So, if you want to take notes, I would only write down what you don't see here.

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, do you remember this?

So, I gave a talk about the Weinstein Conjecture, my first talk, and then Jascha and I were running through Basel, and I think at this point we already had an idea about contact homology, but it took them some time to go through with this. So, of course, there were a certain number of issues.

So, for example, in 1992, nobody knew even how to deal with the transversality issues in Kromoff-Witten and so on, so then we knew that we should perhaps do some version of cylindrical, but it was always not particularly satisfying.

So, then in 1996, 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 felt 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 Hotzky and Sander, we analysed 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 processes 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 strength, you have to deal with all these problems.

And the problems and 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 analysed, 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 Fredholm problem, and what would have happened? Well, if it just be a classical Fredholm problem, then people would have dealt with

this like they deal with Fredholm problems, so once 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 Fredholm problems, so 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 these technical issues wouldn't have arisen, things would have progressed differently.

And now, how you deal with a classical Fredholm problem, if it's actually classical, it's quite easy, it's not difficult, you just take a small perturbation of the section and Sacht's mail tells you, when it's 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 coordinate changes are nowhere differentiable, which of course is the reason why finite dimensional 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. OK, so then at some point, once we found some kind of a setup, 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, dramatic 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 what actually 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 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. And I will talk about precisely how to deal with SFT,

assuming that you would know 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 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 riemann surface, the red things are punctures, this is a building, now we use European notation for the floors,

this is floor 0, because certain things later should have degree 1, floor 1, floor 2, floor 3. So, now, then there are marked points, like this,

then these arrows here. 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 a disk, then you can put corners on the disk which maps 0 to standard disk, which maps 0 to this point,

but when you have this line, you can map 1 in the 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 S1 is 0, the r direction positive or the negative, they hit the same point on the periodic orbit. So, let me say that to the SFT paper, there are some differences,

in this case, for the transversality. I do not number the marked 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 cavern, which actually precisely means putting numbers there, if you want.

So, then... 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,

take a disk map and then compose it with e to 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 a coordinated system of polar coordinates, one positive, one negative, then if you run along zero cross r plus minus, 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, rotate it, and take the rotation class of this. So, there are actually k many choices if the thing is covered k4.

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. 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... Oh, sorry, Hamoud, are you talking possible different maps here, or just maps? I'm not done yet. I'm in the middle of talking. So, now, 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, it could be on the same surface or the other, which you should think of being identified, so the maps will be continuous over those. And then here, in square brackets, that is what is close to Catherine's heart, are maps.

And these 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, many faults, there is a stable Hamiltonian structure, and you have an R action. So, if you take the map defined on these things here,

and you just shift it by an R by constant, it's also, it's an equivalent thing, so this is the equivalence class. You're also pointing out by automorphism of the domain, right? You're also pointing out by automorphism of the domain? No, not yet. At the moment, these are just, so,

remain surface with all these features, 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 maps is chosen so that you can do some analysis. So, what I pick is H3 loc, 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 linear 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 a contact structure, or with a hyperplane bundle in the stable Hamiltonian case, it doesn't have zero in the spectrum,

so each of those guys has a spectral gap, and you have to fix, and of course, since 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. Oh, wait. So, sorry, can you say again what the equivalent terms mean?

Up to r translation. If there is, what is our target here? r cross v. r cross v. v is a stable Hamiltonian, 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, 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. Homotopic to. So it doesn't have to have image r cross f 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 TRIDL because your TRIDL is a comma u, a is the r component, and the thing without TRIDL is the stuff in v.

Then, for connected component 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 to minus i. So therefore, what you could think of is,

if I look at the 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 a principle S1 bundle over the punctures, and you want to have a complex,

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. What is your j part of the data?

At this point, j is fixed, yes. What is j? I don't know, you keep talking. I don't know what you mean by j, there could be possibly two different j's. There could be a j in the target and there could be a j on the domain. Okay.

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 you're talking about, if that's okay with you. Thank you. Yes, you can.

