkan and Sender, we analyzed what are the issues in SFT. And SFT is a really nice problem from basically every aspect you can think about it. Namely, it has a lot of analytical difficulties, but all the wonderful algebraic structure comes, in fact, from all the difficulties you see. So, you can't say, well, you can, but you don't get very far. We put additional hypothesis and some of the difficulties don't arise. And then we can do a little bit and so on. But if you want to do it in full strengths, you have to deal with all these problems. And the problems in the analytical structures is sometimes quite tricky. So, there's a lot to learn if you actually go into details. So, then, ultimately, when this was analyzed, so you could think of how would actually things have developed, so just as a starting point. Suppose a lot of the difficulties wouldn't be there. It would just be a Fredturm problem. And what would have happened? Well, if it just be a classical Fredturm problem, then people would have dealt with this like they deal with Fredturm problems. So, you take a small perturbation and that's the end of the story. Once you put that into this framework, and then, of course, certain things would have done which you don't find in literature. Namely, when you have to deal with isotropy, you have to think about this. Then if you do SFT or this flow type series, then you have a lot of interlocking or depending Fredturm problems. If you perturb here in a particular way, then an infinite number of other problems have to be perturbed in a very particular way as well and so on and so on. So, I think if this technical issues wouldn't have arisen, things would have progressed differently. So, and now, how you deal with the classical Fredturm problem, if it's actually classical, it's quite easy. It's not difficult. You just take a small perturbation of the section and such smell tells you when you set up properly that you can make things generic. So, when we looked at this, we thought maybe one can achieve this as well. So, what are the problems? So, the problems were, of course, that corner changes are nowhere differentiable, which, of course, is the reason why finite dimension reductions were introduced in the first place because you reduce to spaces when you restrict that this transition maps, which are sort of canonical, are smooth. Okay, so then at some point, then this project became, so once we found some kind of a set of what was talked about last week that you can generalize differential geometry where the local models are retract, then this part of the project sort of got alive on its own. So, we did differential geometry, but we also looked at all these cases which would show up when you would actually apply it to SFT. So, that became one project. And there's another part of the project actually providing tools to put a concrete problem into this framework. And there we put also a lot of care into it. So, these are all things which are nearing the end now, namely, one of the annoying features of the analysis when you read any paper is that constructions usually are rather global. So, everything is interlocking. I mean, analytical, topological, geometric aspects are all in this. And then, you know, when you have to write it down, you have to write a lot. So, it's tempting to say one can easily see using standard method that that can be done and you refer to some paper which refers to another paper. And if you follow this, it's not so clear that actually it happened. So, therefore, in principle, when you look at elliptic theory, so you can make local estimates and just take a covering. And if you have estimates on this, the whole thing works. So, in some sense, you want to have this kind of things here as well. So, that also will work. So, okay, so then, ultimately, which was almost forgotten, is that we started out from a problem which now from this perspective is actually, I think, rather transparent. So, and I will talk about precisely how to deal with SFT, assuming that you would know this, this polyphore language, I will explain the things which you have to know. And then, of course, because of last week's mini courses, I think you have an idea about a certain number of the ingredients. Okay, so. So, here is a stable map. So, I want to explain, so it's good to understand what that all means. So, this here is a remnant surface. This, the red thing's a puncture. This is a building. Now, we use the European notation for the floors. This is floor zero, because certain things later should have degree one. Floor one, floor two, floor three. So, now, then there are marked points like this. Then, this arrow's here. So, so this shows you two half lines in tangent space. So, if you have a puncture and a half line in tangent space and you fix the disk, then you can put coordinates on the disk which map zero to standard disk which map zero to this point. But, when you have this line, you can map one in direction of this line. So, it completely determines what you have to do. Now, here, however, what I'm looking at is two arrows up to common rotation. But, you look from right and left, so they actually rotate like this. So, you look up to common rotation. And, what you want is, if you take polar coordinates which are coordinated this way and you run along say on S one is zero, the r direction positive or the negative, they hit the same point on the periodic orbit. Yeah, so I do not, so let me say that to the SFT paper, there are some differences in this case for the transversality. I do not number the mark points. I do not number the punctures. That gives more isomorphisms which, however, have to be compatible. The perturbations have to be compatible with this larger set of isomorphisms. If I number them, then if I take a renumbering, I should perturb it in the same way, it's a little bit more complicated to keep track of. So, if you just throw everything away and do transversality on that level, that's one thing. Then for orientation questions, then you might go to a covering which actually precisely means putting numbers there if you want. So, then. How much? No idea what you said the arrows are. So, here in the tangent space, you take a half, in each tangent space, you look at half lines. And once you fix a half line, then if you fix a disk and take polar coordinates, let's say take a disk map and then you compose it with E to the something, then if you run and you do it and you take the disk map in such a way that the real positive real line in the tangent space is mapped to that line that fixes precisely then polar coordinates. So now, but we don't fix them, but we fix them on two adjacent nodes or punctures and they are allowed to rotate commonly. But the effect of this is then if you fix sort of a coordinated system of polar coordinates, one positive, one negative, then if you run along zero across R plus minus, zero, zero element S1, then they hit the same point on the periodic orbit. So now, if the periodic orbit is simply covered, there's only one choice. But if it's double covered, you have this one, but you can also take this by 90 degrees rotated and takes the rotation class of this. So there are actually K many choices if the thing is covered K for. Is it the same to say that for each puncture, there's an asymptotic marker that you allow basically quotient under S1 action that rotates? The two, common, but yes, that is what you do. Yes, you have a common rotation divided out by the diagonal. So, but one you have to rotate multiplication by J and the other by minus J because of positive and negative. So, I was like, Hamoud, are you talking possible different maps here or just maps? It's just, I'm not done yet. I'm in the middle of talking. So now, so, so we have my, so then here, this blue stuff is what I call nodal pair. So there's on each is a pair one, there are two different points that could be on the same surface on the other, which you should identify, think of being identified. So the maps will be continuous over those. And then here in square brackets, that is what is close to Cartman's heart, are maps. And this maps are not assumed to be pseudo holomorphic, they're just maps. And then the square bracket around it means it's an R class. So because the image lies in R cross, odd dimension manifold, say with a stable Hamiltonian structure and you have an R action. So if you take the map defined on these things here and you was just shifted by an R by constant, it's also, so it's an equivalent thing. So this is the equivalence class. You also put down by a morphism, right? You also put down by a morphism. No, not yet. At the moment these are just, so remain in surface with all these features. Then the, so then of course, at some point you have to do some analysis. So you have to be a little bit specific about what the maps should do. So the quality of the map is chosen so that it can do some analysis. So what I pick is H3 lock, local H3 on this. And near the punctures, they should, if you take polar coordinates, should be of class H3 delta, which we have seen, so this exponential weight. And the exponential weights you pick have to depend, of course, on the periodic orbits. And in order to choose such possible weights, you have to fix an almost complex structure that gives you a linearized asymptotic operator, that gives you an asymptotic operator, which is self-adjoint. And the non-degeneracy assumption means that if you look at the part concerned with the context structure or with the hyperplane bundle in the stable Hamiltonian case, doesn't have zero in the spectrum. So each of those guys has a spectral gap, and you have to fix, and of course, if you have generally infinitely many periodic orbits, the gap might get smaller and smaller if you go to higher and higher periodic orbits. So for each of them, you have to fix something in the spectral gap. A weight. So. Can you say again what the equivalence means? Up to r translation. If there is, what is our target here? r cross v. R cross v. v is a stable Hamiltonian, with many for the stable Hamiltonian structure. So now, there's a stability condition, namely at least one, so each of these things here, so I think of this not being identified in the domain, so each of these things I view as a component of my Riemann surface. So then if you look on one floor, on each floor, there has to be at least one component which is not a trivial cylinder. So what is a trivial cylinder? A trivial cylinder is something like this here, which has a positive and negative puncture, which both are asymptotic to the same periodic orbit, and which is homotopic to a standard geolomorphic cylinder. Just homotopic. Homotopic to. So it doesn't have to have image r cross x for it? No, in between not, but it's asymptotic to the same thing. So if it's homotopic to the other one, the integral u star omega is zero. So I write triddle because your triddle is a comma u, a is the r component, and the thing without triddle is the stuff in v. Then for connected components, c, which is not a trivial cylinder like this one here, we have one of the following is true, either the integral of u star omega is positive, or if it's zero, then you must have enough special points on c, which are the punctures, marked points, points from the nodal pair, and so on. Is it clear what the object is? So this is an object in a category. So this here explains, so what this explains is the following. If you want to be precise here, you have to say that this puncture corresponds to this puncture, and you have to say that you have these two aligned arrows. Now these two aligned arrows, of course, you can give that by complex anti-linear map or complex linear map if you change the structure on one side, two minus i. So therefore what you could think of is, if I look at a punctures and I take the tangent space, then I can take the real projectification of this. It's an S1, but not naturally, but it has a natural S1 action on it. So you have something like principle S1 bundle over the punctures, and you want to have a complex, you want to have a, if you take here the complex structure J and here minus J, you want to have a S1 bundle isomorphism, which gives you a bijection between the underlying punctures, and it says which line corresponds to what line? I'm sorry, what is your J part of the data? J is, at this point, J is fixed, yes. What is J? I don't know, you keep talking. Yeah, okay, so, but I don't know what you mean by J, there could be possibly two different Js, cut, there could be a J in the target, and there could be a J on the domain. What are you actually fixing? Okay, okay, so when, okay, so can you just give me a little bit of a chance here to explain this? Yes, okay, there will be a precise definition, but it would be good before you see letters, you know what we're talking about, if that's okay with you, thank you. So now, yes, you can, but give me at least a chance to finish this, and after this you can ask questions. So there in particular, if I look at the punctures, this would be the positive puncture of the floor called I minus one to the floor, negative puncture on the floor I, then I projectivize the tangent space real, with this structure here, with the minus structure, and this would be the principle bundle, asomorphism covering this bijection. So the bijection would say which puncture corresponds with what, and this would align the arrows. All right, sorry, can you explain what your letters mean? What is S I minus one? What is S I minus one? Okay, okay, okay, Katrin, bear with me. So now, so now this thing which we just saw can be encoded like this. This of course, so we have a sequence of letters. The alpha is one floor, negative punctures on floor number I, S I remnant surface on floor I, J I the J on floor I, M I mark points on floor I, D I nodal pairs on floor, so this would be horizontal things, on floor I, the equivalence class of the map positive punctures. Then this one are precisely the ones which I explained here, which tell you how to align things in between. Sorry, who aligns what? I'm trying to take notes here. I just told you you shouldn't because this is online here. No, Katrin, no, Katrin, this is online, you just open your thing and write onto the text, yeah? Okay, so now. Okay, does anybody else still follow? Yes, yes. Good, so how about you tell me who matches what? What? I mean, this has been explained here. This, I think