Program: Week 1

All talks will be in Cottonwood A, in the Snowbird Center, Level 2.

Lecture Series

  • Emily Clader The Gauged Linear Sigma Model
  • Todor Milanov The Landau-Ginzburg B-model
  • Dimitri Zvonkine Semisimple Cohomological Field Theories

Speakers, Titles, and Abstracts

      • Chen, Qile Compactifying moduli of sections using stable logarithmic maps
        Stable logarithmic maps in the sense of Abramovich-Chen-Gross-Siebert were developed to compactify the space of stable maps tangent to boundary divisors along marked points with fixed multiplicities. In this talk I will first review the theory of stable logarithmic maps. Then I will introduce recent joint work with Felix Janda, Yongbin Ruan, and Adrien Sauvaget on its application to sections of r-spin bundles, aiming at a study of the Gauged Linear Sigma Model of Fan-Jarvis-Ruan via compact logarithmic moduli spaces.
      • Janda, Felix Compactifying the space of maps with a p-field
        I will explain a compactification of the moduli space of stable maps with p-fields (obtained in joint work with Q. Chen and Y. Ruan, and used to compute higher-genus invariants of quintic threefolds), and give an example computation using it.
      • Lee, Y.-P. Gromov-Witten theory under Birational Maps and Transitions
        In this talk I will summarize some of joint work with H.-W. Lin, C-L. Wang and F. Qu on functoriality of Gromov-Witten (GW) theory under birational transformations and extremal transition. In particular, the wall-crossing behavior of Kahler cones under special smooth flops, flips and blow-ups will be discussed. For the Calabi-Yau threefolds under conifold transitions, I will describe a conjectural linking GW invariants and how their wall-crossing might be related to Chern-Simons theory.
      • Liu, Chiu-Chu Melissa Holomorphic Anomaly Equations and Modular Anomaly Equations
        I will survey conjectures and theorems on holomorphic anomaly equations and modular anomaly equations for both open and closed Gromov-Witten invariants.
      • Marcus, Steffen Logarithmic compactification of the Abel-Jacobi section
        Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. This recovers the double ramification cycle, as well as variants associated to differentials.
      • Oh, Jeongseok Localized Chern characters for 2-periodic complexes
        For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of ε-stable quasimaps and moduli of the corresponding LG ε-stable quasimaps, in full generality. The talk is based on joint work with Bumsig Kim.
      • Okounkov, Andrei Monodromy and derived equivalencesIn joint work with R. Bezrukavnikov, we prove that for many classes of equivariant symplectic resolutions, the Bezrukavnikov-Kaledin derived autoequivalences act on K-theory as the monodromy of the quantum differential equation. This talk will be an introduction to this topic and, time permitting, to a further development of these ideas in the context of the quantum difference equation in joint work with M. Aganagic.
      • Webb, Rachel The Abelian-Nonabelian Correspondence for I-functions
        When a complex reductive group G with maximal torus T acts on an affine variety W, one can form two (GIT) quotients: W// G and W// T. With the right hypotheses, W// G and W// T are both smooth projective varieties. The relationship between the cohomology rings of these two varieties is well understood. However, the relationship between their quantum cohomology rings is only known up to a conjectured correspondence of small J-functions (encoding Gromov-Witten invariants with one insertion). With a wall-crossing result of Ciocan-Fontanine and Kim, this last piece becomes a correspondence of small I-functions. I will discuss a proof of the I-function correspondence when W is a vector space and G is connected.

      Program: Week 2

      All talks will be in Cottonwood A, in the Snowbird Center, Level 2.

