 Yes, and thanks for all the fabulous questions. I do need to get to the point landing though So Helmut can take off next week So I would ask that if you have questions that are technical technical or academic academic you ask me after the talk so which means What you should ask me about is how in the world do I apply this to my favorite j-curve space, you know That's what we're really here for right so The why don't you just or is this algebraic geometry questions are good for coffee Good all right, so here's a theorem and today the goal is to actually explain every single word so We want to regularize a modular space which we think of as being cut out as The compact zero set of some section We know by now what an M poly fold is We know essentially what scale smooth is I'm gonna need to tell you what Fred home as we know what a section is Right so the reason we want to model everything on scale hybrid spaces is that we want cut off functions Right exactly Right, but we don't want to worry about that right now So then Really the key is that something is non-empty And what is that so it's a space of other sections that are sc plus in some sense Which I'm gonna need to so if you're seeing sc plus what you should think is compact perturbation that are transverse General position to boundary if there is boundary And I'm gonna be allowing myself to fix a neighborhood of the zero set and the norm And I'm allowing myself to force the section the perturbation to only happen in that neighborhood And to be bounded by that norm since I can scale I'm gonna be able to scale that norm down So I can have a one here So this pretty much means I can make my perturbations as small as I want and that is the reason that I think you Now with these two Riders, I think you don't need any kind of generosity or commiga. You know this gets you to Make your perturbed zero set as close to the unperturbed as you want There is also the Obamacare writer If you like your solutions, you can keep your solutions So What I need to explain at some point is also what an auxiliary norm is what Fred Holm is What else sc plus this is right, so what do I mean by controlling compactness so Right if you just add a perturbation in this infinite-dimensional setting, it's totally not clear that the perturbed zero set Right, which we're going to want here. I Mean it's going to be smooth if it's cut out transversely, but why is it compact? And so we're just going to build this into the definition And then somebody has to you know prove that there are norms and neighborhoods that control compactness so controlling compactness means that if I have any SC plus section that satisfies these two things not necessarily transversality than this set However, a regular it is is compact so Check what's what? Why prime? Oh, right. Why good. Yes. Why one is sort of the quality one fibers And I should say that again at some point that's gonna happen So right so now the implicit function theorem that Joel already told us about gives us Yeah, cuts out nice smooth manifolds and Since I'm asking for general position actually, I know exactly what my boundary in Kona strata are That's important if you sort of do Fleur theory or something Right, you do a one-dimensional moduli space You regularize it and then you're going to get you know that the sum over the Boundary terms is zero because it's a one manifold now You'd like to know that those are actually the once broken trajectories and not the triply broken trajectories or something right, so the actual boundary would be degeneracy one in my perturbed Fleur space and this regularization theorem so says that that's actually exactly where my perturbed zero said hits the degenerate index one Locus in my ambient space and in Fleur spaces degeneracy exactly come from number of breakings So that's the only boundary you get So this is actually has applications Right, so you might ask why in the world did we talk about good enough So the good position is something that happens when you're trying to force more Coherence of your perturbations. You might not be able to make general position But that's because you've already prescribed the section on boundaries, which is something I haven't done here so That's where good position comes from Let's see What else do I need to say right so in order to regularize now I should say that somehow Whatever count I get here is invariant under choices And so I need to particularly well, I really I need to think about what happens when I vary J or whatever in my Setup, but mainly in this theorem. I need to think what it what happens if I take two different perturbations and the claim well Certainly most of these put most of these conditions except for transversality sort of a convex set so I can make a One parameter family and then I wiggle that one parameter family again a little bit to get transversality so Right so I take a pt that goes from p0 to p1 and I built this into Fredholm section over zero one times x So the neat thing is this is all now Fredholm So I don't really have to sort of go and prove again that this is Fredholm What I'm doing here is I'm adding One dimension in the domain That should you know add one to my