1/2 Introduction to Polyfold Regularization
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Wehrheim, Katrin
This lecture will discuss the overall ideas and challenges in regularizing moduli spaces, and introduce the two basic ideas behind polyfold theory: Making reparametrization actions "smooth" and making pregluing a "chart map". [related literature: Sections 2.1 and 3.3 of Polyfolds: A First and Second Look. Related videos: Lecture 8 and Lecture 20 from Wehrheim's special topics course.]
 Awesome. Thank you. And thanks for putting this on. I believe the world will be a better place. I should also give you a renormalization instruction for any criticism that I might utter. If I don't tell you that Ethereum is complete bullshit, I have not read it. And I think it's not interesting. If I do tell you it's complete bullshit, it just means that you should actually understand the proof for yourself and not trust it. And by that sort of criterion, really, I think all proofs in mathematics should be complete bullshit because, well, you should understand them. So I'm actually not going to talk about analysis pretty much at all. I'm going to talk to you about why we need to worry about analysis. So I'm also going to try and sort of ground us in a big picture of what are we actually doing here. So I'm very happy that somehow the term I think that I coined regularization has taken root. But I want to tell you what that actually means from my point of view. So when you regularize a modular space, well, and when I say modular space, I'm usually going to mean pseudo-amorphic curve modular space. So if you're given a compact modular space, which will always be called m bar, usually of j curves, I would like to assign to it in some sort of unique way, well, one or two things. So in the Grom of Whitten world, I would like to actually give it a fundamental class. But then, of course, the question is, what do I mean by fundamental class in the homology of something that might just be a compact metric space? So if this was a manifold, it would have a fundamental class. If it is not a manifold, we don't know what its fundamental class is. But the nice thing with trying to actually construct that right in here is that from the modular space, we have evaluation maps in Grom of Whitten. And so if we had a class in there, we could push it forward directly with evaluation maps. That's why I think that's the cleanest way of saying what we want is to make a homology class straight in there. So this is really the Grom of Whitten case. And pretty much in all other theories, Fleur, SFT, anything, infinity related, you're going to expect your modular space to have boundary. And so really what you want to do is you want to somehow count things and then get a relation from the boundary. And so what we'd like there is some kind of cobaltism class of closed, no, not closed, but compact things. So here you could get this from maybe a compact manifold. Well, at least that would have a homology and how I get it into M-bar is not so clear. But here I'm not going to say closed because they're going to have boundary compact manifolds. Right, in general, they might be weighted branched. And they're going to come from some choices. Knew that Chris Wendell up here had up here, but then unfortunately somebody diligently removed this. So your task for this next hour is also to figure out what's completely different in the philosophy that I project here than it is on Chris's board about abstract perturbations. And I'm happy to discuss that in office hours. So really these manifolds are going to come from some kind of choices of perturbations. Knew that we'll talk a lot more about. The main thing is it's going to have boundary in general and usually also then corners. And the algebraic identities usually come from identifying the boundary with some kind of fiber product of products of the modular space itself. So if this seems totally alien, maybe think about Fleur theory. And what I'm thinking about here is I'm packaging all Fleur trajectory spaces in this M-bar new. So no matter between which critical points and no matter between of what index. And then this just says that somehow the boundary is given by broken ones. So that's not the whole thing is not going to be compact. Yes. Well, indeed. Yes. So that's why this is the philosophical overview. But I could do this on a certain up to a certain energy. And then this would be true. So good. So this is my goal. This is what I mean by regularizing a modular space. Assigning to something that I can count or that I could integrate over and that has some kind of nice boundary structure that gives me the algebraic identities I would like. So well, I don't need to write this. What are my tools here? Right? So in the algebraic geometry, then I describe J curves on my modular spaces are really modular spaces of subvarieties or not manifolds, but they're sort of subsets of an ambient algebraic variety that are cut out by some algebraic equation. And so then I can do algebra to describe my modular space. And the thing that Gromov realized was that, well, if my J isn't integrable, I can't quite do algebra, but maybe I can do exactly the same things that algebraists do with PDE methods. So that's why there's a lot of analysis here, because all the magic that you can do with algebra we now have to do with analysis. However, we always have somehow this intuition that everything that's true in algebraic geometry should be true in symplectic geometry. Because the general Gromov written in this, the quick connection with Gromov written in this was later based on the paper by Gromov. Right. OK. Yeah. So some things flow back. Yes. Good. Catherine, can you stay away from the last four inches? Yes. This. OK. Here. Awesome. Good. Yes, I can. So let's see. What is my intuition here? So well, I should say something. So first of all, I'm going to do an analytic description of M bar, namely in terms of solutions of a PDE, but then there are two other things. And so I'm just going to roughly think about having a fixed Riemann surface and a fixed almost complex structure on a symplectic manifold. But I have to do two more things. I have to mod out reprametrizations. Because as an algebraic geometry, I'd like to actually count the curves, not the maps. So an image in M bar I could parameterize in different ways by maps. And so I want to quote you out by that ambiguity. And then we're always going to want something compact. So I'm adding Gromov compactifications. And so maybe just as a side note to the examples that Duzer gave, right, there were beautiful examples of non-compactness coming from Mark points running together. However, that's not the worst of our problems. So usually really the hard problems with Gromov compactification come from energy concentrating. And that's where you get the sphere bubbles that then have an unstable domain. And this is where really all analytic hell breaks lose. So that's really a PDE blow up situation. So you really do need this Gromov compactification even if we have no Mark points. So if that's how we described M bar, then we can do, well, two things. So first of all, as you've already seen, I think in both talks, there's a more or less well, local or global Fredholm description together with gluing maps, which we'll sort of start talking more and more about. And so very roughly what that does is it somehow, this is a lie, but the intuition is that you're writing your modular space as the zero set of some kind of section. Actually, mainly the section here is a lie. So but let's pretend, right? So at least to some extent, that seemed to be the description. And then the reason you believe that these things should be things that have a fundamental class. I believe is really this finite dimensional regularization theorem, which says, well, let's pretend for the moment that my nice compact modular space of J curves is actually cut out from a finite rank vector bundle over a finite dimensional base. And let's say that's even smooth. I guess let me make it an oriented vector bundle. Then well, first of all, even without the section, these things have an Euler class, right? So that's just the Euler class of the bundle, but you can pull that back and let it sit in the base. And I guess the theorem here, well, you can construct that with algebraic topology or you can construct it by taking a section and perturbing it. So there's various methods of constructing Euler classes even. And so I think the main one that we talk about in some plectic geometry is to do it by perturbations. So I would like to be able to just say that my Euler class is actually maybe the fundamental class of some perturbed section of the zero set. So that would then sit in that homology. I believe I should actually add, oh right, so usually you need the base to be compact to get an Euler class at least. But I believe you can do this as long as the zero set is compact. You mean as a homology class? Right, yeah. Right, otherwise it's just not unique as a homology class. There's no chance you'll ever have that. And also, right, so I'd like B and E to both be finite dimensional. But the finite dimensionality is actually not the problem. I think you could also take them infinite dimensional and then you would just need the section to be Fredholm. And then the same theorem is going to hold with exactly, essentially the same proof. Right. But what do I mean by this really? So how do I construct? What do I mean by having perturbations that give me this Euler class? And this is where, right, this is where the word generic comes in. That's what I was going to strike in big and red from Chris's board. Because really, so people always say take a generic perturbation or a generic J. So if you give me a J, how do I know whether it's generic? Generic is not an adjective that goes with a single element of a set. Right. Somehow we say generic when there is a commiga set and then the elements of that set are the generic elements somehow. And so, yeah. So I think that's one danger because people often say generic in situations where it's not even clear what the Banach space should be from which you're taking something. And also, I mean, I said somehow second category was wrong. But I actually don't quite understand why we're so focused on having sort of a commiga set or a large set or a full measure set. So what I'm going to say is really all you need is a set that's not empty. And in fact, sets of second category are non-empty. So in that sense, that's a perfectly fine thing to prove when everything goes through if you just prove that something is second category. I just don't know how you ever prove that something is second category without proving that it's commiga. But anyways. So there is a non-empty set in the smooth sections. So that for any perturbation in this set, so these are the good