 And thanks to the organizers for inviting me. It's a pleasure to be here. So I'm talking about Fredholm theory and Lean Mumford spaces for something you haven't heard of called witch balls. All of this is joint work with Katzchen Werheim. And you can see what we've written so far in these two preprints. And so here's the plan. So I'm going to give you some motivation. I'm going to tell you what quilts and correspondences are. I'll tell you the blueprint for the algebra that we've come up with for this thing called simp, this two category-like thing. And finally, in the last section, I'll tell you a new result about Fredholmness for quilts with certain kinds of singularities. So four could be called the P's section. So let me recall for you that if we have a symplectic manifold, then under favorable circumstances, we can define the Foucaille category associated to it. So this is an infinity category. The objects are certain kinds of Lagrangians inside of M. Depending on your situation, they'll be required to satisfy some hypotheses. And if you take two morphisms, two Lagrangians, then as long as these guys intersect transversely, the morphisms are defined to be the free vector space generated by their intersections. And in general, it's equal to the third complex. OK. And then the last thing I want to tell you about Foucaille is that there are composition operations that eat D inputs for D, at least one, which is something that any infinity category is required to do. And the D-area composition operation is defined by counting pseudoholmorphic D plus 1 So the images of these pseudoholmorphic polygons in M look something like this. OK. So this is sort of the main thing we care about in the talk. So before I move on, any questions about this? All right. So several years ago, Verheim and Woodward defined something that they called Quilted Flur theory. And the point of Quilted Flur theory is to build functoriality into the Foucaille category. So relate the Foucaille categories of different symplectic manifolds. So in particular, they had this idea that if you take Lagrangian inner product is of symplectic manifolds. So L01 sits inside M0 minus times M1. The minus just means flip the sign of the symplectic form on M0. Then that should give rise to an infinity functor FL01 from Fouc M0 to Fouc M1. And if this looks funny to you, I can at least tell you some motivation for why the right notion of morphism from M0 to M1 should be a Lagrangian like this, which is Weinstein's symplectic creed. Everything is Lagrangian. And therefore, morphisms between symplectic manifolds should be Lagrangians. And I'll tell you more about these Lagrangian correspondences in the next section. But for now, if you want to keep in mind, you can think of the graph of a symplectic morphism from M0 to M1. And that's just a real stupid question. So the four gene complex here for the morphisms, how is it how is that different if I switch L and L prime? I have to have a morphism. So morphism has to go between two objects, so it's directed. So what's the difference between CFLL prime and CFL prime and L prime? They're closely related. So in good circumstances, there's a duality between those things. OK, now it turned out, when they tried to carry out this goal, that this is really a non-trivial thing to do. On the object level, it's not so hard to understand how this functor should work. Well, it's not trivial, but it's not as difficult as what happens on the morphism level where you're counting something pseudo-homorphic, but it's going to be these funny objects with singularities that will be the subject of today's talk. And Verheim-Woodward's resolution was to study some related objects where the analysis was not so hard, but where the result was not quite this. So the goals of Katrin and me are, first of all, to actually do the analysis for these singular objects. So let me call this goal star. And then the second goal is to enlarge this algebraic framework. So their algebraic picture is Lagrangian correspondence should induce an infinity functor. And the bigger algebraic framework is that we actually expect there to be a infinity 2 category. So just think of this as some kind of infinity version of 2 category, where the objects are some large class of symplectic manifolds. The morphisms are Lagrangian correspondences. And the two morphisms are Flurr-coachanes. So the picture is something like m0 and m1 are objects. Lagrangian correspondences give you your morphisms. Your one morphisms, anyway. And your two morphisms are given by Flurr-coachanes. So I'll say more about this algebraic picture in the third section. So what I'll do today is I'll, if I have time, recap the progress we've made. And then the thing that I'll definitely say is I'll explain a new result for fretfulness of the pseudoholmorphic objects used to define the structure maps. Before I move on to section 2, I want to mention the result that Verheim and Oderb were able to prove. So like I said, they worked with some pseudoholmorphic things which were easier to deal with analytically. But they got this result which was a little bit different than what they originally aimed at. So if you assume that you're in the either monotone or exact setting, then they showed that a Lagrangian correspondence gives rise to an infinity functor, but not between fukhaya categories, between these things called extended fukhaya categories, which they invented