 Yesterday, I kind of still was speaking in some kind of generality about how to take some kind of, you know, some kind of Kalaviya 3 situation. You associate it to some modular space of objects. And throughout category she is or something like that. And then to this you associate some kind of you know, Comology theory, which is this fee is some perverse sheath, basically obtained by gluing together vanishing cycles and then you can take its Comology or you can take it stock and kind of study that. One thing I think I forgot to kind of give you know I kind of get some attribution in the middle of it. But I mean just overall this kind of program of kind of going from X, the I mean this was kind of the work of a lot of people but the kind of you know, you know, I was focusing on the contributions of this group at Oxford sort of this kind of Oxford group. But kind of both before and in parallel there was also kind of work of, you know, five barons, you know, already in his original work he was kind of moving in this direction. And then this work of conservation soil man, all basically discussing this kind of process and you know, in various settings or various generals. And today what I'd like to talk to talk about is now to get to the second part of the, of the title which is this notion of Gopu Komarov often during. And at least in the setting that I'm going to be interested in this is really kind of in some sense supposed to be kind of a geometric idea so. So X really will be some kind of you know Columbia threefold, and we'll be interested in maybe some kind of you know curve counting on it. In the sense for instance of, you know, let's say Rahul lectures. I would, you know, for instance, I could, you know, first fix a curve class on X. And then if I'm working for instance and Gromov Witten theory, I would take this kind of, you know, stable mapping space for I could, I would take a genius I would look at the stable mapping space, space of stable mass, look at the corresponding you know, Gromov Witten invariant in this class with this genius which is now some rational number. And then I could, you know, form some kind of generating series for these. So what I've been doing which is more she theoretically as we can consider this kind of PT perspective where I take stable pairs. And so here the object would be maybe it'll call the PT where I take one plus the sum over beta comma and, and I take these PT virtual degrees. So here is indexing the Euler characteristic of the one dimensional sheet. So the idea is, you know, even if I fix beta there's still kind of instantly many numbers associated to both of these theories just because here I have to sum over the genius here I sum over this kind of value of n. And kind of this proposal of Gopal Korn and Vafa from, you know, the 90s was that you know, there should really only be finally many numbers that are kind of controlling these kind of generating series. So they were working this kind of sheet theory perspective wasn't around yet so they were really talking about the Gromov Witten side. And so the way that we kind of think about this. You know, you could try to think about what they were proposing geometrically although this is not really how they phrased it was that you know you have this cloudy at three fold imagine kind of you perturb the complex structure so that all embedded curves and X are kind of you know, smooth and isolated. You can do this but you can kind of imagine so you have maybe curves of different genus in this class beta. But maybe not just class beta maybe all the all the curve classes you know why not. And then imagine that there's some kind of you know counter these you know some kind of little call ng beta some count of these. How, if you were in this ideal situation you could try to ask how would these kinds of embedded curves contribute to each of these, you know, generating functions. For instance on the Gromov Witten side the proposal is the following kind of formula. So what do I mean by contribute well okay if I have this embedded curve, well it'll contribute to you know the corresponding Gromov Witten variant for G and beta but it also contribute to a bunch of other Gromov Witten variants because I could take a higher genus curve and map it onto this one. So there's kind of every curve contributes to you know many many different coefficients by a kind of multiple covers. So if you try to kind of, you know, make this heuristic a little bit more precise. What you get is a kind of proposal, the variables here, expressing this kind of Gromov Witten series in terms of some kind of you know, you know, simpler variance so I'll put, I'll put a GW here in parentheses just to indicate that right now I'm thinking of the Gromov Witten potential. It doesn't really matter so much what the exact formula is. I'll call this maybe equation one. And similarly you know on the on the stable pair side, you could ask the same thing. How does one of these curves contribute to kind of all the different you know stable pair endurance. If you again try to make this heuristic precise you kind of the following kind of expression. And again a single contribute curve contributes not just a class beta but to beta and so on. And that's kind of what this K is doing here. And so in either perspective, this kind of leads you to a conjecture which is that either the series it has an expression like this, where so the kind of conjecture that you're led to is that there exists integers and G beta, which are