Schedule

Tuesday 21 June 2022
All talks in Lecture Theatre C, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

10am - Kazushi Ueda (Tokyo)(online)

11am - Coffee Break

11:30pm - Giulia Gugiatti (ICTP)

12:30pm - Lunch 

2pm - Paul Aspinwall (Duke)(online)

3pm - Coffee Break

3:30pm - Ilka Brunner (LMU Munich)

Wednesday 22 June 2022
All talks in Lecture Theatre C, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

10am - Kentaro Hori (IPMU)(online)

11am - Coffee Break

11:30am - Mark Shoemaker (Colorado State)

12:30pm - Lunch 

2pm - Yongbin Ruan (IASM Zheijiang)(online)

3pm - Coffee Break

3:30pm - Andrei Caldararu (Wisconsin)

Thursday 23 June 2022
All talks in Lecture Theatre B, Watson Building, University of Birmingham. (First floor) - note room change this day
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

10am - Rachel Webb (Berkeley)

11am - Coffee Break

11:30am - Johanna Knapp (Melbourne)

12:30pm - Lunch 

2pm - Ran Tessler (Weizmann)

3pm - Coffee Break

3:30pm - Tarig Abdelgadir (Loughborough)

Friday 24 June 2022
NB: Building Change for all activities! Also, the University will be very busy as it is an open day.
All talks in Room G12, University House, University of Birmingham. (Ground floor)
Coffee Breaks and lunch nearby. 

10am - Hiroshi Iritani (Kyoto)(online)

11am - Coffee Break

11:30am - Livia Campo (Korean Institute of Advanced Study)

12:30pm - Lunch

2pm - Melissa Liu (Columbia)(online)

3pm - Coffee Break

3:30pm - Liana Heuberger (Angers)(online)

Titles and Abstracts

  Tuesday

•  Kazushi Ueda (Tokyo) - Stable Fukaya categories of Milnor fibers
  
  We define the stable Fukaya category of a Liouville domain as the quotient of the wrapped Fukaya category by the full subcategory consisting of compact Lagrangians, and discuss the relation between the stable Fukaya categories of affine Fermat hypersurfaces and the Fukaya categories of projective hypersurfaces. We also discuss homological mirror symmetry for Milnor fibers of Brieskorn-Pham singularities along the way. This is a joint work in progress with Yanki Lekili.

•  Giulia Gugiatti (ICTP) - Hypergeometric LG models and anticanonical log del Pezzo surfaces
  
  I will explain how hypergeometric LG models offer a tool to build Hodge-theoretic mirrors for certain Fano varieties out of the known constructions. Among these Fanos is the Johnson-Kollar series of anticanonical log del Pezzo surfaces. I will describe the Hodge-theoretic mirrors of these surfaces, built jointly with A. Corti, and I will exhibit full exceptional collections for the derived category of the associated stacks. This last result is the first part of a project with F. Rota to study homological mirror symmetry for these surfaces. 
  
  

• Paul Aspinwall (Duke) - An Introduction to the Gauged Linear Sigma Model and the 1-Loop correction
  
  In this introduction for mathematicians to Witten's famous 1993 paper I will review the basic motivation and construction of the Gauged Linear Sigma Model. The importance and application of quantum corrections  will be emphasized.

• Ilka Brunner (LMU Munich) - D-brane transport between phases of GLSMs using defects
  
  In this talk, I will describe how to  construct defects (domain walls) that connect different phases of two-dimensional gauged linear sigma models (GLSMs). Furthermore,  defects can be used to embed those phases into the GLSMs. Via their action on boundary conditions they give rise to functors between the different D-brane categories.

Wednesday

• Kentaro Hori (IPMU) - Grade restriction rule in the Rodland model and its dual
  
  The Rodland model is a GLSM relevant for the Pfaffian/Grassmannian correspondence. I will describe the grade restriction in the model and its dual. As an application, I will describe the monodromy along loops with base points both in the Grassmannian phase and in the Pfaffian phase. Based on a joint work with Richard Eager, Johanna Knapp and Mauricio Romo.
  
  

• Mark Shoemaker (Colorado State) - Towards a mirror theorem for GLSMs
  
  GLSMs provide  broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing the Gromov-Witten theory of complete intersections as well as FJRW theory.  More exotic GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions.  Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out for general GLSMs.  In this talk I will give some interesting examples of more general GLSMs, and describe a new method for computing generating functions of genus zero GLSM invariants. I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants.
  
  

• Yongbin Ruan (IASM Zheijiang) - Logarithmic R-map
  
  The fundamental object of Gromov-Witten theory is the notation of stable map to a target Kahler/Symplectic manifold. In LG A-model in physics, the target carries an additional $C^*$ action called R-charge. It twisted the notation of map in so called gauged linear sigma model. Furthermore, its moduli space is noncompact even with stability condition. In the talk, we will introduce the notation of $R$-map. The effort to compactify it leads to the theory of Logarithmic R-maps. This is a joint work with Qile Chen and Felix Janda.
  