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-1 to the negative puncture on the floor i, then I projectivise the tangent space real with this structure here, with the minus structure

and this would be the principal bundle isomorphism covering this bijection. So the bijection would say which puncture corresponds with what and this would align the arrows. Alright, sorry, can you explain what your letters mean? What is s i minus one? What does s i mean? Okay.

Catherine, 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 i's is one floor negative punctures on floor number i

s i remanent surface on floor i, j i the j on floor i m i marked points on floor i, d i nodal pairs on floor so that 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 Catherine, Catherine, this is online just open your thing

and write on to the text. Okay, so now Okay, so does anybody else still follow? 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. So v is a map which is a projectification. It's a tangent space at the punctures of the remanent surface for this structure. Note here

is a minus sign for the negative structure and this is the principal bundle map. Principal bundle isomorphism. Alright, and so where is that b in your next slide now? This covers that. Oh right, but in the next slide. You see b hat? It's a sign b one hat implies

b. That's supposed to be what's the previous and that tells you how to attach alpha zero to alpha one? Yes, that including the rotation which lines are mapped to what? Good. So what is 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 their map punctures to punctures, the negative punctures to negative punctures. On this part here, S-I-J-I

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 there is a constant on each level, if you shift your U-I, then you get that composition. Sorry, what is the D again?

Nodal pairs. Nodal pairs. Nodal pairs. Plus just means shift in the real direction. Yes, that is on the, so for each floor, you're allowed a common shift of everything. By adding a constant to the R component. Because we remember the objects were

equivalence classes on each floor. So under so these are the morphisms. Does the morphisms have to preserve the matching of the asymptotic marker? Yes, it should. So if you, that's a good question, so if you have T phi I,

it actually gives you, yeah, so it should map equivalence matching to equivalence matching. That's actually quite important. So for example, to answer this question, illustrate a little bit more, so if you have OK, suppose you have

two double covered cylinders, so a building consists of two non-trivial double covered cylinders. Then, so in particular in the middle, I have a double covered periodic orbit and there you have these 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. So if here is 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 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. Okay, 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 biholemorphic 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 finitely

many morphisms. So of course I call it groupoid. So a groupoid is usually a small category which would have this property. I call it groupoidal 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 one the grading would be zero. That means I only see the ground floor if I have a building of height two the d would be one that would be the number of the top floor. So European notation having this grading

my category decomposes and let me artificially find what I mean by the boundary of the category these are all objects in SI for i at least one. 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. You can build on Riemann surface lying in R3 or something. Realize in R3

so it's clearly a set. So the morphisms preserve the matching but not the asymptotic markings? The asymptotic mark is not here. So in contrast as I said in the beginning to what we write in the paper where we have

the asymptotic things. We have them only numbered but we have markers there and we fixed points on the periodic orbit. We don't do that. So that we had before as well but we had on the top our mark points were ordered the punctures were ordered and they were asymptotic markers the punctures which were assumed

to hit a particular point on the periodic orbit. The periodic markers are only the punctures that are matched. I'm confused about what the question was and what the answer was. No, he obviously read the book. So what did he ask? I'm asking about the part which are marked in this

setting. So you have these Bs which match up. Yes, they just match up which puncture to what and then the B hat is the lift of this. It just says it matches up lines up to rotation and then if you take a holomorphic polar coordinates positive negative one adapted to this then if you run along

zero in the direction of plus minus infinity then they hit the same periodic orbit. It's the same point on the periodic orbit. I think the question is does the morphism in your category have to carry this data? Yes, yes, yes. The B hat has to be preserved. the choice of polar coordinates is not necessary. So you can have just two arbitrary different choices

automatically as one. So if you have two of these things which generate this class then if you apply this morphism here phi i minus one and phi i here then it is mapped to a pair of lines which is that associated to the object.

Sorry, can I ask that again? Maybe differently. So let's say you have a morphism from alpha to alpha prime. Each with two levels. So then there is one matching datum.

