abstract: The Allen-Cahn equation is a semilinear elliptic PDE modelling phase transitions. It depends on a small parameter: as this goes to zero its solutions accumulate near minimal surfaces, in a certain weak sense. In this talk I want to discuss some of the mechanisms underlying this phenomenon. The focus will be on variational aspects of the problem, and the Morse index in particular. The eventual aim is to present some results concerning low-energy solutions in the round three-sphere.