OK, so far, so we have the category of stable maps, then we showed that there are certain... Then the subcategory of holomorphic objects, pseudo-holomorphic objects is given by a theta

which just associates the weight 1 to such an object and otherwise 0. And the idea is to deform this theta j into a theta so that certain number of properties hold.

So we discussed that in the first lecture. Then in the second lecture, we showed actually that there is a smooth structure on this category and explained what that means. And what we would like to find are smooth things which do this. And now how do we get them?

And that is going to start to happen in this lecture, is that we will define another category lying over it for which we have the Cauchy-Riemann operator gives a section functor of this so the fibers actually will be Hilbert spaces over each object as a Hilbert space

and the Cauchy-Riemann can be viewed as a section of this. And then the theta will be obtained in such a way for another kind of functors which are this time defined here, which you can view as multi-sections. And I will explain all this.

So here we already know there are some kind of smooth objects, smooth functors. There will be also some kind of smooth functors here. And if you choose this thing in general position satisfying some properties like this but we have to formulize them for these things, then this one will actually have these properties

and will be of a good type, namely it will be of the smooth weighted category type so locally it is represented by many folds divided. So it's good enough that you can actually integrate forms over it and so on

and actually can define SFT. There are some issues, something that we would have to discuss, namely orientation. And orientations are better done to actually go to a covering where one actually introduce numberings of the punctures and so on.

Okay, so that is what we did so far. Okay, so this bundle category. So we have our category of stable maps and we take a functor into Hilbert spaces. So when I formulate things I usually give somewhat general formulation

but for the category of stable maps. So you can think of other categories you put in. The scheme works actually for a lot of series. So we associate to an object in Hilbert space

and in this category the morphisms, so two different objects might get two different Hilbert spaces but the morphisms are actually lifted to linearize the morphisms. So then you can define a new category, namely it takes the object and the vector which lies over that object in the Hilbert space.

And what are morphisms? Well it's a pair of phi. Phi is a morphism in your category of stable maps. E belongs to the Hilbert space lying over the source of phi. So the source of phi is say alpha.

And what is the target of this morphism? It's just the vector obtained by applying the lift to the original vector E. So you lift each morphism as a linear map between the fibers and the objects are the vectors in this fiber

and the target is the image under this linear map. So I wonder who did that? So I don't want to destroy this piece of art.

So here we have the object alpha, alpha prime.

So here's zero, the other zero. There's a morphism phi between these guys. And we have a vector E and the lift mu of phi is a linear isomorphism which would map this here to mu phi of E.

And this thing here, so we can identify then a morphism for this bundle category with the underlying object and the vector E, morphism is phi.

And the source of this is E and the target is the image which you get here.

So what do we do in our case? Well if you have, so that's a building of height one let's say, then what do we do there? Then this, then here we have an equivalence class of maps up to R action. So we take a representative, now you see when you take the tangent space here

then because you have the R action here you can just identify this with R cross the tangent space of the underlying V component of this map. Your twiddle has the R component and the V component and our Hilbert space for this object consists of all maps which are complex anti-linear

from the underlying Riemann surface, that's a point Z, into this thing which is identified since I take a representative of here mod R action but the first factor is independently defined on which thing I take.

And this map should have a certain regularity property namely it should be away from nodes of the class H2 and that is because the Kocherimann section will act on H3 stuff and it goes down to H2. And at the punctures you have to also take exponential decay of this thing

so if you take a puncture and you take cylindrical holomorphic coordinates you want exponential decay of the partial derivatives up to order 2. Which is precisely when you take these stable maps which are asymptotic to cylinders and have quality 3 delta 0

so 3 times partial derivatives with exponential decay and then if you apply the Kocherimann operator they would go to 2 derivatives with the same decay. So that is what you take. So that's the Hilbert space and more generally if you have a building

then over each of those so you have alpha 0 up to alpha N over each of them you take such a thing. It's clear then how this acts namely if you have a 0 of 1 form you just take T so then these morphisms come from biholomorphic maps between the buildings

and you map E to T phi inverse or something like this. E composed with T phi inverse the tangent map of the biholomorphic map inverse. Can I ask a question from this slide?

So you said that you wanted a bundle of Hilbert spaces but in your definition of what it means to be a bundle has not been given yet. So in my talk I first did the underlying space just as a category without smooth structure then we looked at all the relationships

then I put a smooth structure on it could say more about it I put just an algebraic structure on this discuss that and then I put a smooth structure on these two things and at that point you are on the level where you can just unleash some abstract perturbation results which brings things in general position so that's the structure of the talk.

