Schedule

Abstracts for hour-long talks below.

Tuesday 6 August 2024
All talks in Lecture Theatre A, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

9:00 - Registration

9:30 - Rob Silversmith (Warwick) - Stable curves and chromatic polynomials

10:30 - Coffee

11:00 - Andrew Harder (Lehigh) - Tropical geometry of Clarke mirror pairs

12:00 - Junior Talks

• Hannah Dell (Edinburgh) - Stability conditions and group actions

• Thibault Poiret (St. Andrews) - Some natural covers of the space of stable curves

• Luca Giovenzana (Sheffield) - Special stability conditions on local CY 3folds

12:30 - Lunch

2:00 -  Kimoi Kemboi (Princeton) - Full exceptional Collections via Window Categories

3:00 - Junior Talks

• Michela Barbieri (UCL) - Brane Monodromy in Toric GIT

• Peter Spacek (TU-Chemnitz) - Type-independent canonical mirror models for cominuscule homogeneous spaces

3:30 - Coffee

4:00 - Ed Segal (UCL) - The categorical origins of equivariant quantum cohomology

Wednesday 7 August 2024
All talks in Lecture Theatre A, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

9:30 - Junwu Tu (Shanghai) - Towards open-closed categorical enumerative invariants

10:30 - Coffee

11:00 - Alan Thompson (Loughborough) - Mirror symmetry for fibrations and degenerations of K3 surfaces

12:00 - Junior Talks

• Austin Hubbard (Bath) - Crepant resolutions of symplectic singularities

• Prajwal Samal (IMPAN) - Artinian Rings and Calabi-Yau Manifolds

• James Jones (Loughborough) - Type II Degenerations of K3 Surfaces

12:30 - Lunch

2:00 - Elana Kalashnikov (Waterloo) - Sagbi bases and mirror constructions for Kronecker moduli spaces

3:00 - Conference Photo, then coffee

3:30 - Junior Talks

• Daniil Mamaev (LSGNT) - A Relative Mirror to the Affine Plane Blown Up at a Point

• Irit Huq-Kuruvilla (Virginia Tech) - Quantum K-Rings of partial Flags

• Siao Chi Mok (Cambridge) - Expanded logarithmic product

• Francesca Leonardi (Leiden) - Some derived notions of logarithmic schemes

• Robbie Hanson (IST-Lisbon) - BBB-branes in geometric Langlands

• Alexia Corradini (Cambridge) - The Lagrangian Ceresa cycle

4:45-ish to 6:30 - Informal conversations.

7:00 - Conference Dinner

Thursday 8 August 2024
All talks in Lecture Theatre A, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

9:30 - Ailsa Keating (Cambridge) - Homological mirror symmetry for K3 surfaces

10:30 - Coffee

11:00 - Soheyla Feyzbakhsh (Imperial) - Relations of Kuznetsov components of a Gushel-Mukai variety and its hyperplane sections

12:00 - Lunch

2:00 - Marton Hablicsek (Leiden) - A formality result for logarithmic Hochschild (co)homology

3:00 - Coffee

3:30 - Kai Hugtenburg (Lancaster) - Open Gromov-Witten invariants from the Fukaya category

Friday 9 August 2024
All talks in Lecture Theatre A, Watson Building, University of Birmingham. (Ground floor)
Coffee breaks and lunch in the Mathematical Learning Centre, Watson Building, University of Birmingham. (First floor)

10:00 - Renata Picciotto (Cambridge) - Reduced Gromov-Witten invariants via desingularization of sheaves

11:00 - Coffee

11:30 - Jeff Hicks (Edinburgh) - Geometry and topology of Lagrangian lifts of tropical curves

Titles and Abstracts

• Soheyla Feyzbakhsh (Imperial College) - Relations of Kuznetsov components of a Gushel-Mukai variety and its hyperplane sections

Let X be a very general Gushel–Mukai variety of dimension n>3, and let Y be a smooth hyperplane section. There are natural pull-back and push-forward functors between the Kuznetsov components of X and Y. In this talk, I will show that the Bridgeland stability of objects is preserved under both of these functors and discuss some applications of this result. Joint work with Henfei Guo, Zhiyu Liu and Shizhuo Zhang. 

