 John Pardon, who will be our next speaker. He's speaking about representability theorems for the marginalized spaces of pseudo-homomorphic curves. Thank you very much for the invitation to give this talk. About eight years ago, a little over eight years ago, I was at MIT giving a talk on a similar subject. And I made some effort to make it accessible to everyone. And there were lots of questions. People came up and asked lots of questions after the talk. Tom also came up and asked some questions. We talked a little bit and then he said, well, to heck with what anybody else gets out of the talk, he would much prefer I just started at the point I ended my talk and say whatever comes after that. So I hope to fulfill that eight-year-old request now by covering more ground. Okay, so there are three parts. Part one is the transverse case. And my excuse for spending any time at all discussing this is that it's an excuse to set notation for what's going to happen later. So let's consider C, Riemann surface, and we'll assume that it's con and x will be our almost complex manifold. Right, so I'll denote by all Cx is the space of pseudo holomeric maps from Ctex. So a space really here is in quotes here because sort of the whole theme of the talk is to say what exactly do we mean by space of holomeric maps? So quite a few different sort of structures one might attempt to put on this sort of space. One initial thing to say about the space is you can write it as this fiber product. You look at the space of all maps from C to x and we look at the space of maps from C to I guess Tx tensor Rc bar. This is the map associating to a map U, the anti-holomorphic part of this differential and this is the zero map. Oh yeah, so that fiber product is just the locus of maps U with du zero one equals zero and this map I might also call bold D. Okay, so at the most basic sort of structure we could put on everything here is we could say everything is a topological space and we have the topology of C infinity convergence on these spaces. And aside, at least on the space in the upper left-hand corner holomorphic maps from C to X there is also reasonable to complement plate topologies like the CK topology or the C zero topology. Now for case-efficiently large, the CK topology agrees with the C infinity topology just using straightforward elliptic regularity. In fact, the C zero topology also agrees with the C infinity topology. You need some slightly more specific a priori estimates to do that called the Gromov-Schwarzlamma but I don't wanna get into that as far as I know this is the only topology you really ever. Care about on the space and coincidence that can coincides with other topologies. Okay, so what does this space look like locally? Well, locally this fiber product diagram equivalent to one of the form capital F inverse of zero going to a point going to Rn, Rm. So F is just some smooth map from Rn to Rm and this inverse image of zero under F is a local model for this topological space. Now, so I think the audience may correct me but the references I've seen for, this is a very old picture, the references I've seen for this fact that this well really any modular space of solutions to non-linear elliptic PDE has local charts of this form. It's usually attributed to Kuranishi and separately Atiyah, I don't have it on here, Atiyah Singer and Hitchin 65. Yeah, so this map F is often called the Kuranishi map. Great, so there's one situation in which this is sort of easy to understand and that's when the differential is surjective. So if we have U in a holomorphic map, then we can look at the linearization of this map, the D that goes here for star Tx and this is equivalent to the differential of this function for the alpha U from Rn to, okay, so we say U is regular or transverse when du is surjective or equivalently when this finite dimension of the model, finite dimension of the reduction, F is surjective and so then in that case, the implicit function theorem tells me that F inverse of zero is a manifold and so whole Cx is also a manifold. So being regular is open condition, regularity is open so we denote by whole Cx reg open locus of map so the proposition which I'm going to explain what it means is that whole Cx reg representable a smooth manifold. So this is a very, very classical result and I need to tell you what it means as a slightly different way of formulating it than it's usually formulated as. So I'm going to declare that, so let's fix some notation. This will be the category of smooth manifolds and we declare a map from Z to C infinity AB, B smooth if and only if this map from Z cross A to B is smooth. So here Z and A and B are all smooth manifolds so this is my definition of what it means to have a smooth map from Z to this space and then you can use this fiber product description to say what it means to have a smooth map from Z to whole Cx. So whole Cx is now a functor on smooth manifolds and this content of the result is that that function is represented by a smooth manifold. How do you prove it? Well, how is this usually approached? So you choose a Sobolev completion sort of arbitrarily, say HS regularity classes of maps from here and here. Let's see that maps takes a derivative so I guess I need HS here and HS minus one here. The point of doing that is so that now you have Banach manifolds here. These C infinity spaces are for Shea manifolds. You a little bit harder to apply inverse function there for Shea manifolds. So you pick a Sobolev