Fredholm theory and Deligne-Mumford spaces for witch balls
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Bottman, Nathaniel
In work-in-progress with Katrin Wehrheim, we aim to bind together the Fukaya categories of many different symplectic manifolds into a single algebraic object. This object is the "symplectic A-infinity-2-category", whose objects are symplectic manifolds, and where hom(M,N):=Fuk(M-xN). At the core of our project are witch balls - certain pseudoholomorphic quilts with figure eight singularity. I will discuss recent progress: toward the construction of the moduli space of domains on one hand, and toward establishing the Fredholm property on the other.
 Thanks to the organizers for inviting me. It's a pleasure to be here. So I'm talking about Fret Home Theory and Lean Mumford Spaces for something you haven't heard of called Witch Balls. All of this is joint work with Katzchen Verheim. You can see what we've written so far in these two preprints. 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. Finally, in the last section, I'll tell you a new result about Fret Home this 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 A and FinD category. The objects are certain kinds of Lagrangians inside of M depending on your situation, they'll be required to satisfy some hypotheses. If you take two morphisms to 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 is equal to the third complex. Okay. 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 categories required to do. The D-area composition operation is defined by counting pseudoholomorphic D plus 1 gaons. So the images of these pseudoholomorphic polygons in M look something like this. Okay. So this is 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 a Lagrangian in a product 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 F L01 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 Nate, can I ask just a real stupid question? So the four chain complex here for the morphisms, how is that different if I switch L and L prime? They have to have a morphism. So morphism has to go between two objects, so it's directed. So what's the difference between Cf L L prime and Cf L prime L? They're closely related. So in good circumstances, there's a duality between those things. Okay. 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 Cachron 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 there, algebraic picture is Lagrange's correspondence should induce an infinity functor. And the bigger algebraic framework is that we actually expect there to be a infinity two category. So just think of this as some kind of infinity version of two category where the objects are also well are some large classes of symplectic manifolds, the morphisms are Lagrangian correspondences, and the two morphisms are flurco chains. 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 flurco chains. 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 Fred Holmness of the pseudomorphic objects used to define the structure maps. Before I move on to section two, I want to mention the result that Verheim and Oderb were able to prove. So like I said, they worked with some pseudomorphic things which are 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 Lagrangian correspondence gives rise to an infinity function, but not between fukai categories, between these things called extended fukai categories, which they invented for this purpose. The objects of an extended fukai category not being Lagrangians, but by formally composable sequences of Lagrangian correspondences. Yes, thank you. Yeah. Okay. Oh, and final thing is back in the audience. No. Okay. Well, anyway, for the algebraic geometers. Okay. Great. Well, anyway, so if you're an algebraic geometry, then something that may have come to mind is Fourier mucai transforms. So there's this thing that at least formally seems very analogous in algebraic geometry. So if I take an object in dbcoe 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 dbcoe of x to dbcoe of y. Really useful. So the hope is that eventually we'll be able to understand how mirror symmetry intertwines these functors fl01 and Fourier mucai 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? No? I think it said about 938 on that clock. Okay. Great. Thank you. Okay. 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. All right. 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. Okay. 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. The last example, which is the most non-trivial of these, is that Hamiltonian group actions by Lie 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 up by G and you get another symplectic manifold. There's some hypothesis. It turns out that this zero-level set is a Lagrangian correspondence from m to the symplectic quotient. Okay. I'm sorry. Okay. 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. 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. Okay. 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 pretty general hypotheses. 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 pi02 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 the Merce Lagrangians. Okay. Yeah. All right. So before I move to the next section, any questions? Okay. Sorry. I guess this is second half of the second section. So I mentioned these pseudoholmorphic gadgets in the introduction, and now I'll tell you what they are. They're called pseudoholmorphic quilts. 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 pseudoholmorphic curves. What the theory of quilts tells us is that Lagrangian correspondences define scene conditions. So I'll tell you what scene condition is, and I'll also simultaneously define what a pseudoholmorphic quilt is. Okay. So here's Riemann surface. Let's give it some genus. Now let's divide it into two pieces by drawing this circle here. 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 grungians 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 the system of coupled PDEs. Okay. 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. 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 okay 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. I'm not imposing an equation like that. So the crib L01 can somehow go in any way on this. Oh, I'm sorry. You're talking about this seam. Oh, okay. There are conditions and what you mean if you're talking about a non-singular quilt is you don't allow any, let's just say you don't allow any intersections. But for the purposes of this talk, I'm only going to be considering really specific quilt. So we don't think about the question, what's a general well-behaved quilt? I can tell you more afterward. Okay. Great. All right. So now it's time to talk about figure eights and which falls. 