is, it tells you which puncture from the lower floor goes to which, and above is the alignment of the arrows. What's the V is a map? Projectification. It's a tangent space at the punctures of the remnant surface for this structure. Note here is a minus sign for the negative structure, and this is the principle bundle map. Principle bundle isomorphism. All right, and so where's that V in your next slide now? It's this, this covers that. Well, right, but in the next slide. Well, you see V hat? It's the, V hat implies V. That's supposed to be what's the previous, and that what, tells you how to attach alpha zero to alpha one? Yes, that including the rotation, which lines are mapped to what, okay? Good. So what is a morphism? So a morphism, so this is an object in my category. So a morphism, I write like this, so here's the source, the target. It's a sequence of bi-holomorphic maps. So they gave from here to here, which means they're map punctures to punctures, the negative punctures to negative punctures. On this part here, S-I-J-I, it has to be bi-holomorphic. M-I is being mapped to M-I prime, nodal pairs are mapped to nodal pairs and so on. And it should hold that if you, there is a constant on each level. If you shift your UI, then you get that composition. Yeah? What is the D again, nodal pairs? Nodal pairs? Nodal pairs. Plus this means shift in the real direction. Yes, that is on the, so for each floor, you allow it to common shift of everything. Yeah, it's by adding a constant to the R component. Yeah? Because we remember the objects were equivalence classes on each floor, yeah, so that is. So under, so these are the morphisms. Does the morphism have to preserve that matching with the asymptotic marker? Yes, yeah, yeah, okay, yes, it should. So if you, that's a good question. So if you have T-fi-I, it actually gives you, yeah. So it should map equivalence matching to equivalent matching. That's actually quite important. So for example, to answer this question, illustrate a little bit more. So if you have, okay, suppose you have two double-covered cylinders, so a building consisted of two non-trivial double-covered cylinders. Then, so in particular in the middle, I have a say double-covered periodic orbit. And there you have this matching things. So now, this can be isomorphic, it can be isomorphic to a thing which has this shifted by 180 degrees. But this will be a different object. If I have an automorphism of this one, that requires that this matching is being preserved. So for example, if one cylinder is simply covered, but the break is double-covered and the top is double-covered, then this object only has trivial isotropy. Yeah? So if here's simple covered, a double-covered periodic orbit and a double-covered cylinder on top, this has only trivial isotropy because I have to preserve the matching. So I can't really rotate downstairs. But it's isomorphic to another object which has this 180 degrees. So in this business, it will be extremely important not to confuse being the same and being isomorphic because for the counting things. Because when you do in the fine points of perturbation theory, if you be sloppy about it, you start counting wrong things. OK, so this is now a category. And clearly, every morphism is an isomorphism because you can write down the inverse immediately by just reversing the order of alpha alpha prime and inverting all the by-holographic maps. Now, what is not so clear but which is a consequence of the stability condition in the nice exercise is that between two objects, there are only finally many morphisms. So of course, I call it group-oil. So group-oil is usually a small category which would have this property. I call it group-oil because it's not a small category. But then this category admits a grading because I can talk about the top floor number. So if I just have a building of height 1, the grading would be 0. That means ground, I only see the ground floor. If I have a building of height 2, the d would be 1. That would be the number of the top floor. So European notation. Then having this grading, my category decomposes. And let me artificially define what I mean by the boundary of the category. These are all objects in SI for I at least 1. So then there's also an important point. There's also an important point, namely, if you look at the orbit space, orbit class of this category. So the orbit class consists of all equivalence classes of isomorphic objects. Then this is a set. Now you can build on the remain surface lying in R3 or some sense, realize in R3. So it's clearly a set. The morphisms could preserve the matching, but not the asymptotic markings? The asymptotic markers not here. So in contrast, as I said at the beginning, to what we are writing in the paper, we have the asymptotic things. We have them only numbered, but we have markers there. And we fixed points on the periodic orbits. We don't do that. Then these half lines. So that we had before as well. But we had on the top, our mark points were ordered. The punctures were ordered. And there were asymptotic markers as a punctures, which were assumed to hit a particular periodic point on the periodic orbit. Markers are only punctures that are matched. Yeah. I think you've got both what the question was and what the answer was. No, he obviously read the book. Oh, I see. No, somebody asked. I'm asking about the part, the ones which are marked in this setting. So you have these B's, which kind of match up. Yes, they just met up which puncture to what. And then the B hat is a lift of this. It just says it matches up lines up to rotation. And then if you take a holomorphic polar corner, suppose negative 1 adapted to this, then if you run along 0 into direction to plus minus infinity, then they hit the same periodic all. It's the same point on the periodic all. I think the question is, does the morphicine's new category have to carry this data? Yes. Yes. So the B hat. The B hat has to be preserved. So the choice of polar coordinates is necessary. So you can have just two arbitrary different choices and be automatically asymptomorphic. So if you have two of these things which generate this class, then if you apply this morphism here, i minus 1 and 4i here, then it is mapped to a pair of lines which is that associated to the object. This is a new object. So can I ask that again, maybe differently? So let's say you have a morphism from alpha to alpha prime. Right? Each with two levels. So then there's one matching data. So if I fix my matching data B for alpha, are there different matching data for alpha prime that I might have morphism to? Or is the matching data? No. I just said, if you, for example, take. So suppose this is double covered, covering numbers two. And then here I have something where covering number is one, let's say. So this is twofold covered in this one. So then suppose I have this matching here. Then obviously I have also a matching where this, where the one appears here. I don't know if you have colored chalk. But then you could, so one matching is this common rotation. The other would be you put this thing on the other side. This common rotation. Let's say I fixed my matching there. And now I have maps phi that take both the top to something and the bottom part to something. Right. The something is some stuff this has already given a matching. So the lower part maps should, if you fix two lines here, the lower thing should map it to a line here and the upper to a line there, which precisely gives a matching on that object. It's a different object. It has all this data. So it's not so difficult, Katry. So if you. Right. So question. If I fix everything in your morphism, if you go back one page. If I fix everything. If I fix alpha prime. Right. Let's say, so the phi i. If I fix the phi i. Right. I fix phi 1 and phi 2. I fix alpha 0, b hat 1, alpha 1. I fix alpha 0 prime and alpha 1 prime. Does that determine b 1 prime? So the first thing is in a given object, things would be fixed. So now if you want to talk about morphisms between this, you have to preserve this data. Now what you ask is, maybe, that if for my alpha this is fixed and I want to build another object, then in fact, what you can do is you can take different morphisms on each level, map it to something else, and adjust the marker so that this is isomorphic to the thing you started. What I'm asking is whether the phi, together with the choice of source, determine the target. Maybe answer just yes. Yes. The answer is yes. OK. Pelman, in this example, the top cylinder, it's just a trivial mapping cylinder? Or? Yes. No, no, no, no, trivial. Just something which is double covered. Yeah, so this would have an automorphism. And this itself as well. But as a thing with the marker here, the automorphism group is trivial. But this is isomorphic to a different object where this one, let's say, is rotated by 180 degrees. But that is a different object. It has different data. OK, weighted subcategories. So it's something like Facebook, like or not like, but with rational coefficients. So you look at q plus between 0, 1, which is just the identities as morphisms. So you look at a functor there. And you're interested in, and if I'm given such a thing, then I have a full subcategory, which are all the things which I like to some degree, positive. And they are carrying a weight how much I actually like them. Yeah? So example, given J-triple, which is almost, I invent almost complex structure, R cross V, and a compatible with the stable Hamiltonian structure, we can talk about J-holomorphic objects in S, which are the objects that are used to satisfy the Cauchy-Emen equation. So then define this by associating to pseudo-morphism the weight 1 I like and to other, this thing I don't like. So that's our modelized space for the time being. OK. So now the whole theory will consist of basically taking a partition of unity for the one and moving, now each point is represented by several which are fractionally liked, and moving it into a better position that each of these things looks like a manifold. So that needs, of course, some technology to do this. OK. Good? I felt hope. Yeah? Katrin? You said there was something about the aim. About the same of the aim. The aim is, that is what I said before it showed up here. So they form this into a theta with additional structures from which we can derive the data we need to construct SFT. So it turns out that you can later define theta's, you can define differential forms on these categories, and you can integrate them. So there's a really cool, nice theory for this. So OK. Aim clear. Now, so there's more structure here. So when I have an object in my category, let me define a different category P here, which consists out of the following. It consists of maps from finite sets into periodic orbits. And the morphisms are by ejections between these finite sets so that that is true. So this is actually already part in some sense of the data, namely, if I look at the positive punctures, I just take this point here as my point, as my set I. And the evaluation plus map is this map here, which associates to each of these points the puncture, and the same here, at the negative part. Now, this category might, for example, if the different points in this finite set go to the same periodic orbits, you might have different morphisms. For example, if I have two points here and they go to the same orbit, then switching the two points would be more fizzier. Is the point of all this category of the language just to keep track of isotropy? So in this category, I will explain everything to you, and then I will say there's a notion of a smooth structure on this category. And then after this, I'll explain a little bit that there are smooth versions of this like or not like functor. Then once you have a smooth structure, you can talk about differential forms on this category. For example, the pullback by an evaluation map from the many-fold of differential form will be differential from the category. And when you have orientations, you can actually integrate. You never have to leave. So once you have a poly, so it will later call be a polyphonic structure on the category, you never have to leave. It's all kinds of universal constructions. You can do the whole transversality on that level. Yeah. There's nothing. Well, when I make my definition, I assume that it was non-degenerate form. No. Moxbot I haven't dealt with, and I don't want to. So. Oh, yeah. So that is what was discussed last week. So we have an odd dimension manifold with a stable Hamiltonian structure, and that defines the so-called wave vector field, and that has periodic orbits. OK. So now, so now, so we have an input and output. We just look at the restriction to the punctures. We get this map, which says to which periodic orbit it goes. And then we can build chains like this, and it's called weak-fibre products, and they look like this. So. So observe here that such a map bi, which we introduced before, this bi lies below the bi hat. So if you forget about the matching of the lines, is in fact such a morphism. So we look at chains of objects, and this morphism is a category B that is called, and it takes a fiber product. Yeah. So we have an object. We map it forward, which would give the map which associates to the positive punctures of the periodic orbits. Then we have a morphism from there to another such a thing, which comes from the next one, but going down. So we have here different, so these are not floors, so we have here different buildings. Objects, so these are objects in my previous category. I map them down, then there's an, into this category here, and there is a map mapping it to something which came from the negative evaluation. So note that this thing here, it looks like a new element in your category S, but it isn't because the head is missing here. We don't know, we don't have information how the lines are being matched. That is not there. That actually will lead to the fact that there is a relationship in our previous category, and this, we are some category, we are some covering. Functor. Is that clear? Excuse me. Yeah. So it's clear, but I don't understand why you're doing it. Is it to describe the difference? Well, I mean, this is where D squared comes from. But I mean, oh, so you're saying that in the back structure you don't offer you already have matching? No, you don't, because if you, no, in the boundary