for this purpose. The objects of an extended fukhaya category not being Lagrangians, but by formally composable sequences of Lagrangian correspondences. Yes, thank you. Oh, and final thing is Vec in the audience. No. OK. Well, anyway, for the algebraic geometers, yes. OK, great. Well, anyway, so if you're an algebraic geometer, then something that may have come to mind is Fourier-Muci transforms. So there's this thing that at least formally seems very analogous in algebraic geometry. So if I take an object in db-co of a product of smooth projective varieties, x and y, then it's this totally well-known construction that you can produce from that a functor going from db-co of x to db-co of y. Really useful. And so the hope is that eventually we'll be able to understand how mirror symmetry intertwines these functors, f, l, 0, 1, and Fourier-Muci transforms. So that's a really quite distant goal. All right, so before I move to the next section, any questions? Does anyone know when I started? OK, great, thank you. OK, so let me tell you a little bit more about Lagrangian correspondences and give you some examples and that sort of thing. So a bit of notation from now on, I'm going to write Lagrangian correspondence. Instead of writing it as a subset, I'm going to write it as an arrow. So let me give you some examples. If I take a Lagrangian in M0 and a Lagrangian in M1, then I can form their product and that will be a Lagrangian in M0 minus times M1. So it'll be a Lagrangian correspondence. In particular, if M0 is a point, this tells you that Lagrangians induce Lagrangian correspondences from the point to whatever manifold they live inside. OK, something I mentioned before is that if phi is a map from M0 to M1, let's say it's a symplectomorphism, then its graph is a Lagrangian correspondence from M0 to M1. And the last example, which is the most non-trivial of these, is that Hamiltonian group actions by league groups give rise to Lagrangian correspondences. So say that I have G acting on M in a Hamiltonian fashion. So the Hamiltonianness means that that comes with a map called mu from M to the dual of the Lie algebra. There's this construction called symplectic quotient where you take the zero-level set of mu and you quotient it by G and you get another symplectic manifold. Not a semi-apothesis. And it turns out that this zero-level set is a Lagrangian correspondence from M to the symplectic quotient. Now there's this important feature of Lagrangian correspondences, which is that you can compose them. So let's say that I take L01 from M0 to M1 and L12. Then their composition is defined by pretending that they're graphs of functions and doing what we would do to compose those functions. So that is to say, we're first going to form the fiber product of these guys over M1. And then we'll project down to M0 times M2. And so an example of this is that the graph of phi and the graph of psi are going to give you the graph of psi composed phi. OK, now in order for this to be sensible, we have to have some results saying that the compositions of Lagrangian correspondences are, again, Lagrangian. And it turns out that that's true under a pretty general hypothesis. So this is a theorem of Gilman and Sternberg, which says that if the fiber product is cut out transversely, which is to say that the intersection of the product with the diagonal in M1 is transverse, then pi 0, 2 defines a Lagrangian immersion of the fiber product into M0 minus times M2. So important note, there is no similarly general hypothesis you can write down with the property that you'll get a Lagrangian embedding. So if you're going to be in the business of composing Lagrangian correspondences, you have to do stuff with immersed Lagrangians. And all right, so before I move to the next section, any questions? Sorry, I guess this is the second half of the second section. So I mentioned these pseudoholomorphic gadgets in the introduction, and now I'll tell you what they are. They're called pseudoholomorphic quilts. And the way you form these is you understand what rule Lagrangian correspondences should play when they interact with holomorphic curves. So we're all used to saying that if you have a Lagrangian, then that's defining a natural boundary condition for pseudoholomorphic curves. And what the theory of quilts tells us is that Lagrangian correspondences define seam conditions. So I'll tell you what seam condition is, and it'll also simultaneously define what a pseudoholomorphic quilt is. So here's a Riemann surface. Let's give it some genus. And now let's divide it into two pieces by drawing this circle here. OK, and now let's label the chunks that it's been divided into, or the patches, by symplectic manifolds. Let's label the boundary components by Lagrangians, and then let's label these one-dimensional submanifolds by Lagrangian correspondences. So just as when you draw an unquilted normal Riemann surface and label it with m's and l's, that represents PDE with boundary conditions, this represents a system of coupled PDEs. So this represents the following system. We require a map u, which goes from this left patch into m0, u0, a map u1 going from the right patch into m1. Then for all points in the boundary circle here, they're supposed to be mapped by u1 to l1. And finally, the new interesting condition is that if we take any point in this seam and we pair up its image under u0 and under u1, note that they're