So here we inherit a lot of the structures which comes from the stable maps so we had this input out, this evaluation maps functors E, V, plus, minus well, we just compose it with this projection down here

and then we get such an evaluation map for E a grading, the grading we take from the underlying object then we can decompose this

and that is actually as before except that this thing now has a little bit more structure over each object so it has a structure over each object over each object there lies a Hilbert space and the next thing is to lift the data from S to E

and the data from S to E means in particular the covering business which we had so how does it look like? actually rather trivial so in the base we have this chopping functor

and then you just over each of these parts you have the 01 form and you just put it forward and it's linear, it's fiberwise a linear isomorphism so I have this object which is a building and over each of these floors I have a 01 form and I chop it here and just take that forward

and that is an isomorphism on the fibers so it then satisfies precisely the relationships which we had so that's basically completely on the nose so you don't even have to think about it and then there is a functor maybe if alpha is given which consists of different buildings

you just for each map on each building coming from each building you just apply the Kuchariman so that's a functor and so let's now think you might have noticed it looks really rather like this

the color is better that is Joe's color yeah, well, I mean it's sort of better than that one ok, no, let's not elaborate on this any further

so now let's first algebraically discuss what we can do here so the idea is to perturb theta j and theta j limits more options here

so if I put a pseudo holomorphic object into this

then this vanishes and this thing gives to the zero vector weight one and otherwise zero so then that's precisely this one so then we want to perturb this and what we see here this gives just the weight one for the zero section so the idea is now

if this is a zero section locally I want to have a partition of unity of this and just move so I've used a zero section as several zero sections but with rational weight and then I move these individual parts away to achieve transversality

that is what you ultimately want to do so that's the idea of course then you run it so this one will be perturbed let's say but when you move this away you want to keep the symmetries it should stay a functor

for example locally you have the action of the automorphism group this whole thing which you get so it might be so this will be turned into something like this if this is a zero section so here's the base s and here's sort of e it would sort of look like this

and you want to turn it into this but you want that this whole stuff these things each have a fractional weight you want that this is invariant then of course you do this at different places so it becomes a little bit more messy so if you go globally so then the things which are constructed might be then bifurcated further off

and so on so that is what you want to do so you need to develop sort of a machinery to be able to pick such things which sort of so you see if lambda zero is replaced by some lambda

which consists out of this section then this is only positive if the Kocheriman operator perturbed by such a section is zero what does that mean if this is a graph if this puts a weight on a graph of different sections then this here will only only become positive

if you solve Kocheriman of alpha equals one of the things in the graph and this you want to achieve transversally and if this is transversal then that actually will be a smooth object a smooth functor so that's sort of the idea and for this then you have to develop a little bit of machinery

is that clear sort of what the aim is you are the chair you are not asking questions ok you are good it's like the most basic possible question I should ask before what would happen if your perturbations were not continuous well then you lose some symmetry and I think there is still a theory

but it's not the theory you want to do I think you can actually do some really brutal stuff and ignore some of the structures that you can do and that's another way to produce data and then out of this presumably you can produce some invariants

if they are interesting I don't know it's like if you do S1 invariant Morse theory then you just forget the fact that it's S1 invariant and then you have usual Morse theory something like that so it's on that level but if you work locally then what the functor is doing

is really saying you choose something locally and then it's consistent with something else you choose to choose somewhere else so you have to keep that but you might not so in some sense when you patch it together you want that it fits but you might maybe it could fit in some slightly complicated way

it could be so you have to think about it because just saying this should fit with that is easier to say with a whole lot of structure which we have than actually saying I relax it somewhat locally I could say it should not be invariant under the action of the isotropy group

and then say it can match it up globally because the constructions are local so you write as you will see a perturbation as the sum of a lot of perturbations and you construct them locally and then you have to transport them all over the place just by the morphisms the local construction but I think it's possible

at least in a general framework if you have a criterion to forget some of the structure for example you could definitely forget the structure that you want to when you perturb that if you add trivial cylinders to it

that they should just appear through the holomorphic cylinders you could use them in your perturbation so that would stay consistent as long as it's invariant under the morphisms you could also disregard the fact that things are disjoint unions I mean if you have disjoint unions if this is the solution to this you could also go away from this

however I don't think for the latter one you would get a new theory because since you can do it there's at least a cobordism from not doing it to the one doing it and then I think when you arrange the data you might actually get just a lot of constellations but I haven't carried this out so there are a lot of things you can think about

so what are the requirements on lambda guaranteeing the desired properties for theta here they are actually they are even better to formalize than for theta so

so first of all we have to define something on the fiber product so if this on the fiber product I ignore, I don't write down that E0 lies over alpha 0 so then you just take the product of the things you don't want any 0

the 0? the 1 should be 0 oh yes I want everything so this was my keyboard I guess I typed the wrong button so this is a 0 sorry

so this is a restriction then we have this covering function so here's the algebraic version of this which corresponds to the version the first lecture but this thing can be also lined up according to the underlying faces you have which was sort of lecture part 2 with a lot of

confusion which hopefully decreased in the discussion session which would be the same formula so this is the algebraic version so is the right hand side not equal to just taking this maximum breaking of any one object?

