Kwokwai Chan: A local-to-global description of mirror symmetry

Abstract: Mathematically, mirror symmetry is a duality between the symplectic geometry on one Calabi-Yau manifold and the complex differential geometry on another Calabi-Yau manifold, called the mirror. In this talk, I will briefly review the mirror symmetry story and try to explain an emerging picture of a local-to-global description of mirror symmetry which originated from ideas of Fukaya, Seidel, Gross-Siebert and Kontsevich.

Jeongseok Oh: The quantum Lefschetz principle

Abstract: “Quantum Lefschetz” is a pretentious name for understanding how moduli spaces -- and their virtual cycles and associated invariants -- change when we apply certain constraints. (The original application is to genus 0 curves in P^4 when we impose the constraint that they lie in the quintic 3-fold.) When it doesn’t work there are fixes (like the p-fields of Guffin-Sharpe-Witten/ Chang-Li) for special cases associated with curve-counting. We will describe joint work with Richard Thomas developing a general theory.

Hanwool Bae: Calabi-Yau structures on Rabinowitz Fukaya categories. (second talk of a lecture series with Wonbo Jeong)

Abstract: Rabinowitz Floer homology, introduced by Cieliebak-Frauenfelder, is a Floer theoretic invariant of the contact boundary of a symplectic manifold, whose non-triviality implies the non-displaceability of the contact boundary. Furthermore, it has been shown by Cieliebak-Oancea and Cieliebak-Hingston-Oancea that it carries a Frobenius algebra structure. This can be said to be a Floer theoretic enhancement of the classical Poincare duality of the boundary. Recently, GanatraGao-Venkatesh introduced Rabinowitz Fukaya category as a categorification of Rabinowitz Floer homology. In this talk, I will first review the construction of Rabinowitz Fukaya category. Then I will explain that under a degreewise finite dimensionality assumption, the Rabinowitz Fukaya category of a Liouville domain of dimension 2n is (n-1)-CalabiYau. Furthermore, I will explain that this is induced by a relative graded proper Calabi-Yau structure on the natural functor from the wrapped Fukaya category to the Rabinowitz Fukaya category. This talk is based on joint work with Wonbo Jeong and Jongmyeong Kim.

Sam Bardwell-Evans: Introduction to Kuranishi structures II (second talk of a lecture series with Taesu Kim)

Abstract: TBA

Ki Fung Chan: Mirror symmetry of a non-abelian group action

Abstract: It's a joint work with Conan Leung. Teleman conjectured a Hamiltonian T^n action is mirror to a holomorphic fibration to the complexified dual torus. However, the naive generalization of this conjecture to non-abelian group actions is not true. The correct mirror of a Hamiltonian G-action should be a Lagrangian in the BFM space. In this talk, we will provide motivations and definitions of the nonabelian Teleman conjecture, which implies a nontrivial consequence on Lagrangian Floer homology. I will give a sketch of proof of the consequence at the end of the talk, if time permits.

Hyunbin Kim: Morse Superpotential of Log Calabi-Yau Surfaces and Mirror Symmetry

Abstract: In mirror symmetry, the critical points of a Landau-Ginzburg mirror superpotentials play a crucial role. After a brief review of the construction of superpotentials for log Calabi-Yau surfaces, we will explore how tropical geometric methods can effectively identify the precise locations of critical points and determine their degeneracy. Using these tools, we will demonstrate the proof of closed-string mirror symmetry for log Calabi-Yau surfaces.

Sungho Kim: Rabinowitz Floer homology for prequantization bundles

Abstract: TBA

Taesu Kim: Introduction to Kuranishi structures I (first talk of a lecture series with Sam Bardwell-Evans)

Abstract: TBA

Wonbo Jeong: Cluster categories from Fukaya categories (first talk of a lecture series with Hanwool Bae)

Abstract: Let M be a plumbing of cotangent bundles of n-sphere (n>2) along a tree T. We show that the derived wrapped Fukaya category W, the derived compact Fukaya category F and the cocore disks of M form a Calabi-Yau triple. This implies that the quotient category W/F becomes the Amiot-Guo-Keller cluster category. When the tree T is of Dynkin type, we compute the morphism spaces between cocores in W/F as a path algebra of a specific quiver. This is a joint work with Hanwool Bae and Jongmyeong Kim.

Hongtaek Jung: Hitchin components and their symplectic structure

Abstract: Hitchin components are natural higher rank generalizations of the Teichmuller space. After a brief review on Hitchin components, we mainly focus on the Atiyah-Bott-Goldman symplectic structure on Hitchin components. In particular, we show that the PSL(3,R)-Hitchin component admits a global Darboux coordinate system. If time permits, we discuss a recent result on the infinitude of the symplectic volume of higher rank moduli spaces.

Chin Hang Eddie Lam: Equivariant families of mirror symmetry

Abstract: In this talk, we will discuss some formulation and examples of 2d mirror transform in the equivariant fibration settings. This could be suitably interpreted as 3d mirror symmetric statement for pairs of Lagrangians in hypertoric varieties. As a result, we propose another way to construct 3d mirror of hypertoric varieties. The talk is based on the forthcoming work of Ki Fung Chan and Conan Leung.

YuTung Yau: Quantization of Kähler manifolds via branes

Abstract: In their physical proposal for quantization, Gukov-Witten suggested that, given a symplectic manifold M with a complexification X, the A-model morphism spaces Hom(Bcc, Bcc) and Hom(B, Bcc) should recover holomorphic deformation quantization of X and geometric quantization of M respectively. Here, Bcc is a canonical coisotropic A-brane on X and B is a Lagrangian A-brane supported on M.

Assuming M is spin and Kähler with a prequantum line bundle L, Chan-Leung-Li constructed a subsheaf Oqu of smooth functions on M with a non-formal star product and a left Oqu-module structure on the sheaf of holomorphic sections of L twisted by a square root of the canonical bundle of M.

In this talk, I will discuss the relation between (holomorphic) deformation quantizations of M and X so as to verify that Chan-Leung-Li's work provides a mathematical realization of the action of Hom(Bcc, Bcc) on Hom(B, Bcc). I will also explain my recent work - the construction of a Oqu-O-qu bimodule structure on the sheaf of smooth sections of L2 realizing the actions of Hom(Bcc, Bcc) and Hom(Bcc, B-cc) on Hom(B-cc, Bcc) - which is related to the analytic geometric Langlands program.