Xiaohan Yan (UC Berkeley)
Title: Quantum K-theory of flag varieties via non-abelian localization
Abstract: Quantum cohomology may be generalized to K-theoretic settings by studying the "K-theoretic analogue" of Gromov-Witten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (K-theoretic) ”big J-functions”, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutation-invariant big J-function of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finite-difference operators, from the quantum K-theory of their associated abelian quotients which is well-understood. Generating functions of K-theoretic quasimap invariants, e.g. the vertex functions, can be realized in this way as values of various twisted big J-functions. We also discuss properties of the level structures as applications of the method. A portion of this talk is based on a joint work with my advisor Alexander Givental.
Sang-Bum Yoo (Department of Mathematics Education, Gongju National University of Education)
Title: AN INTRODUCTION TO P=W CONJECTURE
Abstract. Let C be a complex smooth projective curve of genus g > 1. The non-Abelian Hodge theorem states that the moduli space M_B of representations of the fundamental group of C into GL_2(C) is naturally diffeomorphic to the moduli space M_Dol of Higgs bundles of rank 2 and degree 0 on C. Since they are not biholomorphic, it is natural to investigate the relation between the mixed Hodge structures on their cohomology groups. In 2012, M.A. de Cataldo, T. Hausel, and L. Migliorini proved that the weight filtration on H^*(M_B) coincides with the perverse Leray filtration on H^*(M_Dol) associated with the Hitchin map on M_Dol, which is called P=W theorem. If GL2(C) is replaced by other reductive groups or if we consider the twisted representations of the fundamental group of the punctured curve C \ p with nontrivial local monodromy, variants of P=W theorem appear. These variants are called P=W conjecture. In this talk, I will explain how P=W conjecture appears in Hodge theory, and then
introduce recent progresses of P=W conjecture done by several mathematicians and propose some problems related to a variant of the conjecture over a singular base curve.
Tasuki Kinjo
Title : Virtual classes via vanishing cycles
Abstract : For a quasi-smooth derived scheme, Behrend-Fantechi and Li-Tian constructed its virtual fundamental class which plays a very important role in enumerative geometry. In this talk, I will explain a new construction of the virtual fundamental class via the perverse sheaf of vanishing cycles on the -1-shifted cotangent space. If time permits, I will discuss a similar conjectural approach to the DT4 virtual class.
Dimitri Zvonkine
Titile: Gromov-Witten invariants of complete intersections
Abstract: Gromov-Witten invariants of a smooth complex variety do not change as the variety is deformed. Moreover, if a variety X degenerates into a union of two smooth varieties X_1, X_2 intersecting along a smooth divisor D, the degeneration formula by Jun Li allows one to recover a part of Gromov-Witten invariants of X from those of X_1, X_2, and D. The degeneration formula applies nicely to complete intersections. Indeed, if f is one of the polynomial equations defining a complete intersection X, we can deform f into a product g_1 g_2 of two polynomials of smaller degrees, thus degenerating the complete intersection into a union of two simpler complete intersections. However, the degeneration formula has a limitation: it only applies to cohomology classes that are well-defined on the total space of the degeneration. Thus, in general, it is inapplicable to the primitive cohomology of complete intersections. On the other hand it does use the primitive cohomology of D to express the answer. In other words, it is not possible to compute Gromov-Witten invariants by induction by just using this formula. In a joint work with Arguz, Bousseau and Pandharipande we solved this difficulty by introducing nodal Gromov-Witten invariants. We constructed an algorithm that computes all Gromov-Witten invariants of all complete intersections.
Mark Shoemaker
Title: A mirror theorem for GLSMs
Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V. This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient. GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out. In this talk I will describe a new method for computing generating functions of GLSM invariants. I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.
Hyeonjun Park
Title: Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds
Abstract: I will introduce a virtual pullback formula between Oh-Thomas virtual cycles on the moduli spaces of sheaves on Calabi-Yau 4-folds. The two main applications are the Lefschetz principle in Donaldson-Thomas theory and the Pairs/Sheaves correspondence. I will explain how they can be applied to the four conjectures in DT4 theory: (1) the Cao-Kool conjecture on the tautological Hilbert scheme invariants, (2) the Cao-Kool-Monavari conjecture on the DT/PT correspondence, (3) the Cao-Maulik-Toda conjecture on the genus zero Gopakumar-Vafa type invariants, and (4) the Cao-Toda conjecture on the JS/GV correspondence.
Michael Wemyss
Title: GW and GV invariants via the movable cone
Abstract: Given any smooth threefold flopping contraction, I will give a combinatorial characterisation of which Gopakumar–Vafa (GV) invariants are non-zero, and also explain how they change under iterated flop. To do this requires us to prescribe multiplicities to the walls in the movable cone. From this, I will draw (!), the critical locus of the associated quantum potential. There are various corollaries, including a visual proof of the Crepant Resolution Conjecture in this context. This is joint work with Navid Nabijou.
Amin Gholampour
Title: 2-dimensional stable pairs on 4-folds
Abstract: I will discuss a 2-dimensional stable pair theory of nonsingular complex 4-folds parallel to Pandharipande-Thomas' 1-dimensional stable pair theory of 3-folds. The stable pairs of a 4-fold are related to its 2-dimensional subschemes via wall-crossings in the space of polynomial stability conditions. In Calabi-Yau case, Oh-Thomas theory is applied to define invariants counting these stable pairs under some restrains. This is a joint work with Yunfeng Jiang and Jason Lo.