 Without further ado, so we're happy to have Chilean Gao from University of Very Noise Urbana-Champaign telling us about curved neighborhoods in a minimal degree. Please take it away, Chilean. All right. Well, thank you for the introduction and invitation. So today I'm going to talk about some joint work with Jian Gao from Harvard and Yibo Gao from Beijing University. Okay. So we considered the complete black variety overseas. The co-modulating of the black variety is a 3D module generated by the Schubert classes. These are the co-modulated classes of opposite Schubert varieties. So the famous Schubert calculus problem asks for a combinatorial interpretation for the structured constant where we take product of two Schubert classes and expand it back into some combination of Schubert classes. So the quantum co-modulated ring qh, which is isomorphic to the co-modulated ring tensor with a polynomial ring over n minus 1 variables. It's a free zq module generated again by these Schubert classes. The product in the quantum co-modulated ring, this star, this quantum product, is a deformation of the usual co-modulated product of the co-modulated ring. So in particular, if we set this q, all of the q's to be zero, then we recover the product, the expansion in the co-modulated ring of black variety. And so, and moreover, if we have all of the d's, here the d is the degree where all d i's are non-nuclear integers, if you have all the d i's to be zero, oh, right, sorry. So the structure of this quantum product is given by something called the Grumov-Witton invariance. So these are, again, non-negative integers where we would recover these Schubert structured constants if we have all the d's to be zero. So there's a natural partial order on these degree d's that appear in a quantum product, meaning that if we have, oh, well, it's given by divisibility of these monomials q to d. And today we'll focus on the smallest q to d that appears in this product. And more importantly, this quantum product in that minimal degree. So the work of Footh and Woodward back in 2004 characterized these minimal quantum degrees that appear in the product in terms of degrees of chains of permutations. The work of Posnickov relates all of the quantum degrees that appear in the product to weights of paths on something called the quantum brouhac graph. And moreover, he proved that there is a unique minimal degree q to d mean, meaning that if we look at this product and look at this expansion and look at all the degrees that appear, there's a unique one where this sequence of integers d i is entry-wise smaller than all of the other degrees that appear. And more recently, the work of Book-Chan Li and Mihao-Chan relate each of the d i's in this minimal q to d to the minimal quantum degree in a quantum chronology of Grasmanian. And the work of Schiffler study these minimal quantum degrees using something called the Maya diagram. Okay, so to study these quantum products and these Gromov-Footh and Neomerians, Book-Chan Li and Mihao-Chan define something called the curve neighborhood. So if we fix two permutations, u and v, and we fix a degree, so this is just a sequence of n minus one non-negative integers, the 2.8-curve neighborhood is the union of all degree-d rational curves passing through the opposite super-varied xu and the super-varied xv. So this is very closely related to this quantum product. So when q to d appears in the product, the Comodicus of our curve neighborhood is in fact 1 over c times the q to d part of the expansion of our quantum product for some positive integers. So one can think of these curve neighborhoods as a quantum version of Richardson varieties. So maybe the name curve neighborhood is foreign to many of you guys, but we've probably all seen some examples of it. For example, in the Grasmanian, the work of Knussel and Lamb-Espire proved that all of these curve neighborhoods are actually instances of positive curve varieties. So these curve neighborhoods are in fact a proper subset of the set of all positive curve varieties. So the work is generalized by book Chaput, Mihalsha, and Perrin to all common skew g-mop, where they show that all of these curve neighborhoods are images of Richardson varieties under the natural projection from g-mop to g-mop. So in the complete flag variety case, if we set d to be, you know, all of di is to be zero, then the curve neighborhood is just the Richardson variety. And if d is almost zero, so everything is zero except one entry, one of di is one, then the work of me and Mihalsha proved that this curve neighborhood is again a Richardson variety with possibly different permutations. So if we set u to be the identity, so the curve neighborhood will be, we'll be looking at the union of all curves passing through this super variety XV. This is in fact also a super variety. This is worked out by Book and Mihalsha. And in all these cases, this constant c in the equation here are all one. So these co-multi-class of the curve neighborhood in fact just is the q to d part or that degree piece of our quantum product. But in general, meaning let's say we, in general in the complete flag variety case, we don't know which flag lies in the curve neighborhood for some arbitrary d. And the goal of the talk today would be, sorry, let me stop the share and share it again. All right, here we go. Yeah, sorry about that. So the goal today would be to give a concrete description where the degree is the unique minimal degree that appears in the quantum part. And we want to do it in terms of certain rank conditions. So before we get into the specific description, let me first introduce something called the shifted Gal order. So