 I want to thank, obviously, the organizers for the chance to lecture here, although I wish the original plan, when this was set up a couple of years ago, it was going to come with my family, and it was going to be this two weeks, and France, and all that. And obviously, this is slightly less than what I was hoping for. So let me begin by giving some motivation about what I want to talk about, which has to do. So for motivation, let's just start off with the case of a smooth curve of genus G. And then I can look at its Hilbert scheme of points, of endpoints, which, since C is smooth, is just the same as taking the symmetric power of the curve. And so there's a very nice formula going back to, well, probably in this version it's older, but on the level of, say, homology, goes back to an old paper of McDonald, which says that if I take the Euler characteristics of these Hilbert schemes, and I sum over n, well, this just has a very nice formula where I just take 1 minus q to the 2g minus 2. I can then kind of make more complicated in kind of a dumb way by writing it as 1 minus q to the 2g over 1 minus q square. And now the numerator of the right-hand side itself has kind of a geometric meaning. If I take the Jacobian of my curve and I take its homology, well, we know what that is. It's just the exterior algebra of h1 of my curve. And so if I take the Poincare polynomial here, the Jacobian, where I'll just introduce a minus sign, that's exactly the numerator of this expression. And so now what I have is I have some kind of slightly silly identity where the left-hand side I'm kind of summing up overall n, these Euler characteristics of the Hilbert schemes. On the right-hand side, I just have a single space, just the Jacobian, but now I'm doing something a little bit more refined. I'm taking the actual homology instead of just the Euler characteristics. And the q kind of has a different meaning depending on which side I'm on. On the left-hand side, the q is just indexing which module I space I'm working with. On the right-hand side, the q is this kind of homological variable. It's helping me keep track of my homological degree of the Jacobian. And so in some sense, one of the things I would want to try to explain in these lectures is kind of a non-sox due to many people. So I'll do the attributions later on when I actually get to it. But the iteration that I'll be talking about has kind of joined with the Toto, you can know with Toto, which basically proposes a way of extending this to kind of much more singular curves. So for instance, pervs in the kind of setting of Kalabi out threefolds. And one of the things that I kind of want to get to hopefully is that, OK, so generally it's a conjecture. And you can believe it or not. But in the cases where we can prove it, already kind of gives you examples where something like this holds for extremely singular curves. And the technique of proof is kind of nice because in some sense what the technique of proof really does is it reduces it to the case of the smooth case, where it originally just McDonald's formula. So there'll be some kind of chain of logic where kind of the final step will just be applying the original identity. So let me give kind of, I'll start talking about this properly in maybe the third lecture. So let me just say now kind of what I'm hoping to kind of cover in these five lectures, which is in the kind of first two lectures. I want to give some kind of overview about Donaldson Thomas theory, some version of which you saw in Richard's lectures last week. But I'm going to focus really again on the setting of Kalabi out threefolds. And in particular, the kind of perspective that you get when you get to in the Kalabi out three setting is that instead of doing things intersection theoretically there's kind of an alternate approach where you work in this kind of constructible world. So constructible functions and then that'll be today and then eventually tomorrow constructible sheath. And then kind of in the remaining lectures, I want to then kind of talk about this kind of picture that I just sketched out, which is this notion of an approach to thinking about what are called Gokomar, Gokomar Vasa invariance, which you should just think of the analog of this as the analog of the right-hand side of this McDonald's formula, some analog of what to put in the numerator in general, where we'll be using the kind of technology developed in the first couple of lectures to pursue that. And then in kind of the last lecture, I want to kind of talk about some related conjectures to this story, in particular a conjecture of Tota in the last couple of years, which is related. These kinds of topics are related to kind of defining right-hand side of this McDonald's formula. Are there any questions before I get started properly? So again, so if there are things that show up in the chat, I'm not really probably going to be able to see them so easily, I think, so hopefully Andre can read them aloud in his dulcet tones. So today, I want to start talking really in some generality about numerical Donaldson-Thomas theory. And so again, the setting for today will be focusing on kind of the geometric setting, and specifically in the case where we start off with some kind of Kalabiyao threefold. It does not necessarily have