yes if you have this property yeah I mean so what this says is basically maybe I should never write this formula so what it says is I guess mainly I'm asking is there some interesting sign that I need to be aware of?

no, it's just a lot of cancellation so ultimately there's one term if you have this let's say lambda is always plus 1 or minus 1 the result here is always going to be plus one so now here that is

now important if I have an object alpha then it has an associated Riemann surface and then the Riemann surface can be decomposed as the component so let's first think a building

of height 1 the parts of the surface which carry the things which are not trivial cylinders and the things which carry trivial cylinders and if you have a building a trivial cylinder building is just a line of those guys and so when you look at this thing

you can see the trivial cylinder buildings and the rest of the components so that's a natural decomposition and you have a forgetful functor namely it forgets the trivial cylinder buildings ok, we heard that already so now there is first of all a Whitney type decomposition of E

so if I look at my Hilbert space and have a 0, 1 form over it I can put this thing 0 on the trivial cylinder building or I can put 0 on the

complement so this here is the part which is defined on the original building but it is 0 over the trivial cylinder component and this one is perhaps non-zero over the trivial cylinder component but it's 0 on its complement so you have this decomposition here it's written

and I have a question so if you have a two level building and if you have a two level building and it has no trivial buildings in it so it's non-trivial on every level and suppose on the bottom level you have a trivial cylinder and something non-trivial then what's the corresponding splitting

is ETC just restricted to trivial cylinder buildings or restricted to all trivial it would be here trivial cylinder buildings so so if I have something

and then what you said I have some non-trivial cylinder here and I have a trivial cylinder there and then I have this which is trivial cylinder so it would put so I have a 01 from over this

so the ETC would just put the value here 0 not here but it turns out when you do your inductive steps for actually constructing lambda then this bit already appeared earlier and then that the thing was already having required properties here to begin with

and the trivial cylinder is something which is in a sort of zero it's homotopic to something to a J holomorphic cylinder so here is the picture so

so here is a lift so here is a lift of the little C it just forgets the underlying trivial cylinder building and just restricts and gets this new object here in E so before you just forgot about part of the stable map now you throw away

part of your object in E so now you have two functors so one is pi pi is just the projection here of this Whitney decomposition on E, on this so in particular it preserves the underlying it covers the identity

on objects but this one C does not cover the identity, it covers the forgetful functor below where you actually throw away trivial cylinder components why are we having such a careful discussion of the trivial cylinder buildings? because of

if you want SFT or want to so if you think of this here as a preparation of producing ultimately data by integrating forms over components and so on then the next step is what can I do with the data can I represent it as a chain complex or something like this

if you want this thing to have certain properties and in this case an algebra property you have to you have the discussion of the cylinders they look of course completely trivial but since you make concatenation you add something into it and so on so they play actually a non-trivial role so that is

why if you would disregard them in some way let's presume you're also some theory but it would be different or possibly different so now this is of course a projection here and here if the underlying object

doesn't have trivial cylinder components then actually you have this identity obviously because there's nothing to put zero all this functor this is also a retraction it's linear on the fibers it covers the other retraction

so that's the structure which we have and this thing this commute in this way and then what is important is that if you restrict C

to the non-trivial cylinder part you get actually fiber and isomorphism, that's actually important that allows me to pull back perturbations by this, because I have the linearity in the fiber isomorphism in the fiber so if you go through this list of things here you find that it's actually

rather trivial in the concrete example but this kind of thing how I write it is actually in all the problems like flow theory and so on so it's just always this kind of structure ok, requirements so there's this pullback operation and here

so what does that say? so lambda e should satisfy this so if lambda e is positive that has to be this, so if this is positive this has to be 1 and what does that mean this is 1? this means that

over the trivial cylinder part the component of e is 0 what that means is I actually don't perturb over trivial cylinders yeah so if you think if you don't perturb over trivial cylinders that means then d bar over trivial cylinder is 0

