Name: Santiago Encinas Title: Algorithmic Equiresolution Abstract: Given an algorithm of resolution of singularities satisfying certain conditions, natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are proposed. These conditions are equivalent. Something similar is done for families of sheaves of ideals, here the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_{j}$, such that the induced family on each part $T_{j}$ is equisolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equiresolvable families.