to be a projective. So just some, you know, smooth algebraic threefold with nowhere vanishing algebraic threefold. And the kind of basic moduli space that you associate here is going to be some moduli space, m of x, which is going to be some moduli of stable coherent sheaves, moduli space of stable coherent sheaves, or more generally, complexes of sheaves, on x with a fixed discrete invariance during classes indexed by some vector v in the cosmology. And so the kind of classic version of this story is where you just take, maybe you fix a polarization, and then the moduli space you look at is just, you know, let's say, you know, geesecker stable sheaves on x to avoid kind of stacky issues or issues with the obstructions that you often will kind of trivial, you know, fix an isomorphism of the determinant of ease. So here I'm giving some kind of examples. For curve counting purposes, it's usually better to work with some kind of variation on these spaces. So, you know, the old version that we kind of, the subject kind of started off with was to work with the Hilbert scheme of curves, or rather, Hilbert scheme of one-dimensional sub-schemes on x. So here maybe I'm gonna fix an element of H2, and then I also fix an Euler characteristic. And so this would parameterize, you know, sub-schemes, one-dimensional sub-schemes, where the support, the kind of one-dimensional piece is last beta, and then the Euler characteristic of the structure sheaves is n. And then one example that, you know, we'll see if I have time to talk about maybe in some, is the special case when beta is zero, and then you're just considering the Hilbert scheme of points on x. Now, in these examples, the way you kind of put it into this framework of modular space of stable sheaves is that instead of thing about the sub-scheme and then the kind of surjection from the structure sheaf, you just remember the ideal sheaf. So you think of this, you look at the ideal sheaf of your sub-scheme, which is a rank one sheaf on x with the trivialization of the determinants, and this turns out to be equivalent to looking at the Hilbert scheme. So the way you put it in this framework of modular space of sheaves is by forgetting about your sub-scheme and instead remembering just the ideal sheaf that cuts it out. The version that's cleaner, and maybe actually Rahul already has spoken about this in his lectures, or if not, he will soon, I'm sure, is a variation. Again, for curve-counting purposes of the Hilbert scheme construction, where you work with what are called stable pairs. And so this theory was developed by Pondre Ponde and Thomas. And so here the modular space, again, you kind of fix a curve class and you fix an integer. And then the data here in this modular space is, there's two pieces of data here. First is a sheaf e, which is one-dimensional support and is pure. Pure meaning has no kind of zero-dimensional sub-sheaves. And then the second piece of data is just a section of the sheaf. And then the stability condition is just the statement that the co-kernel of this section is zero-dimensional. So what does an object of this space look like? So the simplest kind of example is to think about, if you haven't seen this before, oh, I should just say what the discrete invariants are. So again, the support of the sheaf, just the cycle theoretic support is gonna be in class beta. And then the Euler curve is Euler x, is n. So the way to think about what this space looks like to first approximation is, let's say here is x. And then imagine the support of E is, let's say, smooth curves inside of x. And so you could ask, what are all the stable pairs with fixed support like this? Well, the simplest is I just take the surjection, O x to O c. So that's an example of a stable pair. But what I could also do is I could also take, if I give you some line bundle on this curve, I could take E to be its push forward, which is now a coherent sheaf on x with one-dimensional pure support. And then if I just any non-zero section of L will then produce a stable pair on x. So given non-zero section here, it defines for me a section of E and the co-cernel of the corresponding section of E on E will exactly just be the zero locus of the section on the curve. So the way you can think about this is that you have some line bundle on this curve and now I have some section. And so the same curve will contribute many different stable pairs. Just, you take any line bundle with a section or equivalently any collection of points on my curve, there will be a corresponding stable pair on x. So if I, the contribution of C to all of these stable pairs fit ends up looking a lot like just, taking the different symmetric powers of my curve. And so in particular, this will be kind of, the left-hand side of this kind of McDonald inequality in general. And so again, how do we wanna put this in this kind of setting of DT theory? So again, the setting of DT theory that I wrote it is that we're gonna wanna consider a modular space of sheaves or complexes of sheaves on x. And so here I have this kind of two-term complex. I just think of the corresponding object in the drive category. So I take this two-term complex and I think of this as an element in the drive category of coherent sheaves on x. And this is how I'm gonna think about this modular