defined when you're entering the equals zero. And only finitely many are non zero for fixed beta, such that these equations hold. This is the kind of conjecture that you're kind of led to from this perspective. And actually, I mean one thing you can do so the first one is equation one the second is equation two. And you know, one way this kind of subject has developed a, you know, over the years is that you actually could take for instance you could take equation one as a definition of these integers you could kind of formally take this generating series, and you know expanded out like this. When you do that then there's something to check you still you have to check that whichever definition you take satisfies the kind of conditions of this conjecture. So this property here, and then becomes a conjecture, even though from the heuristic this property was kind of automatic. And actually, I mean, in the on the ground with inside so you start off with the ground with potential you kind of take this expansion, you get these quantities. It's not obvious that the quantities are even integers let alone that they have these properties. But this is actually now recently been proven so now this approach, just earlier this year I think this has been announced by. So there I mean this kind of nice picture where you actually have finally many embedded curves is definitely not true. But instead they kind of you know work directly with this kind of definition and prove that it has the integrality properties and the finiteness properties that you want. You could instead kind of take perspective to meaning the stable pair series. So this is a kind of a stronger version. I alluded to this in my lecture yesterday. This is kind of like a stronger version of this rationality theorem that I kind of sketched at the beginning of yesterday's lecture. And in this formulation it's still open this is still open. One thing I like about this conjecture so the conjecture is that you kind of can you could there exists an expand you know you there exists an identity like this. So formally there's always some kind of identity like this, but where you allow genus the genus index to be negative. So there's always a formal identity like this, then you need to show that only kind of non negative genus terms contribute, which is of course what you'd expect if you were supposed to think of these as coming from embedded curves, embedded connected curves of genus G. So this is still open. And again, one thing that's cool about this is that you know the rationality statement from last time would be true if you just took for instance the naive topological order characteristic. This version really requires this kind of you know virtual structure to see this approach is very different from the original paper so what what these guys originally did was that they actually had a direct physical definition as the weight spaces of some representation on the fixed modular space, some modular space and brains. So there's one single modular space, you take it's comology now not some kind of virtual count, and then you kind of you know mess around with it a little and that kind of gives you the finally many integers that kind of govern both of these generating functions. And so I mean this is reminiscent of kind of this McDonald formula that I kind of started on Monday's class with. On the other side you have you know this kind of generating function of Hilbert schemes on a smooth curve. And on the other side you kind of have this kind of comology of a single space just the Jacobian of the curve with some kind of pre factor. And so you know over the years there's been kind of a number of efforts to kind of try to make make this idea precise actually come up with some kind of direct definition that you could then compare with these other enumerative theories. So this is a story that kind of goes back. So I'm, I'm, I'm going to kind of explain today that kind of the latest in a right iteration of this kind of proposal but it goes back to work of. Sonno, Sido. Hashi, and then there was some work of Cam and Lee and then this kind of latest version is, you know, a proposal of myself and you can open. So this is the goal the goal is that you know we have, you know, there's some kind of you know expression like this and we want to kind of directly, you know find a way of studying these kinds of numbers and ideally it should be something close to what their proposal was something kind of So what is the mathematical setup. And so again this is inspired very much from the ideas in their paper where, you know what they were looking at was something that looked like, you know, a one dimensional sheath that curve was smooth they were looking at the Jacobian you know some the Jacobian of the curve. And in general, what we're going to do is we're going to take the module I of one dimensional sheaths. It's going to be like the analog of taking the Jacobian on the right hand side so it's similar to taking stable pairs except without the section, or precisely what I'm going to define is this I'm going to take m beta of x, where it's the module I space of is a one dimensional pure sheath, where again the support of this sheath is in class beta. So what are the characteristics to be equal to one, and then satisfying some kind of stability. And so what is the notion of stability that I want to take. Say that he is stable, but he has other characters one is stable, if for all subsheets, the other characteristic is less