you have, but then now you have to relate it to the full category, which says if you have a building in the boundary, then you have to relate it to S. And in fact, how can you do this? Well, you'd chop your thing in the boundary, you'd chop it apart here, but once you do this, what does that mean? You just took a weak fiber product. You lose the matching condition. Yeah, because in this case, at the positive negative functions, I don't have, I didn't keep track of this. So now in the original version, we had asymptotic markers there, so that information wasn't lost, but then you get a covering in the other direction, because when you have everything marked and then you glue it together, you have to forget where you glue it together at the order of the periodic orbits, then you get a covering there. So there's no free lunch here, but this thing, take all the numbers away, all the orders away, has the advantage that I have a lot of morphisms, and my perturbations have to be compatible with this. They also have to be compatible in the other one, but there it's a little bit hidden, because you have all this additional structure, and then you have to say, if I take here all the numbering away from this object, all the numbering here, they should have been perturbed in the same way. Why not just for the transversality, just take everything away, which actually is irrelevant for that? Okay? Okay, so now that is what we are looking at now. We take fiber products of lengths, so there's two elements, three elements, and so on, this weak fiber product. So now S was graded by D, the top floor number, and now let me look at, take multi-indices, and I is greater or equal to zero. Then if I, because I want to, so this has actually two filtration categories, so then I can talk about the norm of such a multi-indice by just taking some of the numbers, and I can talk about the lengths of this. Can you avoid this problem by taking this S to be category with objects, as they add the, remember these mark points, isomorphisms, turn the circle, and two morphisms between the isomorphisms. So you want to go to a two category? Yeah, I'd ask you a two category. I don't know. I didn't think about it. So then given such a multi-indice S, I can obviously build this product here, so this are degeneracy N0, which is NK, and then SP can be decomposed like this. And there is, okay. Now. Can you say in words what that means in terms of broken terms? Yeah, you see pictures. I will show you in a second. So then there's a degeneracy functor which you can define on this by just taking the degeneracy of the objects, and there's a length functor. Just look how long is that product here. So here's a picture. Oh, maybe. Okay, so here is how you have to understand that. So this here is a building in our original category. It has the top floor number six, and I put a head here. This is the BI, these are the BI heads. So now I give a multi-index 1002. So what does it tell me? It just says I should chop that into four buildings, and the first building should have top floor one. The other should have just gone floors and the other top floor two. So that is a certain requirement on this here, namely the no, namely there should be at least top floor six. So now I chop a building with top floor one, five-word product with ground floor, five-word product with ground floor, five-word product with a thing of this one. This organization is important for the perturbation scheme because you have to work your way up with all these compatibilities on this thing. So you have to know how you chop and what perturb, what perturb, and what perturb on. Is that clear? So when I see a round dot here, I should think one trajectory. A round what? What are the dots? The dots are connected correctly. Where's dot? So this here, this is a building with height seven top floor number six. So this is like the first picture you saw except that had less floors. So if I just want to try and think how to prove this when equal to zero? No, you don't want to prove anything like this. It's a little bit early for this. You will let me, you will see that a little bit already. But the floor is kind of complicated. Each of the floors can be many, many things. So, but it's sort of an easy procedure. So if you have enough of those guys, the right number here and such a multi-index. So the norm is three, the length is four, and then you have to subtract one is six, which is precisely the number here. So if the norm plus the length minus one is precisely this, then you can chop in this way. If I get another one, if I put zero here and one here, then you would have a ground floor here, then this here would be a building of height two and so on. So there are different ways how you can chop this thing. Then the perturbation later, which you would see over this object, could be pulled back from this one and any other way it was cut. That shows you already some compatibility things. So now let me, the notion of a covering functor. So, so finite covering functor. So you have two categories and over each object you have, so it should be surjective on objects and over each object you have lying a finite number of points. And this condition should be true, which looks a little bit difficult at the beginning, but it says the following. If you fix the source of all morphisms, like I fix, so I, I, I am interested in this point. I take a point which lies above it and look at all morphisms which start from this, which is this one, this one, this one, and here's the identity. Then this should be in bijection to all the morphisms which you see here. And so you're writing your compositions with the source on the, the source on the right and the target on the left. What source on the, no, sources, I was writing like this. So this, this here, okay, so this are the morphisms, this is a morphism set of this category. This is a morphism set of this category. Then it has a source map into D and F goes into the objects. So it takes a fiber product here and I do this. So, so now if you fix, if you fix this point and take a pre-image under this covering, then if you look at all morphisms starting here, then you get a morphism to the object starting here. What are the significance of your different typefaces? Because you've got italic typefaces which are the categories. Yeah, so it's okay. So this is maybe I'm too didactical here. So, so, so a normal C would be the objects and this bold face would be the morphisms. Oh, well that's what I want. It's the bold face with more faces. Yeah, so these are objects, these are morphisms. So forget about the definition. Actually, we already saw such a functor, namely something like this. And these are all functors of this kind. So take in your category, so, so we have our category S, we have the degeneracy index and we have the evaluation maps. So now a covering structure for this consists of the following and it's clear what it should be. For integer greater or equal to zero, Fij takes an element of the degeneracy i plus j plus one and it chops it precisely at the right spot to produce a building with top floor i and top floor j. So if I have a building of a certain length, I can say I chop it here, I chop it here, I chop it here, I chop it here. So, and these are actually covering functors. It can just take this definition from here to here. So they're precisely satisfied with the definition. So you see here already that in this formalism, you keep very much track of the isomorphisms, which is actually very important for counting later on. Sorry, so the degeneracy is just the number of flows? The degeneracy is the top floor number using European notation. So it's one less than what you would say the height of the building. It doesn't count in general number of points. No, no, no. So then it's clear if I have something and I chop it here and then I chop it there, you could also chop it first there and chop it there, which is precisely this thing here. So you have a comp... So chopping, if you choose the right indices, it's commutative. So then using this rule, actually you can extend this chopping and it's already the picture inductively for every multi-index. So there is from Si plus this, this product, there is such a chopping. So, and that is precisely this here. So if I take F1002, it is on S6 and it chops it in this way. So these are all covering functors. So you start with an initial so you can check. If you just use, forget anything else, just algebraic, these are all covering functors. Created from a small list of covering functors, name it those guys. You're writing equals everywhere, but you have a bunch of more reasons to categorize. No, that's actually equal. That's actually equal. No, just, you know, if you have three buildings with three things, chop it here, then you get a building of height one, height two, and then chop it here. I could have first chopped it here and then there. So that's precisely this identity. So in terms of actual data, does the F simply forget one V hat and just keep track of the V? Yes, so, yeah, what does this thing actually do? So this here takes the right spot in between, but you have to cut that on one side, top floor is I and the other J, and it forgets precisely the hats. Only the one at the right spot, which is determined by giving I and J. That is what it does. You don't change the object, nothing. You just forget the hats for the B. And that is obviously rather commutative. I forget here and forget there or the other way around. So, okay, so, if you look at the J-trill case, the pseudo-holomorphic case, you see that immediately. So this, if you have a building where each of the functions U is J-holomorphic, if I forget the hat or not, it doesn't change. This thing is stay holomorphic, so obviously that's true. So now we come to some formalism, which is actually quite useful. So if I'm given such a Facebook function here, and then let me define the boundary of this, it's a restriction to the boundary of the category. So now, given this structure here, consisting out of this maps here and then what I can build out of this, I can define that thing here. Okay, look at it, it will be complicated, and you try to understand this. Sit down. It doesn't say more than precisely that this is true for the theta. Forget the J here. So I guess the compatibility that if I pull back theta times theta by Fij, I get theta on the corresponding Sij plus one. So this thing here, that this is equal, just means compatibility with this thing here. So if you have a theta, then obviously define a theta on the fiber product. You just take theta of a fiber product as a product of the citas of the particular values. So I define theta p. And then, so here's a type where it should be a p. Sorry, here should be a p. Then you pull this back, so you get to the boundary of the category, you take the sum here, and you want this condition. It just means that Fij composed with theta p is just theta restricted to Si plus J plus one. That is what it means. So there's a short form. So this quantity means high compatibility with this structure. So this is part of the conditions from which d squared equals zero comes from later. So it looks complicated, but it's a really short form to formulate it. I'd rather have to say for all ij, this and this holds true. Just say that. Now this formula occurs in a lot of way. You can do a Freton series satisfying this formula and so on. So you can include an x-ray in front of that sum. So model this thing that here should be theta sub p. You just build, you see this takes values in zero one, rational, and it just takes the alternating sum, and that's the definition. And what you require, what you say is compatible is, so this thing is defined on the boundary here. Because the Fis come from an Si where i is at least one. So this here defines a new functor on the boundary, and you say it's compatible if this is true. And this is a non-trivial condition because when you look at this here, you take fiber product of length two, then you restrict it, so that theta is given. Then you restrict, then you pull it back, you get something on the boundary, and that was the supposed value there, and you work your way up. So there's a lot of compatibility. So that's, but it doesn't say more than compatibility with the Fijs. So now, trivial cylinder buildings. When we do a perturbation theory later, or for the algebraic structure, I'm going to have a modernized space out of some, a modernized space of j-holomorphic objects. And I multiply on each level, I put say the same j-holomorphic cylinder in. I get a new modernized space, and obviously it should be the same up to that multiplication. It should be identifiable with the other one. Yeah? I was just trying to follow what was going on with the picture. Yeah, no, I haven't explained the picture yet. I haven't even started. So when I do a perturbation theory, and for example, this building of height one would be pseudo-holomorphic, then, so I would have this modernized space of this kind of things. So if I multiply it by some j-holomorphic cylinder, that should also be in my modernized space. Because otherwise, you never get the algebraic structures that you have. So if you are, if I multiply it by several j-holomorphic cylinders, that should also be in the modernized space. It might have different signs and so on. So if I integrate over that modernized space having this, it should be basically like integrating over this modernized space, say, up to sign, for example. Can you just explain that data formula again? For one last page. Suppose I have something with, like, you know, three levels. So there's various things to be assigned. To be assigned product of data is at the three levels. To be assigned data at the first two levels times data at the third. So what is the relation between all of these things? If this is true, it just means that for the two, for things, product of two things, is equal to theta on s i plus j plus one for all i j. That is what it means. It's a convoluted way to say this all at the same time. But it's a good formula. Can you explain the minus one to the length of i? Yeah, so it has something to do with this. Suppose I have a function defined here and a function defined here and I want to extend it over this. Suppose f is here and g is here. This is t and this is s. How do you just define g of s plus f of t minus g of zero? You assume that