which means it's actually the pseudo holomorphic cylinder I don't have to perturb there is that clear? so if lambda e is positive, here I have lambda e lambda e here, then this one has to be 1 so this here is a part of e over a trivial cylinder building and lambda 0 is our original thing

which puts weight 1 on the 0 section and otherwise lambda 0, so if this is 1 this means this is a 0 vector that means the e over a trivial cylinder component is 0 so then the d bar part on this, if d bar

is equal to that e over a trivial cylinder, it's a pseudo holomorphic cylinder so let's see how it already produces one of the properties of our theta I'm going back to that picture for the cylinder that's in the middle of the bottom level can that be perturbed? no, because of the inductive

nature of things so on some levels, like a level 1 building, of course that is something which would satisfy this property so whatever you construct in the perturbation because of this algorithm, it will actually not perturb over trivial cylinders so you would always get pseudo holomorphic cylinders out after the perturbation

so then then of course there's sort of what we had before, if I have two stable buildings and I put them together, I can move them other than you want that property here if I take one of the representatives so

now so what does that now mean? let's just discuss what does that mean if this is positive on an object? so first of all, it means there exists a rational number sigma positive, a vector in the fiber over that object so that lambda of e is e and alpha satisfies

this equation here with a weight sigma sigma is the number associated to this object alpha so now if alpha is actually a building of height k plus 1, so top floor is k then the sigma can be written

as a product of positive rational numbers and each of the alpha i's, and this e then of course is a sequence from e0 to ek and each of those satisfies this equation with a weight sigma i so if you look at

so what's the interpretation of lambda composed with so it means since in the fiber of different vectors that it satisfies one of the equations coming from the vectors with non-zero weight and they have a weight coming from the underlying alpha and so I get a sequence of equations and each of them carries a weight and if I add the

weights all up it's 1, that's precisely the splitting of the 0 section but the equation itself doesn't depend on the weight no no no yes, so ultimately in some sense we count solutions but we don't count them 0 1, we just count them with a rational weight, and of course there might be a sign also

but this contributes so if I have two solutions with each topologically counting one and the equation has weight 1 half then the total thing I see is 1 1 half for that equation plus 1 half on the other so it's a system of equations where if the equation

is true and you have a solution you just it contributes according to its weight it's like the S&P index, how big is the capitalization of a company or something like this for those who are interested in buying stocks it's all this kind

but you're also going to say cap lambda of E equals sigma, that's what that means cap lambda of E, yes so this is of course what happens here which I said somewhere here, lambda of E is sigma so that means if I solve the equation D bar equals E this equation counts, and it's taken

into the general bookkeeping with the weight sigma so then this one here if there's a length, it's decomposed in a certain number of Ei's and we have this product structure and so each equation is this here, has this weight then I can put this together to this one

and the sigma comes from this individual weight of this part so then because of this property here each Ei vanishes on the trivial cylinder components actually on every i because this one was already perturbed and this Ei

so this is a building of height one and this Ei already satisfies the property that over a trivial cylinder component is zero so all the trivial cylinder bits, for example this one here, if you look at the blackboard

all over this, if there are solutions would actually be real J holomorphic cylinders so then if the alpha i has different components so on a so I'm looking on a floor then this then this component

you can decompose it according to the different components and some of them have trivial cylinders in it and each of them actually has a weight and the sigma i would be a product of the weights for the individual components, so this is what the perturbation all does so individual components are perturbed separately

trivial cylinders turn out to be pseudo holomorphic and so on so now we come to the smooth so algebraically I think now it's clear and now we have to put a smooth structure on this thing and see that we can

define what is a smooth lambda and so on and then we are ready for the perturbation Can I ask a question? What's going on? At the very end you end up with something which is a q-linear combination of manifolds or something which is only locally a q-linear combination and unfortunately that means

No, that means, yeah, ok, so so modulo the following is actually one of the things you said, so which I'm explaining now so if I have too many faults, weight of each is one half I could view them as four many faults

by taking a copy of each of them with weight one quarter, that would be considered equivalent so then you can if you have overlaps, so you have here some many faults and here some many faults with weight what does it mean they fit together? basically it means that if you take a certain number of copies here

and a certain number of copies here, you can match them up so that the weights are the same so they are smooth so they fit together Which one of the things which I said is correct? I think the second one it's not a it's in the middle you can't break it up into manifolds and then give each of them a weight

because there may be there's no natural identification you just can identify after a copy locally you can do that but not globally I mean, you can put some artificial structure on it to say what you have to do at any given moment in the overlaps

