 Okay, welcome everyone to the Pi Day edition of the sugar seminar. We're happy to have a way home. Sue X lower you from Rutgers University way home is a student in Rutgers grab the student in Rutgers. And she's going to tell us about quantum k theory of incidence varieties. Please go ahead. Thank you for the introduction. Thank you for the opportunity to speak and special thanks to Anders who has been very helpful throughout this project. Let me put the slides in the chat. Just in case you want to look at it during the talk. Okay, you should have it now. If you have any questions, please feel free to unmute yourself at any time. Here's the plan for today. First, I will say a little bit about the background and introduce our objects of study. Next, I will state the main geometric result, which I will then use to compute some geometric invariance on the incidence variety, called T. Equivariant k theoretic Gromov written invariance. These invariants are closely related to the equivalent quantum k theory ring of the incidence variety. Knowing these invariance means we know how to multiply in this ring. In particular, I will state some very explicit formulas, namely an equivalent Chevrolet formula and a non-equivariant little wood Richardson rule. Afterwards, I will revisit the geometric result and say a little bit about its proof. In the end, if time permits, I will also state a conjectured Chevrolet formula for the quantum k theory ring of an arbitrary type A flag variety. Here is the setup. Let G be a complex semi-simple linear algebraic group. As usual, we will fix a maximum tourist T inside a borale subgroup B inside a parabolic subgroup P. And we will let X be the flag variety G mod P, which naturally admits a G action in type A. G is just SLCN. P can be taken to be the subgroup of block upper triangular matrices with some fixed block sizes. B can be the subgroup of upper triangular matrices and T can be the diagonal matrices. For this talk, it's fine to just think about type A. We will fix an effective degree D in the second homology group of X. And we will write M D for the cons, cons of each modular space of genus zero three pointed degree D stable maps to X, which is just the compactification. Here, we have the set of maps from P1 to X, such that the push forward of the fundamental class of P1 is equal to D. Here, this compactification means we are allowing for degenerate maps whose source is no longer P1, but a tree of P1 with three mark points. The maps need to satisfy a stability condition that if a component is mapped to a point, then there are at least three special points on the component, including nodes and mark points. The modular space M D comes with three evaluation maps, EV1, EV2, and EV3. In the interior, you're just given by evaluating F at 01 and infinity on P1. Schubert varieties are sub varieties of X that are stable under the action of a certain Borrel subgroup. We will consider different Borrel subgroups because we want the Schubert varieties to be in general position. In particular, we will use lower indices to index Schubert varieties that are stable under the standard Borrel B and upper indices to index Schubert varieties that are stable under the opposite Borrel B minus. If B is the subgroup of upper triangular matrices of SLCN, then B minus is the subgroup of lower triangular matrices. Let me draw a cartoon picture. So in X, we have XU and XV. We will consider the two-point Gromov-Rieten variety M D of XU, XV. It's the sub-variety of M D consisting of maps with the first mark point sent to XU and the second mark point sent to XV. When we take the image of a degree D stable map, we get a degree D stable curve. And if we take the union of all the degree D stable curves, meaning XU and XV, we get the two-point curve neighborhood gamma D of XU, XV. The sub-variety of X given by the image of the two-point Gromov-Rieten variety under XV3. If we do this with just one Schubert variety, we get the degree D curve neighborhood of that Schubert variety. It's not hard to show that this will again be a Schubert variety stable under the action of the same Borrel. Here is everything we need from T equilibrium K theory for this talk. First, let's recall the non-equivariant K theory. So the ordinary K theory of X is the free abelian group generated by isomorphism classes of algebraic vector bundles on X, the equivalent relation given by short exact sequences. Multiplication is simply given by tensor product of vector bundles. In our case, X is a non-singular projective variety, which implies every coherent sheaf on X has a finite resolution by locally free sheafs. We can use such a resolution to define the class of a coherent sheaf as this alternating sum inside the K theory ring of X. And this definition is independent of the choice of the resolution. Pulling back vector bundles along the structure morphism makes the K theory ring of X an algebra over the K theory ring of a point. The structure sheaves of Schubert varieties and opposite Schubert varieties give two basis for the K theory ring of X over the K theory ring of a point, which is of course just a Z. Since T acts on X, it's natural to consider the free abelian group generated by T-equivariant vector bundles on X modulo the equivalence relation given by T-equivariant short exact