• Marton Hablicsek (Leiden University) - A formality result for logarithmic Hochschild (co)homology

Hochschild homology is a foundational invariant for associate algebras, schemes, stacks, etc. For smooth and proper varieties X over a field of characteristic 0, Hochschild homology and its variants, like cyclic homology, are closely related to Hodge cohomology and to de Rham cohomology. In this talk, via a geometric approach, we extend Hochschild homology to logarithmic schemes, in particular to compactifications, i.e, to pairs (X,D) where X is a smooth and proper variety and D is a simple normal crossing divisor. This geometric approach allows us to extend well-known facts about Hochschild homology and its variants to a logarithmic setting, in particular, (1) we generalize the celebrated HKR theorem to relate logarithmic Hochschild homology to logarithmic differential forms, (2) we define and provide a description of logarithmic cyclic homology, (3) and we compute log Hochschild (co)homology of logarithmic orbifolds. This is a joint work with Francesca Leonardi and Leo Herr.

• Andrew Harder (Lehigh University) - Tropical geometry of Clarke mirror pairs

I'll explain how to construct a graded sheaf of vector spaces on the tropicalization of any nondegenerate toric hypersurface whose cohomology recovers the orbifold Hodge numbers of the original hypersurface. This partially generalizes work of Itenberg, Katzarkov, Mikhalkin, and Zharkov. As an application, we prove that a stringy Hodge-number duality statement holds for a large and mysterious class of toric mirror pairs called Clarke mirror Landau-Ginzburg models. Using this, we recover a famous result of Batyrev and Borisov, a result of Krawitz, and prove a conjecture of Katzarkov, Kontsevich, and Pantev. Time permitting, I'll mention an unexpected relationship between certain maximally mutable Laurent polynomials and the Deligne-Hodge numbers of singular toric varieties. Joint with Sukjoo Lee (Edinburgh).

• Jeff Hicks (University of Edinburgh) - Geometry and topology of Lagrangian lifts of tropical curves

Given a tropical curve that admits a pair-of-pants decomposition in the base of a Lagrangian torus fibration, Mikahlkin constructed a Lagrangian submanifold whose projection under the Lagrangian fibration approximates the original tropical curve. Under appropriate conditions, this Lagrangian submanifold is mirror to the algebraic-geoemtric realization of the underlying tropical curve.

In this talk, we will employ tools from geometric topology to demonstrate that these Lagrangian submanifolds admit a metric with non-positive curvature in dimension three. Additionally, I will explore the algebraic-geometric implications of this structure as viewed through the mirror. Time permitting, I will also discuss connections to non-superabundance and Floer-theoretic unobstructedness. 

• Kai Hugtenburg (University of Lancaster) - Open Gromov-Witten invariants from the Fukaya category

Open Gromov-Witten invariants count holomorphic curves with boundary on a Lagrangian submanifold. In this talk I will explain how one can extend enumerative mirror symmetry to include open Gromov-Witten invariants, and how one can obtain open Gromov-Witten invariants from the Fukaya category, thus realising part of Kontsevich's proposal that homological mirror symmetry implies enumerative mirror symmetry.

• Elana Kalashnikov (University of Waterloo) - Sagbi bases and mirror constructions for Kronecker moduli spaces

One way of constructing candidate mirror partners to Fano varieties is via toric degenerations. The case in which this is best understood is the Grassmannian, using the well-known SAGBI basis of the Plucker coordinate ring indexed by semi-standard Young tableaux (SSYT). The mirror construction goes back to work of Eguchi‑Hori‑Xiong, however its geometry and combinatorics still plays an important role in current mirror constructions. In this talk, I will give an overview of this story, then turn to the question of what can be generalized for Kronecker moduli spaces. Like Grassmannians (which they generalize), Kronecker moduli spaces are high Fano index Picard rank 1 smooth Fano varieties. I will introduce linked SSYT pairs, which play the analogous role of SSYT for Grassmannians in understanding the coordinate ring of the Kronecker moduli space. This is joint work with Liana Heuberger. 