Fredholm index And then I'm adding a smooth section. That's also compact So that shouldn't nearby sort of there should be Fredholm stability So this shouldn't change my Fredholm index at all. And so, you know all those sort of basic Fredholm Facts I survived in poly folds so This is going to be poly fold Fredholm It's going to be transverse after I wiggle enough by using that theorem again, and then I get a smooth cubordism out Which goes exactly well the only if there was no boundary before then the only boundary I have here comes from the interval And so then it's a cubordism from whatever happens at zero and whatever happens at one So All right, I wanted to write a box except some somehow statements of theorems nowadays are harder than proving theorems Good. Yes Well, and then you have to wiggle a little to get transversality Good so any questions about the statement any non-academic questions good so Right, I want to what I should say what a strong tame M-polyphold bundle it actually is and in particular need to fix notations. I can actually tell you what a Fredholm section is But I'm going to just tell you about this special case that so far has sufficed in all in All applications, so I'm going to say and the nice thing is that this special case will be things that are Automatically strong and tame so I want to not just talk about general Tractions, I'm just going to talk about splicings So an M-polyphold bundle of splicing type is what? Well, so I start with a something that should be a bundle. So I need some kind of suggestion between two polyfolds That are well that are modeled on on splicings Which you may not really know yet, but it's going to become evident once I write down the local tribulations So Together with So it like this to be a vector bundles. I'd better put a real vector bundle Structure on each fiber vector space structure So that's going to be oh goodness Right why sub X you put a superscript T here If you were wondering these if you perturb you might have to actually go into a different no this is that's totally a lie Bundle doesn't change when I change my perturbations. Sorry It just does if I change J, right? Okay, good. So these are actual fibers here, right and I need local trivializations So you see the exercise in writing the poly fold book is to really take your differential geometry book and Just sort of write SC or poly fold in front of everything So Right, yeah at some point You have to worry a little so right, so there are splicings that are just sort of the M-poly fold models and now for a bundle. I want a bundle splicing. So what is that? Well, it happens over an open set in the base I Would like the local bundle to Be SC defiomorphic on fibers To a certain model R, which is again a retract, but it's a specific retract. So I'm gonna at some point probably forget to specify open subsets of Retracts of splicing. So if I do that you can fill them in or you can just ignore this so So whatever this retract is it's gonna have a very nice structure. So there's gonna be one parameter that parameterizes families of projections Which give you the base and the fiber so this whole thing so my My parameter is allowed to That's where all the boundary comes from. So this is where V is some finite dimensional parameter, and that's where all the boundary comes from and then I have a family of projections on a Scale Hilbert space E and one on a scale Hilbert space F so These two are So in particular linear projections Which means also I don't have to worry about scale smoothness of them by themselves I just have to worry about scale smoothness with respect to that parameter. These are usually the gluing parameters so And now I'd only have to write this once because they don't know whether this is a small pie or a large pie so but there are two different families here and on this Right, and this is well, this would be the retraction actually let me write the retraction here Drag and drop So the parameter space of V is here So then the retraction just goes to V Projection on E capital Pi projection on F This is not algebraic geometry. No, sorry First you take the projection and then you build a bundle out of things you projected and then you get a section Why do you do that instead of taking an honest SC infinity vector of one or whatever then taking a section and then taking a projection at the end? Because that's what you have Really if you think about it well ask me that again once I've given you the real example I think that's Or even the toy example, so So you're saying it's an SC splicing because these projections are linear, right? Yeah, and so the nice thing is right So so the all the weirdness now comes from jumps in dimension of the fibers right of these images of the projections so Right. Yes. So the thing is that what I'm asking to be scale smooth is this map I'm not asking Pi V as an operator to depend continuously on V Hmm, it will not in fact well it might but then everything is boring So if it if it varies continuously with V then sorry if the operator Topology is continuous then the dimension of the fibers doesn't change But in infinite dimensions you can let the dimension of the fibers change That was exactly