Consider the set of valuations v: C[[x,y]] -> R \cup \{\infty\}, normalized so that min(v(x), v(y)) = 1. We'll show that this set naturally has the structure of a metric tree. This insight allows us to study the singularities of ideals, graded systems of ideals, and plurisubharmonic functions in a unified way.

This talk will serve as an introduction to the valuative tree, following Favre-Jonsson. We'll focus on the basic structure of the tree and numerical invariants of valuations with examples.