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 zero, 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? All right. 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 Fukaya 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 and one output. And out of that, he got this amazing algebraic structure called the Fukaya category. Right, so then as a naive graduate student, I thought, let's try to do this with a 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 Fukaya did, why don't we put some mark points on the seams? And since the south pole is what attaches to the face quilt over in the situation where the figure eight emerged, let's regard the south pole as the output. OK. 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, 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. Sorry, like corresponding to the picture, shouldn't that just be m1 and m2? Like just from the picture. Oh, yeah. You know, I probably should have reversed these. I mean, it doesn't matter, but I'll get confused otherwise. No, I just want to move it. OK, and then we know that this sort of input mark point should eat a Flurr code 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 Flurr code 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? I think you decide to group it like that because you can prove that that's where the limit will live in that Flurr code chain group. Does it have to do with the fact that 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 code 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 structures should we get? Let's do it over here. So there's some heuristic, heuristics 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. Right. 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, Cf L12 prime, Cf L12 prime, Cf L01 prime, two Cf L01 compose L12, L01 compose L12 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 of the count over the zero dimensional strontum 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, you know, 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, Foucaille M1 minus times M2, Foucaille 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. Right, 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 the immersed Lagrangian. Yes. Saying something is an infinity bifunctor 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 it's 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 seamen 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. OK. 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. So if the two circles come together, 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. If 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 if that is to say you can have energy concentration at either of these guys. OK. 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 disk 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 C2y2 C2y1. This one here is mu2 of C2 of y1, y2. This is C2 of y2u1 of y1. And then this one is C2 of u1 of y2y1. So anyway, the fact that these guys arise as the boundary of this one-dimensional watch-less space, or they're expected to anyway, tells us that these algebraic expressions should sum to zero. And this is one of the infinitely many relations that must be satisfied in order for C2 to be an infinity by functor. OK. 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 seems. The structure maps in the Thukai category of a product you can represent as counts of spheres with one circle as seems. So why don't we study circles with arbitrarily many circles as seems? 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 Thukmd-1- times md through Thukm0- times m1 to Thukm0- times md. And I should say that it's only for d is equal to 2 that this thing should be a a-infinity multi-funkter, 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 Thukm0-1- 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 Thukm category. So this is exactly equivalent to saying that Thukm0- times m1 is an infinity category. So you see this by like taking the sphere and folding it into a disk across that one circle. Yes. OK. The second relation is equivalent to saying that c2 is an infinity bifunctor. 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, gosh, let's see, is that, well, let's see. So I'll say that this thing I mentioned in the title, this expected infinity two category, is just, we plan to get it by u. Using these cd's as a structure maps. I'll tell you one special case of this whole construction, which is when d is equal to two and let's see, m0 is equal to a point. In this case, the fact that c2 should be an infinity bifunctor means that if you fix l12, then that as a formal corollary of the expectation is that it applies to an infinity from fukm1 to fukm2 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 eight quilts, these three patch spheres, and 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, Bairam and Woodward did is they straightened out the seams in the neighborhood of this singular point. So today 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 fukai category, you get the infinity operations because your spaces of pseudohormorphic curves live over the operative associate hydra. And the infinity operations reflect 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-Mumpard space of quilted disks. And the different faces represent co-dimension one degenerations, which are indicated here. So the edges are co-dimensioned two degenerations and so forth. And it's now time to play a game. What's someone named two adjacent faces? Alpha, beta. Good choice. The fact that they're adjacent should mean that alpha and beta have a common co-dimension one degeneration. So let's see if I can figure out what that is. I promise I haven't practiced. Okay. 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... So I hope that that's convincing. Any questions about... Can you just explain that again? You're blocking the view. Oh, yeah, I'm sorry. Okay, 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 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. Okay. 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 associahedra. You can also specialize it to the multiplahedra, and furthermore to the bi-associahedra and bimultiplahedra. So it seems to be a pretty rich object. It's sort of like a two-categorical version of the associahedra. Okay, 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 three. Am I reading that right? Yes. Is there a simple example that could be done where you can see... because you said this is supposed to be a convex polytope and R3, so I was just curious if you could give us 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 afterward, 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, the number of nodes gives you the degeneracy index. So why do we have three nodes on a boundary face? Right, okay, suggestive question. Yeah, 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 eight 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 until 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 do 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, Genshin. 