Hurwitz trees and Berkovich curves

The Hurwitz tree is a combinatorial object attached to a cover of curves over a discrete valuation ring in mixed characteristic. It measures the ramification of the cover in characteristic zero and the properties of its reduction in positive characteristic. Berkovich spaces are analytic spaces on a complete non-Archimedean field that enjoy nice topological properties.

In this talk, I will introduce carefully these two notions, and explain how Hurwitz trees are used to solve instances of lifting problems. Then I will present a formalism to describe such trees as non-Archimedean analytic objects, in the sense of Berkovich.