both defined at that point, then we land in l01. So that's what I meant by seam condition. Is everyone OK with that? So first of all, let me say an important thing I didn't say, which is that u and u1 are supposed to be j-holomorphic, of course. No, I'm not imposing an image like that. So the crib L01 can somehow go in any way on this surface? Oh, I'm sorry. You're talking about this seam. Oh, OK. Yeah, there are conditions. And what you mean if you're talking about a non-singular quilt is you don't allow any intersections. But for the purposes of this talk, I'm only going to be considering really specific quilts. So we don't want to think about the question, what's a general well-behaved quilt? I can tell you more afterward. OK, great. So now it's time to talk about figure eights and witch balls. So what a figure eight bubble is, well, it's a new kind of singularity that Verheim and Woodward discovered when they were studying strip shrinking. So the situation that they were looking at is they were considering sequences of maps whose domain looks like this. So this is defining a quilt problem. And they noticed that, well, if you have a family of quilts with this as the domain, and if the width of the middle strip is shrinking, then this funky thing can happen. So like usual, you can have bubbling wherever you like on this quilt. And in particular, it could happen in the middle of this shrinking strip. So let me try to draw what's going to happen at an intermediate step. So we're puffing out a bubble. But we'd better be carrying the seams along with us when we blow this bubble out. So on the base quilt, the seams now look something like that. But the bubble is puffing them out. And if the rate of bubbling is proportional to the rate of the delta going to 0, you're going to get something interesting happening in the limit. So in the limit, you'd expect to see a quilted sphere sitting on top of a double strip where the seams on the sphere look like this. So this sphere has two circles as seams. This one here, this one here. And they meet at the south pole tangentially. Does that make sense to everyone? OK, and now it's time for the philosophy portion of this talk. So in the late 80s, Fleur was studying Fleur strips, though I suppose you just call them strips. And he noticed that you can have this funky thing happen where bubbles can occur. In particular, you can have disc bubbling on the boundary. And he said, whoa, let's see what hypotheses we can put on our situation so that we can avoid that. But then Kenji Fukai comes along and he says, well, Fleur strips are pseudomorphic discs, more or less, or inhomogeneous ones with one input and one output. And disc bubbles are also discs with zero inputs and one output. So instead of regarding them as a bugaboo, let's study pseudomorphic discs with arbitrarily many inputs in one output. And out of that, he got this amazing algebraic structure called the Fukai category. Right, so then as a naive graduate student, I thought, let's try to do this with the figure eight bubble. And it turned out that an interesting algebraic structure seems to emerge from that. And by the way, the reason that this is called the figure eight bubble is that if you look at it from the south pole, then the two seems look like a figure eight. All right, great. So then the idea that comes out of that philosophy is to count quilts like the figure eight bubble, though let's call them figure eight quilts, since they're no longer bubbles. They're really the primary object of study. And just like Fukai did, why don't we put some mark points on the seams? And since the south pole is what attaches to the base quilt over in the situation where the figure eight emerged, let's regard the south pole as the output. Now, before we can do anything, we have to have some idea of what do the inputs want to eat. So what should this be defining a map between? And the way you can understand that is let's look at a little neighborhood of one of these mark points. So we get this little disk mapping into m1 and m0, with these seam conditions L01 to L01 prime. OK, and then you can see more or less trivially that if you fold this thing across the center line, this is equivalent to a little half disk mapping into m0 minus times m1 with honest Lagrangian boundary conditions in L01 and L01 prime. Corresponding to the picture, shouldn't that just be m1 and m2? Just from the picture, there's no light from there. Oh, yeah. I probably should have reversed these. I mean, it doesn't matter, but I'll get confused otherwise. OK, and then we know that this sort of input mark point should eat a FLIR co-chain in CF L01 L01 prime. OK, so now we know what sort of input this guy should produce. And well, a removal of singularity theorem that I might not talk about today tells us that we'd expect that this output mark point would produce something in, well, I'd better give things names. Let's see, so this is L01 prime. This is L12 prime prime L12 prime L12 L01. So this guy should spit something out in CF. L01 compose L12. L01 prime compose L12 prime prime. And so this is a FLIR co-chain group in m0 minus times m2. At that point, you've got four Lagrangian's plane. How do you decide to group them one way rather than another? Does it not matter, or is that? I think you decide to group it like that, because you can prove that that's where the limit will live in that FLIR co-chain group. Does it have