 So as I was told it's a discussion, which I guess starts with you asking a question So good loser, what about you? Loosen up the crowd here. Yes, so Are there so okay, so I mean so you saw the whole thing and you I think it presumably triggers a certain number of questions You can ask does this project have a future or something. I mean what what you think Directions one should develop or what do you think one can do with this? I mean, you know there or could you explain a certain specific aspect, so I Mean or Can you explain this? Even is a even bigger context in the background So I only talked actually about about one sort of So what you heard about that that should explain he's great Yes No, no, no, no, no, no, no, so Life sometimes tries to be kind but I think there's generally no free rights. So if the complexity goes up From the rise and so so so what what this so what what we have seen so far is That it explains how to perturbed the Kusher even operator in the case R cross V which I was let's say referred to as a homogeneous case to some modernized space which is Not a manifold doesn't stand for manifold on all before which can happen in certain situations But to something which is a little bit wilder. Yes, it's represented by smooth many falls with weights which globally sort of fit together and that allows you to do Integration theory over it and it allows you triangulations So that that is what you can do on this level So then so so then you want to produce numbers out of this then you have to deal with orientation questions So so you can't take the shortcut that I just look at zero dimensional spaces in a count and plus minus one Because you destroyed that structure already by using this weighted branched submit faults You're dealing with rational numbers points of having rational weights. So there's no way to avoid orientation questions at that point so so then you start happily orienting and then You see that you have actually a lot of morphisms there because I Did not so I did not Ma I did not order the marked points I did not order the positive punctures and I neither did I order the negative punctures when you look at the SFT paper This is given. We gave this additional structure So then you also heard about bait bad orbits. That's also an issue. So So it turns out that You can go to a covering. So basically you take that solution space which you have and you go to a covering where you take each object, but with different numberings of the mark of the mark points and with different numbering top and bottom and also introducing that what you have at the intermediate phases namely there we had this Matching asymptotic markers up to rotation. So you put on top and bottom additional markers and then you put Randomly on the set of periodic orbits on each periodic orbit you put a point So the mark are then on top should be if you take the associated Holographic polar coordinates if you run along r cross zero r cross s1 and zero is sort of the one element It's a zero element Then you you hit that mark point then if it's multiple covered k times you have k possibilities to do so top and bottom so then So then at this point you can take A bourgeois moon and they had they had some ideas about orienting this now when they wrote this the modelized space Technically speaking was just a letter because nobody really knew what it was. So there So so they are orient this letter M But but when you look in the proof so they say some of the right things so you can take that you can turn that into a proof I think that has to preserve No, no, no, so so now it depends now on the so for the grading or so if you make the right Gratings with the corner if you take the right definition of cornered center index then And the right definition should have the property that if you if you go in the context structure at the periodic orbit and takes a return map and Linearize it so then you can look at Of course the linearized thing minus the identity and that is either Plus minus this is positive. It's non degenerate. So so I take the I take the linearized return of in the contact manifold and then You can look at this and and then you say It's an even or odd depending on the sign and the normalization of the current center index should be also so that the module reduction is Precisely that but it's one of the things Yeah, so we can talk about even an odd current center indices and an even content center index Corresponds actually to precisely the things which have a plus sign so then then if you interchange The now so you suppose you have a numbering and that is a model our space is oriented and then you change the number of two punctures If this both belong to odd content center and this is orbit the sign Will have to change of the orientation So here you want to know a little bit about how one oriented stuff? Okay So So the starting point is you want of course to mumble as much as possible Yeah, and leave So so you want to have a minimalistic approach to also So you take you take a periodic orbit. It's non degenerate Then J and all we have all this data fixed Then we know that there is an asymptotic self-adjoint operator sitting over it So you you so you trivialize this Some way, but you can think of doing the following things. So just Put abstractly a cap there With the kosher even type operator, which has precisely this asymptotics, which I have So you choose an orientation of this thing so you do this for all periodic orbits You know, so this are on half So on on disc with cylindrical end and having this operator Okay, so so this is a choice you make yeah, so here me so you have there's a You have a freedom operator there Yeah, you linearize so one can say a little bit about that and you have a determinant bundling