 Thank you, Jessica. Thanks very much for the invitation to speak. Can't be there in person. Let me know if you have any trouble hearing me and also first let me know if you have any questions. So, here's a drawing provided by my daughters. If you zoom in you can see some equations. So here's a plan for the these lectures. In the beginning, I'll be discussing the general concept of symplectic resolutions. This plan roughly corresponds to the days of the talks but maybe not precise. Then I'll talk about symplectic duality. Then I'll speak about the government. Then I'll speak about the special case of number three generalize. And then time permitting. Cool own branches and in particular of these generalized African grass mania slices. So which goes into the name of truncated shift to the unions. So there's a plan. And as I said, we'll roughly correspond to the lectures but maybe not precisely. And there'll be two question answers. The first one on Wednesday, we hosted by Michael McRee. And the topic of that. And the second one on Thursday will be hosted by eHow. Sorry, if I misspelling it last name you have. And the topic of this session will be on connections with physics. And specifically also about these generalized African grass mania slices. So let's begin today with the synthetic resolution. So, there's a famous quote from a kunkab from about 10 years ago and he said something like symplectic resolutions are the le algebra of the 21st century. And hopefully I'll convince you today that there are very interesting objects of study. They have been quite extensively studied in the 21st century. So, let me begin the definition of first glance looks a little complicated, a lot of structures. So what is a symplectic resolution. So in fact, we're going to play the restricted class called chronicles. It's a first of all, morphism of two complex algebraic varieties X and Y. There's something there's some feedback. I did hear also a little bit of echo is it. Maybe now people have muted themselves and it's better sounds better to me now so okay great. Okay, so the conical symplectic resolution is a morphism of algebraic varieties. This map pies is a resolution of algebraic variety so it should be irrational and projective. And why is smooth and usually I'll work over the complex numbers all over algebraic varieties over the complex. Why is smooth and carries the algebraic symplectic structure. Okay, and now echo seems to be gone. Okay. Okay, well, let me know if anyone has any questions. Okay, great. So let's go again. So kind of a symplectic resolution we have it's a by ration. It's a two algebraic varieties over the complex numbers map between them which is a resolution of singularities so it's by rational and projective. The space Y is smooth and symplectic. And the space X is a fine. Poisson. So maybe I just remind you that this particular this means that the coordinate ring of X is a plus on algebra. That means in addition to multiplication, it carries a plus on bracket, which which is something which satisfies Jacobi identity, and also the liveness identity with respect to the multiplication. Okay. And more structure. This is okay so that up till now that's what it means to be conical. I mean a symplectic resolution now I'm going to add the adjective conical. That means I have an action of C star on Y and on X so it acts on both varieties and the morphism is compatible with the action. And the, the, this action of C star and X gives rise to a grading z grading on that algebra functions. And I demand that the nth graded piece is zero. If n is less than zero, and is C if n equals zero. So it's strictly not it's not negatively graded with C and degree zero. So this, this fact is equivalent to the geometric fact that the fixed points for this C star action is just a single point. And that the action of C star contracts X to this fixed point. So that's another piece of structure. And now yet one more piece of structure where somehow is not quite mentioned in the adjective sometimes people might impose a different word but let me just say, I'll put another piece of structure without giving it a name exactly. I want another action of a Taurus. So a Taurus another algebraic Taurus, which will also act on the morphism X and Y. I should have mentioned here before that this scales the symplectic form. In terms of this plus on bracket, it means that this plus on bracket has degree one or degree two. So we can consider conical actions of weight one or conical actions weight two. I'm not going to be too specific about. Okay, but now I have another action of a Taurus, which is now preserves the symplectic form. In fact, it's a Hamiltonian action. And I'll demand, not that this, not that this contracts the space to a point that wouldn't be possible for action which preserves the symplectic form. But I will ask is that there's finding the many fixed points for this Taurus action on Y. So this is a condition, which is not always true, but often is true. Okay, so that's a lot of structure. The definition took a whole page. So let's see some examples. So even though it seems like a long list of structures these things which satisfy this list of axioms is are quite common and very interesting in doing major representation there. So here's the first example is we take the cotangent bundle of P1 mapping to the nilpone