 Let me start the seminars. So today we're happy to have Jakub Konski from Warsaw telling us about stable envelopes. Yeah, both summaries and resolutions. Please take it away, Jakub. Thank you very much. Everything I will tell you today is based on the joint work with Andrzej Weber, my supervisor, and maybe let's start with some setting. I will consider only complex algebraic varieties. And by torus, I mean complex algebraic torus, which is just C star to some power. And I want to talk about characteristic classes in the equivalent k theory. So let me first introduce equivalent k theory and the equivalent k theory. So if we have a variety X, then the standard k theory is a free a billion group generated by algebraic vector bundles over X divided by some relation. Relation is given by short exact sequences for every such exact sequence. We say that the middle term is in the relation with the sum of the left one and the right one. And we can repeat this definition in the equivalent setting. If we have T variety X, by which I mean variety equipped with an action of a torus, then the equivalent k theory is just free a billion group generated by classes of equivalent vector bundles over X divided by exactly the same relation given by short exact sequence of equivalent vector bundles. And I want to think about k theory as some sort of cohomology theory. So at least I want to have some punctuality. And for every map of varieties, we have the pullback map from the k theory of target to k theory of source. Exactly as in the case of standard cohomology, this map is given by standard pullback. Okay, this is due to the fact that the pullback is exact. If we have a short exact sequence of vector bundles, then the pullback sequence is also exact. So the relation defining k theory is preserved. Okay, but there is second harder map. This is forward. If we have a proper map between varieties with the smooth target, then we may define a map from the k theory of the third to the k theory of the target. And we cannot give such simple definition as in the pullback case because there are two problems. First, push forward is not exact. And second, if we push forward the vector bound we obtain a coherent shift, not necessarily a vector bundle. Okay, it's possible to circumvent both of these problems to go around the first one, we may take some alternating sum of derived punctures and to go around the second, we may use the fact that way is smooth and every coherent shift has some finite resolution by vector bundles. So we can define still some element in k theory. But I want to only emphasize that this map is hard. Like it's hard to define because one need to and hard to compute explicitly because one needs to know this higher derived factors and one needs to know this the resolution. So there are lots of complicated things, you know. But in some cases, this map is really nice. If we have the map, which sends the whole projective variety to a point, then the push forward is given just by the other characteristic. So when we think of this push forward as some relative version of other characteristic and I want to say that in the equivalent setting the situation is somehow easier. We can define an element in this abstract ring in the k theory by giving some bunch of polynomials and this push forward will be defined by some combinatorial operations on this polynomial. So I will tell you a little about localization theorem. If we have first what is the source of polynomials, if we have any representation of a torus, it decomposes as a sum of one dimensional representation. So if torus acts on vector space, linearly then we have may decompose it into one dimensional representations. One dimensional representation corresponds to characters which are homomorphisms from torus to C star. And action on C C I is given by the following formula. Data on C C Y, data in spow. We apply the character C I to element of torus and use complex multiplication. Okay, this allows to have following isomorphism. Equivalent k theory of a point is just lowering polynomials in rank T variables. Okay, this T I corresponds to some special characters. They are just projection to each factor, rank T to C star. And then it's easy to show that any character is Laurent polynomial in T I's. And if we have any vector space, it belongs to this equivalent k theory of a point, then we may assign to it some of characters C I, which corresponds to irreducible factors. Okay, so equivalent k theory of a point is just Laurent polynomials. So now we want to generalize, have something similar for a variety. So if we have a fixed point, then the inclusion map is T-equivariant. Therefore, we may have pullback map from the k theory of whole variety to the k theory of a fixed point, which is just Laurent polynomial. Okay, this map is given by take the class of vector bundle and assign to it a class of fiber, which is just representation of the torus. And then do the same as above decompose it into one dimensional representation. And roughly speaking, localization theorem tells us that if we consider all polynomials corresponding to different fixed points, they determine this class in the equivalent k theory. More formally, if suppose that x is smooth projective to variety and that the fixed point set is finite, then the Biaonisky-Biroula theorem and localization theorem imply that the equivalent k theory of the x is a subring of a ring of k equivalent k theory of fixed points, which is just some number of copies of Laurent polynomials links. So if we have a class here, then it is better mind by a collection of polynomial describing the fibers, okay, these are polynomials. This is really nice from the computational point of view, because if we, for example, want to have tensor product here of two classes, then we may just multiply polynomials point-wise and take some, which is easy. And moreover, these two maps pull back and push forward can be computed on the level of polynomials, okay, without talking about some global information in vector boundaries. So first pullbacks because this is easier if we have any map and the class E in the equivalent k theory of the target, okay, then by localization theorem, this is just determined by some bunch of polynomials, one for every fixed point, okay. And we want to say what is the class of the pullback in the equivalent k theory of x. And okay, this is pretty simple. We, the map on the varieties induces the map on the fixed point set. And we just look how this map looks, okay. These two points go over the first and rewrite the polynomials around the arrows. Okay, and this is no hard theorem. This is just functionality of pullbacks if you look on how localization theorem works. So the pullback map is actually simple and combinatorial. What is more interesting probably is that push forward can be also computed on this level but it's more complicated formula. We have something called Leicester-Trimmann-Roch. We said that if we have a map between the smooth projective varieties and the fixed point sets are isolated and we have some elements indicate your x, then we have formula for the push forward, okay. In the equivalent k theory