space. It's a modular space of a certain kinds of two-term complexes. If you, again, so one thing that's kind of instructive to do is, if x is projected then this space of pairs is also projected. And so kind of a fun thing to try to do is just to understand what certain limits look like in this space. So for instance, maybe exercise. This is a local question. So it doesn't really matter what the ambient space, what the ambient three-fold is, but imagine I have two lines that are kind of colliding, two skew lines that are kind of about to collide in some limit. And then we know from Hartron or something that the limit in the kind of Hilbert scheme of one-dimensional subschines is you get two intersecting now coplanar lines and a little fat point there. But this kind of limit isn't allowed in the stable pair space because the support of the sheaf because the sheaf isn't pure. And so instead what you can try to work out is what the limit of this kind of thing in the stable pairs moduli spaces where the support of the sheaf is still going to be these two lines but now there's going to be what used to be a kind of a fat point now gets replaced with some kind of non-zero co-kernel. So these are kind of, the examples of the kinds of moduli spaces, geometric moduli spaces that one can look at. But actually everything I'll be talking about today it doesn't really have to be a geometric setting. So they're kind of the kind of main example of non-geometric examples to look at or come from looking at representations of quivers with potential. So everything I'll be saying today and tomorrow kind of makes sense in that setting. In particular, there's a lot of the material in Marcus Reineke's lectures I think will be relevant. So these are the examples I want to look at. And so as always, we're interested in some kind of virtual structure in these moduli spaces. And so in the context of Richard's talks last week the kind of initial piece of data that you want to understand is something about the deformation theory. So you have a deformation and obstruction theory for understanding these moduli problems which because I'm just working with moduli spaces of sheaves or maybe complexes of sheaves they're given by X groups. So the deformation space, the tangent space is given by the self X one of whatever your sheave for complex is. If I fix the determinant then we usually do some kind of traceless thing. And then the obstruction space is then given by next two. And so again, my understanding is Richard kind of talked in more detail about how these show up for moduli of sheaves. And so already the first nice thing that happens in the Calabi out three case. So this is of course, so far everything is completely general which is you get to apply serduality. So if I take the dual of the obstruction space I get X one with this twist by the canonical bundle of X but because I'm in the Calabi out three setting this is just trivial. So I get exactly the deformation space. And so in particular it means that the virtual dimension of my moduli space was just the difference in dimensions is zero. And so if X is proper or at least if my moduli space is proper that's really all I need. I have this virtual class which is a zero cycle and under the properness hypothesis I can take its degree which will give me a number. I'll call this kind of the virtual number associated to my moduli space and it's just some integer. And the procedure for doing this again this is something that Richard sketched out is the way you produce this virtual class from all this data is basically by using some techniques from intersection three. So that's kind of the world where these constructions live most naturally. Okay, so what do I want to explain first then is again in the Calabi out setting what another way of thinking about what these numbers are. So this is this notion of what we now call the Baron functions. The setting here is that let's say I have some, I have some moduli space M and I've equipped it with this kind of perfect obstruction theory meaning I have some two term complex that calculates the deformations and the obstructions to my moduli problem. We say that E is symmetric I have a quasi isomorphism between E and it's shifted dual. So you should think of E as being kind of supported in degrees negative one and zero. And then E dual is going to be shifted supporting degrees zero and one and then I shift it back. So it's again, supporting degrees negative one and zero. Well, I have some isomorphism like this with the symmetry condition. So such that this isomorphism itself has some kind of self duality property. The easiest way to get a kind of data like this is that for instance, if I have what they have F as a vector bundle with a symmetric bilinear form alpha then you can produce an example of a complex with this kind of symmetry just by taking a map from F dual to F and then you have using this bilinear form you can produce a map like this which exactly has this kind of symmetry. And so the baby example of a moduli space with a two term perfect obstruction theory with this kind of symmetry is where your obstruction theory is kind of dumb. So let's say M is smooth and then your obstruction theory basically consists of the map from the tangent bundle of M the cotangent bundle of M, which