TITLE: applications of Berkovich spaces
QUESTION [12 upvotes]: What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$

REPLY [7 votes]: Let me add a few more applications to what has already been mentioned.

Relation with Bruhat-Tits buildings (Berkovich, then Rémy/Thuillier/Werner). If $G$ is a reductive group over a non-archimedean valued field, then the Bruhat-Tits building $\mathscr{B}(G)$ of $G$ embeds into the analytification $G^{an}$ of $G$. If you choose a parabolic subgroup $P$, you have a map to $(G/P)^{an}$, which is the analytification of a proper variety, hence a compact space. This can help you compactify the building, describe the strata of the compactification, etc.
Complex dynamics. Favre and Jonsson used Berkovich spaces (actually some instance of it that they call "valuative tree") in order to study the dynamics of a polynomial endomorphism of $\mathbb{C}^2$ near a superattracting fixed point or at infinity.
Resolution of singularities (Thuillier). Let $X$ be an algebraic variety over a perfect field $k$ with an isolated singular point $x$. Let $f \colon Y \to X$ be a resolution of it such that $f^{-1}(x)$ is a normal crossing divisor $E$. Then the homotopy type of the incidence complex of $E$ is independent of the choice of the resolution. (Remark here that $k$ is any perfect field and that the Berkovich spaces that come up in the proof are over the field $k$ endowed with the trivial valuation.)
$p$-adic differential equations. André used Berkovich spaces in order to prove a conjecture of Dwork on the logarithmic growth of the solutions of $p$-adic differential equations (he actually needs Berkovich spaces only when the base field is not locally compact). In a different direction, Baldassarri (then Pulita and myself, and also Kedlaya) took up the study of $p$-adic differential equations on Berkovich curves. For instance, in my work with Pulita, we manage to give conditions for the finiteness of the de Rham cohomology of a curve with coefficients in some module with a connection. (Here, my point is that even if you can easily state the results in any theory you like, rigid geometry for instance, you will have a hard time proving them without Berkovich spaces.)