than or equal to zero. And it turns out and this is just kind of a fortunate accident of looking at kai equal to one that this is equivalent to just you know, he secures stability for any. He secures stability to pick like a polarization or something. And it turns out in the case of kai equals one it's just independent of the polarization. That's the nice thing about picking kai equals one. Oh, you can ask what happens for other values of pie and that's what I'll talk about later on. And so again you know heuristically you know if I fix a curve then the contribution of that curve to this modular space is just going to be the know the rank one sheaths on that curve push forward. So in particular this modular space is going to be a quick with the map to the chow variety of that so the chow variety, the chow variety, which a parameterizes effective one cycles, meaning a point in this variety here corresponds to just some formal linear combination of curves with with non negative coefficients. So this map is just I take a sheaf and I just send it to its cycle theoretic support. Now this kind of this is I should say this is kind of a weird thing in the sense that you know, all the modular spaces we generally look at or you know, like modular spaces they represent some functor on steam and so on. The chow varieties and really like that it's kind of constructed, you know, classically as just as a variety so even really thinking of this as a scheme. Recent work of Barnosky which maybe does something like this but in general, it's not something we have access to. But for now this is you know this is really the best we can do we want to take a sheaf and we just remember want to remember the support of it. So in particular if I fix a point in the chow variety corresponding to a smooth smooth curve. So the advantage in this modular space is exactly the kind of degree D. Sorry, the degree G line bundles, but then if I have a much more complicated one cycle the fiber of this map is going to be crazy. I'll try to get some examples. And so okay so the idea and again. This is the motivated by kind of go this original paper is that we want to kind of you know, decompose something like the comology. Then beta. According to the contributions of curves of different genius, so if I have a you know a curve of genius G let's say smooth curve, then it's going to contribute kind of this Picard. And it contributes basically the factor of the Jacobian, which to the comology is going to be just this you know comology of a Torah so one plus widely to G, and I'm going to put a little shift here to make it so it's actually symmetric and around wider to zero. It's like a palindromic polynomial. And then if I had let's say let's say I had a kind of a family of curves or some kind of you know, this is a pretty unlikely situation. Let's say I have kind of a family of curves. I'm kind of constant family curves inside of X. Then the corresponding when I look at m beta. You know, I get some kind of you know, pick G cross be mapping to be. Yeah, so maybe so the idea is that just based on the fact that a curve of genius G should kind of contribute something like this to the comology. What I want to do is kind of decompose homology of M as a sum of terms like this. And then the kind of coefficient of you know this term in the expansion would be like the genius G contribution. And so then if I had for instance just a family a constant family of this curve of genius G in the kind of modular space I would end up with a constant family of this Jacobian, and I would like to wait this by just the Euler characteristic of the base. So the idea is that something like this would contribute this factor is comology of this chorus, weighted by the Euler characters to the base. All right, so let's see how we can kind of try to execute that. So the kind of the first case which is wildly unrealistic is when kind of everything inside is smooth. So let's say m beta smooth. And this map is smooth. Okay, but maybe the child variety is disconnected so has like pieces parameterizing curves of genius G and then another parameterizing curves of genius, you know, H. So there's still something going on here. And again, if I take a point the fiber of it is going to be this part. I kind of, you know, mimic what I did in this kind of constant case what I can do is I can just take the constant sheath on M. Push it forward, my child variety which everything is smooth. So this thing if I look at the comology sheath of it. This is just going to be like a local system over this child variety. And then I can take it's Euler characteristic, which is kind of what I was doing here when I waited the constant family by the other characters to the base. This local system would just be constant. And then if I decorate this with, you know, why to the power representing the homological degree, some kind of appropriate shift so it's centered around zero. This is going to be a symmetric sending why goes to one over why. Again, this is just, you know, by the, you know, relative left chefs. And so in particular, I can expand it. We have a question from here. What's up with the high around sheaths on the curve. Oh, yeah, right. So if I have a higher ranked sheaths supported on the curve, the corresponding cycle would be like two times the curve. And so something like that could, in fact, contribute here. Was that your question. Yeah, so I'm still right now in this kind of ideal situation where I have like, I'm thinking about the curve