to do with a FLIR? There's this tangency condition? Yes, that's right. Yeah, and maybe I should give a zoomed-in view near the south pole. So let me cut out a little disk near the south pole and then look at it from below, and I'll show you what it looks like. Yeah, so anyway, yeah, the tangency is what makes the result be a co-chain in here. And I'll come back to this, but I'll also say that the reason that the analysis of these figure eight quilts is difficult is exactly because of this local picture here. So if the seams were coming into the output point, like that, then you'd have no trouble at all. And that's pretty much what Verheim and Woodward and Mao did. OK, great. So what kind of algebraic structure should we get? I'll do it over here. So there's some heuristic for gluing that tell us what we should expect. And I don't think I'll get into the heuristics, but I'll tell you the result. So based on what I was saying over here, counting these figure eight quilts of the precise type that I wrote down with two mark points on the left seam and one on the right, that we expect to define a map from, let's see, CFL12 prime, CFL12 prime, prime, tensor CFL12 L12 prime, tensor CFL01 L01 prime, two CFL01 compose L12 L01 compose L12 prime, prime, prime. And just to make it really clear what I mean when I say that counting these quilts should give you this map, what I mean is that if I call this map C2 and if I feed in Y2, Y1 into the left seam and X1 into the right seam, then I'm defining the output to be the sum, overall, Z in the intersection of the composed Lagrangians, the count over the zero dimensional stratum of figure eight quilts passing through those specified co-chains, and that's the coefficient in front of C. So putting all of the different maps that you get out of this process, for however many mark points on the left seam and however many on the right, what you expect to get is something called C2, which has input these two Foucaille categories, Fouc M1 minus times M2, Fouc M0 minus times M1, and it maps to the Foucaille category of M0 minus times M2. So what do I mean by this? Well, what I mean is on the object level, it sends a pair of Lagrangians to a single Lagrangian, which when L01 and L12 have good composition is just the composition of them. And if they don't, then you are going to have to do some work. And on the morphism level, your font size is getting smaller and smaller. OK, I'll try to remedy that. Thank you. On the morphism level, it's going to input however many morphisms you like in each of the Foucaille categories and spit out a single morphism, defined exactly by this process. And I said that we have some expected relations for these maps, and in the case of C2, what that relation is is that C2 is expected to be an infinity bifunctor, which is essentially the same thing as saying that it is expected to be an infinity functor from the tensor product of these two Foucaille categories to this Foucaille category. And let me draw the picture that gives us that expectation. So these are all Foucaille categories of immersed Lagrangian? Yes. Saying something is an infinity bifunctor does it mean that you have to know a whole lot of other higher CIs to make that definition? Or does it mean that all of the structure tells you what the thing is? You don't need to know any higher CIs. So to be an infinity bifunctor, that's defined just in terms of the infinity operations on the three infinity categories involved. And the CIs are not the same thing as those infinity operations. I'll say it in a second, but the infinity operations are the same thing as C1. Right. OK. So here's an example of that gluing heuristic that I mentioned. Let's consider a one-dimensional logicalized space of figure eight quilts with two mark points on the left seam and zero mark points on the right seam. So let's think about how this thing can degenerate, i.e. let's think about what the boundary of this should be. So one thing that can happen is these two mark points can come together. So when these two mark points come together, we'll get a two-patch sphere sitting on a figure eight quilt. So the two-patch sphere is divided into two by this circle here. Now other stuff can happen. For instance, the two seams can come together and collide. And depending on where the mark points are, when that collision happens, you're going to get different bubbles resulting. So for instance, if the two mark points do not come together as the seams collide, then you'll get two figure eight quilts sitting on a two-patch sphere. Or they could come together as the circles come together, in which case you'll have a single figure eight bubble sitting on a two-patch sphere. OK. Have I missed anything? Oh yeah, I've missed two things, which is that you can have floor breaking at either of these two mark points. So that is to say you can have energy concentration at either of these guys. And now let me write down the algebraic expressions that correspond to these sets. So if you count quilts like this, oh, and I didn't write the mark points on it. So this thing, so this sphere with two patches, if you fold across the seam, you see that this is just the same thing as a pseudo-holomorphic disc mapping to M1 minus times M2. So this corresponds to doing the C2 operation to the product of two morphisms in M1 minus times M2. And this bar here divides the things that you're feeding into the left seam and the