and cone of SL2. So what is that well in pictures it looks like this nuclear reactor in this ice cream cone. So this morphism collapses the P1 sitting in here. In matrices, so I think of the nilpone and cone of SL2 as two by two matrices of trace zero and determinant zero. Two by two matrices trace a zero and determinant a zero. In terms of equations, those matrices look like w u v negative w. Already we get trace zero and determinant zero. We get w squared equals u v. Maybe w squared plus u v equals zero. And this is the same as the as an algebraic variety to the quotient of C2 by the action of Z mod two. And we think of T star P1 as a pair consisting of a matrix like this and a line. So LA is a matrix and L is a line point in P1. And the condition is that A of C2 is contained inside of L and A of L is zero. That's just a way of describing this cotangent bundle. And so the fiber of this map. Well, if a is a non zero matrix, there's a unique such L just it's kernel. And if a is a zero matrix that's here, this is a zero matrix sitting here. So it is zero matrix. Well, there's no any L will work. So we get this P1 fiber here. That's our Y. That's our X. We have this C star action, the conical action. Joella. Yeah, we have a question. The question of the torus and the C star should be compatible or not. Yeah, they commute. Any other question. Okay, so we have in this case the conical C star action just comes from scaling of the matrix. Or equivalently scales and cotangent fibers. And, and we have this Hamiltonian. Torus T which, well, you could say it's C star squared or you could just say it's C star because part of the C star square will actually anyway this Hamiltonian C star squared action this. It acts just by acting on C2. Therefore on P1 and on the, and on the no one con. That's our first example. So this example of course admits many generalizations. So the first one maybe they'll be like really relevant for us will be the cotangent bundle of a project of space to PN minus one, which maps to the square zero. And by an matrices of rank less than or equal to one. So that's our X. That's our why it's resolution. And again we have conical C star action Hamiltonian torus action. That's before. Of course this is itself admits many generalizations. We could put here cotangent bundle of some flag variety. This is all under number two years we could put here some G mod P. And it would resolve P of G here we need semi simple group or the complex numbers and P parable example. And this would resolve a no put an orbit closure. I won't actually talk so much about this kind of thing so I'm not going to go into much detail about it so let me just put it in a bracket. Something that will be a bit more special case of this is more relevant for us is the cotangent bundle of the flag variety of SLN. So right that in more detail, which resolves the no potent cone of SLN. And here we have all n by n no potent matrices. And here the resolution is by a pair of no potent matrix and a flag. In the n dimensional vector space. So the flag in a n dimensional vector space. And this matrix and the flag are compatible in this way. So that's one. And once again we have, maybe it's worth saying here we have C star conical, which is given by scaling, and Hamiltonian tourist action, well in this case like the tourists or whatever group we had or in this case he started this C star n minus one, this acts on the, on the ambient CN. Another way this first example generalizes. So I pointed out that it was that this affine algebraic variety W square plus UV equals zero can be thought of as C2 mods Z mod two. So another way we could generalize that is we can consider C2 mod gamma as our X. Take gamma to be a finite subgroup of SL2. And then we have this client in singularity C2 mod gamma. That's our X, and has a well known resolution C2 mod gamma. That's our Y. So if we take, if we take the zero point here. This, this, this map is well it's by rational. And it's in fact an isomorphism away from this zero point, and over the zero point is a chain of P ones. And these finite subgroups of SL2 C are classified by simply lays thinking diagrams. And so the, and the, the thinking diagram tells you the pattern of this P one so this, this example here I'm trying to show a pattern of P ones, like, well I know how many drew seven or something. So this is a six connected together in a chain. So this is of course one to the basics thinking diagram, according to the graph. And actually, if you look at my, the conditions that I wanted to satisfy, the only way to get the Hamiltonian tourist action as if gammas is the group Z mod in so to get some kind of tourists acting here with finding many fixed points. So it would not satisfy the last condition that I impose the vision, if we didn't have, if we didn't take some of it. So mostly we'll consider the case of C2 mod Z mod n type a singular. Another way to generalize the cotangent bundle of P one, and the cotangent bundle of other projector spaces is the theory of hyper toric varieties. So this is a beautiful combinatorial way of constructing many solutions. And it's a main source of like intuition in the in the subject. So I'll explain a little bit about this, but for more detail. This will be the topic of Michael session on Wednesday. So, how many hyper toric varieties work. So I'm going to