and choose an orientation Okay, so then Then of course I can do the opposite. I put top caps here So so let me let me write like this so they are so they are oriented because I made choices So these are all the choices I make where right oriented So now I put this caps on there of course now you can do a gluing construction and Get a compact surface on which you can actually destroy everything you have you have a kosher iman type operator by the gluing Now kosher iman type operators can always be homotopped to complex linear operators and That is a canonical orientation. They have canonical orientation So so this glute thing has a canonical orientation, which is the complex orientation So now you only have to make one Sure, you know, you have to make some conventions. So you just say so let's call this lower cap. Yeah, so L Let's let me write L plus because positive function L minus and now you have only to think about in which order You want to write them? So let's say I write this first and I take D debt of L plus tensor debt of L minus goes into debt of glute But this has a complex orientation and this is oriented This induces an orientation here So so in the literature I will see this is naturally isomorphic Of course, it would be also naturally as a morphic if it were interchange the order But then you might get different science later on so so so one important thing is whenever you see an orientation See it's naturally and they don't give you the convention actually Alexi Singer studied the the higher order theory of orientations. So so you can have many natural Conventions so it's good to know which one you take. So so I let me say I write this first and then You don't actually have discs in your mouth. No, but it's abstractly. I just have to know I have to just trivialize this here And then built an abstract operator ending to build an abstract operator. Yes, and so you do that if you choose that below and above and then And they and they have sort of there's a trivialization so they have together with some identification here which you fix Okay, and now you can orient everything so so now here is For the energy surface Well, so I put my so let's make this red here So now I glue my abstractly red stuff in here and then I glue my abstractly oriented Other stuff in here And now I have to just so but now Now you see Suppose my punctures were numbered. Now you see What should be my convention because I have to relate it with the determinant of this determinant of this Determine of this determinant of this this this this and this and the whole thing has a complex orientation Huh, so in principle you would say this all should do the same then here But I glue this in yeah, so so you glue this stuff in then it's just complex surf is a surface It's kushariman you take it's complex orientation, but how do you do this? So so you have to come up with a convention In which order to write my determinants first the lower ones in which order the top So so the easiest way of course is if actually the punctures are ordered if this is puncture one two three four and this is puncture Positive puncture here say one two three four, let's say and Then you make a convention Namely so and this is why I have to go to the covering So this would be my modelized space and it's a covering there are numbers So now I make just a convention that I first write down the lower ones So so I write first down which has numbered one So Determinant of L one Tensor the term plus two plus Determinant four plus times determinant of L Okay, so okay, so let's see This is so just have to get the So I have to put the surface somewhere So I Okay, this then I could write here Determinant of let me call this w. That's that operator You see so I would and then you could write down the determinant of L1 minus that would be for example this one Then then this one and so on and then if you glue this this goes to the determinant of the glute and This has a natural orientation this have orientations this have orientations this orient that But you see I could also say I put all this garbage here So then you get different sizes. So so here It's actually so if you write this here and then you have a building but there's a gluing Then you then this guys here are matching sort of with those guys and you just have to figure out in which order they eat each other And that is what the science come from Also, so that's the orientation business. So you just have to know to figure out how to Science change Okay, so now This only works nicely if there's no bad orbit. So you have to Otherwise So There are okay, so so let's say the basics of orientation, but there are there are some issues with With the asymptotic operators. So Yes, okay, so how was how did that go? If you have Okay, so when I do this trivialization in particular I have to say where I start trivializing So so the marked points also play around the asymptotic markers and if I have If I have an asymptotic operator Which is a which is associated to bad orbit. So bad orbit is is an even iterate of Simple periodic orbit which has an odd number of eigenvalues where the linearized map has an odd number of eigenvalues Between minus one and zero so then if you If you take a trivialization for for the next marker and do this whole thing you actually figure out you get You get a different sign So when you do this But it turns out that if you if you use any other orbit you can you can trivialize starting from any marker You have and you always get a consistent orientation so Yeah, so So the result of this is but but the basic features is but when you carry it out There are certain ambiguities to encounter, but they only kill you in the case of bad or even iterates of bad orbits Okay, so once once we have this