 Can you go ahead? So next, I'm going to recall what's known about the homology of real flag varieties. So there's two classical descriptions of complex prismanians, so the homology of complex prismanians. The first one is in terms of generators and relations. So these are the characteristic classes of the topological bundles on a prismanian. So the homology of the complex prismanians generated by the trend classes of the topological and quotient bundle, yes, and the quotient bundle on the prismanian. And the relation is given by the Whitney-Sumformin. So the other description, which is more close to the topic of this seminar, is in terms of the basis and structure constants. So we have a complex cell decomposition of the prismanian, which means that we only have even-dimensional cells. And therefore, the boundary map is topologically zero, so there's no one-dimensional cells. And therefore, the integer coefficient chronology of the prismanian is generated by the fundamental classes of the Schubert varieties. Again, parameterized by those that fit into a k times the minus k rectangle. And the structure constants of these Schubert varieties are given by little with riches and coefficients, so there's many combinatorial ways to compute these. Now, in the situation of real-grass manias, again, the situation is less simple. First of all, it's a fact, as we already saw in the examples before, that these contain a lot of two-torsion classes. So as a first approximation, we might try to understand what happens with the rational coefficients. So in the case when both the dimension of the subspace and the ambient space is even, we have a similar situation that we saw earlier, namely, the chronology is generated by characteristic classes. Now these characteristic classes are point-training classes, since these are real vector bundles. And the situation is similar, so these generate the chronology, and the relation is given by the Whitney-Sumformer. Now in the situation when the dimension of the subspace is odd, there is, again, a subring, which is generated by characteristic classes. However, now there is an additional element, which cannot be expressed in terms of characteristic classes. So the easy way to see this is that these point-training classes all have degrees divisible by 4, and this is an odd-degree class. So this class turns out to be the fundamental class of a smaller Grassman, and it generates an exterior algebra. For the other description, again, we have a cell decomposition into real cells and a boundary map, which is now no longer 0, since we have odd-dynamic sheen cells. And again, we have to compute the chain complexes that we saw earlier in order to determine which Schubert triates have cycles, which Schubert triates have fundamental classes. And it turns out that in the even-even case, these are generated, additively, by so-called double Schubert triates. So by this, I mean the following. So if you have a young diagram lambda, then you can associate to it its double, which just means that you take the same young diagram and subdivide each square into four smaller sub-squares. So this is a double of a young diagram. And the structure constants of the Schubert triates turn out to be the same little with Richardson structure constants that we saw earlier. So now I would like to say a few words about what this additive structure looks like for flag varieties. So I should first introduce some notation. So to a sequence of positive integers, one can associate the partial flag variety, which consists of flags where, so this is just a sequence of subspaces embedded in each other, where the dimension of the jumps is just the fixed number of VI. So in other words, these are the dimensions of the quotients. So these flags also have cell decomposition into Schubert cells. And these are parameterized by coordinate flags. So by coordinate flag, I just mean that all of these subspaces are generated by standard bases. So this gives a parametrization of the Schubert cells. So for example, in the case of flag 223, so this means that we have two planes sitting inside of four planes sitting inside in R7. We can take this specific coordinate flag and we can parameterize these by the new basis elements that appear in the given subspace. So for example, here we have two four. So these are the first two basis elements. And then the new basis elements that appear in the next one are three and seven. So these are the next two indices. And then the rest of the indices appear here. So one way I like to think about this is putting numbers in some boxes. So we have numbers one through N. And we have some boxes of some given size. And we have to put these numbers into boxes, such that each box contains the appropriate number of elements. So these maps are also called order set partitions. And the number of such order set partitions is equal to the multinomial coefficient. So in this way, we get parametrization of the Schubert cells in the partial flag variety. So for example, when there's two boxes, then that means that we are in the situation of the Crestmanian. And there's n choose k Schubert cells in that case, as we know. And in the case when m is equal to the number n, then we get that the permutations index the Schubert cells in the complete flag variety. So now we know what the generators of this chain complex are. And the incidence coefficients turn out to be 0 or plus minus 2 in this chain complex. And this depends on somehow the combinatorics of these order set partitions. So the additive structure of the cohomology has a nice description in the so-called even cases. So by this, I mean that you just take a sequence of integers. And by 2D, I'm just going to denote that you double each integer. And for these 2D, you have a doubling map. So from the order set partitions of type D to the