is just the zero map. So this is like, I have an obstruction space but it those obstructions are all unrealized because the space itself in fact happens to be smooth. And in this case, if you calculate what the virtual class is so it should be zero dimension. It's a zero dimensional virtual class is the definition of obstruction in dimension. It's just going to be, it'll just given me that the Euler class of the cotangent bundle, which is my obstruction bundle. And if I take its degree, I just get up to assign the topological Euler characteristic of M. And so the first observation everything here I should say is I'll be saying is due to hi Darren except for the name. He didn't of course name it after himself but at some point it caught on. So in the Calabi out three settings all the examples that I said before again, just the same seriality calculation I did before tells you something a little stronger. It tells you that the obstruction theories. And so let me give the kind of key local example of one of these symmetric obstruction theories which we'll kind of use again next time which is an imagine I have some ambient smooth space V which is just a fine space and I have some function on it. And then the kind of space that I'm looking at my actual modular space is just the zero locus of all the partial derivative just the critical locus of this function F. So because it's some, it's a space cut out by a bunch of equations and in particular has a nice two term of obstruction theory and then you can just write down what the obstruction theory is in this case and it's basically determined by taking the Hessian matrix or F you take this kind of symmetric matrix given by taking the partial, the second partial derivatives. And though this defines exactly that kind of a symmetric obstruction theory and this will be kind of the main example for us. So this kind of baby case is the case where the function was zero and it's kind of dumb but in general it's more interesting. So this is the framework for us. We have a modular space we have this kind of two term obstruction theory and it has this symmetry property. So definition, a function, if I give you any kind of complex scheme, a function from the complex points to the integers is constructible if the set of points where the function has some value. So new inverse of A, this is a constructible set. I have some variety M, there's gonna be some open set where it has some value zero and then maybe there's some locally closed set where it has value one and then maybe some other stratum where it has value negative one. And then given one of these constructible functions I can kind of use some version of integrating it. It's a discrete version of integrating it where what I'm gonna do is I'm gonna sum, I'm gonna look at all the kind of strata where the function has some value and I'm just going to add up those strata weighted by the, add up the Euler characteristics of the strata, weighted by the value of the function there. For instance, if I just had the constant function one I would just be getting the top logical Euler characteristic of M but of course in general I'll get something else. And then in particular I can look at, for instance just a billion group of constructible functions Z value of constructible functions in M. And one way of thinking about this is if I just look at characteristic functions this is a basis indexed by irreducible subrides. Do you only assume for actually many non-empty fibers? Yeah, yes, that's right. I'm sorry, M is gonna be just a finite type thing. So that's right, only finally many. And so the key theorem that kind of kicks off, for me at least this is the whole direction of the subject is that if I give you M with a symmetric obstruction theory in particular any moduli space of, she's or whatever on a clobby at three fold, there exists associated to M and E there exists a constructible function on M such that if M is proper in a virtual number of my moduli space, meaning in the sense of taking the degree of a virtual class is the same as what you get by integrating this constructible function. And so what it means is that, this intersection theoretic quantity, it means that you can kind of study it using ideas from kind of constructible geometry or micro local geometry. So let me just say a couple of remarks about this. I'm gonna say something about why this is true in a second, but let me just kind of say what's kind of so interesting about this. It allows you to do a couple of things that you couldn't really make sense of intersection theoretically. So for instance, this virtual cycle, it's really a cycle class. And so if I give you some subset of M, it doesn't really make sense to talk about, what is the contribution of Z to the virtual class? Cause you can't really localize. There's not a clean way of localizing this kind of zero cycle class to all the different, along some stratification of M. On the other hand, if I give you a constructible function, it's very easy to do it because I can just restrict my constructible function to Z and I can integrate it there. The right-hand side makes sense even if M is not proper. So usually the left-hand side unless you're in some kind of equilibrium setting, like in Richard's lecture, if I have a non-compact modular space, it doesn't