perturbations. The perturbed section is not just transverse, but its zero set is still compact. Yes, we'll do. Thank you. So that tells me that it gets a smooth structure. And if that's compacted as a fundamental class, and then I can embed that into B, and so I get a homology or I get a cycle there. But now what if I take two different perturbations? I would like them to actually have the same fundamental class. And usually we prove that by proving that the two zero sets are co-bordant. So people often talk about the transversality problem in holomorphic curves. And transversality is really just this bit. Whereas somehow regularization for me, when I say regularization, I mean the whole package. I mean transversality, preserving compactness, and then having uniqueness up to co-bordism. So if all you care about is that this set p is non-empty, how do you know there are two sort of disjoints at it? Right. Yes. So it's sort of canonical. Right. I mean, I want to, so usually actually it's not, so usually the statement is not there exists the p, right? Usually the statement is p, which I define as something. So it's a specific set. And I don't need to know that that's commiga, I just need to know that that set is not empty. Right. So for example, in Chris's theorem, I think the statement is going to be that it's a set of j's in this curly j. And the j's are defined exactly by some transfer, by regularity, right? By subjectivity of the linearized operators. Can I make one comment about this? So it's not that I disagree, but when I define what conditions those j's have to satisfy, it's almost always going to be that it satisfies a countable list of conditions. Right. And so for each of those individual conditions in the list, it is pretty important to know that there's a commiga set, because I need to know that the intersection of those sets is not empty. At the end, I only care that it's not empty, but how else would I possibly know that? Yes. Is it true? Sometimes you have some kind of special, like if you want to prove like most people's homology kind of things, you have something that you know, and you want to deform that thing that you know to, and then you want to have some kind of complex, almost complex structure that's near a given possibly degenerate. Right. Sure. Sometimes you'd like to know that kind of density. Yeah. This is just for existence of a possible definition. Yeah. Yeah. But you could see you can write, I mean you can also, again, you don't need this to be dense. You just need to have it be dense near a given point. Right. So you could say, take a set and say, I fixed the support and I fixed some sort of size of the perturbation. And no matter how small the epsilon and delta are, the set is always non-empty. Right. So since which point, you don't know which point you're going to meet in the beginning of the course. Right. Yes. But I mean, so it's right, so the statement that given any J there isn't arbitrarily close by J that is regular is also different from proving that something is commiga. Right. So. I've been, well you're explaining philosophy. Yes. Why do you take a perturbative definition of the order class as opposed to like pull back from the grass, might end or. Right, because I'm going to want to generalize this in such a way that it applies to holomorphic groups. I mean that definition would not be anything, right? It could be a topological space for that definition. Let's discuss that once we're sort of deeper down in the actual application. And actually, right, the challenges are going to come momentarily and then we can talk about whether different approaches will solve that. What was I going to say, P? Right. So, in this case here already, I think the set, so I think Assad's male theorem here would tell you that, you know, the set of ones that, so if I defined this set by the ones that are transverse to zero, I would indeed get a commiga subset in there. However, it's not clear to me that for all the perturbations for which I have transversality, I'm also going to preserve compactness. Right. So, in fact, I think I could cook up a counter example. And then it's also not clear that you're always going to have uniqueness up to cubotism. So it's often you have to sort of fiddle some extra things in here in order to get these two other items. And that's the magic of perturbing J, somehow. Right. No matter how you perturb J, you're always going to have grom of compactness. And somehow between any two J's, there's always sort of a nice family for which you can run another grom of modulized space problem and get this cubotism. So you already see that somehow perturbations of J are nicer even than perturbing finite rank bundles. So, right. So the history of the subject really is that people understood, okay, you have these local Fretholm descriptions. We have this perturbative description. In fact, we have a perturbative description of the Euler class of an Orbi bundle. That's what a lot of the papers took a long time to explain. And then how hard can it be to patch this thing together? And this is, that's literally, how hard can it be that's literally as much as there was written in literature. If there's any justice in the world, then this directly generalizes in such a way that it applies to holomorphic curves. So, and I have to say, I believe that, you know, I really didn't think it would be that hard. However, there are a couple of challenges here. So one is sort of more a comment on the Quranishi approach. So if you really sort of forget about the sort of big D bar operator, you just remember that you have local Fretholm descriptions, then you can write your modulized space as kind of a union of zero sets. Well, let me drop any kind of isotropy here for the second. Modulose some kind of transition maps. And so there's a whole lot of local Fretholm operators, right, who's zero sets will be some part of my modulized space. And once they're Fretholm sections, I can even take a finite dimensional reduction. So I could either make these Fretholm sections or if you don't know what a finite dimensional reduction of a Fretholm section is, you should ask someone this afternoon. Means you can replace, if this is Fretholm, you can replace the whole thing for this purpose of the zero set by a finite rank bundle over a finite dimensional space manifold. And right, gluing maps also fall into sort of the finite rank sections picture. Right, so you have local nice descriptions and you have some kind of overlaps, right? And you would think that you could make the Euler class locally and then somehow patch the Euler classes together. Right, how hard can it possibly be? Or you can think a little bit more globally and that's what both Chris and Duzer did is you can actually look at the zero set of the D bar operator on some space of maps. Right? So here the base was maybe maps from sigma to M, there's some bundle over it and D bar is a section that cuts out most of my modular space except, right, then I need to quotient by the isomorphisms of my base of my domain. And then this is usually still not compact so I should somehow take a compactification in some Gromov topology. Right, so again I'm seeing the zero set of a Fretholm section here and with this sort of Sartre-Smale theorem, you know, I can perturb, I can make this a manifold but as someone asked in Chris's talk, that does not mean that this isomorphism group still acts on it. Right, if I put in some kind of random differential equation perturbation here, it's not clear that, you know, that's going to be invariant under reparameterization. And it's also not clear that after I've perturbed this, I still have a nice compactness result in the Gromov topology. Right, so you add some first order operator to this and who knows what might happen. Right, so the cool thing with sort of perturbing J is that no matter how you perturb J, it's always going to be a covariant. So this isomorphism group is always going to act on the zero set and for every J you have the Gromov compactification. So it really only works when you perturb J. If you can get the transversality by perturbing J. So let's see, right, so I want to point out it's important to say always that this is actually an equivalent Fredholm section. So that's this, right, so really what we would, you know, in our dreams we can generalize this to get a covariant transversality while preserving compactness. And that's just simply wrong. There's sort of two dimensional counter examples to that. So here in our dreams we can somehow get, you know, transverse perturbations in each of the little charts that fit together under the transition maps. But the transition maps are merely somehow continuous maps that factor through this totally irregular space. So it's not at all clear how you would iteratively construct that. So we simply cannot, you know, once you stare at this situation, long enough you realize that it's really a little bit of a fundamentalist religion to believe that that theorem should have a straightforward generalization to J curves. So general M bar, we can't expect to be regularized along the finite dimensional regularization approach. Can I ask why the covariance was so important? Why what? The equivalence. Right, so my compact modular space, what I expect to have a fundamental class is really this, right? It's a zero set modulo, a finite dimensional group, and then compactified, right? So if I now perturb this by something to make it transverse, fine. Then I have a manifold here, but it doesn't even act on it anymore, right? So this is not gone there and so... Less zeros then? I mean, it's like more spot you perturb, you get the homology of the more spot critical. No, this has nothing to do with more spot. Yeah, yeah, it doesn't. So see, I would want to write... So I need to write down a prescription for how to get from this to a fundamental class. And I would like to write down a prescription that gives me the expected thing if the unperturbed one is by itself already a manifold. Right, so if the unperturbed one is a manifold, then it is d bar inverse of zero, mod this group and then compactified is a manifold, I'm taking its fundamental class. So now if this isn't transverse, I'm supposed to, okay, add a perturbation, but if I now simply forget this quotient, I'm going up by like six dimensions, so I'm getting... Even if this somehow magically was compact, I had a compactification, it would suddenly live in a different dimension. So it's clearly not the right answer. So the point is that there are some cases in which it's transverse. Right, yes. Right, right, and so whatever description, whatever definition of this regularization I write down ought to generalize the transverse case. That should be one of my axioms for the fundamental class. So if m bar is already a manifold, it ought to actually be the fundamental class. So this is