 So if you want to take notes, I would only write down what you don't see here, so okay Yeah, so Yeah, so this this SFT business actually maybe most people are not aware started in December 1992 So it was in Basel and I gave a talk you remember this So so I gave a talk about the Weinstein conjecture my first talk and Then yasha and I were running it 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 now Yeah, so So in so of course there was a certain number of issues so for example in 1992 nobody knew even how to deal with the transversality issues in Gromach, Witten and so on So then we knew that we should perhaps do some version of cylindrical But it was always uns not particularly satisfying so so then in 96 and there was 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 so in 2000 I was convinced that I could do SFT in the thousand something pages in full detail but but right from the beginning outside of the feeling it would be sort of the wrong methods to use and If you have to write a lot of things and you're never convinced about the project. It's a really hard so so then Then this results can send up We analyzed sort of what are the issues in SFT and SFT is a really nice problem from 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 so you can't say we put 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 strengths You you have to deal with all these problems and the problems and the analytical structures is It's sometimes quite tricky. So there's a lot lot to learn if you actually go into details so then ultimately When this was analyzed so you could think of how 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 threat on problem and What would have happened? Well, if it just be a classical threat on problem then people would have dealt with this like that Problem 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 then you have a lot of interlocking or depending threat on problems if you perturb here in a particular way then an Infinite me my other an infinite number of other problems have to be perturbed in a very particular way as well and so on and so on So so so I think if this technical issues wouldn't have arisen Things would have progressed differently. So and Now how you deal with a classical threat on problem if you if it's actually classical is quite easy It's not difficult. You just take a small perturbation of the section and such smell tells you when you set up properly That you make can make things generic so So when we looked at this we thought maybe One can achieve this as well. So but what are the problems? So the problems were of course that Corner changes are nowhere differentiable Which of course is the reason why finite dimension reductions were introduced in the first place because you reduce to spaces When you restrict that this transition maps which are sort of canonical are smooth Okay, so so then at some point and this project became so once we found some kind of a set up What was talked about last week that you can generalize differential geometry? Where the local models are retract then then this part of the project sort of got a life on its own So we 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 into this framework and There we put also a lot of care into it So so these are all things which are nearing the end now namely one of the annoying features of the analysis when you read any paper is that Constructions usually are rather global so everything is interlocking. I mean analytical Topological geometric aspects are all in this and then you know when you have to write it down you have to write a lot So it's tempting to say one can easily see Using standard method that that can be done and you refer to some paper Which refers to another paper and if you follow this it's not so clear that actually happened So so so therefore What what in principle when you look at elliptic theory? Yeah, 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 you want to have this kind of things here as well So that also will work so okay, so then Ultimately, which was almost forgotten is that we started out from a problem which now from this perspective is actually I Think rather transparent. So and I will talk about precisely how to deal with SFT assuming that you would know this This polyfoil language, I will explain the things which you have have to know and Then of course because of last week's Many courses. I think you have you have an idea About a certain number of the ingredients Okay, so So here is a stable Map so I want to explain so it's good to understand what that all means so This here is a remnant surface. This are part the red things are punctures. This is a Building now we use European notation for the floors This is floor zero Because certain things later should have degree one floor one floor two floor three so now Then there are marked points like this Then this arrows here so So this shows you two half lines in tangent space so if if you have a puncture and a half line in tangent space and you fix a disc then You can put corners on the disc which map zero to standard disc which map zero to this point But when you have this line you can map one in direction of this line So it completely determines what you have to do now here However, what I'm looking at is two arrows up to common rotation Yeah, but you look from right and left so they actually rotate like this but So you look at 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 zero the r direction positive or the negative They hit the same point on the periodic orbit Yeah, so I do not so let me say that to the to the sft paper There are some differences in this case for the transversality. I do not number the mark