things you're feeding into the right seam. Now what's this? So this is going to be mu2 of C2 y2 C2 y1. This one here is mu2 of C2 of y1 y2. This is C2 of y2 mu1 of y1. And then this one is C2 of mu1 of y2. So anyway, the fact that these guys arise is the boundary of this one-dimensional modulus space where they're expected to anyway tells us that these algebraic expressions should sum to 0. And this is one of the infinitely many relations that must be satisfied in order for C2 to be an infinity bifuncture. OK, so I put witch balls in the title. So what's a witch ball? Well, the answer is figure eight quilts have two circles as seams. The structure maps in the Tokaya category of a product you can represent as counts of spheres with one circle as seams. So why don't we study circles with arbitrarily many circles as seams? So what we call those quilts is witch balls, d-patch witch balls. And counting them is going to give rise to some operation called Cd going from Foucq Md minus 1 minus times Md through Foucq M0 minus times M1 to Foucq M0 minus times Md. And I should say that it's only for d is equal to 2 that this thing should be a A infinity multifuncture, which is exactly analogous to the fact that the only A infinity operation, which is a chain map, is mu2. Great. Now, for every d we have, or for some d we have predicted relations. And we expect predicted relations for all d. We're just working on the algebra. So the first relation is that if you sum up all ways of sticking C1 into itself, you should get 0. Now, C1 goes from Foucq M0 1 minus times M1 to itself. And as it turns out, basically what I said before is that C1 is exactly defined using the infinity structure maps in that Foucquia category. So this is exactly equivalent to saying that Foucq M0 minus times M1 is an infinity category. I could see this by taking the sphere and folding it into a disc across that one circle. Yes. OK, the second relation is equivalent to saying that C2 is an infinity bifuncture, so that you get by sticking C1 into C2, either on the left or the right, and also adding in the result of sticking C2 into C1, however many times as you like. So the dot, dot, dot here means fill up C1 with C2's. So we know what R1 and R2 are, and we have an idea of what all the other expected relations should be, but exactly the algebra, exactly what it should be. We're working out. And the last thing I want to say before I move on is that, well, let's see. So I'll say that this thing I mentioned in the title, this expected infinity 2 category, is just we plan to get it by using these CDs as a structure maps. I'll tell you one special case of this whole construction, which is when D is equal to 2 and let's see, M0 is equal to a point. In this case, the fact that C2 should be an infinity by functor means that if you fix L1, 2, then that as a formal corollary of the expectation is rise to an infinity functor from Fouk M1 to Fouk M2 defined by counting quilted disks that look like that. The reason that I say disks and not spheres is that a priori it's defined using these figure 8 quilts, these three-patch spheres, except one of the patches is mapping to a point. So you can just delete that patch, flatten it out to a disk, and this is what you get. And let me note now that what Mao Behrman Woodward did is they straightened out the seams in the neighborhood of this singular point. So they studied quilted disks where the seam is this teardrop thing. So these seams come into the output point in this sort of transverse fashion. All right. And then really the last thing I'll say before I move on to the new result is that in the Foukai category, you get the infinity operations because your spaces of pseudohormorphic curves live over the operative associate hedra. And the infinity operations reflect the algebraic structure of that operad. So we expect that these spaces of curves live over another operad. And these relations exactly come from that operad structure. Specifically, the operad where a typical space is the moduli space of d-patch witch bolts. So I'll show you an example. So one of these spaces called P300 is the space of thrice quilted disks with three boundary mark points. The example is an example of disks and not spheres because they're easier to draw. And it turns out that what you get is, well, so take this picture, cut it out along the edges. There are certain edge identifications that you have to make that I haven't indicated in this figure. But the point is that it'll glue up to a polytope inside of R3. And what that polytope represents is this compactified deline-mumpered space of quilted disks. And the different faces represent codimension 1 degenerations, which are indicated here. So the edges are codimension 2 degenerations and so forth. And it's now time to play a game. Well, someone named two adjacent faces. Good choice. The fact that they're adjacent should mean that alpha and beta have a common codimension 1 degeneration. So let's see if I can figure out what that is. I promise I haven't practiced. So what the common degeneration is, if the seam in this component here gets larger and larger until it hits the boundary. So it's going to force two patch disks to bubble off. And here what's happening is the two attachment points of these guys are going to come together. So what you'll get is I hope that that's convincing. Any questions about? Can you just explain that again? You're kind of blocking