1/3 Classical transversality methods in SFT
Wendl, Chris
I will give a quick review of the Sard-Smale theorem and the universal moduli space approach to transversality, discuss the relative merits of classical vs. inhomogeneous perturbations, and sketch proofs of the standard theorems stating that the moduli space of regular J-holomorphic curves in symplectic cobordisms is a smooth orbifold, and that all somewhere injective curves are regular for generic J.
 Thanks, Joe. I'm going to start on this board with just a preface of generalities. Let's call this the archetypal moduli problem, by which I mean from the perspective of global analysis or differential topology or simplexed topology as distinct from algebraic geometry, where they also like to talk about moduli problems. So for me, a moduli problem means this kind of thing. You have some space which I'm calling suggestively J. But right now, I'm just going to call it the space of data, which has some kind of topology on it. And for each choice, which I will suggestively call also J, in this space, we get some moduli space, M of J, which also has a natural topology. And of course, we would like to say that it has some much nicer structure than a topology, like being a smooth manifold. Or if we can't achieve that, maybe at least a smooth orbifold. And over the course of these two weeks, I'm sure you'll also hear things about weighted branched orbifolds with boundaries in corners, though not for me. From my perspective, well, the first thing to say about this moduli space is that locally, you can identify it with some kind of zero set, which I'll write sigma J inverse of 0, except not precisely that. Often you have to divide it by some kind of symmetries. So here, sigma J is some section of a Banach space bundle. And we already heard DUSA sketch one example of this. Because you can express all sorts of geometric PDEs in this way as preferably smooth sections of Banach space bundles in an infinitesimensional context. But I'm saying this is true only locally in general. So DUSA described the situation with holomorphic spheres where you can actually do this globally. But in my experience working with J-holomorphic curves, that's rather the exception to the rule. And so I will talk about some more general situations where you can do this but only locally. And these symmetries can cause quite a lot of headaches. But the first problem, of course, is you want to be able to say that this zero section here is a smooth object. And the implicit function theorem ought to tell you that if you do things correctly. But there are two general ways that you can approach that. First, the so-called abstract approach where you perturb the right-hand side of the equation. You can say something like define the moduli space m tilde, which depends on the data J, and some extra auxiliary choice that are called nu. That's the space of all u in this Bonnock manifold, such that sigma J of u matches nu of u. So this is for some other section nu of that same Bonnock space bundle. And the idea is if we choose that generically, then this set of solutions will be a smooth object simply by the Sard-Smale theorem. That's kind of equivalent to the statement just expressed in local trivializations that a generic point in the target of a smooth map is a regular value of that map. So this is a generic section. It also intersects this section transversely. So for generic choices of this auxiliary nu, our special section sigma J is transverse to that, which means this moduli space of perturbed objects I define is a manifold. And I even want to say more because if we're talking about elliptic problems, this section is not just a smooth section, but also is Fredholm in the sense that its linearization at any point is a Fredholm operator. So when we have these kinds of transversality conditions, we get just not just a manifold, but a finite dimensional manifold because of finite dimensional kernels of the linearization. So that's, of course, very nice when that happens. The problem is I perturbed my equation to a different one, which might not be the one I'm actually interested in. That's problem number one. There are more problems, which I think you'll hear more about over the course of the two weeks because this abstract approach is sort of the raison d'etre for the whole polyfield project. But it's difficult. And part of the reason it's difficult is that, first of all, this correspondence between some zero section and my actual moduli space of interest is in general only local, and it also has these symmetries. You have to deal with both of those in a sensible way when choosing these perturbations. And it means just choosing some generic section of a Bonoq space bundle is usually not enough. You have to choose it satisfying some conditions. And maybe those conditions are too stringent to actually apply the Sards-Mail theorem in this way. So do you want to be like invariant under the symmetry to something? Something like you want various things of that sort, yes. I reserve the right to not talk about that anymore because other people will. And I'm going to talk about the alternative, which is in the title of my mini course, so-called classical, or what many people call geometric, perturbation methods, which is the notion that if you take a generic choice of the data in the space of data, then your actual section will be transverse to the zero section, which means, of course, the zero set of your section is a smooth manifold and finite dimensional as well, because we're in the Fredholm context. And well, the space we're actually interested in is that divided by some kind of symmetry. That, in general, will be an orbifold. So an orbifold is something that looks locally like Euclidean space divided by a finite group action. And that's what you get if you're taking a finite dimensional manifold and dividing it by the kind of symmetry groups that we're interested in. So that is minor headache number one to be aware of. We won't get manifolds in general, but we could get orbifolds if we're lucky. Much larger headache number two is that often we're not lucky and this condition simply isn't true. And Duce already talked about that a little bit. I'm going to talk about that some more. So that's all I will say about generalities. I'm going to focus on this second, this classical approach over three talks. And in a particular context, I want to discuss symplectic field theory. So in the original propaganda paper in 2000 by Elishberg Giventhal and Hofer, you have symplectic field theory presented as a very general way of defining invariance of contact manifolds and symplectic abortisms between contact manifolds. And well, one can do it even a little bit more generally than that, which is sort of nice for certain purposes. So I'm going to talk about stable Hamiltonian structures. So first of all, if m is an odd dimensional manifold, the stable Hamiltonian structure H on m is going to be a pair capital omega lambda, which consists of the following. We have first omega is a closed two form of maximal rank. So you should imagine that as something that arises on any hyper surface in a symplectic manifold. You just take the symplectic form and restrict it to a hyper surface, it's going to be a closed two form of maximal rank. If I omit the word stable, that's what we call just a Hamiltonian structure. Now stable comes from this additional one form lambda, which satisfies following properties. First, lambda wedge, the top dimensional power of omega, must be a positive volume form. So this implies in particular that you can take omega and restrict it to the kernel of lambda. Lambda is nowhere zero. So its kernel is a hyper plane distribution, which I'm going to call psi, as I always do with contact structures, because that's one important example. Could be a contact structure, but in general, it's just some hyper plane distribution. And omega makes that hyper plane distribution into a symplectic vector bundle. Gives us a symplectic bundle structure. And then the other condition, since omega has maximal rank on an odd dimensional manifold, it has a kernel that's always one dimensional. That's the so-called characteristic line field, the one dimension of directions in which omega is degenerate. So what I'm going to require is that d lambda also vanishes on that same kernel. Another way of saying it is that the kernel of omega is contained in kernel of d lambda. OK, so this is a definition which has been around for several years. And as stated, it's not too hard to understand, but I think a lot of people don't really understand what it means. So let me talk a little bit about that. First observation is that if I now take this odd dimensional manifold and look at a small cylinder constructed out of it, so some small interval times m, this thing now inherits a natural symplectic form. Let's call the real coordinate little r and define little omega to be d of r lambda plus big omega. And that's an easy exercise. That's going to be a symplectic form. As long as this cylinder is sufficiently small, it might even be symplectic for epsilon larger than that. But in general, I can only guarantee this is true for epsilon small. Moreover, what does the characteristic line field look like? If you restrict, you've got a one parameter family of hyper surfaces. Let's say m sub r is the hyper surface r times m sitting inside that cylinder. The characteristic line field of omega restricted to m sub r is independent of r. So it's the same characteristic line field for all r, all parameters in my parameterized family of hyper surfaces. That's really the original definition, which I think appeared first in the book by Hofer and Sander of a stable hyper surface. That's where this idea comes from. It's the kind of situation in which you would expect to be able to prove something like the Weinstein conjecture that asserts existence of closed characteristics of this characteristic line field. Because if you have them on some hyper surfaces in the vicinity, you have them on all of them. Excuse me. What do you mean with the first symbol of the last line? The first symbol? That's an and, and percent. So in order to get the stable condition that you want this last condition at the kernel of omega containing the kernel under, that's what implies. That's what implies this fact about the characteristic line fields, exactly. And conversely, if you are given a one parameter family of hyper surfaces that has this property, this stability of the characteristic line field, then you can always find a stable Hamiltonian structure that produces that. So that's kind of an exercise using the Moser deformation trick. How much will I lose if I just ignore this definition pretend you said contact structure? Nothing really. That's a perfectly good point. If you don't like this definition for some reason, because it's new, just think lambda is a contact structure and big omega equals d lambda. I was going to say that soon anyway. So there is a, from this data, we have also a canonical generator of the characteristic line field, which by analogy with contact forms we'll call this the rape vector field. So my notation will be r sub h. And it's determined by the conditions. Omega annihilates the direction of the rape vector field. And then we normalize it by lambda. Is there any reason d lambda couldn't be zero? No. There's no reason at all why d lambda couldn't be zero. I will also say that again in a moment. So we have a stable Hamiltonian structure. It defines this hyperplane distribution psi, which might be a contact structure in certain examples, but it doesn't have to be. Could also be a foliation. We have this rape vector field. Another thing we can do is, since we have this small symplectic cylinder we built out of it, we can also just arbitrarily expand that to r times m and rewrite my symplectic structure there in the form omega phi. That's going to be d of phi of r lambda plus omega. So what's phi? Phi is going to be in the space of smooth functions taking r to that same small epsilon interval with strictly positive derivative. So this is a symplectic form also. OK? It's symplectic to the one I wrote down before. I've just stretched things. But the point is that r cross m is the kind of setting where I would like to talk about j-holomorphic curves. And I can do that now by saying, well, the space j dependent on the stable Hamiltonian structure is going to be the space of almost complex structures, j on r cross m, which I will assume always to be r invariant. In other words, there's this r translation action on r cross m. I want that to preserve all my j's. I also want to say that j maps the unit vector in the r direction to the rate vector field. And I want to say that j preserves my hyperplane distribution as well as the symplectic structure. It's compatible with the symplectic structure on that. So let's say j restricted to psi is omega compatible. OK? So here's a lemon which is very easy to prove and helps justify all these definitions. That is that for all choices j in this space of translation invariant, almost complex structures, and all functions phi, as I indicated above, that are mapping to that small interval with positive derivative, it's not just that omega phi is symplectic, but in fact, it's compatible in the usual sense of Gromoth with this choice of j. So I can fix j and I can then allow any choice of phi I want. And that's an important detail for defining things like the energy of a j-holomorphic curve, because we then take that to be the supremum of symplectic area defined with respect to all these different choices of symplectic forms. So let me give two examples. They've already sort of been mentioned. First of all, if alpha is a contact form, we get a stable Hamiltonian structure d alpha, comma alpha. And then this symplectic manifold r cross m with this kind of symplectic structure I've been talking about. I'm not going to try to say these two precisely. You have to make some kind of condition on how you define phi in order to say this. But I'm just going to say one can do it. So this is what's called the symplectization of the contact manifold. I really should relax the epsilon's in order to say that. I should actually, should at least make plus epsilon to be infinity. Let's not worry about this right now. Yeah. Can you quickly write down what the symplectization is? No. I'm going to say it's this. Yes? So what's the condition where you can integrate this hyperplane distribution to a symplectic manifold? Where you can integrate the hyperplane distribution. Yeah, if you integrate it, and then you find a sub-manifold, which has that as its tangent form. Well, for the hyperplane distribution to be integrable, I think that's equivalent to saying that d lambda vanishes on psi. So certainly it's sufficient of d lambda vanishes altogether, which is my next example. Already in the first row, we think that the word compatible sort of looks like compacted. Well, I'm sorry. I'll take this opportunity to make a blanket apology for my handwriting, as I customarily do. There isn't that much I can do about it. OK, compatible. I just said, i actually looks like compacted. Compacted. All right, second example. If m equals the product of s1 with some other manifold w, and I take w to be a symplectic manifold. So let's say I have a symplectic form little omega on w, and then set capital omega to be that plus dt wedge dh. So here, t is the s1 coordinate, and I'm choosing some function h, a Hamiltonian function, which is dependent on time in a periodic way. So this is capital omega, and I take lambda to be just dt. This is a case where it's closed. d lambda vanishes entirely. So now one can check. This also is a stable Hamiltonian structure. And it has a meaning which a lot of people will probably like. You can compute the rate vector field for this. It's simply, it's the vector field in the t direction, plus a vector field in the w direction, which is precisely the Hamiltonian vector field. So this is a way to turn questions about Hamiltonian vector fields on symplectic manifolds, even time dependent ones, into questions about a time independent, rave like vector field on an odd dimensional manifold. And it gets better because for the class of almost complex structures that we've talked about, we can use the same method. So we can use the same method. We can use the same method. We can use the same method. We can use the same method. We can use the same method. So the class of almost complex structures that we've talked about, a j-holomorphic curve in r times this, is basically equivalent to a solution of the Flur equation in w. So this is a way that you can do Flur homology, Hamiltonian Flur homology, from the perspective of symplectic field theory. OK. So now the setting in which I really want to think about holomorphic curves is also a bit more general than this. So let's take a symplectic manifold. So here I'm talking about co-boardisms with stable boundary. So one can define stable boundary in various ways. The main thing I want to say is there exists near the boundary of one parameter family of hyper surfaces that all have the