We can do the following We can actually orient everything which on top only has good or has no bad orbits on top and no bad orbits on button so all the modern spaces They are oriented even the broken ones and even if the If so you can do it first for buildings of height one and then This orientation actually can be moved to the boundary But in the boundary what will happen is that you might have bad orbits occurring in the boundary But but the whole is but then this is a broken thing But the whole thing itself the modern space is oriented Consistently with the interior Okay, so that's of course not good for algebra. Yeah So so we have everything oriented then the broken thing itself is oriented But in order to have a homology theory, we should have compared with orientations of the pieces You're like if you do more theory then each Grade in line between most things difference one should have an orientation Then the modernized space in between has an orientation And then you figure out what the science are right and left and this things break So here they can break where the individual things don't have orientations however The bad what was excluded were even iterates of bad orbits So assume we have a bad orbit and the bad is the second iterate of a bad orbit so Then it breaks apart there then you couldn't then you just move the real so it's in particular double covered then you move the one of the markers by 180 degrees and You get a new object. It's a new object But now the boundary orientation changes and you can glue this together like this and it continues So it's like billiard. So it's something like this So you you have ends in this picture There's something runs out and this would be connected with bad orbits and then somewhere else it continues And there's they always appear in pairs they cancel out So if you disregard them consistently in the theory it does not destroy d squared equals zero Because just that it gets stuck on the boundary doesn't mean it's actually stuck It's just like a tunneling to a different place where it continues So that is with a bad orbit so so they are actually not boundary points They just get stuck on the boundary But you could think of that there's something else from the other side glued that continues Your strength construction of SFT is How much is actually written down not only for contact You know for courtesans, but what about you know surgery formula? Okay, okay, so So I answer this question and Joel keeps track of this so even Can make additional remarks, okay, so So what I did what I described today was Making generic perturbations the situation which would allow me to define a boundary operator along this lines So then of course if I made a different perturbation I get another boundary operator And he would be interested if for example the homologies are the same So we would actually have to study a homotopy from one to the other Which keeps the structure so so that was actually one of my possible endings for this course But I'm usually over optimistic so it's didn't happen. So this So so this has been worked out then of course what also has then to be done is You want to know that on saying homology it's independent of the homotopy So you have actually to consider two different homotopies and have to study a homotopy between homotopies But I think the the amount of difficulty So the amount of difficulty for the transversality today was actually surprisingly little yeah So I mean it was basically I mean you have to check certain things, but it was rather short the the transversality for homotopping between Two such perturbations is a little bit more involved, but but it's also There's a certain number of things one has to point out You cannot achieve general position and so on you create some singularity set So there's a certain one of discussion one has to go through then I think Homotopy of homotopy is quite similar then the next thing is you want to have cobalt disons and the cobalt disons Would be also a category like this where you have positive and negative ends and then two of such categories are operating from the right And the left by concatenations but there the analysis so the analysis completely the same then in that case and Then the transversality I think is is less than what we have seen That's what we what we could have seen if I would have been able to finish the lecture So it would be more on the level of homotopy between two such perturbations So so it gets a little bit more complicated than today, but not much so So that that is this picture then you do this orientation business and then you do so I think you can basically do whatever we promised in in the SFT paper differential equation You can do means there is someone who is writing it and yeah, yeah like me Yeah, yeah, okay, so are you you're more interested in time? Yeah, okay, okay good. Yeah, I'm sure I mean given the experience. I might be dead at the time and it should be coming out but okay, so Okay, so here here's the current state so So there will be two research papers one constructs the poly faults for for all the categories occurring it also has a list actually useful number of Techniques in it to build such spaces for example One of one of the criticisms one can have about the analysis with the with the usual gluing constructions is That you have to do a gluing construction at once. There's no local version of this you have to set up this thing then you do the gluing and We know that this thing is very long. So you do five-dimensional reductions. It's not natural So and then you have to do there's a lot of things you have to do and It's usually so long that and also tries to avoid actually writing it down unless a referee forces You know so then then then there are different