important, right? So if any of you don't understand why we need equivalents, squeeze people. Sorry? But the group like an E and B, right? Yes, yeah. Right, so this is really... Right. There was a question here, yeah. Are you saying that by instead changing the j, the capital J, that the term that you get like in the difference that makes it equitarian? Yes, right, that's the beauty. No matter how I change j, the capital J, it's right, the reprametrization group is always going to act on it. Right, yes. So if I understand correctly the problem is if you perturb in some arbitrary way, when you act by the reprametrization, you get some different perturbation. Right, well I'm solving a different PDE, right? So in the Ken and Festello's treatment of gauge theory, you end up with some similar problem where you try and truncate it some energy, but then the gauge group's action doesn't preserve that. And so you construct some sort of sensor which is preserved by some group action of the homotopy. Is that hopeless here or is that...? Yes. Yeah, let me leave it at that and we can talk more. Yeah. Maybe, right, so the main thing is in gauge theory, the gauge group actually acts smoothly and so you can do some of the things I'm about to do and then the reprametrization group does not act in any pleasant way. Right, so we're in deep doo doo. What are we going to do about that? Wow, I am in doo doo here. Okay, all right, let's see. Well, I mean mainly this is to energize you why we're spending these two weeks on this. So I guess we've already talked quite a bit about the geometric approach. So the intuitive way it doesn't go through, we need to do something. And so there was the geometric approach that you've seen this morning which has prose, it works. It does all this magic, so in particular it gives a covariant transversality and preserves compactness. Moreover there's something really nice somehow it has automatic coherence which is no matter what J I take, so if my boundary or perceived boundary of the modulized space is this fiber product, right, and that comes from some kind of bubble tree or breaking compactification, then no matter how I perturb J it's always going to be like that. So. Do you just have a terminal logical question, geometric approach means only perturb J? It means, well, it really for me means find some geometric way of perturbing your equations that does these two things. And really you're going to need geometry in order to preserve aquavariance and compactness. So there are various aspects of this. And maybe I should say, right, I mean there's work by Chile but Monke makes sort of more general perturbations of J that to me falls squarely under geometric approach. You call it Hamiltonian perturbations. Right, Hamiltonian perturbations exactly. So the corner of this is it just doesn't always works, right? So it requires some kind of geometric control of curves, namely some kind of injectivity. And right, another side note here was that right people were first thinking about closed curves and their loss of injectivity means you're actually dealing with a multiply covered curve and then you get worried about its self symmetries that Chris called it, automorphism group I'm going to call it isotropy I guess, which means you get into this OB Fold world and this is why people got worried about how do you actually make the Euler class of an OB bundle. However, we're not going to talk about isotropy this week in Joel and my course at all because that's really not the problem, right? So lack of injectivity is the problem and if you still think that not somewhere injective always means multiple covered asks someone about the lantern example. So for disks you have holomorphic curves that are nowhere injective but also not multiply covered and so it's really the and for those transversality is really impossible to achieve pretty much. So it's not the self symmetries that are the problem. It's the really you need some kind of magic in order to get this going. So right and maybe right. So really the paper to read for this I would say is Fleurhofer Salomon. And of course the yes. Yeah, right. Yeah. And of course, McDuff Salomon, the book, the Bible, I guess the big book. But also right and I think right. So Chile but more because also I'm only going to write things on where I believe that if you have questions about it, I can answer them. So and I'm going to write that means I need to not write names on the board right now. So so I could not answer any questions about Foucaia's work or Tian at all. I believe I can answer most questions about Zebert's version. So and this is also the most readable of all these accounts. And because it was the most readable, it also was the only people in which mistakes were actually found. And that's why this is the one paper that didn't get published. However, I believe that's actually the one that's closest to being fixable. So or at least I understand more of it than. Well, anyways, so so there are essentially two versions of this. Foucaia at all is somehow a Kuranishi approach. And these two do something that I would call we I think I've called obstruction bundle before but I really think one should call it a stabilization approach which both somehow look easy. But the con is they either aren't easy or they are easy but they don't apply to J curves. So there's a whole bunch of papers about abstract ways of if you have a section of this kind of structure, then it has an Euler class. And I think those are all correct except I don't think any J, modular space of J curves ever gets cut out by a section like that. So that's when this piece of criticism applies. So however this stabilization here really I think is the right idea. And this is very close to sort of a more conceptually good definition of an Euler class rather than by perturbing. So the polyfold approach, I believe Siebert should have discovered polyfolds 20 years ago if he'd done things carefully at the point where mistakes happened. However, so the polyfold approach as of now talks a lot about perturbing. I believe one could make Siebert's approach rigorous by sort of using polyfolds and stabilizing and then there's less perturbations in the picture. So what happens here? We still are going to use this right idea except we're actually going to do it. And dot every i. So the idea is to really get ourselves rigorously into the setting where we have just one section of one bundle. So that was the idea here. Really I would have liked to describe my modular space by just one section and then I would have liked to apply my regularization theorem. And the question is what kind of bundle. So and what do I then mean by Fredholm? So and that's where language is going to come in. So these are things we're going to need to define. The key is that these things have global smooth structure. Whereas the stabilization approach of Siebert only had local smooth structures. And what else? Right. Just like before I should assume that the zero set is. So if you have a topological space you could give it local smooth structures by just putting charts like local homomorphisms to our end. And if you don't assume that the transition maps are smooth then you have local smooth structures but they don't match. And that's exactly what happens in all these papers. They say always there are smooth structures locally in which the operator is smooth and it's Fredholm they just don't match. So whereas probably there are a good number of compact metric spaces that you can't even find local homomorphisms to our end. So once we have this set up as just one section of one bundle there is the rough form of the theorem that hopefully we will not prove but. Now let me state this in a similar way as the regularization theorem up there. So pretty much we're just going to need to define things here in such a way that A they apply to homomorphic curves and B I have a regularization theorem. That's the goal here. So and this is the regularization theorem. So I'm going to write some set of perturbations and roughly speaking this will be compact perturbations supported near the zero set with some measure. And for that non-empty set of perturbations again I have two properties. For all p I have some notion of transversality and I still have compactness. And then for all pairs I have a co-bordism. What should I oops right so now I need to cheat. So I am going to only talk about the M polyfold version of things. And when this is the M polyfold version then these two here imply that the zero set is in fact a finite dimensional manifold compact. Might have boundary and corners but then now I could say what it means for two zero sets to be co-bordant. So pretty much exactly the same theorem except now generalized to some notion of bundle which hopefully applies to J curves. And so the obvious pro is that this seems to be a very clean way of actually getting regularized modulized spaces. And the obvious con is that there are words here that you are going to need to know. So you need to there is some language overhead but that is why you are here. So yeah. Just so we have an example or two to keep in our heads can you give like a sort of minimal example of something where you need polyfolds not to do much of a question. Whoa. Yes let's see right so if you have Gromov if you have a spherical Gromov Wittner and Varen and there is negative Schoen class spheres then do so explain why multiple covers will no matter what J you have will always give you things that are cut out non-transversely. So also if you kind of don't want to worry about the moment you don't know whether your curves are somewhat injective really. Right yes I call that somewhat injective right. So okay yeah right so the moment there is bubbling so soon as there is bubbling you can't actually use domain dependent almost complex structures anymore because the bubble is not part of your original domain so on the bubble somehow automatically J is not domain dependent and so if you can't exclude say sphere bubbling you're forced to deal with well the question of somewhere injectivity. You can use a Chilean Rontkopf but that's more sophisticated. Maybe right but then but then you can only do that if you don't have boundaries and it's not clear that you can I need they I don't think they can do SFT so. I can't do it in general right. Goodness I am totally all right core points hold on yes. Oh yes this is Hofer Bizotzki it's in there yes. And yeah we'll have complete references by the end of this week so the other con that is the coherence so that the boundary of one perturbed thing is still some kind of fiber product of these modular spaces is no longer automatic. So that is going to require some hands on construction. So the co-priced identification is not the same. Right yes right so pretty much right so this kind of says that really the perturbation on the boundary of my