In this talk we will investigate the following question : "Given a closed orientable riemmanian surface (possibly with singularities), how much can two curves of a given lenght intersect ?". It appears that this question is very hard to answer in general, and we will focus on the case of translation surfaces, which are examples of zero curvature surfaces with finitely many singularities. More particularly, we will try to explain what happens in the example of the double pentagon translation surface.

This is joint work with E. Lanneau and D. Massart.