make sense to take the degree of a zero cycle class on it, but you can always just integrate this constructible function. On the other hand, the left-hand side, of course, because it's defined intersection theoretically is deformation at least in the proper situation, I mean, which is the only time it makes sense. And something like the right-hand side, if I just take for instance, the actual topological Euler characteristic, that is as M varies in flat family, that's certainly not going to be a deformation invariant. And so there's something going on with this kind of specific choice of constructible function that's kind of correcting for the failure of the Euler characteristic deformation. This is really not at all obvious from how it's defined. And so this ends up being an extremely useful theorem. Let me just say, maybe I won't break this down. One way that gets used a lot is when you study how these invariants change under change of stability and wall crossing and so on, which is that when you kind of cross some kind of a wall and your stability condition changes, your moduli space usually changes maybe by some kind of flip or flop or something like that. And so understanding how the zero cycle trans, the cycle class transforms might be kind of delicate, but this kind of weighted Euler characteristic, if there's some open part where the two moduli spaces are just the same, then you can just throw it out because the contribution to this kind of integral is going to be the same. And you can just focus on the kind of the actual strata where the stability is changing. So this ends up being kind of an extremely powerful tool for those kinds of analyses. So let me sketch the proof of this result. And it goes into how this kind of virtual class is defined. Which again, I believe Richard covered in his first couple of lectures. Well, the idea that if you have M embedded in some kind of smooth space, let's say, then the way you get this virtual cycle is that you have M sitting inside V and then there's some kind of vector bundle over V. And then there's some kind of cone with multiplicity sitting inside of F. This is kind of a conical cycle, not a cycle class, an honest to God cycle inside of this vector bundle. And then when I intersect it with the zero section, I get exactly this virtual class. So this is true just in general. But what Kai showed in his paper is that if you now add the condition that the obstruction theory is symmetric, again, like in the Kalabiyaev situation, you can actually refine this picture so that this vector bundle F is actually the total space of the cotangent bundle of V. And this conical cycle inside of the cotangent bundle is not just a cone, it's a Lagrangian cone. This cotangent bundle has a natural symplectic form. And when I say this cone is Lagrangian, I mean that every, the smooth locus of every irritable component of this cycle is Lagrangian in the usual sense. The symplectic form restricts to zero and it has middle dimension. So why is that so special? Well, so there's a natural, there's a isomorphism between on the one hand instructible functions on V and this free, this free building group of conical Lagrangian cycles, which is known as the characteristic cycle map. It takes a constructible function here and sends it to what's called the construct of the characteristic cycle of this function. And this is defined in some sense via some kind of a Morse theory type construction. The fact that there is an isomorphism like this shouldn't be kind of super surprising. Each of these spaces has a basis that's indexed by these irreducible sub varieties. So I already talked about how irreducible sub varieties just by taking the characteristic function defines a basis here. Similarly, if I give you an irreducible sub variety I can take the smooth part and I could take its conormal bundle which is the Lagrangian side of here and I can take its closure. And so that gives me a natural basis here but that identification is not what's used to find this isomorphism. It's a little bit more subtle than that. But what's great about this construction is that on each side, there's an evaluation map to the integers. On the left-hand side, when I take the constructible function I can just integrate it. And on the right-hand side, if I give you a conical cycle I can take the degree of its intersection with the zero section. And the way this characteristic cycle construction goes is that this diagram can use. This is what's called the index formula. I don't know, due to many people. So maybe that's in. So this is just a very general statement about constructible functions on V and Lagrangian cycles of the cotangent bundle that you can kind of set up an isomorphism which makes this diagram commute. I'm not gonna, I won't actually, if I had more time I would actually, I had a, I'm sketching a proof of this kind of index formula. There are a lot of proofs. The one I like the most is in a paper of Schmitt and Volonen. Or basically they just reduced to the case of understanding kind of past of the real analytic world. And then you reduced to the case of understanding like a tiny ball. And so this is great. You see the right-hand side is exactly what we want to define the virtual class, to the degree of the virtual number. The left-hand side