On the geometry of spaces of filtrations on local rings

We study the geometry of spaces of fitrations on a Noetherian local domain. We introduce a metric $d_1$ on the space of saturated filtrations, inspired by the Darvas metric in complex geometry, such that it is a geodesic metric space. In the toric case, using Newton-Okounkov bodies, we identify the space of saturated monomial filtrations with a subspace of $L^1_\mathrm{loc}$. We also consider several other topologies on such spaces and study the semi-continuity of the log canonical threshold function in the spirit of Koll\'ar-Demailly. Moreover, there is a natural lattice structure on the space of saturated filtrations, which is a generalization of the classical result that the ideals of a ring form a lattice.