Multiplicities and Volumes of Filtrations

In this article, we survey some aspects of the theory of multiplicities of mR-primary ideals in a local ring (R,mR) and the extension of this theory to multiplicities of graded families of mR-primary ideals. We first discuss the existence of multiplicities as a limit. Then, we focus on a theorem of Rees, characterizing when two mR-primary ideals I subset of J have the same multiplicity, and discuss extensions of this theorem to filtrations of mR-primary ideals. In the final sections, we give outlines of the proof of existence of the multiplicity of a graded family of mR-primary ideals as a limit, with mild conditions on R, and the proof of the extension of Rees' theorem to divisorial filtrations.