points I do not number the punctures that gives more Isomorphisms which are ever have to be compatible Is it perturbation have to be compatible with this larger set of isomorphisms if I number them Then if I take a renumbering I should perturb in the same way is a little bit more complicated to keep track of So so if you just throw everything away and do transversality on that level That's one thing then for orientation questions Then you might go to a covering which actually precisely means putting numbers there if you want so 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 disc and take polar coordinates That's it take a disc map and then the composites e to the something then if you run And and you do it and you take the disk name 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 Punctures and they are allowed to rotate commonly But the effect of this is then if you fix sort of coordinated system of polar coordinates one positive one negative Then if you run along zero cross r plus minus zero Zero element s1 then they hit the same point on the periodic orbit So now if the periodic orbit is simply covered there's only one choice But if it's double covered you have this one But you can also take this by 90 degrees rotated and takes a rotation class of this So so there are actually k many choices if the thing is covered k for The same to say that for each puncture. There's an asymptotic marker that you allow Basically quotient under s1 action that rotates the two common, but Yes, that is what you do. Yes, you have a common rotation divided out by the diagonal So but one you have to rotate multiplication by j and the other by minus j because positive and negative so Maps here or just maps. It's just I'm not done yet. I'm in the middle of talking so so now So so we have my so then here this blue stuff is what I call nodal pair So there's on each is a pair one There are two different points that could be on the same surface on the other which you should identity 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 our maps and this maps are not assumed to be pseudo holomorphic They're just maps and then the square bracket around it means it's an R class. So because the the image lies in our cross stay Dimension manifold save as a stable Hamiltonian structure and you have an R action So if you take the map defined on these things here and you was just shifted by an R by constant It's also so it's an equivalent thing. So this is the equivalence class No, not yet at the moment these are just so Remain surface with all these features Then the quality 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 Yeah, so the quality of the map is chosen so that you can do some analysis. So what I pick is H3 lock Yeah, 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 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. So it gives you a linearized Asymptotic operator that gives 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 this is a play Piper plane bundle in the stable Hamilton case doesn't have zero in the spectrum So each of those guys has a spectral gap and you have to fix And and of course if you 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 the spectral gap await So, so sorry, can you say again what they put on the screen up to our translation? If there is what is our target here our cross V? our cross V V is a stable Hamiltonian With many for the stable Hamiltonian structure. So now there's a stability condition Name me at least one so the each of these things here So I think of this not I being identified in the domain So each of these things I view as a component of my remain in 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 Geomorphic cylinder homotropic to No in between not but it's asymptotic to the same thing. So if it's homotopic to the other one the integral u star Omega is Zero so I write triddle because Your triddle is a comma u a is the r component and the single is our triddle is a stuff in V Then for a 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 nodal points from the nodal pair and so on Is it clear what the object is? So this is an object in a category. So so this here explains so 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 there's two aligned arrows Of course you can give that by complex under linear map or complex linear map if you change the structure on one side to minus I So so therefore what you could think of is if I look at a punctures and I take the tangent space Then I can take the real projectification of this. It's an S1 Yeah, but not naturally, but it has natural S1 action on it. So you have something like Principle S1 bundle over the punctures Yeah, and you want to have a complex you want to have a If you take the take here is a complex structure J and here minus J you want to have a S1 bundle isomorphism which gives you a bijection between the underlying punctures and it says Which line corresponds to what line I'm sorry, what is your J part of the data J is at this point J is fixed What is J? I don't know you keep talking. Yeah, okay, so, but I don't know even what you mean by J There could be possibly two different J's Catherine that could be a J in the target and they could be a J on the domain Okay Okay, so when Okay, so I can you just give me a little bit of a chance here to explain this Yes, okay, there will be precise definition But it would be good before you see letters. You know what we're talking about if that's okay with you. Thank you so now Yes, you can but Give me at least a chance to finish this and after this you can ask question. So so there in particular If I look at the punctures This would be the positive puncture of the floor called I minus