same dynamics on them, same characteristic line fields. And that's equivalent to saying that there exists this collar neighborhood near m plus that looks like a small interval times m plus, where the symplectic structure is just the kind that I wrote down already, d of t lambda plus plus omega plus for some stable Hamiltonian structure, omega plus lambda plus on m plus. Of course, I'm going to do the same thing on m minus. But the collar neighborhood will look like an opposite interval. OK? So this is the definition of a symplectic co-boardism with stable boundary. And having done that, I can add cylindrical ends to this compact object and get a non-compact object. That's what we call the completion. So I'll denote that by w hat omega hat. It's going to depend on some choice of function phi of the sort that I used to define symplectic structures on simpletizations. And that choice of phi is not particularly meaningful. It's just nice to know that I can do it. So the definition is we take a negative cylindrical end, glue that along m minus to w, glue that along m plus to positive cylindrical end. And the symplectic structure here, we take omega phi, omega phi on each of the ends. And the original omega on w, you can choose the function phi such that, so remember, I still have this definition on the board somewhere. Middle board top right. Middle board top right. Very good. d phi times lambda plus omega. If I choose phi so that it equals just the identity map close to 0, then I can obviously glue these cylindrical ends smoothly together with these collar neighborhoods. That's what I'm doing. OK. So with that choice of symplectic structure, and I don't really have to specify phi, I just know that I can choose one. Are you using the same phi on the both ends? I don't care. Actually, I specifically want to not specify phi and allow myself the freedom to change it. So let's say j of omega and the two stable Hamiltonian structures. This is going to be the space of all almost complex structures, j on the completion w hat, which I assume to be everywhere compatible with, well, let's just say, compatible with omega on the original compact object and belonging to these spaces with respect to the stable Hamiltonian structures that I already defined. So the spaces of translation in very almost complex structures on the ends. So that now is going to be an almost complex structure that's compatible with omega hat phi no matter what phi I choose. OK. So now to start talking about the actual J-Hollomarket curves. Any questions so far? OK. So this you require that J's, what does it say? Translation. Compatible. Translation. Translation very on the end. So what about in the co-boardism part? I mean, don't you have a factor r there anyway? When the co-boardism part, I don't have a factor of r except close to the boundary. Yeah. OK. Yeah. And I don't make any requirement there except compatible with the given symplectic form. OK. So the kinds of holomorphic curves we're going to talk about are as follows. We take sigma J to be a closed Riemann surface. Let's actually say there's going to be two finite disjoint ordered subsets. And in particular, sigma dot always denotes sigma with the first finite ordered subset removed. That's a punctured Riemann surface. We take that, also this subset. I don't have to worry about it too often, but it gets partitioned into the so-called positive and negative punctures. Each of those could be empty, but each puncture belongs to one or the other of those. And now I think about maps from this Riemann surface with the various points removed going into my completed symplectic co-boardism. And I want them to be what I call asymptotically cylindrical, which means, well, let's not write down the precise definition. Let's just say asymptotic near each positive or negative puncture z to R times a Raib orbit in m plus minus. Chris, you're missing a hole somewhere. I'm missing a hole. That's true. Could be worse. The opposite would sound like a medical emergency. So what I mean is essentially, for this kind of almost complex structure I'm considering, with the translation invariant thing satisfying these conditions on the ends, I have j maps the vector in the r direction to the Raib vector field, which means r times a Raib orbit is always a holomorphic cylinder in the end. So those are special holomorphic curves that always exist, at least in that region, maybe not globally. And I want to consider punctured holomorphic curves whose behavior near each of those punctures approximates one of those cylinders better and better as you go toward the puncture. So to make it precise, I would say you can choose some holomorphic cylindrical coordinates near each puncture. And in those coordinates, as you go further up toward infinity on the cylinder, your curve looks more and more like one of these trivial orbit cylinders. OK, maybe it's time to recover this. Yeah? In your picture, theta is empty or what's happening? Theta, I haven't really said. Theta doesn't have to be empty. It's just some finite subset right now. But in the picture, it doesn't appear in the picture because it doesn't affect the picture in any way. It's just points. So theta are meant to be marked points. All right, so the actual moduli space looks like the following. It's going to be a space of tuples sigma j gamma theta u, where, well, sigma j is a closed Riemann surface, as I wrote above. Gamma and theta are these disjoint ordered finite subsets. So let me not rewrite that. Mainly, I need to say u maps the punctured surface to the completion w hat j holomorphically and asymptotically, solidically. And then I just need to say what my equivalence relation is. Deuza basically already said it. You could deduce completely the right answer to this from what Deuza said. I will say a tuple sigma j gamma theta u is equivalent to a tuple sigma prime phi star j phi inverse of gamma preserving the ordering phi inverse of theta u composed with phi for any diffeomorphism phi from sigma prime to sigma. But doesn't the ray orbits allowed to be arbitrarily terrible? Yeah, this definition is still OK if I allow the ray orbits to be arbitrarily terrible, but I won't have any theorems. Yeah. I do have, so finally to make some use out of the mark points, I have an evaluation map sending this moduli space to w hat m where theta, the mark points, are labeled zeta 1 through zeta m. So this just evaluates u at those points. And it's well defined because of the way that I define my equivalence relation. These diffeomorphisms map the set of marked points to the set of marked points. All right, so let's finally state a couple of theorems. So theorem one says, if I have, I'm just going to abbreviate elements of this moduli space by u instead of writing out the entire tuple and saying equivalence class and everything. So if u is what I call Fredholm regular, that's a condition I'll have to explain a bit, and it also includes the condition that the asymptotic ray orbits are not arbitrarily terrible. Then a neighborhood of u in the moduli space, is a smooth finite dimensional orbital with isotropy at u, isomorphic to the automorphism group of u. So I didn't write down this definition yet. The automorphism group of u is simply the space of all bi-holomorphic maps of its domain, which fix the punctures and the marked points and satisfy u equals u composed with phi. So unless u is constant, that's always going to be a finite group. And for somewhere injective curves, it's going to be a trivial group. But for non-somewhere injective curves, so for multiply covered curves, if we get this Fredholm regularity condition, whatever it is, we're going to have to deal with an orbifold rather than a manifold because of this isotropy. Is this going to be a centric of tautology on the word regular? In some sense, this is a tautology on the word regular. I mean, one does have to work a little bit to see why you can make this local identification of the moduli space with something that looks like an orbifold even if you have regularity. And I will at least have time to get to that in this talk, I think. Therm two, I'm going to probably postpone the proof of this for tomorrow. But I'll state it now. Let's say let's fix an open subset u with compact closure. Then I'm going to say there exists a bare subset called j u reg. Well, it's a subset of the space of all. So I needed to fix some more data in advance. So fix also an almost complex structure, j0 of the type I've described. Then there exists a bare subset. So this is a subset of the space of all. J's of the type I've described with the extra condition that j matches j0 everywhere outside of this special subset u. So this special subset u we can call the perturbation domain. It's where we're going to allow ourselves to change the almost complex structure, but we leave it fixed everywhere else. So such that for all j in this j reg u, let's finish it over here, every somewhere injective curve u in the moduli space with respect to j, passing through my special subset capital u is thread home regular. But here bare subset is again a commuter. Bare subset literally means commuter, yes. So what did I call residual? Yeah, I never understand residual. It sounds to me like it means the opposite, so I don't use the word. Some people call it second category, which I think is also not technically right if you look up the actual definition of second category. Yeah. Thank you for that input, Catherine. So yeah. Well, someday some kind of sensible general habit will prevail, hopefully. I say bare subset, and most people don't seem to mind. So the literal meaning is countable intersection of open and dense sets. What contains, actually, if you're being strict, it contains. It contains a countable intersection of open and dense sets. Very good. Maybe more useful input is though somehow you, I think you're assuming here now that the Hamiltonian structure is sort of regular. Well, that's in, oh, in theorem two. I am. That's true. Thank you. OK. So the board's out of reach, so I'm not going to write it. Let's just say this. One additional condition you need in theorem two is that, well, let's say all the closed Ray-Borbets for both stable Hamiltonian structures are non-degenerate. So under those conditions, theorem two holds. Otherwise, you could have trouble. And at least for if your stable Hamiltonian structures are contact forms, that's a condition which is generic. That's a standard result. Can you also do this if you're Ray-Borbets or more spot? More spot is also fine. Yeah. More spot or non-degenerate are perfectly fine. OK. So let's talk a little about the functional analytics setup for a neighborhood of, let's say I have a specific element of the modularized space with complex structure, J0, and the map is called U0. Just to pretend the statement of theorem two. You're not asking that it be somewhere injected inside U. It has to be somewhere injected somewhere and pass through U. Those two things are equivalent, which is not so hard theorem. So what I really should ask for is that there is an injective point mapped into U. And it's a good point because if I were working in other settings where I have boundary with Lagrangian boundary conditions or something, then it's not necessarily true that just because the curve is somewhere injective that its injective points are dense. So then I would have to be careful. But here, injective points are dense if the curve is somewhere injective. I think you've hidden the end of the statement. Can you rise the board and choose? Ah. The statement is on the sideboard. Oh, it's this one. It's this one. Then you can block that board so it's up with the theorem. This is called democratic blackboard technique. All right. So I need to actually write down some kind of Bonnock manifold of maps that will contain the solutions I'm interested in. This is something that's kind of standard in the case of closed maps from closed Riemann surfaces. For punctured curves, I guess it's considered sufficiently standard that nobody really feels they need to go through the details when they write papers. But it means there is no paper where anyone does the details really as far as I think. In Matthias Schwartz's thesis, I think there may be details about this, and that's possibly the only place. There's also this paper by the problem of traveling. We know how to paper this. No. No. Well, I mean, what he does in there is basically just lifted from Schwartz's thesis. OK. So I'm going to talk about a space of sub-elevated class maps from the punctured surface to W hat, class WKP. So I want to have k derivatives that are of class LP locally. Locally, that would be the definition, and it's not hard to express. But globally, what does this mean? Easiest way to say is U is of the form x along f of h. h, where f is going to be some smooth map, which is not just asymptotically but literally cylindrical near the ends. Well, I'll say that in a moment. h is going to be some section of class WKP locally along the pulled back tangent bundle along f. So in compact subsets, this is a sufficient definition, but I need to say more specifically what happens at the ends. So I just have confidence. So the j zero that you started with is regular, right? Or that? No. I don't care. It doesn't need to be regular. So are we requiring anything for j zero to begin with? I don't have to require really anything about j zero because I'm going to perturb it in a certain region, and I'm only looking at holomorphic curves that pass through that region. So near each puncture, in holomorphic cylindrical coordinates, I can choose coordinates st, which are in a half cylinder. I'm going to just call it a positive half cylinder if it's a positive puncture, negative half cylinder if it's a negative puncture. And then the requirements are that the map f just looks like T s plus a constant gamma of T t plus a constant in the cylindrical end r plus minus times m plus minus. So here, c1 and c2 are just some real constants. And what do you mean by gamma plus a constant? I haven't. I'm about to say that. Oh, sorry. Well, gamma will be valued in s1. You can add constants in s1. I thought gamma was an m. Very good point. Is that better? Thank you for that. So gamma is going to be a t periodic orbit of the ray vector field. And then what I say about h is simply that in these cylindrical coordinates, where also I'm assuming I've chosen some kind of sensible asymptotically translation invariant trivialization of this tangent bundle, I want to have h on that end be of class w, k, p on the cylinder, the half cylinder. And so I haven't said anything about what k and p are. I'm not really going to make any requirement about that, except k times p should be greater than 2. Because then the Sobolev embedding theorem tells me w, k, p injects continuously into c0. So these maps are at least always going to be continuous. If they weren't continuous, I'd have real troubles making this a good definition. So you don't require any kind of exponential decay for h or anything just? I'm getting to that. I'm getting to that. So I'm confused because if we're looking at a neighborhood of u0, shouldn't we be writing things like neighborhood for x sub u0, like something? Yeah, but I'm going to, well, u0 is going to be a map of this form. And I'm going to consider other maps of this form that are close to u0. I mean, I haven't localized yet. This is still global. Yeah, the bracket's still open. Oh, very good. Wouldn't want to have some sort of general way tech error. Can you explain what's being ruled out by this list of conditions? But I'm ruling out maps that don't have decay at infinity to some very sort of, well, f is a very controlled map. It has this precise cylindrical form. I'm only considering maps that have asymptotic decay to a map of that form. So I don't want them to move around non-trivial amounts arbitrarily far away. OK. All right. Agucin kind of preempted the next little variation I have to put on this. And I won't be able to explain why just yet, but let's just say it. For a parameter delta, I can let, I'm going to call this b. This is the actual bonnack manifold that I'll work with. So this is going to be, again, u equals x belong f of h as above. But the condition on h will be slightly different. I'm going to require that the norm of h, not the norm of h itself, but the norm on the half cylinder of e to the delta s times h is finite. So that enforces some exponential decay in addition to the decay that the wkp norm already requires. So I'm not going to tell you yet why this is necessary. I will tell you that eventually. I will tell you why it's possible, which is called this fact, though I guess it's sort of a hard theorem. First version of it proves by Hofer-Visatzian-Seindor. And there's been other versions in various contexts since then that if the asymptotic orbits are non-degenerate, then for all delta sufficiently small, all asymptotically cylindrical, or let's just say, all curves in the moduli space I've defined belong to this Bonac manifold. That is, the map involved in defining that curve belongs to that manifold. It has the asymptotic exponential decay, as