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. Those two are sort of, because they're on the band, 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 that 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 a seam, the same seam, then they all know about each other, which, you know, 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 know, 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, no. No, I wanted to get... I thought I was going to have 20 minutes. You know, 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 I will see if I have time to do all of that. Right, so this section is about fredholiness. So first of all, let me mention the general strategy for proving fredholiness of a linearized del bar operator. And let's say that this linearized del bar operator goes from H1 to H0. Whoops. You know, you can do... you can prove the same fredholiness 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 fredholm, which is by definition saying that the kernel is finite dimensional and the range is closed. Okay, and then the next step is to identify the co-kernel of du with weak solutions of du star is equal to zero. 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 that H1 norm of C is bounded up to a constant by the H0 norm of duC plus the H0 norm of C itself. Okay, and now, I should have said this, we're going to observe that there's a theorem saying that semi fredholmness is equivalent to the condition that there exists in operator K going from H0 to E, which is compact such that you can bound C H1 by duC H0 norm plus the norm of K C. Sorry for the font size. So anyway, this is 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 fredholmness on a closed Riemann surface. Now, right, 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 fredholmness 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 duC plus the H0 norm of C. So that holds, you know, 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 fredholm. 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 linearized 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 resolution in the quilted case, it turns out you can make it work, but you can't make it work trivially. You have to 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 markpoints on there? And we would like to know that this defines a Fred Holm problem. So the linearized Del Mar operator in this case is Fred Holm. In order for that to be true, we certainly need to have some kind of chance or salad 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, well, 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 diffeomorphisms 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 patches shrinking to zero. And 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 diffeomorphisms, which are going to be tweaking the complex structure. So that's all I'll say about Fredholmness. 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. I was yaying the Fredholm list. 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. Anyone else who can do the math, you've got analysis after this, so we can go. 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 Fredholmness. Are you working in a polyfoil context, as you said, Fredholm? That's the goal, but we haven't achieved that yet. So when I said Fredholm, I meant classically Fredholmness. Classically Fredholmness for standard h1 to standard h2. Yes. We would really like h0 interseq. We would really like something like hk intersect wk minus 1, 4 to the analog 1 level of differentiation down. So we would like to 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 sort of partway done, you know, just because of what I was talking about with DUSA, that we'd like this in a 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. And is that something you expect to be polyfolded to accomplish? Well, no, I'd say that we need to figure out 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? I think I can do it classically. It's a way to try and turn what you can do in a 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 generally you could do that. But the thing we need to do is polyfold gluing, which we have not so much of an idea of how to do. And what was the importance of having the actual witch ball and so this teardrop? If you use the teardrop, then what you'll get is maps between extended Foucaille categories. So these are Foucaille 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 Foucaille 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, how to say it, then you actually compose those things using the one map, do you get the same as using your map to begin with? 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, Dusse? Yeah, I had a question about that. I can't quite see the colors there, but it looks as though some of those, you had bubbling both in the level of blue and of the level of green. As far as I would understand, then that would be sort of co-mentioned too. So you're talking about sigma, for instance. It's something, as I say, I can't quite see, but it looks. Yeah, or what's the simplest example of this? Well, OK, 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. You're right. Can't answer that very well in the time I have, but the idea is that the Deline-Muffered space I wrote down was not the Deline-Muffered 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 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 functors corresponding to different Lagrangian correspondences. And Katrin and Chris Woodward and Sikkimedium out 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 geometric applications that this sort of machinery might allow you to tackle, or as opposed to the usual infinity structure. 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 Fouc of those two symplectic manifolds. So how can you relate Fouc of m to Fouc of m mod g? My understanding is 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. For instance, Rucier 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 junk? Sid Rucier has what? I mean, he makes two categories out of the representation theory of the upgrade representation theory to one category that we'll love. And so in place of the kind of way of the presentations, you've got two categories of categorical representations or whatever. I do not have anything sensible to say about that. We should talk afterward. I think we're a respectable, slightly more than 10 minutes behind schedule. So let's have the next talk start at 10 past 11. Sure. And let's all express our appreciation for the Fetholm property and the big name. Thank you.