the view. Oh, yeah, I'm sorry. OK, so what's happening is the degeneration of alpha is that this seam here, so the border of this green disk is expanding until it hits the boundary of the disk. The mark points in this degeneration have not come together simultaneously. So by definition of this space, when that happens, when that collision of the seam with the mark points occurs, you're forced to bubble off a quilted disk. So that's what these guys are. And after this degeneration happens, this formerly quilted disk becomes an unquilted disk. The way that happens with beta is that these two attachment points have moved together. And so you're forced to bubble off a disk. And that's those two guys. So I don't have a proof, but I expect that it will always be a polytope, which a convex polytope. And I can say that in particular, you can specialize it in two different ways to the associohedra. You can also specialize it to the multiplahedra and, furthermore, to the bi-associohedra and bimultiplahedra. So it seems to be a pretty rich object. It's sort of like a two-categorical version of the associohedra. OK, so any more questions before I move on? So in your picture here, it looks like you've got some points with valence higher than 3. Am I reading that right? Or is there a simple example that can be done where you can see? Because you said this was supposed to be a convex polytope and R3. So I was just curious if I could see an example of a corner which had more than three faces coming in to define it. There is a simple example, but I better show you afterwards, because it'll take up too much of the time. But that even happens with the multiplahedra. And I claim that these are generalizations of the multiplahedra, so you're forced into that. I have a question. Yes. Usually, number of nodes gives you the degeneracy index. So why do we have three nodes on a boundary place? Right, OK, suggestive question. So this has to do with these gluing heuristics that I alluded to earlier. And the answer is that there's this funny thing that happens with figure 8 bubbling. So suppose that we have this picture, so a disk with two patches, and let's look at the degeneration when the seam gets larger and larger till it hits the boundary. So we'll get two quilted disks stuck onto an unquilted disk, like that. So you can ask, what's the co-dimension of that degeneration? So that is to say, how many gluing parameters are there for this situation? And you might think that there are two gluing parameters. There's one for this node and one for this node. And therefore, this stratum should not show up in the algebra. But as it turns out, that's wrong. The reason it's wrong is that there's no way to only glue this node based on the seam structure. So the right way to think about it is in order to glue up this picture to something smooth, you're first going to have to sort of glue this unquilted disk to something that's quilted with a very thin outer patch. So just something like that. And then you're going to have to glue the bubbles in. And it turns out that there's a relation between those three gluing parameters, the two neck lengths at the nodes and the width of this very thin patch that you've introduced. And in fact, that width determines what the neck lengths must be. Satisfactory answer, can't you? OK, great. So today, your co-dimension is, I mean, you have a sort of central bubble, and then it's the number of things going out to the band. I mean, those two are sort of, because they're on the band with the same bubble have the same doing parameter. Exactly right, yeah. So each kind of seam has its own. Yeah, that's right. And so it's sort of like the usual situation you're used to is that if you have some nodal curve, then the different nodes don't know about each other. You can do whatever you want to any one of them locally. But here, if you have a bunch of different bubbles sitting on the same seam, then they all know about each other, which that seems simple enough, at least in this example. But when you have many different seams and many mark points, it turns out to you have to think carefully to figure out which bubbles know about which other bubbles. Any other questions? All right, so time to move on to the next section, which I had intended as a sort of, no, no, no, no. No, I wanted to get, I thought I was going to have 20 minutes. So the theme of this conference is sort of especially educational, so I wanted to do a little bit of a summary of a standard technique and then note how it gets adapted to this situation. So we'll see if I have time to do all of that. So this section is about fretfulness. So first of all, let me mention the general strategy for proving fretfulness of a linearized del bar operator. And let's say that this linearized del bar operator goes from H1 to H0. You can prove the same fretfulness results for lots of different function spaces, but this is the easiest situation. So the procedure has three steps. The first step is to prove that du is semi-fretholm, which is by definition saying that the kernel is finite dimensional and the range is closed. And then the next step is to identify the co-kernel of du with weak solutions of du star is equal to 0. So this is the formal adjoint that Chris mentioned. And then the final thing is you have to prove a regularity result to show that you can identify weak solutions of du star with strong solutions. So here's an