order set partitions of type 2D. So this is just a mapping of numbers 1 through n into some boxes of some given sizes. And you can replace this by replacing each number by the corresponding pair of numbers. So you would replace the first number with 1, 2, the second number with 3, 4, et cetera. So for example, when you have this permutation 2, 1, 3, then the order set partition of type 2, 2, 2 will just be that you take the second pair of numbers, first pair of numbers, and the third pair of numbers. So if you unravel what this means in terms of partitions on the crest manian, then this corresponds exactly to the operation where you take the double of a partition. And then the theorem says that the rational coefficient cohomology of these even flagrities are generated by these Schubert varieties. So the proof consists of two simple steps. First, we show that these Schubert varieties are cycles. They are linearly independent. So this is some easy combinatorics on some parity arguments. And compare this to the dimension of the cohomology of the flag variety and show that this indeed forms a basis of the flag variety. So let me emphasize that rational coefficients here are important because, again, these cohomology rings also contain a lot of two torsion elements. But with rational coefficients, the situation is really just as simple. Surprisingly, the not so nice cases are the complete flag varieties. So here you can see the bruyagraph. So part of the bruyagraph of the complete flag variety. So what the edges mean here, they mean that the incidence coefficient between two Schubert cells are plus minus 2. And the color of the edge means the sign of the incidence coefficient. So red is plus 2 and blue is minus 2. So if you compute the cohomology of this chain complex, then this is the cohomology that you get. And there's a few interesting phenomenon here. So for example, here, what happens is that these two Schubert cells, neither of them generate z-modules or neither of them are cycled. But if you take their sum, then they give you one of the generators of one of the z-modules. So this is sort of similar situation when you have to glue together different Schubert cells in order to get fundamental class. And here something similar, I mean, something different happens. So these two classes, either of them, generates a free z-module. But when you take their sum, their sum gives a two-torsion module. So actually, as the generator of the other z-module, you can either pick this class or this class. And either of them gives you a generator of the z-module. And their difference is a two-torsion class. So some complicated phenomena start occurring when you consider flagrities. And as you go to higher dimensions, the situation gets worse. So it might happen that you have to sum together different Schubert cells. So it might happen that you have to sum together several Schubert cells. And it would be an interesting challenge to find some nice description of these homology classes. So these are maybe the right basis isn't in terms of Schubert cells, but maybe there are some nice subvarieties in the flag varieties, which could represent generators. So, additively, this is an interesting question, I think. So next I would like to discuss the multiplicative structure in the homology ring. So I haven't spoken yet about the Mottu homology of Chris Mannions. The reason is that these are much more simple. So with Mottu coefficients, if we consider the same chain complexes, since all the incidence coefficients are zero plus minus two, modulo two, all of these are just zero. So every Schubert cell represents a fundamental class with Mottu coefficients. So additively, we have a nice description of the homology of Chris Mannions. And in order to get a multiplicative description, there is this nice theorem of Borel and Hefliger from the 60s, which says that if you have complexification of a real variety, which is smooth, and the homology of the complexification is additively generated by fundamental classes of complexifications, then the conclusion is that there is a degree doubling ring isomorphism from the homology of the real points to the homology of the complex points. And this map is given by the complexification of cycles. So if you have fundamental class of a real sub variety, you take its complexification and this map is an isomorphism. So applying this to the case of Chris Mannions and flag varieties, this implies that the Mottu, little with Richard's and coefficients are the same. So little with Richard's and coefficients over Chris Mannions are just integers. You can take their Mottu reduction and these are going to give you the same structure constants of the corresponding Schubert varieties over the reals. So there's a nice modern point of view on this theorem and this was introduced by Hausmann-Holland-Puppet. So this is the theory of conjugation spaces. So the observation is that you have a Z2 action acting on the total space of such a complexification with the fixed point set or the set of real points. So for example, there's a Z2 action on complex Chris Mannion with fixed point set, the real Chris Mannion. So this sort of suggests that one should use Z2 equivariant cohomology to study this question. So the simplest example you can draw down is the case of CP1, where complex conjugation acts by reflecting to the plane through the equator and the equator is just RP1. So here you see explicitly that the fixed point set of this Z2 action is just the real projective line. So what's important for us is that conjugation spaces are topological class of Z2 equivariant spaces which have the property that they have a degree helping isomorphism from the cohomology of the total space to the cohomology of the fixed point set. So this is the inverse of the map that we saw earlier which maps the