they coincide here. That is a smooth extension of this. So it's an included expression. Yeah, so if you do this now for higher things, you get this alternating sign. So that is precisely, you see, if you have this theta p defined, there are a lot of points where they have multiple definitions but they coincide there. So if you write this down, you get this. So that's the underlying idea. OK, so now, trivial cylinder buildings. So first of all, there are... So let me first say, a trivial cylinder building would be one. So if I have height three here, would we say something like this where top and bottom have the same periodic orbit and each of them is homotopic to a standard geolomorphic cylinder. That's a trivial cylinder building. And geolomorphic cylinder building would be where each of this is actually geolomorphic. So there's one more question for this theta. If the theta is like zero or one, then to say theta on a building is one means it's one on each piece of the building. So if theta is zero one and there is in the product, so then if I have a product and one in both of them would be one, then the building above for where it comes from would be one with the number one. Then it's not, isn't it? Then it would be disliked. So, OK. So in the perturbation theory, if this is a liked object and I add geolomorphic cylinders to it, a geolomorphic trivial cylinder building, should be liked as well. So in the perturbation, if I like this building, so here let's say geolomorphic cylinders, and now I add a geolomorphic building to this, then if this was liked, this should be liked. It's the same weight. That is, otherwise you never get any algebra structure. Otherwise you can't get the multiplicative structure in SFG. So these objects are different though. They are different, but in some sense they have to count the same way up to sine. So up to sine. So how can we formalize this? If alpha and alpha prime differ by geolomorphic cylinder building, then theta value coincides. So here you have to notice that there is not a unique way to put a geolomorphic cylinder to it, because you fix an equivalence class here and you have a map on it and you can shift it. So there is on each level at least an R parameter. So that is indicated by this. But I think everybody understands sort of, when I say add a geolomorphic trivial cylinder to it. But there's not a unique way. There's a family. This is just explained. I just said that you add to alpha a holomorphic cylinder building, a trivial holomorphic cylinder building. Is TCB a set or a one? Trivial cylinder building geolomorphic. So it's a set of the geolomorphic. It's just a line. If this building has height k, it's a line of k cylinders, which are holomorphic and trivial. Yes, but there's not a unique way to add it to it. You get families. For each level you get a family, an R family. So the TCB there knows how many levels alpha has and that's how many. Yeah, so this is a short way to write it. So I added trivial cylinder building to it. And then for any representative of this, because it's not unique, the value should be the same. So you would write times R times the set of cylinders. What times R? No, it's R to the k because for every... Yeah, so if there's building of height one, you have just a one parameter family of building of height two for each of the single level R. So I'm just keeping it simple by this. Just add. So whatever this symbol means, alpha, district, union, TCB, so... So you have some set of your object category. So here I added... So I would add one trivial cylinder, but even here it's already not unique because here you have an equivalence class of maps. So you have to choose a representative, take this to this and divide out by the R action, and you get an R family. Then if I have another level, I do the same. Okay, so my requirement is that theta should have this property. Then if alpha contains a trivial cylinder building which is not necessarily holomorphic, but if theta of alpha is positive, then the trivial cylinder component are all j-holomorphic. So on that level, it's a little bit more complicated to formulate than later on a linearized level where you can actually write down some functors, and then that's true. So okay, so we asked this again. When you write QCB, really you mean to just put one chain of trivial cylinder? Right, yes, and then you can do it again and put another chain to it and so on. And so if it contains a trivial cylinder building, which might not be j-holomorphic, but if it's liked, then the trivial cylinder components are all j-holomorphic. Okay, so these are properties. So just I should, yeah, you used up your time, I think, so if you want to say anything in the next minute. I know it flew by for all of us. When did you actually start? You started at 30 after. You finished up very briefly another minute or so. Yeah, let me. You have other lectures coming up this week. I know, but this is frightening. So, time-wise. Okay, so let me just... So now I'm using a theorem of Joel Fish. No, okay, so... No, okay. Okay, so give me a few minutes here and then I'll stop. So now, so if you have two non-trivial components, each building level has a non-trivial component, which is not trivial cylinder-like. Then you can move them against each other, creating, again, a family on each level, an R family. If you have several components, you can move them against each other. And what you want is, if you have two stable, so if alpha and alpha prime are stable, and you take sort of this union of things, so on each level you're allowed to shift them against each other, then the theta of alpha should be the product. So you have to think of this. Theta, in some sense, would say find pseudo-holomorphic things. And if it's pseudo-holomorphic and it is disconnected on each level, then each thing individually already existed, and that's about to happen, and so on. Okay, so summary. Okay, so summary. Okay, so then I'm stopping. So we are interested in deformation theta of theta-trivial. This is the original function which identified geolomorphic objects. We want to deform it to a theta, having the following properties. This is true. Then this is true if we add just a trivial holomorphic cylinder building. If this is positive, then this has to be true. Then this thing actually had to be pseudo-holomorphic. And if I've stable things and move them against each other, this has to be true. So SFT algebraically comes from this. That's all there is. Okay. This is more or less the same as saying the whole theta is determined by the value of theta on one floor thing with one connected component. Is that the true statement? Yeah, that's a good one floor. Okay, so let me not buy into this here. So I have to look at the algorithm. So the algorithm is a little bit... I'll give you the algorithm how to perturb. So the algorithm would precisely... Yes, so in the first time to perturb, you have disconnected objects. So in the hierarchy concerning some complexity or energy on the bottom floor, if there are things cannot degenerate further, there are disconnected things. Connected things which are perturbed. Yeah, it comes from there. All right, well, why don't we thank all the friends? So that is what I wanted to do in the first 15 minutes. So are there any brief questions for Helmut? Yeah. I'd expected you to write down a brief list of axioms in which all but one were manifestly satisfied by the sort of defining property for the pseudo-holomorphic case with which you start and which you just decide one the holomorphic things and the other would be the thing you need to achieve by the definition. Is it easy to... So if you take this properties here, you can see the pseudo-holomorphic. Precisely. But of course, we don't want to take the trivial deformation. So you want the deformation which happens by these and which lasts and so on. Okay, so that's a good... So my... As we've just reminded, I have four lectures. Of course, it's not so clear if I get to the very end of it. But at this point, what we have done is we... I have sort of in this category setting said what things should do. So now, the next thing is there's a notion of a smooth structure on a category but you can only do this for... when you have this polyphonic theory because other things you cannot do smooth. Then you can start talking about smooth objects of this kind. And smooth objects of this kind... So these are sort of smooth sub-categories and what are they? So they actually occur in different flavors. They are manifold type, orbifold type, branched orbifold type and so on. And these things then will be constructed by a Friton theory. So you can transpose these properties into the language of what you should do on the perturbation level, on the linear level. And there's a nice functorial description. So this is obviously sort of functorial here. This is a little bit odd, but there's a nice formulation on the linear level for this. And then you define a bundle over this. You have the Kuschelrimann section functor and you want to perturb that one. And then if you... and that is how it looks like at the end. So you have... So this is basically over each object here. The fiber is... are the zero one forms along the underlying map. So here you have the Kuschelrimann functor. And then you define something like this Facebook thing here. It's... it's... but it's a section. It's... it delivers on the fibers and it associates to each fiber to a certain number of points, a rational number, adding up to one. And then this here is your theta, if this is properly chosen. If you put here theta zero, which just is the zero... which just gives the weight one to the zero section and to everything else zero, here is actually our theta j-tridle. So if this here is theta zero, which just puts weight one on the zero vector and weight zero on the other, then this gives you... this comes first with this gives you that one which you already saw. So here is a small perturbation of this. And since in the fiber everything adds up to one, what you literally do is you take a partition of unity for your theta j-tridle for each object and you move the objects. Now you have several one with fraction of eight in different directions and they line up to many fold pieces locally. That is what they do. All right, so I'd like to make sure we don't have enough time to get enough coffee. So I think we're going to cut it off here if you have any further questions for Helmut if he's around. And we will attempt very strongly to meet back here at 11 for the next talk. So let's thank Helmut one more time. Thank you.
 So if you want to take notes,  I would only write down what you don't see here.  So, okay. Yeah.  So, this SFT business,  actually, maybe most people are not aware,  started in December 1992.  So, it was in Basel,  and I gave a talk, you remember this?  So, I gave a talk about  the Weinstein Conjection, my first talk,  and then Yasha and I were running through Basel.  I think at this point,  we already had an idea about contact homology,  but it took them some time to go through business.  So, now, yeah.  So, of course, there was  a certain number of issues.  So, for example, in 1992,  nobody knew even how to deal with  the transversality issues in Gromov Witten and so on.  So then, we knew that we  should perhaps do some version of cylindrical,  but it was always uns,  not particularly satisfying.  So, then in 96,  and there were certain solutions proposed  for dealing with transversality issues.  And then, I think at that point,  it became clear that in one way or the other,  you could do things, but it wasn't so clear  if the methods would actually be so great.  So, I think in 2000,  I was convinced that I could do  SFT in a thousand-something pages in full detail.  But right from the beginning,  I thought of the feeling it would be  sort of the wrong methods to use.  And if you have to write a lot of things  and you're never convinced about the project,  it's really hard.  So, then, with Sotzkan and Sender,  we analyzed what are the issues in SFT.  And SFT is a really nice problem from basically  every aspect you can think about it.  Namely, it has a lot of analytical difficulties,  but all the wonderful algebraic structure  comes, in fact, from all the difficulties you see.  So, you can't say, well, you can,  but you don't get very far.  We put additional hypothesis  and some of the difficulties don't arise.  And then we can do a little bit and so on.  But if you want to do it in full strengths,  you have to deal with all these problems.  And the problems in the analytical structures  is sometimes quite tricky.  So, there's a lot to learn  if you actually go into details.  So, then, ultimately, when this was analyzed,  so you could think of how would actually  things have developed, so just as a starting point.  Suppose a lot of the difficulties wouldn't be there.  It would just be a Fredturm problem.  And what would have happened?  Well, if it just be a classical Fredturm problem,  then people would have dealt with this  like they deal with Fredturm problems.  So, you take a small perturbation  and that's the end of the story.  Once you put that into this framework,  and then, of course, certain things would have done  which you don't find in literature.  Namely, when you have to deal with isotropy,  you have to think about this.  Then if you do SFT or this flow type series,  then you have a lot of interlocking  or depending Fredturm problems.  If you perturb here in a particular way,  then an infinite number of other problems  have to be perturbed in a very particular way as well  and so on and so on.  So, I think if this technical issues wouldn't have arisen,  things would have progressed differently.  So, and now, how you deal with the classical Fredturm problem,  if it's actually classical, it's quite easy.  It's not difficult.  You just take a small perturbation of the section  and such smell tells you when you set up properly  that you can make things generic.  So, when we looked at this, we thought maybe  one can achieve this as well.  So, what are the problems?  So, the problems were, of course,  that corner changes are nowhere differentiable,  which, of course, is the reason why finite dimension reductions  were introduced in the first place  because you reduce to spaces when you restrict  that this transition maps, which are sort of canonical,  are smooth.  