is being reduced. But if I have a cycle that's like two times C, then they're absolutely situations were like a rank two bundle on that curve would couldn't would just contribute and the point is that the corresponding cycle theoretic support would be two times the currency. And those will kind of contribute to these numbers. Now you're taking just primitive. Well, right now I mean right now, yeah, right now beta. Yeah, right now everything here is in this case one which is the kind of an ideal situation. I'm thinking of this child variety is just parameterizing smooth curves inside of that. Okay, so not with no multiplicities in particular primitive, and then I'll kind of, you know, work my way up to the more general case. Yeah, it's possible that trying to do it slowly like that make them more confusing, but right now this is kind of a, this case one is basically, you know, to naive. But I want to just kind of spell out what it means to kind of take this kind of series and expand it out in terms of the homology of these different Torah. And so, because this Laurent, this Laurent polynomial is symmetric and goes in why those the one of Roy, I can just expand it out in a base in a basis of these contributions of Torah. So this will be some finite some these will be integers. They will only kind of live when the genius is greater than equal to zero. And then this is in some sense the kind of definition. And this was, you know, this was again proposed originally by kind of the sonos I don't talk about she to handle this kind of case and this in, you know, in the very few nice cases where you actually are in the situation. This does in fact match up with the predictions coming from the Northwood and so the next case which already introduces a wrinkle is when let's say the modular space is smooth. Let's say the chow variety is smooth, but the map is no longer smooth. Maybe the map is some singularities. This would happen, for instance, if I had a family of curves that develop singularity, which almost always happens. And you know this proposal here doesn't really make any sense because these comology shoes aren't. I don't have the symmetry anymore. So this is not a talent drama. First of all, they're not going to be local systems. Maybe that's not a big deal, but I needed the symmetry in order to expand it out in terms of the comology of a Torah. So here's kind of just a simple example of the breakdown on this of the symmetry that I want to write down, which is that let's say I had like a just a constant family E cross P one over P one, and then I blow it up at a point. And so, so the original projection is pi blow up map is tau and this composition will call pi tilde. I just want to indicate, you know, the failure of this kind of the failure of this relative lecce theorem, the failure of the symmetry, and then we'll see how to fix it. So if I calculate the push forward of the constant chief upstairs the smooth surface all the way down to P one push forward in two steps. And the first step when I just kind of blow, just do the blowdown map. This push forward the constant chief well there's the turns up there's a constant chief on my original surface, and then I have that extra P one that extra. The exceptional locus gives me an extra class and degree two. So if I think about the she's theoretically I get a skyscraper she's at the point that I'm blowing up on, but I have to put it in degree two. And then if I push this forward all the way down to P one. What do I get I get kind of Q and degree zero. I guess he was an elliptic curve so I get q. Two and degree one. And then in degree two I get, you know, two things, which violates the symmetry so the ideally once I shifted everything, you know this first term and this last term should be the same. And they're not. And if I took Euler characteristics then I would not get something that I could expand out. And so instead, one solution is not clear that this is the right solution in this case, but one thing that would at least solve that problem is to take. Instead of taking the kind of homology she's, I could take the perverse homology she's after pushing for. So in this example, for instance what happens is if I take this perverse homology she's. So now these are no longer she's a perverse she's. And I kind of, you know, shift, instead of pushing with the constant she if I push forward the shift by two so this thing is also perverse. These again satisfy the symmetry that I want, they satisfy. And so K and minus K will give me the same object. And so how does that work in this example well, what happens in this example. So it's the same push forward I've already worked out what the answer is, but now I have to kind of, you know, regroup them according to their kind of perverse degrees. So the q gets shifted by to this is. And I get, you know, this and then this. And then what happens is that this term is the kind of degree negative one term. So these two terms together, give me the degree zero term and then finally this term is in degree plus one. And so now it's again kind of symmetric around zero. Why is that well it's just that on on on this base p one q isn't perverse q shifted by one is perverse, and then q p without the shift. And so that automatically kind of takes this kind of skyscraper she that was causing me problems and moves it into the middle for storing the symmetry. And so that in general so if I take, again, I'm in the situation when M is smooth, but the map is singular. This is originally