completion, they become Banach manifolds. So ejectivity of the linearization now implies that this fiber product here, this is a smooth manifold, smooth sub manifold of HS. So that's a, gives you a chart for whole Cx from an open set in RN. And I claim it represents this factor. And that's not, that has some content in it because our prior, well, what it means to be smooth to that is to sort of have a family like this where in the A direction it's smooth in the Z direction but HS regularity in the A direction. Okay, but elliptic regularity is gonna tell us that in the A direction it's in fact smoothed. Okay, and I wanna make a point here which is that, okay, I have this topological space. Suppose I wanna prove like it's a topological manifold. Well, that's a property. And so the argument I just gave, it produces a locally Euclidean chart for this. So it proves it's a topological manifold. Now, suppose I wanna end out with a smooth structure. Well, a smooth structure is extra structure. And so it's not enough to just specify it locally. I also need to show that the local charts, the transition maps between them are smooth. However, if I first define whole CX-reg as a functor on the category of smooth manifolds, then representability is a local property now. It's a local property. It's not structure, it's an extra, it's just a property. So I can verify it locally using this. And I don't argue what I just said in words and I don't have to check that the transition maps are smooth. Right, so there's a family version of the same thing. So if B is a smooth manifold, I can look at, well, let's just keep things simple. C to B is a family, smooth family of remount surfaces, proper compact fibers. And then I can define parametrized modular functor by saying the map from Z to this space. It's the same thing as a diagram. So you choose this map from Z to B, you pull back family and then you choose a map like this. So you choose this bottom map here and you choose this map here, which is supposed to be through the holomorphic fiber. And again, you can describe this by a corresponding fiber product diagram where you use parametrized smooth maps in all those cases. Okay, so basically the same argument applies to show that if you take this regular locus in the parametrized, I think this is also representable. Okay, so you can also B be any smooth stack such as maybe you want a stack of all modularized stack of smooth remount surfaces. Okay, so in this case there's a universal family over it. So this parametrized modularized space. Now what is it? It's maps up to re-parameterization. And this is important, because that's usually what we actually want to study. It's rare that you want to actually consider it's just the spaces through the holomorphic maps. You want the quotient by some equivalence relation. What is U? It's the universal family over B. I could just call it C. Let me just call it C in every mark. This C to B is the universal family. Yeah, so if C is a category and I have a functor from C up to sets, then it's called representable when it's isomorphic as functors to this functor of HOMS to some object. You can't see it in the definition, but it's a few lines argument to see that this object X representing F is unique up to unique isomorphism. So if I tell you the functor with the property being representable, then I've also told you the object. Okay, so a corollary is that regular locus is also smooth when B is a smooth stack. And the proof is, well, let's take B bar to B to be an atlas, meaning B bar is a smooth manifold and this map F is a rejective submersion. Then the formation of this modular space is sort of for trivial reasons compatible to pull back. So this is a fiber square. And moreover, the inverse image of the regular locus here is exactly the regular locus here. So since this map is a rejective submersion, so is this one, this one's representable. So that's an atlas for this. Okay, now you might complain a little bit and say, well, a smooth stack is a little bit too general of an object. They can be pretty bad. I prefer my moduli spaces to be orbifolds. So there's a theorem and this was conjectured by Weinstein and proved sometime later by Zeng. So this is a non-trivial statement. So the statement is the smooth proper diagonal locally isomorphic to V mod G for G unpack Lee group and defined dimensional representation of G. My proper diagonal is an analog of the house darkness condition. So in any modular space that you're going to really be doing anything with it all, you probably have. So the statement, if you think about what it means, your first guess is that, well, it sort of has a similar flavor to this, you know, just averaging argument. So, you know, if you have a section of a vector bundle and the base space, and it's an equivariate vector bundle, but the section might not be equivariate, then you can average over the action of G to get an equivariate section, the sort of thing. That's the sort of argument you'd like to execute to prove that result, but actually it's in fact, not just something. Okay, so part two. Part two is about gluing into corners. Okay, so if we have a pseudo holomorphic curve, the holomorphic map, which looks like this. So there's two points in the domain, which maps to the same point in the target, then up to finite dimensional construction, you can glue them to obtain something like that. And the local model of the node of that gluing is, is this space, the multiplication map. X, Y, lambda. So