one to the floor negative puncture on the floor I Then I projectivize the tangent space real with this structure here with the minus structure and this would be the principle bundle Isomorphism covering this bijection so the bijection would say which puncture corresponds with what and this would align the arrows Sorry, can you explain what your letters mean? What is si minus one? Okay, okay, okay, Katrin bear with me so now So now this thing which we just saw can be encoded like this This of course, so we have a sequence of letters the alpha is is one floor negative punctures on floor number I S I remain in surface and floor I J I the J on floor I mi mark points on floor I di Nodal pairs on floor so that would be horizontal things on fly the equivalence class of the map positive punctures Then this one are precisely the ones which I explained here which which tell you how to align things in between Who I'm trying to take notes I just told you you shouldn't because this is online here No, Katrin. I'm not Katrin. This is online. You just open your thing and write on to the text. Yeah, okay, so now Yes You tell me who matches why 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. It's the alignment of the arrows Projectification It's a tangent space at the punctures of the remnant surface for this structure Note here is a minus sign for the negative structure and this is the principle bundle map principle bundle isomorphism And so where's that where's that be in your in your next slide now in between it is this this covers that Well, right, but in the next slide It's a song it's be be one had implies be supposed to be what's the previous And not what tells you how to attach alpha zero to alcohol one Yes, that including the rotation Which which lines are mapped to what okay? Good So what is a morphism? So a morphism. So this is an object in my category So a morphism I write like this. So here's a source of target. It's a sequence of by holomorphic maps So they gave from here to here, which means they're map punctures to punctures the negative punctures to negative punctures on This part here si ji it has to be by holomorphic Mi is being mapped to mi prime nodal pairs are mapped to nodal pairs and so on and It should hold that if you there is a constant on each level if you shift your UI, then you get that composition Yeah, I'm with side. What is the D again? Nodal pairs plus this means shift in the real direction. Yes, that is that is on the so for each floor You allow it a common shift of everything Yeah, it's by adding a constant to the r-component Yeah, because we remember the objects were equivalence classes on each floor so under So these are the morphisms Yes, yeah, yeah, okay. Yes, it showed so so if you there's a good question. So if you have t-fire t-fire I it actually Gives you yeah, so it it it should map equivalence matching to a current matching. That's actually quite important So for example, so to answer this question Illustrated a little bit more. So if you have Okay, I suppose you have two double-covered cylinders so a building considered of two nontrivial double-covered cylinders then So in particular in the middle, I have a say double-covered periodic orbit and there you have this matching things So now this can be isomorphic It can be 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 brake is double-covered and the top is double-covered Yeah, then this object only has trivial isotropy So if if here's simple covered a double-covered periodic orbit in a double-covered cylinder on top This has only said trivial isotropy because I have to preserve the matching So I can't really rotate downstairs, but it's isomorphic to another object which has this 180 degrees So in this business it will be extremely important not to confuse being the same and being isomorphic Because for the counting things because when you do in the fine points of perturbation theory if you Be sloppy about it. You start counting wrong things Okay, so so this is now a category and Clearly every morphism is an isomorphism because you can write down the inverse immediately by just in by just Reversing the order of alpha alpha prime and inverting all the by holomorphic maps Now what is not so clear? But which is a consequence of the stability condition in the nice exercise is that between two objects There are only finally many morphisms Now so of course, I call it group portal. So a group point is usually a small category Which would have this property. I call it group portal because it's not a small 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 the building of height one the grading would be zero That means ground I only see the ground floor if I've building of high two the D would be one That would be the number of the top floor So European rotation though then let then having this grading My category decomposes and let me artificially find what I mean by the boundary of the category This are all as objects in si for I at least one So then There's also Important point There's also an important point. Namely if you look if you look at the orbit space Orbit class of this category. So the orbit class consists of all equivalence classes of isomorphic objects Then this is a set Now you can build on remain surface lying in R3 or something realized in R3. So it's clearly a set The asymptotic markers not here so in contrast as a beginning to what we are right 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 orbits 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 There are asymptotic markers as a punctures which will assume to hit a particular periodic point point on the periodic No, he obviously read the book read No, so what he asked In