 All right. Yeah. We are very happy to have Till Verhan speaking about the Chevrolet Monk formulas for both varieties. Please go ahead. Yes. Thank you very much for giving me the opportunity to speak here. So I will start by giving a rough motivation by considering a bigger picture. So by introducing the theory of stable envelope, Molykino-Kunkow provided a way to assign to a very rich family of symplectic varieties with torus action and some additional properties, an integral system. And the family of Bohr varieties is also very general family of varieties living inside these varieties that fit into the framework of Molykino-Kunkow. And one nice property of them is that they naturally generalize the co-tension bundles of flag varieties. And in the case of co-tension bundles of flag varieties, the integral model side is very well known. In this case, we recover the XXX model. But also, of course, the co-tension bundles of flag varieties just predict down to the flag varieties. And in this case, we have these rich combinatorics of the intersection theory, which is of course known as Schubert calculus. And our approach to get an understanding of the theory for Bohr varieties is to take very, very known results from the XXX model and Schubert calculus and try to find out how they generalize to the world of Bohr varieties. So next, let us maybe consider the case of the co-tension bundles of flag varieties and the XXX model in a little bit more detail. Okay, now I cannot, what is happening? Why can't I? Okay, it's going that way. Okay, so here we have the co-tension bundles of flag varieties and the XXX model. And this theory comes with two important homology classes. On the one hand, we have the stable envelope classes or stable envelope basis elements. And on the other hand, we can relate them to the XXX model via the base change matrices. So the base change matrix of different stable envelope basis gives the R matrix of the XXX model. And having an R matrix always gives us a hopf algebra. And in this case, it's the very well-known Jungian of GLN. Now, there's also a second important family of co-mology classes in the theory, namely the first-journ classes of tautological bundles. And they all also have an interpretation on the XXX model side. Namely, they are naturally algebra generators of the maximum commutative subalgebra called the Gelfonsetlin subalgebra of the Jungian. And this algebra was very well studied by Therazov and Narazov. And so this well-established picture in particular implies that we have a very nice formula for the multiplication of turned classes of tautological bundles with respect to the stable envelope basis. And now I want to describe this formula in the special case of full flag varieties and the co-tension bundles. So at first I want to talk about stable envelopes for co-tension bundles of full flag varieties. So fn should be the full flag variety parameterizing full flags in c to the n. And t star fm should be the co-tension bundle of fn. Now, by construction, this variety comes with an action of a rank n plus 1 torus. So the first factor is just inherited from the torus action on c to the n. And since the co-tension bundle is a vector bundle, there's also always a torus action given by selling the fibers. So this is this additional torus here. Now, stable envelopes, they are maps from the t-fix locus to the equivalent co-molegering of the co-tension bundle of the full flag variety. And they depend on a choice of a generic co-character of the rank n torus. Now, since, yeah, they are uniquely determined by certain stability conditions, namely normalization, a support under the doggie condition. And since the t-fix points of the co-tension bundle of a flag variety adjust the symmetric group, we can also view stable envelopes as a map from the symmetric group into this equivalent co-molegering. Now, these stability conditions, they are pretty similar to the stability conditions that appear in the equivalent Schubert calculus. And actually, stable envelopes and Schubert classes are pretty close connected as stable envelopes can be interpreted as one parameter definitions of equivalent Schubert classes. So, via a certain limit argument, the stable envelope classes specialize to the equivalent Schubert classes on the baseband, a base base, the full flag variety. So, the multiplication of tautological bundle with respect to the stable envelope basis can be deduced from the work of Molokin and Kunkhoff. So, if I have my ice tautological bundle, then the multiplication is can be described as follows. So, we have churn class of the ice tautological bundle and we take its dual times the stable envelope class corresponding to a permutation is equal to the following sum. So, we have a diagonal term where the coefficient is just the equivalent multiplicity of my churn class at this point. And then we have an interesting off diagonal term. And this off diagonal term are precisely those permutations that are obtained from W by a transposition tjk such that j is smaller or equal to i and k is strictly larger than i and we demand that the Brieher length goes up by one. And then we have, yeah, and all these permutations contribute to the off diagonal terms, and then they all have the same coefficient, namely the equivalent parameter corresponding to the torus rank one torus that scales the fiber. So, here we have it. And for this formula to hold we choose our co character to be this basic choice to be T goes to T T squared and so on. Yes, and as I said the stable envelope classes degenerate to the Schubert classes. And therefore, this limit gives us back the classical Shiva limon formula in Schubert calculus, which describes the churn