take a picture of a tourist acting on a vector space. CN. This is just some other big tourist of rank, like less, less than or equal to n. I assume it acts effectively this purpose. And then we formed the cotangent bundle of CN, which is the same as CN, plus a dual copy of CN. It has a moment map to the dual of the the algebra of our tourists. So, in coordinates. If this, we can say that this choose coordinates on CN. Well, it has natural coordinates. So this action is given by say some weights kind of one. So it's torus T. And then you can use that we can write down this, this moment. And then we take the, this will be a moment. Then we take the zero pre image of the moment map. And we can take a GT quotient for the action of the tourists. And in fact, we can take two different kind of GT quotient we can take a affine GT quotient or a project of the IQ quotient. So, if we take the affine GT quotient, we'll get the, the space X. And if we take the kind of project of GT quotient. Then, sorry, maybe I'll call this one Kai. I shouldn't have put these kinds. Some weights. That's not worry about them. And then, and then I'll get the space Y. That's our X, and that's our Y. Let's, so let's see an example of this. So, so sub example of these hyper dark writers. So the first example I want to take, as I take C star acting on CN. So just a tourist just one dimensional, and this action just by scaling. So then here, I have T star CN. Again, it's like CN plus CN. And here the action will be like scaling of weight one, and here scaling of weight minus once my tourists will act in opposite ways in the two copies. And I have this moment map which is just given by taking the A, B to summation of a IVI. And then if I take the zero level of this movement. And I divide by this project of GT quotient, well as Kai is just like the identity map. Maybe I should have pointed out. I want to find this project of GT quotient, but it depends on Kai, which is a character of the tourist morphism from the tourists to see star. If I don't get any map, then I will get the cotention bundle of projectors. And no map to the affine GT quotient, which will be this rank square zero rank, square zero rank less than one matrices. In fact this matrix A is made from these two vectors and it's like the, the product of a and B in the opposite order. So that's, that's, that's one example. And then the dual example. So this is our first appearance of symplectic duality. So what would be dual to see star acting on CN by scaling. Well, I mean, you'd never heard of this duality might not, might not be obvious but here's the dual is you instead of taking. Again acting on CN, but rather than just see star you take a big tourist like this set of T1, TN such that the product is one. So rank and minus one towards here at a rank one towards here I have a co-dimension one towards I take this big towards acting on CN. So I do the same construction, I formed the cotention bundle of CN, and it's have its moment map so it's moment map. This time will land again in the dual to the algebra here so not right. But I have a moment here to the algebra, and then I take this Hamiltonian direction. And that is our old friend C2 mod Z mod n results this resolution of the financing. So here we see a way of producing the cotention bundle of PN, and the resolution of the client is singularity using Hamiltonian reductions of an affine space by tourist action. And that's so hyper torque variety is just the generalization of this so you start with any tourist action on your vector space, take the cotention bundle, and take the Hamiltonian reduction. And sort of the ways of doing I mean, not always when you get a smooth space, but the classification of these actions the classification which ones give you smooth things, and the study of the resulting things. So places those that's all can be done combinatorially in this theory of hyper torque rights. So that's the next, that's the end. And even if you don't know what this duality means yet, which you shouldn't because I haven't explained yet, you can sort of see there's something dual about these two things here have a rank one tourists and here I have a rank and minus one to us. So one more major class of examples, which will be important there's two more major class of examples but I'm going to pause on introducing the, the last class of examples, which are the affine gross money and slices and go right here to the equipment varieties. So, for varieties here, I was taking reductions of actions of Tori. Now instead of just Tori I'm going to generalize it to products of GLN so Tori are products of C stars. Now I'm going to consider products of GLNs. And how am I going to do that, I'm going to start with a framed graph. So what is that, and draw this picture. It looks a little like this. So it's a graph, a directed graph. Some of the vertices are circles, some of the vertices are squares important, and all the vertices have numbers. So what I do with this graph, this quiver, I associate first a group, and my group is the product of the GLs of the circled vertices. So these are circles, and vi refers to the number in the vertex so it's just, just think of these numbers as being vector space of those dimensions. So in this case I have GL3, GL4 and GL5. And then I build a big vector space which I call M for some reason. So