Alejandro Jose Vargas De Leon (Università di Bologna)

We start by introducing tropical geometry and its "ragion d'essere": a bridge from algebraic varieties to polyhedral complexes. We focus on r-dimensional linear spaces over K, quite humble on the classical setting, they reveal different guises and shapes tropically, encoded by the combinatorics of valuated matroids. This is because tropicalization of a variety depends highly on the choice of embedding. Wherever choice is involved, so the question arises: is there an object B_r(K) that encodes all possibilities (possibly initially obtained by gluing)? The answer is yes, by joint work with Battistella, Kuehn, Kuhrs, and Ulirsch. After the break we switch to a more technical perspective, that of Berkovich geometry: every algebraic variety X over a non-archimedean field K has an associated analytification X^an that allows analytical techniques in this setting where the topology of K is totally disconnected. We argue there is a sublocus in the huge space P^an that again recovers the affine building B_r(K). From bottom to top, from top to bottom, we arrive at the same answer. How do these stories relate?