 We're good. OK. So I just start talking. So with that introduction, so with that introduction, first, I'm going to say one of the questions that we're actually trying to answer when we're talking about which nilpotenthesenberg varieties are GKM. So which tori are acting on nilpotenthesenberg varieties? That's one question. I've already sort of put nilpotent in parentheses because, well, I guess because I am about to outline, that the question for semisimplehesenberg varieties is both sort of answered and also actually not as complicated in a certain sense. So if we start with this question, which tori act, and then sort of kick the can down the road on the question of, is that action GKM? We focused on subtory of the diagonal matrices for the purposes of the rest of the talk, and I'm going to put t with a bracket to just be my matrix and keep the convention of having the ti's represent the variables. And I'm going to go back to what is the Hessenberg condition. So when I started, I sort of had this condition x of vi is contained in the h of i. So then I also represented my flag with a matrix. So then I said, oh, OK, this is the i-th column of this matrix. And when I say that x times the i-th column of a matrix is contained in, well, I'm really saying the span of the first h of i columns of the matrix. So there's a span of first h of i columns. I can write this as a matrix equation. As long as I let h be this sort of linear subspace of matrices where I have h of 1 free entries in the first column, h of 2 free entries in the second column, h of 3 free entries in the third column, and so on. So we can reframe the sort of flag type defining characteristic of Hessenberg conditions as a statement that these vectors, or in fact, this one particular matrix, xg, has to be contained in a particular subspace of matrices. And really, this is sort of like what we were doing when we did an actual calculation of take a Schubert cell and act on it by x and then start testing whether each column in the image was in the span of a first set of columns that you started with. So what we want is, so this first condition, xg in gh, so this means that somehow g represents a flag in Hessenberg variety. And we want this to mean that if I take the image of this flag under any action of the torus, so this is sort of where the torus can send g, so this should still be in the Hessenberg variety. So then this final part of the equation is saying that t of g is also in the Hessenberg variety. And sort of getting back to Anders' question, so if t and x can mute, then x tg is just txg, so if this part, so if that red part xg is in g of h, then this whole thing is going to be in th, which is what we want. And in general, this is sort of like suggesting that we can start focusing on things like what happens when you take your matrix x and you start conjugating by a torus element, sort of like a good place to start, let's say. So if x starts diagonal, then it's going to commute with t all the time. So this is why we're going to sort of eliminate the semi-simple matrices of the semi-simple x and the semi-simple Hessenberg varieties from our analysis, because actually people have already done that because there it is, the whole torus acts on it. And the GKM condition is sort of nicely hereditary in many situations, so the fact that all of these things are sitting inside the flag variety means that, for instance, if you have a full torus action and on a sub-variety, then it will also inherit the GKM properties. Now, here's an example, sadly, a situation where the full torus, so the full torus does not act on this matrix x, where x is nil-potent, it is like regular nil-potent, it's got these ones just above the diagonal. On the other hand, this particular matrix K, the collection of matrices, almost commutes with x. So just in the sense that I multiply it this way. So multiplying on the left is going to scale the, it's going to act on the rows. Multiplying on the right is going to act on the columns. The non-zero entries are always in the i-throw and i-plus first column, so it's going to end up actually scaling this whole thing by the parameter T inverse. So they don't commute, but they commute up to this constant, like the projectively commute. So in fact, whenever you have a nil-potent that is strictly upper triangular and it's non-zero only in rook positions, meaning a non-zero entry only in a single row, one entry of each row, one entry of each column, then we can actually find similarly another torus that acts in this almost commuting multiplies by a scalar fashion. And then we can start extending this same kind of analysis to just a sort of more general set of matrices. So like if G satisfies the Hessendberg condition, so XG is in GH, then what about if we act on X by some torus element? Is does that same flag still satisfy the Hessendberg condition? And really, in a sense, like what we did at the very beginning of the talk was we looked at the column XG and then we compared it to the bunch of particular columns sort of go. So you can look at the first H of I columns here and just see whether the rank lines up when you just look at the image X of the I column on the right-hand side. And each sort of move through all of the columns. And