this vector space is just the direct sum of all arms along all edges in this graph. So which we could split in terms of Vs and Ws. And my convention is that there's only arrows from squared ones to circle ones. So it ends up looking like this. So that's my end. And then I do something like what I did a few minutes ago with the hyper torque varieties, I double the vector space I take its cotangent bundle. So again it's just a vector space so it's just n plus n dual. And you can think of and the dual lens is thinking of, you could think of it as doubling the graph, but arrows in both directions. And then this thing admits a moment map to the algebra this tourist. If you think about this as being doubling the arrows, this moment map is just about composing all possible, following one arrow and falling opposite arrow back. So if I put in this here, and maybe give this guy some names maybe I'll call a this will be. So if this map is just the bias bias. I j j or something, then this map is something like the sum over j of a i j b i b j i plus a j i. I mean, no. It's just the sum over always are going out and back from your vertex so either you follow this one out and back you go. Either you go out and back like this, or you go out and back like this. So just some boiling all the ways out and back. Okay, so that's the moment map. So we construct this Nakajima program writing. So this construction is due to Nakajima. We take the zero level of this moment map. And we take the project of g i g quotient by this product of gels, and here Kai, again it's a map from G to see stars given by the product of determines. So that's a special case. There are a couple of facts about what one general fact when they know, yeah, one general fact and then it's a special case I just got us. So, these w vertices, they didn't participate. They were just sitting on the sidelines, but they sort of come into play right now that this guy will retain an action of the general linear group of those w vertices in particular, we can consider the action of the tourists inside there. This tourist flag, and that's the that's going to be our Hamiltonian Taurus that acts on this question. Sorry, is there a question. What is the notation w I for so I should maybe should emphasize that these dimension vectors are called w eyes. These ones are called the eyes. It's just a convention. I mean, this is just the integers I wrote over there and the integers are so in this case, these are three four and five and W's is to maybe zero and seven. Here's a special case, which is very nice. So if we just take the following quiver. One, two, one, two, three and so on up till in minus one, and then put it in dimensional framing over there is that quiver, the quiver variety produced by this construction is actually just our old friend the cotangent bundle of the flag. And how does it work. Well, it seems at first glance, like a miracle like here I had some matrices how did I get this flag. Well, you just compose everything up into up into this framing vector space. So, maybe, maybe I'll do it in a shorter example, then you'll see exactly it works. In fact, let me just think slightly. Let me do it slightly. So that's my example. I want to explain that one. But I explained this one works pretty much the same way. So I put just K, and then frame the vertex. So then I have matrix a. So my end here just on CK to CN. So just a K by matrix. Then when I double it. I have two matrices. Another one matrix going backwards be. And if I take this Hamiltonian reduction. Then I then my momo map condition is just the product. So, G, okay, takes a B. In the right order. So first I want to use major A and then the matrix B to get back to CK. And then if I take the zero level. So then I'll be studying matrices like that with product zero in this order. And then when I take this project of GIT quotient by GLK, so G here is just GLK, because I only have one circle of the vertex. If I take in my notation before V would be K and W would be N is only one of each of them. If I take the, this Hamiltonian reduction. Sorry, if I take this project of GIT quotient, then it's the same as restricting to Paris AB, like this. Such that a is injected. This may be not very obvious fact, but it's a true back. And then mining out by the action of GLK. So if I restrict to the locus AB or is injected, then the image of a will be a K dimensional subspace of CN. And I also remember the AB and I'll be my animal. So this base here is is more to the potential bundle of a grass mining. So, and again I think of this potential bundle of grass mining as a pair, W, C, so W is a subspace of CN and degree K, I mean dimension K, and C is this matrix. CN, W, C, zero CN. So, backwards. There is a question in the Q&A. Great. One second, C, W, zero and C of CN is contained in W. And this, to go from this pair AB to this W, CN, this W is the image of a, and this C is the composition of A and B this way around. So that's first B and then A. Go ahead, no question. Oh, I mean, I guess I need to read it. Sorry. This is the direction from square N to circle N. So I was maybe a little like imprecise. It actually doesn't really matter what directions you put there on the arrows. In the sense it matters for the construction of this vector space N. So the definition of this vector space. The definition of this