Abstract: Introduced by Chi Li, the local volume of a klt singularity plays an important role in the study of K-stability of Fano varieties, their moduli spaces and boundedness of singularities. The notion is closely related to graded sequences of ideals, or filtrations, on the singularity by the work of Yuchen Liu. In this talk I will introduce a metric on the space of filtrations on a general local ring and discuss some basic properties. On a klt singulatiry, the geometry of spaces of filtrations under this metric is related to the stable degeneration theorem, which is the fundamental problem of local volumes. Such relations depict a picture similar to the question of existence and uniqueness of Kähler-Einstein metrics on a Fano variety. This talk is based on joint work of Harold Blum and Yuchen Liu and some ongoing work.