So if I fix my matching datum B for alpha are there different matching data for alpha prime that I might have a morphism to? No. What I just said if you for example take so suppose this is

double covered covering number is two and then here I have something where a covering number is one let's say. So this is twofold covered and this one. So then suppose I have this matching here then obviously I have also

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. The something is some stuff which has already given a matching. So the lower part maps 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 country. So if you 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 the phi I. If I fix the phi I I fix phi one and phi two I fix alpha zero b hat one alpha one I fix alpha zero prime and alpha one prime.

Does that determine b one 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 the answer is just yes. The answer is yes. Okay. So Helmut in this example the top cylinder it's just a trivial mapping cylinder or or yes no no no not trivial it's just something which is double covered.

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 180 degrees but that is a different object

it has different data. Okay. Weighted subcategories. So it's something like Facebook like or not like but with rational coefficients. so

so you look at Q plus between 0 and 1 which is just the identities as morphisms so you look at a functor there and you are interested in and if I am 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. So example given j-triddle which is almost complex structure r cross v and compatible with the stable Hamiltonian structure we can talk about j-holomorphic objects in S which are the u's the objects where the u's satisfy the Cauchy-Riemann equation. So then define

this by associating to the weight one I like and to the other this thing I don't like. So that's our modelized space for the time being. 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. Okay. Good.

I felt hope. Yeah. Katrin? You said that there was something about the aim The aim is that is what I had said before

it showed up here. So deform 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 thetas, you can define differential forms on these categories and you can integrate them. So there's a really cool nice theory for this. Okay.

Aim clear. So now so there's more structure here. So when I have an object in my category let me define a different object, 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 positive punctures I just take these points 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.

Yeah. For example if I have two points here and they go to the same orbit then switching the two points would be a morphism. Is the point of all this categorical 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

a manifold of a differential form will be a differential form of the category and when you have orientations you can actually integrate. You never have to leave so once you have so it will later be a polyfold structure on the category. You never have to leave this. There are all kinds of universal constructions. You can do the whole transversality on that level here.

At this moment you don't assume anything about this? There's nothing. No, I mean about the structure theory we quoted. Well when I make my definition I assume that it was a non-degenerate form. Not most what? No. Most what I haven't dealt with and I don't want to. Let me take a word for you.

Oh, yes. That was discussed last week. So we have an odd dimension manifold with a stable Hamiltonian structure and that defines the so-called ray vector field and that has periodic orbits. 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 fibered products and they look

like this. So observe here that such a map b i which we introduced before this b i lies below the b hat. So if you forget about the matching of the lines is in fact is in fact such a morphism. So

so we look at chains of objects and these morphisms in category b that is called and we take the fibered product. 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 into this category here and there is a map mapping 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 hat 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 but we are some category we are some covering functor. Is that clear? So it's clear

but I don't understand why you're doing it. Is it to describe the Well I mean this is where d squared comes from. So you're saying that in the back structure you don't already have matching No you don't because in the boundary you have but 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 chop your thing in the boundary you chop it apart here but once you do this what does that mean? You just took a weak fiber product. You don't

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 when 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 where I 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 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. Ok, so now that is what we are looking at now. We take five products of length so with two elements, three elements and so on this weak viable product. So now S was graded by

D the top floor number and now let me take multi indices and I is greater equal to zero. Because I want to so this has actually two filtrations in this category so then I can talk about

the norm of such a multi index but just taking the sum of the numbers and I can talk about the length of this. Can you avoid this problem by taking this S to be a category with objects remember these points isomorphisms turn the circle and two morphisms between

the isomorphisms. So you want to go to a two category? I don't know. I didn't think about it. So then given such a multi index S I can obviously build this product here. So these are degeneracy n0 nk and then

sp can be decomposed like this. And there is now Can you say in words what that means? Yeah you see pictures. I will show you in a second. So then there is a

degeneracy functor which you can define on this by just taking the degeneracy of the objects and there is a length functor just look how long is that product here. So here is a picture. Maybe. OK 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 hat here. These are the BI hats. So now I get 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 ground floor and the other top floor two. So that is a certain requirement on this here. Namely namely there should be at least top floor six so now I chop a building with top floor one fiber product with ground floor fiber product with ground floor, fiber 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 and so on.