polyfold should be sort of the product of other perturbations but the thing is when you're just allowed to somehow construct a perturbation just like a usual function it doesn't mean that on some stratum it's given by the product of some lower strata somehow that's it that's something you need to just do hands on and that means you come out need to define these perturbations sort of iteratively you know complexity by complexity upwards and sometimes that's possible and sometimes it's actually simply not possible and then you need to sweat a little more. So interesting things happen. So that's the master approach right and so evidently right now I need to explain to you how this fits in with what we think is actually the modular space right we know how to describe the modular space as a zero set mod something and then compact define we don't know how to describe it as a zero set of something right. So there's really a total paradigm shift happening here in that right from describing the modular space as this to just one zero set right what happens here is I first perturb J then I quotient and then I sort of add in bubbles breaking and the like. So here right we realize I can't get a quverin transversality so the perturbation somehow needs to happen after the quotienting and it actually also needs to happen after adding bubbles and breaking. So the order in polyfold is that we first quotient then add bubbles and breaking and eventually we perturb so but that means there's something weird going on right so on this side I have a bundle I have my D bar operator and the whole thing is a quverient under my isomorphism group and so on this side what I should sort of do is I should divide first hope that that gives me a bundle hope that somehow my section still makes sense but not just that I kind of need to actually sort of add the broken things before I write my section and now I don't take this overline as I should maybe make it a wiggly line so this is not a closure in any topology because those are never going to be compact right they're going to be very infinite dimension this is a function space though but I need to add all the stuff that I eventually want in my compactified moduli space before I solve the equation. Do you have the numbers correct on the right side? I mean we don't divide all actually. Yes we do. Well in my interpretation of what you do you do. So well let me say what I'm. No. No. No. No. No. No. But never mind. So what happens here right and this is the core ideas of polyphol theory so my theory has always been that if I explain to some graduate some reasonably smart graduate student these two ideas then I can lock you up in a dark basement for a year and you're going to come up with polyphol theory. Yeah. So let's see so what do we need right so first of all we need a smooth structure on a space of maps modulo the isomorphism group of my surface right. Despite the fact that this action is pretty much never differentiable. So and that's what you'll hear from Joel this afternoon why this is never differentiable. So pre composition with maps is really really bad. So so this is pretty much so the only ways in which this is differentiable is what either your isomorphism group is actually finite which doesn't hold if sigma j is for example the s and s two or its r times s one with standard J or yeah a torus all of these will not have finite isotropic groups isomorphism groups and the other way is if map if this is actually a smooth if this is a finite dimensional manifold that sits in the smooth maps then it's also classically differentiable otherwise I challenge you to find any reasonable for the space topology on an infinite dimensional space of maps in which this is an even one point differentiable. So is this why algebraic geometry is easier because I can put map in exactly yeah this yeah this is exactly why algebraic geometry is easier. One of the reasons because you don't need to quotient right because instead of this you just take curves in M. You still take maps don't you? Maps is finite and how does it now maps is this inside infinity and finite dimensional. Oh because your analytic maps right yes yes maybe right oh right analytic maps might be yes yes right and so the idea is here whoops so the idea is defined so the idea is to pretty much define a notion of scale smoothness and prove that this action is scale smooth right which then gives me a scale smooth structure on this space so in that sense I would say we quotient out first. No that's not what we wrote because you guys should have quotienting out last. Okay goodness I'm glad we talked about it. Good okay. Boy do I have two minutes all right good so this is a reasonable idea to have the second idea is really not reasonable that sort of right so I ought to define to describe the neighborhood of a broken curve or nodal curve as if it was sort of in a Banach manifold right so somehow I want to I'm going to revert to Joel's example right two spheres right requires somehow maps from S2 times to M times maps from S2 to M that's the data I need here for my Banach manifold but somehow nearby I'm going to have smooth curves and really here I just need once maps from S2 to M and this is a different Banach manifold from that in particular really the neighborhood is described by something called free gluing and that should become a short map of the polyfold of the underlying polyfold or M polyfold B so which fortunately also gets me out of the problem of fix of matching notation with Joel's because I'm not going to give you notation. So what happens here you're going to somehow describe nearby curves by a pre gluing map so I'm going to take right so how do I describe all the curves so let's see this is now a point in my big base space right and nearby there are some curves that are still nodal but I've perturbed both maps but there's also non-nodal curves nearby so how do I describe them I actually write there's a rotation parameter which I'm not going to introduce and then there's going to be some gluing parameters which could be infinity and that's not good let me forget about the S1 here for a second so I have a nearby map and another nearby map and some gluing parameter and I'm going to make a pre-glued curve nearby which I get essentially from writing V1 with a long neck and V2 with a long neck and then I'm going to forget V2 beyond here and V1 beyond here and I'm going to interpolate on this little strip and glue the two curves together so that's evidently not going to be geomorphic anymore but this is actually the answer to what is the topology so this tells you what a neighborhood is. You do have a rotation parameter. Right in this case there should be a rotation parameter indeed yes which I just end of lecture great good so the key observation here is that this map is supposed to be a chart map but it is clearly not injective right because I'm forgetting data here and so the idea is to say well it factors through something that is a really weird subset here but then this is it factors through something where I have a local homeo here and this here is something like a retraction so it's a subset in here and I have a controlled way of getting from a general point here into this set and that's really the key to happiness here is because while this is a weird space it has an ambient topology so this ambient topology is matrisable and it has a smooth structure and everything is good and unlike this microphone so I shall stop here. We have questions for Catherine? You mentioned before a big philosophical difference between this idea and the kind of abstract perturbation that I sketched. Right because what you sketched sort of said I perturb here but then I have to preserve the covariance and compactness. So the point is the first step here is figuring out how to describe the whole argyllite space literally as globally as zero section of something. Correct yes right which means I need to figure out what my ambient space is right I want an ambient space for my moduli space that the compact moduli space actually sits in and that's actually that's sort of a general problem that's exactly that problem the problem of an ambient space is what you end up having to deal with with coronishi structures what you also end up having to deal with the Tian and Zebert approach it's always what is the right ambient space in which I could describe things as just one zero set. Yeah so what exactly do we expect from smoothness in this picture? Like there should be some kind of implicit function here in this class but then. Sots mail. But then in the end what we use is this classification of one dimensional manifolds right. Oh if you do one dimensional manifolds yes but how even a one dimensional manifold in order to get something from a Fredholm section that's one dimensional you need an implicit function theorem right you need a notion of transversality so you probably get away without smoothness you could say maybe C1 just for the implicit function theorem but okay it's going to get technical now so Sots mail is the reason I believe you really need smoothness and not CK. More questions? Why is why do you think scale smoothness is such an obvious idea that a graduate student could come up with it? I refer to Joel. No I don't think that's what she claimed she claimed I think. No no no I think if you explain to a student my claim was if you give somebody the definition of scale smoothness and the definition of what these retracts are then. Yes that's not quite what you said. No I know. Well my graduate students of course. Alright. Why did I? I didn't I didn't want to have to deal with trying to explain the rotation because also right so technically if I I shouldn't write S1 times this because at infinity there's no S1 parameter. So what I do at infinity here is I just don't glue and when I don't glue I don't have an S1 choice and that in fact is the reason why this is not a boundary point it's completely an interior point so you're going to see I think tomorrow Joel is going to explain to you that really for nodal curves really the gluing parameters are just a disc and where the middle point is infinity. So here pretty much if you think of these as discs right then you don't have an S1 parameter and then you see where the boundary comes from. But when there is an S1 parameter then infinity should come without that. And I'll we'll have any anyone with like philosophical questions of like why can't you just I am most happy to talk to you all week and you know particularly maybe we'll have a session this evening where we think about why can't we just. The more questions. Why can't we just thank you. Yeah. I can only ask you for my hand. Go to my mobile phone. Hello. Let me hear you.
