Skip to main content

Abstracts

Thomas Prince
Title: Smoothing del Pezzos with cyclic quotient singularities via the Gross-Siebert reconstruction algorithm
Abstract: Recent work of Coates, Corti, Kaspryzk et al. aims to classify orbifold del Pezzo surfaces admitting toric degenerations using a conjectural correspondence with Laurent polynomials in two variables coming from mirror symmetry. We explain how this project lies within the Gross-Siebert program. In particular, we describe a 'tropical version' of a Q-Gorenstein smoothing and show how to apply the Gross-Siebert reconstruction algorithm to build (a formal version of) the desired toric degeneration. Given time we shall indicate how tropical disc (broken line) counts should give rise to the Laurent polynomial mirrors.

Balazs Szendroi
Title: Purity in cohomological DT theory
Abstract: Cohomological Donaldson-Thomas theory is the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. After a brief introduction, I will explain the role of purity in the theory and some applications and open questions.

Ivan Smith
Title: Quiver algebras as Fukaya categories

Ailsa Keating
Title: Homological Mirror Symmetry for singularities of type Tpqr
Abstract: We present some homological mirror symmetry statements for the singularities of type $T_{p,q,r}$. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E.
We will consider some symplectic invariants of the real four-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space P2, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different ''flavours" of invariants (e.g., versions of the Fukaya category) match up on both sides.

Diego Matessi
Title: Lagrangian submanifolds in local Calabi-Yaus and mirror symmetry
Abstract: Mirror symmetry of toric Calabi-Yau threefolds can be described in terms of SYZ duality of Lagrangian torus fibrations. From this point of view, we will describe a construction of Lagrangian spheres and sections inside the mirrors of toric Calabi-Yau threefolds. Some results on the topology of these Lagrangian submanifolds also suggest an explicit conjectural homological mirror symmetry correspondence between sections and line bundles and between spheres and sheaves supported on the compact toric divisors. One evidence for this correspondence is that it naturally provides an An configuration of sheaves which should be mirror to the vanishing cycles of An singularities. This is joint work with M. Gross.

Gabriel Kerr
Title: Homological mirror symmetry for affine and quasi-affine toric varieties
Abstract: I will discuss a generalization of homological mirror symmetry for proper toric varieties to the affine and quasi-affine cases. The A-model category mirror to the derived category of coherent sheaves is a partially wrapped category whose wrapping Hamiltonian is the kinetic energy relative to a mirror Lagrangian skeleton. After describing this category and contrasting it with alternative models, I will sketch the proof of this version of the HMS conjecture for the simple case of the punctured affine plane. If time permits, I will discuss the A-model version of GIT constructions.

Sean Keel
Title: Tyurin's conjecture for polarized K3 surfaces
Abstract: I'll explain my construction, joint with Gross, Hacking, and Siebert, of a canonical toroidal compactification of the moduli space of polarized K3 surfaces, together with an extension of the universal polarized family endowed, near the boundary, with canonical "theta functions", a basis of sections of powers of the polarisation and a formula for the associated structure constants for multiplication in the homogeneous coordinate ring in terms of counts of (a combinatorial version of) holomorphic discs.

Jingyu Zhao
Title: Periodic Symplectic Cohomologies
Abstract: Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplectic cochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result is that there is a localization theorem for the finitely supported periodic symplectic cohomology.

Paul Hacking
Title: Tyurin's conjecture for polarized K3 surfaces 2
Abstract: Continuation of Sean Keel's talk.

Tyler Kelly
Title: Special Linear Systems and Toric Mirror Constructions
Abstract: Given a Calabi-Yau complete intersection in a toric variety, certain mirror constructions give different mirrors. Whether they give the same answer depends on if one's complete intersection lies in a special anti-canonical linear system instead of the full linear system. We will explain this phenomenon in the context of homological mirror symmetry. We will focus on studying this phenomenon in the context of some of our favourite examples. This talk is based on joint work with C. Doran and D. Favero (Alberta).

Daniel Pomerleano
Title: Symplectic cohomology in the topological limit
Abstract: Let M be a projective variety equipped with an ample normal crossings divisor D. In this talk, we describe an approach to computing the symplectic cohomology of the complement M\D which is modeled after the PSS isomorphism for compact symplectic manifolds. The method applies to so called "topological pairs", which include the Kahler pairs of Sheridan and surfaces of log-Kodaira dimension 2. This is joint work in progress with Sheel Ganatra.

Mohammed Abouzaid
Title: Family Floer cohomology and mirror symmetry
Abstract: I will begin by recalling Kontsevich-Soibelman's point of view on the SYZ program, whose goal is to associate, to every Lagrangian torus fibration, an analytic space whose points correspond to fibres equipped with appropriate local systems. Starting with an idea of Fukaya, I will explain how every Lagrangian gives rise to a (complex of) coherent sheaf on the mirror side, that this correspondence extends to a functor from the Fukaya category to the derived category of coherent sheaves, and indicate some ideas about the proof that this functor is faithful.

Andrew Harder
Title: Toric degenerations and Laurent polynomials related to Givental's Landau-Ginzburg models
Abstract: The mirror of a Fano variety X is a variety equipped with a regular function called a Landau-Ginzburg model. It is expected that there is a close relationship between degenerations of X to toric varieties and open subvarieties of the Landau-Ginzburg model of X which are biregular to an algebraic torus.
I will discuss recent results in the case where X is a toric complete intersection which support this expectation.

Matt Ballard
Title: Derived categories of moduli spaces of sheaves on rational surfaces
Abstract: We show how wall crossing in Gieseker stability yields semi-orthogonal decompositions relating derived categories in adjacent chambers. The intervening pieces come from derived categories of products of Hilbert schemes of points. The method of establishing these SOD's involves a improvement on the technology that goes under the name, windows.
This is of independent interest. If times allows, applications to wall crossing in Bridgeland stability and birational geometry will be discussed.

Paul Horja
Title: Singularities and mirror symmetry
Abstract: In this talk, a categorical point of view of some classical singularity theory concepts will be described. The proposed constructions are conjectural but are supported by homological mirror symmetry considerations as well as some examples.

Rina Anno
Title: Twisted line bundles over DG categories
Abstract: This talk is based on a joint work in progress with Timothy Logvinenko. In 2005, Huybrechts and Thomas introduced P-objects that induce autoequivalences of triangulated categories (P-twists) via a certain double cone construction. In order to generalize this to a proper notion of P-functors we need to take DG enhancements of the triangulated categories in question, and since these constructions are deeply rooted in geometry, we would like to develop DG representation theory parallels to certain geometric constructions. Line bundles are one of them.