Fenglong You (Nottingham) - Intrinsic mirror symmetry and theta functions via root stacks

Mirror symmetry predicts that the symplectic geometry of a manifold is equivalent to the complex geometry of the mirror manifold. In particular, enumerative mirror symmetry relates quantum cohomology (Gromov-Witten invariants) of a manifold with period integrals of the mirror. A fundamental question in mirror symmetry is to construct mirrors. I will explain Gross-Siebert's intrinsic mirror symmetry which gives a mirror construction using relative quantum cohomology. More precisely, one considers a maximally unipotent monodromy degeneration of a Calabi—Yau manifold and take proj of the degree zero part of a relative quantum cohomology ring associated with the degeneration. Theta functions form a canonical basis of the degree zero part of the relative quantum cohomology ring. Theta functions can be defined in terms of punctured Gromov-Witten invariants of the broken line type. I will explain a variant of intrinsic mirror symmetry using orbifold Gromov-Witten invariants of root stacks, theta functions for general snc pairs and possible applications to questions in mirror symmetry.