the example that Nate worked out yesterday And so you have this sort of bundle With varying dimensions of fibers and and really what happens is see these retracts. That is your space right, that's where you have the sections. It's just like Kuranishi structures that the sections happen over the sort of small little things and you don't have a Canonical extension to anything bigger. So the the bigger stuff here Sort of ambient spaces just exist locally and nobody says that they fit together in a nice way So I don't usually have an extension of the section to the bigger stuff. This is not like What I'm talking about here s is not just a restriction of some section in a banach bundle to some weird subspace Right. So the local sort of banach ambient spaces are just local Does that make sense? Good All right So Yeah, ah, yes, I should okay Did I say I didn't did not even say this so Right, so this is the this is the base and this is the fiber. So Really right. So what I should say search that Fibers go to Images and in F. So really right. So this is the base That's where boundary and corners happen and F for fiber And here now it's just it's an N gluing parameters here So that's yeah, it's just some gluing parameters. Give me boundary and some don't Yeah Random choice of words I'm not even going to say why this is tame, but it's automatically strong because Here's the why one bundle so strong means I have a meaningful bundle of Better quality fibers and in this case the fibers are simply given by right. This is my local model and I take Everything in the base but in the fiber. I only take the fiber of quality one so this is This this one is that one. So good. So example, right toy example, right? So this is a an M poll of poly fold So what you see here is an open ball that I've attached sort of two-dimensional Saturn ring to Again with open boundary here, and then I've attached just one interval So that's sort of one-dimensional two-dimensional three-dimensional pieces of my x and in order to for a you know Transverse Section to cut out something smooth here, right? So what I would like to be able is to say Okay, I have a Fredholm index one section over this You know that goes into this two-dimensional domain smoothly and then maybe even into the three-dimensional domain to cut out something Smooth, right? So the goal is to not get a sub poly fold, but it's actual manifold So if that is to be the case I'd better have fibers of different dimensions and The fibers better jump sort of at the same rate as The base dimension jumps and that's sort of nice to explain with this placing. I find right so here in this case Y over x right if I want one dimension here, I should have sort of No vector space over that everything is zero over these points sort of yx should have One and over these points I should have a two-dimensional fiber and so really at some point The condition is going to be really that in this bundle splicing somehow the fibers jump at the same rate But we're going to build that into our definition of Fredholm operator So that's not in here yet. No, right now, right? These could be these ill two things that this jumps at zero and that jumps at one Right, that would not be good news Homework right imagine how Nate's example gave you a little bit of Transition map or a little bit of a chart for a piece here Right and then you might have to might want to put you know another chart That's just two-dimensional and then you'd have sort of an overlap But it just goes from this two-dimensional piece to that two-dimensional pacing here You then need somehow our definition of scale smoothness to say in what sense that's scale smooth What I do right now I do want that Yeah When you say so Fredholm section is gonna mean something that imposes Jumps at the same time. Yeah, actually, maybe let me write this down and then right so So here's a little chart that I would like to so my chart map is gonna be to feel more fake to some open subset of This ambient Splicing so now right I better have a topology in which this thing is open but That's how we built that and What is this right so this here this ambient splicing was the union over? Also are is this direction here? and Each fiber here was some line in L2 so the funny thing is just that these lines in L2 are not constant they sort of Turn into all the infinite dimensions because It's this family of Bump functions that somehow weakly converges to zero because it just gets pushed out to infinity as You as you get closer to that point here so Right so here so what I'm actually going to take here Right, so I'm going to take the projections to be the same Projection on L2 to the same little bump function. That's just zero for V Lesser equal to zero and it's a bump centered at Something like each of the one over V When V is positive So right so so far this is just here. This is just the image of the little pie Because that's just my base Right, but now My fiber over this is going to be the same thing except it uses the same base gluing parameter again so You know over this I'm going to have the same fibers and here right at every point here the fiber Is zero Right, so that's the same thing. I'm going to