Algebraic varietes over fields equipped with non-Archimedean valuations can be studied from the perspective of analytification (in the sense of Berkovich) and tropicalization. The purpose of this course is to provide a gentle introduction to Berkovich’s theory through the lens of tropical geometry. Special emphasis will be given on computationally effective tools to examine these complicated spaces of valuations through their (easier) tropical shadows.

A central theme of the course will be the notion of faithful tropicalization: when the fixed tropical variety best reflects the topology of the analytic space. The guiding questions will be: is a tropicalization faithful, and if not, can we fix it? We will illustrate the approach to tackle these questions via two examples with rich combinatorics: (hyperelliptic) curves and Grassmannians.

The first two talks will be devoted to a hands-on introduction to Berkovich spaces and its connection to tropicalization, following the work of Payne. In the last two talks we will focus on the concrete examples mentioned earlier and provide some open questions.