Oberseminar Kombinatorische Algebraische Geometrie

Kevin Kühn, Arne Kuhrs (Frankfurt)

10:00 - 11:00: Kevin Kühn, Buildings, valuated matroids, and tropical linear spaces

We first recall some basic notions about tropical linear spaces and valuated matroids. Then, we take a look at Affine Bruhat--Tits buildings, which 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 that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings of P^n in P^m 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.

11:00 - 12:00: Arne Kuhrs, The Signed Goldman-Iwahori space and real tropical linear spaces

In the second talk we describe a signed analogue of the story we outlined in the first talk. We define for a real closed field K, e.g. the Puiseux series over the field of real numbers, a signed analogue of the Goldman--Iwahori space consisting of signed seminorms on a finite-dimensional vector space over K. This space can be seen as a linear analogue of the real analytification of projective space over K which was introduced by Jell, Scheiderer, and Yu. We recall their notion of a real analyitfication of a variety over K and real tropicalizations. Then, we show that there is a natural real tropicalization map from the signed Goldman--Iwahori space to a real tropicalized linear space and that this space is the limit of all real tropicalized linear subspaces of rank n (as the linear embedding and the dimension of the ambient projective space vary). While in the first talk tropical linear spaces where combinatorially described by valuated matroids, the combinatorics of the real tropical linear spaces showing up in this talk are goverend by oriented (valuated) matroids. We recall these notions from a perspective of matroids over hyperfields. Finally, we observe that the signed Goldman-Iwahori space is the real tropical linear space associated to the universal realizable oriented valuated matroid.

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. They play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We introduce these objects and discuss their main properties, including dimension, degree, singular locus and defining equations, as well as some computational experiments.