versions of shortcuts by saying a slight modification of any pointed in other paper where then you look at it's actually not there and he also points to something And so so that's so that is Basically in the nature of things so what what we are coming up with is something like this You want to introduce for example the notion of a pre-freton theory? that means if I would know that I have say so if I would know for some reason because somebody at some point proved this That if I have a kosher email operator on something without Actually having boundary condition and it satisfies certain things It's cannot be fret on because I'm a boundary conditions then if and but but that I can put a stamp on it It's pre-freton Then if I have some other situations and this is pre-freton and I can sort of identify these things I could say this is pre-freton Then if I could take this stuff and glue it on to this one and there's no free-end anymore I could conclude its fret on on the non-linear level So the pictures then is actually drawing so if you study things like this then you could say There are sort of regions of identification and you could represent this by a graph like this so This thing like a graph like this and the other like a graph like this where you can identify this here So you could draw this picture and there's no friend. It's it's fret on and if you have so you could think of Pictures like this where each of this is some piece of non-linear analysis of of freedom analysis for example this gluing Gluing near node. Yeah, it's a freedom operator nearby So if I understand this and I understand the tube, you know, which is obviously quite trivial Then if I take a collar here and I can identify this and I can identify this then I have already a theory for a Sphere with sort of a note Yeah, so so then once you have this you actually You can build a library of pre fret on see You can build a library where if person proves a new piece of analysis And it is it is built up to certain specifications. You can plug that into the machinery and it works So you can start recycling things so for example We start all this and then somebody does this here this There's a piece of arc and on this you have coacherima and this you have the gradient flow and proves its pre fret home Then you can plug that into the game Yeah, if you saw that certain constructions like PSS morphism or something like this Yeah But not on a satisfactory level so so on the level of the operators I could but it takes a little bit of a while But so still complaining about what should be there should be it's pretty certain. There's an abstract theory So basically pre fret on is something that you can glue something to it that it's fret on Plus plus some small print so that This is so suppose you have operate on this and you say it's pre fret on because I can glue some because for example I can put Lagrangian boundary conditions on that boundary and Then suppose We have something here and this also for a time because I can put Lagrangian boundary conditions on it and then Currently we are building a switch and the switch is something like this so this here so so we sort of Suppose this area looks pretty much like this area Then we then then you want to switch from gluing this part together with that part and this Part together with that part and that would be a product of two freedom operators And you would have proved that if this is freedom is Lagrangian this then this here to glue together Together with this with this one with the Lagrangian boundary condition here and here the product is for a time So therefore that is for a time So this is actually known on the on the linear level so the excision formula and at your signal index and such But I think there is there's a nonlinear version of this But we're getting carried away now Okay, so and so that would be one thing for the photon theory and the other is Actually constructing spaces out of small things so when I construct poly faults So I can construct a poly fault for that piece a poly fault for this piece And then that the things overlap in this harmless piece is a fiber product condition So there's also a theory putting this things together So at the end of so at the end of the day there's I mean now There's still some work to be done But if you so you look at the problem you have to develop sort of an idea of what are some of the moving parts and then you can Put this into small pieces build fiber products and you get you get the uniformizers Then you look at the kosher iman you chop it into pieces if then each identifiable piece is pre-fret harm And is in your library then that thing is for a time And then again the big picture then the expectation is once once is allowed there that people will start Making versions of like certain paper and things like that using that language right because I think one of the advantages of the theory is actually That it that it can be chopped into smaller pieces. I think that that is actually a useful feature and And so then then the pieces of our not since you only you know in principle Then if you have such a situation and you and you just cover it by open sets and for each open set You could say the restrictions prefer time and then if it's all covered then that the thing has to be fret on if it would go in this direction Then then yeah, then on small patches I mean you may always assume the images are to end so there's some usually some estimate For something on a small disc or near a node Into some aren't and you have to prove something about this and then it gets then if this is proof Then it gets a stamp pre fret on which is a list a catalog of properties And if you have another one on this you can start gluing these things together So you'll be able