use maybe I should say here right over V the fiber of My base splicing is zero So the fiber of my bundle splicing is also going to be zero and at a point here this fiber is One-dimensional and so over that point Here my fiber is also going to be our beta V So the local trivialization or the bundle Why over you is? Simply going to be well again this open subset. What is it? Right I could take the points V comma X or V E That lie in this thing here open subset oh right and Over each of them the fiber is again our beta V So these are my fibers What about right so what happens at the interesting point right so the Fiber of my splicing is still just zero Right, and so the fiber of my bundles also just going to be zero It's just that then because the fiber of the base jumps I'm going to have the fiber of the Bundle although that I'm going to have the fibers also jump So that way the jump dimension jumps at you know equally instead of my fret home index actually stays constant Which is a good thing? It's not built in here. No, it's going to be built into the definition of a I should but one could say okay, so Let's actually do this here. So why is fillable if for all V I think the kernel of Pi V This isomorphic to the kernel of capital Pi V So that's the complementary And we're going to have to fill at some point Strong just means that this this nice bundle why one is defined Which in this case is sort of automatically defined because we have these nice fibers So we I could just take the fibers of quality one in the in the ambient Buna space It's a property of a general and poly fold bundle it is automatic for and poly fold bundles of splicing type Right and what you want to think of is when you have this right? Where's my right the real example, right? It is time I think too much both infinite in need and therefore I mean, what do I mean by this right? In some sense, I'm going to need if can you just trust me that this is going to be built into the definition of Fred homefilling Great. Yes, please complain. I guess you out of all have the right to complain Anyway, so the way you should really think about this strong and why one is because you have fibers age three over age three maps So Really what you should think about strong is that? age three in a Pullback tangent bundle makes sense When you was in age three But so far usually our fiber over a Point of quality age three is usually just age two roughly speaking So they're sort of yeah Better quality. Well, it makes sense and it's sort of invariant under coordinate changes accepted Right, so if I wanted to make sense of age four over an age in an age three pullback tangent bundle and that is Probably going to depend on some choices of the trivialization Right, but when the age three, it's it's going to be independent. So That's Let's see good Okay, so this was the baby example and this was already sort of halfway to the real example So, ah, it's going to be very sloppy So I want to think about the bundle near a k-fold broken Fleur trajectory Because I'm lazy and I don't want to say SFT you could think, you know k-fold F a building right this a k-fold broken Fleur trajectory is the same thing as a k plus one floor building Because if you want to break k times you need k plus one trajectories and The first place where I'm going to be totally sloppy is that of course Fleur trajectories I have to mod out by r at least and I'm not going to do that. So Yeah The gluing parameters right yes, so for every Floor then when I glue together I need one gluing parameter exactly. Yes Yes, so K is in there in this. Yeah Well, no actually hard. No, it's the right K Because these come from nodes interior So every actual breaking every Fleur breaking that's what gives me the boundary and Yes Yeah, and then internal nodes are just going to give me right internal I mean I could have written C here actually fight Let me not do that right so Good so what is my bundle splicing here, right so so I'm going to have K gluing parameters I need some base space And some fiber space and I'm just going to Tell you what the projections are But first I should tell you what these Scare spaces are right. So what do I need to parameterize? Fleur trajectories right I need to vary these so really For every trajectory there I need to have a some section of a pullback tangent bundle and then Once I've varied them and I apply the Fleur Operator the Pater Kushi Riemann operator to it. I should end up in the fiber So these fibers then are just going to be same kind of product except age two and the fiber because I'm thinking about an operator of order one And then right I'm going to be lazy for now and just write epsilon E F goes to pi epsilon of E Hi epsilon of F and now I need to tell you what the projections actually are Right so really and each of these is a tuple right this K tuple and these are K plus one tuples so pi epsilon is the projection in the sort of Retract that Joel Defined at length so it should be the projection to The kernel of an anti-gluing along The kernel of the plus gluing right, so whenever I see Variations of the flirt trajectories right really I'm thinking okay a very and then I glue it together Right, that's