sequences. Again, we can define multiplication using the tensor product. This gives us the T-equivariant K theory ring of X. There is a notion called a T-equivariant coherent sheaf. I will not give the technical definition. We just need to know two things about them. The first is that they come with finite resolutions by T-equivariant vector bundles. Just like before, we can use such a resolution to define the class of an equivariant coherent sheaf in our equivariant K theory ring. The other thing we need to know about equivariant coherent sheaves is that whenever we have a T-stable sub-variety, its structure sheaf will be T-equivariant. In particular, Schubert varieties and opposite Schubert varieties are both T-stable, so their structure sheaves are examples of T-equivariant coherent sheaves. Just like before, pulling back along the structure morphism makes the T-equivariant K theory ring of X an algebra over the T-equivariant K theory ring of a point. The T-equivariant K theory ring of a point is also known as the representation ring of T because the T-equivariant vector bundle over a point is just a T representation. Since the torus T is commutative, every irreducible representation is one dimensional given by a character epsilon. We will write C epsilon for such a representation and its class lives in the representation ring of T. Just like before, Schubert classes and opposite Schubert classes give two bases for the equivariant K theory ring of X over the equivariant K theory ring of a point. Gromov-Wieten invariants are closely related to three-point Gromov-Wieten varieties. These are sub-varieties of M-D consisting of maps with the markpoints sent to three Schubert varieties stable under different boroughs. Let's assume that these Schubert varieties are in general position. Then the cohomological Gromov-Wieten invariant simply counts the number of points in the Gromov-Wieten variety. If this count is finite, in case the Gromov-Wieten variety has positive dimension, the cohomological invariant is zero. The K-theoretic Gromov-Wieten invariant computes the Schief order characteristic of the structure Schief of the Gromov-Wieten variety. So when the Gromov-Wieten variety is zero dimensional, we recover the cohomological invariant. But the K-theoretic invariant can still be non-zero even when the Gromov-Wieten variety has positive dimension. The K-theoretic invariant can also be computed by pulling back Schubert classes via evaluation maps from the K-theory ring of X to the K-theory ring of M-D, multiplying them and then pushing forward to the K-theory ring of a point, which is Z. Since everything is T-equivariant, we can suit this up to get the T-equivariant version of the K-theoretic Gromov-Wieten invariant, which lives in the equivariant K-theory ring of a point, aka the representation ring of T. To recover the non-equivariant case from the equivariant case, we simply need to set all the equivariant parameters to one. Of course, we are very interested in computing these Gromov-Wieten invariants. But it suffices to compute the Gromov-Wieten invariants defined by Schubert classes because these invariants are linear in each entry and the Schubert classes form a basis for the equivariant K-theory ring. These invariants are relatively well understood when X is co-mini-skew. Co-mini-skew varieties are a nice subclass of flat varieties. We have Picard rank one, and in type A, they are just the familiar grass manians. A result of Boug, Capu, Mihao Chi, and Pohang says that when X is co-mini-skew, the restricted EV3 from a two-point Gromov-Wieten variety to the corresponding two-point curve neighborhood is co-homologically trivial. This means that the push forward of the structure sheaf of the source equals the structure sheaf of the target, and all higher push forwards of the structure sheaf of the source vanish. In K-theory, the push forward of a class is defined as the alternating sum of the classes of higher push forwards. Therefore, when a map is co-homologically trivial, the push forward of the K-theory class of the source is just the K-theory class of the target. As a corollary, we get this nice formula that says equivalent K-theoretic Gromov-Wieten invariants can be computed in the equivalent ordinary K-theory ring of X. Unfortunately, when X is not co-mini-skew, this nice quantum equals classical formula doesn't always hold. Here is a counterexample from a paper by Buch-Presch and Tandakis. In the full-flat variety FL5, there are two rational curves of degree D equals 2332 passing through three general points. This means the Gromov-Wieten invariant associated to three points is equal to two. Now, since a degree D curve meeting two general points can reach a third general point, this two-point curve neighborhood is the entire flat variety, and this K-theory class is one. So the right-hand side is just computing the Schiff-Euler characteristic of the structure Schiff of a point, which is one. Here, the formula fails because the restricted EV3 from the two-point Gromov-Wieten variety to the corresponding two-point curve neighborhood is a two-to-one cover, which is not co-homologically trivial. However, when X is of type A and one of the Schubert varieties is a divisor, we still expect the restricted EV3 to be co-homologically trivial. This is a