Noémie Combe — How many Frobenius manifolds are there?

In this talk an overview of my recent results is presented. In a joint work with Yu. Manin (2020) we discovered that an object central to information geometry: statistical manifolds (related to exponential families) have an F-manifold structure. This algebraic structure is a more general version of Dubrovin’s Frobenius manifolds, serving in the axiomatisation of Topological Field Theory. This unexpected result enters the scene of tetralogy of Frobenius manifolds involving some of the deepest domains of the last past decades: quantum cohomology (related to Gromov—Witten invariants), Saito manifold (unfolding of singularities) and solutions to Maurer—Cartan (appearing in Barannikov—Kontsevitch’s theory).

These statistical manifolds turn out to have incredibly rich algebraic and geometrical properties. Moreover, it can be shown that classes of Frobenius manifolds have deep connections. Recently I proved the existence of statistical Gromov--Witten invariants for statistical manifolds.