vector space N, that depends on the direction of the arrow. But once you go to this cotangent bundle, you double the arrows anyway. So it doesn't actually matter the direction of the arrow. Well, sometimes people usually fix on convention and stick to it, but I don't seem to be doing that. So it doesn't, it doesn't matter too much, maybe usually better to write from Bs to Ws, but it doesn't really matter which direction. So here I wrote from B to W. Here again I guess I wrote from B to W. Any other questions? So, that's it. I mean, it's not even the talk, but that's it for the examples of simplified resolution. So let me just remind you that there's a long list of examples. So, so again, I'll just summarize this example so things to always keep in mind so cotangent bundle of P1 that's like the basic example. And this generalizes to cotangent bundle of flag varieties. It generalizes to C2 mud gammas. So, I need resolution planning. So it generalizes to hypertorics. And it generalizes to. Joel, there's a big problem here. Your screen is completely messed up. So you should just wait a few minutes. Sorry. It just got better. Is it okay now? No, no, it didn't stay better for very long. Wait, what's the writing is messed up? There's a local problem on our side. Just a screen projector is messed up. I turned on the microphone in the room so you can stop. We are very sorry. The story is that our video is messed up so nobody in the room can see what you're writing because you know the mathematical symbols. So this is the right time to ask questions. You know, in order to avoid more awkwardness. The people in Zoom, can they see if everything is fine? Yeah, they can see. Yes. So give us the microphone, it's on the right. No one wants to ask a question also here. Okay, now maybe, no, now it's started. No, but if it moves, I think it's going to mess up. Can you please scroll? No, it's completely frozen. Yeah, you see it's frozen. Oh, it's for the one that's the same. Someone else at the distance, it's the same on your computer? Yeah, but my computer is perfect. Yeah, right here. Oh, so you may see Bonjour, but but there is another question. Let's see. Yeah, no, this was. Can you, yeah, but I can transfer, can you share? Maybe we found a solution. So join me. That's, that's all for that. Can I ask a question? Yes, please go ahead. So what is the relationship between the synthetic reality and maybe it's murals? Are there any relationships between them? Like homology? Sorry, between synthetic reality and what? You know, for example, for the synthetic reality. The relationship between, for example, homology between the space and its mural. Oh, well, that's going to be the subject of like the next few hours. So many, many, the short answer is there's many, many relationships, but we'll see some soon, although I don't think I'll talk about synthetic reality until next time. But yeah, we'll see lots of relationships. So there was a question, what does this tilde mean here? So we're just a shorthand. So there's this client in singularity C2 my gamma. I suppose I just go back up. So there was a question, what is this C2 my gamma tilde. So it just means that there's this client in singularity C2 my gamma. Well, just to find as the affine GAT quotient of C2 my gamma. In other words, C2 my gamma is by definition spec of invariant functions. And then it's a singular variety and it has a kind of unique minimal resolution, which is very well studied. So I just call it C2 my gamma tilde. It can be constructed in other ways. For example, I mean, can be constructed sort of. Yeah, more representation theoretically, let's say. And for example, in C2 my Z money and it's constructed as this hyper torque right here. C2 my Z money until this. There is also another question. And then you can start because now we solve our problem. Right. And then the last question is, you may see the Q and F. Oh yeah, the circles and squares I think exactly come from the physicists. Yes. My last point was going to be there's one more class of examples which I'm not going to talk about just yet, which are called affine gross money and slices. And maybe I should mention any. I didn't emphasize it so much but there's also these cotangent bundles of other flag rights which I'm in general not really going to talk about much in this series of stocks. Okay, so. So we define this general notion of conical synthetic resolutions. I give you lots of examples. By the way, I don't know if it's exactly considered a problem to try classify them but maybe it's not believed that there's very many examples besides the ones I've listed. There's some kind of complete list of examples as a big statement but I don't know of any precise statement like that. One thing you might you might ask what about this way off topic but people have tried to like classify all possible. There's a few specific resolutions which are of this form so CN mud or T star CN or something my gamma gamma is fine it so that that's something. That's something people have studied tried to classify and I think I think the classification of those those ones is complete. Okay, there's a few besides these two markets. So there's a few more things like that which don't fit on this. So definitely does not a complete list but maybe that's close to a complete list. Okay, so these are some examples so what are