 Awesome.  Thank you.  And thanks for putting this on.  I believe the world will be a better place.  I should also give you a renormalization instruction for any criticism that I might utter.  If I don't tell you that Ethereum is complete bullshit, I have not read it.  And I think it's not interesting.  If I do tell you it's complete bullshit, it just means that you should actually understand  the proof for yourself and not trust it.  And by that sort of criterion, really, I think all proofs in mathematics should be complete  bullshit because, well, you should understand them.  So I'm actually not going to talk about analysis pretty much at all.  I'm going to talk to you about why we need to worry about analysis.  So I'm also going to try and sort of ground us in a big picture of what are we actually  doing here.  So I'm very happy that somehow the term I think that I coined regularization has taken  root.  But I want to tell you what that actually means from my point of view.  So when you regularize a modular space, well, and when I say modular space, I'm usually  going to mean pseudo-amorphic curve modular space.  So if you're given a compact modular space, which will always be called m bar, usually  of j curves, I would like to assign to it in some sort of unique way, well, one or two  things.  So in the Grom of Whitten world, I would like to actually give it a fundamental class.  But then, of course, the question is, what do I mean by fundamental class in the homology  of something that might just be a compact metric space?  So if this was a manifold, it would have a fundamental class.  If it is not a manifold, we don't know what its fundamental class is.  But the nice thing with trying to actually construct that right in here is that from  the modular space, we have evaluation maps in Grom of Whitten.  And so if we had a class in there, we could push it forward directly with evaluation maps.  That's why I think that's the cleanest way of saying what we want is to make a homology  class straight in there.  So this is really the Grom of Whitten case.  And pretty much in all other theories, Fleur, SFT, anything, infinity related, you're going  to expect your modular space to have boundary.  And so really what you want to do is you want to somehow count things and then get a relation  from the boundary.  And so what we'd like there is some kind of cobaltism class of closed, no, not closed,  but compact things.  So here you could get this from maybe a compact manifold.  Well, at least that would have a homology and how I get it into M-bar is not so clear.  But here I'm not going to say closed because they're going to have boundary compact manifolds.  Right, in general, they might be weighted branched.  And they're going to come from some choices.  Knew that Chris Wendell up here had up here, but then unfortunately somebody diligently  removed this.  So your task for this next hour is also to figure out what's completely different in  the philosophy that I project here than it is on Chris's board about abstract perturbations.  And I'm happy to discuss that in office hours.  So really these manifolds are going to come from some kind of choices of perturbations.  Knew that we'll talk a lot more about.  The main thing is it's going to have boundary in general and usually also then corners.  And the algebraic identities usually come from identifying the boundary with some kind  of fiber product of products of the modular space itself.  So if this seems totally alien, maybe think about Fleur theory.  And what I'm thinking about here is I'm packaging all Fleur trajectory spaces in this M-bar  new.  So no matter between which critical points and no matter between of what index.  And then this just says that somehow the boundary is given by broken ones.  So that's not the whole thing is not going to be compact.  Yes.  Well, indeed.  Yes.  So that's why this is the philosophical overview.  But I could do this on a certain up to a certain energy.  And then this would be true.  So good.  So this is my goal.  This is what I mean by regularizing a modular space.  Assigning to something that I can count or that I could integrate over and that has some  kind of nice boundary structure that gives me the algebraic identities I would like.  So well, I don't need to write this.  What are my tools here?  Right?  So in the algebraic geometry, then I describe J curves on my modular spaces are really modular  spaces of subvarieties or not manifolds, but they're sort of subsets of an ambient algebraic  variety that are cut out by some algebraic equation.  And so then I can do algebra to describe my modular space.  And the thing that Gromov realized was that, well, if my J isn't integrable, I can't  quite do algebra, but maybe I can do exactly the same things that algebraists do with PDE  methods.  So that's why there's a lot of analysis here, because all the magic that you can do with  algebra we now have to do with analysis.  However, we always have somehow this intuition that everything that's true in algebraic geometry  should be true in symplectic geometry.  Because the general Gromov written in this, the quick connection with Gromov written  in this was later based on the paper by Gromov.  Right.  OK.  Yeah.  So some things flow back.  Yes.  Good.  Catherine, can you stay away from the last four inches?  Yes.  This.  OK.  Here.  Awesome.  Good.  Yes, I can.  So let's see.  What is my intuition here?  So well, I should say something.  So first of all, I'm going to do an analytic description of M bar, namely in terms of solutions  of a PDE, but then there are two other things.  And so I'm just going to roughly think about having a fixed Riemann surface and a fixed  almost complex structure on a symplectic manifold.  But I have to do two more things.  I have to mod out reprametrizations.  Because as an algebraic geometry, I'd like to actually count the curves, not the maps.  So an image in M bar I could parameterize in different ways by maps.  And so I want to quote you out by that ambiguity.  And then we're always going to want something compact.  So I'm adding Gromov compactifications.  And so maybe just as a side note to the examples that Duzer gave, right, there were beautiful  examples of non-compactness coming from Mark points running together.  However, that's not the worst of our problems.  So usually really the hard problems with Gromov compactification come from energy concentrating.  And that's where you get the sphere bubbles that then have an unstable domain.  And this is where really all analytic hell breaks lose.  So that's really a PDE blow up situation.  So you really do need this Gromov compactification even if we have no Mark points.  So if that's how we described M bar, then we can do, well, two things.  So first of all, as you've already seen, I think in both talks, there's a more or less  well, local or global Fredholm description together with gluing maps, which we'll sort  of start talking more and more about.  And so very roughly what that does is it somehow, this is a lie, but the intuition is that you're  writing your modular space as the zero set of some kind of section.  Actually, mainly the section here is a lie.  So but let's pretend, right?  So at least to some extent, that seemed to be the description.  And then the reason you believe that these things should be things that have a fundamental  class.  I believe is really this finite dimensional regularization theorem, which says, well,  let's pretend for the moment that my nice compact modular space of J curves is actually  cut out from a finite rank vector bundle over a finite dimensional base.  And let's say that's even smooth.  I guess let me make it an oriented vector bundle.  Then well, first of all, even without the section, these things have an Euler class,  right?  So that's just the Euler class of the bundle, but you can pull that back and let it sit  in the base.  And I guess the theorem here, well, you can construct that with algebraic topology or  you can construct it by taking a section and perturbing it.  So there's various methods of constructing Euler classes even.  And so I think the main one that we talk about in some plectic geometry is to do it by perturbations.  So I would like to be able to just say that my Euler class is actually maybe the fundamental  class of some perturbed section of the zero set.  So that would then sit in that homology.  I believe I should actually add, oh right, so usually you need the base to be compact  to get an Euler class at least.  But I believe you can do this as long as the zero set is compact.  You mean as a homology class?  Right, yeah.  Right, otherwise it's just not unique as a homology class.  There's no chance you'll ever have that.  And also, right, so I'd like B and E to both be finite dimensional.  But the finite dimensionality is actually not the problem.  I think you could also take them infinite dimensional and then you would just need the  section to be Fredholm.  And then the same theorem is going to hold with exactly, essentially the same proof.  Right.  But what do I mean by this really?  So how do I construct?  What do I mean by having perturbations that give me this Euler class?  And this is where, right, this is where the word generic comes in.  That's what I was going to strike in big and red from Chris's board.  Because really, so people always say take a generic perturbation or a generic J. So if  you give me a J, how do I know whether it's generic?  Generic is not an adjective that goes with a single element of a set.  Right.  Somehow we say generic when there is a commiga set and then the elements of that set are  the generic elements somehow.  And so, yeah.  So I think that's one danger because people often say generic in situations where it's  not even clear what the Banach space should be from which you're taking something.  And also, I mean, I said somehow second category was wrong.  But I actually don't quite understand why we're so focused on having sort of a commiga  set or a large set or a full measure set.  So what I'm going to say is really all you need is a set that's not empty.  And in fact, sets of second category are non-empty.  So in that sense, that's a perfectly fine thing to prove when everything goes through  if you just prove that something is second category.  I just don't know how you ever prove that something is second category without proving  that it's commiga.  But anyways.  So there is a non-empty set in the smooth sections.  So that for any perturbation in this set, so these are the good perturbations.  The perturbed section is not just transverse, but its zero set is still compact.  Yes, we'll do.  Thank you.  So that tells me that it gets a smooth structure.  And if that's compacted as a fundamental class, and then I can embed that into B, and so I  get a homology or I get a cycle there.  But now what if I take two different perturbations?  I would like them to actually have the same fundamental class.  