this is the fiber over zero, this is the fiber over. Okay, now the sort of inverse operation where you have a sequence of curves, which approach something like this by having a neck stretch is called bubbling. I guess maybe bubbling is only the right term when one of the components is a sphere. But anyway, this often happens in modular spaces. You need it modular space to be compact. That's due to moving back in a general setting and then grow them off or see the holomorphic curves specifically. So we'd like to understand modular space near the holomorphic maps near a point like this. And this is the cause of many headaches it's not clear what exactly the structure should be. So I want to talk about the notion of a log smooth manifold to capture what's going on here. And this is a notion which all the essential components of the definition are due to Melrose, although he didn't write down the definition. I'm gonna write down. It was clear I think from the beginning that the definition Melrose gave is relevant to this sort of problem. Parker definitely wrote down the class of smooth functions to irrelevant and Joyce made basically the definition I'm about to say. Okay, so let's say P is a real polyhedral cone. And then XP is this harm from P to positive real numbers. This is a sort of local model of a real torque variety. If you pick some vectors which generate P then and you try to write down what the space is in terms of that basis, well not the basis but vectors generating P then you'll realize it's a subset of an orphan to cut up by some anomaly quitter. So just to understand sort of topologically what's going on. In fact XP homomorphic to P this is unnaturally as you might expect it's sort of a dual but it's homomorphic to P unnaturally homomorphic. And this is as stratified spaces as spaces stratified by the faces. So that tells you a little bit about what it looks like. Okay, so now the key notion is what it means for a function on one of these things to be smooth. So to each P in P you get a one form which I guess I should write as D of log X of P. So X is the coordinate on X of P. P is the element of P. So any element of P gives me a function X of P that's a function from XP to the positive the non-negative real is just the evaluation of X. I take log of it and D of that. So for example on the half line is the X associated to the half line. Then we get DX over X. And that's the only thing you get. And we just declare that these are basis of the cotangent bundle. So making this declaration, we get a notion, there is a notion of smoothness. So if you have a map, we can pull back one forms and say dualize to define a map from TXP to TXQ. I just defined for you the 10 space. So I say F is CK if and only if TF exists and is CK minus one. C zero just means continuous. So we have a notion of smoothness. This is getting very abstract. So let's give an example. So my notation for the space associated to a half line is this R greater than or equals zero with a prime just indicates I'm regarding it as a log smooth manifold. So if I have a map like this, I'll call this map F. Yes, this is a map F. It's a map of log topological spaces. Okay, so a map like this, I'd like to write it in log coordinates. So R union minus infinity R union minus infinity, perhaps my coordinates are X here and S in here. So I guess that makes S log X and X e to the S. Okay, so I can get my map here, G. So here, so F is smooth. The output of this definition is that F is smooth if and only if G of S is equal to A times S plus little over one infinity. So at A is just a non-negative real number. This is as S goes to minus infinity. Okay, so you can also take products like you cross this and this is a model for like a cylindrical end model on U. And then you can ask for what are the class of smooth functions on this space? Well, they're the functions which are smooth and which converge as you take S to infinity to some C infinity converge to some C infinity function on U. Now if you want, you can also just change the definition to require this little low of one to be say, big O of e to the minus delta S for some delta if you like. That's relevant at some points. Okay, so what's, how do I want to use this? So local model or nodal resolution would be S1 times two half lines going to a single half line. And this is the map X, Y goes to X times Y. And the thing to know this about this map is that its derivative sends X dx to lambda d lambda and Y dy to lambda d lambda. So it's surjective on tangent spaces despite the fact that it's not a trivial vibration, locally trivial vibration, okay. So it seems sort of like a magic trick, but okay, one of the things this means is that this vibration, which looks like, which is not locally trivial, it has a connection. I can lift the vectors on them. There's a, I can choose a way of lifting vectors on the base to vectors on the total space. And that's what you need in order to define a smooth structure on one of these parameterized modular spaces that you need to be able to differentiate in the base direction. And that's the thing which you could not do if you just viewed this as a sort of map of smooth manifolds. Okay, so proposition would be that if B is a log smooth manifold and C to B, the family of Riemann surfaces with simple breaking, which means locally modeled on this, then this parameterized modularized space is representable. It's a log smooth manifold. Okay, and once you set up the definitions, there's somehow