the general properties why is this interesting things to study these like resolution so. So I'm going to try to explain some general theorems about on a person like the resolutions. And basically these results are due to collated. And then the color from roughly speaking, the early 2000s, 2000 to 2010, maybe some of the results were in a bit early. Okay, so the first result is that X has a partition into finite finite partition. And these. Okay, so over some pieces x alpha right. So this is finite. What it what it what's, what's good. These x alphas are all simplected. So there's a plus on structure on x, and these x alphas are the symphonic leaves of that plus on structure. And they're also smooth, which is that before I said there was somebody. And also this map pie from y to X is. Maybe they're very least. But it's like locally trivial over x alpha. That's exactly true, but they're very least the push forward of the constant shape is constant over each other. Let me not write this let me write something more precise. Okay, so if we can check the push forward of this constant sheet. It will decompose in the following way. Well it's going to be. Let me ignore this. Let me come back to this slowly. Number two. The map from y to X is semi small. So semi small what does it mean. It means that the dimension of the fiber product, why was why over X is equal to the dimension with Y or X. So that means that the fibers are not too big. In particular, the dimension of the fiber over some point x alpha, little x that's supposed to be a little x alpha and big x alpha. And that fiber is equal to one half the co dimension of the stratum. I think in most cases, it's actually. And together what one and two will mean is that if I take it that I can take the push forward of the constant sheet on why. And this is the complex of constructible sheaves on X, which captures the information of the cosmology of the fibers of the map from my next. And this will decompose into a direct sum over the alphas of the IC sheaf of X alpha. The topology of the fiber alpha. Your alpha is fine. And I should say one thing here I'm assuming really I should be right a bit more complicated statement but just to make my notation simpler. And since it's true in the most examples I'm interested in. I'm assuming that actually X alpha is simply connected. And this is, for example, not true for the cotangent bundle of the flag variety outside of typing, but it is true for the potential model of flag variety in type. So we have a decomposition of the push forward of this. And I'm sorry to be a bit more precise expositions for the homological shift. Excuse me. Can you take the closure when you take IC? Well by the IC sheaf of some stratum I mean the intermediate extension of its constant sheet. So yes, you could say it's the same as the closure. It's just, yeah. Yeah, you, you, you asked for I saw for to be smooth. That's why I asked. Yeah, so I really mean the, we could put a bar here. I'm just simplifying the notation of it but yeah I mean the intermediate extension of constant sheet on the X. Okay, so. So we'll come back to this decomposition shortly. But when I'm done, but you should take away from the part that has beautiful properties that's not from what. Okay. Next point is the existence of a deformation. So our original map Y to X. So it's a family. Script the X. How does it work so this script II Y maps to the vector space, which is, which is identified with H two of Y second called module Y with complex coefficients. So script the X maps to H two of Y modular the action of a finite group W which is called the Nami color while group in such a way that the fibers of these maps over zero give you X and Y. This seems a little weird but let me explain an example will become clear. So here we take the cotangent bundle of five variety. Mapping to no bone and cone. And as I mentioned before, these are pairs. Such that a of the eye is contained in the I minus one is no potent. And then I can construct, I can embed this in a bigger space and this will become my scripty Y which is in this case usually called g tilde. And in this case this deformation is called the growth in the simultaneous resolution usually. So here I put pairs a V, where a is not necessarily no potent, and I just have the condition a of vi is contained in the eye. So then I can just remember the matrix a. Sorry, then I can management. Yeah, that's this here. And this no point of course is contained inside the whole the algebra GLM. So all matrices. So this map takes a V to a, and the space of matrices maps to the T my W, which is just taking the characteristic polynomial matrix. And this thing maps to T, which is Cm, which is just recording the eigenvalues of a but we get to actually record them in order, because we can just take the action of a on each portion. So I have a flag, which is invariant under a. So able act by a scalar on each quotient vi over vi minus one. So that's this growth in the simultaneous resolution. And that's what happens in general so my original symplectic resolution is here. And it extends to this family. And if I put, take the fiber, if I take post zero, zero polynomial, of course I get back no potent comb. And that's the general picture. So why, when you see this you might ask where did this come from seems a little surprising. But he