Is that clear? So when I see a round dot here I should think of one trajectory. A dot? Or one of the dots here? Yeah dots. Where is 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 it had less floors. So if I just want to try and think how to prove D squared 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 floors are kind of complicated each floor can be many many things. So it's an easy procedure 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 one 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 final 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 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 is 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 right and the target on the left. What source on the, no source is always writing like this.

So this here okay, so 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 the final product here and I do this. So now 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, bijection to the object starting here. What is the significance of your different typefaces because you've got italic typefaces which are the categories.

Okay, so this is maybe I'm too didactical here. So a normal C would be the object and this bold face would be the morphism. Ah, well that's what I want the bold face would be morphism. 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 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 equals zero fij takes an element of 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 or I chop it here or I chop it here or I chop it here. And these are actually covering functors so you can just take this definition from here to here. So that precisely satisfies 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 floors. 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 internal nodal points.

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 chopping if you choose the right indices

is commutative. So then using this rule actually you can extend this chopping and we saw already the picture inductively for every multi-index. So there is from s i plus this this product, there is such chopping

and that is precisely this here. So if I take f 1 0 0 2 it is on s 6 and it chops it in this way. So these are all covering functors. So you start with initials so you can check if you just use anything else just algebraic

these are all covering functors. Created from a small list of covering functors just name me those guys. You're writing equals everywhere but you have a bunch of more different categories. No that's not, that's actually equal. That's actually equal. No just you know if you have three buildings

chop it here then I get a building of height 1, height 2 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 kick back on the v?

Yes so what does this thing actually do? So this here takes the right spot in between where you have to cut that on one side top floor is i and the other j and it forgets precisely the heads. 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 heads for the b. And that is obviously rather commutative if I forget here and forget there or the other way around. So if you look at

the j-trill case, the pseudo holomorphic case you see that immediately. if you have a building where each of the functions u is j-holomorphic if I forget the head or not it doesn't change that the things stay holomorphic. Obviously that is true.

So now we come to some formalism which is quite useful. If I am given such a Facebook functor here and then let me define the boundary of this. It is 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. It looks a little complicated and you try to understand this. Sit down. It doesn't say more than precisely that

this is true for this theta. Forget the j here. It is just 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 you can obviously define a theta on the fiber product. You just take theta of a fiber product as a product of the thetas of the particular values. So I define theta p.

And then so here is a type here should be p. Sorry, here should be a p. Then you pull this back so you get to the boundary of the category build this sum here and you want this condition. It just means

that fij composed with theta p is just theta restricted to i plus j plus one. That is what it means. So there is 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. Otherwise I have to say for all ij this and this holds true. Just say that. Now this formula occurs in a lot of ways. You can do a

photon theory satisfying this formula and so on. Is this an inclusion exclusion formula? So model this thing that here should be theta p. You just build, you see this takes values in zero one rational and you just take 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 fi's come from an s s i 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. But it doesn't say more than compatibility with the fij's. So now trivial cylinder buildings. When we do a perturbation theory later

for the algebraic structure when I have a modelized space a modelized space of J-holomorphic objects and I multiply on each level I put say the same J-holomorphic cylinder in. I get a new modelized space and obviously it should be the same up to that multiplication.

It should be identifiable with the other one. 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. when I

do a perturbation theory and for example this building of height 1 would be pseudo-holomorphic. Then so I would have this modelized space of this kind of thing. Then if I multiply it by some J-holomorphic cylinder

that should also be in my modelized space. Because otherwise you never get the algebraic structures which you have. So if I multiply it by several J-holomorphic cylinders that should also be in the modelized space. It might have different signs and so on. So if I integrate over that modelized space having this it should be basically like integrating over this modelized space

say up to sine for example. Can you explain that theta formula again? We're on the last page. Suppose I have something with like three levels. So the various things could assign to it. It could assign product of theta of the three levels. It could assign