you can say you put the structure on top and it says you have to take so many copies and that you have to identify with this one which is actually you have to do when you do extension when you do extension properties like for sections I mean the same thing for sections because when you look at how

if I have a section defined over the boundary how do I extend it to the interior well, the only method is you make a local extension take a partition of unity but if I don't know what to identify with what what do I add actually up then so you need that structure of this identification extend and then according to this identification you glue these things together

to get an extension near the boundaries thank you, good question any more? okay so so now the theorem is there exists a natural here star means up to fixing some discrete set of data a strong bundle structure for this thing here

so it is basically like the polyfold structure for this except that we here have a strong bundle lying over O we have the associated translation group part so the objects fibering so the objects on top are

vectors in a strong bundle over E then we have an action by the isotropy group on this and here it covers this thing, this is one of the size from the polyfold structure on S and we have this in a coherent way and then if you have two of those guys then we get the transition set

and that transition set is actually bundled over the transition set here and this has a strong bundle structure as it was defined in one of the lectures last week so this would be the generalization of the structure which we had here

and at this point you can start talking about smooth multisection functors so now it takes a Kocher-Riemann functor and it goes from here to here and actually

just look at this composition here then it lies in the image of this one and you get a local representative and it turns out it's S C Fretan which was defined by Cutrin and Jo and the modeler category so that is where this is zero

has a property that its orbit space intersected with each connected component in the underlying space the orbit space of this is compact that's gamma of compactness so now we are

in a smooth setting the Kocher-Riemann section is an S C smooth section that is what that property means, that this is Fretan then if this thing vanishes this means we have pseudo holomorphic objects which is sort of this

associated modeler category to this one, if I take this isomorphism class is intersected with a connected component of the orbit space, it's compact it's gamma of compactness that's by definition what it means you have a Fretan functor I have forgotten what O and S

O and S so S stable maps, good that is what we are talking about for a while now then O this functor defines the polyfold structure so these are these are the things which are injective on objects, if you pass to orbit space

you get so there is a point here, an object here which is mapped to the original given object alpha and so on but O is a retract well, you just take an M polyfold so then this is a strong bundle of an M polyfold that was also introduced, so that is the model

so is the statement that if I take E over S is some bundle it's a statement that whenever you pull back to a polyfold bundle a strong polyfold bundle is that what it means? so that means that given any object here there is a selection of a set of

those guys and if you put that in then that is sort of the local structure near the object alpha so is del bar phi bar defined by pullback? so in this case then for this construction which you have, if you take any of

those guys it is a freedom operator in the sense as we have discussed so the pictures here when I described before, you have the smooth functor theta and I put my hand in then I see sort of manifolds, now I have a bundle over this lying here and when I see the Kocherimann functor, the trace what it does here and where it's mapped to

that's actually a real freedom operator and that I can put anywhere and this structure gives if I know something here I can always transport it to a neighborhood of any isomorphic object so that is what it means what it means so now here is the polyfold

packaging of the SFT data so we have a strong bundle structure over a polyfold the Kocherimann section functor is SC smooth and fertile we have SC smooth covering functors with compatibility and some additional stuff where we have for each phase, so I haven't defined this but it's all clear, we defined it for

on the level for S so if a phase here then this here is just the stuff of E lying over S theta and this was a covering functors then there were a certain number of compatibility conditions which we discussed in the last lecture and also yesterday in the discussion so you have these diagrams of these things and you have that

so I suppressed here the moving of components against each other then out of this data then one can write down which we did before a requirement for the perturbation you want to do

but that is basically sort of the smooth packaging of the data which you need to do, to produce the data which you need for SFT the last diagram also is commutative if you replace P by G bar all these functors commute

in the first two which you talk about this or this or this so there are three diagrams and there are two equations below that which I read as if you change the arrows down and label the P to arrows up and label the D bar so

if you put the Kocheriman section here, the log representative so here this is a restriction of the Kocheriman and the other one everybody is asking why there are three equalities on the bottom of three diagrams ok

so that's good because I forgot to write them so if you apply if you apply so what do I want to say this more controls this one so if I you see here here is the identity so there is not too much happening

with respect to D bar so it is controlled by the C but the C has certain properties with respect to the P I mean that's what you want to say here if you have but it is true that if P composed with D bar is simply zero no

if you have a solution on the solution set what you are interested in is identity minus pi composed with D bar would be zero which means on the trivial cylinders you would be pseudo holomorphic identity minus pi composed with D bar equals zero means on the trivial cylinders you are pseudo holomorphic right so that's exactly the equation here right