do the sonos I don't talk to Hashi. I can just take what I did originally in the kind of case where everything is smooth and just replace taking the homology she's these perverse homology she's. And so again I have to do some shifting to get things centered around zero. The first statement is because of this, you know, perverse hard left this is now again, palindromic and this is, you know, this is symmetric around with why goes to one over why. And so just formally I can always expand it in terms of the, the homology of my billion varieties. And this already I mean what's striking here is the use of kind of having to take kind of these kinds of perverse homology she's, and this is already kind of works in lots of interesting cases. So, so I was going to give a couple of examples one of some of these I think I was going to ask Jun Liao to do in the Q&A session tomorrow, just because they're they're nice. You know, it's not it's always nice to do some computation. So for instance one kind of basic example is where you take X over ass is a elliptic. You have three folds of ass is some surface. And you know the easiest to work in the case where you have integral fibers, and for beta we could just take the class of a fiber. In that case if you look at what the modular spaces. So again it'll be one dimensional sheaves scheme theoretically supported on the fiber. It ends up being the same as taking picking a elliptic fiber and taking a rank one torsion free sheaf on it and pushing it forward. So this ends up, you know, because the fibers are all genus one curves, the modular space of rank one she's on it just recovers the curve so the modular space as I vary which fiber I'm looking at is just isomorphic to us. The chow variety in this case, which is you just forget the sheath just remember the sport is just telling me which fiber it is. So the chow variety is just s. And this Hilbert chow map is just the original elliptic vibration. And so you can then just do the calculation here so that you calculate this push forward what you end up getting, you know in the different perverse degrees is, you know, the perverse homology and degree negative one and one is just the constant sheaf. And then you get something in the middle, you can you know something but all you need care about really the kind of Euler characteristics. And so when you feed this into the definition what you get is the genus one Gokumar of often variant from this definition that involves these kind of outer terms. And so this just gives you the other characteristic of the base. The gene is zero is kind of determined you know it's kind of what's left over, and it ends up being just the other characteristic of the total space. You see if you look at this expression is not totally ugly. If I plug in y equals negative one, only the genus zero term lives on the right and so on the left, you just get the other characteristic of the sheaf upstairs so this, this kind of thing is always true. And so the thing to check that this actually matches what you expect from the groom of witnesses, promote inside so this matches. This is not a hard calculation. There's a more kind of striking check for me, which I'll kind of talk about next time tomorrow, where you really have to kind of, you know, use this kind of perverse filtration to get the right answer, which is where you take X in this case would be the total space of the canonical bundle. So if you if you think about it, you know, stability forces the sheaves to be really scheme theoretically supported on the zero section. So, and beta of X really is just some modular space of one dimensional sheaves on P two. So the chow variety is just mapped to the like the linear system. And so if you at least restrict to the locus of let's say reduced curves, this kind of perverse in a homology is exactly what you need to match the contribution from the curve counting theory. I'll explain this next time I just want to kind of indicate there's really there really situations where you actually really need the perverse homology to get the right answer and then it kind of matches up beautifully. So the case has, you know, kind of a restriction on it, which is that I assume you have a lot of smoothness still. And so then kind of, you know, case three, and beta singular. And you know, then again everything, what I said so far breaks down so you again, don't have this is kind of now really a strange object you see before I was using the fact that the constant chief after a shift is perverse on a smooth curve. And so here this is no longer going to be true. This is no longer symmetric. And also it's kind of the wrong thing you want something that you know detects the virtual structure. And so the idea is, well, maybe not so surprising because we spent the whole lecture developing it is to use this sheaf from yesterday. So this was again this was a perverse sheaf on my modular space. This is a self dual, this is some property just a vanishing cycles. I mean if I take the very dual of fee I get fee again. And so why is that important well it means that if I look at this proper map from and beta to Chow beta. It means that the symmetry is going to be kind of guaranteed for me, the symmetry that I want that just kind of follows from, basically just follows from properties of this dual. And then again now I can just take the same definition that I did in case to but replace the constant sheaf on am with this new thing. So this is the kind of definition, which is still slightly