to use this machinery in the more less Exumatic way without actually having to understand all the technicalities, or will you actually have to All the technicalities to use it. Well, I think the technicalities on this microscopic level usually rather similar I mean like gluing at nodes gluing at periodic orbits So I think it's good in in life to see at once what enters. Yeah, but then I think Things are quite similar, but I think on the level of consumer of such a technology You know, I would definitely look a little bit into the machinery I mean like at a car I would open the hood and see at least a little bit what is in there I mean, maybe I don't want to know too much about the electronics, but you know, I want to see want to see some parts and then so so that then my So for example, we wrote this Gorma of Witten's thing So it's considered which is sort of written not in this language. It is written Browser down to us. It was a little bit of this polyphonic technology It has a certain links, but it definitely addresses all the issues So you see what the problems are how you address it and then you get a solution So so it's sort of good to at least see it once so so the SFT paper now Which is sort of the next thing which comes out just doesn't take any prisoners. Yeah, we just use that technology Construct the things so there are two papers. This is what I just said about construction of the spaces And then then you start precisely where my lecture starts. I have these categories There are smooth structures on this and I just talk about the The algorithms for the perturbations and how orientations and and do the and show what what kind of invariance you can do with this Yeah, so just then deliver SFT but what but it's also interesting to note that From this level of generality you see that there should be other things you should be able to do because on this category For example, I have SC differential forms Example of this are the pullbacks by evaluation once I have this mark points numbered I can pull back so I have to take a covering of the category here by adding this numberings and so on which is also as a Smooth structure, so then you start pulling back differential forms from The contact manifold or you you have a map into the Lin-Mumford space by forgetting the map then throwing away the unstable domain components and you can pull back differential forms From the delin-Mumford stack of Stable human surfaces there will be examples of differential forms on the category and Then using those you construct this most general form of SFT, but you could also think well, why do I not just Take the diram complex on the category and integrate Now this forms will have less symmetry properties because when I pull back at mark points The other forms are now sort of when I remember the mark points and pull this back in different order I just have a sign change which I can predict you have an I integrate this forms but But this you will not have if you do other things But it could very well be that this differential forms that there are plenty of them Which just discover something of the topology of the spaces of the underlying orbit spaces and That are more than this this forms used so far You know, so I think it is a good viewpoint that one should separate out the analysis You do this then you have smooth structure on the category and then you do just a lot of General procedures on that level that produces you some numbers at the end and then some Some numerical object algebraic objects and then you start thinking about how can I represent this data in some algebraic fashion? And from that perspective contact to homology and cylindrical and so are Possibilities you have in particular circumstances to do but but of course when we wrote this papers we will be sort of Brainwashed by life. Yeah, so so like there was Fleur theory and then when once Fleur theory was there Everybody who saw a problem that there should be Fleur theory has immediately the immediate right instinct what to do precisely More or less this idea and take it But I think one should one should view it from today's perspective since you can separate all out There might be a lot of other algebraic things you can do so So then you can think about other structures on the category. So I have this Evaluate this evaluation maps plus minus you could have maybe categories which have more things coming out So you can glue different categories in different ways together. I know just a plus and a minus Just combinatorial configurations of categories to do certain things. I'm pretty sure they're of course Anyway, so I'm sure their application for this kind of things I haven't checked all the series but So, yeah So I think I mean I think I could There's no analytic obstruction for not doing studying Lagrangian boundary conditions and put that into a frame and Sort of exhibiting this kind of structure to King different ends can glue at all different places and so on So I think there are a lot of different structures on the level of categories You can think of and there might be realizations as this kind of model I model our problems Also, I think a lot of mileage and symplectic geometry you get by actually artificially creating new type of operators and things just to To prove some geometric fact by counting some weird modelized space So so if you see sort of that that you can put the things in smaller pieces So it becomes more like a puzzle that you put things together and so on So so at least the language gives us a freedom. I don't I mean I haven't thought too much about it except carrying out this SFT So But but but I say the optimist I am so I would think that that it should be some kind of like domino pieces You can put them together and blah blah