Okay, so then at some point, then this project became,  so once we found some kind of a set of what was talked about  last week that you can generalize differential geometry  where the local models are retract,  then this part of the project sort of got alive on its own.  So, we did differential geometry,  but we also looked at all these cases  which would show up when you would actually apply it to SFT.  So, that became one project.  And there's another part of the project  actually providing tools to put a concrete problem  into this framework.  And there we put also a lot of care into it.  So, these are all things which are nearing the end now,  namely, one of the annoying features of the analysis  when you read any paper is that  constructions usually are rather global.  So, everything is interlocking.  I mean, analytical, topological,  geometric aspects are all in this.  And then, you know, when you have to write it down,  you have to write a lot.  So, it's tempting to say one can easily see  using standard method that that can be done  and you refer to some paper which refers to another paper.  And if you follow this, it's not so clear  that actually it happened.  So, therefore, in principle,  when you look at elliptic theory,  so you can make local estimates and just take a covering.  And if you have estimates on this, the whole thing works.  So, in some sense, you want to have  this kind of things here as well.  So, that also will work.  So, okay, so then, ultimately,  which was almost forgotten,  is that we started out from a problem  which now from this perspective is actually,  I think, rather transparent.  So, and I will talk about precisely  how to deal with SFT,  assuming that you would know this,  this polyphore language,  I will explain the things which you have to know.  And then, of course, because of last week's mini courses,  I think you have an idea  about a certain number of the ingredients.  Okay, so.  So, here is a stable map.  So, I want to explain, so it's good to understand  what that all means.  So, this here is a remnant surface.  This, the red thing's a puncture.  This is a building.  Now, we use the European notation for the floors.  This is floor zero,  because certain things later should have degree one.  Floor one, floor two, floor three.  So, now, then there are marked points like this.  Then, this arrow's here.  So, so this shows you two half lines in tangent space.  So, if you have a puncture and a half line in tangent space  and you fix the disk,  then you can put coordinates on the disk  which map zero to standard disk  which map zero to this point.  But, when you have this line,  you can map one in direction of this line.  So, it completely determines what you have to do.  Now, here, however, what I'm looking at is  two arrows up to common rotation.  But, you look from right and left,  so they actually rotate like this.  So, you look up to common rotation.  And, what you want is,  if you take polar coordinates which are coordinated this way  and you run along say on S one is zero,  the r direction positive or the negative,  they hit the same point on the periodic orbit.  Yeah, so I do not, so let me say that to the SFT paper,  there are some differences in this case  for the transversality.  I do not number the mark points.  I do not number the punctures.  That gives more isomorphisms  which, however, have to be compatible.  The perturbations have to be compatible  with this larger set of isomorphisms.  If I number them, then if I take a renumbering,  I should perturb it in the same way,  it's a little bit more complicated to keep track of.  So, if you just throw everything away  and do transversality on that level,  that's one thing.  Then for orientation questions,  then you might go to a covering  which actually precisely means putting numbers there  if you want.  So, then.  How much?  No idea what you said the arrows are.  So, here in the tangent space,  you take a half, in each tangent space,  you look at half lines.  And once you fix a half line,  then if you fix a disk and take polar coordinates,  let's say take a disk map  and then you compose it with E to the something,  then if you run and you do it  and you take the disk map in such a way  that the real positive real line in the tangent space  is mapped to that line that fixes precisely  then polar coordinates.  So now, but we don't fix them,  but we fix them on two adjacent nodes or punctures  and they are allowed to rotate commonly.  But the effect of this is then if you fix  sort of a coordinated system of polar coordinates,  one positive, one negative,  then if you run along zero across R plus minus,  zero, zero element S1,  then they hit the same point on the periodic orbit.  So now, if the periodic orbit is simply covered,  there's only one choice.  But if it's double covered, you have this one,  but you can also take this by 90 degrees rotated  and takes the rotation class of this.  So there are actually K many choices  if the thing is covered K for.  Is it the same to say that for each puncture,  there's an asymptotic marker that you allow  basically quotient under S1 action that rotates?  The two, common, but yes, that is what you do.  Yes, you have a common rotation divided out by the diagonal.  So, but one you have to rotate multiplication by J  and the other by minus J  because of positive and negative.  So,  I was like, Hamoud, are you talking  possible different maps here or just maps?  It's just, I'm not done yet.  I'm in the middle of talking.  So now,  so, so we have my, so then here,  this blue stuff is what I call nodal pair.  So there's on each is a pair one,  there are two different points that could be  on the same surface on the other,  which you should identify, think of being identified.  So the maps will be continuous over those.  And then here in square brackets,  that is what is close to Cartman's heart, are maps.  And this maps are not assumed to be pseudo holomorphic,  they're just maps.  And then the square bracket around it means  it's an R class.  So because the image lies in R cross,  odd dimension manifold, say with a stable Hamiltonian  structure and you have an R action.  So if you take the map defined on these things here  and you was just shifted by an R by constant,  it's also, so it's an equivalent thing.  So this is the equivalence class.  You also put down by a morphism, right?  You also put down by a morphism.  No, not yet.  At the moment these are just,  so remain in surface with all these features.  Then the, so then of course,  at some point you have to do some analysis.  So you have to be a little bit specific  about what the maps should do.  So the quality of the map is chosen  so that it can do some analysis.  So what I pick is H3 lock, local H3 on this.  And near the punctures, they should,  if you take polar coordinates, should be of class H3 delta,  which we have seen, so this exponential weight.  And the exponential weights you pick have to depend,  of course, on the periodic orbits.  And in order to choose such possible weights,  you have to fix an almost complex structure  that gives you a linearized asymptotic operator,  that gives you an asymptotic operator,  which is self-adjoint.  And the non-degeneracy assumption means that  if you look at the part concerned with the context structure  or with the hyperplane bundle in the stable Hamiltonian case,  doesn't have zero in the spectrum.  So each of those guys has a spectral gap,  and you have to fix, and of course,  if you have generally infinitely many periodic orbits,  the gap might get smaller and smaller  if you go to higher and higher periodic orbits.  So for each of them,  you have to fix something in the spectral gap.  A weight.  So.  Can you say again what the equivalence means?  Up to r translation.  If there is, what is our target here?  r cross v.  R cross v.  v is a stable Hamiltonian,  with many for the stable Hamiltonian structure.  So now, there's a stability condition, namely at least one,  so each of these things here,  so I think of this not being identified in the domain,  so each of these things I view as a component  of my Riemann surface.  So then if you look on one floor,  on each floor, there has to be at least one component  which is not a trivial cylinder.  So what is a trivial cylinder?  A trivial cylinder is something like this here,  which has a positive and negative puncture,  which both are asymptotic to the same periodic orbit,  and which is homotopic to a standard geolomorphic cylinder.  Just homotopic.  Homotopic to.  So it doesn't have to have image r cross x for it?  No, in between not, but it's asymptotic to the same thing.  So if it's homotopic to the other one,  the integral u star omega is zero.  So I write triddle because your triddle is a comma u,  a is the r component,  and the thing without triddle is the stuff in v.  Then for connected components,  c, which is not a trivial cylinder like this one here,  we have one of the following is true,  either the integral of u star omega is positive,  or if it's zero,  then you must have enough special points on c,  which are the punctures, marked points,  points from the nodal pair, and so on.  Is it clear what the object is?  So this is an object in a category.  So this here explains,  so what this explains is the following.  If you want to be precise here,  you have to say that this puncture corresponds  to this puncture,  and you have to say that you have these two aligned arrows.  Now these two aligned arrows,  of course, you can give that by complex anti-linear map  or complex linear map if you change the structure  on one side, two minus i.  So therefore what you could think of is,  if I look at a punctures and I take the tangent space,  then I can take the real projectification of this.  It's an S1, but not naturally,  but it has a natural S1 action on it.  So you have something like principle S1 bundle  over the punctures,  and you want to have a complex,  you want to have a,  if you take here the complex structure J and here minus J,  you want to have a S1 bundle isomorphism,  which gives you a bijection between the underlying punctures,  and it says which line corresponds to what line?  I'm sorry, what is your J part of the data?  J is, at this point, J is fixed, yes.  What is J?  I don't know, you keep talking.  Yeah, okay, so,  but I don't know what you mean by J,  there could be possibly two different Js, cut,  there could be a J in the target,  and there could be a J on the domain.  What are you actually fixing?  Okay, okay, so when,  okay, so can you just give me a little bit of a chance  here to explain this?  Yes, okay, there will be a precise definition,  but it would be good before you see letters,  you know what we're talking about,  if that's okay with you, thank you.  So now, yes, you can,  but give me at least a chance to finish this,  and after this you can ask questions.  So there in particular, if I look at the punctures,  this would be the positive puncture of the floor  called I minus one to the floor,  negative puncture on the floor I,  then I projectivize the tangent space real,  with this structure here, with the minus structure,  and this would be the principle bundle,  asomorphism covering this bijection.  So the bijection would say which puncture corresponds  with what, and this would align the arrows.  All right, sorry, can you explain what your letters mean?  What is S I minus one?  What is S I minus one?  Okay, okay, okay, Katrin, bear with me.  So now, so now this thing which we just saw  can be encoded like this.  This of course, so we have a sequence of letters.  The alpha is one floor, negative punctures on floor number I,  S I remnant surface on floor I,  J I the J on floor I,  M I mark points on floor I,  D I nodal pairs on floor,  so this would be horizontal things,  on floor I, the equivalence class  of the map positive punctures.  Then this one are precisely the ones which I explained here,  which tell you how to align things in between.  Sorry, who aligns what?  I'm trying to take notes here.  I just told you you shouldn't because this is online here.  No, Katrin, no, Katrin, this is online,  you just open your thing and write onto the text, yeah?  Okay, so now.  Okay, does anybody else still follow?  Yes, yes.  Good, so how about you tell me who matches what?  What?  I mean, this has been explained here.  This, I think is, it tells you which puncture  from the lower floor goes to which,  and above is the alignment of the arrows.  What's the V is a map?  Projectification.  It's a tangent space at the punctures  of the remnant surface for this structure.  Note here is a minus sign for the negative structure,  and this is the principle bundle map.  Principle bundle isomorphism.  All right, and so where's that V in your next slide now?  It's this, this covers that.  Well, right, but in the next slide.  Well, you see V hat?  It's the, V hat implies V.  That's supposed to be what's the previous,  and that what, tells you how to attach alpha zero to alpha one?  Yes, that including the rotation,  which lines are mapped to what, okay?  Good.  So what is a morphism?  So a morphism, so this is an object in my category.  So a morphism, I write like this,  so here's the source, the target.  It's a sequence of bi-holomorphic maps.  So they gave from here to here,  which means they're map punctures to punctures,  the negative punctures to negative punctures.  On this part here, S-I-J-I, it has to be bi-holomorphic.  M-I is being mapped to M-I prime,  nodal pairs are mapped to nodal pairs and so on.  And it should hold that if you,  there is a constant on each level.  If you shift your UI, then you get that composition.  Yeah?  What is the D again, nodal pairs?  Nodal pairs?  Nodal pairs.  Plus this means shift in the real direction.  Yes, that is on the, so for each floor,  you allow it to common shift of everything.  Yeah, it's by adding a constant to the R component.  Yeah?  Because we remember the objects were equivalence classes  on each floor, yeah, so that is.  So under, so these are the morphisms.  Does the morphism have to preserve that matching  with the asymptotic marker?  Yes, yeah, yeah, okay, yes, it should.  So if you, that's a good question.  So if you have T-fi-I, it actually gives you, yeah.  So it should map equivalence matching  to equivalent matching.  That's actually quite important.  So for example, to answer this question,  illustrate a little bit more.  