 Thanks to the organizers for inviting me.  It's a pleasure to be here.  So I'm talking about Fret Home Theory and  Lean Mumford Spaces for something you haven't heard of called Witch Balls.  All of this is joint work with Katzchen Verheim.  You can see what we've written so far in these two preprints.  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.  Finally, in the last section,  I'll tell you a new result about Fret Home this 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 A and FinD category.  The objects are certain kinds of  Lagrangians inside of M depending on your situation,  they'll be required to satisfy some hypotheses.  If you take two morphisms to 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 is equal to the third complex.  Okay. 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 categories required to do.  The D-area composition operation is defined by  counting pseudoholomorphic D plus 1 gaons.  So the images of these pseudoholomorphic polygons  in M look something like this.  Okay. So this is 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  a Lagrangian in a product 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 F L01 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 Nate, can I ask just a real stupid question?  So the four chain complex here for the morphisms,  how is that different if I switch L and L prime?  They have to have a morphism.  So morphism has to go between two objects,  so it's directed.  So what's the difference between Cf L L prime and Cf L prime L?  They're closely related.  So in good circumstances,  there's a duality between those things.  Okay. 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  Cachron 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 there, algebraic picture is Lagrange's correspondence should induce an infinity functor.  And the bigger algebraic framework is that we actually  expect there to be a infinity two category.  So just think of this as some kind of  infinity version of two category where the objects are  also well are some large classes of  symplectic manifolds, the morphisms are Lagrangian correspondences,  and the two morphisms are flurco chains.  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 flurco chains.  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 Fred Holmness of  the pseudomorphic objects used to define the structure maps.  Before I move on to section two,  I want to mention the result that Verheim and Oderb were able to prove.  So like I said, they worked with  some pseudomorphic things which are 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 Lagrangian correspondence  gives rise to an infinity function,  but not between fukai categories,  between these things called extended fukai categories,  which they invented for this purpose.  The objects of an extended fukai category not being Lagrangians,  but by formally composable sequences of Lagrangian correspondences.  Yes, thank you.  Yeah. Okay.  Oh, and final thing is back in the audience.  No. Okay. Well, anyway, for the algebraic geometers.  Okay. Great.  Well, anyway, so if you're an algebraic geometry,  then something that may have come to mind is Fourier mucai transforms.  So there's this thing that at least formally seems very  analogous in algebraic geometry.  So if I take an object in dbcoe 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 dbcoe of x to dbcoe of y.  Really useful.  So the hope is that eventually we'll be able to understand  how mirror symmetry intertwines these functors  fl01 and Fourier mucai 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? No?  I think it said about 938 on that clock.  Okay. Great. Thank you.  Okay. 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.  All right. 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.  Okay. 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.  The last example, which is the most non-trivial of these,  is that Hamiltonian group actions by  Lie 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 up by  G and you get another symplectic manifold.  There's some hypothesis.  It turns out that this zero-level set is a Lagrangian correspondence from m to the symplectic quotient.  Okay. I'm sorry.  Okay. 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.  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.  Okay. 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 pretty general hypotheses.  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 pi02 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 the Merce Lagrangians.  Okay.  Yeah. All right.  So before I move to the next section, any questions?  Okay. Sorry.  I guess this is second half of the second section.  So I mentioned these pseudoholmorphic gadgets in the introduction,  and now I'll tell you what they are.  They're called pseudoholmorphic quilts.  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 pseudoholmorphic curves.  What the theory of quilts tells us is that Lagrangian correspondences define scene conditions.  So I'll tell you what scene condition is,  and I'll also simultaneously define what a pseudoholmorphic quilt is.  Okay. So here's Riemann surface.  Let's give it some genus.  Now let's divide it into two pieces by drawing this circle here.  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  grungians 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 the system of coupled PDEs.  Okay. 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.  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 okay 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.  I'm not imposing an equation like that.  So the crib L01 can somehow go in any way on this.  Oh, I'm sorry. You're talking about this seam.  Oh, okay. There are conditions and what you mean if you're talking about a non-singular quilt is you don't allow any,  let's just say you don't allow any intersections.  But for the purposes of this talk,  I'm only going to be considering really specific quilt.  So we don't think about the question, what's a general well-behaved quilt?  I can tell you more afterward.  Okay. Great.  All right.  So now it's time to talk about figure eights and which falls.  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 zero,  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?  All right.  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 Fukaya 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 and one output.  And out of that, he got this amazing algebraic structure called the Fukaya category.  Right, so then as a naive graduate student, I thought,  let's try to do this with a 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 Fukaya did, why don't we put some mark points on the seams?  