so pi composed with D bar is in fact equal to D bar it would be well you might have a non-trivial cylinder which is not holomorphic in S you are a non-trivial non-holomorphic cylinder in S

and then the non-holomorphic cylinder would not be zero no then it would not be zero I just said only on the solution set ultimately I think you can write certain things under the so identity minus pi D bar equals zero provided the

provided actually the equal zero means well that is precisely what you can say identity minus pi composed with D bar equals zero precisely means the trivial cylinders which you see are pseudo holomorphic but that is exactly your right hand diagram

couldn't you say ok good ok so something in that direction so let's put a weight on this 0.1 so there's some truthiness to it so now constructions of SC plus multi-section functors which are a particular

class of those guys here so this is an important class these are multi-section functors which you can view as multi-sections of compact perturbations of this so multi-section functors is

of that particular kind provided it has the following properties so if you take this uniformizer and the underlying thing Q0 would be the object where you're looking at then this composition here is

a count of the number of indices for for having the number of indices where this age satisfies this so you go into the base ph so age lies in k

ph lies in O these things are defined on O and if SI of the underlying base point is equal to the vector you put in you count the number of indices and divide by the number of indices you had and these things here should be local SC plus sections and let me

remind you what that was this strong the strong bundle comes with a double filtration namely it made sense to talk about M k and here M where k

is less than or equal to 0 less than or equal to M plus 1 so in particular you have a k you have a k 0 1 lying over an O 0 and the SC plus sections are actually going from here to here and lie in the fiber over M

in M comma M plus 1 so the SIs so the SIs go from so they are defined on O 0 but they go into k M M plus 1 and then of course what is important that is why I said is compact

if you go this is a fiber regularity and if you view them with respect to the different norm that is the compact inclusion that is what I call this compact perturbation so these are some kind of sections but they are constrained by having this property and I think you talked about that or it was mentioned maybe

last week I don't understand what the definition is there saying that there exists SIs yes so there exists finally many SIs indexed by the set I and you look at the coincidences so basically the picture here is if this is O here and this

is a fiber then locally so you have a certain number SIs I and I and each of them carries the weight 1 over I 1 over the number of elements in this thing and you just look at

this vector here how often on many how many graphs are there in which it lies so you have this vector this is an E here we have a definition of a multi-section function the only difference now is that we require the things that we locally represented by the S

right so locally in a chart or uniformizer so first of all the multi-section functions were in each fiber there were a finite number of vectors which having weights adding up to 1 so now if I put the chart in then I have this of course on the image but this difference should lie on graphs

of an SC plus section is that clear? so if you put your hand in and you see in the fiber the different points they line up as lying on a graph of SC plus sections so now we want to and this section should be sort of compatible

with the group action and that's sort of the compatibility so there is an action of our automorphism group on the set I and you have the orbits under the conjugation by this thing lying in there so let me first say certain properties

what you can do with these guys so you can build this sum here which is sort of a convolution and this is smooth so if each of those guys is an SC smooth or SC plus so I forgot the plus here so if this is SC plus then this is SC plus because what is the representation

what is the local section structure of this thing you just have the section structure Si for one and Ti for the other and you just take all possible additive things and just take as a weight as a way to take one over the number of the indices here times the indices of the other are you using a lambda 1 and lambda 2

to set up different bundles? no, no, they are for our bundle E why do you call this a sum and not a product? it's a convolution that's better because on the section structure you take basically all possible things how you can add up things so if locally so if locally

the first one is given by Si and the other by Si prime and this index set i index i prime then the sum is given locally by taking all these combinations here but where the index set is actually

i cross i prime so plus because of that so so then this one here well just replaces

the sections locally by T times the section so of course it's 1 over T on the other side so this is a smooth family so if T is 0 you just get lambda 0 so I put the lambda the 0 up here what is lambda 0? the 0 should be up here

it's a section which is weight 1 on the 0 section I mean this is a smooth procedure so this one here what does that mean? the indicator function here just means that the local structure is T times Si

and then if T goes to 0 then you get the 0 section so lambda 0 of E is 0 unless E is 0 in which case it's 1 yeah and here oh that works exactly because the total sum of weights over any fiber is 1

so that is actually a smooth family if you change T so then if you have an SC smooth functor into R so it's clear what that means that means if you compose it with a uniformizer as SC smooth then you can put that in front of it

so you can use partitions of unity to cut off such multi-sections smoothly then this makes sense as long as locally near a point in a uniformizer that the family is locally finite

so if you take a point and then you have only finitely many non-zero vectors there and you just add them all up in this way then this is also again a good section so these are important facts for actually constructing perturbations this allows you to construct things locally and then just