So if you have, okay, suppose you have two double-covered  cylinders, so a building consisted  of two non-trivial double-covered cylinders.  Then, so in particular in the middle,  I have a say double-covered periodic orbit.  And there you have this matching things.  So now, this can be isomorphic, it can be isomorphic  to a thing which has this shifted by 180 degrees.  But this will be a different object.  If I have an automorphism of this one,  that requires that this matching is being preserved.  So for example, if one cylinder is simply covered,  but the break is double-covered and the top is double-covered,  then this object only has trivial isotropy.  Yeah?  So if here's simple covered, a double-covered periodic orbit  and a double-covered cylinder on top,  this has only trivial isotropy because I  have to preserve the matching.  So I can't really rotate downstairs.  But it's isomorphic to another object which has this 180 degrees.  So in this business, it will be extremely important  not to confuse being the same and being isomorphic  because for the counting things.  Because when you do in the fine points of perturbation theory,  if you be sloppy about it, you start counting wrong things.  OK, so this is now a category.  And clearly, every morphism is an isomorphism  because you can write down the inverse immediately  by just reversing the order of alpha alpha prime  and inverting all the by-holographic maps.  Now, what is not so clear but which  is a consequence of the stability condition  in the nice exercise is that between two objects,  there are only finally many morphisms.  So of course, I call it group-oil.  So group-oil is usually a small category  which would have this property.  I call it group-oil because it's not a small category.  But then this category admits a grading  because I can talk about the top floor number.  So if I just have a building of height 1,  the grading would be 0.  That means ground, I only see the ground floor.  If I have a building of height 2, the d would be 1.  That would be the number of the top floor.  So European notation.  Then having this grading, my category decomposes.  And let me artificially define what  I mean by the boundary of the category.  These are all objects in SI for I at least 1.  So then there's also an important point.  There's also an important point, namely,  if you look at the orbit space, orbit class of this category.  So the orbit class consists of all equivalence classes  of isomorphic objects.  Then this is a set.  Now you can build on the remain surface lying in R3 or some  sense, realize in R3.  So it's clearly a set.  The morphisms could preserve the matching,  but not the asymptotic markings?  The asymptotic markers not here.  So in contrast, as I said at the beginning,  to what we are writing in the paper,  we have the asymptotic things.  We have them only numbered, but we have markers there.  And we fixed points on the periodic orbits.  We don't do that.  Then these half lines.  So that we had before as well.  But we had on the top, our mark points were ordered.  The punctures were ordered.  And there were asymptotic markers as a punctures,  which were assumed to hit a particular periodic point  on the periodic orbit.  Markers are only punctures that are matched.  Yeah.  I think you've got both what the question was  and what the answer was.  No, he obviously read the book.  Oh, I see.  No, somebody asked.  I'm asking about the part, the ones which  are marked in this setting.  So you have these B's, which kind of match up.  Yes, they just met up which puncture to what.  And then the B hat is a lift of this.  It just says it matches up lines up to rotation.  And then if you take a holomorphic polar  corner, suppose negative 1 adapted to this,  then if you run along 0 into direction to plus minus infinity,  then they hit the same periodic all.  It's the same point on the periodic all.  I think the question is, does the morphicine's new category  have to carry this data?  Yes.  Yes.  So the B hat.  The B hat has to be preserved.  So the choice of polar coordinates is necessary.  So you can have just two arbitrary different choices  and be automatically asymptomorphic.  So if you have two of these things which generate this class,  then if you apply this morphism here, i minus 1 and 4i here,  then it is mapped to a pair of lines  which is that associated to the object.  This is a new object.  So can I ask that again, maybe differently?  So let's say you have a morphism from alpha to alpha prime.  Right?  Each with two levels.  So then there's one matching data.  So if I fix my matching data B for alpha,  are there different matching data for alpha prime  that I might have morphism to?  Or is the matching data?  No.  I just said, if you, for example, take.  So suppose this is double covered, covering numbers two.  And then here I have something where covering number is one,  let's say.  So this is twofold covered in this one.  So then suppose I have this matching here.  Then obviously I have also a matching where this, where  the one appears here.  I don't know if you have colored chalk.  But then you could, so one matching  is this common rotation.  The other would be you put this thing on the other side.  This common rotation.  Let's say I fixed my matching there.  And now I have maps phi that take both the top to something  and the bottom part to something.  Right.  The something is some stuff this has already given a matching.  So the lower part maps should, if you fix two lines here,  the lower thing should map it to a line here  and the upper to a line there, which  precisely gives a matching on that object.  It's a different object.  It has all this data.  So it's not so difficult, Katry.  So if you.  Right.  So question.  If I fix everything in your morphism,  if you go back one page.  If I fix everything.  If I fix alpha prime.  Right.  Let's say, so the phi i.  If I fix the phi i.  Right.  I fix phi 1 and phi 2.  I fix alpha 0, b hat 1, alpha 1.  I fix alpha 0 prime and alpha 1 prime.  Does that determine b 1 prime?  So the first thing is in a given object,  things would be fixed.  So now if you want to talk about morphisms between this,  you have to preserve this data.  Now what you ask is, maybe, that if for my alpha this is fixed  and I want to build another object, then in fact, what  you can do is you can take different morphisms on each  level, map it to something else, and adjust the marker  so that this is isomorphic to the thing you started.  What I'm asking is whether the phi, together with the choice  of source, determine the target.  Maybe answer just yes.  Yes.  The answer is yes.  OK.  Pelman, in this example, the top cylinder, it's just a  trivial mapping cylinder?  Or?  Yes.  No, no, no, no, trivial.  Just something which is double covered.  Yeah, so this would have an automorphism.  And this itself as well.  But as a thing with the marker here, the automorphism group  is trivial.  But this is isomorphic to a different object where this  one, let's say, is rotated by 180 degrees.  But that is a different object.  It has different data.  OK, weighted subcategories.  So it's something like Facebook, like or not like, but  with rational coefficients.  So you look at q plus between 0, 1, which is just the  identities as morphisms.  So you look at a functor there.  And you're interested in, and if I'm given such a thing, then  I have a full subcategory, which are all the things which  I like to some degree, positive.  And they are carrying a weight how much I actually like them.  Yeah?  So example, given J-triple, which is almost, I  invent almost complex structure, R cross V, and  a compatible with the stable Hamiltonian structure, we can  talk about J-holomorphic objects in S, which are the  objects that are used to satisfy the  Cauchy-Emen equation.  So then define this by associating to pseudo-morphism  the weight 1 I like and to other, this thing I don't like.  So that's our modelized space for the time being.  OK.  So now the whole theory will consist of basically taking  a partition of unity for the one and moving, now each point  is represented by several which are fractionally liked, and  moving it into a better position that each of these  things looks like a manifold.  So that needs, of course, some technology to do this.  OK.  Good?  I felt hope.  Yeah?  Katrin?  You said there was something about the aim.  About the same of the aim.  The aim is, that is what I said before it showed up here.  So they form this into a theta with additional structures  from which we can derive the data we need to construct SFT.  So it turns out that you can later define theta's, you can  define differential forms on these categories, and you  can integrate them.  So there's a really cool, nice theory for this.  So OK.  Aim clear.  Now, so there's more structure here.  So when I have an object in my category, let me define a  different category P here, which consists out of the  following.  It consists of maps from finite sets into periodic orbits.  And the morphisms are by ejections between these finite  sets so that that is true.  So this is actually already part in some sense of the data,  namely, if I look at the positive punctures, I just take  this point here as my point, as my set I.  And the evaluation plus map is this map here, which  associates to each of these points the puncture, and the  same here, at the negative part.  Now, this category might, for example, if the different  points in this finite set go to the same periodic orbits,  you might have different morphisms.  For example, if I have two points here and they go to the  same orbit, then switching the two points would be more  fizzier.  Is the point of all this category of the language just to  keep track of isotropy?  So in this category, I will explain everything to you, and  then I will say there's a notion of a smooth structure on  this category.  And then after this, I'll explain a little bit that there  are smooth versions of this like or not like functor.  Then once you have a smooth structure, you can talk about  differential forms on this category.  For example, the pullback by an evaluation map from the  many-fold of differential form will be differential from  the category.  And when you have orientations, you can actually integrate.  You never have to leave.  So once you have a poly, so it will later call be a  polyphonic structure on the category, you never have to  leave.  It's all kinds of universal constructions.  You can do the whole transversality on that level.  Yeah.  There's nothing.  Well, when I make my definition, I assume that it was  non-degenerate form.  No.  Moxbot I haven't dealt with, and I don't want to.  So.  Oh, yeah.  So that is what was discussed last week.  So we have an odd dimension manifold with a stable  Hamiltonian structure, and that defines the so-called  wave vector field, and that has periodic orbits.  OK.  So now, so now, so we have an input and output.  We just look at the restriction to the punctures.  We get this map, which says to which periodic orbit it goes.  And then we can build chains like this, and it's called  weak-fibre products, and they look like this.  So.  So observe here that such a map bi, which we introduced  before, this bi lies below the bi hat.  So if you forget about the matching of the lines, is in  fact such a morphism.  So we look at chains of objects, and this morphism  is a category B that is called, and it takes a  fiber product.  Yeah.  So we have an object.  We map it forward, which would give the map which  associates to the positive punctures of the periodic  orbits.  Then we have a morphism from there to another such a  thing, which comes from the next one, but going down.  So we have here different, so these are not floors, so we  have here different buildings.  Objects, so these are objects in my previous category.  I map them down, then there's an, into this category here,  and there is a map mapping it to something which came  from the negative evaluation.  So note that this thing here, it looks like a new element in  your category S, but it isn't because the head is missing  here.  We don't know, we don't have information how the lines  are being matched.  That is not there.  That actually will lead to the fact that there is a  relationship in our previous category, and this, we are  some category, we are some covering.  Functor.  Is that clear?  Excuse me.  Yeah.  So it's clear, but I don't understand why you're  doing it.  Is it to describe the difference?  Well, I mean, this is where D squared comes from.  But I mean, oh, so you're saying that in the back  structure you don't offer you already have matching?  No, you don't, because if you, no, in the boundary you  have, but then now you have to relate it to the full  category, which says if you have a building in the  boundary, then you have to relate it to S.  And in fact, how can you do this?  Well, you'd chop your thing in the boundary, you'd  chop it apart here, but once you do this, what does  that mean?  You just took a weak fiber product.  You lose the matching condition.  Yeah, because in this case, at the positive negative  functions, I don't have, I didn't keep track of this.  So now in the original version, we had asymptotic  markers there, so that information wasn't lost, but  then you get a covering in the other direction, because  when you have everything marked and then you glue  it together, you have to forget where you glue it  together at the order of the periodic orbits, then you  get a covering there.  So there's no free lunch here, but this thing, take all  the numbers away, all the orders away, has the  advantage that I have a lot of morphisms, and my  perturbations have to be compatible with this.  They also have to be compatible in the other one,  but there it's a little bit hidden, because you have all  this additional structure, and then you have to say, if I  take here all the numbering away from this object, all  the numbering here, they should have been perturbed  in the same way.  Why not just for the transversality, just take  everything away, which actually is irrelevant for that?  Okay?  Okay, so now that is what we are looking at now.  We take fiber products of lengths, so there's two  elements, three elements, and so on, this weak  fiber product.  So now S was graded by D, the top floor number, and  now let me look at, take multi-indices, and I  is greater or equal to zero.  