long as delta is sufficiently small. Hey, Chris, I'm in the back row. I can't quite see. Are there absolute value signs around that s in the exponential? No, but there should be. Thank you. Let's make it a plus or minus. I want a minus sign if I'm at a negative puncture. The point is that it enforces asymptotic decay, or exponential decay, rather than exponential growth or something. What can I sensibly say in the zero amount of time I have left? You have like three minutes. My question stands. I need to. Sorry. The more force you have. That would be remarkable. So in that last actually, is it during the delta? Doesn't it depend on the components of MLK? Yeah, it does. Yeah, to say this properly, I would have to say that given a specific set of asymptotic orbits, there is a delta sufficiently small so that all curves that are asymptotic to those orbits satisfy the decay condition. OK. So let me try and at least get to the definition of Fredholm regularity in some form. What I'm going to do now, I've defined a class of maps I want to work with. I also have to worry about the complex structure and the domain, because that's not fixed. It's not some sort of God given data to work with, but I allow it to vary in different points in this modular space of different complex structures. Now, I'm looking at just a neighborhood of some u0 with complex structure j0, which is it still visible somewhere on the board? No. There it is. So let's just worry about the complex structures that are close to j0. And one way to do that, I don't have to consider everything in this infinite dimensional space of complex structures. But if I want to cover all of them that are not equivalent by diffeomorphisms, I can think about Teichmühler space. That's at least a nice smooth manifold. So what I'm going to do is say let t be a smooth finite dimensional family of complex structures on sigma, which I can, if I want to, I can even assume that they are fixed near gamma and theta. Another thing, I can just take a model surface sigma and fix punctures in one particular place, or several particular places, fix also the marked points in particular places. I don't need to allow them to move around. Because if I want to get all the different conformal structures up to equivalents with these punctures and marked points, all of them can be represented by some complex structure that have the punctures and the marked points in that particular place. That's just a question of choosing a diffeomorphism. So I'm fixing gamma and theta in place on sigma, but I'm letting the complex structures vary. And this is going to be a smooth finite dimensional family that parameterizes a neighborhood of the equivalence class of J0 in Teichmühler space. What's that? I don't understand what you just said. Suppose that it is like w, it ends at empty, and sigma is the sphere. Yes. Then you're not varying the complex structure at all, because you can, but you say no to forget any configuration points that you want. Yes, I can. But I have to vary the complex structure. Because a complex structure is just a section of the tangent bundle, all the morphisms of the tangent bundle are all diffeomorphic. But you can certainly vary them. And also, they're not all diffeomorphic if there are more than three marked points. No, right. So even on the sphere, I have to do this sometimes. So I'm parametrizing a neighborhood in Teichmühler space, which is the space t sigma dot of theta of all complex structures on sigma divided by the identity component of the diffeomorphism group preserving or fixing punctures and marked points. So it's a classical fact that this is a finite dimensional manifold. So I can choose some sort of finite dimensional space of complex structures parametrizing it. And there are various sensible ways to do this. I'll talk about tomorrow if I have time. There's a quick lemma. Let's also assume, just for simplicity at the moment, that the Euler characteristic of the punctured surface minus the marked points is negative. That's, Dusser also talked about this a little bit. That's the stable situation. So that's the situation in which the automorphism group of the domain with its punctures and marked points is finite. It's not an absolutely necessary assumption for what I'm talking about, but it does make a few things easier. So I can, in particular, say there exists a choice of this object, T. I'm going to, in the future, call this a Teichmiller slice since it parametrizes a neighborhood in Teichmiller space. So that is invariant under the action of the automorphism group of the domain. So having made this choice, let me finally write down the point of theorem one. We now can define a smooth section of some Bonnack space bundle. The domain is going to be the product of a Teichmiller slice with my Bonnack manifold of maps. Section is called B bar j. So the fiber over j comma u is going to be the space of wk minus 1p sections of the bundle of complex anti-linear maps from T sigma j to pull back tangent bundle. I hope that's the right number of parentheses. If not, somebody will surely let me know. And we're going to take j comma u to the obvious thing, tu plus big j composed with tu, composed with little j. If you tell me I need to put a 1 half in there somewhere, I will tell you to get a life. And so finally, so. How about changing the T for a D? My own PhD students need to be especially careful in moments like that. So. Analyzing notation using T. That's fine. I go back and forth. So the lemma is that the automorphism group of u acts on the zero set of this section by the obvious phi star j u composed with phi such that a neighborhood of u in the modular space is in bijective correspondence with a neighborhood of the equivalence class. Sorry, I should have said neighborhood of u naught in the modular space with the equivalence class in this automorphism group of u naught, j naught u naught in the zero set modulo that action by the automorphism group. So now what does this really say? By the implicit function theorem, mj is going to be an orbital near u naught with isotropic group equal to the automorphism group. If that zero section is a manifold, which is going to be true if the linearization of the section is subjective. So that's some linear operator from the direct sum of a finite dimensional space tangent to a space of complex structures with, well, let's just write it as tangent space to the Bonnack manifold, which is some space of sections of the pullback tangent bundle of class wkp. Yes? Yes. It's probably delta. Sorry? It should be a delta on your e in the definition of e. Oh, you're right. Thank you very much. If you're told he wants to have a life, it's lunch. This is the moment where I say I thought you people cared about mathematics. I've already said so. OK. Basically, so I wanted to at least end with having written down this operator. So Fredholm regularity means, specifically, I need to finish the sentence and say is surjective. So Fredholm regularity means just that. This operator is surjective. I'll talk more tomorrow about what one needs to do to make sure that that operator is surjective. The result is then we identify local neighborhoods in the modular space with this thing, which is a finite dimensional manifold divided by a finite group. I'm not going to say anything about the proof of the lemma except that if you stare at it long enough, it becomes obvious. It's not that hard. So I'll stop there for now. So there is office hours. So I again suggest longer questions, get postponed until then. So anyone have a short question that they're willing to ask? Everyone? So good to practice. I'm going to show that something is regular, right? What do you need to take into the slides and bury your complex structure? Right. OK. It's a good question. If you want to show that something is regular, well, the answer is I've never been in a situation where I had to actually choose a type of slice concretely. But it has to be there in the theoretical setup, because if it's not there, right? I mean, it would obviously be sufficient if the linearization of this operator, just mapping this space of sections to this space of sections, is surjective. That would suffice. That would imply that this one is surjective. The problem is that that's not true as often as you want it to be. You have this additional finite dimensional stuff in the domain, which sometimes actually makes the difference between an operator with a finite dimensional co-coronal and one that's surjective. So in fact, so if I have time to talk about automatic transversality tomorrow, hopefully, it's a really crucial detail there. The criterion that used to prove automatic transversality does not work if you leave out this whole discussion of type familiar space, because you need to have this freedom to vary the complex structure in order to get as much transversality as you need. Do you have an actual example of the curve which is regular in this sense and not regular if I take that particular space? I could probably come up with that. Ask me in office hours. And you have a discussion tomorrow. And I have a discussion tomorrow. In addition to your lecture. You can certainly find examples if you take chloride on stuff with a pi of t. This will definitely need to vary. Ties will especially vary. What is the fact that you take a finite dimensional balance just all by a new? If I took all of them, this operator would not be Fred Holm. That would be bad. So here you're thinking of fixing big j, which is the complex structure on the target. So far, I'm fixing big j. And your space of j here. I was using a curly j to mean variations of j on the target space. But you're fixing that and making your j as a finite dimensional. I have two curly j's. Well, you have two curly j's. Only one on this board. You see, your curly j there is a type of a slice. But you could, in principle, vary the other j on the target space if you want it to. Yes. You mean when I'm talking about finite dimensional families or you're just generic? I mean, in year 0 of 1, you were saying that you were taking generic elements where you were allowed to vary in the target in the open subset, u of your target, right? Yeah, that's right. That curly j is not actually in this limit. Here you're sometimes assuming you fix that. Right. No, I mean, the question of why you get transversality for simple curves for generic j, I will address tomorrow. So far, all I've talked about is what criterion actually gives you smoothness of the moduli space. And how does one go about proving that? One thing to say is why do you need to vary j and type on the spaces? For example, if you look at the space of tori in CQ2, and look at, you say, you know, the class of DQC, embedded tori. You look at the induced complex structure on the tori. That changes. So if you fix the complex structure on the tori, it's not necessarily going to be regular. Because when you move your points around, you leave the color on j on the domain, naturally change. So you have to have that variation in. Without further ado, I propose we break for lunch. And at 2 PM, I'll make some short remarks about organizational things like finances and stuff. So I'll see you at 2. Thank you.
 Thanks, Joe.  I'm going to start on this board with just a preface  of generalities.  Let's call this the archetypal moduli problem, by which  I mean from the perspective of global analysis  or differential topology or simplexed topology  as distinct from algebraic geometry, where they also  like to talk about moduli problems.  So for me, a moduli problem means this kind of thing.  You have some space which I'm calling suggestively J.  But right now, I'm just going to call it the space of data,  which has some kind of topology on it.  And for each choice, which I will suggestively call also J,  in this space, we get some moduli space, M of J, which also  has a natural topology.  And of course, we would like to say  that it has some much nicer structure than a topology,  like being a smooth manifold.  Or if we can't achieve that, maybe at least a smooth orbifold.  And over the course of these two weeks,  I'm sure you'll also hear things about weighted branched  orbifolds with boundaries in corners, though not for me.  From my perspective, well, the first thing  to say about this moduli space is that locally,  you can identify it with some kind of zero set, which  I'll write sigma J inverse of 0, except not precisely that.  Often you have to divide it by some kind of symmetries.  So here, sigma J is some section of a Banach space bundle.  And we already heard DUSA sketch one example of this.  Because you can express all sorts of geometric PDEs  in this way as preferably smooth sections of Banach space  bundles in an infinitesimensional context.  But I'm saying this is true only locally in general.  So DUSA described the situation with holomorphic spheres  where you can actually do this globally.  But in my experience working with J-holomorphic curves,  that's rather the exception to the rule.  And so I will talk about some more general situations  where you can do this but only locally.  And these symmetries can cause quite a lot of headaches.  But the first problem, of course,  is you want to be able to say that this zero section here  is a smooth object.  And the implicit function theorem  ought to tell you that if you do things correctly.  But there are two general ways that you can approach that.  First, the so-called abstract approach  where you perturb the right-hand side of the equation.  You can say something like define the moduli space m tilde,  which depends on the data J, and some extra auxiliary choice  that are called nu.  That's the space of all u in this Bonnock manifold,  such that sigma J of u matches nu of u.  So this is for some other section nu of that same Bonnock  space bundle.  And the idea is if we choose that generically,  then this set of solutions will be a smooth object simply  by the Sard-Smale theorem.  That's kind of equivalent to the statement just  expressed in local trivializations  that a generic point in the target of a smooth map  is a regular value of that map.  So this is a generic section.  It also intersects this section transversely.  So for generic choices of this auxiliary nu, our special section  sigma J is transverse to that, which  means this moduli space of perturbed objects I define  is a manifold.  And I even want to say more because if we're  talking about elliptic problems, this section  is not just a smooth section, but also is Fredholm  in the sense that its linearization at any point  is a Fredholm operator.  So when we have these kinds of transversality conditions,  we get just not just a manifold, but a finite dimensional  manifold because of finite dimensional kernels  of the linearization.  So that's, of course, very nice when that happens.  The problem is I perturbed my equation  to a different one, which might not be the one I'm actually  interested in.  That's problem number one.  There are more problems, which I think  you'll hear more about over the course of the two weeks  because this abstract approach is sort of the raison  d'etre for the whole polyfield project.  But it's difficult.  And part of the reason it's difficult  is that, first of all, this correspondence  between some zero section and my actual moduli space of interest  is in general only local, and it also has these symmetries.  You have to deal with both of those in a sensible way  when choosing these perturbations.  And it means just choosing some generic section of a Bonoq space  bundle is usually not enough.  You have to choose it satisfying some conditions.  And maybe those conditions are too stringent to actually apply  the Sards-Mail theorem in this way.  So do you want to be like invariant under the symmetry  to something?  Something like you want various things of that sort, yes.  I reserve the right to not talk about that anymore  because other people will.  And I'm going to talk about the alternative, which  is in the title of my mini course, so-called classical,  or what many people call geometric,  perturbation methods, which is the notion  that if you take a generic choice of the data  in the space of data, then your actual section  will be transverse to the zero section, which  means, of course, the zero set of your section  is a smooth manifold and finite dimensional as well,  because we're in the Fredholm context.  And well, the space we're actually interested in  is that divided by some kind of symmetry.  That, in general, will be an orbifold.  So an orbifold is something that looks locally  like Euclidean space divided by a finite group action.  And that's what you get if you're taking  a finite dimensional manifold and dividing it  by the kind of symmetry groups that we're interested in.  So that is minor headache number one to be aware of.  We won't get manifolds in general,  but we could get orbifolds if we're lucky.  Much larger headache number two is  that often we're not lucky and this condition simply  isn't true.  And Duce already talked about that a little bit.  I'm going to talk about that some more.  So that's all I will say about generalities.  