Alessandro Chiodo: From mirror symmetry to spin curves

In this talk we will move from two examples of dual varieties in the direction of mirror symmetry. More precisely we will present pairs (X,Y) of Calabi-Yau three-folds related by the symmetry h^{p,q}(X) =h^{3-p,q}(Y). We will cast these examples into a general statement, conjectured by Berglund and Hübsch and proven by Ruan and myself. The proof allows to present a new approach to the geometry of Calabi Yau varieties and to their higher genus invariants from the point of view of Landau-Ginzburg models and spin curves.