 All right. Hello, everybody. Welcome to the first round of early career talks for this semester. We will have some more on December 11, but we have a tight program with 20-minute talks. So without further ado, we are very happy to have Wei Hongsu for the first talk, which will be about quantum K-Witney relations for partial flap varieties. Please go ahead. Thank you for the opportunity to speak. This is joint work with Leonardo and physicists Weigu, Eric Sharpe, Hao Zhang and Hao Zou. Well, there haven't been that many quantum K-theory talks at the seminar lately, so let me start with what is quantum K-theory? Well, for a smooth, projective variety X, the quantum K-theory ring is a deformation of the ordinary K-theory ring, which is additively generated by algebraic vector bundles over X, modulo short exact sequences. Now, if you have coherent sheaves, such as structure sheaves of sub varieties, because they have finite resolutions by vector bundles, they also live in the K-theory ring as alternating sums of vector bundles. In this talk, X will be the partial flag variety, and additively, quantum K-theory ring is just the ordinary K-theory ring tensored with this formal power series ring, where the Q1 through QK are quantum parameters, as many as the number of steps in our partial flag. In particular, we still have our favorite Schubert basis in quantum K-theory, now over this power series ring. The product in quantum K-theory is a deformation of the tensor product of vector bundles. They are defined using K-theoretic Bromow-Wittern invariants. Each such invariant is determined by a couple of K-theory classes, here I denoted them as sigma, and an effective degree D, which is an invariant of a curve inside X. In our case, this degree is a K-tuple of non-negative integers measuring how many times this curve intersects the K-Schubert divisors when they are put in general positions. And when the sigmas are given by structure sheaves of subvarieties in X, this invariant measures the arithmetic genus of the family of degree D curves in X passing through these fixed varieties when they are put in general position. And if there are finitely many such curves, then this invariant is equal to the cohomological Bromow-Wittern invariant, which just count the number of curves. And finally, because there is, because there are group actions over X, it makes sense to consider equivariant vector bundles, and this equivariant refinement can be carried over to quantum K-theory. In case you're more familiar with quantum cohomology, I'd like to point out that quantum K-theory is more complicated in several ways. First of all, there is no divisor axiom to help us compute K-theoretic Bromow-Wittern invariants. And moreover, the structure constants in quantum K-theory are no longer a single K-theoretic Bromow-Wittern invariant, and they are not enumerative in general. And finally, when you multiply two K-theory classes in quantum K-theory and expand in your K-theory basis, it priori the coefficients are going to be formal power series in Q rather than polynomials. But these complications tend to be controllable for GMOD P. It was proved by Anderson Chen and Zeng that when you multiply Schubert classes and expand in quantum K-theory, there will only be finitely many Q-powers. For co-mini-sku-flat varieties, this was proved by Buk, Kapu, Mihao-chi, and Bechen, and Kato proved it for complete flat varieties. Even though the structure constants are no longer enumerative, we still expect Schubert structure constants to be positive in the appropriate sense to have positivity properties. I proved this for incidence varieties, and for mini-sku-flat varieties and quadrics, this was proved by Buk, Kapu, Mihao-chi, and Bechen. And right now, I'm working with Benedetti and Bechen, improving this for this type C analog of incidence varieties. So I've said that there is no divisor axiom in general in quantum K-theory, but we have the following conjecture of Buk and Mihao-chi, which can be viewed as a divisor axiom for type A-flags. So here, this oxi is the K-theory class given by the structure sheaf of the i-th Schubert divisor. And in the case, notice that in the case where the divisor and the and the degree paired to zero, which is the second case, the K-theoretic gromo-viton invariant may not necessarily vanish. So this is also a difference from co-homological gromo-viton invariance. This conjecture was proved by Buk and Mihao-chi for basmanians, and I proved it for incidence varieties. Now, we may ask for a presentation for the equivalent quantum K-theory ring by generators and relations. We can use the recipe, which was used by a Gu Mihao-chi, a sharp and so for basmanians. First we start with a presentation for equivalent ordinary K-theory. Then we look at the relations in this presentation and study how they deform in the quantum case. And finally, we use some Nakayama type argument to prove that the deformed ring is a presentation for quantum K-theory. And then, yes, the presentation for ordinary K-theory that we start with is the Whitney presentation, which I will describe next. So whenever you have a short exact sequence of vector bundles on X, going from E prime to E to E double prime, then the Whitney relations say that the total turn class of E is equal to the product of the total turn classes of E prime and E double prime. This happens in cohomology. Now to get the K-theoretic analog, you just need to replace the total turn class by its K-theoretic analog, which is the lambda y class defined in this way. In particular, the ith turn class of a vector bundle is replaced by the K-theory class of the ith wedge of that vector bundle. And here the y is a formal variable. And these Whitney relations are equivariant. Over our partial flag variety X, there is a nested sequence of tautological vector bundles. At each step of this flag, we have a short exact sequence of this form, giving rise to these Whitney relations in equivariant cohomology and equivariant K-theory. So the Whitney presentation for cohomology has generators, the turn classes of these tautological bundles and their successive quotients, and the relations are given by Whitney relations. Similarly, in K-theory, the generators are wedges of the tautological bundles and their successive quotients, and the relations are the K-theoretic Whitney relations. Next we need to study how these Whitney relations deform. For Grassmanians, Wheaton gave the answer in quantum cohomology. So the deformation is this last term involving Q. In quantum K-theory, this was a recent result of Gumi, Haucha, Sharpe, and so, where you can see that the deformation is slightly more complicated. For general partial flags, we can capture that the Whitney relations at each step get deformed in a way that is very similar to the Grassmanian case. We can formalize these relations by introducing formal variables for the K-theoretic turn routes of the tautological bundles and their successive quotients. We let S be the algebra generated over the equivariate K-theory of a point by elementary symmetric polynomials in these formal variables. And we let, we can write IQ for the ideal inside the formal power series ring over S generated by the coefficients of Y in these deformed relations, written in terms of our formal variables. Then our first result says that if the deformed relations hold in quantum K-theory, then they form a complete set of relations in the sense that this formal power series ring over S modulo the ideal IQ is isomorphic to the equivariate quantum K-theory ring. We also prove our conjecture for incidence varieties. This uses the quantum K-devisor axiom that I proved earlier. For the complete flag variety, we also prove that the quantum K-or conjectured quantum K-Whitney relations follow from the quantum K-devisor axiom. Finally, I'd like to say a few words about the physics computation that inspired our conjecture. This is in the case of FL3. So our physics collaborators handed us with this twisted