I'm going to focus on this second, this classical approach  over three talks.  And in a particular context, I want  to discuss symplectic field theory.  So in the original propaganda paper in 2000  by Elishberg Giventhal and Hofer,  you have symplectic field theory presented  as a very general way of defining  invariance of contact manifolds and symplectic  abortisms between contact manifolds.  And well, one can do it even a little bit more generally  than that, which is sort of nice for certain purposes.  So I'm going to talk about stable Hamiltonian structures.  So first of all, if m is an odd dimensional manifold,  the stable Hamiltonian structure H on m  is going to be a pair capital omega lambda, which  consists of the following.  We have first omega is a closed two form of maximal rank.  So you should imagine that as something  that arises on any hyper surface in a symplectic manifold.  You just take the symplectic form  and restrict it to a hyper surface,  it's going to be a closed two form of maximal rank.  If I omit the word stable, that's  what we call just a Hamiltonian structure.  Now stable comes from this additional one form lambda,  which satisfies following properties.  First, lambda wedge, the top dimensional power of omega,  must be a positive volume form.  So this implies in particular that you  can take omega and restrict it to the kernel of lambda.  Lambda is nowhere zero.  So its kernel is a hyper plane distribution,  which I'm going to call psi, as I always do with contact  structures, because that's one important example.  Could be a contact structure, but in general, it's just  some hyper plane distribution.  And omega makes that hyper plane distribution  into a symplectic vector bundle.  Gives us a symplectic bundle structure.  And then the other condition, since omega  has maximal rank on an odd dimensional manifold,  it has a kernel that's always one dimensional.  That's the so-called characteristic line field,  the one dimension of directions in which omega is degenerate.  So what I'm going to require is that d lambda also  vanishes on that same kernel.  Another way of saying it is that the kernel of omega  is contained in kernel of d lambda.  OK, so this is a definition which  has been around for several years.  And as stated, it's not too hard to understand,  but I think a lot of people don't really  understand what it means.  So let me talk a little bit about that.  First observation is that if I now  take this odd dimensional manifold  and look at a small cylinder constructed out of it,  so some small interval times m, this thing now  inherits a natural symplectic form.  Let's call the real coordinate little r  and define little omega to be d of r lambda plus big omega.  And that's an easy exercise.  That's going to be a symplectic form.  As long as this cylinder is sufficiently small,  it might even be symplectic for epsilon larger than that.  But in general, I can only guarantee  this is true for epsilon small.  Moreover, what does the characteristic line  field look like?  If you restrict, you've got a one parameter family  of hyper surfaces.  Let's say m sub r is the hyper surface r times m  sitting inside that cylinder.  The characteristic line field of omega restricted to m sub r  is independent of r.  So it's the same characteristic line field for all r,  all parameters in my parameterized family of hyper surfaces.  That's really the original definition, which I think  appeared first in the book by Hofer and Sander  of a stable hyper surface.  That's where this idea comes from.  It's the kind of situation in which  you would expect to be able to prove something  like the Weinstein conjecture that  asserts existence of closed characteristics  of this characteristic line field.  Because if you have them on some hyper surfaces in the vicinity,  you have them on all of them.  Excuse me.  What do you mean with the first symbol of the last line?  The first symbol?  That's an and, and percent.  So in order to get the stable condition  that you want this last condition at the kernel of omega  containing the kernel under, that's what implies.  That's what implies this fact about the characteristic line  fields, exactly.  And conversely, if you are given a one parameter family  of hyper surfaces that has this property,  this stability of the characteristic line field,  then you can always find a stable Hamiltonian structure  that produces that.  So that's kind of an exercise using the Moser deformation  trick.  How much will I lose if I just ignore this definition  pretend you said contact structure?  Nothing really.  That's a perfectly good point.  If you don't like this definition for some reason,  because it's new, just think lambda is a contact structure  and big omega equals d lambda.  I was going to say that soon anyway.  So there is a, from this data, we  have also a canonical generator of the characteristic line  field, which by analogy with contact forms  we'll call this the rape vector field.  So my notation will be r sub h.  And it's determined by the conditions.  Omega annihilates the direction of the rape vector field.  And then we normalize it by lambda.  Is there any reason d lambda couldn't be zero?  No.  There's no reason at all why d lambda couldn't be zero.  I will also say that again in a moment.  So we have a stable Hamiltonian structure.  It defines this hyperplane distribution psi, which  might be a contact structure in certain examples,  but it doesn't have to be.  Could also be a foliation.  We have this rape vector field.  Another thing we can do is, since we  have this small symplectic cylinder we built out of it,  we can also just arbitrarily expand  that to r times m and rewrite my symplectic structure there  in the form omega phi.  That's going to be d of phi of r lambda plus omega.  So what's phi?  Phi is going to be in the space of smooth functions taking r  to that same small epsilon interval  with strictly positive derivative.  So this is a symplectic form also.  OK?  It's symplectic to the one I wrote down before.  I've just stretched things.  But the point is that r cross m is the kind of setting  where I would like to talk about j-holomorphic curves.  And I can do that now by saying, well, the space  j dependent on the stable Hamiltonian structure  is going to be the space of almost complex structures,  j on r cross m, which I will assume always to be r invariant.  In other words, there's this r translation action on r  cross m.  I want that to preserve all my j's.  I also want to say that j maps the unit vector  in the r direction to the rate vector field.  And I want to say that j preserves my hyperplane  distribution as well as the symplectic structure.  It's compatible with the symplectic structure on that.  So let's say j restricted to psi is omega compatible.  OK?  So here's a lemon which is very easy to prove  and helps justify all these definitions.  That is that for all choices j in this space of translation  invariant, almost complex structures,  and all functions phi, as I indicated  above, that are mapping to that small interval  with positive derivative, it's not just  that omega phi is symplectic, but in fact,  it's compatible in the usual sense of Gromoth  with this choice of j.  So I can fix j and I can then allow any choice of phi I want.  And that's an important detail for defining things  like the energy of a j-holomorphic curve,  because we then take that to be the supremum of symplectic area  defined with respect to all these different choices  of symplectic forms.  So let me give two examples.  They've already sort of been mentioned.  First of all, if alpha is a contact form,  we get a stable Hamiltonian structure d alpha, comma alpha.  And then this symplectic manifold r cross m  with this kind of symplectic structure  I've been talking about.  I'm not going to try to say these two precisely.  You have to make some kind of condition on how you define phi  in order to say this.  But I'm just going to say one can do it.  So this is what's called the symplectization  of the contact manifold.  I really should relax the epsilon's in order to say that.  I should actually, should at least make plus epsilon  to be infinity.  Let's not worry about this right now.  Yeah.  Can you quickly write down what the symplectization is?  No.  I'm going to say it's this.  Yes?  So what's the condition where you can integrate this hyperplane  distribution to a symplectic manifold?  Where you can integrate the hyperplane distribution.  Yeah, if you integrate it, and then you  find a sub-manifold, which has that as its tangent form.  Well, for the hyperplane distribution to be integrable,  I think that's equivalent to saying that d lambda  vanishes on psi.  So certainly it's sufficient of d lambda vanishes  altogether, which is my next example.  Already in the first row, we think that the word compatible  sort of looks like compacted.  Well, I'm sorry.  I'll take this opportunity to make a blanket apology  for my handwriting, as I customarily do.  There isn't that much I can do about it.  OK, compatible.  I just said, i actually looks like compacted.  Compacted.  All right, second example.  If m equals the product of s1 with some other manifold w,  and I take w to be a symplectic manifold.  So let's say I have a symplectic form little omega on w,  and then set capital omega to be that plus dt wedge dh.  So here, t is the s1 coordinate, and I'm  choosing some function h, a Hamiltonian function, which  is dependent on time in a periodic way.  So this is capital omega, and I take lambda to be just dt.  This is a case where it's closed.  d lambda vanishes entirely.  So now one can check.  This also is a stable Hamiltonian structure.  And it has a meaning which a lot of people will probably like.  You can compute the rate vector field for this.  It's simply, it's the vector field in the t direction,  plus a vector field in the w direction, which is precisely  the Hamiltonian vector field.  So this is a way to turn questions about Hamiltonian vector  fields on symplectic manifolds, even time dependent ones,  into questions about a time independent,  rave like vector field on an odd dimensional manifold.  And it gets better because for the class of almost complex  structures that we've talked about,  we can use the same method.  So we can use the same method.  We can use the same method.  We can use the same method.  We can use the same method.  We can use the same method.  So the class of almost complex structures that we've talked  about, a j-holomorphic curve in r times this,  is basically equivalent to a solution of the Flur  equation in w.  So this is a way that you can do Flur homology,  Hamiltonian Flur homology, from the perspective  of symplectic field theory.  OK.  So now the setting in which I really  want to think about holomorphic curves  is also a bit more general than this.  So let's take a symplectic manifold.  So here I'm talking about co-boardisms  with stable boundary.  So one can define stable boundary in various ways.  The main thing I want to say is there  exists near the boundary of one parameter family of hyper  surfaces that all have the same dynamics on them,  same characteristic line fields.  And that's equivalent to saying that there exists  this collar neighborhood near m plus  that looks like a small interval times m plus,  where the symplectic structure is just the kind that I wrote  down already, d of t lambda plus plus omega plus  for some stable Hamiltonian structure, omega plus lambda  plus on m plus.  Of course, I'm going to do the same thing on m minus.  But the collar neighborhood will look like an opposite  interval.  OK?  So this is the definition of a symplectic co-boardism  with stable boundary.  And having done that, I can add cylindrical ends  to this compact object and get a non-compact object.  That's what we call the completion.  So I'll denote that by w hat omega hat.  It's going to depend on some choice of function  phi of the sort that I used to define  symplectic structures on simpletizations.  And that choice of phi is not particularly meaningful.  It's just nice to know that I can do it.  So the definition is we take a negative cylindrical end,  glue that along m minus to w, glue that along m plus  to positive cylindrical end.  And the symplectic structure here,  we take omega phi, omega phi on each of the ends.  And the original omega on w, you can choose the function phi  such that, so remember, I still have this definition  on the board somewhere.  Middle board top right.  Middle board top right.  Very good.  d phi times lambda plus omega.  If I choose phi so that it equals just the identity map  close to 0, then I can obviously glue these cylindrical ends  smoothly together with these collar neighborhoods.  That's what I'm doing.  OK.  So with that choice of symplectic structure,  and I don't really have to specify phi,  I just know that I can choose one.  Are you using the same phi on the both ends?  I don't care.  Actually, I specifically want to not specify phi  and allow myself the freedom to change it.  So let's say j of omega and the two stable Hamiltonian  structures.  This is going to be the space of all almost complex structures,  j on the completion w hat, which I assume to be everywhere  compatible with, well, let's just say,  compatible with omega on the original compact object  and belonging to these spaces with respect  to the stable Hamiltonian structures that I already  defined.  So the spaces of translation in very almost complex structures  on the ends.  So that now is going to be an almost complex structure  that's compatible with omega hat phi no matter what phi I choose.  OK.  So now to start talking about the actual J-Hollomarket curves.  Any questions so far?  OK.  So this you require that J's, what does it say?  Translation.  Compatible.  Translation.  Translation very on the end.  So what about in the co-boardism part?  I mean, don't you have a factor r there anyway?  When the co-boardism part, I don't have a factor of r  except close to the boundary.  Yeah.  OK.  Yeah.  And I don't make any requirement there  except compatible with the given symplectic form.  OK.  So the kinds of holomorphic curves  we're going to talk about are as follows.  We take sigma J to be a closed Riemann surface.  Let's actually say there's going to be two finite disjoint  ordered subsets.  And in particular, sigma dot always denotes sigma  with the first finite ordered subset removed.  That's a punctured Riemann surface.  We take that, also this subset.  I don't have to worry about it too often,  but it gets partitioned into the so-called positive  and negative punctures.  Each of those could be empty, but each puncture  belongs to one or the other of those.  And now I think about maps from this Riemann surface  with the various points removed going  into my completed symplectic co-boardism.  And I want them to be what I call asymptotically cylindrical,  which means, well, let's not write down  the precise definition.  Let's just say asymptotic near each positive or negative  puncture z to R times a Raib orbit in m plus minus.  Chris, you're missing a hole somewhere.  I'm missing a hole.  That's true.  Could be worse.  The opposite would sound like a medical emergency.  So what I mean is essentially, for this kind of almost  complex structure I'm considering,  with the translation invariant thing satisfying  these conditions on the ends, I have  j maps the vector in the r direction to the Raib vector  field, which means r times a Raib orbit is always  a holomorphic cylinder in the end.  So those are special holomorphic curves  that always exist, at least in that region, maybe not globally.  And I want to consider punctured holomorphic curves  whose behavior near each of those punctures  approximates one of those cylinders better and better  as you go toward the puncture.  So to make it precise, I would say  you can choose some holomorphic cylindrical coordinates  near each puncture.  And in those coordinates, as you go further up  toward infinity on the cylinder, your curve  looks more and more like one of these trivial orbit cylinders.  OK, maybe it's time to recover this.  Yeah?  In your picture, theta is empty or what's happening?  Theta, I haven't really said.  Theta doesn't have to be empty.  It's just some finite subset right now.  But in the picture, it doesn't appear in the picture  because it doesn't affect the picture in any way.  It's just points.  So theta are meant to be marked points.  All right, so the actual moduli space  looks like the following.  It's going to be a space of tuples sigma j gamma theta u,  where, well, sigma j is a closed Riemann surface,  as I wrote above.  Gamma and theta are these disjoint ordered finite subsets.  So let me not rewrite that.  Mainly, I need to say u maps the punctured surface  to the completion w hat j holomorphically  and asymptotically, solidically.  And then I just need to say what my equivalence relation is.  Deuza basically already said it.  You could deduce completely the right answer to this  from what Deuza said.  