------------------------------------------------------- Title: Extended r-spin theory and the mirror symmetry for the A-singularity Abstract: The Landau-Ginzburg mirror symmetry for the A-singularity says that the associated Saito Frobenius manifold is isomorphic, after a certain change of coordinates, called the mirror map, to the Frobenius manifold structure controlling the intersection theory on the moduli space of r-spin curves. In the talk, I will show that the mirror map, described traditionally in terms of the singularity theory, also has an elegant description in terms of the geometry of the moduli space of r-spin curves. ---------------------------------------------------------