for integer a in 1 through n, we define this total ordering of 1 through n where we set a to be the smallest one and a minus 1 to be the largest one. So it goes a, a plus 1, all the way to n and then which is less than or equals to 1 and which is less than or equals to all the way to a minus 1. So the shifted Gal order on the set of k element subsets of 1 through n is defined by if we take i and j, we reorder their elements in terms of this total ordering of n. We say i is less than or equals to j under this shifted order given by a if i is entry-wise less than or equals to j. So i is less than or equals to j under this total ordering given by it. So it is a fact that for any two given permutations, u and v, there always exists a sequence a consisting of n minus 1 integers in 1 through n such that if we, such that u k which is the first k element of the first k numbers in the one line notation of u is less than or equals to the first k numbers in one line notation of v under this, this shifted Gal order given by a k. In this case, we'll write u less than or equals to v under this order given by the sequence a. So this, this ordering actually comes from the combinatorics of something called the tilted bra order but we won't get into that in today's talk. So for example, if we set u to be the permutation for 3, 2, 1, v to be the permutation 3, 1, 4, 2, we can just set a to be the sequence 4, 2, 2, and we can check. So u1 is 4, we want 3. Under this total ordering where 4 is the smallest, we have 4 less than or equals to 3. So if we look at the first two elements, we have 3, 4, and 3, 1. Under the total ordering where 2 is the smallest or we have 3 equals 3 and 4 is less than 1. And similarly, if we have, if we look at the first three numbers, we have 2, 3, 4, comparing with 3, 4, 1, and we can check that 2 is less than 3, 3 is less than 4, and 4 is less than 1 in this shifted order given by 2. Okay, so motivated by this partial order with Jian and Yibo, we define something called, we define the open and closed tilted Richardson varieties. And as the name suggests, it's fairly similar to Richardson varieties. So if we take a invertible ambient matrix and think of the associated flag, which is where the case flag is given by the column span of the first K column vectors, the Richardson varieties are defined in terms of certain southwest and northwest and southwest rank conditions on these invertible ambient matrices. So the two Richardson varieties are defined fairly similarly, but with a twist. And I'll explain that with an example. So let's take U to be permutation 431. So this is the coordinate flag given by this permutation matrix. We said V to be 3142. Remember, we're looking at these flags in terms of column spans. So that's 3142. And as before, we have a sequence A to be 422. And we represented using this red lines. And we think of it as if we're looking at the first column, we make the fourth row to be the top row and the third row to be the bottom row. So if we're looking at the first two columns, this segment of the red line tells us that the second row, we should think of the second row as the top row and the first row to be the bottom row. And similarly for this segment, where the second row is the top row, the first row is the bottom row. So if we fix k equals 2, the rank condition defining two Richardson varieties, there will be six rank conditions concerning the first two columns that defines this TUBA. So we would ask this 1 by 2 sub matrix, this region green, to have rank at most the number of stars, which is number of these represented by the first two numbers of U. Less than equals to the number of stars in this green area, which is zero, want the rank of this two by two matrix to be less than equals to the number of stars, which is one. And this two by three matrix to have to rank at most two. So these are the sort of corresponding Northwest rank conditions in terms of similar to the Richardson varieties. And similarly we would ask this one by two matrix to have rank at most number of these blue dots, which is one in this case. And this two by two matrix in yellow to have to rank at most one. And this three by two matrix to have to rank at most two. So there are a total of six rank conditions for each k equals one, two, or three. And so in total there will be 18 rank conditions that defines this two Richardson varieties. And if we replace this less than equals to with qualities in this rank condition, this gives us the definition of the open two Richardson varieties. Okay, so our main theorem. So first of all, both the open and closed two Richardson varieties are independent of the choice of A, as long as U is less than equals to B under this order given by A. So we'll remove this A in the notation and just denote them as GUB and GUB CERC. So we in fact have a stratification of these two Richardson varieties given in terms of these open ones. So here the description is in terms of these partial orders given by A, there's a more natural interpretation in terms of subintervals in the two-tiered Bruja order. And I'm supposed to give you a three-minute warning. Okay, thank you. Thank you. And in fact, the two-tiered Richardson variety GUB is the closure of the open two-tiered Richardson variety GUB CERC. So we show that these two-tiered Richardson varieties are irreducible and they are in fact just the CERC neighborhood in a minimal quantum degree. And moreover, we show that the Comodic class is in fact just the Q2D mean degree expansion of the quantum product. So the 1 over C in the previous slide, so the C in a previous slide is just 1. And I will stop here. Thank you very much for a wonderful talk.