Séminaire de géométrie - Salle de conférences du CMLS - 14h
12 nov. 2025
14h – Arne Kuhrs (Université de Paderborn, Allemagne)
Titre: Buildings, valuated matroids, and tropical linear spaces
Abstract: Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups.
The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne’s result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid, extending a result of Dress and Terhalle. This is joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch and Alejandro Vargas.
The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne’s result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid, extending a result of Dress and Terhalle. This is joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch and Alejandro Vargas.
Les séances suivantes du séminaire seront le 26 novembre (orateur : Souvik Goswami) et le 12 décembre (Pietro Piccione).