And since the south pole is what attaches to the face quilt over in the situation where the figure eight emerged,  let's regard the south pole as the output.  OK.  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, 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.  Sorry, like corresponding to the picture, shouldn't that just be m1 and m2?  Like just from the picture.  Oh, yeah.  You know, I probably should have reversed these.  I mean, it doesn't matter, but I'll get confused otherwise.  No, I just want to move it.  OK, and then we know that this sort of input mark point should eat a Flurr code 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 Flurr code 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?  I think you decide to group it like that because you can prove that that's where the limit will live in that Flurr code chain group.  Does it have to do with the fact that 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 code 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 structures should we get?  Let's do it over here.  So there's some heuristic, heuristics 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.  Right.  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, Cf L12 prime, Cf L12 prime,  Cf L01 prime, two Cf L01 compose L12, L01 compose L12 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 of the count over the zero dimensional strontum 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, you know, 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,  Foucaille M1 minus times M2, Foucaille 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.  Right, 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 the immersed Lagrangian. Yes. Saying something is an infinity bifunctor 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 it's 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 seamen 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.  OK. 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.  So if the two circles come together, 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. If 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 if that is to say you can have energy concentration at either of these guys.  OK. 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 disk 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 C2y2 C2y1.  This one here is mu2 of C2 of y1, y2.  This is C2 of y2u1 of y1. And then this one is C2 of u1 of y2y1.  So anyway, the fact that these guys arise as the boundary of this one-dimensional watch-less space, or they're expected to anyway, tells us that these algebraic expressions should sum to zero.  And this is one of the infinitely many relations that must be satisfied in order for C2 to be an infinity by functor.  OK.  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 seems.  The structure maps in the Thukai category of a product you can represent as counts of spheres with one circle as seems.  So why don't we study circles with arbitrarily many circles as seems?  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 Thukmd-1- times md through Thukm0- times m1 to Thukm0- times md.  And I should say that it's only for d is equal to 2 that this thing should be a a-infinity multi-funkter,  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 Thukm0-1- 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 Thukm category.  So this is exactly equivalent to saying that Thukm0- times m1 is an infinity category.  So you see this by like taking the sphere and folding it into a disk across that one circle.  Yes.  OK.  The second relation is equivalent to saying that c2 is an infinity bifunctor.  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, gosh, let's see, is that, well, let's see.  So I'll say that this thing I mentioned in the title, this expected infinity two category, is just, we plan to get it by u.  Using these cd's as a structure maps.  I'll tell you one special case of this whole construction, which is when d is equal to two and let's see, m0 is equal to a point.  In this case, the fact that c2 should be an infinity bifunctor means that if you fix l12, then that as a formal corollary of the expectation is  that it applies to an infinity from fukm1 to fukm2 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 eight quilts, these three patch spheres,  and 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, Bairam and Woodward did is they straightened out the seams in the neighborhood of this singular point.  So today 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 fukai category,  you get the infinity operations because your spaces of pseudohormorphic curves live over the operative associate hydra.  And the infinity operations reflect 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-Mumpard space of quilted disks.  And the different faces represent co-dimension one degenerations, which are indicated here.  So the edges are co-dimensioned two degenerations and so forth.  And it's now time to play a game.  What's someone named two adjacent faces?  Alpha, beta.  Good choice.  The fact that they're adjacent should mean that alpha and beta have a common co-dimension one degeneration.  So let's see if I can figure out what that is.  I promise I haven't practiced.  Okay.  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...  So I hope that that's convincing.  Any questions about...  Can you just explain that again?  You're blocking the view.  Oh, yeah, I'm sorry.  Okay, 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 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.  Okay.  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 associahedra.  You can also specialize it to the multiplahedra,  and furthermore to the bi-associahedra and bimultiplahedra.  So it seems to be a pretty rich object.  It's sort of like a two-categorical version of the associahedra.  Okay, 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 three.  Am I reading that right?  Yes.  Is there a simple example that could be done where you can see...  because you said this is supposed to be a convex polytope and R3,  so I was just curious if you could give us 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 afterward, 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, the number of nodes gives you the degeneracy index.  So why do we have three nodes on a boundary face?  Right, okay, suggestive question.  Yeah, 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 eight 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 until 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 do 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, Genshin.  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.  