of y is fully determined by the restrictions, by localization theorem. So we only need formula for these polynomials. So we take polynomial with some polynomials corresponding to all pre-image, all fixed points which lie in the pre-image of all fixed points which lie in the pre-image of our troubles in y and multiply each of these things by some correction. This correction may look scary, but it's nothing hard. It depends on tangent representations of x to x and y, okay. The tangent space to y, it's the fixed point y and tangent space to x and the fixed point x representation of the tori, we decompose them to one-dimensional representation and we take such factor one minus one over x for any weight course in the two representations. Okay, but what is important in this formula is that it uses only local information, like only some information about representation of the torus, not some global information about vector bundles. So all these things can be really easily, usually can be easily computed there. This is just some operation on rational functions, nothing more. Okay, and I have very silly example to show you how this stuff works. I will compute other characteristic of p1 in o minus one. This is of course zero, but I will use this left shift formula, Riemann-Röck formula to do this. So first, we need some action, torus action to use left shift Riemann-Röck. So let's c star act on c2 by t times xy equals t xy. This induces action on p1 and on p2. o minus one, because o minus one, it's just some total space of o minus one, it's just some subset of p1 times c2. Okay, the six point set is just two points, one zero and zero one. And they consider the map which sends whole p1 to a point, okay? Then the other characteristic in which I'm interested is by definition, just push forward of the class of shift o minus one. And I have written here what the left shift Riemann-Röck formula tells us in this case. To compute push forward, we need to fact two summands for each point, and the correction is really simple because we have one, this corresponds to a fact that tangent space to point at point is zero dimensional. And here is factor corresponding to tangent space to p1. Well, I only need to compute these two things, okay? And this is pretty simple, okay? O minus one at x, pp1 at x, okay? Here we have t because there was added by t on the first coordinate. Here we have one, because there was don't act on the second coordinate. Here there is one over t and here there is t. So left shift Riemann-Röck, that we have first summand for one over one zero, it's t times one, one minus t, okay? This t here is this t here. And this one minus t is one minus one over this thing. And for the second point, we have one times one, one minus one over t. And we may simplify and we obtain t minus t by one minus t, which is zero, okay? So everything works. And I probably know that if I change this thing here, then I only need to change this little calculation here to obtain new results. This probably is not very exciting for p1, but in general, this is like really powerful thing to in computational sense. Okay, so I hope to conclude maybe, I hope this shows that the second variant category is nice computational tool. And if I have a class in the equivalent category of some variety, then I think it's given in explicit way. Like if I, in fact, can determine the polynomials, okay, which are, which define this class. Like if I can determine the restrictions to every fixed points, because then I can use all this localization machinery to work with this class, okay? I don't have something abstract, I have some bunch of polynomial. And I want to present some very vague way how to define some such classes for close subvarieties, okay? So suppose that we have a close subvariety of some ambient smooth variety m. And we want to define some characteristic class in equivalent k2r of m, somehow describing connected with geometry of x. For example, fundamental class or some deformation of it. So how to do this? One of the ideas is to find a resolution of singularities of x. By resolution of singularities, I mean that z2x is irrational and proper and that is smooth. And of course, I assume that fixed point sets of, that everything is the equivalent and the fixed point set of x, m and z are finite. Then we may define class in the k2r of z in some explicit way, right? And by this, I mean that we know the restriction to the fixed points and push forward it to m. If we know the whole geometry of z, then the push forward can be computed using CLEF3 mad rock formula and we obtain all the information about this class also. Okay, and many classes are defined in such way. For example, x has rational singularities, then the fundamental class is defined by taking a equal one and other classes, for example, the challenge for the McPherson class in equivariance k-tory, what do you return class in k-tory or elliptic class of Boris Epleb and Leibbub in elliptic theory. Okay, and I want to use this machinery to some very concrete varieties to Schubert varieties in flag varieties. I probably don't need to say anything about Schubert varieties in this seminar, but just for the sake of completeness, let me read definitions one more. I assume that G is semi-simple, simply connected algebraic group with a chosen Borel subgroup and Maximo Torres. Then I consider the generalized flag variety, G mod B, which is just this portion. This is smooth and projective and if we consider natural action of Maximo Torres on this variety, then the fixed point said G mod B, T is finite, this is in the bijection with bangles. Okay, and for every fixed point, I consider Schubert cell, which is of this point, which is just B orbit. And it turns out that this is either more to a fine space and the Schubert variety is the closure of Schubert cell and this is possibly singular. This may be singular. And it turns out that Schubert variety have very nice resolution of singularities, which allow to define a lot of characteristic classes by the procedure given in the previous slide. Okay, and maybe this will be the last slide before the break. If we have, okay, so there is something called Botsamersen resolution. If we have some fixed point, which is W in G mod B, T, then this is, in fact, a veil group element and we may write it as a word in a simpler reflection. And for every such decomposition, we obtain a resolution of singularities. I don't, if I have time, I will give you definition later how to construct this thing, but I probably don't have time. But I want only to say that this resolution is very nice, that the fixed point, it's the equivalent. The fixed point set is fine, right? We know that the, okay, it's isomorphic over the Schubert cell, like if we define the interior of Botsamersen as the pre-image of Schubert cell, then this map is an isomorphism. And we can describe, say, control boundary, like if we define boundary as the exceptional locals, then this is everything other than pre-image of Schubert cell, then this is a T invariant divisor with smooth components and these components intersect nicely. Okay, and I think this is a good point for a break. All right.