Title: Mirror symmetry, geometry, and topological recursion
Abstract: Mirror symmetry was originally discovered by physicists. It reflects the symmetry in string theory, where two different versions of string theory called type IIA and type IIB string theory give rise to the same physics. Mathematicians became interested in this relationship around 1990 when Philip Candelas,Xenia de la Ossa, Paul Green, and Linda Parks showed that it could be used as a tool to count rational curves in the quintic 3-fold. In this talk, I will first introduce the most classical mathematical formulation of mirror symmetry which is about a mirror pair of Calabi-Yau 3-folds. The A-model is given by genus 0 Gromov-Witten invariants of the Calabi-Yau 3-fold, which can be constructed by either algebraic geometry or symplectic geometry. The B-model is given by the variation of Hodge structures which is related to complex geometry. Then I will move on to the genus 0 mirror symmetry for toric manifolds/orbifolds. In this case, we consider the Landau-Ginzburg B-model which is constructed by singularity theory. Then I will talk about my work (joint with Bohan Fang and Melissa Liu) on the proof of the Remodeling Conjecture, which can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3-orbifolds. It relates the higher genus open-closed Gromov-Witten potential of a toric Calabi-Yau 3-orbifold to the higher genus B-model potential which is obtained by applying the Eynard-Orantin topological recursion to the mirror curve. In this case, both algebraic geometry and symplectic geometry are used on A-model. If time permits, I will discuss some nice applications of the Remodeling Conjecture such as the holomorphic anomaly equation, modularity, and the crepant resolution conjecture.