• Ailsa Keating (University of Cambridge) - Homological mirror symmetry for K3 surfaces

Joint work with Paul Hacking (U Mass Amherst). We first explain how to prove homological mirror symmetry for a maximal normal crossing Calabi-Yau surface Y with split mixed Hodge structure. This includes the case when Y is a type III K3 surface, in which case this is used to prove a conjecture of Lekili-Ueda. We then explain how to build on this to prove an HMS statement for K3 surfaces. On the symplectic side, we have any K3 surface (X, \omega) with \omega integral Kaehler; on the algebraic side, we get a K3 surface Y with Picard rank 19. The talk will aim to be accessible to audience members with a wide range of mirror symmetric backgrounds.

• Kimoi Kemboi (Princeton University) - Full exceptional Collections via Window Categories

This talk will focus on a particular structure of derived categories called a full exceptional collection. We will discuss the landscape of full exceptional collections and its connections to geometry, then explore how to produce them for linear GIT quotients using ideas from window categories and equivariant geometry. As an example, we will consider a large class of linear GIT quotients by a reductive group, where this machinery produces full exceptional collections consisting of tautological vector bundles. This talk is based on joint work with Daniel Halpern-Leistner.

• Renata Picciotto (University of Cambridge) - Reduced Gromov-Witten invariants via desingularization of sheaves

Gromov-Witten invariants are related to counts of curves in X of genus g and class d, but they often encode contributions from degenerate maps with reducible domains. I will explain in this talk how to define invariants that capture the least degenerate stable maps. This is achieved via a more general construction: desingularization of coherent sheaves and abelian cones on integral Artin stacks. I will give a construction that resolves the total cone of a coherent sheaf into a union of vector bundles supported on substacks by a combination of blow-ups. I will outline how this can be applied in Gromov-Witten theory to separate contributions to genus g invariants coming from different irreducible components of the moduli space. We use this to define reduced Gromov-Witten invariants in all genera, extending various previous constructions for genus 1 and 2 in the literature. This talk is based on arXiv:2310.06727, a work joint with A. Cobos-Rabano, E. Mann and C. Manolache. 

• Ed Segal (University College, London) - The categorical origins of equivariant quantum cohomology

Given a Hamiltonian U(1) action on a symplectic manifold one can try to generalize the Fukaya category to a `U(1) equivariant Fukaya category'. I will describe the (2-)categorical structure that is believed to arise when we do this, and how it should be possible to recover equivariant quantum cohomology from this structure. I'll also discuss the mirror picture and show how to calculate the mirror to equivariant quantum cohomology in many examples. 

• Rob Silversmith (University of Warwick) - Stable curves and chromatic polynomials

For any finite simple graph G, we introduce an associated natural class of intersection numbers on moduli spaces of stable curves. We prove a surprising formula for the intersection numbers in terms of the chromatic polynomial of G, establishing a new connection between algebraic graph invariants and algebraic curves. I'll discuss the result and it's proof, and also discuss some related questions and speculations. Joint with Bernhard Reinke.

• Alan Thompson (University of Loughborough) - Mirror symmetry for fibrations and degenerations of K3 surfaces

I will describe recent progress, joint with L. Giovenzana, on the problem of mirror symmetry for Type II degenerations of K3 surfaces. I will give a lattice-theoretic definition for when a Type II degeneration of K3 surfaces and an elliptically-fibred K3 surface, with an appropriate splitting of the base, form a mirror pair. I will then explain how this definition is compatible with lattice polarised mirror symmetry for K3 surfaces and with Fano-LG mirror symmetry for (quasi) del Pezzo surfaces. The upshot will be a concrete mirror symmetry conjecture for these objects. Finally, I will describe recent joint work with C. F. Doran and E. Pichon-Pharabod which allows this conjecture to be checked in explicit examples.

• Junwu Tu (Shanghai Tech) - Towards open-closed categorical enumerative invariants

In this talk, we discuss some work-in-progress to construct open-closed enumerative invariants from Calabi-Yau A-infinity categories. We shall focus on the genus zero case, and discuss new phenomena in order to define these invariants. (This is a joint work with Lino Amorim from Kansas State University.)