A codimension one foliation of a 3-manifold is a partition of the manifold by immersed surfaces which locally pile up nicely like parallel affine planes but which can have a very complicated global behavior. I will start by presenting a beautiful proof by Thurston that any plane field on a 3-manifold can be homotoped to one that is tangent to a foliation (such a plane field is called integrable). In order to understand the space of all foliations existing on a given manifold, the next natural question is: If two foliations have homotopic tangent plane fields, can they be connected by a path of integrable plane fields? I will explain how this question can be reduced to the following problem of one-dimensional dynamics: Can any two pairs of commuting diffeomorphisms of the interval be connected by a path of such pairs? If time permits, I will unveil some of the subtleties hidden behind this still open question. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.