 Thank you very much for listening. It's great to be here again after two years, because two years before I attended the workshop and summer school in also STB, but the other building, not the building. So today I'm going to give a talk on the room of the invariant of the compact Lagrangian Grammian. So first of all, let me start with some notation and what the Lagrangian Grammian is. So let me start with V, is a symmetric vector space over C, a field of compact number and dimension to time n. So here by the symmetric vector space, I mean vector space endow with a symmetric form, say omega from V times V to C. So which is a symmetric form, which is valine, non-degenerator, and skew symmetry. So for example, you can choose standard symmetric form. So and then let's take a subspace of V. It's said to be astrophic, if the omega rigid to W, we identify with zero. And it's said to be Lagrangian, if it's astrophic and maximum dimension, so that means dimension of W will be n. And now the Lagrangian Grammian, which is denoted by Lz of n in this talk, will be defined by the set up on a Lagrangian subspace of V. Some basic property of Lz of n is, so first of all Lz of n, so will be a smooth sub variety ordinary Lagrangian, which is without symmetric form. And the dimension, the compact dimension of Lz of n equal to n times n plus one. And yet other properties, Lz of n can be realized as a homogenous space, so which Lz of n can be realized as equation of a symmetric group of a unitary group. So this is a homogenous space. So here Lz of n is a symmetric group, so which is a league group of Thaisi and U of n is unitary group. This compact symmetric group. So let me define superclasses and also remind some basic facts on homogenous Lz of n. So let me denote d of n here, the set up on a strict partition. Let's say lambda, a sequence of integral number, lambda one up to lambda l, n bigger than or equal to lambda one bigger than lambda two, lambda n bigger than zero. So this will be a set up in the set up of the homology, a basic of homology ring of Lz of n. So now let me fix some technical definitions. I'll fix a flag of an atomic subspace. The basis of v, say zero, f1, f2, fn in v and the dimension of f i equal to i, four on i, and fn is must be actual big. So now for, or is lambda in dn, so we, one can define super priority which is denoted by s lambda. It must be depend on the choice of flag, but I didn't write, I'm not going to write. So this will be the set up element in Lz of n, such that the dimension of the intersection w, the intersection between w and fn plus one minus lambda i is must be bigger than or equal to i, four on i. So this here will be a super priority on Lz of n. And then we take the Bunker classes, let me denote it by sigma lambda, lambda will be the class of each lambda. So this is the line of the homology, say Lz of n, take the integral coefficient. So here by absolute of lambda, I mean the sum of, so absolute of lambda will be lambda one up to, and this is a super class. And there's a fact, say that the set up all of super classes where lambda run on, run in d of n. So we'll form z basic, the homology ring of Lz of n. So this is some fact on homology ring of Lz of n. And now let me move to, the main goal today is to remove the invariant of Lz of n. The main goal today will be Gromopiton, say again. Yeah, in some sense. So I only focus on Zinu 0, K Makinborn. Okay, let me fix some, let me remind the definition of Gromopiton invariant first. Okay. Okay, let's take lambda one up to lambda K in d of n, and take the sigma lambda one up to sigma lambda K. So this is super classic with reference to lambda n. So this you will be. And the definition of Gromopiton invariant. Okay, let me write out the definition. The Zinu 0 associate sigma lambda one up to lambda K. So I didn't notice by, which I did notice by sigma lambda one up to sigma lambda K. Yeah, of course, I should take this natural number. So by definition, so this will be the integral over Molas Bayer stable map. Let's say M. So Zinu 0 means the Zinu 0 K. And K is, we are considering the K Makinborn and Mz of n. So it must be a degree D. Yeah. So integral E V one, the boom back up sigma lambda one, couple of those with E V K sigma lambda K. So let me have, so well, okay, I still have a lot of time. Let's say again degree two, this one. No, we take any degree. The dimension may be dimension. I think so you're thinking of dimension. I'm drawing D. Yes. In the literature, they say the class beta and beta is D times L or something like this. So, yeah, in some sense, you can think of D just replay D by beta is like this. No, what do you mean? Yeah, it's similar. And maybe you can write P equal D times L. L is the hyperplane class, the class of hyperplane or class of flyer or something. Okay, so where M bar 0 K, Lz of n is a conspic Molas Bay stable map degree. So it's mean, while a stable map, I mean a map from C with the K Makinborn, say P 1 up to P K to Lz of n and C smooth with another curve and P 1 up to P key, B B K is a smooth boy on C and stability mean the automorphism group of F smooth B finite. And D is equal degree of the good for work of class C something. Evie, I mean the evolution map between L which is sending a stable map C P 1 up to P K, sending to the value of F at the boy P i. So which is so good at evolution. Yeah, basically, in general, we have to take the vital fundamental class rather than the Molas Bay stable map. But in this case, where Lz of n is a homogenous bay, so the vital fundamental class is coincided with the Molas Bay. So in this case, everything is okay. And then magically, so the group movement invariant in this case is, in fact, it's counting a number of pressure no curve on Lz of n certify some geometric condition, for example, counting the number of pressure no curve up to 3D intersecting with some subclassic, correct more than subclassic. Subclassic, correct more than subclassic. So in our own, so my goal is, so how to compare young or like, so by some dimension condition, so we have a remark. So the group movement B is 0 and let absolute lambda 1 equal n times n plus 1, which is the dimension of Lz of n plus D times n plus 1 plus K minus 3. So this is exactly the dimension of the Molas Bay stable map. And so by the quantum classical principle, due to under birth, in 2003, they blew the group movement invariant sigma lambda 1 up to sigma lambda K. K can be computed as classical intersection number over actual big graph mania. So we spin integral over IC n minus D to n. So here sigma lambda 1D. And here by IC of n minus D to n, it's been, I mean, actual big graph mania. So this is the isotropic, just a set of all of isotropic subs bay of V. We don't need the maximum dimension. Deep power, yeah. No, no, no, deep power, just notation. Just notation, yes, yes, yes. And of course, sigma lambda I D is a super. I'm not going to define in more detail because it takes time. So, and my result is, so, because I somehow I can relate the integral over actual big graph mania with the, I mean, we can relate the intersection number over actual big graph mania to, to know over classical graph mania. And then we can, what do we get it? So sigma lambda 1 can be computed as intersection number over graph mania. But here we have to define new, new classes over classical graph mania. So we spin theta, I did it by theta. And some extra superclasses here and plus D. So here I mean theta lambda I D is a theta class on D to N. So this is defined, well, basically, defined, I rely on the Zambali type formula for, for ashric graph mania due to birth grade and type of this. So D K T in 2007. And sigma theta N blood D is a super normal, just a standard superclasses over classical graph mania. So why, theta N blood D is mean just a stack or just a stack, stack K, what is it? So I think I should stop here. Yeah, I stop here. Yeah.