Then if I, because I want to, so this has actually  two filtration categories, so then I can talk about  the norm of such a multi-indice by just taking some  of the numbers, and I can talk about the lengths of this.  Can you avoid this problem by taking this S to be  category with objects, as they add the, remember these  mark points, isomorphisms, turn the circle, and two  morphisms between the isomorphisms.  So you want to go to a two category?  Yeah, I'd ask you a two category.  I don't know.  I didn't think about it.  So then given such a multi-indice S, I can obviously  build this product here, so this are degeneracy N0,  which is NK, and then SP can be decomposed like this.  And there is, okay.  Now.  Can you say in words what that means in terms of  broken terms?  Yeah, you see pictures.  I will show you in a second.  So then there's a degeneracy functor which you can  define on this by just taking the degeneracy of the  objects, and there's a length functor.  Just look how long is that product here.  So here's a picture.  Oh, maybe.  Okay, so here is how you have to understand that.  So this here is a building in our original category.  It has the top floor number six, and I put a head here.  This is the BI, these are the BI heads.  So now I give a multi-index 1002.  So what does it tell me?  It just says I should chop that into four buildings, and the  first building should have top floor one.  The other should have just gone floors and the other top  floor two.  So that is a certain requirement on this here, namely the  no, namely there should be at least top floor six.  So now I chop a building with top floor one, five-word  product with ground floor, five-word product with ground  floor, five-word product with a thing of this one.  This organization is important for the perturbation  scheme because you have to work your way up with all  these compatibilities on this thing.  So you have to know how you chop and what perturb, what  perturb, and what perturb on.  Is that clear?  So when I see a round dot here, I should think one  trajectory.  A round what?  What are the dots?  The dots are connected correctly.  Where's dot?  So this here, this is a building with height seven  top floor number six.  So this is like the first picture you saw except that had  less floors.  So if I just want to try and think how to prove this  when equal to zero?  No, you don't want to prove anything like this.  It's a little bit early for this.  You will let me, you will see that a little bit already.  But the floor is kind of complicated.  Each of the floors can be many, many things.  So, but it's sort of an easy procedure.  So if you have enough of those guys, the right number here  and such a multi-index.  So the norm is three, the length is four, and then you  have to subtract one is six, which is precisely the number  here.  So if the norm plus the length minus one is precisely this,  then you can chop in this way.  If I get another one, if I put zero here and one here, then  you would have a ground floor here, then this here would be  a building of height two and so on.  So there are different ways how you can chop this thing.  Then the perturbation later, which you would see over this  object, could be pulled back from this one and any other  way it was cut.  That shows you already some compatibility things.  So now let me, the notion of a covering functor.  So, so finite covering functor.  So you have two categories and over each object you have,  so it should be surjective on objects and over each object  you have lying a finite number of points.  And this condition should be true, which looks a little bit  difficult at the beginning, but it says the following.  If you fix the source of all morphisms, like I fix, so I,  I, I am interested in this point.  I take a point which lies above it and look at all morphisms  which start from this, which is this one, this one, this one,  and here's the identity.  Then this should be in bijection to all the morphisms which  you see here.  And so you're writing your compositions with the source on the,  the source on the right and the target on the left.  What source on the, no, sources, I was writing like this.  So this, this here, okay, so this are the morphisms,  this is a morphism set of this category.  This is a morphism set of this category.  Then it has a source map into D and F goes into the objects.  So it takes a fiber product here and I do this.  So, so now if you fix, if you fix this point and take a pre-image  under this covering, then if you look at all morphisms starting here,  then you get a morphism to the object starting here.  What are the significance of your different typefaces?  Because you've got italic typefaces which are the categories.  Yeah, so it's okay.  So this is maybe I'm too didactical here.  So, so, so a normal C would be the objects and this bold face  would be the morphisms.  Oh, well that's what I want.  It's the bold face with more faces.  Yeah, so these are objects, these are morphisms.  So forget about the definition.  Actually, we already saw such a functor, namely something like this.  And these are all functors of this kind.  So take in your category, so, so we have our category S,  we have the degeneracy index and we have the evaluation maps.  So now a covering structure for this consists of the following  and it's clear what it should be.  For integer greater or equal to zero, Fij takes an element  of the degeneracy i plus j plus one and it chops it precisely  at the right spot to produce a building with top floor i  and top floor j.  So if I have a building of a certain length, I can say I chop it here,  I chop it here, I chop it here, I chop it here.  So, and these are actually covering functors.  It can just take this definition from here to here.  So they're precisely satisfied with the definition.  So you see here already that in this formalism,  you keep very much track of the isomorphisms,  which is actually very important for counting later on.  Sorry, so the degeneracy is just the number of flows?  The degeneracy is the top floor number using European notation.  So it's one less than what you would say the height of the building.  It doesn't count in general number of points.  No, no, no.  So then it's clear if I have something and I chop it here  and then I chop it there, you could also chop it first there  and chop it there, which is precisely this thing here.  So you have a comp...  So chopping, if you choose the right indices,  it's commutative.  So then using this rule, actually you can extend this chopping  and it's already the picture inductively for every multi-index.  So there is from Si plus this, this product, there is such a chopping.  So, and that is precisely this here.  So if I take F1002, it is on S6 and it chops it in this way.  So these are all covering functors.  So you start with an initial so you can check.  If you just use, forget anything else, just algebraic,  these are all covering functors.  Created from a small list of covering functors, name it those guys.  You're writing equals everywhere,  but you have a bunch of more reasons to categorize.  No, that's actually equal.  That's actually equal.  No, just, you know, if you have three buildings with three things,  chop it here, then you get a building of height one,  height two, and then chop it here.  I could have first chopped it here and then there.  So that's precisely this identity.  So in terms of actual data, does the F simply forget one V hat  and just keep track of the V?  Yes, so, yeah, what does this thing actually do?  So this here takes the right spot in between,  but you have to cut that on one side, top floor is I and the other J,  and it forgets precisely the hats.  Only the one at the right spot, which is determined by giving I and J.  That is what it does.  You don't change the object, nothing.  You just forget the hats for the B.  And that is obviously rather commutative.  I forget here and forget there or the other way around.  So, okay, so, if you look at the J-trill case,  the pseudo-holomorphic case, you see that immediately.  So this, if you have a building where each of the functions U is J-holomorphic,  if I forget the hat or not, it doesn't change.  This thing is stay holomorphic, so obviously that's true.  So now we come to some formalism, which is actually quite useful.  So if I'm given such a Facebook function here,  and then let me define the boundary of this,  it's a restriction to the boundary of the category.  So now, given this structure here,  consisting out of this maps here and then what I can build out of this,  I can define that thing here.  Okay, look at it, it will be complicated,  and you try to understand this.  Sit down.  It doesn't say more than precisely that this is true for the theta.  Forget the J here.  So I guess the compatibility that if I pull back theta times theta by Fij,  I get theta on the corresponding Sij plus one.  So this thing here, that this is equal,  just means compatibility with this thing here.  So if you have a theta,  then obviously define a theta on the fiber product.  You just take theta of a fiber product as a product of the citas of the particular values.  So I define theta p.  And then, so here's a type where it should be a p.  Sorry, here should be a p.  Then you pull this back, so you get to the boundary of the category,  you take the sum here, and you want this condition.  It just means that Fij composed with theta p is just theta restricted to Si plus J plus one.  That is what it means.  So there's a short form.  So this quantity means high compatibility with this structure.  So this is part of the conditions from which d squared equals zero comes from later.  So it looks complicated, but it's a really short form to formulate it.  I'd rather have to say for all ij, this and this holds true.  Just say that.  Now this formula occurs in a lot of way.  You can do a Freton series satisfying this formula and so on.  So you can include an x-ray in front of that sum.  So model this thing that here should be theta sub p.  You just build, you see this takes values in zero one, rational,  and it just takes the alternating sum, and that's the definition.  And what you require, what you say is compatible is,  so this thing is defined on the boundary here.  Because the Fis come from an Si where i is at least one.  So this here defines a new functor on the boundary,  and you say it's compatible if this is true.  And this is a non-trivial condition because when you look at this here,  you take fiber product of length two, then you restrict it,  so that theta is given.  Then you restrict, then you pull it back, you get something on the boundary,  and that was the supposed value there, and you work your way up.  So there's a lot of compatibility.  So that's, but it doesn't say more than compatibility with the Fijs.  So now, trivial cylinder buildings.  When we do a perturbation theory later, or for the algebraic structure,  I'm going to have a modernized space out of some, a modernized space of j-holomorphic objects.  And I multiply on each level, I put say the same j-holomorphic cylinder in.  I get a new modernized space, and obviously it should be the same up to that multiplication.  It should be identifiable with the other one.  Yeah?  I was just trying to follow what was going on with the picture.  Yeah, no, I haven't explained the picture yet.  I haven't even started.  So when I do a perturbation theory, and for example, this building of height one  would be pseudo-holomorphic, then, so I would have this modernized space of this kind of things.  So if I multiply it by some j-holomorphic cylinder, that should also be in my modernized space.  Because otherwise, you never get the algebraic structures that you have.  So if you are, if I multiply it by several j-holomorphic cylinders,  that should also be in the modernized space.  It might have different signs and so on.  So if I integrate over that modernized space having this,  it should be basically like integrating over this modernized space, say, up to sign, for example.  Can you just explain that data formula again?  For one last page.  Suppose I have something with, like, you know, three levels.  So there's various things to be assigned.  To be assigned product of data is at the three levels.  To be assigned data at the first two levels times data at the third.  So what is the relation between all of these things?  If this is true, it just means that for the two, for things, product of two things,  is equal to theta on s i plus j plus one for all i j.  That is what it means.  It's a convoluted way to say this all at the same time.  But it's a good formula.  Can you explain the minus one to the length of i?  Yeah, so it has something to do with this.  Suppose I have a function defined here and a function defined here  and I want to extend it over this.  Suppose f is here and g is here.  This is t and this is s.  How do you just define g of s plus f of t minus g of zero?  You assume that they coincide here.  That is a smooth extension of this.  So it's an included expression.  Yeah, so if you do this now for higher things, you get this alternating sign.  So that is precisely, you see, if you have this theta p defined,  there are a lot of points where they have multiple definitions  but they coincide there.  So if you write this down, you get this.  So that's the underlying idea.  OK, so now, trivial cylinder buildings.  So first of all, there are...  So let me first say,  a trivial cylinder building would be one.  So if I have height three here,  would we say something like this where top and bottom have the same periodic orbit  and each of them is homotopic to a standard geolomorphic cylinder.  That's a trivial cylinder building.  And geolomorphic cylinder building would be where each of this is actually geolomorphic.  So there's one more question for this theta.  If the theta is like zero or one,  then to say theta on a building is one means it's one on each piece of the building.  So if theta is zero one and there is in the product,  so then if I have a product and one in both of them would be one,  then the building above for where it comes from would be one with the number one.  Then it's not, isn't it?  Then it would be disliked.  So, OK.  So in the perturbation theory,  if this is a liked object  and I add geolomorphic cylinders to it,  a geolomorphic trivial cylinder building,  should be liked as well.  