I will say a tuple sigma j gamma theta u  is equivalent to a tuple sigma prime phi star j phi inverse  of gamma preserving the ordering phi inverse of theta u  composed with phi for any diffeomorphism phi from sigma  prime to sigma.  But doesn't the ray orbits allowed to be arbitrarily terrible?  Yeah, this definition is still OK if I  allow the ray orbits to be arbitrarily terrible,  but I won't have any theorems.  Yeah.  I do have, so finally to make some use out of the mark  points, I have an evaluation map sending this moduli space  to w hat m where theta, the mark points,  are labeled zeta 1 through zeta m.  So this just evaluates u at those points.  And it's well defined because of the way  that I define my equivalence relation.  These diffeomorphisms map the set of marked points  to the set of marked points.  All right, so let's finally state a couple of theorems.  So theorem one says, if I have, I'm just  going to abbreviate elements of this moduli space by u  instead of writing out the entire tuple  and saying equivalence class and everything.  So if u is what I call Fredholm regular,  that's a condition I'll have to explain a bit,  and it also includes the condition  that the asymptotic ray orbits are not arbitrarily terrible.  Then a neighborhood of u in the moduli space,  is a smooth finite dimensional orbital with isotropy at u,  isomorphic to the automorphism group of u.  So I didn't write down this definition yet.  The automorphism group of u is simply  the space of all bi-holomorphic maps of its domain,  which fix the punctures and the marked points  and satisfy u equals u composed with phi.  So unless u is constant, that's always  going to be a finite group.  And for somewhere injective curves,  it's going to be a trivial group.  But for non-somewhere injective curves,  so for multiply covered curves, if we get this Fredholm  regularity condition, whatever it is,  we're going to have to deal with an orbifold rather  than a manifold because of this isotropy.  Is this going to be a centric of tautology  on the word regular?  In some sense, this is a tautology on the word regular.  I mean, one does have to work a little bit  to see why you can make this local identification  of the moduli space with something  that looks like an orbifold even if you have regularity.  And I will at least have time to get to that in this talk,  I think.  Therm two, I'm going to probably postpone  the proof of this for tomorrow.  But I'll state it now.  Let's say let's fix an open subset u with compact closure.  Then I'm going to say there exists a bare subset called  j u reg.  Well, it's a subset of the space of all.  So I needed to fix some more data in advance.  So fix also an almost complex structure,  j0 of the type I've described.  Then there exists a bare subset.  So this is a subset of the space of all.  J's of the type I've described with the extra condition  that j matches j0 everywhere outside of this special subset  u.  So this special subset u we can call the perturbation domain.  It's where we're going to allow ourselves  to change the almost complex structure,  but we leave it fixed everywhere else.  So such that for all j in this j reg u,  let's finish it over here, every somewhere  injective curve u in the moduli space with respect to j,  passing through my special subset capital u is  thread home regular.  But here bare subset is again a commuter.  Bare subset literally means commuter, yes.  So what did I call residual?  Yeah, I never understand residual.  It sounds to me like it means the opposite,  so I don't use the word.  Some people call it second category, which I think  is also not technically right if you look up  the actual definition of second category.  Yeah.  Thank you for that input, Catherine.  So yeah.  Well, someday some kind of sensible general habit  will prevail, hopefully.  I say bare subset, and most people don't seem to mind.  So the literal meaning is countable intersection  of open and dense sets.  What contains, actually, if you're being strict,  it contains.  It contains a countable intersection  of open and dense sets.  Very good.  Maybe more useful input is though somehow you,  I think you're assuming here now  that the Hamiltonian structure is sort of regular.  Well, that's in, oh, in theorem two.  I am.  That's true.  Thank you.  OK.  So the board's out of reach, so I'm not going to write it.  Let's just say this.  One additional condition you need in theorem two  is that, well, let's say all the closed  Ray-Borbets for both stable Hamiltonian structures  are non-degenerate.  So under those conditions, theorem two holds.  Otherwise, you could have trouble.  And at least for if your stable Hamiltonian structures  are contact forms, that's a condition which is generic.  That's a standard result.  Can you also do this if you're Ray-Borbets or more spot?  More spot is also fine.  Yeah.  More spot or non-degenerate are perfectly fine.  OK.  So let's talk a little about the functional analytics  setup for a neighborhood of, let's say  I have a specific element of the modularized space  with complex structure, J0, and the map is called U0.  Just to pretend the statement of theorem two.  You're not asking that it be somewhere  injected inside U. It has to be somewhere  injected somewhere and pass through U.  Those two things are equivalent, which is not so hard theorem.  So what I really should ask for is  that there is an injective point mapped into U.  And it's a good point because if I  were working in other settings where  I have boundary with Lagrangian boundary conditions  or something, then it's not necessarily  true that just because the curve is somewhere  injective that its injective points are dense.  So then I would have to be careful.  But here, injective points are dense  if the curve is somewhere injective.  I think you've hidden the end of the statement.  Can you rise the board and choose?  Ah.  The statement is on the sideboard.  Oh, it's this one.  It's this one.  Then you can block that board so it's up with the theorem.  This is called democratic blackboard technique.  All right.  So I need to actually write down some kind of Bonnock  manifold of maps that will contain the solutions  I'm interested in.  This is something that's kind of standard in the case of closed  maps from closed Riemann surfaces.  For punctured curves, I guess it's  considered sufficiently standard that nobody really  feels they need to go through the details when they write papers.  But it means there is no paper where anyone does the details  really as far as I think.  In Matthias Schwartz's thesis, I think there may be details  about this, and that's possibly the only place.  There's also this paper by the problem of traveling.  We know how to paper this.  No.  No.  Well, I mean, what he does in there  is basically just lifted from Schwartz's thesis.  OK.  So I'm going to talk about a space of sub-elevated class  maps from the punctured surface to W hat, class WKP.  So I want to have k derivatives that are of class LP locally.  Locally, that would be the definition,  and it's not hard to express.  But globally, what does this mean?  Easiest way to say is U is of the form x along f of h.  h, where f is going to be some smooth map, which is not just  asymptotically but literally cylindrical near the ends.  Well, I'll say that in a moment.  h is going to be some section of class WKP locally  along the pulled back tangent bundle  along f.  So in compact subsets, this is a sufficient definition,  but I need to say more specifically what happens at the ends.  So I just have confidence.  So the j zero that you started with is regular, right?  Or that?  No.  I don't care.  It doesn't need to be regular.  So are we requiring anything for j zero to begin with?  I don't have to require really anything about j zero  because I'm going to perturb it in a certain region,  and I'm only looking at holomorphic curves that  pass through that region.  So near each puncture, in holomorphic cylindrical  coordinates, I can choose coordinates st, which  are in a half cylinder.  I'm going to just call it a positive half cylinder  if it's a positive puncture, negative half cylinder  if it's a negative puncture.  And then the requirements are that the map f just  looks like T s plus a constant gamma of T t plus a constant  in the cylindrical end r plus minus times m plus minus.  So here, c1 and c2 are just some real constants.  And what do you mean by gamma plus a constant?  I haven't.  I'm about to say that.  Oh, sorry.  Well, gamma will be valued in s1.  You can add constants in s1.  I thought gamma was an m.  Very good point.  Is that better?  Thank you for that.  So gamma is going to be a t periodic orbit  of the ray vector field.  And then what I say about h is simply  that in these cylindrical coordinates, where also I'm  assuming I've chosen some kind of sensible asymptotically  translation invariant trivialization  of this tangent bundle, I want to have h on that end  be of class w, k, p on the cylinder, the half cylinder.  And so I haven't said anything about what k and p are.  I'm not really going to make any requirement about that,  except k times p should be greater than 2.  Because then the Sobolev embedding theorem tells me  w, k, p injects continuously into c0.  So these maps are at least always going to be continuous.  If they weren't continuous, I'd have real troubles  making this a good definition.  So you don't require any kind of exponential decay for h  or anything just?  I'm getting to that.  I'm getting to that.  So I'm confused because if we're looking  at a neighborhood of u0, shouldn't we  be writing things like neighborhood  for x sub u0, like something?  Yeah, but I'm going to, well, u0  is going to be a map of this form.  And I'm going to consider other maps of this form that  are close to u0.  I mean, I haven't localized yet.  This is still global.  Yeah, the bracket's still open.  Oh, very good.  Wouldn't want to have some sort of general way tech error.  Can you explain what's being ruled out  by this list of conditions?  But I'm ruling out maps that don't have decay at infinity  to some very sort of, well, f is a very controlled map.  It has this precise cylindrical form.  I'm only considering maps that have  asymptotic decay to a map of that form.  So I don't want them to move around non-trivial amounts  arbitrarily far away.  OK.  All right.  Agucin kind of preempted the next little variation  I have to put on this.  And I won't be able to explain why just yet,  but let's just say it.  For a parameter delta, I can let,  I'm going to call this b.  This is the actual bonnack manifold that I'll work with.  So this is going to be, again, u equals x  belong f of h as above.  But the condition on h will be slightly different.  I'm going to require that the norm of h,  not the norm of h itself, but the norm on the half cylinder  of e to the delta s times h is finite.  So that enforces some exponential decay  in addition to the decay that the wkp norm already requires.  So I'm not going to tell you yet why this is necessary.  I will tell you that eventually.  I will tell you why it's possible,  which is called this fact, though I guess  it's sort of a hard theorem.  First version of it proves by Hofer-Visatzian-Seindor.  And there's been other versions in various contexts  since then that if the asymptotic orbits are non-degenerate,  then for all delta sufficiently small,  all asymptotically cylindrical, or let's just say,  all curves in the moduli space I've defined  belong to this Bonac manifold.  That is, the map involved in defining that curve  belongs to that manifold.  It has the asymptotic exponential decay,  as long as delta is sufficiently small.  Hey, Chris, I'm in the back row.  I can't quite see.  Are there absolute value signs around that s  in the exponential?  No, but there should be.  Thank you.  Let's make it a plus or minus.  I want a minus sign if I'm at a negative puncture.  The point is that it enforces asymptotic decay,  or exponential decay, rather than exponential growth  or something.  What can I sensibly say in the zero amount of time I have left?  You have like three minutes.  My question stands.  I need to.  Sorry.  The more force you have.  That would be remarkable.  So in that last actually, is it during the delta?  Doesn't it depend on the components of MLK?  Yeah, it does.  Yeah, to say this properly, I would  have to say that given a specific set of asymptotic orbits,  there is a delta sufficiently small  so that all curves that are asymptotic to those orbits  satisfy the decay condition.  OK.  So let me try and at least get to the definition  of Fredholm regularity in some form.  What I'm going to do now, I've defined a class of maps  I want to work with.  I also have to worry about the complex structure  and the domain, because that's not fixed.  It's not some sort of God given data to work with,  but I allow it to vary in different points in this  modular space of different complex structures.  Now, I'm looking at just a neighborhood of some u0  with complex structure j0, which is it still visible  somewhere on the board?  No.  There it is.  So let's just worry about the complex structures that  are close to j0.  And one way to do that, I don't have  to consider everything in this infinite dimensional space  of complex structures.  But if I want to cover all of them  that are not equivalent by diffeomorphisms,  I can think about Teichmühler space.  That's at least a nice smooth manifold.  So what I'm going to do is say let t  be a smooth finite dimensional family of complex structures  on sigma, which I can, if I want to,  I can even assume that they are fixed near gamma and theta.  Another thing, I can just take a model surface  sigma and fix punctures in one particular place,  or several particular places, fix also the marked points  in particular places.  I don't need to allow them to move around.  Because if I want to get all the different conformal structures  up to equivalents with these punctures and marked points,  all of them can be represented by some complex structure  that have the punctures and the marked points  in that particular place.  That's just a question of choosing a diffeomorphism.  So I'm fixing gamma and theta in place on sigma,  but I'm letting the complex structures vary.  And this is going to be a smooth finite dimensional family  that parameterizes a neighborhood of the equivalence  class of J0 in Teichmühler space.  What's that?  I don't understand what you just said.  Suppose that it is like w, it ends at empty,  and sigma is the sphere.  Yes.  Then you're not varying the complex structure at all,  because you can, but you say no to forget any configuration  points that you want.  Yes, I can.  But I have to vary the complex structure.  Because a complex structure is just a section  of the tangent bundle, all the morphisms of the tangent  bundle are all diffeomorphic.  But you can certainly vary them.  And also, they're not all diffeomorphic  if there are more than three marked points.  No, right.  So even on the sphere, I have to do this sometimes.  So I'm parametrizing a neighborhood  in Teichmühler space, which is the space t sigma  dot of theta of all complex structures on sigma divided  by the identity component of the diffeomorphism group  preserving or fixing punctures and marked points.  So it's a classical fact that this  is a finite dimensional manifold.  So I can choose some sort of finite dimensional space  of complex structures parametrizing it.  And there are various sensible ways to do this.  I'll talk about tomorrow if I have time.  There's a quick lemma.  Let's also assume, just for simplicity at the moment,  that the Euler characteristic of the punctured surface  minus the marked points is negative.  That's, Dusser also talked about this a little bit.  That's the stable situation.  So that's the situation in which the automorphism group  of the domain with its punctures and marked points is finite.  It's not an absolutely necessary assumption  for what I'm talking about, but it does make a few things easier.  So I can, in particular, say there exists a choice of this object,  T. I'm going to, in the future, call this a Teichmiller slice  since it parametrizes a neighborhood in Teichmiller  space.  So that is invariant under the action  of the automorphism group of the domain.  So having made this choice, let me finally write down  the point of theorem one.  We now can define a smooth section of some Bonnack space  bundle. The domain is going to be the product of a Teichmiller  slice with my Bonnack manifold of maps.  Section is called B bar j. So the fiber over j comma u  is going to be the space of wk minus 1p sections of the bundle  of complex anti-linear maps from T sigma j to pull back  tangent bundle.  I hope that's the right number of parentheses.  If not, somebody will surely let me know.  And we're going to take j comma u to the obvious thing,  tu plus big j composed with tu, composed with little j.  If you tell me I need to put a 1 half in there somewhere,  I will tell you to get a life.  And so finally, so.  How about changing the T for a D?  My own PhD students need to be especially careful  in moments like that.  So.  