class multiplication of tautological bundles with respect to the Schubert basis. The actual big difference in this formula is just that the length here just has to go up by one when the stable envelope world. The length just should increase. Yes. And since we can get back to classical Shiva limon formula. And view this theorem here as as a generalization of this formula to the quotient and bundles of black variety. Okay, having said a lot about the world of quotient and bundles of black varieties. Let us now move to the world of both varieties. And I want to say first some general remarks, why they're interesting, and why one could expect that we should also a formula in this business. So first of all, they are a family of smooth symplectic varieties with a torus action and they fit into the framework of the work of more than the conco, which in particular gives us that the stable base stable envelopes exist. Moreover, this family is very rich. There are lots of interesting varieties in it. So, it contains type and a kajima career varieties, as well as a 3d Google spaces, which are the so called Coulomb branches. And from their construction. It's really surprising that there is one family of varieties that contains both of them. Moreover, they also come by construction with a family of tautological bundles. And they have finally many torus fixed points and the fixed point combinatorics of both varieties naturally extends the fixed points combinatorics of black varieties and the cotangent bundles. So, all together, these facts give that it's reasonable to ask if we can generalize the previous formula, which we had in the world of the cotangent bundle of black varieties to the world of both varieties. Okay. Now I want to say a short word about the construction of four varieties, which is due to Nakajima and Takayama. And I will use the combinatoric language of the kajimani. So the input of both variety is a brain diagram and the brain diagram is an object like this. So we have at first a bunch of red slashes, then we have a couple of blue backslashes and between them. We have horizontal lines which carry a label, a non-negative natural number. Now the construction of both varieties takes this input, a brain diagram, and the first step is to assign to it a space of clever representation. And this is the certain symplectic variety to which we can apply Hamiltonian reduction. And this then gives us the bow variety, which is denoted by C to the D. Now, by the construction of a bow variety, each of these horizontal black lines gives us a tautological bundle. However, the tautological bundles in the blue part, they are just trivial. So all interesting tautological bundles that can appear on the red part of this brain diagram. Next, I want to continue with saying a few words about the fixed point combinatorics of bow varieties. Bow varieties also come by construction with a torus action which scales the symplectic form. And the fixed points have been described by Nakajima and also by Ricca Trimani and Shu in a very convenient way. Namely, the fixed points of a bow variety correspond to certain combinatorial data attached to the brain diagram D, which are called tie diagrams. Now, a tie diagram is an object like this. We extend our brain diagram by adding ties between red and blue lines. And we demand that the number of ties covering a horizontal black line is always equal to the label of this horizontal black line. Yes, so this here is an example of a tie diagram. And in particular, if our brain diagram looks like this, so we have red slashes and blue back slashes. And we start with one two up to N and then we go down N and minus one and minus two up to one. Then my bow variety turns out to be the cotangent bundle of a flag variety. And the tie diagrams correspond to elements of the symmetric group SN. So in this case, we recover the fixed point combinatorics of flag varieties. Now the main result is that via the following combinatorial moves, we get also a formula for the cotangent for the multiplication of first joint classes of tautological bundles with respect to the stable envelope basis. So this combinatoric move is what I called a simple move. So we have a configuration like this, where we fix a horizontal line and call it X. And we have two red lines, one to the right, one to the left and two blue lines. And we demand that the red line is connected to the first of the blue lines, but not to the second. And the red line at the left is connected to the last of the blue line, but not the first. And then we say we apply a simple move when we just swap the endpoint. Then this will again give us back a tie diagram. And the theorem is that when I take a black line in the red part, so the interesting part. And I also consider the corresponding tautological bundle. Then the multiplication of this turn class with respect to the stable envelope basis looks at follows. So, C one of Chi X times stable envelope basis element labeled by the tie diagram D is again equal to the following sum. We have a diagonal term, which is just a fixed point restriction of this turn class. And then we have an off diagonal term. And the off diagonal term are precisely those tie diagrams that we can obtain from D via a simple move over X. And then we have again this equivariant factor which corresponds to scaling the symplactic form. And it appears up to a sign that we can explicitly compute. So altogether, this is basically what we could hope for, namely we have a very nice generalization of the Chevrolet Monc formula from the co tangent world to the world of both varieties. And this is all I wanted to say. Thank you very much. Thank you very much for a very nice talk to tell.