fundamental class of a real cycle to its complexification. And later Van Hommel showed that under the conditions of the Poirot and Heftiger's theorem, if one takes the fixed point set of the complexification that, so that this, sorry, this map copper maps the fundamental class of the complexification to the fundamental class of the real points. So this gives a modern point of view on the Poirot and Heftiger's theorem and an alternative proof. So we're interested in the multiplicative structure of even flag varieties, for example, even Grassmanians. And we can consider the following identification. So pick CN and forget the complex structure and just retain the U1 action given by the complex structure. So you can think of CN as R2N with the U1, a real linear U1 action. And this, any such linear action on R2N induces an action on the Grassmanian. So this action induces an action, a U1 action on the Grassmanian. And the second observation is that if you have a real subspace in CN which is even dimensional, then it is complex if and only if it is invariant under this U1 action. Well, indeed, because by definition, the subspace is complex if and only if it is closed under complex multiplication. So by this observation, we get that the fixed point set of this U1 action can be identified with a smaller complex Grassmanian since the invariant subspaces just correspond to fixed points on the Grassmanian. So let me point out that so we have a U1 action on the double-sized, even Grassmanian, a real Grassmanian, the fixed point set of the smaller complex Grassmanian. And the dimension of this reverse Manian is double the dimension of the complex Grassmanian. So this sort of indicates that one might look for some U1-equivariant analog of these conjugation spaces. And also if one chooses the flags defining the Schubert varieties carefully, then the fixed point set of the Schubert varieties turn out to be the smaller complex Schubert varieties in the complex Grassmanian. And there is a topological class of U1 spaces with this property. So now the Z2 action is replaced by a U1 action and F2 coefficients are replaced with rational coefficients. So there's a topological class of U1 spaces with such a multiplicative degree of isomorphism which have the property that for some nice sub varieties they map them to the fundamental class of the fixed point set. And the theorem says that these even flag varieties are circle spaces with a fixed point set the smaller complex flag varieties. And in particular, this degree-halfing map maps for the fundamental class of such a double Schubert variety to the fundamental class of the smaller complex Schubert variety. In particular, this tells us what the ring structure is on the even dimensional, on these even flag varieties namely the little with Richardson coefficients describe the structure constants of these double Schubert varieties. A question when you say circle space does the same thing as U1 space, right? Not entirely. So that's a more stringent definition. So U1 spaces are, so circle spaces are particular class of U1 spaces which have, I didn't say what the definition is. So the definition involves equivangent cohomology. What's important for us is really this degree-halfing ring isomorphism property. So using this, we can conclude what the cohomology of the even flag varieties are, even real flag varieties are since we know the cohomology ring structure of the complex flag varieties. So this ties back then to our original motivation namely obtaining lower bounds in Schubert calculus. So in the end, what we get is that any double Schubert problem has a lower bound the half size complex Schubert problem. So if you have such a Schubert problem, which is real and indexed by these double partitions or doubled order set partitions, then there is a lower bound given by a corresponding complex Schubert problem. So here are some examples. So in the case of this Schubert problem, which is a problem in Grassman 8R16, we get an upper bound by doing the computation in the complex Grassmanian. So this gives an upper bound 70. And to get the lower bound, we can consider a smaller Schubert problem, smaller Grassmanian do the computation there and that gives a lower bound to this enumerative problem. In particular, this is one of the instances of the balance of space problem that I mentioned earlier where a complete list of the possible solutions appears. So as you can see here, there's some larger jumps. And however, this is more general. So this doesn't restrict to the problem of the subspaces, namely, for example, we can consider this problem in Grassman 8R16 and do the computation in the complex Grassmanian which gives an upper bound of this number, do the computation in a smaller Grassmanian which gives a lower bound of this number and in between the number of solutions are not understood. So we get a lower bound and upper bound and some solutions in between. So even, I would like to emphasize that this approach has its severe limitations. So not every Schubert variety has such a chronological interpretation. So many Schubert varieties do not have fundamental classes at all. And however, the Schubert problems without such a chronological counterpart are still very meaningful and challenging. So Frank Sotil and his collaborators did many simulations for example, using computer and they observed different possible solutions. So none of these problems, for example, have any chronological interpretation. However, as you can see, there's many possible solutions appearing for these Schubert problems. And there's also some interesting phenomena, some interesting lower bounds, some jumps. So it would be really nice to have a general theory for these real Schubert problems. So that's what I wanted to say. Thank you. Thank you. Let's thank the speaker. Very nice talk. And