We will discuss the dynamics of isoperiodic foliations of strata of holomorphic 1-forms (or translation surfaces). Leaves of this foliation are navigated by varying a holomorphic 1-form without changing its integrals along closed loops. Roughly speaking, this is done by moving the zeros of the 1-form relative to each other. In the generic stratum, the dynamics of this foliation are well-understood due to a close connection with homogeneous dynamics. In this talk, I will discuss a classification of leaf closures for this foliation in all strata, outside of a high-codimension subvariety, in the spirit of Ratner's orbit closure theorem.