Abstract: Graded sequences of ideals, or filtrations, on a local ring, play an important role in the theory of singularities and birational geometry. In this talk I will focus on the space of a class of filtrations, and show that such spaces have nice structures.

The first result is that the space of saturated filtrations with positive multiplicity is a geodesic metric space, where the metric is a local, non-archimedean analogue of the Darvas metric in complex geometry. The relation between this metric space and local volume of singularities depicts a picture similar to the question of existence and uniqueness of Kähler-Einstein metrics in Tian's properness conjecture. In another direction, such filtrations form a distributive lattice, generalizing a classical result that the ideals of a ring form a modular lattice.

As an application, we get some continuity properties of log canonical thresholds under different topologies, in the spirit of Demailly-Kollár.