here's an explanation. So if we pick. If we take theta in each to why. And we have a good name, but we just temporarily I'll call it C or something. And then we take C inverses. Why theta so the fiber over this data. So essentially why theta is isomorphic to why, well, not as algebraic variety, but as a, as a smooth manifold, not as an algebraic variety. In fact, in fact, for most, for most values of theta why theta is actually affine is an algebraic right. But in any case, it is a smart because a smooth manifold. And we can use that to identify the second homology of why theta with the second homology of why. And inside the second homology of why theta, we have we can consider the symplectic form of why theta, and thereby, we can map it to a class in each to a why. The, like, compatibility of this diagram is that this class of the second homology of this, the class of this effective form is actually there. So for example, if we took theta to be zero, the class of the symplectic form on why is zero, which is not surprising because I didn't say it's above but this this why only its homology is only in that it only has KK homology only down the center of the high diamonds. So this symplectic form automatically would be in the two zero part of the homology. So that's somehow this that's somehow the reason for this appearance of this H2 of y. So for this reason, this map is sort of a period map, maybe precisely a period. I don't know much about the period maps that is mentioned. So this is another beautiful feature of symplectic resolutions they always come with this deformation of the big varieties. And the another feature that they come with is that they always come with. How much. I guess there was a couple of interruptions so maybe I go another five minutes or something is that is that good Francesco. Maybe also 10 minutes. Yeah, you can do it even more. Yeah, I mean we interrupt you so many times. I guess 10 minutes is a side to know. Yeah, 10 minutes. Okay, I think. Great. So the next feature is the existence of the universal quantization. So I think this will be the last thing I'll talk about, but it will probably take attendance. Okay, so what does it mean quantization. So I mentioned before that this, let's start just algebra purely algebraically. So this coordinate ring of X, I mentioned before this is a question algebra. So a quantization of X is over a polynomial ring and some variable usually called each bar quantization parameter. And sometimes this would be the power series in each bar, but one slight advantage of this adding this axiom of it being a conical symplectic resolution is that we can, it's quantization will actually live over polynomials in CH, but it's not a very big deal. So quantization of X is a CH bar algebra a such that if I take the quotient of I specialize each bar to zero to aim on each bar a. This will be isomorphic to see a X as a person algebra. To make this precise, I need to tell you why the left hand side is a person algebra. So it's an algebra just from this algebra structure. Now this a is going to be a non-community of algebra, and this the plus instruction left side is going to represent the first order non-community of algebra. So the plus and bracket of a bar B bar. So here in the elements in a bar and B bar elements in a mod H a. This is by definition, we form the commutator of a and B in a, and then because divide by each bar. Okay. Another way of saying this is like, sorry, this thing is this this commutator is necessarily divisible by H bar, because AB minus BA are equal. So, so this in C since CX is commutative, and the products are equal in CX. So the first order of non-community activity is divisible by H bar. And then, then we take the class of that and we get this plus on bracket. So that's the definition of the plus on bracket. And I would like that this, and that this plus on bracket agrees with my original plus on back and CX. Okay, so that's the notion of quantization guy. And then we could, this could be a quantization of why we would define a similar way but just in the sheath of algebras. Why. In this sense that this is is more of a structure. So then, so then I would like to say that there exists a unique universal quantization, just like our universal deformation. So this again will be parameterized by the again, parameterized by this H2. Or if we work, maybe I'll just write the statement for X is slightly simpler. So it's used for quantization of X parameterized by the same age to Y mod this W. So this is it. So this can be an algebra a and not too many different kinds of script. Okay, this will be another script. It will contain the coordinate ring of this H2 of Y mod W. So as a center is the center they and for any respect of the center which is the same as H2 of Y mod W. I can take the reduction by this ideal journey by this data. And this is a quantization of X. And more over every quantization of X horizons way. So I guess to be a little bit more precise. I should say every quantization of X that extends to a sheet of quantizations on Y horizons. So that's like an example again. And we'll take again this potential bundle the five varieties are our example. So, so X is this no quote and cone. Why is the potential bundle of this library. So then let's say G is the algebra GLN or SLN. SLN for simplicity. Then this universal quantization