sort of my chart map But that has ambiguity so what I want to do is I want to sort of reduce the ambiguity That's why I go to the kernel of the anti-gluing But by reducing the ambiguity I should like somehow not change what The point actually is in X that I have in mind. That's why I go along the kernel of the Gluing so I don't actually want to change the point that I have in mind Right and now you're going to say okay, there's Evidently Gluing parameters missing and I think it's just worthwhile to say this again. So So what are these so these are Auntie or pre-gluing off the Right of really the sections. I'm gonna I think probably Well, there's a lot of fuzziness here, right? So you have to ask yourself. Can I actually Glue these and then I should do this with some gluing parameters and those I like to be somehow large So there's always going to be some gluing profile in the picture That takes small parameters to large parameters and the choice of this gluing Profile is sort of that is that is one of the foundational choices in this whole subject There's pretty much just well, there's one for which you get smoothness and that's unfortunately not the sort of Obvious one you choose from Dylan Mumford space. So Eventually somebody needs to prove that the invariance that we get out of this are independent of the choice of gluing profile So yeah, yes I think it's pretty clear that something should work But then there's a question whether there's an actually in total order on gluing profiles Yes But it's basically like it if somebody gives you are the real line And you don't know what is and then you have to choose a chart So you take the identity or you take X goes to X So the difficulty is only one point Yeah Possibly yes, we don't know that Yeah Why right let's let's keep that for next week Right so Let's see What happens right so strictly speaking I told you how to glue in the base, but you can do exactly the same thing in the fiber You so for the fiber and this again goes back to sort of the vex question of what do I actually have a section off, right? I don't have a section that's just lives over sort of the broken flirt trajectories I wanted also sort of encode my section that lives over the unbroken flirt trajectories, right? And that takes a glued flirt trajectory and applies the flirt operator to it So you get just one section of one You know something that lives over a glued Curve and so in the fiber I kind of need to have the same domains that I have or the sort of glued domains Right I glue the cylinders so I need the same domain in the fiber that I have in the base So that's why somehow the Dylan Mumford parameter, maybe you could also think of the Epsilon's here on the splicing right the parameters are sort of the Dylan Mumford parameters and they affect The maps and the sort of zero one forms in the same way Right so why is that not defined on the whole space before I could define it on the whole space except There's nothing to do with Fleur theory So I should write I should maybe write this down right so Let's see. What do I have here? So my Fleur idea Right, so what is my D bar operator? really in terms of this pre-gluing and things really what it it sort of takes a big pre-glued Curve to D bar Off that pre-glued curve Right on most of it, right? It's just one D bar. It's not broken Right, so they open or the the main stratum of X is the unbroken one So this is the main stratum of my operator, right? Except when I'm near a broken one this sort of I want to pull all of this back somehow to Something here that should be in the fiber and something here that should be in the base right, so somehow this comes from a choice of gluing parameters and All these Fleur trajectories right so that I glue together to parameterize this point Then I apply the D bar operator and then I need to go back and so I kind of take the What what what am I doing here, right? I'm going to the fiber so to get to the fiber I need to take plus and minus inverse of Exactly this thing and I force minus gluing to be zero. So this then sits in that fiber Yes, right So that way I can this way I'm pulling back the D bar operator that just acts on this on a single Cylinder I pull it back to this sort of tuple of cylinders here by sort of first gluing Applying D bar and then sort of undoing the gluing so Right. However, as you can see right if this is a Fredholm operator, right? I'm Losing a lot of dimension here, right? So I have a massive kernel Then this is Fredholm and then I go back and I have a massive Well, this is never going to be surject if there's a lot of things I don't map to so I have Infinite kernel here and infinite co-kernel here and so the idea is going to be to Cancel the kernel here by the co-kernel there I'm not sure I'm going to get there So but this is so this is my real-life example and again, right? So here Y1 now comes from the local things that are V e and then The f's that are actually already in H3 and not just in H2 So that's really what I should be thinking of as that bundle Y1 So yes, right? Yeah, I fear that would come right so Yes, someone could say that But I haven't said what filling is so ask me that again once I've said filling