add things up and then a good fact is if I give you any so it should be smooth a smooth object and a smooth vector then there is actually such a lambda of SC plus multi-section functor where lambda

of E is positive so I'm going to show you this how to prove this so now for example locally remember when Katrin was describing the transversality result and perturbation result so if you have a freedom or even in finite dimensions if you have a section

of a vector bundle and you want to make it transverse by small perturbation what you take is you add to it some Ti times the perturbation to fill up the co-kernel then you solve this with respect to the additional parameter you get a manifold and then you project onto the parameters you add it and

take a regular value and for every regular value that is a good perturbation so now what do we have locally locally so what we want to achieve locally is we want to break the symmetry that is generally what we have to do to achieve transversality of course sometimes we can avoid this then we take the orbit

of this perturbation which is also transversal and then we have maybe some more perturbations but for each of these problems it's precisely that argument so when you look at this you just have to make sure that one of the local problems is transversal you get a set of full measure for the perturbation you take intersection and then you take some of the values there so that's the only additional

complication but otherwise you use precisely this thing so what that means is that actually rather than taking local sections you construct local multi-sections and take that sum and each local multi-section depends on a few real parameters T you take the direct sum so that

it fills up the co-kernel and then among all this then you get sort of this branched manifold and then you have a projection on T and then for each piece of manifold you require that that projection is regular now these are countable conditions and so you find regular things so that's the only difference so it's a straight forward

thing coming from there then you can even go further for example when you have a boundary point and the kernel lies a little bit stupid with respect to the boundary like it's tangential to the boundary

then you could then if you introduce multi-sections who have a particular linearization you can actually tilt the kernel into the manifold to make it transversal but that is also a little bit, so for this you only have to construct a section which takes enough values to fill up the co-kernel

here you have to think about that it has a specific, it might have say value zero there but it should have a particular derivative which together with the linearized Cauchy-Riemann operator has a certain thing but that's the same problem like in finite dimensions so there's no, nothing new I mean, it's of course not so surprising because Fredholm theory is locally

a finite dimension problem times something you don't have to care about and for that finite dimension thing this perturbations are as rich as in the finite dimension theory ok, so so let me just explain you how I construct a section

so I want to construct at alpha in the neighborhood so I want to construct a section which has a certain property at an object alpha Didn't you just explain to us how you construct a section? So now I do it no

let me on some level now I give you on a precise level and still I have ten minutes ok five, ok good so you know you just have to put something on the table and then you get a good answer

so I want to construct something at the smooth object alpha with a given smooth vector over it, so what do I do? so I take such a uniformizer so here's a picture the underlying thing

so this is an orbit space that would be psi the image of this one if I pass to orbit space would be this red stuff so there's a point somewhere here which corresponds to the object, I take a neighborhood u there so now what do I need?

so in Hilbert spaces you always have smooth bump functions but on certain Banach manifolds as well but unfortunately on certain Banach manifolds or Banach spaces where you don't so I think c alpha does not have smooth bump functions so there was a study thirty years ago

given the interest in it, so there's a lot of literature which banner spaces have smooth bump functions and so on. But SC smooth bump functions are a little bit more there because it's a bigger requirement. But in any case, on hitabout spaces where I'll set up, you don't have to worry. So here is my, here is the set O, and here is sort of a neighborhood. So now just here say this is a point

which goes to the object alpha, now you just construct with a bump function this object here. So you just take a bump function, which at this object corresponding to alpha takes this value e0, which corresponds to the given thing in your fiber.

You take the support in the small set, and then you rotate it around by the action. So now you have a local thing. So this is now on the image of psi bar of k, so that then we define it by this formula, which is precisely the definition. And now we extend it to the whole category.

Namely, if you have any vector, then if there is no morphism which brings the underlying base point into the image of O, you just say it's the multi-section which has one on the zero section. And if you actually can reach this patch here, then you just define it by this, by what you reach.

And that is a smooth functor. Because if I go from one to the other, I have this smooth transition. So if I have a local section structure here, I just can move it over there. So that is a local construction. So now you can take a finite number of those parameterized

by p to fill up the co-kernel at alpha. Then the photon property actually guarantees that it nearby is also the case. You do this at different spots covering the compact solution space, and then you have enough things to do precisely what Catherine says. Said some time ago, OK?

So now I generously got five minutes, and I only use three of them. Stop here. Otherwise, this. Are there any questions for our speaker?