      Lecture Series

      • Li, Wei-Ping p-fields and cosection localization
      • Ross, Dusty Higher genus computations in FJRW theory
      • Shoemaker, Mark Matrix Factorizations and the virtual class

      Speakers, Titles, and Abstracts

      • Bertram, Aaron Crossing Bridgeland Stability Walls
        When we vary Bridgeland stability conditions on the moduli of vector bundles (or complexes of vector bundles) with a fixed Chern class, we may see birational transformations of moduli spaces, or even the creation (destruction) of entire components. I will illustrate this phenomenon with examples using the projective plane and projective three-space, in which the wall-crossing behavior is agrees with the variation of a geometric invariant theory quotient.
      • Blankers, Vance Descendent Potentials for Hassett Spaces with Diagonal Weights
        Hassett spaces, or moduli spaces of weighted stable curves, are a family of compactifications of Mg,n associated to rational weight data (a1,…, an). When all weights are 1, we recover the usual Deligne-Mumford compactification Mg,n We discuss the descendent potential for Hassett spaces with weights an = 1/q and relate it to the Deligne-Mumford descendent potential. As a corollary, we obtain a new proof of a result of Manin and Zograf connecting the Witten potential to the kappa-class potential on Mg,n via a change of variables.
      • Dumitrescu, Olivia Interplay between CohFT and Topological Recursion
        We study a few examples of Cohomological field theories (CohFT) defined over a 1-dimensional Frobenius algebra. We aim to develop a formalism to recover the spectral curve of topological recursion using the classification theorem of CohFT for semi-simple Frobenius algebras. We present the relation between Hurwitz numbers and spectral curves, and we study intersection numbers of mixed kappa- and psi-classes on the moduli spaces of curves. This is work in progress with Motohico Mulase.
      • Mi, Rongxiao Quantum D-modules and Extremal Transitions
        Extremal transitions are a contract-deform surgery, which play a significant role in classifying Calabi-Yau threefolds. In this talk, I will outline a framework that relates two quantum D-modules under extremal transition. Then I will give several examples in the toric case (such as cubic and degree-four extremal transitions) and explain how it works.
      • Priddis, Nathan Borcea-Voisin Mirror Symmetry for LG models
        Borcea and Voisin described a particular version of mirror symmetry involving the quotient of the product of a K3 surface and an elliptic curve by a certain involution. This is often referred to as Borcea-Voisin mirror symmetry. Using the ideas of wall-crossing, we obtain a corresponding Landau-Ginzburg model (FJRW theory). In this talk I will introduce the construction, the corresponding LG model and show how Borcea-Voisin mirror symmetry translates into the LG theory.
      • Whitcher, Ursula Hypergeometric structure of K3 quartic pencils
        We exploit intuition from the BHK mirror symmetry construction to study hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We write the point counts over finite fields in terms of explicit, hypergeometric global L-functions, each associated to a Picard-Fuchs differential equation. The computation gives a complete description of the motives for these pencils in terms of hypergeometric motives. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight.
      • Zhang, Ming Verlinde algebra and quantum K-theory with level structure
        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 Verlinde numbers. Based on joint work with Yongbin Ruan.
      • Zhou, Jie Mirror symmetry for elliptic curves and modular forms
        Modular forms are interesting and useful tools in the studies of enumerative geometry of varieties related to elliptic curves. They also provide concrete formulae to test new ideas and techniques.In this mostly expository talk, I will try to discuss some interesting modular phenomena arising from the mirror symmetry of elliptic curves, part of which can hopefully be generalized to more general targets. I will first start with the Gromov-Witten theory for elliptic curves and explain why the generating series of Gromov-Witten invariants should be related to modular forms at all from the perspective of mirror symmetry. Then I will discuss the sheaf-theoretic formulation of quasi-modular forms and explain why the generating series should be quasi-modular as opposed to being honestly modular. After that I will discuss some concrete examples in which modular transformations relate different enumerative theories. Finally, if time permits, I will try to say a few words about the non-holomorphicity in the modular completed generating series.
      • Zhou, Zijun GW/DT correspondence for local gerby curves with transversal An singularity
        In this talk, I will discuss an ongoing joint project with Zhengyu Zong on the Gromov-Witten/Donaldson-Thomas correspondence for local gerby curves, as an orbifold generalization of the corresponding work on the smooth case by Bryan-Pandharipande and Okounkov-Pandharipande. By applying degeneration formulas, the correspondence can be reduced to the case of [ℂ2/ℤn+1] × ℙ1, where the 3-point relative GW/DT invariants can be related to the quantum multiplication by divisors for Hilb([ℂ2/ℤn+1] ) and Sym([ℂ2/ℤn+1] ). This is a crepant resolution counterpart to the work of Maulik-Oblomkov and Cheong-Gholampour.