nothing to prove. Great, so okay, so part for any structures. Okay, so SM was the category of smooth manifolds. So there's a category called category of Koreanishi spaces and it's an enlargement of the category of smooth manifolds. And the purpose of this category of Koreanishi spaces is all fiber products of smooth manifolds and that they should remember their fiber product presentation locally, at least modulo transverse fiber products of smooth manifolds. So you'll recall at the beginning, I said modularized space of pseudo holomorphic curves is locally described by Koreanishi charts. Those Koreanishi charts are well defined up to transverse fiber products of smooth manifolds. So, okay, so I should write some names here. I guess the first name is Koreanishi, which I already wrote. There's also Kaya Ono, who gave a very complicated definition for what you could possibly mean by gluing together agreement of Koreanishi charts developed later by Kaya Ono. But it was Joyce who really defined this as a category and we really need it as a category if we're gonna be taking fiber products or trying to represent functors. Okay, so let me skip the definition of a Koreanishi space. If someone asks me later, I can give it. So, okay, we can define this parameterized modularized space as before, as this fiber product where for now I define these C infinity mapping spaces as functors on Koreanishi spaces. In the natural map from Z to C infinity AB is simply map from Z cross A to B, smooth manifolds A and B. That defines a functor on smooth manifolds. I take the fiber product of all of these, that's holomorphic maps. Okay, and the proposition is that this is representable. So it is a Koreanishi space. Now, this was conjectured by Joyce. In fact, really the conjecture of Joyce is the two things, so how that there should be a way of defining a, he didn't define the mod sly functor in this way. Conjecture there should be a some way of defining a mod sly functor on Koreanishi spaces such that this is representable and it is the thing we want. Okay, let me give a proof. So it's, I mean, once you have the category of Koreanishi spaces set up, it looks like the proof is going to be basically trivial. And the reason is the following. Well, okay, so first of all, as I remarked earlier, it's a local assertion. Okay, so all I have to prove is that this local Koreanishi chart you get from the classical theory actually represents this functor. How would we do that? So let's fix a U and prove it locally. So what I'd like to do is expand, just expand the parameter space. So this is the thing that you always do in the situation. So this is B and this is B prime. So we're in smooth manifolds here. B is a smooth manifold. You extend your parameter space and you vary the almost complex structures on X and C over B prime in such a way to guarantee that this point U is when regarded in this bigger mod sly space is regular. This is sort of trivial because you can define, you can just choose, you can choose B prime arbitrarily. So you can choose it so that variations in the direction normal to be fill out that extra fine dimensional co-kernel with this linearized operator. Okay, so then as I remarked before, formation of this is compatible with pullback. And so we have this hyper diagram. This is a regular locus represented by with manifold by what we, I've sort of trivial classical fact that I mentioned earlier. And now we have a, now this is a fiber product of smooth manifolds, smooth manifolds, smooth manifolds, smooth manifolds that we're done, except there's a big except. So I slipped something past you here, which is that earlier in the talk, I showed that the regular locus as a functor on smooth manifolds is represented by a smooth manifold. And here, what I need in this proof is that the regular locus is represented by a smooth manifold as a functor on the category of Kearney spaces. These are different statements except, I need to check, we take this regular locus as a functor on smooth manifolds and I push it forward to the category of Kearney spaces that that is the same thing as regular locus as a functor on Kearney spaces. And this requires quite a bit of work to actually prove you have to understand what exactly goes on when you extend the functor from smooth manifolds to Kearney spaces in this formal con extension way. You have to prove that this preserves finite products and this requires some geometry of Kearney space. Well, you sort of wanna take the push out of parts two and part three over part one. Yeah, so one thing you wanna do, for example, is well, if all points were regularly you'd have a smooth manifold and maybe the boredism class of that smooth manifold is a good invariant of, now in most, in too many cases, you cannot ensure that the regular locus is everything. But this Kearney space, whatever it is, you can form boredism as the same as boredism of smooth manifolds or sometimes it's the same, sometimes it's a little bit more interesting but it's an algebra topological invariant which you couldn't get just from say topological space structure. I guess the gauge group action probably would correspond to diffimorphism of C. Maybe if I took sort of the space of all almost complex structures on C and then had the gauge group acting. I guess the most I have here are five-dimensional groups acting. Yeah, I'm sort of taking as input that there's a nice five-dimensional smooth stack of Riemann surfaces which is where the gauge group is living.