Analyzing notation using T. That's fine.  I go back and forth.  So the lemma is that the automorphism group of u  acts on the zero set of this section  by the obvious phi star j u composed with phi such that a  neighborhood of u in the modular space  is in bijective correspondence with a neighborhood  of the equivalence class.  Sorry, I should have said neighborhood of u naught  in the modular space with the equivalence class  in this automorphism group of u naught, j naught u naught  in the zero set modulo that action by the automorphism group.  So now what does this really say?  By the implicit function theorem,  mj is going to be an orbital near u naught  with isotropic group equal to the automorphism group.  If that zero section is a manifold,  which is going to be true if the linearization of the section  is subjective.  So that's some linear operator from the direct sum  of a finite dimensional space tangent to a space  of complex structures with, well,  let's just write it as tangent space to the Bonnack  manifold, which is some space of sections of the pullback  tangent bundle of class wkp.  Yes?  Yes.  It's probably delta.  Sorry?  It should be a delta on your e in the definition of e.  Oh, you're right.  Thank you very much.  If you're told he wants to have a life, it's lunch.  This is the moment where I say I thought you people cared  about mathematics.  I've already said so.  OK.  Basically, so I wanted to at least end  with having written down this operator.  So Fredholm regularity means, specifically,  I need to finish the sentence and say is surjective.  So Fredholm regularity means just that.  This operator is surjective.  I'll talk more tomorrow about what one needs to do  to make sure that that operator is surjective.  The result is then we identify local neighborhoods  in the modular space with this thing, which  is a finite dimensional manifold divided by a finite group.  I'm not going to say anything about the proof of the lemma  except that if you stare at it long enough,  it becomes obvious.  It's not that hard.  So I'll stop there for now.  So there is office hours.  So I again suggest longer questions, get  postponed until then.  So anyone have a short question that they're  willing to ask?  Everyone?  So good to practice.  I'm going to show that something is regular, right?  What do you need to take into the slides  and bury your complex structure?  Right.  OK.  It's a good question.  If you want to show that something is regular, well,  the answer is I've never been in a situation  where I had to actually choose a type of slice concretely.  But it has to be there in the theoretical setup,  because if it's not there, right?  I mean, it would obviously be sufficient  if the linearization of this operator, just mapping  this space of sections to this space of sections,  is surjective.  That would suffice.  That would imply that this one is surjective.  The problem is that that's not true as often as you want it  to be.  You have this additional finite dimensional stuff  in the domain, which sometimes actually makes  the difference between an operator  with a finite dimensional co-coronal and one that's  surjective.  So in fact, so if I have time to talk about automatic transversality  tomorrow, hopefully, it's a really crucial detail there.  The criterion that used to prove automatic transversality  does not work if you leave out this whole discussion of type  familiar space, because you need to have this freedom  to vary the complex structure in order  to get as much transversality as you need.  Do you have an actual example of the curve  which is regular in this sense and not regular  if I take that particular space?  I could probably come up with that.  Ask me in office hours.  And you have a discussion tomorrow.  And I have a discussion tomorrow.  In addition to your lecture.  You can certainly find examples if you take  chloride on stuff with a pi of t.  This will definitely need to vary.  Ties will especially vary.  What is the fact that you take a finite dimensional  balance just all by a new?  If I took all of them, this operator would not be Fred Holm.  That would be bad.  So here you're thinking of fixing big j, which  is the complex structure on the target.  So far, I'm fixing big j.  And your space of j here.  I was using a curly j to mean variations of j  on the target space.  But you're fixing that and making your j as a finite  dimensional.  I have two curly j's.  Well, you have two curly j's.  Only one on this board.  You see, your curly j there is a type of a slice.  But you could, in principle, vary the other j on the target  space if you want it to.  Yes.  You mean when I'm talking about finite dimensional families  or you're just generic?  I mean, in year 0 of 1, you were saying that you were taking  generic elements where you were allowed to vary in the target  in the open subset, u of your target, right?  Yeah, that's right.  That curly j is not actually in this limit.  Here you're sometimes assuming you fix that.  Right.  No, I mean, the question of why you get transversality  for simple curves for generic j, I will address tomorrow.  So far, all I've talked about is what criterion actually  gives you smoothness of the moduli space.  And how does one go about proving that?  One thing to say is why do you need to vary j and type  on the spaces?  For example, if you look at the space of tori in CQ2,  and look at, you say, you know, the class of DQC,  embedded tori.  You look at the induced complex structure on the tori.  That changes.  So if you fix the complex structure on the tori,  it's not necessarily going to be regular.  Because when you move your points around,  you leave the color on j on the domain, naturally change.  So you have to have that variation in.  Without further ado, I propose we break for lunch.  And at 2 PM, I'll make some short remarks  about organizational things like finances and stuff.  So I'll see you at 2.  Thank you.
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Thanks, Joe. I'm going to start on this board with just a preface of generalities. Let's call this the archetypal moduli problem, by which I mean from the perspective of global analysis or differential topology or simplexed topology as distinct from algebraic geometry, where they also like to talk about moduli problems. So for me, a moduli problem means this kind of thing. You have some space which I'm calling suggestively J. But right now, I'm just going to call it the space of data, which has some kind of topology on it. And for each choice, which I will suggestively call also J, in this space, we get some moduli space, M of J, which also has a natural topology. And of course, we would like to say that it has some much nicer structure than a topology, like being a smooth manifold. Or if we can't achieve that, maybe at least a smooth orbifold. And over the course of these two weeks, I'm sure you'll also hear things about weighted branched orbifolds with boundaries in corners, though not for me. From my perspective, well, the first thing to say about this moduli space is that locally, you can identify it with some kind of zero set, which I'll write sigma J inverse of 0, except not precisely that. Often you have to divide it by some kind of symmetries. So here, sigma J is some section of a Banach space bundle. And we already heard DUSA sketch one example of this. Because you can express all sorts of geometric PDEs in this way as preferably smooth sections of Banach space bundles in an infinitesimensional context. But I'm saying this is true only locally in general. So DUSA described the situation with holomorphic spheres where you can actually do this globally. But in my experience working with J-holomorphic curves, that's rather the exception to the rule. And so I will talk about some more general situations where you can do this but only locally. And these symmetries can cause quite a lot of headaches. But the first problem, of course, is you want to be able to say that this zero section here is a smooth object. And the implicit function theorem ought to tell you that if you do things correctly. But there are two general ways that you can approach that. First, the so-called abstract approach where you perturb the right-hand side of the equation. You can say something like define the moduli space m tilde, which depends on the data J, and some extra auxiliary choice that are called nu. That's the space of all u in this Bonnock manifold, such that sigma J of u matches nu of u. So this is for some other section nu of that same Bonnock space bundle. And the idea is if we choose that generically, then this set of solutions will be a smooth object simply by the Sard-Smale theorem. That's kind of equivalent to the statement just expressed in local trivializations that a generic point in the target of a smooth map is a regular value of that map. So this is a generic section. It also intersects this section transversely. So for generic choices of this auxiliary nu, our special section sigma J is transverse to that, which means this moduli space of perturbed objects I define is a manifold. And I even want to say more because if we're talking about elliptic problems, this section is not just a smooth section, but also is Fredholm in the sense that its linearization at any point is a Fredholm operator. So when we have these kinds of transversality conditions, we get just not just a manifold, but a finite dimensional manifold because of finite dimensional kernels of the linearization. So that's, of course, very nice when that happens. The problem is I perturbed my equation to a different one, which might not be the one I'm actually interested in. That's problem number one. There are more problems, which I think you'll hear more about over the course of the two weeks because this abstract approach is sort of the raison d'etre for the whole polyfield project. But it's difficult. And part of the reason it's difficult is that, first of all, this correspondence between some zero section and my actual moduli space of interest is in general only local, and it also has these symmetries. You have to deal with both of those in a sensible way when choosing these perturbations. And it means just choosing some generic section of a Bonoq space bundle is usually not enough. You have to choose it satisfying some conditions. And maybe those conditions are too stringent to actually apply the Sards-Mail theorem in this way. So do you want to be like invariant under the symmetry to something? Something like you want various things of that sort, yes. I reserve the right to not talk about that anymore because other people will. And I'm going to talk about the alternative, which is in the title of my mini course, so-called classical, or what many people call geometric, perturbation methods, which is the notion that if you take a generic choice of the data in the space of data, then your actual section will be transverse to the zero section, which means, of course, the zero set of your section is a smooth manifold and finite dimensional as well, because we're in the Fredholm context. And well, the space we're actually interested in is that divided by some kind of symmetry. That, in general, will be an orbifold. So an orbifold is something that looks locally like Euclidean space divided by a finite group action. And that's what you get if you're taking a finite dimensional manifold and dividing it by the kind of symmetry groups that we're interested in. So that is minor headache number one to be aware of. We won't get manifolds in general, but we could get orbifolds if we're lucky. Much larger headache number two is that often we're not lucky and this condition simply isn't true. And Duce already talked about that a little bit. I'm going to talk about that some more. So that's all I will say about generalities. I'm going to focus on this second, this classical approach over three talks. And in a particular context, I want to discuss symplectic field theory. So in the original propaganda paper in 2000 by Elishberg Giventhal and Hofer, you have symplectic field theory presented as a very general way of defining invariance of contact manifolds and symplectic abortisms between contact manifolds. And well, one can do it even a little bit more generally than that, which is sort of nice for certain purposes. So I'm going to talk about stable Hamiltonian structures. So first of all, if m is an odd dimensional manifold, the stable Hamiltonian structure H on m is going to be a pair capital omega lambda, which consists of the following. We have first omega is a closed two form of maximal rank. So you should imagine that as something that arises on any hyper surface in a symplectic manifold. You just take the symplectic form and restrict it to a hyper surface, it's going to be a closed two form of maximal rank. If I omit the word stable, that's what we call just a Hamiltonian structure. Now stable comes from this additional one form lambda, which satisfies following properties. First, lambda wedge, the top dimensional power of omega, must be a positive volume form. So this implies in particular that you can take omega and restrict it to the kernel of lambda. Lambda is nowhere zero. So its kernel is a hyper plane distribution, which I'm going to call psi, as I always do with contact structures, because that's one important example. Could be a contact structure, but in general, it's just some hyper plane distribution. And omega makes that hyper plane distribution into a symplectic vector bundle. Gives us a symplectic bundle structure. And then the other condition, since omega has maximal rank on an odd dimensional manifold, it has a kernel that's always one dimensional. That's the so-called characteristic line field, the one dimension of directions in which omega is degenerate. So what I'm going to require is that d lambda also vanishes on that same kernel. Another way of saying it is that the kernel of omega is contained in kernel of d lambda. OK, so this is a definition which has been around for several years. And as stated, it's not too hard to understand, but I think a lot of people don't really understand what it means. So let me talk a little bit about that. First observation is that if I now take this odd dimensional manifold and look at a small cylinder constructed out of it, so some small interval times m, this thing now inherits a natural symplectic form. Let's call the real coordinate little r and define little omega to be d of r lambda plus big omega. And that's an easy exercise. That's going to be a symplectic form. As long as this cylinder is sufficiently small, it might even be symplectic for epsilon larger than that. But in general, I can only guarantee this is true for epsilon small. Moreover, what does the characteristic line field look like? If you restrict, you've got a one parameter family of hyper surfaces. Let's say m sub r is the hyper surface r times m sitting inside that cylinder. The characteristic line field of omega restricted to m sub r is independent of r. So it's the same characteristic line field for all r, all parameters in my parameterized family of hyper surfaces. That's really the original definition, which I think appeared first in the book by Hofer and Sander of a stable hyper surface. That's where this idea comes from. It's the kind of situation in which you would expect to be able to prove something like the Weinstein conjecture that asserts existence of closed characteristics of this characteristic line field. Because if you have them on some hyper surfaces in the vicinity, you have them on all of them. Excuse me. What do you mean with the first symbol of the last line? The first symbol? That's an and, and percent. So in order to get the stable condition that you want this last condition at the kernel of omega containing the kernel under, that's what implies. That's what implies this fact about the characteristic line fields, exactly. And conversely, if you are given a one parameter family of hyper surfaces that has this property, this stability of the characteristic line field, then you can always find a stable Hamiltonian structure that produces that. So that's kind of an exercise using the Moser deformation trick. How much will I lose if I just ignore this definition pretend you said contact structure? Nothing really. That's a perfectly good point. If you don't like this definition for some reason, because it's new, just think lambda is a contact structure and big omega equals d lambda. I was going to say that soon anyway. So there is a, from this data, we have also a canonical generator of the characteristic line field, which by analogy with contact forms we'll call this the rape vector field. So my notation will be r sub h. And it's determined by the conditions. Omega annihilates the direction of the rape vector field. And then we normalize it by lambda. Is there any reason d lambda couldn't be zero? No. There's no reason at all why d lambda couldn't be zero. I will also say that again in a moment. So we have a stable Hamiltonian structure. It defines this hyperplane distribution psi, which might be a contact structure in certain examples, but it doesn't have to