this Yes, they just met up which puncture to what and then the behead is a lift of this it just says it matches up lines up to rotation and then if you if you take a Holographic polar coordinates positive negative one adapted to this then if you run along zero in Interaction up plus plus minus infinity then they hit the same periodic all it's the same point on the periodic The question is just the more the morphisms in your category have to carry this. Yes The behead has to be preserved so so So if you have two of these things which generate this class Then if you apply this morphism here five minus one and fire here Then it map is mapped to a pair of lines, which is that associated to the object So can I ask that again Maybe differently, so let's say you have a morphism from alpha to alpha prime With each with two levels So then there's one matching data Sorry, so if I fix my matching data B for alpha are the different matching data for Alpha prime that I might have morphism to or is it matching? No, I just said if you if you for example take So suppose this is double-covered Covering numbers to and then here I have something Where a covering number is one let's say Yeah, so this is to fold cover than this one So then suppose I have this matching here, then obviously I have also matching where this The one appears here, I don't know if color short but Then you you could so so one matching is this this common rotation the other would be you put this thing on the other side This common rotation, yeah Well, let's say I fixed my matching there and now I have maps fee That take both the top to something at the bottom part to something Well, the some the something is some stuff. This has already given a matching So the lower part maps should if you fix two lines here The lower thing should map it to a line here in 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 the person is not so difficult country So if you write so question if I if I fix everything in your morphism if you go back one page If I fix everything If I fix alpha prime Right, let's say so the PI if I fix the PI right. I fix fee one and be to I Fix alpha zero be had one alpha one. I fix alpha zero prime and alpha one prime Does that determine be one one so Okay, so the first thing is in and 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 of course 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 fee Together with the choice of source determine the target. Yes. Yes Helmet in this example the top cylinder. It's just a triple mapping cylinder or Yeah, so no, no, no, no trivial just something which is double covered yeah, so so this would have an automorphism and This itself as well But as a thing with a with a marker here is the automorphism group is trivial But this is isomorphic to a different object where this one let's say is rotated behind at 80 degrees But that is a different object has different data Okay Vated subcategories, so it's something like Facebook like or not like but with rational coefficients, so So So you have you look at q plus between zero one, which is just the identities as morphisms So you look at a functor there and you're interested in and and if I'm given such a thing then I have a full subcategory which are all the things which I like to some degree positive and they are Have carrying a weight how much I actually like them. Yeah So example Given j-tribble, you know, which is an almost I invent almost complex structure R cross V and a compatible with Stable Hamilton structure we can talk about j holomorphic objects in s which are the use the objects The use that is was a kosher human equation So then define this by associating to pseudo morph the weight one I like and to other this thing I don't like yeah, so that's our modernized space. Yeah For the time being okay, so so now the whole theory will consist of basically taking a partition of unity for the one and Moving now now each point is represented by several which are fractionally liked and moving it into a better position That each of the things look like a manifold So that needs of course some technology to do this Okay, good. I I I felt hope Yeah, Katrin The aim is that is what I had said before it showed up here So they form this into a C type is additional structures from which we can derive the data We need to construct SFT So it turns out that you can you can later define cheetahs you can define differential forms on these categories You can integrate them. So there's a really cool nice theory for this. So Okay, aim clear so Now so there are small structure here So when I have an object in my category, let me define a different object a different category P here Which which consists out of the following it consists of maps from finite sets into periodic orbits And the morphisms are by ejections between this finite set so that that is true So this is actually We are already party in some sense of the data namely if I look at the positive punctures I just take this 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 of puncture and The same here at the negative part now this category might for example If if the different points in this finite set go to the same periodic orbits, you might have different morphisms Yeah Yeah, for example, if if I have two points here and they go to the same orbit then switching the two points Would be a morphism It is the point of all this kind of language just to keep track of isotropy. Is that 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 and 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 up by an evaluation map from the from a manifold of differential form will be differential from the category And when you have orientations, you