And usually we prove that by proving that the two zero sets are co-bordant.  So people often talk about the transversality problem in holomorphic curves.  And transversality is really just this bit.  Whereas somehow regularization for me, when I say regularization, I mean the whole package.  I mean transversality, preserving compactness, and then having uniqueness up to co-bordism.  So if all you care about is that this set p is non-empty, how do you know there are  two sort of disjoints at it?  Right.  Yes.  So it's sort of canonical.  Right.  I mean, I want to, so usually actually it's not, so usually the statement is not there  exists the p, right?  Usually the statement is p, which I define as something.  So it's a specific set.  And I don't need to know that that's commiga, I just need to know that that set is not empty.  Right.  So for example, in Chris's theorem, I think the statement is going to be that it's a set  of j's in this curly j.  And the j's are defined exactly by some transfer, by regularity, right?  By subjectivity of the linearized operators.  Can I make one comment about this?  So it's not that I disagree, but when I define what conditions those j's have to satisfy,  it's almost always going to be that it satisfies a countable list of conditions.  Right.  And so for each of those individual conditions in the list, it is pretty important to know  that there's a commiga set, because I need to know that the intersection of those sets  is not empty.  At the end, I only care that it's not empty, but how else would I possibly know that?  Yes.  Is it true?  Sometimes you have some kind of special, like if you want to prove like most people's  homology kind of things, you have something that you know, and you want to deform that  thing that you know to, and then you want to have some kind of complex, almost complex  structure that's near a given possibly degenerate.  Right.  Sure.  Sometimes you'd like to know that kind of density.  Yeah.  This is just for existence of a possible definition.  Yeah.  Yeah.  But you could see you can write, I mean you can also, again, you don't need this to be  dense.  You just need to have it be dense near a given point.  Right.  So you could say, take a set and say, I fixed the support and I fixed some sort of size  of the perturbation.  And no matter how small the epsilon and delta are, the set is always non-empty.  Right.  So since which point, you don't know which point you're going to meet in the beginning  of the course.  Right.  Yes.  But I mean, so it's right, so the statement that given any J there isn't arbitrarily close  by J that is regular is also different from proving that something is commiga.  Right.  So.  I've been, well you're explaining philosophy.  Yes.  Why do you take a perturbative definition of the order class as opposed to like pull  back from the grass, might end or.  Right, because I'm going to want to generalize this in such a way that it applies to holomorphic  groups.  I mean that definition would not be anything, right?  It could be a topological space for that definition.  Let's discuss that once we're sort of deeper down in the actual application.  And actually, right, the challenges are going to come momentarily and then we can talk about  whether different approaches will solve that.  What was I going to say, P?  Right.  So, in this case here already, I think the set, so I think Assad's male theorem here  would tell you that, you know, the set of ones that, so if I defined this set by the  ones that are transverse to zero, I would indeed get a commiga subset in there.  However, it's not clear to me that for all the perturbations for which I have transversality,  I'm also going to preserve compactness.  Right.  So, in fact, I think I could cook up a counter example.  And then it's also not clear that you're always going to have uniqueness up to cubotism.  So it's often you have to sort of fiddle some extra things in here in order to get these  two other items.  And that's the magic of perturbing J, somehow.  Right.  No matter how you perturb J, you're always going to have grom of compactness.  And somehow between any two J's, there's always sort of a nice family for which you  can run another grom of modulized space problem and get this cubotism.  So you already see that somehow perturbations of J are nicer even than perturbing finite  rank bundles.  So, right.  So the history of the subject really is that people understood, okay, you have these local  Fretholm descriptions.  We have this perturbative description.  In fact, we have a perturbative description of the Euler class of an Orbi bundle.  That's what a lot of the papers took a long time to explain.  And then how hard can it be to patch this thing together?  And this is, that's literally, how hard can it be that's literally as much as there was  written in literature.  If there's any justice in the world, then this directly generalizes in such a way that  it applies to holomorphic curves.  So, and I have to say, I believe that, you know, I really didn't think it would be that  hard.  However, there are a couple of challenges here.  So one is sort of more a comment on the Quranishi approach.  So if you really sort of forget about the sort of big D bar operator, you just remember  that you have local Fretholm descriptions, then you can write your modulized space as  kind of a union of zero sets.  Well, let me drop any kind of isotropy here for the second.  Modulose some kind of transition maps.  And so there's a whole lot of local Fretholm operators, right, who's zero sets will be  some part of my modulized space.  And once they're Fretholm sections, I can even take a finite dimensional reduction.  So I could either make these Fretholm sections or if you don't know what a finite dimensional  reduction of a Fretholm section is, you should ask someone this afternoon.  Means you can replace, if this is Fretholm, you can replace the whole thing for this purpose  of the zero set by a finite rank bundle over a finite dimensional space manifold.  And right, gluing maps also fall into sort of the finite rank sections picture.  Right, so you have local nice descriptions and you have some kind of overlaps, right?  And you would think that you could make the Euler class locally and then somehow patch  the Euler classes together.  Right, how hard can it possibly be?  Or you can think a little bit more globally and that's what both Chris and Duzer did is  you can actually look at the zero set of the D bar operator on some space of maps.  Right?  So here the base was maybe maps from sigma to M, there's some bundle over it and D bar  is a section that cuts out most of my modular space except, right, then I need to quotient  by the isomorphisms of my base of my domain.  And then this is usually still not compact so I should somehow take a compactification  in some Gromov topology.  Right, so again I'm seeing the zero set of a Fretholm section here and with this sort  of Sartre-Smale theorem, you know, I can perturb, I can make this a manifold but as someone  asked in Chris's talk, that does not mean that this isomorphism group still acts on  it.  Right, if I put in some kind of random differential equation perturbation here, it's not clear  that, you know, that's going to be invariant under reparameterization.  And it's also not clear that after I've perturbed this, I still have a nice compactness result  in the Gromov topology.  Right, so you add some first order operator to this and who knows what might happen.  Right, so the cool thing with sort of perturbing J is that no matter how you perturb J, it's  always going to be a covariant.  So this isomorphism group is always going to act on the zero set and for every J you  have the Gromov compactification.  So it really only works when you perturb J. If you can get the transversality by perturbing  J.  So let's see, right, so I want to point out it's important to say always that this is  actually an equivalent Fredholm section.  So that's this, right, so really what we would, you know, in our dreams we can generalize  this to get a covariant transversality while preserving compactness.  And that's just simply wrong.  There's sort of two dimensional counter examples to that.  So here in our dreams we can somehow get, you know, transverse perturbations in each  of the little charts that fit together under the transition maps.  But the transition maps are merely somehow continuous maps that factor through this totally  irregular space.  So it's not at all clear how you would iteratively construct that.  So we simply cannot, you know, once you stare at this situation, long enough you realize  that it's really a little bit of a fundamentalist religion to believe that that theorem should  have a straightforward generalization to J curves.  So general M bar, we can't expect to be regularized along the finite dimensional regularization  approach.  Can I ask why the covariance was so important?  Why what?  The equivalence.  Right, so my compact modular space, what I expect to have a fundamental class is really  this, right?  It's a zero set modulo, a finite dimensional group, and then compactified, right?  So if I now perturb this by something to make it transverse, fine.  Then I have a manifold here, but it doesn't even act on it anymore, right?  So this is not gone there and so...  Less zeros then?  I mean, it's like more spot you perturb, you get the homology of the more spot critical.  No, this has nothing to do with more spot.  Yeah, yeah, it doesn't.  So see, I would want to write...  So I need to write down a prescription for how to get from this to a fundamental class.  And I would like to write down a prescription that gives me the expected thing if the unperturbed  one is by itself already a manifold.  Right, so if the unperturbed one is a manifold, then it is d bar inverse of zero, mod this  group and then compactified is a manifold, I'm taking its fundamental class.  So now if this isn't transverse, I'm supposed to, okay, add a perturbation, but if I now  simply forget this quotient, I'm going up by like six dimensions, so I'm getting...  Even if this somehow magically was compact, I had a compactification, it would suddenly  live in a different dimension.  So it's clearly not the right answer.  So the point is that there are some cases in which it's transverse.  Right, yes.  Right, right, and so whatever description, whatever definition of this regularization  I write down ought to generalize the transverse case.  That should be one of my axioms for the fundamental class.  So if m bar is already a manifold, it ought to actually be the fundamental class.  So this is important, right?  So if any of you don't understand why we need equivalents, squeeze people.  Sorry?  But the group like an E and B, right?  Yes, yeah.  Right, so this is really...  Right.  There was a question here, yeah.  