example of number one. So let's say that sigma is a closed Riemann surface. Then the standard elliptic estimates tell you that you have an inequality of the form. The H1 norm of c is bounded up to a constant by the H0 norm of du c plus the H0 norm of c itself. And now, I should have said this, we're going to observe that there's a theorem saying that semi-fretholiness is equivalent to the condition that there exists an operator k going from H0 to E, which is compact such that you can bound c H1 by du c H0 norm plus the norm of k c, sorry for the font size. So anyway, this exactly satisfies that condition because the embedding of H1 into H0 on sigma, which is closed, is compact. So that's great. So that's how you prove semi-fretholness on a closed Riemann surface. Now, I'm going to skip some stuff because I don't have so much time. But let me just say that another example of proving semi-fretholness is the case that sigma is not closed. It's equal to the cylinder. So let's call that cylinder c. Then there's this problem. The problem is you still have elliptic estimates. So you have that the H1 norm of c is bounded by the H0 norm of du c plus the H0 norm of c. So that holds these function spaces are on c. But the problem is that relics theorem no longer applies because we're not on a compact Riemann surface. And therefore, H1 embedding into H0 is not compact. So we don't yet know that this operator is semi-frethol. The reason it's not compact, if you haven't seen it before, is because you can just think of a sequence of bump functions which are scooting off to infinity. So let me just say that the resolution is to study the asymptotic operator, which is the limit as s goes to plus or minus infinity of the linear as Del Mar operator. And then the asymptotic operator is, of course, s invariant. And so you can argue that there's an injectivity estimate for it. It's actually an isomorphism. And that's what allows you to fix this problem. You can therefore replace this H0 norm on c with an H0 norm on a compact subcylinder. So rather than saying anything more about that, about what you do about that resolution in the quilted case, it turns out you can make it work, but you can't make it work trivially if you do some work. Instead of doing that, I think I'll just say what the analog of the elliptic estimates are in the quilted case. So here's what I mean by the quilted case. I apologize in advance for going about three minutes over. So in the quilted case, let's say we want to understand the figure eight quilt. Why don't we put some mark points on there? And we would like to know that this defines a fret home problem. So the linearized Del Mar operator in this case is fret home. In order for that to be true, we certainly need to have some kind of transversality conditions on the correspondences. But let me not say anything about that. It's exactly what you'd expect. So to get the argument started, we need elliptic estimates, which you can get locally. So away from this bad point, you can get these elliptic estimates just from the standard unquilted elliptic estimates. But then you have this bad point down here. So let's figure out what happens there. So let's cut out a disk centered at that point. And let's go into cylindrical coordinates on that disk. So then what you get is a quilted cylinder with these non-straight seams. So how the heck are you going to get elliptic estimates on this thing, given this weird structure of the seams? And the answer is, look at my thesis. Let's look at these chunks. So these are constant width, constant height rectangles. We're looking at their translates. And we can note that you can try straightening out the seams in each of these rectangles. So what I mean by that is choosing diffe morphisms of these rectangles with pictures that look like that. So what you see is a sequence of quilted squares with three patches, where the width of the middle patch is shrinking to zero. So Verheim Woodward came up with an estimate for quilts with domains like that. And I upgraded that in my thesis to, in particular, include the case when the domain complex structure is not standard, which will happen here because you're using these diffe morphisms, which are going to be tweaking the complex structure. All right, so that's all I'll say about fretfulness. I'll just finish by saying that we now know as a theorem that, in this case, du is fredholm as an operator from h1 to h0. Yay. No, I'm not done. Oh, I was getting the fredholm list. Yeah, I thought people were starting to find it. Right, so the last thing I want to do is I just want to mention the laundry list of analysis goals that we started with. And I'll tell you which ones we've checked off. So the first analysis goal was a removal of singularity for figure eight quilts and, more generally, witch balls. So that's done. The next thing is a compactness theorem for modular spaces of such things. OK, the next thing is fredholness. And are you working in the polyfoil context that you said, fredholm? That's the goal, but we haven't achieved that yet. So when I said fredholm, I meant classically fredholment for standard h1 to standard h0. Yes. We would really like h0 intersect. We would really like something like hk intersect wk minus 1, 4 to the analog 1 level of differentiation down. I think we can get that. And then there's still some gluing stuff we'll have to do in order to put that into the polyfold