can actually integrate you don't never have to leave the so once you have a policy So it will later call be a polyphonic structure on the category never have to leave this are all kinds of universal constructions You can do the whole transversality on that level There's nothing Well When I when I Make my definition I assumed that it was non degenerate form Most what I haven't dealt with and I don't want Oh, yeah, so that is was discussed last week so So we have on we have an odd dimension manifold with a stable Hamiltonian structure and that defines the so-called rape vector field And that has periodic orbits Okay, so So now So now so so we have an input and output Yeah, we just look at the restriction to the punctures. We get this map where some which says to which periodic orbit it goes and Then we can build chains like this and it was called weak fiber products and they look like this so So observe here that such a map bi which we introduced before this bi lies below the behead Yeah, so if you forget about the matching of the lines is in fact Is in fact such a morphism so So we look so we look at chains of objects and this morphisms in category B that is called and we takes a fiber product yeah, so We have an object we map it forward Which would give the map which associates to the positive points 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 So we have here Different bills so these are not floors. So we have here different buildings Objects so these are objects in my previous category. I map them down Then there's an into this category here and there is a map mapping it to something which came from the negative evaluation So note that this thing here It looks like a new element in your category S But it isn't because the head is missing here We don't know we don't have information how the lines are being matched. That is not there That actually will lead to the fact that there is a relationship in our previous category and this But we are some category we are some covering Fungta, is that clear? Yeah, so it's here, but I don't understand why you're doing I mean this is where D squared comes from but I mean No, you don't because if you know in the boundary you have But then now you have to relate it to the full category Which it says if you have a building in the boundary then you have to relate it to S And in fact, how can you do this? Well, you'd shop your thing in the boundary You'd chop it apart here, but once you do this. What does that mean? You just took a weak fiber product. You don't you lose the matching condition. Yeah, because In in this case at the 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 Because when you have everything marked and then you glue together You have to forget where you glue together at the order of the periodic orbits and you get you get a covering there So so there's no free lunch here But this thing that I take all the numbers all the numbers away all the orders away Has the advantage that I have a lot of isomol a lot of morphisms and my perturbations have to be compatible with this They also have to be compatible in the other one there But that'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 number here They should have been perturbed in the same way Why not just for the transversality just take everything away, which actually is irrelevant for that Okay Okay, so now That is what we are looking at now. We take fiber products of lengths. So this two elements three elements and so on This week by about product so now s was graded by D the high is a top floor number and now let me Defy look at take multi indices Yeah, this n is greater equals zero Then if I because I want to so this has it has actually two filtration this category So then I can talk about the norm of such a multi index by just taking the Sum of the numbers and I can talk about the length of this Can you avoid this problem by taking this as to be a category where objects, you know As they add the remember these mark points isomorphisms turn the circle and two morphisms Between the isomorphisms, so you want to go to a two I don't know good. I didn't think about it So then given such a multi index si I can obviously build this product here. Yeah, so this are degeneracy and zero okay, and then SP can be decomposed like this and there is Now can you say in words what that means in terms of broken? Yeah, you see pictures. I will show you in a second so So then there's a degeneracy function which you can define on this by just taking the degeneracy of the objects Yeah, and there's a length function. We just look how long is that product here? So here's a picture. Oh Maybe Okay, so 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. This is the bi these are the bi hats So now I give a get a multi index one zero zero two 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 floors in the other top floor two So that is a certain requirement on this here. Namely the know that 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 way work your way up with all this Compatibilities on this thing. So you have to know how you chop and what perturb what and so on Is that clear? When I see a round dot here, I should think one There's dot So the oh so so so this year are This is a building with height seven top floor number six So this is like the first picture you saw except that had less floors So if I just want to try and think how to prove this one equal to zero No, you don't want to prove anything like this. It's a little bit early for this You will you will let me you will see that a little bit Each of law can be can be many many things Yeah, so but it's it's not an easy procedure It just 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 