theta of the first two levels times theta of the third. So what is the relation between all of these things? If this is true it just means that for the product of two things

is equal to theta on s i plus j plus one for all ij. 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. Well 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 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. Now trivial cylinder

buildings. First of all let me first say a trivial cylinder building would be one so if I have height three here it would be 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 a geolomorphic cylinder building would be where each of this is actually geolomorphic. So one more question on this theta. If the theta is like zero 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 both of them would be one then the building above where

it comes from would be one with the number one. Then it's not then it would be disliked. in the perturbation theory if this

is a liked object and I add geolomorphic cylinders to it a geolomorphic trivial cylinder building it 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 with the same weight. That is

otherwise you never get any algebra structure. Otherwise you can't get the multiplicative structure. So these objects are different though? They are different but in some sense they have to count the same way 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 holomorphic 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

least an R-parameter. So that is indicated by this. But I think everybody understands, sort of, when I say add a holomorphic trivial cylinder to it. But there's not a unique way. There's a family. What do you mean by disjoint union sub R? That is just explained. I just said that you add to alpha a holomorphic cylinder building, a trivial holomorphic cylinder

building. So is TCB a set or a one? Trivial cylinder building, J-holomorphic. So it's a set of the J-holomorphic? It's just a line. If this building has height k, it's a line of k cylinders, which are holomorphic and

trivial. I know. Yes, but there's not a unique way to add it to it to 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 add a trivial cylinder building to it, and then for any representative of this,

because it's not unique, the value should be the same. So I would write times R, times the set of two-line cylinder building. What times R? No, it's R to the k, because for every... Yeah, so if there's a building of height one, you have just a one-parameter family of

building of height two. For each of these things, you have a level R. So I'm just keeping it simple by this. Just add. So if I... So whatever this symbol means, alpha disjoint union TCB, it's a subset of your object

category. So here 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. Okay, so let me ask this again. So when you write TCB, really you mean to just put one chain of trivial cylinders?

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.

What? Then you actually start. Yeah, let me...

I know, but this is frightening. So, time-wise. Okay, so let me just... So now I'm using a theorem of Joel Fish. Okay, so give me a few minutes here and then I'll stop.

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's disconnected on each level, then each thing individually already existed and was perturbed and so on. Okay, so summary.

Then I'm stopping. So we are interested in deformation theta of theta twiddle. This is the original function which identified geolomorphic objects. We want to deform it to a theta, having the following properties. This is true if we add just the triple 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 have stable things and move them against each other, this has to be true. So SFT algebraically comes from this.

That's all there is. 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 a true statement? Yeah, that's a good one.

Okay, so let me not buy into this. So I have to look at the algorithm. So the algorithm is a little bit… I will give you the algorithm how to perturb. So the algorithm precisely…

So in the first time to perturb you have disconnected objects. So in the hierarchy concerning some complexity or energy, on the bottom floor, where things cannot degenerate further, there are disconnected things. Connected things which are perturbed. Yeah, it comes from there.

So that is what I wanted to do in the first 15 minutes. So are there any brief questions for Helmut?

So 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 in which you just assign one to holomorphic things and the other would be the thing you need to achieve by the deformation.

Is it easy to… So if you take these properties here, you see immediately that they are satisfied by the pseudo-holomorphic. Precisely. So of course we don't want to just take the trivial deformation. So you want deformation which satisfies these plus something. Okay, so that's a good…

So my… As we were 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 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 in a category but you can only do this for… when you have this polyphony 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 subcategories

and what are they? So they actually occur in different flavors. There are manifold type, orbifold type, branched orbifold type and so on. And these things then will be constructed by a freedom 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 of this. You have the Kosher-Riemann 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 are the zero-one forms along the underlying map. So here you have the Kosher-Riemann functor

and then you define something like this Facebook thing here. But it's a section, 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 gives the weight one to the zero section and to everything else zero, then this here is actually our theta J-twiddle. 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, of course, 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-twiddle for each object and then you move the objects now you have several ones with fraction of weight

in different directions and they line up to manifold pieces locally. That is what they do.