provisional, for a reason I'll explain in a second. This is this, this is kind of proposed in this work I did with the UK no boo couple years ago. And that I again take with generating functions of you know, Euler characteristics is a Laurent polynomial symmetric with y goes to one over why, and then I can just expand it out into the contributions of Jacobians. And again just completely just on just general grounds, these numbers are you know integers because they're coming basically by alternating. And then your combinations of these Euler characteristics, since everything here is a Laurent polynomial only finally many are non zero for fix beta. Again if I stick in y equals negative one that picks out the genius your term on the right and then the left I just get the Euler characteristic if I plug in negative one I'm just getting the Euler characteristic of the sheaf upstairs, which by how we were you know thinking this is just the virtual number for my modular space of one dimensional shoes. And this is again this is something that we expected this is expected. You know if you will kind of believe all these conjectures that I stated at the beginning of my lecture, then this is this kind of thing is expected by by the other objectors that this should calculate the genus zero go from our Boston drain. And this is kind of something that was proposed by Sheldon cats, and before, before these other approaches were developed. We have a question. So, so these are the graded pieces of the thermos filtration on the common sheet of the vanishing sheaf right. Yeah, I mean, I don't know but yes, graded pieces yes. Yes, there's also this hodge filtration on this on this common sheet of the vanishing sheaf. Is there some relation expected. No, I mean in general they're just different. I mean what you see what's interesting. What's interesting is that you can try to take. Yeah, you can say you can try to take a definition that's more hodge theoretic. So in general in general they're just different I think, but you know you can ask why are we taking the perverse homology sheaves as opposed to something that's like involves like the weight filtration or the hodge filtration something that's a little bit more like hodge theoretic. And the short answer is it just gives the wrong answer. So, so you know I mentioned before this kind of can lead proposal that can lead proposal is basically the kind of do something with like the weight filtration on this and that's easier to calculate but it just gives the wrong answer and in kind of various cases. Alright, so did I don't know that answer your question. Yeah, thanks. That is that you Georg. Yeah, it's me. Okay, um, so let me just, I mean, so I'm going a little slower than I anticipated so what I'll do is. There's another example which I guess I will leave this for Junlian to do also. So there's no information contained in this filtration that is not seen by this mix hot structure whatever. Yeah, well this for I mean depends what you mean by the mix hot structure but if you forget homology and only remember like, you know, like HPQ or something like that, then, then you lose the information yeah as far as we can tell we tried we tried for a while to come up with the definition along those lines. And it, you know, there's, there's just there are some examples where it just gives the wrong answer. And so maybe I maybe I guess I'll end up doing an example of that at the beginning of tomorrow's lecture. So one example where you can kind of see, which is useful for seeing how the kind of vanishing cycles kind of helps out is where you take a nodal curve like a nodal rational curve genus one, and I embed it in kind of some rigid way into a sort of a clobby out three fold. And then I consider just this, you know, see what this definition spits out. In this case the modular space. Again, this is, you're looking things are essentially rank one torch and free sheets on this curve push forward, so at least set theoretically. You get the curve again, but actually there's some kind of non reduced structure at the point so there's some kind of non reduced. And if you then feed this kind of, you know, definition, you get, you know, what you get is you get the genus one invariance is one, and then the genus zero invariance is also one. And then this is a calculation you can do and again this, you can do the corresponding stable pairs calculation because you just using the Baron function or something like that. And you recover exactly this and if you did just the constant sheaf you get kind of a different answer. Okay, so okay, let me just mention one thing though which is that I stated this is a provisional definition and why is it provisional definition. Well, it's because of something I said last time which is that to define this sheaf, you don't just need the modular space you also need an orientation on it. Skip that. And it really matters actually if you pick the different orientation you would get different invariance like so for instance in this example here this rigid nodal curve example. You have another choice of orientation, and I picked one to give you the right answer. And if you pick the other one you'll just get a different answer which is incorrect. And so how do we pick it well okay so this is kind of again something that you know was needed just experimentally. In the modular space we say an orientation is quality out if it's restriction to kind of the fibers of this map to the child variety are triggered conjecture