number here So if the norm plus the length minus one is precisely This then you can chop in this way if I if I take if I get another one if I put zero here and one here Then you would have a ground floor here then this year 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. Yeah, so now Let me the notion of a covering functor. So So finite covering functor. So 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 fixed 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's the identity then This should be in bijection to all the morphisms, which you see here So you're writing your compositions in the end with the source on the Right in the target on the left with sauce on the nose sauce is always writing like this So this this year is okay. So this are the more fit. This is a morphism set of this category This is a morphism set of this category Then it has a source map into D and F goes into the objects. So it takes a fiber product here And I do this so so now if you fix if you fix this point and take a pre-image under this covering Then if you look at all morphisms starting here Then you get a morphism to the object starting here as a get bijection to the object starting What are the significance of your different type faces because you've got a talent type faces, which is a category Yeah, so it's okay. So this is maybe I'm too did didactical here So so 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 of a morphism. Yeah, so these are objects It's almost business so forget about the definition actually we already saw such a functor namely something like this And these are all functons of this kind. So take in your category So so we have our category S. We have the degeneracy index and we have the evaluation maps So now a covering structure for this consists of the following and it's clear what it should be for Integers greater equals zero Fij takes an element of degeneracy i plus j plus 1 And it chops it precisely at the right spot to produce a building with top floor what I and top floor J So if I have a building of a certain lengths, I can say I chop it here I chop it here. I chop it here. I chop it here. So and this are actually covering functors You can just take this definition from here to here. So they're precisely satisfied the definition So you see here already that in this formalism you keep very much track of the isomorphisms Which is actually very important for counting later on Sorry, so the degeneracy is just the number of flows The degeneracy is the top floor number using European notation. So it's one less than that what you would say the height of the building It doesn't count in general no no points. No, no So so then it's clear if I have something and I chop it here And then I chop it there you could also chop it first there and chop it there Which is precisely this thing here So you have a comp so chopping if you choose the right indices Social it is commutative. Yeah, so 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 si plus this this product there is Such a chopping so and that is imprecisely this here. So if I take f 1 0 0 2 It is on s 6 and it chops it in this way Yeah, so this are all covering functors Yeah, so you start initials you can check if you just use forget anything. I just algebraic. These are all covering functors Yeah, create it create it from a small list of covering functors. Name is those guys You're writing equals everywhere, but you have a bunch of more persons of categories. No, that no, there's not that that's actually equal That's actually equal No, just you know if you have three Building with three things chop it here Then they get a building of hide one hide two and then chop it here I could have first chopped it here and then there so that's precisely this identity so in terms of actual Data does the f simply Forget one we have Yes, so Yeah, 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 be Yeah, and that is obviously rather commutative I forget here and forget there or the other way around so Okay, so so if you if you look at the J. Trill case of pseudo holomorphic case you see that immediately so This if you have a building where each of the functions you it is J holomorphic Yeah, if I forget the head or not doesn't change that this is things stay holomorphic Obviously, that's true so now Now we come to some formalism which actually quite useful so if if I'm given such a Facebook functor here and Then let me define the boundary of this 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 I can define that thing here? Okay, look a little bit complicated, and you try to understand this sit down It is doesn't say more than precisely that this is true for the seat of forget the J here It's just the compatibility that if I pull back Cita times Cita by Fij I get Cita on the corresponding si J plus one so so this thing here that this is equal Just means compatibility with this things here. So if you have a Cita You can obviously define a Cita on the fiber product You just take Cita of a fiber product as a product of the Citas of the particular values. So I define Cita P and then So so here's a here's a here's a type where it should be a P Sorry, here should be a P Then you pull this back so you you get to the boundary of the category Build the sum here, and you want this condition It just means that Fij composed with Cita P is just Cita restricted to si plus J plus one That is what it means. So there's a short form to So it's so so this quality means high compatibility with this structure So so this is Part of the conditions from which d squared equals zero comes from later Yeah, so it looks complicated But it's a really short form to formulate it rather