be. Could also be a foliation. We have this rape vector field. Another thing we can do is, since we have this small symplectic cylinder we built out of it, we can also just arbitrarily expand that to r times m and rewrite my symplectic structure there in the form omega phi. That's going to be d of phi of r lambda plus omega. So what's phi? Phi is going to be in the space of smooth functions taking r to that same small epsilon interval with strictly positive derivative. So this is a symplectic form also. OK? It's symplectic to the one I wrote down before. I've just stretched things. But the point is that r cross m is the kind of setting where I would like to talk about j-holomorphic curves. And I can do that now by saying, well, the space j dependent on the stable Hamiltonian structure is going to be the space of almost complex structures, j on r cross m, which I will assume always to be r invariant. In other words, there's this r translation action on r cross m. I want that to preserve all my j's. I also want to say that j maps the unit vector in the r direction to the rate vector field. And I want to say that j preserves my hyperplane distribution as well as the symplectic structure. It's compatible with the symplectic structure on that. So let's say j restricted to psi is omega compatible. OK? So here's a lemon which is very easy to prove and helps justify all these definitions. That is that for all choices j in this space of translation invariant, almost complex structures, and all functions phi, as I indicated above, that are mapping to that small interval with positive derivative, it's not just that omega phi is symplectic, but in fact, it's compatible in the usual sense of Gromoth with this choice of j. So I can fix j and I can then allow any choice of phi I want. And that's an important detail for defining things like the energy of a j-holomorphic curve, because we then take that to be the supremum of symplectic area defined with respect to all these different choices of symplectic forms. So let me give two examples. They've already sort of been mentioned. First of all, if alpha is a contact form, we get a stable Hamiltonian structure d alpha, comma alpha. And then this symplectic manifold r cross m with this kind of symplectic structure I've been talking about. I'm not going to try to say these two precisely. You have to make some kind of condition on how you define phi in order to say this. But I'm just going to say one can do it. So this is what's called the symplectization of the contact manifold. I really should relax the epsilon's in order to say that. I should actually, should at least make plus epsilon to be infinity. Let's not worry about this right now. Yeah. Can you quickly write down what the symplectization is? No. I'm going to say it's this. Yes? So what's the condition where you can integrate this hyperplane distribution to a symplectic manifold? Where you can integrate the hyperplane distribution. Yeah, if you integrate it, and then you find a sub-manifold, which has that as its tangent form. Well, for the hyperplane distribution to be integrable, I think that's equivalent to saying that d lambda vanishes on psi. So certainly it's sufficient of d lambda vanishes altogether, which is my next example. Already in the first row, we think that the word compatible sort of looks like compacted. Well, I'm sorry. I'll take this opportunity to make a blanket apology for my handwriting, as I customarily do. There isn't that much I can do about it. OK, compatible. I just said, i actually looks like compacted. Compacted. All right, second example. If m equals the product of s1 with some other manifold w, and I take w to be a symplectic manifold. So let's say I have a symplectic form little omega on w, and then set capital omega to be that plus dt wedge dh. So here, t is the s1 coordinate, and I'm choosing some function h, a Hamiltonian function, which is dependent on time in a periodic way. So this is capital omega, and I take lambda to be just dt. This is a case where it's closed. d lambda vanishes entirely. So now one can check. This also is a stable Hamiltonian structure. And it has a meaning which a lot of people will probably like. You can compute the rate vector field for this. It's simply, it's the vector field in the t direction, plus a vector field in the w direction, which is precisely the Hamiltonian vector field. So this is a way to turn questions about Hamiltonian vector fields on symplectic manifolds, even time dependent ones, into questions about a time independent, rave like vector field on an odd dimensional manifold. And it gets better because for the class of almost complex structures that we've talked about, we can use the same method. So we can use the same method. We can use the same method. We can use the same method. We can use the same method. We can use the same method. So the class of almost complex structures that we've talked about, a j-holomorphic curve in r times this, is basically equivalent to a solution of the Flur equation in w. So this is a way that you can do Flur homology, Hamiltonian Flur homology, from the perspective of symplectic field theory. OK. So now the setting in which I really want to think about holomorphic curves is also a bit more general than this. So let's take a symplectic manifold. So here I'm talking about co-boardisms with stable boundary. So one can define stable boundary in various ways. The main thing I want to say is there exists near the boundary of one parameter family of hyper surfaces that all have the same dynamics on them, same characteristic line fields. And that's equivalent to saying that there exists this collar neighborhood near m plus that looks like a small interval times m plus, where the symplectic structure is just the kind that I wrote down already, d of t lambda plus plus omega plus for some stable Hamiltonian structure, omega plus lambda plus on m plus. Of course, I'm going to do the same thing on m minus. But the collar neighborhood will look like an opposite interval. OK? So this is the definition of a symplectic co-boardism with stable boundary. And having done that, I can add cylindrical ends to this compact object and get a non-compact object. That's what we call the completion. So I'll denote that by w hat omega hat. It's going to depend on some choice of function phi of the sort that I used to define symplectic structures on simpletizations. And that choice of phi is not particularly meaningful. It's just nice to know that I can do it. So the definition is we take a negative cylindrical end, glue that along m minus to w, glue that along m plus to positive cylindrical end. And the symplectic structure here, we take omega phi, omega phi on each of the ends. And the original omega on w, you can choose the function phi such that, so remember, I still have this definition on the board somewhere. Middle board top right. Middle board top right. Very good. d phi times lambda plus omega. If I choose phi so that it equals just the identity map close to 0, then I can obviously glue these cylindrical ends smoothly together with these collar neighborhoods. That's what I'm doing. OK. So with that choice of symplectic structure, and I don't really have to specify phi, I just know that I can choose one. Are you using the same phi on the both ends? I don't care. Actually, I specifically want to not specify phi and allow myself the freedom to change it. So let's say j of omega and the two stable Hamiltonian structures. This is going to be the space of all almost complex structures, j on the completion w hat, which I assume to be everywhere compatible with, well, let's just say, compatible with omega on the original compact object and belonging to these spaces with respect to the stable Hamiltonian structures that I already defined. So the spaces of translation in very almost complex structures on the ends. So that now is going to be an almost complex structure that's compatible with omega hat phi no matter what phi I choose. OK. So now to start talking about the actual J-Hollomarket curves. Any questions so far? OK. So this you require that J's, what does it say? Translation. Compatible. Translation. Translation very on the end. So what about in the co-boardism part? I mean, don't you have a factor r there anyway? When the co-boardism part, I don't have a factor of r except close to the boundary. Yeah. OK. Yeah. And I don't make any requirement there except compatible with the given symplectic form. OK. So the kinds of holomorphic curves we're going to talk about are as follows. We take sigma J to be a closed Riemann surface. Let's actually say there's going to be two finite disjoint ordered subsets. And in particular, sigma dot always denotes sigma with the first finite ordered subset removed. That's a punctured Riemann surface. We take that, also this subset. I don't have to worry about it too often, but it gets partitioned into the so-called positive and negative punctures. Each of those could be empty, but each puncture belongs to one or the other of those. And now I think about maps from this Riemann surface with the various points removed going into my completed symplectic co-boardism. And I want them to be what I call asymptotically cylindrical, which means, well, let's not write down the precise definition. Let's just say asymptotic near each positive or negative puncture z to R times a Raib orbit in m plus minus. Chris, you're missing a hole somewhere. I'm missing a hole. That's true. Could be worse. The opposite would sound like a medical emergency. So what I mean is essentially, for this kind of almost complex structure I'm considering, with the translation invariant thing satisfying these conditions on the ends, I have j maps the vector in the r direction to the Raib vector field, which means r times a Raib orbit is always a holomorphic cylinder in the end. So those are special holomorphic curves that always exist, at least in that region, maybe not globally. And I want to consider punctured holomorphic curves whose behavior near each of those punctures approximates one of those cylinders better and better as you go toward the puncture. So to make it precise, I would say you can choose some holomorphic cylindrical coordinates near each puncture. And in those coordinates, as you go further up toward infinity on the cylinder, your curve looks more and more like one of these trivial orbit cylinders. OK, maybe it's time to recover this. Yeah? In your picture, theta is empty or what's happening? Theta, I haven't really said. Theta doesn't have to be empty. It's just some finite subset right now. But in the picture, it doesn't appear in the picture because it doesn't affect the picture in any way. It's just points. So theta are meant to be marked points. All right, so the actual moduli space looks like the following. It's going to be a space of tuples sigma j gamma theta u, where, well, sigma j is a closed Riemann surface, as I wrote above. Gamma and theta are these disjoint ordered finite subsets. So let me not rewrite that. Mainly, I need to say u maps the punctured surface to the completion w hat j holomorphically and asymptotically, solidically. And then I just need to say what my equivalence relation is. Deuza basically already said it. You could deduce completely the right answer to this from what Deuza said. I will say a tuple sigma j gamma theta u is equivalent to a tuple sigma prime phi star j phi inverse of gamma preserving the ordering phi inverse of theta u composed with phi for any diffeomorphism phi from sigma prime to sigma. But doesn't the ray orbits allowed to be arbitrarily terrible? Yeah, this definition is still OK if I allow the ray orbits to be arbitrarily terrible, but I won't have any theorems. Yeah. I do have, so finally to make some use out of the mark points, I have an evaluation map sending this moduli space to w hat m where theta, the mark points, are labeled zeta 1 through zeta m. So this just evaluates u at those points. And it's well defined because of the way that I define my equivalence relation. These diffeomorphisms map the set of marked points to the set of marked points. All right, so let's finally state a couple of theorems. So theorem one says, if I have, I'm just going to abbreviate elements of this moduli space by u instead of writing out the entire tuple and saying equivalence class and everything. So if u is what I call Fredholm regular, that's a condition I'll have to explain a bit, and it also includes the condition that the asymptotic ray orbits are not arbitrarily terrible. Then a neighborhood of u in the moduli space, is a smooth finite dimensional orbital with isotropy at u, isomorphic to the automorphism group of u. So I didn't write down this definition yet. The automorphism group of u is simply the space of all bi-holomorphic maps of its domain, which fix the punctures and the marked points and satisfy u equals u composed with phi. So unless u is constant, that's always going to be a finite group. And for somewhere injective curves, it's going to be a trivial group. But for non-somewhere injective curves, so for multiply covered curves, if we get this Fredholm regularity condition, whatever it is, we're going to have to deal with an orbifold rather than a manifold because of this isotropy. Is this going to be a centric of tautology on the word regular? In some sense, this is a tautology on the word regular. I mean, one does have to work a little bit to see why you can make this local identification of the moduli space with something that looks like an orbifold even if you have regularity. And I will at least have time to get to that in this talk, I think. Therm two, I'm going to probably postpone the proof of this for tomorrow. But I'll state it now. Let's say let's fix an open subset u with compact closure. Then I'm going to say there exists a bare subset called j u reg. Well, it's a subset of the space of all. So I needed to fix some more data in advance. So fix also an almost complex structure, j0 of the type I've described. Then there exists a bare subset. So this is a subset of the space of all. J's of the type I've described with the extra condition that j matches j0 everywhere outside of this special subset u. So this special subset u we can call the perturbation domain. It's where we're going to allow ourselves to change the almost complex structure, but we leave it fixed everywhere else. So such that for all j in this j reg u, let's finish it over here, every somewhere injective curve u in the moduli space with respect to j, passing through my special subset capital u is thread home regular. But here bare subset is again a commuter. Bare subset literally means commuter, yes. So what did I call residual? Yeah, I never understand residual. It sounds to me like it means the opposite, so I don't use the word. Some people call it second category, which I think is also not technically right if you look up the actual definition of second category. Yeah. Thank you for that input, Catherine. So yeah. Well, someday some kind of sensible general habit will prevail, hopefully. I say bare subset, and most people don't seem to mind. So the literal meaning is countable intersection of open and dense sets. What contains, actually, if you're being strict, it contains. It contains a countable intersection of open and dense sets. Very good. Maybe more useful input is though somehow you, I think you're assuming here now that the Hamiltonian structure is sort of regular. Well, that's in, oh, in theorem two. I am. That's true. Thank you. OK. So the board's out of reach, so I'm not going to write it. Let's just say this. One additional condition you need in theorem two is that, well, let's say all the closed Ray-Borbets for both stable Hamiltonian structures are non-degenerate. So under those conditions, theorem two holds. Otherwise, you could have trouble. And at least for if your stable Hamiltonian structures are contact forms, that's a condition which is generic. That's a standard result. Can you also do this if you're Ray-Borbets or more spot? More spot is also fine. Yeah. More spot or non-degenerate are perfectly fine. OK. So let's talk a little about the functional analytics setup for a neighborhood of, let's say I have a specific element of the modularized space with complex structure, J0, and the map is called U0. Just to pretend the statement of theorem two. You're not asking that it be somewhere injected inside U. It has to be somewhere injected somewhere and pass through U. Those two things are equivalent, which is not so hard theorem. So what I really should ask for is that there is an injective point mapped into U. And it's a good point because if I were working in other settings where I have boundary with Lagrangian boundary conditions or something, then it's not necessarily true that just because the curve is somewhere injective that its injective points are dense. So then I would have to be careful. But here, injective points are dense if the curve is somewhere injective. I think you've hidden the end of the statement. Can you rise the board and choose? Ah. The statement is on the sideboard. Oh, it's this one. It's this one. Then you can block that