which in general we don't know how to show is that these exist. If they exist actually doesn't matter which one you pick you'll get the same answers, but then. The correct definition is you know define the Gokul Kumar of often variants. Just as I did before, but using the sheaf vanishing cycles for a cloudy orientation. And so when I do some examples, tomorrow, you know all the examples that we know have this property, we have to make sure you choose the right one so like for instance, and this kind of rigid curve example, the child variety in this case is just a point. The other thing is that you have to pick your orientation so that it's just, you know, trivial so it's you know basically. Oh, and if you instead picked a different two torsion line bundle you would just get something else. And so then okay so then what is the conjecture so now the kind of key conjecture is that you know, this is computing what we wanted to compute, which is that if you like the kind of these integers define using this kind of comology of them beta agree with the kind of stable pair theory again, as a reminder this was defined by taking the logarithm of this kind of generating function and expanding it out. And so for okay so in general is because of this logarithm it looks a little complicated in that if you specialize to the case when beta is like irreducible. Then you know then this identity becomes a little easier to understand on the on the stable pair side. I might be in my notes I go I was a little sloppy with a plus or minus q so I think I think this is how I should do it. And if it is irreducible then I don't really have to worry about this logarithm so much and then the statement becomes kind of clean. What it's saying is I'm going to take on the one hand I take the pt series. On the other hand I take this kind of Gokomar boss expression, and then the thing on the numerator is just this original kind of perverse. So the idea is that if I have an irreducible curve I take the pt series on the left hand side, and on the right hand side I'm just getting in the numerator, this, this thing obtained by taking the perverse comology of the sheet of vanishing cycles. And in particular is the engine beta here. Say what the engine data in this formula is the engine beta pt. Yeah, sorry GV. Yeah, maybe this. Well, I mean the conjecture is that they're the same. So, so the idea is that. And so this is again this really is if I kind of specialized even further. So for instance in the situation, I just have a smooth curve. And you know all of these complexities that we had to introduce for the virtual structure go away so let's say that the Baron function is identically you know one. And this chief of vanishing cycles is just the constant chief up to a shift. Then this is really is this is this really is just the McDonald formula, where I'm taking on the left hand side the oiler characteristics of the Hilbert scheme of points, and the right hand side the comology of the Jacobian. And so in general, this is kind of what this is kind of what I meant with the beginning of my first lecture is that you know this picture gives some way of, you know, promoting this to study, you know, much more singular curves. And so I guess I'll stop here what I'll do next time is give, you know, some examples in just the examples where we can actually kind of prove this. And in particular, there's some very general class of examples where we can prove this kind of this conjecture. And so I'll try to talk about that in particular say something about the techniques of proof that let you actually study something like this. When you look at it right now it seems kind of hopeless because you both have this very mysterious sheet of vanishing cycles which is very hard to get your head around. You also have to take this kind of perverse homology on the chow variety which is also kind of hard to get your head around. And so at least in the cases where we understand well what's going on. It's just that those constructions while kind of mysterious also have good formal properties that let you actually prove this kind of statement in many cases. So I guess that's what I'll do tomorrow. Let's take the question in the Q&A first. Is there a way to see or reason to explain that these cobalt number numbers are deformation invariant. Oh, yeah, absolutely. That is I think that that's like the big mystery. Yeah, so so I'll give an exact. Yeah, so the short answer is that's like I think if that were true then this conjecture would definitely be true. But in fact we don't know how to see that. And so I in fact, it's possible that maybe it's not true. There'll be one kind of example that you can open up like a lot where you actually see the deformation invariance kind of, you know, by hand, or you just calculate it and kind of interesting examples, and it's a really non trivial statement. And so I should say with this conjecture, I mean, you know, I, my feeling about this conjecture is that like for local geometries, I have a pretty good amount of confidence in the conjecture for compact geometries like the Wintek threefold, then it's less clear what's going on. And, you know, the thing is that like, you know, this is already the third iteration of this kind of, you know, of this conjecture. So, you know, it would be a little, you know, it would be, I think a little naive to assume that maybe this is the final settled form of it. That said, you know, in the in these kinds of local geometries then a much more