Are you saying that by instead changing the j, the capital J, that the term that you get  like in the difference that makes it equitarian?  Yes, right, that's the beauty.  No matter how I change j, the capital J, it's right, the reprametrization group is always  going to act on it.  Right, yes.  So if I understand correctly the problem is if you perturb in some arbitrary way, when  you act by the reprametrization, you get some different perturbation.  Right, well I'm solving a different PDE, right?  So in the Ken and Festello's treatment of gauge theory, you end up with some similar  problem where you try and truncate it some energy, but then the gauge group's action  doesn't preserve that.  And so you construct some sort of sensor which is preserved by some group action of the homotopy.  Is that hopeless here or is that...?  Yes.  Yeah, let me leave it at that and we can talk more.  Yeah.  Maybe, right, so the main thing is in gauge theory, the gauge group actually acts smoothly  and so you can do some of the things I'm about to do and then the reprametrization group  does not act in any pleasant way.  Right, so we're in deep doo doo.  What are we going to do about that?  Wow, I am in doo doo here.  Okay, all right, let's see.  Well, I mean mainly this is to energize you why we're spending these two weeks on this.  So I guess we've already talked quite a bit about the geometric approach.  So the intuitive way it doesn't go through, we need to do something.  And so there was the geometric approach that you've seen this morning which has prose,  it works.  It does all this magic, so in particular it gives a covariant transversality and preserves  compactness.  Moreover there's something really nice somehow it has automatic coherence  which is no matter what J I take, so if my boundary or perceived boundary of the modulized  space is this fiber product, right, and that comes from some kind of bubble tree or breaking  compactification, then no matter how I perturb J it's always going to be like that.  So.  Do you just have a terminal logical question, geometric approach means only perturb J?  It means, well, it really for me means find some geometric way of perturbing your equations  that does these two things.  And really you're going to need geometry in order to preserve aquavariance and compactness.  So there are various aspects of this.  And maybe I should say, right, I mean there's work by Chile but Monke makes sort of more  general perturbations of J that to me falls squarely under geometric approach.  You call it Hamiltonian perturbations.  Right, Hamiltonian perturbations exactly.  So the corner of this is it just doesn't always works, right?  So it requires some kind of geometric control of curves, namely some kind of injectivity.  And right, another side note here was that right people were first thinking about closed  curves and their loss of injectivity means you're actually dealing with a multiply covered  curve and then you get worried about its self symmetries that Chris called it, automorphism  group I'm going to call it isotropy I guess, which means you get into this OB Fold world  and this is why people got worried about how do you actually make the Euler class of an  OB bundle.  However, we're not going to talk about isotropy this week in Joel and my course at all because  that's really not the problem, right?  So lack of injectivity is the problem and if you still think that not somewhere injective  always means multiple covered asks someone about the lantern example.  So for disks you have holomorphic curves that are nowhere injective but also not multiply  covered and so it's really the and for those transversality is really impossible to achieve  pretty much.  So it's not the self symmetries that are the problem.  It's the really you need some kind of magic in order to get this going.  So right and maybe right.  So really the paper to read for this I would say is Fleurhofer Salomon.  And of course the yes.  Yeah, right.  Yeah.  And of course, McDuff Salomon, the book, the Bible, I guess the big book.  But also right and I think right.  So Chile but more because also I'm only going to write things on where I believe that if  you have questions about it, I can answer them.  So and I'm going to write that means I need to not write names on the board right now.  So so I could not answer any questions about Foucaia's work or Tian at all.  I believe I can answer most questions about Zebert's version.  So and this is also the most readable of all these accounts.  And because it was the most readable, it also was the only people in which mistakes were  actually found.  And that's why this is the one paper that didn't get published.  However, I believe that's actually the one that's closest to being fixable.  So or at least I understand more of it than.  Well, anyways, so so there are essentially two versions of this.  Foucaia at all is somehow a Kuranishi approach.  And these two do something that I would call we I think I've called obstruction bundle before  but I really think one should call it a stabilization approach which both somehow look easy.  But the con is they either aren't easy or they are easy but they don't apply to J curves.  So there's a whole bunch of papers about abstract ways of if you have a section of this kind  of structure, then it has an Euler class.  And I think those are all correct except I don't think any J, modular space of J curves  ever gets cut out by a section like that.  So that's when this piece of criticism applies.  So however this stabilization here really I think is the right idea.  And this is very close to sort of a more conceptually good definition of an Euler class rather than  by perturbing.  So the polyfold approach, I believe Siebert should have discovered polyfolds 20 years  ago if he'd done things carefully at the point where mistakes happened.  However, so the polyfold approach as of now talks a lot about perturbing.  I believe one could make Siebert's approach rigorous by sort of using polyfolds and stabilizing  and then there's less perturbations in the picture.  So what happens here?  We still are going to use this right idea except we're actually going to do it.  And dot every i.  So the idea is to really get ourselves rigorously into the setting where we have just one section  of one bundle.  So that was the idea here.  Really I would have liked to describe my modular space by just one section and then I would  have liked to apply my regularization theorem.  And the question is what kind of bundle.  So and what do I then mean by Fredholm?  So and that's where language is going to come in.  So these are things we're going to need to define.  The key is that these things have global smooth structure.  Whereas the stabilization approach of Siebert only had local smooth structures.  And what else?  Right.  Just like before I should assume that the zero set is.  So if you have a topological space you could give it local smooth structures by just putting  charts like local homomorphisms to our end.  And if you don't assume that the transition maps are smooth then you have local smooth  structures but they don't match.  And that's exactly what happens in all these papers.  They say always there are smooth structures locally in which the operator is smooth and  it's Fredholm they just don't match.  So whereas probably there are a good number of compact metric spaces that you can't even  find local homomorphisms to our end.  So once we have this set up as just one section of one bundle there is the rough form of the  theorem that hopefully we will not prove but.  Now let me state this in a similar way as the regularization theorem up there.  So pretty much we're just going to need to define things here in such a way that A they  apply to homomorphic curves and B I have a regularization theorem.  That's the goal here.  So and this is the regularization theorem.  So I'm going to write some set of perturbations and roughly speaking this will be compact  perturbations supported near the zero set with some measure.  And for that non-empty set of perturbations again I have two properties.  For all p I have some notion of transversality and I still have compactness.  And then for all pairs I have a co-bordism.  What should I oops right so now I need to cheat.  So I am going to only talk about the M polyfold version of things.  And when this is the M polyfold version then these two here imply that the zero set is  in fact a finite dimensional manifold compact.  Might have boundary and corners but then now I could say what it means for two zero sets  to be co-bordant.  So pretty much exactly the same theorem except now generalized to some notion of bundle which  hopefully applies to J curves.  And so the obvious pro is that this seems to be a very clean way of actually getting  regularized modulized spaces.  And the obvious con is that there are words here that you are going to need to know.  So you need to there is some language overhead but that is why you are here.  So yeah.  Just so we have an example or two to keep in our heads can you give like a sort of minimal  example of something where you need polyfolds not to do much of a question.  Whoa.  Yes let's see right so if you have Gromov if you have a spherical Gromov Wittner and  Varen and there is negative Schoen class spheres then do so explain why multiple covers will  no matter what J you have will always give you things that are cut out non-transversely.  So also if you kind of don't want to worry about the moment you don't know whether your  curves are somewhat injective really.  Right yes I call that somewhat injective right.  So okay yeah right so the moment there is bubbling so soon as there is bubbling you  can't actually use domain dependent almost complex structures anymore because the bubble  is not part of your original domain so on the bubble somehow automatically J is not domain  dependent and so if you can't exclude say sphere bubbling you're forced to deal with  well the question of somewhere injectivity.  You can use a Chilean Rontkopf but that's more sophisticated.  Maybe right but then but then you can only do that if you don't have boundaries and it's  not clear that you can I need they I don't think they can do SFT so.  I can't do it in general right.  Goodness I am totally all right core points hold on yes.  Oh yes this is Hofer Bizotzki it's in there yes.  And yeah we'll have complete references by the end of this week so the other con that  is the coherence so that the boundary of one perturbed thing is still some kind of fiber  product of these modular spaces is no longer automatic.  So that is going to require some hands on construction.  So the co-priced identification is not the same.  