blah and then you build certain things out of this And then you get some numbers you represent something then you sort of look it represents some geometric fact And and the constructions will be much faster than if I have such complicated thing I have going through all kinds of other things in particular. I think when you have this prefer to on theory So, yeah, I think that sorry just to finish answering your question as you asked a few I think so I thought my understanding is that if it's in sort of SFT propaganda papers then, you know Helmet and co-authors will make sure that it's backed up with all the appropriate theory So that will be done if you want additional theories to come out of HWC or a door or helmet and company Then then You don't expect that so if you want a more spot theory I wouldn't expect to see that to come out of helmet company, right? I do other stuff I'm sure you can do it, but you have to do a little piece, but also here So let me just finish. So Right, so so they I think helmet really wants to make everything in that prop again paper absolutely rigorous And then sort of move on to other more interesting projects But part of doing that means, you know, so in SFT you have so one unfortunate bit is they have at least two different types of Degenerations that can happen no little sort of degenerations that can happen to grandma with me That's sort of breaking like you might see in floor And so when it comes down time to build charts, you don't want to have to keep track of these things You know very separately and rewrite new charts from the beginning So part of the abstract package is to localize that analysis so that once you have two little two bits of pieces You can fit them together very rapidly and then sort of extend the theory on to sort of other things And and my understanding is that the hope is that this will be done to sort of such a level that if you sort of say Ah, I have this new problem It hasn't been solved before that there's a small bit of sort of standard analysis that will have to be done sort of Essentially understanding breaking or noting a new phenomena that hasn't been studied. This is sort of step one So you do a bit of new analysis It's not so difficult and then you have to sort of then after you've done that The polyphol for your problem should effectively be done your site appropriate results. It's just boom suddenly You've got this big ambient space where you can return But then I think there is sort of hard work to be done Relatively hard work to be done which is to say, okay You know now I want to see an algebra or something to pop out of this And so if I and getting that algebra to sort of fall out of your for your preferred modular spaces Like the way that this will occur is by making sure that you can choose your Protervations in such a way that everything is compatible and this takes a bit of work not is not an analysis work It's like the sort of work in the past several lectures, right? It's the sort of thing that you can you can work on and think about without having to work and worry about the analysis, right? Directly and that I think is accessible to you know a fairly large range of people once you at least seen it done You know in one instance and then you might have to be creative to set it up in a very other in a very different framework, right? but Not so much analysis, right? Yeah, I think it's a clean way to think about it If you don't like then like the methods and you come up with your own methods You definitely if you're not so so the thing is it's like Domino you put things together and Behind there should be a see-way. Now if you don't believe this at least you can write it down and see what the results should be And then you prove it by a different method But the one but the question is it really Then you have to make a decision do I want to use a different method or just learn the language here That's presumably the easier way to do it But sorry, so sorry if I know what what algebraic structure should be getting out of whatever construction I'm doing Should this perturbation? Yes, you will need to find that you'll need to find the right sort of you have the sort of the general mindful sort of framework But I mean as you saw you know helmet for instance in the SFT stuff You have to deal with these sort of covering funtions, right? Yeah, these covering funtions because sometimes you want to keep track of Aseptic markers and so forth and there inevitably is going to be something sort of similar in whatever approach you want to deal with Right, so you need to find that you know the right sort of the right sort of poly fold to build But you'll end up building it out of these parts that have already been better sort of come for free And then once you've made this big construction Then you have to sort of construct your yeah you had to construct the perturbations to make sure that you get the algebra out that you want right and that's that's sort of on you to manage but that's But but Angel vz should guarantee it should effectively guarantee sort of the transfer salary part for you But you know you're still gonna have to construct these perturbations so they satisfy all the appropriate compatibility conditions which guarantee your algebra I mean you have all this well you have this inter Well, I mean here so when you look at the algorithm, so there are a lot of Things which if I perturb here like this then this piece of perturbation appears all up everywhere because whenever I go to a new phase It appears as a product and then later. It's being reused and so on so so that might in in some problems lead to Generosity issues, you know, I mean here it doesn't yeah for the D off for the boundary operator But you could imagine that that there