conjecture of Buch and Michauci. What I did was considering the simplest type A flat variety that is not co-mini-skew, namely the incidence variety. The incidence variety is a special two-step flat variety. It consists of pairs of vector subspaces of CN, U, V such that U sits in V, the dimension of U is one, and the dimension of V is M minus one. Taking out the U gives us a point in the projective space, and taking out the V gives us a point in the dual projective space. This incidence relation translates into this equation where X1 through Xn are the coordinates of CN and Y1 through Yn are the dual coordinates. In the GMOGP description, G is SLCN and P is the subgroup of upper triangular matrices with, sorry, block upper triangular matrices with three blocks sized one M minus two one. Schubert varieties in X stable under some fixed borough are indexed by W upper P. This consists of pairs of unequal integers between one and N. It sits inside the vial group as minimal coset representatives of W mod the vial group of P. So each element in W upper P corresponds to a Schubert variety and an opposite Schubert variety. They are both, they are each given by the vanishing of some coordinates. Containment relationships of these Schubert varieties are governed by the Bruha order on W upper P, which is inherited from the vial group W. For example, the element of one N is the smallest element in W upper P and X lower one N is just a point while X upper one N is the entire X. There are two opposite Schubert divisors, which we denote by D1 and D2. They are cut out by X1 equals zero and Yn equals zero respectively. And their classes generate the P card group. Finally, we will simplify notations for Schubert classes by dropping some square brackets and some letters in this way. Our main geometric result says that the general fiber of this restricted EV3 is rationally connected. You can think of the general fiber as the set of degree D stable maps with the first mark point sent to a Schubert variety. The second mark point sent to an opposite Schubert divisor and a third point sent to some fixed point that is as general as it can be. Using a result of color, this implies that the same map is co homologically trivial. As a corollary, we get the same quantum equals classical formula, except that one of the Schubert classes comes from a divisor. Note that the right hand side is very easy to compute because what is this two point curve neighborhood. Well, when the pairing of the degree D and the divisor DK is positive. Every curve of degree D is guaranteed to meet the divisor DK. But this two point curve neighborhood is just the curve neighborhood of the Schubert variety, which is again a Schubert variety. When the pairing of the degree D and the divisor DK is zero. Every curve of degree D is either disjoint from DK or contained in DK. The two point curve neighborhood is the intersection of the curve neighborhood of the Schubert variety with the divisor DK, which means we need to multiply the corresponding K theory classes. In order to state the final result, I need to introduce a little bit of notation. Recall that W upper P is the set of pairs of unequal integers between one and N, and it is the indexing set for Schubert varieties and Schubert classes. As it turns out, our formulas become much simpler if we index Schubert classes and degrees at the same time. This can be achieved by extending the set W upper P to the set W upper P total, which consists of pairs of integers that are not congruent to each other mod N. Each such pair ij has a unique reduction mod N, i bar j bar, which lives in WP with the property that i bar is congruent to i, j bar is congruent to j. What we're doing here is nothing but a shifted version of taking the remainder. By keeping track of how many copies of N, we need to subtract from i, and how many copies of N we need to add to j to get our reduction mod N. We can also recover a degree, which lives in the second homology group of X, which is Z2 in our case. When we write a Gromov-Wieten invariant, we will suppress the degree by allowing the index of the last Schubert class to vary in WP total. So doing reduction mod N to this index will recover the degree and an actual Schubert class. Again, T is the maximum torus of diagonal matrices inside SLCN, and we will write epsilon i for the character of T that records the i-diagonal entry. Here is what we get for the equivalent Gromov-Wieten invariants if we actually carry out the computation using our quantum equals classical formula. And as a corollary, the associated three-point Gromov-Wieten varieties have arithmetic genus zero, whenever they are non-empty. This can be seen by specializing these Gromov-Wieten invariants to the non-equivarian case, which amounts to setting these equivalent parameters to one. We see immediately that the non-equivarian invariants are either zero or one. And when they are zero, it's not hard to show that the corresponding Gromov-Wieten variety must be empty. Therefore, when the Gromov-Wieten variety is non-empty, the Euler, the Schief Euler characteristic of its structure Schief is one, which is the same thing as saying the variety has arithmetic genus zero. I think this is a good place to take a break. Please let me know if you have any questions. All right, very nice.