is the kind of thing that Barron's theorem is about. So to produce this kind of Barron function in the statement of this theorem, I'm gonna take the conical Lagrangian cycle that's associated my obstruction theory and just move it over to the left. So this is now what we now call the Barron function is just it's whatever constructible function maps to this obstruction cone under this characteristic cycle map. And so that's exactly how you kind of go from the intersection theory to roll over to this kind of constructible world. So then this begs the question, what do we know about this Barron function? How do we, this is kind of a somewhat abstract statement. How do we kind of compute it in any examples? And in general, it's quite hard. So I would say if I give you a kind of a random moduli problem and I give you some random point in the moduli space, it's not so easy to kind of compute this thing. But some cases, we have some kind of statement. So for instance, the easiest case is when M is smooth and it's smooth, you can again kind of put the stupid, you know, this is the baby example where I just have the zero function, the zero section defines for me a symmetric construction theory and then the Barron function in this case is just constant, negative one to the dimension. And then Barron's theorem says exactly what I wrote before. If I integrate the Barron function, I'm getting negative one to the dimension times the topological Euler characteristic of M, which is the degree of this kind of virtual cycle that we associated it before. What about this local example I did up here? So here I kind of wrote down this key local example where I take the critical locus of a function. So M is the critical locus of a function S and N variables. And so in this case, if I give you some point on M, the value of the Barron function, again, it's something pretty nice. This is related to what's called the Milner number, maybe the reduced Milner number of my function at P. So it's some kind of notion and singularity theory, which let me just state what it is. So given a function and some point in the critical locus, I can take the Milner fiber, which is just I take a tiny, I take a ball of a closed ball of some tiny radius around my point P, and then I intersect it with the fiber. So let's assume that F of P is zero. Make my life easier. And then I just intersect it with a nearby fiber of my function. So here epsilon is much less than delta, much less than one. And so then what the Barron function is in this case is I'm just taking, again, up to a sign, taking one minus the Euler characteristic of this Milner fiber. So this isn't super explicit, but it's again, something familiar from singularity theory. So there's a question in the chat. Do you assume that the singularity is isolated and what if it's not? Oh yeah, I am not assuming the singularity is isolated. You can still, this definition makes sense in general and it makes it a little harder to think about, but this definition is still not. So for the whole thing, you wrote X is in V, do you mean X is in AM? Oh, sorry, yeah, V was AM, that's right, yeah. So let me give a more complicated example where you get to kind of see this. So let's say I wanna, so I'm gonna take the following kind of Kalabi out three-fold, I'm gonna take a three-three hyper surface inside of P2 cross P2, which if I kind of pre, this is a Kalabi out three-fold, complete intersection, I project onto P2 and I get an elliptically fibered Kalabi out three-fold. I'm going to pick it, pick the defining equations such that, this is some elliptic. So that one of the singular fibers of this vibration is sitting inside of P2, just given by X squared times Y. So one of the fibers is this reducible, non-reduced cubic inside of P2. So this fiber looks like maybe two C1 plus C2. And so I'm gonna look at the following, modular space of sheaths, I'm gonna look at sheaths, which are one-dimensional sheaths where the support is C1. So that kind of, I take the non-reduced component and I just take the underlying reduced curve, which is just a P1. And I basically, I'm just gonna set my discrete invariance to be whatever the train character of the structure sheath of this curve is. So the support of the sheath is C1 and the Euler characteristic is one. And so I can look at the corresponding modular space of sheaths on X and set theoretically, it's just a point, this is the only object in it. But it turns out, and this is the calculation, this is from Richard 30 years ago or something, is that this modular space scheme theoretically is non-reduced. So you can see what this is, is first of all, it's cut out by the equations U squared, V squared. So this is the same as looking at the critical locus of the function U cubed plus V cubed, because these up to three, these are the partial derivatives. And so, okay, so then, so this is a pretty explicit function, you can work out what this, what this Milner number calculation gives you. So if you, well, it's just a point. So the value of the baron function at this unique point ends up being negative one squared, one minus negative three. So this negative three is exactly this Euler characteristic of the Milner fiber in this case, which is value four. On the other hand, this thing is zero dimensional. So the virtual