are certain things that you have so many conditions that you run into difficulties Doing so so then it could be that at the end then you who have to allow for certain mistakes Yeah from you have to allow for more general perturbations and then you get some kind of higher structure Which keep track of the lack of being actually able to be keeping all the structures being transverse. Yeah, you get say this higher level structures where The mistake you have to make to get a theory on the level at least so much transfer salary But you need to do something might still need to break the symmetry But then there are different ways to break the symmetry and you keep track of the different ways to break the symmetry And then you go to the next level and that is an algebraic object, you know Yeah, so I think you you know when you when you want to intersect the diagonal or what you just breaks a symmetry also I think I mean like like the Ram see you the Ram. Comology is on the level of of the code chains Is a product where a singular isn't? They have to perturb and keep track of it So so so I don't know here in this way, but but I could imagine I mean if I just put so much structures on this I cannot just achieve standard regularities. So so I have to break To break I have to give up some of the structures But but I think in general then there's an there's there could be in the background an algebraic theory of in which way You can give up this structure and then you keep track of this and you build your way up and you have a new mathematical object Yeah, so so in most bot then so so here in this case Of course one part is the gluing at periodic orbit So I have to understand that picture and the analysis and glue this in so if it's the most bot Well then in the neighborhood, so if this is a most bot manifold you would at least describe some analysis here So it depends now how you want to deal with this. Yeah, so some people like to put a mox function on this Yeah, but you could also I Guess have this ends running around and build homology classes by evaluation at the ends and they have to intersect with the other Something like this. So so it depends how you want to approach this But but the end result is personally building So so this here becomes a little black box which looks like this And then if I have something else which I worked on I glue this together Something here and build this isn't the other thing So so you have to build a black box here, which does sort of the right thing inside and then glue it into the other Things will be explained on examples how to do this black box business So I'm still a bit unclear about the role of SC calculus in all this Okay, so okay, there's the reparameterization action problem and the retracts. Yeah, so Okay, so So the okay, so So when we decided to deal with a setup on which we want to Fritum set up on which we want to work Then our decision was that on the domains and everywhere. We want to have local constructions very local, I mean you could imagine if you have a space of Maps from surfaces which also Can be glued you could think of putting a structure on there which is being Obtained by solving some PDEs and so on which it would not be of local nature So What we want to have is that that There's a locality to this if I know the local properties around the nodes everywhere. I can recover from this structure of the space That is what it makes it possible that you can chop it into pieces It's completely. These are just always point-wise things Then near near the nodes because of course glue and we have to deal with a problem that we get different domains So so then when you when you want to have this Function spaces or this topological space which has this very local property then Then there seems to be no banner space set up for modeling it Even if I would only have interior nodes and no breaking This is a space of maps which are defined on a note And you want to describe the neighborhood they glue the cylinders As far as I I can see cannot be modeled on an on an open-setting about a space So so therefore So many so so then the in our so by exploring this and there was the idea of Putting this entire gluing in and then you you got some kind of a messy subspace on which on which you could Define a chart now, but if it then later turned out to be the retract and The retract then turned out for the sd smooth structure to be a smooth map for the new smooth for the sd calculus So we had actually a smooth model for the spaces, but it was actually retract So the retract is also not smooth in the classical sense No, no, absolutely not if the if there would be a smooth snap onto this retract This retract would actually be a bannach sub manifold, but it has actually varying dimensions So that is so so in the classical theory that's actually our carton's last serum in the Comte-Rondue So it was definitely his last mathematically published work Which however, I only learned later from by Etangis so so he proved that if you have if you have In a bannach space a map R which is smooth from the open set into itself and ask are composed with ours equal R that the image is a bannach space is a bannach sub manifold so that is what he proved and What we had figured out is also we had figured this also out because we realized first There was a retraction and it was differentiable for for this sd calculus, but then of course a natural question What is what is it actually if it's classically smooth and we also had also figured out that it has to be a bannach sub manifold but But in for the sc retractions in general it is not the case so It is a wilder space, but then and then of course once you know this that if you have smooth retractions For any notion of differentiability and you have the