Let me just quickly say what what can I say now, right? So let me go down my laundry list actually, right? I needed to tell you what an auxiliary norm is so What this is is a continuous function on this better quality bundle That's a complete norm in each fiber. So classically continuous norm in each fiber and in the example well You take The norm at a point. What is this V e f to be? F H3. Oh boy, and now I'm That's probably an exponential weight on here Because if I want this to be a scale space, right? If this is f0, then this is supposed to be f1 and so that needs to be Compact in f0 there was an extra as an extra condition on the norms Because the others can get some digital if If you have a sequence of vectors The base converges to x and the limit super is going to zero Okay, good. Okay. Yes. There's something. Okay. Good. Okay Small print There's also right so I should say right so obviously you can see all the small print in the hoof of is that skits in the papers And if you're trying to find where that specific small print is I would recommend looking at the what we used to call the user's guide poly folds at a glance or two that We also posted on the on the web page that that is lacking this Small print. However, it always precisely sites the place in the publications where you can see all the wait First and second glance, right? Yes, because it became too long, right? so I Need to tell you what an sc plus section is so a section Well, first of all a section is something that gives you the identity when projected down and it's sc plus if It actually takes values in these better quality fibers and is sc infinity as a section of that better quality bundle Which is exactly you know a first order. So what you should think if here is a zeroes order operator compact operator Maybe compactly supported would be really nice The note here is that the classical perturbations if I change J this is something like J minus J prime DT That is evidently first order. So this is not sc plus So wiggling J is not what happens in this regularization theorem however a Homotopy of J's fits into this cubordism argument and so that makes sort of Changes of J built into the theory after all Well, it's first order so in particular this needs to take age three to age three And it yes, right. Yes Good, okay, so now I can Right, so now the I think the only thing left to say is what is Fred home am I missing anything else? But we know what compact means And I explained what controlling compactness means so So first things first so Now this is the this is the place where I think a grad student Hold up in a basement would have a little bit of trouble making up the whole theory So Fred home is not quite obvious. So one thing I Need as I need it to be regularizing which means that if I'm in the quality If my section takes values in the quality k fiber Then I actually want to have the base also quality k which is like saying if D bar u is in HK Then u is an HK plus one also known as elliptic regularity So that's something that's sort of true for the operators that we usually look at and then the key point is this filling that we've Alluded to and this at last count you needed a filling somehow strangely not just at Solutions but at everything that's sort of smooth. That's of quality infinity So near each Point I need a local trivialization and a filling and I'm going to say what the filling is So that the filled section is Fred home ha ha so this Fred home However, it's going to be easier to define than that Fred home Because this Fred home happens on splicings and retracts and this Fred home is going to be a map between Or map germ of maps right between banner spaces or SC banner spaces or SC who but spaces in this case So I'm going to leave that for the moment undefined and tell you what the filled section is so so in a local Chart Right. I'm going to this bundle splicing This sits in and now I'm going to forget that there are open sets. I don't want to write another one in here so that sits in a splicing that sits in here and this bundle splicing sits in the bundle splicing that sits in here and Right, so what I would really like to do is I would like to fill up my section here So that I have any chance of this being Fred home. So let's see. What could I do? Right. So first of all my problem is that this e is not this ve is not necessarily in O But I can map it down with my retraction and Then apply s which sits in here. So That's not so bad. Um, ooh, let me I Should say what this s actually does Yes, so I want to write so the section takes Ve obviously to ve and then something happens in the fiber. So I'm just going to write f for whatever s does in the fiber and So then I can apply f f is only defined on Oh, so I need to throw in a projection here. So I can do that But then I'm only ever going to hit the The splicing within this fiber never going to hit the complement. So I want to add something and I'm going to let whatever happened They're just depend on the complement of e so I can write each e as Pi ve plus 1 minus pi ve That's the splitting Here and I'm going to use the same splitting in F This is the splicing and the complementary splicing. I'm going to