Can you go back just to the last slide? My computer is very entertaining. OK. Can you go through this again and tell me again

which spaces are which in this picture? OK. So in this argument, actually, it wasn't so apparent. It's actually important that the underlying space is actually at least power compact. So I took a look at psi of O. It takes the associated isomorphism class, which

is sort of this red stuff. Then that is here. So then in this isomorphism class is the class of the original object alpha given. And I take a neighborhood around this. And then the thing is that this is a metrizable space,

so it's normal. So I can actually find a small neighborhood around it that the closure and the whole space is still contained in it. That's important because otherwise, that thing will actually not become even continuous. So then you take the preimage of U in O, which is this blue thing. So this red stuff is O, and this is a preimage of this U.

So now in this one, you take a bank function which has support in this. And here somewhere is a point which corresponds to the object alpha, which lies here. Now it's the object alpha. This is a category. There's the object alpha somewhere here. It comes from a point which lies in the blue region. So over the blue region, there's

this point representing alpha, say q0. And over this alpha, there was this fiber. There was a vector e, which corresponds to some vector lying over this q0 in the bundle k. So now you just take a bump function, which is 1 in the neighborhood or at this point here times this vector and extend it.

So I haven't talked about extension results, but they are on the portfolio level quite easy. So you can ask me maybe on Friday, and I can show you how to construct them there. So anyway, so there is a section with support in the blue thing. And now you want to construct a functor. So what I do is I transport this section

around by conjugation. So and then I define, then I get as many, of course, some of it could be if you have a symmetric section, then some of them are the same. But that doesn't matter. This is your index set. And you give each of them the weight one over the order of the group.

One over the order of the group. So this definition of fg, suppose this q0 that you had isotropic, I mean, you can still. Yeah, yeah, yeah. So this is actually coming from the isotropic group of this element. So this is, so if it has a large isotropy, then in general, I would, for example, construct something like this f,

which achieves some transversality nearby. Then since the bile is a functor, then if I conjugate the functor, it doesn't change. But this then, so then the perturbation by this one is a conjugation of the perturbation by g. So it's also transversal, at least in the region where I want.

So moving this around doesn't destroy transversality. And then, of course, in general, you might see some other sections often coming from the overlaps or so, yeah. That's the point. But for the construction, that is sort of the minimalistic thing you have to do. And then you give each of them the weight one over the number of elements in the group. So that's precisely the requirement.

So that means now on that slice, when you look at what happens, so the section is now defined on that thin slice here. So now I have to extend it. So then I get an object here, here with some vector. If there's no morphism from this one, which reaches a point which lies in here,

so if I cannot, if I, so if, so if, so either I can reach this slice or not. If I can reach this slice by morphism, then I, if I cannot reach this slice, I put the weight on the zero section, one.

And if I can reach this slice by morphism, then I define it like this. So I look at what point is there and I give it the same value. And that is an SC plus section now defined globally. So it's, you know, it's not very difficult. It's just really always the local constructions.

And the language, I mean, the language is sort of so high level that you basically always see everything on the notes. So you don't have to go in complicated coordinates and say what it actually means, what you're doing. So that makes it, of course it could and it would be an equivalent theory, but the language level is much easier

if you stay on that high level. In particular, since on that high level, every information is there and there are abstract results who produce whatever you want. Okay, so, good? So I will ask a question that maybe I kind of know the answer, but then I'm not sure.

Okay. So what is, what exactly, what is the reason that we need to go to MPolyfolds instead of working with retracts as a local model for this kind of category? Well, you can, you could put retracts there if you want.

You can also put, you know. I mean, there are, so this is a little bit larger. So MPolyfolds is locally modeled on retracts. So rather than taking just something which has one chart by a retract, then it could replace this one chart by the actual retract, I do this.

It also has some advantage when I do cover, when I discuss the coverings. So what is a local model? So at some point, of course, in the whole thing which has been suppressed, I have to give a definition what is actually a covering functor, yeah? In the whole thing. And then I have to give a local model for this. Then on the top, you generally have more points.

So it's actually more union of retracts going down to the other thing. So the things are easier, but one could, but I think it would be unnecessarily restrictive to say that.

I mean, it's a fair question. I mean, it's like, I mean, the equivalent would be in MPolyfolds that are defined in MPolyfolds as something which locally has charts as a morphic to an open set in Rn. And I just defined, this is a manifold because it's locally homomorphic to some manifold, to some smooth manifold, it's a transition-smooth manifold.

Yeah, so. The manifold is like locally a manifold, I mean. No, no, no, no, no, no. Right, that's what this is. Also, one star is a category and the other is actually some well-defined smooth kind of object.