Abstract: “Allen-Cahn minimal hypersurfaces” are obtained as limits of nodal sets of solutions to the Allen-Cahn equation. Understanding the local picture of this convergence is a fundamental problem. For instance, can we avoid the situation of a nodal set looking like a multigraph over the limit hypersurface? Examples of this phenomenon, known as “multiplicity” or “interface foliation”, are only known when the limit hypersurface is unstable. Together with A. Neves and F. Marques we proved that generically (and in all dimensions) stable Allen-Cahn minimal hypersurfaces can only occur with multiplicity one. We will discuss this and other topics.