have right this guy here and What do I want right so? so F prime of v comma dot right maps the kernel of little pi v to the kernel of capital pi v and I'd like this in an appropriate sense to be an s c infinity family of Isomorphisms The idea being that I do need to soup up kernel and co-kernel to make s twiddle Fred Holm, but I don't actually want to change the zero set. So that's the main remark as twiddle inverse of zero is s inverse of zero and also kernel and the image of the linearizations of ds twiddle Isomorphic to the That's that is wrong kernel and co-kernel image perp are isomorphic to the co-kernel and co-kernel of ds So that allows me to dream of Fred Holm theory. So that is the theory Yes isomorphism, yeah, yes Yeah, yeah, I mean in my splice, I mean this is a simple if I mean I think so well So here's the example, right? So in the example actually listen one example is as good as all yeah Right so the filler is the linearized operator Where you may ask right? It's it's on the anti-gluing so So overall the filled section So if you look up there, I wrote down what actually the section is in Local coordinates so the section went from epsilon e To what was that? I just wrote down the fiber plus minus inverse of plus of e and Zero right so that was my that was the section and In order to get the filled section all that I'm doing is I'm writing down the minus gluing here By complete Breakdown, of course, there was a d bar operator here and so the minus gluing of e Always lives in So if h3 of r times s1 it comes from data that is very close to the breaking so It just lives over a fixed Hamiltonian orbit. Well, it's really a product right because they have various breakings so Point being so there's no r dependence here. And so when I take the linearized D bar operator on this There's a general theory that says that Our invariant operators here, you know if they are useful if they're fret home at all then they're actually isomorphisms so This is of the form DS plus a and Then you read my favorite paper by robin Salomon about I think spectral flow right and I Have successfully sidestep the question of what a fret home map between scale Hilbert spaces is Which is evidently in my notes so I'm just going to hand wave for like one minute and then you can ask me about details if you want them So the problem is really when you're proving Invisit function theorem that you need to do Newton iteration and the Newton iteration sort of comes from a contraction and When you write down the contraction that you get from a scale smooth family of from a scale smooth What what what do you actually mean by a fret home section, right? That's nonlinear usually we say well the linearizations better be fret home, right? so that's something that makes sense but the other thing we need is that Newton iteration works and that requires a certain continuity of the linearizations and Unfortunately, there's this scale shift and so you get a contraction property, but the Contraction goes sort of down in levels by one So if you start iterating you go further and further down in levels that does not bode well for any convergence So you kind of need to sort of force yourself back up So we need to add Something called a contraction germ property that sort of implements this Contraction So really what this is is a contraction a level preserving contraction in All but finitely many dimensions and so then you ask yourself well, how do you ever prove this? well in that sort of comes if Your fret home section is actually right. It's scale smooth That means you always have that loss of levels, but if you have classically see one in all But finitely many dimensions Then you can prove that that implies this contraction property well plus some small print and so that's this can be found in the Grim of Witten paper and Somehow I independently figured well but there should be a better definition of fret homelessness and I'm currently revising that paper in which I also attempt to prove the fret home property for flirt trajectories and I would end by saying that poly folds are awesome Because you actually get referee reports So if the referees are in the audience, thank you You know They did not complain of me failing to spread peace in harmony They actually wrote the read the paper and you know font mistakes and pointed out things and so I'm revising and that is Yes, beautiful thing. So I Wish all referee reports were like this and maybe I just need to keep doing poly folds So Yeah, well, I'm I'm going to scan these Yes, and then put them on. Yeah. I mean, however, I would say I mean all of this is sort of essentially just skimmed from the user's guide Yes, yeah, yeah, if you want to there's more than everything in there I Right he's asking me and I'm asking you because I don't know what the small print it is so I thought you just use cutter functions That works right it satisfies whatever small print you have awesome good Ah As the as the index of the linearized Fethom operator. Yeah, and then somewhere it's it's built in that that's actually constant Yes, and then you could ask how do you define orientations, right? So you need to construct the determinant line bundle and yeah