  

• Andrei Caldararu (Wisconsin) - Categorical Enumerative Invariants of Matrix Factorizations
  
  The theory of Categorical Enumerative Invariants (CEIs) is a non-commutative generalization of the theory of Gromov-Witten invariants. Such CEIs are associated to an arbitrary Calabi-Yau category and a splitting of its Hodge filtration. I will present a general overview of this theory, introduced by Costello, Tu, and I a few years ago. I will also outline specific problems and results that arise when applying this theory to categories of matrix factorizations, an approach that conjecturally leads to a definition of B-model FJRW theory in arbitrary genus, even in the presence of an arbitrary symmetry group.

Thursday

• Rachel Webb (Berkeley) - Virtual Cycle on the Moduli Space of Maps to a Complete Intersection
  
  A driving question in Gromov-Witten theory is to relate the invariants of a complete intersection to the invariants of the ambient variety. In genus-zero this can often be done with a ``twisted theory,'' but this fails in higher genus. Several years ago, Chang-Li presented the moduli space of p-fields as a piece of the solution to the higher-genus problem, constructing the virtual cycle on the space of maps to the quintic 3-fold as a cosection localized virtual cycle on a larger moduli space (the space of p-fields). Their result is analogous to the classical statement that the Euler class of a vector bundle is the class of the zero locus of a generic section. I will discuss work joint with Qile Chen and Felix Janda where we extend Chang-Li's result to a more general setting, a setting that includes standard Gromov-Witten theory of smooth orbifold targets and quasimap theory of GIT targets. This work is joint with Qile Chen and Felix Janda.
  
  

• Johanna Knapp (Melbourne) - GLSMs of genus 1-fibered CY 3-folds
  
  Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this talk we will discuss examples and properties of a class of Calabi-Yau threefolds with 5-sections. These Calabi-Yaus cannot be constructed by means of toric geometry. One way to obtain them is as vacuum manifolds of GLSMs with non-abelian gauge groups. This approach makes it possible to find connections between different genus one fibrations with 5-sections that fit into the framework of homological projective duality. Based on a concrete example, we will showcase how GLSM methods can be used to study such Calabi-Yaus and their moduli spaces. This is joint work with Emanuel Scheidegger and Thorsten Schimannek.
  
  

• Ran Tessler (Weizmann) - Open FJRW theory and Mirror symmetry
  
  FJRW theory is a very rich geometric theory defined by Fan, Jarvis, and Ruan as a cohomological field theory generalizing Witten's r-spin construction. In my talk I'll review these two theories, and describe a generalization of some cases to the open setting, that is, to an intersection theory on the moduli space of FJRW discs. I will then state an open mirror theorem for the resulting invariants. If time permits I will also describe the proof ideas. This is based on a joint work with Mark Gross and Tyler Kelly.
  
  

• Tarig Abdelgadir (Loughborough) - The McKay correspondence via VGIT
  
  For a Kleinian singularity, the McKay correspondence famously relates the orbifold cover of the singularity to a crepant resolution. In type A, both are toric and it is easy to write down a GIT problem which produces both the orbifold and the geometric resolution as possible quotients. However, no such construction seems to be known for types D and E. I'll describe some work-in-progress with Ed Segal where we aim to fill this gap using a construction inspired by Tannaka duality. 

Friday

• Hiroshi Iritani (Kyoto) - Fourier transformation in GLSM
  
  I will explain how the Fourier transformation connects various solutions (e.g. hemisphere partition functions, I-functions, oscillatory integrals) of GLSM. I will give an application of this idea to quantum cohomology of projective bundles. This talk is based on joint works with Sanda and Koto. 
  
  

• Livia Campo (Korean Institute of Advanced Study) - Non-solidity of uniruled varieties
  
  In this talk we will establish sufficient conditions for a uniruled variety to admit a birational map to a strict Mori fibre space. As a consequence we will examine non-solidity of Fano 3-folds having Fano index 2. This is based on a joint work with Tiago Guerreiro.
  
  

• Melissa Liu (Columbia) - Higgs-Coulomb correspondence for abelian gauged linear sigma models
  
  The input data of a gauged linear sigma model (GLSM) consist of a GIT quotient of a complex vector space V by the linear action of a reductive algebraic group G (the gauge group) and a G-invariant polynomial function on V (the superpotential) which is quasi-homogeneous with respect to a C∗-action (R symmetries) on V. The Higgs-Coulomb correspondence relates GLSM invariants which are virtual counts of Landau-Ginzburg quasimaps (Higgs branch) and Mellin-Barnes type integrals (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work in progress with Konstantin Aleshkin.
  
  

•  Liana Heuberger (Angers) - Q-Fano constructions using Laurent Inversion
  
  Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial.  Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial.  We apply this to construct 100 examples of previously unknown Fano 3-folds with terminal quotient singularities.  A Laurent polynomial f determines, through its Newton polytope P, a toric variety $X_P$, which is in general highly singular. Laurent inversion constructs, from f and some auxiliary data, an embedding of $X_P$ into an ambient toric variety Y.  In many cases this embeds $X_P$ as a complete intersection of line bundles on Y, and the general section of these line bundles is the Q-Fano 3-fold that we wanted to construct, i.e. the mirror of f. We discuss two detailed examples of such constructions.
  
  
  

This conference is supported by the following bodies: