David Favero

Title: Enumerative Invariants of GLSMs via matrix factorizations
Abstract: I will discuss the construction of enumerative invariants for certain pairs (X,w) consisting of a target space X (which is a GIT quotient of affine space) together with a global function w.   For such spaces, I will discuss the construction of a "fundamental matrix factorization", which is a object in a certain category whose chern character, more or less, plays the role of the virtual fundamental class.  In fact, there is a concrete way in which this can be used to recover both Gromov-Witten theory for GIT quotients of affine space and the algebraic FJRW invariants defined by Polishchuk-Vaintrob.  This talk is based on joint work with I. Ciocan-Fontanine, J. Guere, B. Kim, and M. Shoemaker.

 

 

 

Takeshi Ikeda

Titile: Relativistic Toda lattice and K-theoretic Peterson isomorphism
Abstract: The relativistic Toda lattice is the group version of the Toda lattice. We will discuss how unipotent solutions of the integrable system connect the K-homology of the affine Grassmannian and the quantum K-theory of the flag variety (in type A). Our map gives an explicit relation between Lenart-Maeno's quantum Grothendieck polynomials and a modified version of Lam-Schilling-Shimozono's K-theoretic k-Schur functions. We should note that the existence of such a map in torus equivariant setting for a semisimple group G was conjectured by Lam, Li, Mihalcea, and Shimozono independently from our construction in type A, and recently, Kato proved the conjecture in full generality. The talk is based on my joint work with Iwao and Maeno.

 

 

Syu Kato

Title: Equivariant $K$-theory of semi-infinite flag manifolds and its relation to the quantum $K$-theory of flag manifolds

Abstract: We begin by what we mean by semi-infinite flag manifolds, and explain some of the main ideas on the definition of the equivariant $K$-theory of semi-infinite flag manifolds. Then, we construct an isomorphism between the equivariant $K$-theory of semi-infinite flag manifolds and equivariant quantum $K$-theory of flag manifolds. In case we have time, I will also explain how this naturally explains the connection between the equivariant $K$-theory of affine Grassmannian and equivariant quantum $K$-theory of the corresponding flag manifolds.

 

 

Young-Hoon Kiem 

Title: Localizing virtual structure sheaves by cosections

Abstract: I will discuss a joint work (arXiv:1705.09458) with Jun Li in which we construct a cosection localized virtual structure sheaf when a Deligne-Mumford stack is equipped with a perfect obstruction theory and a cosection of the obstruction sheaf.

 

 

Changzheng Li

Title: Euler characteristics in the quantum K-theory of flag varieties. 
Abstract: In this talk, we will discuss the sum of the Schubert structure coefficients in the equivariant quantum K-theory of flag varieties G/P. We will show that the sheaf Euler characteristic of the equivariant quantum K-product of a Schubert class and an opposite Schubert class is equal to q^d, where d is the smallest degree of a rational curve joining the two Schubert varieties. Along the way, we provide a description of the smallest degree d in terms of its projections to flag varieties de fined by maximal parabolic subgroups. This is my joint work with Anders Buch, Sjuvon Chung and Leonardo Mihalcea. 

 

 

Sanghyeon Lee

Title : Virtual pull-backs via semi-perfect obstruction theory.

Abstract : In this talk, I will discuss how to construct virtual pull-backs of Chow groups for a morphism of Deligne-Mumford stacks equipped with compatible semi-perfect obstruction theories. This is a generalization of the result of Huai-Liang Chang and Jun Li (https://arxiv.org/abs/1105.3261). Also, I will simply talk about virtual structure sheaves via semi-perfect obstruction theories.

 

 

Yuan-Pin Lee

Title: Towards a quantum Lefschetz hyperplane theorem in all genera
Abstract: I will explain simple ways to derive an algorithm of determining Gromov--Witten invariants of ample hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space. This is a joint work with Honglu Fan.

 

 

Leonardo Mihalcea

TALK I

Title: K-theoretic Gromov-Witten invariants and quantum K theory
Abstract: The goal of this introductory talk is to define K-theoretic Gromov-Witten (KGW) invariants and the (small) quantum K theory ring for a projective manifold X, focusing on the case when the appropriate moduli space is nice, i.e. no virtual cycles are needed. The main example is provided when X is a flag manifold. Our main references are Givental and Y.P. Lee early works on this subject. Time permitting, I will include some explicit calculations of KGW invariants and structure constants in the quantum K theory ring of a projective space.

 

TALK II

Title: Positivity in the quantum K theory of Grassmannians
Abstract: It is expected that the Schubert structure constants in the quantum K theory ring of any flag manifold are alternating, i.e. they have predictable signs. This generalizes the similar result from the ordinary K theory ring, proved by Buch for Grassmannians and by Brion for any flag manifold. I will discuss this positivity property in the case of the quantum K theory ring of a (cominuscule) Grassmannian. This is based on joint work with Buch, Chaput and Perrin.

 

 

Satoshi Naito

Title: Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds 
Abstract: We give a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of a semi-infinite flag manifold, which describes the (tensor) product of the class of a line bundle associated to an arbitrary anti-dominant weight with the class of the structure sheaf of a semi-infinite Schubert variety, in terms of quantum Lakshmibai-Seshadri paths. As an application, we obtain a Monk formula in the K-group above, which describes the product with the class of the structure sheaf of a semi-infinite Schubert variety of codimension one. On the basis of the isomorphism between the K-group above and the torus-equivariant (small) quantum K-group of an ordinary flag manifold, which has recently been established by Syu Kato, our results yield an explicit description of the quantum multiplication by a line bundle associated to an anti-dominant fundamental weight; in particular, we can verify a conjectural Monk formula (presented by Lenart and Postnikov) in the quantum K-theory of a flag manifold. This talk is based on a joint work with D. Sagaki and D. Orr. 

 

 

Jeongseok Oh

Title: Localization by 2-periodic complexes in K-theory
Abstract: We discuss the properties of the operator in K-theory constructed by a given 2-periodic complex on a space. As an application, we compare the virtual structure sheaves of moduli spaces. This talk is based on the joint work with B. Sreedhar.

 

 

Feng Qu

Title: Virtual pullbacks and applications
Abstract: In these three lectures, I plan to explain intersection theory arguments involved in establishing properties of virtual classes and virtual structure sheaves.
Topics include
(1) Cosection localized virtual pullbacks, their bivariance and functoriality.
(2) CohFT axioms in GW theory and quantum K-theory.
(3) The virtual localization formula.  

 

 

Bhamidi Sreedhar 

Titile: $K$-theory and Chow Groups of Quotient Deligne-Mumford stacks.

Abstract:

TALK I: This will be a survey talk on equivariant algebraic $K$-theory and equivariant Chow groups. We shall briefly recall the definition of equivariant (higher) Chow groups as defined by Edidin-Graham and shall discuss the equivariant Riemann-Roch theorem proved by Edidin-Graham for $G_0$ and for higher $K$-theory by Krishna.

TALK II: In this talk we shall outline how to use equivariant methods to prove a Grothendieck-Riemann-Roch type theorem for quotient Deligne-Mumford stacks over $mathbb{C}$. This talk is based on joint work with A. Krishna.

 

 

Hsian-Hua Tseng

TALK I

Title: On the D_q-module structure in quantum K-theory

Abstract: We give an introduction to the finite-difference module structure in quantum K-theory, following the work of Givental-Tonita and Iritani-Milanov-Tonita

 

TALK II

Title: On some analytic property of K-theoretic J-function of G/B

Abstract: We discuss certain vanishing property of the K-theoretic J-function of flag manifold G/B. This is based on joint work with D. Anderson and L. Chen.

 

 

Ming Zhang

TALK I, II

Title: Permutation-equivariant quantum K-theory following Givental

Abstract: This is an expository talk on permutation-equivariant quantum K-theory following a series of papers by Givental.

 

TALK Ⅲ: Verlinde algebra and quantum K-theory with level structure

Abstract: In this talk, I will first introduce the level structure in quantum K-theory. When the target is the Grassmannian, I will explain the wall-crossing approach to obtain the relation between quantum K-invariants with level structure and GL Verlinde numbers.