a will be the H bar version or the resalgebra of the universal envelope the algebra SLN. This is like the resalgebra of the usual universal envelope the algebra SLN, or if you like, it's the resalgebra generated by X by elements of SLN and then relations X, Y, Y, X is equal to H bar. So if we in this thing contains the center of the universal envelope in the SLN, which can be identified with the team with the carton model. So, so in particular, if we specialize to the Anyway, you can see if you set H bar to zero, then, then a will become this universal commutative deformation. The coordinate ring will become this same aspect of the symmetric algebra SLN, which is just this universal coordinate, which is just SLN itself, which is the universal commutative deformation SLN. Okay, so this I could have said that here as well that for this universal quantization a, if we take the any fiber of it, if we specialize data, we get a quantization of X and if we specialize if we don't specialize data and just take amod HA for the whole a, then it's the coordinate ring of this X tilde that I wrote about. Now, the last point I just want to make before we stop is that so there's like two kind of notions of quantization this thing here which called a formal quantization. So this CH bar version. But that's called usually a formal quantization. And then there's also a filtered quantization. So the first thing we specialize H bar to be one, so we take a what H bar minus one, then we get the filtered quantization. So we get instead of filtered algebra. It was associated graded is our original personal. So for example. In the example, we could take this you G and you H bar G and then we take this one. And then we get just the usual you G units. So oftentimes we'll study these filter quantizations instead of these formal quantizations but usually you can pass back and forth between the two things. Yeah, I'll stop there. Any questions from the audience or from the people over zoom. Yes, I have a question. When you define Nakajima with a varieties, you took some kind, which was a product of determinants. So if you choose another guy, another character for the group. Yeah. So, it's a good question. It's actually sort of a good general question. Okay, so let me let me actually answer the question sort of more generally and then come to this because I think so. So I started with X. Why is my like starting data. What's the solution. One interesting thing one can do is to start with just X. And then, and then try to find all possible wise. Maybe multiple different symbolic resolutions of X. One reason why you might want to do this is because we'll see soon that this data of these different symbolic resolutions has a sort of meaning on the symbolic dual side. For that reason, it's kind of interesting. And so in the case of an X expected as a Nakajima quiver variety. Or more generally as any kind of Hamiltonian reduction. So I'll write, I didn't actually introduce this location, but let me write it here for the first time. All right, three lines that I mean that this symplectic reduction. So what I mean by that is I take the in the mode map image and then take the, the genetic portion. That means and actually this zero in fact. When I write this notation, it's maybe helpful to write two pieces of notation here maybe. Hi, which could be well which would be this zero. It's just zero zeros. Don't need to invent some new letter. So I just take this. And then as you, as you said, we could take and change the, the this guy so this would be take the zero level the moment still, but then take the different character guy so so as I mentioned before usually this would be the product of determinants, but you could do something and yeah, if you do that in the case of Nakajima for writers are more generally for these symbiotic reductions like this. Yes, you could obtain other synthetic resolutions. And certainly there's some examples where you get interesting spaces. This way I'm not I mean, I know the people in the audience much more expert than me. But for example, in the case where you are considering Hilbert schemes, maybe Hilbert schemes of points on on on C2 my gammas, you can get these different versions of the Hilbert scheme this way is by varying this, this guy. And this brings up a different point which maybe I should have mentioned earlier, you can also vary that this parameter, the level of the movement, so which I'm writing here, here, here, you can vary that thing. So this is going to be responsible for this deformation that I explained earlier. So this deformation of the cover variety this universal deformation from the theorem of Kaleida and Amakawa. That's going to come from very bad that that that zero to other to other. So thanks for the question, but yes this study of different possible synthetic resolutions of a given synthetic singularity. This exit called a synthetic singularity. We have also a question and a window. So if you can check it or I can read it for you. Yeah, does the deformation in part three is a theorem carry a personal structure with the leaves given by the various white seated. Yeah, I think so yes. These are global plus on structure on the whole thing. Because I was, but yeah, I think so, yes. Any other question or comment. And no one is also. Okay, so maybe we can thank. You can join again.