context. So this is partway done just because of what I was talking about with DUSA that we'd like this in the polyfold context. We'd like it with slightly different function spaces. And then the final thing that's really not done is gluing to undo strip shrinking. Is that something you expect to be polyfolded to accomplish? Well, no, I'd say that we need to figure out four in order to be able to put it into a polyfold setting. Once we have four done, I think we're really ready to turn the polyfold crank. So anyway, four is next up on the docket. Thanks for your attention. Any further questions? Just really quick on this last point. You need classical gluing to undo strip shrinking. What's next on the docket? Polyfold splitting. OK, so you can do it classically. It's just a way to try and turn what you can do when it sort of hands on application into the polyfold package. Something like that. So Virheim and Woodward did classical gluing in one situation. I don't know if you could generalize that classical gluing to sort of witch balls. And I don't know how general you could do that. But yeah, the thing we need to do is polyfold gluing, which we have not so much of an idea of how to do. I think I didn't understand what was the importance of having the actual witch balls at this teardrop. If you use the teardrop, then what you'll get is maps between extended focaya categories. So these are focaya categories where your objects are formal sequences of Lagrangian correspondences, which is formally somewhat easier to deal with, but is quite a bit further removed from geometry. So that's why we want these things amongst focaya categories. Furthermore, even if you work with a teardrop, if you have bubbling, the bubble is not going to have that teardrop structure. It's going to have this witch ball structure. So you would only be able to try to do things with teardrops when you could exclude all kinds of, every kind of bubbling that happens when strip shrinking also happens. If you use this teardrop map, and then you use the, I would say, and then you actually compose those things using your map, do you get the same as using your map to begin with? Um, I, I, probably, but it's a little bit of a perverse thing to do, because in order to prove something like that, you have to completely understand the witch balls, and therefore why are you thinking about the teardrops? Do you have a question to say? Yeah, I had a question about, I can't quite see the colors there, but if not just some of those, you had bubbling both from the level of blue and of the level of green. As far as I would understand, then that would be sort of code mentioned too. So you're talking about a sigma, for instance. Something, I can't, as I say, I can't quite see, but it looked. Yeah, or like, what's the simplest example of this? Well, okay, yeah, let's look at row. So here we have a whole bunch of nodes, and they're not all nodes of the same type. Um, can't answer that very well in the time I have, but the idea is that the Deline-Mufford space I wrote down was not the Deline-Mufford space that you would write down on your first try. The thing you'd write down on your first try would have some strata, which are, it's not clear how to cast them in terms of algebra. So we've stuck on some additional cells in order to make every strata correspond to an algebraic expression, and the ones that you're confused about are exactly the new faces that you get. And a pleasing thing about this picture, so the picture I wrote down is exactly the polytope that tells you that C3 will define a homotopy between the functor's corresponding to different Lagrangian correspondences. Um, and Catrin and Chris Woodward, they, they, and, and Sikkimedia Mao, proved such homotopy statements, though they had to do some funny analysis with delay functions because they hadn't done this resolution. So a pleasing thing about this resolution is that you won't have to do such funny mucking around with delay functions. You have an idea of the flavor of, you know, geometric applications that this sort of machinery might allow you to tackle, or as opposed to the usual infinity structure. What, what other things might you be able to do? I think an interesting goal would be, take one of these correspondences coming from symplectic quotient. So going from m to m mod g, and then this is expected to give you a functor between fuc of those two symplectic manifolds. So how can you relate fuc of m to fuc of m mod g? Which is, my understanding is that's, that's an interesting question in algebraic geometry, so you'd think it'd be interesting here too. More generally, I have some ideas, but I'd rather not speculate in public. So, like, for instance, Rukie has endless two categorical representations of this type of thing. Do you have ideas about what symplectic manifold you might use to get those out of symplectic jumps? Said Rukie has what? I mean, you know, he makes two categories out of the representation theory, the upgrade representation theory to one category that will up. And so in place of the two categories of representations, you've got the two category of categorical representations. I do not have anything sensible to say about that. We should talk afterward. Spectable, slightly more than 10 minutes behind schedule. So let's have the next class start at 10 past 11. Sure. And let's all express our appreciation for the Fetholm property and the big names.