board so it's up with the theorem. This is called democratic blackboard technique. All right. So I need to actually write down some kind of Bonnock manifold of maps that will contain the solutions I'm interested in. This is something that's kind of standard in the case of closed maps from closed Riemann surfaces. For punctured curves, I guess it's considered sufficiently standard that nobody really feels they need to go through the details when they write papers. But it means there is no paper where anyone does the details really as far as I think. In Matthias Schwartz's thesis, I think there may be details about this, and that's possibly the only place. There's also this paper by the problem of traveling. We know how to paper this. No. No. Well, I mean, what he does in there is basically just lifted from Schwartz's thesis. OK. So I'm going to talk about a space of sub-elevated class maps from the punctured surface to W hat, class WKP. So I want to have k derivatives that are of class LP locally. Locally, that would be the definition, and it's not hard to express. But globally, what does this mean? Easiest way to say is U is of the form x along f of h. h, where f is going to be some smooth map, which is not just asymptotically but literally cylindrical near the ends. Well, I'll say that in a moment. h is going to be some section of class WKP locally along the pulled back tangent bundle along f. So in compact subsets, this is a sufficient definition, but I need to say more specifically what happens at the ends. So I just have confidence. So the j zero that you started with is regular, right? Or that? No. I don't care. It doesn't need to be regular. So are we requiring anything for j zero to begin with? I don't have to require really anything about j zero because I'm going to perturb it in a certain region, and I'm only looking at holomorphic curves that pass through that region. So near each puncture, in holomorphic cylindrical coordinates, I can choose coordinates st, which are in a half cylinder. I'm going to just call it a positive half cylinder if it's a positive puncture, negative half cylinder if it's a negative puncture. And then the requirements are that the map f just looks like T s plus a constant gamma of T t plus a constant in the cylindrical end r plus minus times m plus minus. So here, c1 and c2 are just some real constants. And what do you mean by gamma plus a constant? I haven't. I'm about to say that. Oh, sorry. Well, gamma will be valued in s1. You can add constants in s1. I thought gamma was an m. Very good point. Is that better? Thank you for that. So gamma is going to be a t periodic orbit of the ray vector field. And then what I say about h is simply that in these cylindrical coordinates, where also I'm assuming I've chosen some kind of sensible asymptotically translation invariant trivialization of this tangent bundle, I want to have h on that end be of class w, k, p on the cylinder, the half cylinder. And so I haven't said anything about what k and p are. I'm not really going to make any requirement about that, except k times p should be greater than 2. Because then the Sobolev embedding theorem tells me w, k, p injects continuously into c0. So these maps are at least always going to be continuous. If they weren't continuous, I'd have real troubles making this a good definition. So you don't require any kind of exponential decay for h or anything just? I'm getting to that. I'm getting to that. So I'm confused because if we're looking at a neighborhood of u0, shouldn't we be writing things like neighborhood for x sub u0, like something? Yeah, but I'm going to, well, u0 is going to be a map of this form. And I'm going to consider other maps of this form that are close to u0. I mean, I haven't localized yet. This is still global. Yeah, the bracket's still open. Oh, very good. Wouldn't want to have some sort of general way tech error. Can you explain what's being ruled out by this list of conditions? But I'm ruling out maps that don't have decay at infinity to some very sort of, well, f is a very controlled map. It has this precise cylindrical form. I'm only considering maps that have asymptotic decay to a map of that form. So I don't want them to move around non-trivial amounts arbitrarily far away. OK. All right. Agucin kind of preempted the next little variation I have to put on this. And I won't be able to explain why just yet, but let's just say it. For a parameter delta, I can let, I'm going to call this b. This is the actual bonnack manifold that I'll work with. So this is going to be, again, u equals x belong f of h as above. But the condition on h will be slightly different. I'm going to require that the norm of h, not the norm of h itself, but the norm on the half cylinder of e to the delta s times h is finite. So that enforces some exponential decay in addition to the decay that the wkp norm already requires. So I'm not going to tell you yet why this is necessary. I will tell you that eventually. I will tell you why it's possible, which is called this fact, though I guess it's sort of a hard theorem. First version of it proves by Hofer-Visatzian-Seindor. And there's been other versions in various contexts since then that if the asymptotic orbits are non-degenerate, then for all delta sufficiently small, all asymptotically cylindrical, or let's just say, all curves in the moduli space I've defined belong to this Bonac manifold. That is, the map involved in defining that curve belongs to that manifold. It has the asymptotic exponential decay, as long as delta is sufficiently small. Hey, Chris, I'm in the back row. I can't quite see. Are there absolute value signs around that s in the exponential? No, but there should be. Thank you. Let's make it a plus or minus. I want a minus sign if I'm at a negative puncture. The point is that it enforces asymptotic decay, or exponential decay, rather than exponential growth or something. What can I sensibly say in the zero amount of time I have left? You have like three minutes. My question stands. I need to. Sorry. The more force you have. That would be remarkable. So in that last actually, is it during the delta? Doesn't it depend on the components of MLK? Yeah, it does. Yeah, to say this properly, I would have to say that given a specific set of asymptotic orbits, there is a delta sufficiently small so that all curves that are asymptotic to those orbits satisfy the decay condition. OK. So let me try and at least get to the definition of Fredholm regularity in some form. What I'm going to do now, I've defined a class of maps I want to work with. I also have to worry about the complex structure and the domain, because that's not fixed. It's not some sort of God given data to work with, but I allow it to vary in different points in this modular space of different complex structures. Now, I'm looking at just a neighborhood of some u0 with complex structure j0, which is it still visible somewhere on the board? No. There it is. So let's just worry about the complex structures that are close to j0. And one way to do that, I don't have to consider everything in this infinite dimensional space of complex structures. But if I want to cover all of them that are not equivalent by diffeomorphisms, I can think about Teichmühler space. That's at least a nice smooth manifold. So what I'm going to do is say let t be a smooth finite dimensional family of complex structures on sigma, which I can, if I want to, I can even assume that they are fixed near gamma and theta. Another thing, I can just take a model surface sigma and fix punctures in one particular place, or several particular places, fix also the marked points in particular places. I don't need to allow them to move around. Because if I want to get all the different conformal structures up to equivalents with these punctures and marked points, all of them can be represented by some complex structure that have the punctures and the marked points in that particular place. That's just a question of choosing a diffeomorphism. So I'm fixing gamma and theta in place on sigma, but I'm letting the complex structures vary. And this is going to be a smooth finite dimensional family that parameterizes a neighborhood of the equivalence class of J0 in Teichmühler space. What's that? I don't understand what you just said. Suppose that it is like w, it ends at empty, and sigma is the sphere. Yes. Then you're not varying the complex structure at all, because you can, but you say no to forget any configuration points that you want. Yes, I can. But I have to vary the complex structure. Because a complex structure is just a section of the tangent bundle, all the morphisms of the tangent bundle are all diffeomorphic. But you can certainly vary them. And also, they're not all diffeomorphic if there are more than three marked points. No, right. So even on the sphere, I have to do this sometimes. So I'm parametrizing a neighborhood in Teichmühler space, which is the space t sigma dot of theta of all complex structures on sigma divided by the identity component of the diffeomorphism group preserving or fixing punctures and marked points. So it's a classical fact that this is a finite dimensional manifold. So I can choose some sort of finite dimensional space of complex structures parametrizing it. And there are various sensible ways to do this. I'll talk about tomorrow if I have time. There's a quick lemma. Let's also assume, just for simplicity at the moment, that the Euler characteristic of the punctured surface minus the marked points is negative. That's, Dusser also talked about this a little bit. That's the stable situation. So that's the situation in which the automorphism group of the domain with its punctures and marked points is finite. It's not an absolutely necessary assumption for what I'm talking about, but it does make a few things easier. So I can, in particular, say there exists a choice of this object, T. I'm going to, in the future, call this a Teichmiller slice since it parametrizes a neighborhood in Teichmiller space. So that is invariant under the action of the automorphism group of the domain. So having made this choice, let me finally write down the point of theorem one. We now can define a smooth section of some Bonnack space bundle. The domain is going to be the product of a Teichmiller slice with my Bonnack manifold of maps. Section is called B bar j. So the fiber over j comma u is going to be the space of wk minus 1p sections of the bundle of complex anti-linear maps from T sigma j to pull back tangent bundle. I hope that's the right number of parentheses. If not, somebody will surely let me know. And we're going to take j comma u to the obvious thing, tu plus big j composed with tu, composed with little j. If you tell me I need to put a 1 half in there somewhere, I will tell you to get a life. And so finally, so. How about changing the T for a D? My own PhD students need to be especially careful in moments like that. So. Analyzing notation using T. That's fine. I go back and forth. So the lemma is that the automorphism group of u acts on the zero set of this section by the obvious phi star j u composed with phi such that a neighborhood of u in the modular space is in bijective correspondence with a neighborhood of the equivalence class. Sorry, I should have said neighborhood of u naught in the modular space with the equivalence class in this automorphism group of u naught, j naught u naught in the zero set modulo that action by the automorphism group. So now what does this really say? By the implicit function theorem, mj is going to be an orbital near u naught with isotropic group equal to the automorphism group. If that zero section is a manifold, which is going to be true if the linearization of the section is subjective. So that's some linear operator from the direct sum of a finite dimensional space tangent to a space of complex structures with, well, let's just write it as tangent space to the Bonnack manifold, which is some space of sections of the pullback tangent bundle of class wkp. Yes? Yes. It's probably delta. Sorry? It should be a delta on your e in the definition of e. Oh, you're right. Thank you very much. If you're told he wants to have a life, it's lunch. This is the moment where I say I thought you people cared about mathematics. I've already said so. OK. Basically, so I wanted to at least end with having written down this operator. So Fredholm regularity means, specifically, I need to finish the sentence and say is surjective. So Fredholm regularity means just that. This operator is surjective. I'll talk more tomorrow about what one needs to do to make sure that that operator is surjective. The result is then we identify local neighborhoods in the modular space with this thing, which is a finite dimensional manifold divided by a finite group. I'm not going to say anything about the proof of the lemma except that if you stare at it long enough, it becomes obvious. It's not that hard. So I'll stop there for now. So there is office hours. So I again suggest longer questions, get postponed until then. So anyone have a short question that they're willing to ask? Everyone? So good to practice. I'm going to show that something is regular, right? What do you need to take into the slides and bury your complex structure? Right. OK. It's a good question. If you want to show that something is regular, well, the answer is I've never been in a situation where I had to actually choose a type of slice concretely. But it has to be there in the theoretical setup, because if it's not there, right? I mean, it would obviously be sufficient if the linearization of this operator, just mapping this space of sections to this space of sections, is surjective. That would suffice. That would imply that this one is surjective. The problem is that that's not true as often as you want it to be. You have this additional finite dimensional stuff in the domain, which sometimes actually makes the difference between an operator with a finite dimensional co-coronal and one that's surjective. So in fact, so if I have time to talk about automatic transversality tomorrow, hopefully, it's a really crucial detail there. The criterion that used to prove automatic transversality does not work if you leave out this whole discussion of type familiar space, because you need to have this freedom to vary the complex structure in order to get as much transversality as you need. Do you have an actual example of the curve which is regular in this sense and not regular if I take that particular space? I could probably come up with that. Ask me in office hours. And you have a discussion tomorrow. And I have a discussion tomorrow. In addition to your lecture. You can certainly find examples if you take chloride on stuff with a pi of t. This will definitely need to vary. Ties will especially vary. What is the fact that you take a finite dimensional balance just all by a new? If I took all of them, this operator would not be Fred Holm. That would be bad. So here you're thinking of fixing big j, which is the complex structure on the target. So far, I'm fixing big j. And your space of j here. I was using a curly j to mean variations of j on the target space. But you're fixing that and making your j as a finite dimensional. I have two curly j's. Well, you have two curly j's. Only one on this board. You see, your curly j there is a type of a slice. But you could, in principle, vary the other j on the target space if you want it to. Yes. You mean when I'm talking about finite dimensional families or you're just generic? I mean, in year 0 of 1, you were saying that you were taking generic elements where you were allowed to vary in the target in the open subset, u of your target, right? Yeah, that's right. That curly j is not actually in this limit. Here you're sometimes assuming you fix that. Right. No, I mean, the question of why you get transversality for simple curves for generic j, I will address tomorrow. So far, all I've talked about is what criterion actually gives you smoothness of the moduli space. And how does one go about proving that? One thing to say is why do you need to vary j and type on the spaces? For example, if you look at the space of tori in CQ2, and look at, you say, you know, the class of DQC, embedded tori. You look at the induced complex structure on the tori. That changes. So if you fix the complex structure on the tori, it's not necessarily going to be regular. Because when you move your points around, you leave the color on j on the domain, naturally change. So you have to have that variation in. Without further ado, I propose we break for lunch. And at 2 PM, I'll make some short remarks about organizational things like finances and stuff. So I'll see you at 2. Thank you.
J-holomorphic curves, that's rather the exception smooth object simply by the Sard-Smale theorem. approach is sort of the raison d'etre for the section is a smooth manifold and finite is sort of regular. Well, that's in, oh, in somewhere injective that its injective points are just say, all curves in the moduli space I've space of sections to this space of sections, is for simple curves for generic j, I will address