otherwise have to say for all ij this and this holds to just say that Now this formula occurs in a lot of way you can do a Freton theory satisfying this formula and so So so model this thing that you should be Cita sub P You just build you see this is takes values in zero one rational It just takes the alternating sum and that's the definition and What you clay what you require what you say is compatible is if so this thing is defined on the boundary here because the f i's come from an s an s si where i is at least one So this here defines a new functor on the boundary and you say it's compatible if this is true and this is a non-trivial condition because When you look at this here You take fiber product of lengths to 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 a supposed value there and you work your way up So there's a lot of compatibility So that's that's but it doesn't say more than compatibility this obviously f ij's so now Trivial cylinder buildings When we do a perturbation theory later or for for the algebraic structure when I have a modern life space out of a modern life space of j holomorphic objects and I multiply on each level I put say the same J-holomorphic cylinder in I get a new modern life space and obviously it should be the same up to that Multiplication should be identifiable with the other one I was trying to follow Yeah, no, I haven't explained the picture yet. I haven't even started so when when so when I Do a perturbation theory And for example this building of height one would be pseudo holomorphic Then so so I would have this modernized space of this kind of things then if I multiply it By some J-holomorphic cylinder Yeah, that should also be in my model of space Because otherwise you never get the algebraic structures that you have So if you are if I multiply it by several J-holomorphic cylinders that should also be in the modern life space It might have different signs and so on so if I integrate over that modern life space having this It should be basically like integrating over this modern life space Say up to sign for example Explain a beta formula again Let's suppose I have something with like, you know, three legs So there's various things to design doing to design product affairs of the three levels To design data of the first two levels times a third. So what is the relation between all these? If this is true, it just means that For the two for four things product of two things is equal to CETA on Si plus j plus one for all ij that is what it means It's a it's a convoluted way to say this all at the same time But it's a good formula Yes, so It has something to do with this Suppose I have a function defined here and a function defined here, and I want to extend it over this Suppose F is here and G is here. This is T and this is S How do you just define G of S plus F of T minus G of 0? You assume that they're coincide here But that is a smooth extension of this Yeah, so so if you do this now for higher things you get this alternative Science so so that is precisely you see if you have this CETA p defined There's a lot of points where they have multiple definitions, but the coincide there So if you write this down you get this so that's the underlying idea okay so Now triple cylinder buildings so first of all they are So let me first say what so a trivial cylinder building would be one So if I have hide three here would be say something like this Well top and bottom have the same periodic orbit and each of them is homotopic to a standard Geomorphic cylinder that's a trivial cylinder building and Geomorphic cylinder building would be where each of this is actually Geomorphic One more question is that the thing 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 CETA is zero one and there is in the product So then if I have a product and one in both of them would be one then this is a building above For bed comes from would be one with the number one And then it's not send it then it would be disliked So, okay, so In the perturbation theory if this is a liked object and I add Geomorphic cylinders to it a Geomorphic trivial cylinder building it should be liked as well. So in the perturbation if I like this building so here, let's say Geomorphic cylinders and Now I add a Geomorphic 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 they are different but in some sense they have to count the same way up to sign so up to sign So how can we formalize this? If alpha and alpha prime differ by J. Holographic cylinder building then theta value coincides So here you have to notice that there is not a unique way to put Holographic cylinder to it because you fix an equivalence class here and you have a map on it And you can shift it so there's on each level at least an hour parameter Yeah, so that is indicated by this But I think everybody understands sort of when I say edit edit a holomorphic trivial cylinder to it Yeah, but there's not a unique way. There's a family That is just explained I just said that you add to alpha a Holographic cylinder building a trivial holomorphic cylinder building Travel trivial cylinder building J. Holographic It's just a line if this building has height K. It's a line of K cylinders Which are holomorphic and trivial Yes, and but there's not a unique way to add it to it to get families for each level you get a family And our family so the TV there knows how many levels alpha has and that's how yeah So this is short way to write it. So I added trivial cylinder building to it and Then for the for any representative of this because not unique the value should be the same So I would write times are times the set of What times are No, it's R to the K because yeah, so yeah, so if