Primtive form of singularity and mirror symmetry

In 1980s, K. Saito introduced the primitive forms in his study of periods as to generalize the elliptic period integral theory to higher dimensions. An important consequence was that there exists a flat structure on the space of universal unfolding of a singularity. Later after Witten proposed his famous conjecture relating the intersection theory on moduli space to the integrable hierarchy, this flat structure was re-discovered by Dubrovin, and nowadays it is referred as a Frobenius mafniold structure. Frobenius mafniolds provide a convenient tool for formulating the mirror symmetry, which was a special type of duality among string theories orginally studied by physicits. In this talk, I will explain the related constructions, and show by examples how the study of primitive forms gives prediction of geometric quantities such as Gromov-Witten invariants.