in fact, if we do the analysis that we were doing at the beginning, then we actually looked for pivots. Like when we look at a specific Schubert cell, we looked at specific pivots. And we kind of narrow down our analysis further to sort of say, as long as the pivots in XGI are in certain locations, then the answer is like, yes, we can solve this equation. If not, well, then as long as we have the ability to impose adequate conditions on the entries of the Schubert cell, then we will still be able to satisfy this condition. So really sort of partitioning the matrix into blocks and then analyzing conditions within each of those blocks. And then what happens when you add this sort of condition here? Well, so changing, so changing from X to something like T inverse XT will rescale some entries. So it won't change the pivot and non-pivot conditions, but it will start moving around some of the other entries and possibly screwing up if there were conditions on the entries like we had an equation that looked like A equals C early on. So if you start scaling those by different entries, that could get screwed up. So essentially, this is the kind of analysis that we do. What are the takeaways? There's an important interplay between what sub-tourists you pick and X. And I guess I could also add to that. It also depends on the incorporating what sort of conditions H imposes adds to that interplay. Furthermore, concrete linear algebra analysis will allow us to identify many cases in which to write do or do not act on a given X. So rather than specifically tell you theorems, I'm going to give you the sort of spirit of the results that we have. So we have, for instance, if X is skeletal nil potent, what are some conditions that guarantee that the Hessenberg variety admits an action of the torus? So for instance, one kind of condition, H of 1 equals 1, H of 2 equals n, another kind of condition, H of 1. And so H of 1 equals n minus 1, H of 2 equals n minus 1, dot, dot, dot, H of n minus 1 equals n minus 1, and H of n equals n. So these are actually pretty restrictive conditions. This is sort of what we expect on some level because having a full torus action is hard to do. So we sort of think that it should not be the case that a lot of nil potent Hessenberg varieties have a full torus action. So we also sort of fully characterized for some special kinds of X when the Hessenberg variety has a T action. And so in that case, we are set up and have some sort of preliminary work for thinking about the equivariant co-homology. Oh, and I think this is a place to say, this is sort of like building on, or like this. So it might be like to address those in the room who wonder why to think about just specific X. In addition to the fact that specific X, like the Peterson variety or the regular nil potent X, or for that matter, the regular semi-simple X, have a really richly developed theory. Abe and Crooks have an example. So Abe and Crooks analyze the particular nil potent X where you just have a one in the top left corner and zero everywhere else. And so we were sort of thinking a lot about extending this, extending that sort of block a little bit further and have some very complete results when you have a two by two block in the upper right. So for the same matrix, in fact, so we can not only tell you when the Hessenberg variety has that full torus action, in which case it's GKM there, but also tell you when this sort of almost commuting sub-torus endows it with a GKM action. So when it is GKM with respect to this, almost commuting sub-torus and sort of giving you a schematic of the kinds of Hessenberg varieties that work. You sort of either, you have your upper triangular matrices and you can get just a teeny little bit more other than that sort of upper triangular matrix case of H of I equals I or Springer fiber. So those red blocks can sort of be shifted up and down on the diagonal, but you cannot add any more red blocks to those diagrams. And finally, I think in the last two minutes or so, so there's some interesting implications of these connecting back to sort of Schubert varieties. So one thing that we observed is that of Hessenberg variety admits a full torus action, then the way that it's going to intersect the Schubert cells is by imposing conditions, imposing conditions just of the form certain entries are zero. So it does not need to contain the entire Schubert cell, but it will not impose any super complicated conditions or even conditions like we saw before where A equals C. And I'm also going to add results of trying to alpha the tires on the fly. Lara and Martha and John have some like also interesting results describing when Hessenberg varieties are Schubert varieties. Which connects to this idea of what exactly are the Hessenberg varieties intersecting Schubert cells in. But we could further actually say that like not like even if your Hessenberg variety has a full torus action and intersects Schubert cells with a sort of like nice conditions, that does not actually mean that they are unions of Schubert varieties. So it really is a sort of like distinct set of conditions are like related to being a Schubert varieties, but different, truly different from. And I think on that note, I'm going to stop, thank you.