Right yes right so pretty much right so this kind of says that really the perturbation  on the boundary of my polyfold should be sort of the product of other perturbations but  the thing is when you're just allowed to somehow construct a perturbation just like a usual  function it doesn't mean that on some stratum it's given by the product of some lower strata  somehow that's it that's something you need to just do hands on and that means you come  out need to define these perturbations sort of iteratively you know complexity by complexity  upwards and sometimes that's possible and sometimes it's actually simply not possible  and then you need to sweat a little more.  So interesting things happen.  So that's the master approach right and so evidently right now I need to explain to you  how this fits in with what we think is actually the modular space right we know how to describe  the modular space as a zero set mod something and then compact define we don't know how  to describe it as a zero set of something right.  So there's really a total paradigm shift happening here in that right from describing  the modular space as this to just one zero set right what happens here is I first perturb  J then I quotient and then I sort of add in bubbles breaking and the like.  So here right we realize I can't get a quverin transversality so the perturbation somehow  needs to happen after the quotienting and it actually also needs to happen after adding  bubbles and breaking.  So the order in polyfold is that we first quotient then add bubbles and breaking and  eventually we perturb so but that means there's something weird going on right so on this side  I have a bundle I have my D bar operator and the whole thing is a quverient under my isomorphism  group and so on this side what I should sort of do is I should divide first hope that that  gives me a bundle hope that somehow my section still makes sense but not just that I kind  of need to actually sort of add the broken things before I write my section and now I  don't take this overline as I should maybe make it a wiggly line so this is not a closure  in any topology because those are never going to be compact right they're going to be very  infinite dimension this is a function space though but I need to add all the stuff that  I eventually want in my compactified moduli space before I solve the equation.  Do you have the numbers correct on the right side?  I mean we don't divide all actually.  Yes we do.  Well in my interpretation of what you do you do.  So well let me say what I'm.  No.  No.  No.  No.  No.  No.  But never mind.  So what happens here right and this is the core ideas of polyphol theory so my theory has  always been that if I explain to some graduate some reasonably smart graduate student these  two ideas then I can lock you up in a dark basement for a year and you're going to come  up with polyphol theory.  Yeah.  So let's see so what do we need right so first of all we need a smooth structure on  a space of maps modulo the isomorphism group of my surface right.  Despite the fact that this action is pretty much never differentiable.  So and that's what you'll hear from Joel this afternoon why this is never differentiable.  So pre composition with maps is really really bad.  So so this is pretty much so the only ways in which this is differentiable is what either  your isomorphism group is actually finite which doesn't hold if sigma j is for example  the s and s two or its r times s one with standard J or yeah a torus all of these will  not have finite isotropic groups isomorphism groups and the other way is if map if this  is actually a smooth if this is a finite dimensional manifold that sits in the smooth maps then  it's also classically differentiable otherwise I challenge you to find any reasonable for  the space topology on an infinite dimensional space of maps in which this is an even one  point differentiable.  So is this why algebraic geometry is easier because I can put map in exactly yeah this  yeah this is exactly why algebraic geometry is easier.  One of the reasons because you don't need to quotient right because instead of this  you just take curves in M.  You still take maps don't you?  Maps is finite and how does it now maps is this inside infinity and finite dimensional.  Oh because your analytic maps right yes yes maybe right oh right analytic maps might be  yes yes right and so the idea is here whoops so the idea is defined so the idea is to pretty  much define a notion of scale smoothness and prove that this action is scale smooth right  which then gives me a scale smooth structure on this space so in that sense I would say  we quotient out first.  No that's not what we wrote because you guys should have quotienting out last.  Okay goodness I'm glad we talked about it.  Good okay.  Boy do I have two minutes all right good so this is a reasonable idea to have the second  idea is really not reasonable that sort of right so I ought to define to describe the  neighborhood of a broken curve or nodal curve as if it was sort of in a Banach manifold  right so somehow I want to I'm going to revert to Joel's example right two spheres right  requires somehow maps from S2 times to M times maps from S2 to M that's the data I need here  for my Banach manifold but somehow nearby I'm going to have smooth curves and really  here I just need once maps from S2 to M and this is a different Banach manifold from that  in particular really the neighborhood is described by something called free gluing and that should  become a short map of the polyfold of the underlying polyfold or M polyfold B so which  fortunately also gets me out of the problem of fix of matching notation with Joel's because  I'm not going to give you notation.  So what happens here you're going to somehow describe nearby curves by a pre gluing map  so I'm going to take right so how do I describe all the curves so let's see this is now a  point in my big base space right and nearby there are some curves that are still nodal  but I've perturbed both maps but there's also non-nodal curves nearby so how do I describe  them I actually write there's a rotation parameter which I'm not going to introduce and then  there's going to be some gluing parameters which could be infinity and that's not good  let me forget about the S1 here for a second so I have a nearby map and another nearby  map and some gluing parameter and I'm going to make a pre-glued curve nearby which I get  essentially from writing V1 with a long neck and V2 with a long neck and then I'm going  to forget V2 beyond here and V1 beyond here and I'm going to interpolate on this little  strip and glue the two curves together so that's evidently not going to be geomorphic  anymore but this is actually the answer to what is the topology so this tells you what  a neighborhood is.  You do have a rotation parameter.  Right in this case there should be a rotation parameter indeed yes which I just end of lecture  great good so the key observation here is that this map is supposed to be a chart map  but it is clearly not injective right because I'm forgetting data here and so the idea is  to say well it factors through something that is a really weird subset here but then this  is it factors through something where I have a local homeo here and this here is something  like a retraction so it's a subset in here and I have a controlled way of getting from  a general point here into this set and that's really the key to happiness here is because  while this is a weird space it has an ambient topology so this ambient topology is matrisable  and it has a smooth structure and everything is good and unlike this microphone so I shall  stop here.  We have questions for Catherine?  You mentioned before a big philosophical difference between this idea and the kind of abstract  perturbation that I sketched.  Right because what you sketched sort of said I perturb here but then I have to preserve  the covariance and compactness.  So the point is the first step here is figuring out how to describe the whole argyllite space  literally as globally as zero section of something.  Correct yes right which means I need to figure out what my ambient space is right I want  an ambient space for my moduli space that the compact moduli space actually sits in  and that's actually that's sort of a general problem that's exactly that problem the problem  of an ambient space is what you end up having to deal with with coronishi structures what  you also end up having to deal with the Tian and Zebert approach it's always what is the  right ambient space in which I could describe things as just one zero set.  Yeah so what exactly do we expect from smoothness in this picture?  Like there should be some kind of implicit function here in this class but then.  Sots mail.  But then in the end what we use is this classification of one dimensional manifolds right.  Oh if you do one dimensional manifolds yes but how even a one dimensional manifold in  order to get something from a Fredholm section that's one dimensional you need an implicit  function theorem right you need a notion of transversality so you probably get away  without smoothness you could say maybe C1 just for the implicit function theorem but  okay it's going to get technical now so Sots mail is the reason I believe you really need  smoothness and not CK.  More questions?  Why is why do you think scale smoothness is such an obvious idea that a graduate student  could come up with it?  I refer to Joel.  No I don't think that's what she claimed she claimed I think.  No no no I think if you explain to a student my claim was if you give somebody the definition  of scale smoothness and the definition of what these retracts are then.  Yes that's not quite what you said.  No I know.  Well my graduate students of course.  Alright.  Why did I?  I didn't I didn't want to have to deal with trying to explain the rotation because also  right so technically if I I shouldn't write S1 times this because at infinity there's  no S1 parameter.  So what I do at infinity here is I just don't glue and when I don't glue I don't have an  S1 choice and that in fact is the reason why this is not a boundary point it's completely  an interior point so you're going to see I think tomorrow Joel is going to explain  to you that really for nodal curves really the gluing parameters are just a disc and  where the middle point is infinity.  So here pretty much if you think of these as discs right then you don't have an S1 parameter  and then you see where the boundary comes from.  But when there is an S1 parameter then infinity should come without that.  And I'll we'll have any anyone with like philosophical questions of like why can't you just I am  most happy to talk to you all week and you know particularly maybe we'll have a session  this evening where we think about why can't we just.  The more questions.  Why can't we just thank you.  Yeah.  I can only ask you for my hand.  Go to my mobile phone.  Hello.  Let me hear you.
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