This lecture is a survey of the minimum dilatation problem for pseudo-Anosov mapping classes. Pseudo-Anosov mapping classes are isotopy classes of self-homeomorphisms of an oriented surface with a well-mixing property, and the dilatation is a measure of how fast the mixing occurs. Given a fixed surface, an open problem is to find mapping classes that have well-mixing, but small rate of mixing. In the talk, we will describe a way studying minimization problem using the geometry of 3-manifolds to interpolate between mapping classes on different surfaces. We end with a conjecture about the general structure of mapping classes with asymptotically small dilatations relative to the Euler characteristic of the surface.