We propose to discuss (a selection of) the following items:
An introduction to the basic ideas of Berkovich spaces and why they are a powerful tool in non-Archimedean geometry and in particular in the topic of p-adic differential equations.
Discussing the importance of radii of convergence of solutions to p-adic differential equations and their implications.
Presenting some local and global decomposition theorems that apply to p-adic differential equations. This is similar to the decomposition theorem of a differential equation over C((x)) by the slopes of its Newton Polygon. In fact, this is a particular case of a more general theorem.
An overview of the finite-dimensionality results for the de Rham cohomology associated with these equations. The fact that contrary to the algebraic case, they do not always have a finite dimensional de Rham cohomology and the information about the cohomology is encoded in the radii.ns.