So in the perturbation,  if I like this building,  so here let's say geolomorphic cylinders,  and now I add a geolomorphic building to this,  then if this was liked, this should be liked.  It's the same weight.  That is, otherwise you never get any algebra structure.  Otherwise you can't get the multiplicative structure in SFG.  So these objects are different though.  They are different, but in some sense they have to count the same way up to sine.  So up to sine.  So how can we formalize this?  If alpha and alpha prime differ by geolomorphic cylinder building,  then theta value coincides.  So here you have to notice that there is not a unique way to put  a geolomorphic cylinder to it,  because you fix an equivalence class here  and you have a map on it and you can shift it.  So there is on each level at least an R parameter.  So that is indicated by this.  But I think everybody understands sort of,  when I say add a geolomorphic trivial cylinder to it.  But there's not a unique way.  There's a family.  This is just explained.  I just said that you add to alpha a holomorphic cylinder building,  a trivial holomorphic cylinder building.  Is TCB a set or a one?  Trivial cylinder building geolomorphic.  So it's a set of the geolomorphic.  It's just a line.  If this building has height k, it's a line of k cylinders,  which are holomorphic and trivial.  Yes, but there's not a unique way to add it to it.  You get families.  For each level you get a family, an R family.  So the TCB there knows how many levels alpha has and that's how many.  Yeah, so this is a short way to write it.  So I added trivial cylinder building to it.  And then for any representative of this,  because it's not unique, the value should be the same.  So you would write times R times the set of cylinders.  What times R?  No, it's R to the k because for every...  Yeah, so if there's building of height one,  you have just a one parameter family of building of height two  for each of the single level R.  So I'm just keeping it simple by this.  Just add.  So whatever this symbol means, alpha, district, union, TCB,  so...  So you have some set of your object category.  So here I added...  So I would add one trivial cylinder,  but even here it's already not unique  because here you have an equivalence class of maps.  So you have to choose a representative,  take this to this and divide out by the R action,  and you get an R family.  Then if I have another level, I do the same.  Okay, so my requirement is that theta should have this property.  Then if alpha contains a trivial cylinder building  which is not necessarily holomorphic,  but if theta of alpha is positive,  then the trivial cylinder component are all j-holomorphic.  So on that level, it's a little bit more complicated  to formulate than later on a linearized level  where you can actually write down some functors,  and then that's true.  So okay, so we asked this again.  When you write QCB, really you mean to just put one chain of trivial cylinder?  Right, yes, and then you can do it again  and put another chain to it and so on.  And so if it contains a trivial cylinder building,  which might not be j-holomorphic,  but if it's liked, then the trivial cylinder components are all j-holomorphic.  Okay, so these are properties.  So just I should, yeah, you used up your time, I think,  so if you want to say anything in the next minute.  I know it flew by for all of us.  When did you actually start?  You started at 30 after.  You finished up very briefly another minute or so.  Yeah, let me.  You have other lectures coming up this week.  I know, but this is frightening.  So, time-wise.  Okay, so let me just...  So now I'm using a theorem of Joel Fish.  No, okay, so...  No, okay.  Okay, so give me a few minutes here and then I'll stop.  So now, so if you have two non-trivial components,  each building level has a non-trivial component,  which is not trivial cylinder-like.  Then you can move them against each other,  creating, again, a family on each level,  an R family.  If you have several components,  you can move them against each other.  And what you want is, if you have two stable,  so if alpha and alpha prime are stable,  and you take sort of this union of things,  so on each level you're allowed to shift them against each other,  then the theta of alpha should be the product.  So you have to think of this.  Theta, in some sense, would say find pseudo-holomorphic things.  And if it's pseudo-holomorphic and it is disconnected on each level,  then each thing individually already existed,  and that's about to happen, and so on.  Okay, so summary.  Okay, so summary.  Okay, so then I'm stopping.  So we are interested in deformation theta of theta-trivial.  This is the original function  which identified geolomorphic objects.  We want to deform it to a theta,  having the following properties.  This is true.  Then this is true if we add just a trivial  holomorphic cylinder building.  If this is positive, then this has to be true.  Then this thing actually had to be pseudo-holomorphic.  And if I've stable things and move them against each other,  this has to be true.  So SFT algebraically comes from this.  That's all there is.  Okay.  This is more or less the same as saying  the whole theta is determined by the value of theta  on one floor thing with one connected component.  Is that the true statement?  Yeah, that's a good one floor.  Okay, so let me not buy into this here.  So I have to look at the algorithm.  So the algorithm is a little bit...  I'll give you the algorithm how to perturb.  So the algorithm would precisely...  Yes, so in the first time to perturb,  you have disconnected objects.  So in the hierarchy concerning some complexity  or energy on the bottom floor,  if there are things cannot degenerate further,  there are disconnected things.  Connected things which are perturbed.  Yeah, it comes from there.  All right, well, why don't we thank all the friends?  So that is what I wanted to do in the first 15 minutes.  So are there any brief questions for Helmut?  Yeah.  I'd expected you to write down a brief list of axioms  in which all but one were manifestly satisfied  by the sort of defining property  for the pseudo-holomorphic case with which you start  and which you just decide one the holomorphic things  and the other would be the thing you need  to achieve by the definition.  Is it easy to...  So if you take this properties here,  you can see the pseudo-holomorphic.  Precisely.  But of course, we don't want to take the trivial deformation.  So you want the deformation which happens by these  and which lasts and so on.  Okay, so that's a good...  So my...  As we've just reminded, I have four lectures.  Of course, it's not so clear if I get to the very end of it.  But at this point, what we have done is we...  I have sort of in this category setting said  what things should do.  So now, the next thing is there's a notion  of a smooth structure on a category  but you can only do this for...  when you have this polyphonic theory  because other things you cannot do smooth.  Then you can start talking about smooth objects of this kind.  And smooth objects of this kind...  So these are sort of smooth sub-categories  and what are they?  So they actually occur in different flavors.  They are manifold type, orbifold type, branched orbifold type  and so on.  And these things then will be constructed by a Friton theory.  So you can transpose these properties  into the language of what you should do on the perturbation level,  on the linear level.  And there's a nice functorial description.  So this is obviously sort of functorial here.  This is a little bit odd,  but there's a nice formulation on the linear level for this.  And then you define a bundle over this.  You have the Kuschelrimann section functor  and you want to perturb that one.  And then if you...  and that is how it looks like at the end.  So you have...  So this is basically over each object here.  The fiber is... are the zero one forms along the underlying map.  So here you have the Kuschelrimann functor.  And then you define something like this Facebook thing here.  It's... it's... but it's a section.  It's... it delivers on the fibers  and it associates to each fiber to a certain number of points,  a rational number, adding up to one.  And then this here is your theta,  if this is properly chosen.  If you put here theta zero,  which just is the zero...  which just gives the weight one to the zero section  and to everything else zero,  here is actually our theta j-tridle.  So if this here is theta zero,  which just puts weight one on the zero vector  and weight zero on the other,  then this gives you...  this comes first with this gives you that one  which you already saw.  So here is a small perturbation of this.  And since in the fiber everything adds up to one,  what you literally do is  you take a partition of unity for your theta j-tridle  for each object  and you move the objects.  Now you have several one with fraction of eight  in different directions  and they line up to many fold pieces locally.  That is what they do.  All right, so I'd like to make sure  we don't have enough time to get enough coffee.  So I think we're going to cut it off here  if you have any further questions for Helmut if he's around.  And we will attempt very strongly  to meet back here at 11 for the next talk.  So let's thank Helmut one more time.  Thank you.
Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_0_947.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_948_2339.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_2340_4187.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_4188_5903.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_5904_6071.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_6072_6515.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_6516_10739.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_10740_14099.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_14100_14183.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_14184_14459.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_14460_16859.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_16860_17015.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_17016_17243.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_17244_17999.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_18000_18335.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_18336_18383.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_18384_18791.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_18792_23159.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_23160_31931.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_31932_32399.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_32400_32807.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_32808_35291.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_35292_35483.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_35484_37643.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_37644_37967.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_37968_38051.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_38052_38231.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_38232_39251.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_39252_41183.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_41184_42359.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_42360_48947.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_48948_49031.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_49032_49823.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_49824_50015.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_50016_51899.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_51900_55271.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_55272_57467.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_57468_58091.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_58092_58559.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_58560_59435.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_59436_61271.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_61272_62111.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_62112_62183.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_62184_63635.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_63636_66191.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_66192_69575.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_69576_71075.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_71076_72863.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_72864_74159.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_74160_74231.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_74232_74687.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_74688_75035.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_75036_75347.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_75348_76451.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_76452_77711.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_77712_80447.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_80448_80855.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_80856_82175.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_82176_83231.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_83232_83675.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_83676_84683.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_84684_84863.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_84864_85439.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_85440_85511.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_85512_85955.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_85956_86135.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_86136_86987.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_86988_87947.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_87948_92447.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_92448_92555.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_92556_93107.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_93108_98399.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_98400_99731.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_99732_102611.png Helmut_HOFER_-_1_4_Polyfolds_and_the_construction_of_Symplectic_Field_Theory_102612_105768.png