Mirror symmetry for Looijenga interiors and beyond

This is a hybrid conference that will take place on July 25-29, 2022. The preceeding week, on July 18-22, there will also be a summer school, mainly targeted towards local participants but open to everyone. The physical location for both events was going to be IMBM - please click on the link to see what went wrong. Now the summer school will be held in Feza Gursey Institute located in the Kandilli campus of Bogazici University and the conference in Bogazici University Math Department located in the Guney campus of Bogazici University.

Here is the registration form for students. Students will be housed in the dormitory of Feza Gursey Institute. We are getting in touch with people who request lodging using the form but there is limited space.

Please register before 15th of July here to receive the Zoom link.

Below is the schedule for the conference using Istanbul time. Talks are 1 hour each. Click for abstracts.

Monday (25.07)

13:30 - Bernd Siebert - Talk 1: Punctured Invariants / Talk 2: Canonical wall structure

These talks will be followed by Mark Gross' talk.

15:00 - Philip Engel - Compact moduli of K3 surfaces

Abstract: The moduli space of K3 surfaces of genus g is the quotient of a symmetric space of complex dimension 19 by the action of an arithmetic group. In this capacity, it has a natural infinite class of so-called "toroidal" compactifications. Is any one of these toroidal compactifications distinguished, in the sense that it parameterizes some generalized or "stable" K3 surfaces? I will describe joint work with V. Alexeev giving an affirmative answer to this question. The key combinatorial object encoding the stable K3 surfaces over the boundary is a family of "polarized integral-affine spheres" over a fan in the positive cone of R^{1,18}.

17:00 - Tony Yue Yu - TBA

TBA

Tuesday (26.07)

11:30 - Abigail Ward - TBA

TBA

13:30 - Bernd Siebert - Talk 1: Punctured Invariants / Talk 2: Canonical wall structure

These talks will be followed by Mark Gross' talk.

15:00 - Andrew Hanlon - Monomial admissibility for Looijenga pairs

Abstract: We will discuss an approach to homological mirror symmetry for Looijenga pairs that is closely tied to the tropical perspective. Namely, we will see how to construct a monomially admissible Fukaya category on the mirror to a Looijenga pair and tropical Lagrangian sections in this category in bijection with line bundles. Moreover, generators of the Lagrangian Floer cohomology of certain sections can be seen to correspond to integral points of polytopes that encode theta functions on the mirror. This is joint work with Abigail Ward.

16:30 - Ian Le - TBA

TBA

Wednesday (27.07)

13:30 - Mark Gross - Intrinsic mirror symmetry

This talk will be a sequel to Bernd Siebert's talks.

15:00 - Yoel Groman - TBA

TBA

16:30 - Yu-Shen Lin - Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry

Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. I will also explain various applications of the SYZ fibrations. The talk is based on joint works with T. Collins, A. Jacob, S.-C. Lau and T.-J. Lee.

Thursday (28.07)

13:30 - Hulya Arguz - The higher dimensional tropical vertex

Abstract: Given a log Calabi--Yau pair (X,D), consisting of a smooth projective variety X together with a normal crossings anti-canonical divisor D, we first provide a combinatorial algorithm for solving the enumerative problem of computing rational stable maps to (X,D) touching D at a single point. We then explain how to use the solution to write explicit equations for mirrors to such pairs at arbitrary dimensions. Part of this is joint work with Mark Gross.

15:00 - Joe Brendel - TBA

TBA

16:30 - Travis Mandel - TBA

TBA

Friday (29.07)

13:30 - Pierrick Bousseau - Fock--Goncharov dual cluster varieties and Gross--Siebert mirrors

Abstract: Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock--Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi--Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross--Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross--Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock-Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram defined by Gross-Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz.

15:00 - Dan Pomerleano - The quantum connection, Fourier-Laplace transform, and families of A_\infty categories

Abstract: A well-known conjecture states that the (one-parameter) quantum connection on a Fano manifold M has a singularity of unramified exponential type at q=\infty. This conjecture is a pre-requisite to being able to state the Gamma conjectures or Dubrovin's conjecture and is also part of the conjecture that QH^*(M) carries a non-commutative Hodge structure.
Suppose M carries a smooth ample anti-canonical divisor D and let U=M\D be its complement. I will explain how to reduce the above conjecture on the quantum connection to an S^1-equivariant version of the conjecture of Borman-Sheridan-Varolgunes that "QH^*(M) is a deformation of SH^*(U)." Part of the argument is purely categorical and uses the recent result of Petrov, Vaintrob, and Vologodsky that the Gauss-Manin connection on the periodic cyclic homology of a smooth family of categories over C[t] has regular singularities. This is joint work in progress with Paul Seidel.

16:30 - Hang Yuan - Family Floer mirror space for local SYZ singularities.

Abstract: We aim to give an explicit example for a Floer-theoretic T-duality. More importantly, the dual analytic fibration, including the singular fibers, will be explicitly presented in my talk. Its construction modifies a non-archimedean singular model of Kontsevich-Soibelman and also follows the work of Gross-Hacking-Keel. Surprisingly, we will see the dual singular fiber can be larger than the Maurer-Cartan set. We will start with a review of the family Floer mirror construction and briefly explain why the ideas and technologies in my thesis are necessary. Also, we will explain the motivations for our explicit dual analytic fibration and propose a concrete SYZ mirror statement. If time allowed, we will describe a version of SYZ converse and make some computations for a folklore conjecture for the critical values of the mirror Landau-Ginzburg superpotential.

The schedule for the summer school is as follows in Istanbul time. Lectures are 1 hour each.

Day 1 (18.07)

- 11:30 - Introduction to mirror symmetry (Mark Gross)
- 13:30 - Log CY varieties (Ozgur Kisisel)
- 15:00 - Weinstein manifolds and their invariants (Dogancan Karabas)
- Discussion session

Day 2 (19.07)

- 11:30 - Introduction to microsheaves (Vivek Shende)
- 13:30 - Symplectic structures and Lagrangian fibrations on Looijenga interiors (Umut Varolgunes)
- 15:00 - Classification of Looijenga interiors I (Travis Mandel)
- 16:30 - Classification of Looijenga interiors II (Travis Mandel)

Day 3 (20.07)

- 11:30 - Looijenga conjecture (Philip Engel)
- 13:30 - The tropical vertex (Hulya Arguz)
- 15:00 - Hacking-Keating theorem (Umut Varolgunes)
- Discussion session

Day 4 (21.07)

- 11:30 - Shende-Treumann-Williams construction (Dogancan Karabas)
- 13:30 - Cluster varieties and Fock-Goncharov mirror symmetry (Pierrick Bousseau)
- 15:00 - Examples (Ozgur Kisisel)
- Discussion session

Day 5 (22.07)

- 11:30 - Mirror symmetry and microsheaves (Vivek Shende)
- 13:30 - Floer theory based mirror constructions (Yoel Groman)
- 15:00 - GHKS construction (Hulya Arguz)
- Discussion session

Organizational committee : Dogancan Karabas, Ozgur Kisisel, Ferit Ozturk, Umut Varolgunes

The event will be partially supported by TUBITAK grant number 121C034.