optimistic. I mean those are the cases where we can really prove some stuff. And so, I mean, this, this conjecture, for instance, even if you take local P2, it's open. I'm very confident in that case, but we can only prove it in some kind of for some open locus of curves and local P2. And I think even understanding what's going on in that geometry, I think it's really interesting. Some more questions here. Can this conjecture of upper form of upper Pt be proven under ideal Calabria three geometry. What do you mean, sorry, in which geometry. So, so if you assume that every smooth curves are embedded and isolated then can we prove the conjecture of GV Pt correspondence. Yeah, that's, yeah, that's right. I mean, in that case, if you were in that really nice situation. Then I, you know, then I think it's basically you have some assumption about the normal bundles. I have to think about that I mean it's not clear. Yeah, I mean, to the extent to which it makes sense, I think you can prove it, but you have to be a little careful because that would be some kind of likely non outbreak deformations of talking about the sheath theory is a little delicate maybe. But that certainly is, I think basically, in that case, you know the child varieties, you know just a bunch of points. So on the this kind of calculation on the, what I'm calling the GV side is you know there's nothing going on so you have to somehow argue that on the the Pt side that there's no kind of, you know, secret, you know, barren stuff kind of throwing up showing up. But if you're kind of willing to make that assumption that I think it's okay. Yeah. Sorry, that's not a really satisfying assumption. I think I think it's not entirely a well posed question because those kinds of that ideal situation that's going to have some kind of non algebraic features. But not to do that. We probably also need to consider a higher rank stable bundles over a smooth embedded curves. Yeah, right. So right. So usually what'll contribute. So right if the curve is like sufficiently rigid then if you look at like two times the curve. Yeah, actually that's a good point. So if the curve is sufficiently rigid, or you know have some kind of generic normal bundle or something like that you would expect to see higher bundles on the curve contributing like to two times the curve. And so where that would fit in in this picture. Yeah, so I guess even in that case it would not be obvious. Thank you. So yeah, maybe this question was already answered so someone supplanted jump to be also at the scope of a lot of engineering so yeah, you expected your definition to kind of make sense there. No, definitely not. Yeah, I would say no yeah. I mean, I mean, of course I haven't really studied this so I'm still sharing the screen right so I mentioned this, this very recent work. I haven't really studied this paper. I mean, I talked to Alex about it like a year ago but I haven't really studied the, what, what they've done so I mean it'd be interesting to see exactly, you know, if there are techniques in this paper that would for instance, you know, for instance even just understanding the kind of PT version of this conjecture, where you have to prove that there are no mis-contributions. Now that was proving that via this kind of you know, symplectic approach would be really interesting. And that's something that don't and while Puski I think have been thinking about like how exactly can one make sense for instance of a symplectic version of PT invariance and they have some ideas about that. Apologies if you said this already but what's the process connection between the PT numbers and the Grimow-Witton numbers? Oh yeah, actually I didn't say that. I mean, of course I sort of assumed that Rahul will talk about that or something like that. But so, you know, one way of thinking about it is that, you know, if these lower ends should be equal to these lower ends, and so therefore that implies some relationship between the Grimow-Witton invariance and the PT invariance. And that's, you know, that was kind of this, you know, original conjecture relating the kind of sheath theory approach to curve counting with the kind of Grimow-Witton approach to curve counting. So up to some, after some change of variables, these two things are supposed to be the same. And that corresponds to the fact that, you know, this underlying integral and these underlying integers that kind of control this generating function are equal to the, should be equal to the underlying integers that control this generating function. Can I ask one more question? So if you study two-dimensional sheaths in PT theory, like should one also expect some sort of Kupakumar-Waffa theory there or like have some integral structure in the PT invariance? Yeah, that's maybe different, right? So what you're asking about is if I take two-dimensional sheaths on the three-fold. Is that what you're asking about? Sorry. I'm sorry, I didn't hear, I didn't hear half of what you said. Sorry, could you say that again? I'm saying, I'm saying what's special about curves, you know, like we could take other sheaths, you know, they're not good. Oh, yeah. I don't know. That's, I mean, yeah, that's, I mean, of course, as you know, there's some kind of, there is some kind of expected structure theory for these two-dimensional invariance. But it looks really different from this kind of thing, right? Yeah, that's true, yeah. Yeah, that's a good question. All right. Any more wishes? Let's spend the wish again.