this If there's building of height one you have just a one parameter family of building of height two for each of the thing 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 district union TCP It's okay, so So Here I added so I would add one trivial cylinder, but even here it's already not unique because here you have an equivalence class of maps So you have to choose a representative take this to this and divide out by the R action and you get an R family Then if I have another level I do the same Okay, so my requirement is that theta should have this property Then if alpha contains a trivial cylinder building, which is not necessarily holomorphic But if theta of alpha is positive then the trivial cylinder component are all J holomorphic So on that level it's a little bit more complicated to formulate then later on a linearized level where you can actually write some down some Functors, and then that's true. So, okay, so we asked us again So when you write to see be really you mean to just put one chain of traversal Right. Oh, 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 like to 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 No, okay Okay, so give me a few minutes here and then I stop so now So if you have two non-trivial components each each building level has a non-trivial component, which is not trivial cylinder like Then you can you can move them against each other creating again a family on each level Yeah, and our family if you have several components that can move them against each other and What you want is if you have to 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 seat of alpha should be the product Yeah, so you have to think of this seetha in some sense would say fine pseudo holomorphic things And if it's pseudo homomorphic and it's disconnected here on each level then each thing individually already existed I was perturbed and so on okay, so summary, okay, so summer okay, so Then I'm stopping so we are interested in deformation seetha of seetha twiddle. This is the original Identificate Functor which identified J holomorphic objects. We want to deform it to a seetha having the following properties. This is true Then this is true if We add just a twiddle holomorphic cylinder building If this is positive then this has to be true then this thing actually had to be pseudo homomorphic and If I've stable things and me moves against the other this has to be true So SFT algebraically comes from this. That's all there is This is more or less the same is saying the whole theta is determined by the value of theta on one One floor thing with the right connected component that the true statement Yeah, that's a good one floor. Okay, so Let me not buy into this here, so So I have to look at I have to look at the algorithm so so the algorithm is a little bit I would give you the algorithm how to put up. Yeah, so it so the algorithm with precisely Yes, so in the first the first time to perturb you have Disconnected objects. So in the hierarchy concerning some complexity or energy on the bottom floor there things cannot degenerate further They are disconnected things This 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 helmet? Yeah, so I'd expect if 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 shooter holomorphic case with which you start in which you just decide one the holomorphic things and the other would be the Thing you need to achieve by the definition. Is it easy to? so So if you if you take this property here, you see immediately that they are satisfied by the sort of more precisely so Sorry, but of course we don't want to take the trivial information. So you want the information which happens by these Okay, so that's a good so so my my As we were just a minor four lectures, of course, it's not so clear if I get to the very end of it, but At this point what we have done is we I have sort of in this category setting Said what things should do, okay? So now the next thing is there's a notion of a smooth structure on the category, but you can only do this for When you have this polyfoil theory because other things you cannot do smooth Then you can start talking about smooth objects of this kind and smooth objects of this kind So these are sort of smooth sub Categories and what are they? So they actually occur in different flavors. There are many full type Or be full type branch or be full type and so on and This things then will be constructed by a fritton theory. So so you can Transpose this 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 then You define a bundle over this you have to push a 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 is are the zero one forms along the underlying map so here you have to push a Riemann functor and Then you define Something like this Facebook thing here. It's it's but it's a section It's delifts on the fibers and it's associates to each fiber to a certain number of points a rational number adding up to one and Then this here is your theta if this is properly chosen if you put here Theta zero which just is the zeros which just gives a weight one to the zero section into everything else zero Then this here is actually our Ceta J. Triddle So if this year's Ceta zero which just puts weight one on the zero vector and weight zero on the other then this gives you This comes with this gives you that one which you already saw So here is a small perturbation of this and since in the fiber everything adds up to one What you literally do is you take a partition of unity for your your Ceta J. Triddle for each object And then you move the objects now you have several one with fraction weight in different directions and they line up to manifold pieces locally