chain rule so that you can do this Then you can build a theory based on this you can take the images of the retraction as a local model So there might be actually even different forms of differentiability. It is what you would like to try this Yeah, so so then it then it so once you had so so this concept was born out looking at periodic Or by looking at periodic orbits at notes and so on and it turned out to be rather useful and I think I made that remark before that That maybe even classical differential geometry should be made this retracts because a lot of stuff is Simple so if I so for example when I have So nobody would really like to define this as a sub manifold Classically because the positions is so if you if you take a differential geometry book They definitely want to avoid to say that there's a sub manifold But this yellow subset is a local retract around each point. You have a retraction So so the better definition would be to say a sub manifold is one which is a is the image of a classically smooth retraction and then This is all fine. It has a tangent space It's it's a manifold structure the only thing what you can say in general in higher dimensions You cannot say that the boundary is a sub manifold because it might be rather wild However, if you know that on the boundary the tangent space is a nice position to the whole thing Then it has a nice structure. So you can say if it for example cuts it like this or the higher dimensional versions Then you know the induced thing is a manifold with boundary and corners So retracts are actually useful So if you build it on this and and then if you want to build many fault of maps that goes really fast so So suppose So I have the bonus is not a homology group. That's a bonus space here a space of Maps into some Rn and and the only thing I have to know is If I have a smooth map from Rn into RM It this is a fact here. Then of course you get a smooth map by composition into this So that's a classical theory so so now Support so I claim so and this is fun tutorial Yeah, so to every index and you have a Hilbert space and for smooth map from Rn into RM You get an induced smooth operator between this Hilbert space This thing has a canonical extension to many faults. So what do you do? Yes, oh or some say women in surface So so now suppose I have a manifold You can embed the manifold into a sufficiently large Rn So suppose a compact manifold if not you have to properly embed it, but let's say compact manifold Here's my Rn Here's a manifold Take a two-way neighborhood here Whitney to your neighborhood and then you there's a retraction from this neighborhood into itself There's the image as a manifold So now this one by restricting it a little bit smaller you can extend to a map So by cutting it near the bow and So by restricting it to a slightly smaller neighborhood you may assume this is a restriction from a map from Rn into Rn So so now you Take the open subset of this thing here where the image lies in that small neighborhood So so takes a subset So take all use in H3 of s Rn where the image of s lies in that neighborhood This is an open subset of the space and define our triddle by Triddle of u is r composed with u But since this is a restriction of a smooth map that is a smooth map and it has a property r squared equals zero And what are the what are in the images are all maps which go into M So therefore there's at least a polyfoam an M polyfoam so So now take a different embedding into some other Rn then On the two embeddings you have a diffimorphism and you extend it in both direction to a smooth map So you see that the two things are actually a seed if you morphic So therefore the the abstract space of all maps into M Which together with an embedding R in R3 has a well-defined M polyfoam structure and then you think a little bit and you see it's actually a C many fall structure that rather than taking retracts you can take a C spaces as a model so No, no actually in this case it's in classically smooth on each level So so so in this case actually on H3 so so here I don't even have a scale here This is classically smooth. So this is a smooth Hilbert space. All right. It's actually it's you see It's actually a classical It's a polyfoam structure for classically retraction and by cartoon theorem It is actually a Hilbert sub manifold So so one thing I said if it might depend on the embedding if you take a different embedding then of course The two embedded things by this map are diffimorphic then for each of them you take it take a smooth extension Just as a smooth map right and left No, no, no, no, no, I didn't say it was no it's not so so here. This is just classical smoothness so forget Yeah, so yeah, so so so what this proves is that the that the space of There's a subset of maps from S into our end of class H3 which has the image in M is a classical retract for the classical differential of structure and therefore Hilbert space manifold and This definition doesn't depend on the embedding you just look at the maps into M which together with an embedding have this property And if you take a different embedding They are there the structures are diffimorphic. So it's a canonical instruction Yeah, so so in the in the literature usually people write more about it So that's I think the shortest proof of actual putting structure on this Yeah, so but anyway, so retractions is good stuff Three minutes to closing Last round, thank you It's on the scientific committee and organizers, so I think at the end we should give really Big applause to the RCS who made this possible and all the help we achieved also to our organizers We are only see so Joe this oh Dan and Joe and tools to Joe Joe who did a lot of work as a chair of the committee and Had to go to Paris