Mirror symmetry and relative Gromov--Witten theory
In enumerative mirror symmetry, a mirror theorem can be stated as the relation between generating functions ($J$-functions) of Gromov--Witten invariants and period integrals (or the $I$-functions) of the mirror. Given a smooth projective variety $X$ and a smooth divisor $D \subset X$, relative Gromov--Witten theory is an enumerative theory of counting curves in $X$ with tangency conditions along $D$. I will explain how to obtain a mirror theorem for the pair $(X,D)$, i.e. a relative mirror theorem, via orbifold Gromov--Witten theory. This is based on joint work with Honglu Fan and Hsian-Hua Tseng.