(Quasi-)conformal methods in some two-dimensional free boundary problems

In this talk I will explain how to obtain precise informations on the structure of the free-boundary to $2$-dimensional solutions of the one and two phase problems at so-called branching points using the theory of (quasi-)conformal maps. The talk is based on joint work with G. De Philippis (Courant) and B. Velichkov (Pisa).