dimension equals the axle dimension. So the virtual class in this case, if I just calculated it, this is just the length of the zero dimensional scheme, which is four. All right, so this is the main theorem. So what I'd like to kind of do in the, I guess I'll start this now and then I'll continue this tomorrow, is I wanna kind of sketch how this, give some indication about how this theorem gets used. This ends up being a really useful theorem for understanding, for calculating these numbers. And so, okay, so what I'll maybe do is I'll just do one example now, and then I'll just, I'll write it now and I'll kind of finish it tomorrow. So let me state this theorem. And so this is gonna be a theorem about these, you know, the stable pair invariance. This was this space of stable pairs. And then we can define a generating function where I fix beta and then define what I'll call the PT series. We get virtual number of these spaces, summed over two. Which again is kind of, should be reminiscent of the kind of generating function that I started this talk with, where I took the summation of the Euler characteristics of all the Hildred schemes for a six curve. So this is gonna be the kind of general version for an arbitrary Calabi-I-3 function. And so the theorem is that, so again, the X here was my Calabi-I-3 function, is that this generating function is the Laurent expansion, Q of the rational function, symmetric with respect to Q goes to Q inverse. In other words, it's basically built out of things that look like, you know, Q to the one minus R, one plus Q to the two R minus two. You can express this generating function in terms of the specific rational function. And so, okay, so what I wanna do is I wanted to kind of sketch the proof of this in the special case when data is invisible to prove it's due to Ponder, Ponder and Thomas. And what's kind of nice about this, so I won't do it now, I'll do that tomorrow. But let me just say why this is a nice result, which is that, you know, this rationality is something that we expect for any threefold. But right now we can only prove it, because this is expected. And we can prove it, you know, for things like, you know, complete intersections and so on. But in terms of a really general, this is a really general statement. If we, the only case where we can prove it, really in some kind of generality without knowing something really specific about the geometry of X, is in this Kalabiow setting. This is not only known, but for general, in general, we don't know. What's special about the Kalabiow case is precisely we get to use this kind of constructual approach to prove this kind of theorem. So I'll maybe say, like, you know, a few lines about this argument tomorrow. Let me, thank you. In the examples you wrote down in the Baron function, is it worked out by going through this sketch you outlined, or are the functions found differently? You mean the examples of what the Baron function is? Yeah. Yeah, so right, so in the examples I wrote down, you can calculate, this number number is something that you can calculate. We have techniques for calculating it. And so that's basically how you can do it. So like, for instance, it turns out that finding the Milner fiber, the Milner number for like, you can, that is something you can more or less do by hand. And then you just work it out and you get this negative three popping out. And in general, there is kind of a procedure for calculating it. If you have like a function and you have like a lot of time, then there is like an algorithm for calculating what the Milner number is. But for, you know, usually the geometries we're interested in get larger and larger dimensional. And so getting your hands on it isn't really feasible in fact. If you wanna do like, so an example of, here's an example of something that I, you know, that we like is to, you know, if you do like the Hilbert scheme of points on C3, the function that you would wanna find them, you know, the Milner numbers for is in some sense pretty explicit, but involved, you know, three end-by-end matrices. And so if you wanted to actually kind of, you know, do that for any given point in the Hilbert scheme, this is actually quite difficult. More questions? Yes, we have one. So if I take an opinion scheme or a fed point, is it true that the bearing function is bounded above by the length of this fed point? Sorry, say that again? If I take a fed point, so a spec of an arachnian ring. So I have only one close point like in the example by Richard. So is it true that the length of this fed point is an upper bound for the bearing function? Yeah, I mean, again, up to a, I mean, so I think, so I mean, if the, you know, if it's coming from one of these, maybe it's just equal to it even in general. I, it's definitely equal to it. If you know it's coming from a critical locus, that's just what the theorem says. But if it's not coming from a critical locus, there's still a definition of the bearing function and then, you know, maybe it's still equal to it in that case. But in the cases that show up naturally, it'll always just equal that length. Right, certainly it can be smaller. I was wondering if it can be bigger as well, probably not. No, I think that's right. Thank you. Any other questions? No, the last thing. Wish again. Thank you.