Those two are sort of, because they're on the band, 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 that 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 a seam, the same seam, then they all know about each other,  which, you know, 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 know,  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, no.  No, I wanted to get... I thought I was going to have 20 minutes.  You know, 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 I will see if I have time to do all of that.  Right, so this section is about fredholiness.  So first of all, let me mention the general strategy for proving fredholiness  of a linearized del bar operator.  And let's say that this linearized del bar operator goes from H1 to H0.  Whoops.  You know, you can do... you can prove the same fredholiness 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 fredholm,  which is by definition saying that the kernel is finite dimensional  and the range is closed.  Okay, and then the next step is to identify the co-kernel of du  with weak solutions of du star is equal to zero.  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 that H1 norm of C is bounded up to a constant by the H0 norm of duC  plus the H0 norm of C itself.  Okay, and now, I should have said this, we're going to observe that there's a theorem  saying that semi fredholmness is equivalent to the condition that there exists  in operator K going from H0 to E, which is compact such that you can bound C H1  by duC H0 norm plus the norm of K C.  Sorry for the font size.  So anyway, this is 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 fredholmness on a closed Riemann surface.  Now, right, 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 fredholmness  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 duC plus the H0 norm of C.  So that holds, you know, 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 fredholm.  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 linearized 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 resolution in the quilted case,  it turns out you can make it work, but you can't make it work trivially.  You have to 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 markpoints on there?  And we would like to know that this defines a Fred Holm problem.  So the linearized Del Mar operator in this case is Fred Holm.  In order for that to be true, we certainly need to have some kind of chance or salad 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, well, 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 diffeomorphisms 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 patches shrinking to zero.  And 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 diffeomorphisms,  which are going to be tweaking the complex structure.  So that's all I'll say about Fredholmness.  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.  I was yaying the Fredholm list.  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.  Anyone else who can do the math, you've got analysis after this, so we can go.  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 Fredholmness.  Are you working in a polyfoil context, as you said, Fredholm?  That's the goal, but we haven't achieved that yet.  So when I said Fredholm, I meant classically Fredholmness.  Classically Fredholmness for standard h1 to standard h2.  Yes.  We would really like h0 interseq.  We would really like something like hk intersect wk minus 1,  4 to the analog 1 level of differentiation down.  So we would like to 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 sort of partway done, you know, just because of what I was talking about with DUSA,  that we'd like this in a 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.  And is that something you expect to be polyfolded to accomplish?  Well, no, I'd say that we need to figure out 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?  I think I can do it classically. It's a way to try and turn what you can do in a 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 generally you could do that.  But the thing we need to do is polyfold gluing, which we have not so much of an idea of how to do.  And what was the importance of having the actual witch  ball and so this teardrop?  If you use the teardrop, then what you'll get  is maps between extended Foucaille categories.  So these are Foucaille 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 Foucaille 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,  how to say it, then you actually compose those things  using the one map, do you get the same as using your map  to begin with?  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, Dusse?  Yeah, I had a question about that.  I can't quite see the colors there,  but it looks as though some of those,  you had bubbling both in the level of blue and of the level  of green.  As far as I would understand, then that  would be sort of co-mentioned too.  So you're talking about sigma, for instance.  It's something, as I say, I can't quite see, but it looks.  Yeah, or what's the simplest example of this?  Well, OK, 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.  You're right.  Can't answer that very well in the time I have,  but the idea is that the Deline-Muffered space I wrote down  was not the Deline-Muffered 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 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 functors corresponding  to different Lagrangian correspondences.  And Katrin and Chris Woodward and Sikkimedium  out 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 geometric applications  that this sort of machinery might allow you to tackle,  or as opposed to the usual infinity structure.  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 Fouc of those two symplectic manifolds.  So how can you relate Fouc of m to Fouc of m mod g?  My understanding is 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.  For instance, Rucier 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 junk?  Sid Rucier has what?  I mean, he makes two categories out  of the representation theory of the upgrade representation  theory to one category that we'll love.  And so in place of the kind of way of the presentations,  you've got two categories of categorical representations  or whatever.  I do not have anything sensible to say about that.  We should talk afterward.  I think we're a respectable, slightly more than 10 minutes  behind schedule.  So let's have the next talk start at 10 past 11.  Sure.  And let's all express our appreciation  for the Fetholm property and the big name.  Thank you.
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