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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 $P$, ...


Jan Kiwi (Duke Journal) used Berkovich spaces to give the first proof of my conjecture that any sequence of quadratic rational maps which divergence in moduli space (of Mobius conjugacy classes) has at most two "rescaling limits".


I would first recommend the paper of Antoine Ducros (Espaces analytiques $p$-adiques au sens de Berkovich, Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with applications. Here is a list of applications which I find striking, starting from those mentioned by Ducros's survey. Étale cohomology. Berkovich developed a good theory ...


I have found the following: Walter Gubler applied it to the Bogomolov conjecture. Jérôme Poineau applied it to the inverse Galois problem: http://arxiv.org/abs/0809.2880


Here is one of the reasons (maybe the only reason) for the name. The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\operatorname{Hom}_H(H,\tau^{-n}H)$ where $\tau$ is the Auslander-Reiten translate. As an $H$-module it is isomorphic to the direct sum of all the ...

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