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.

In this talk, we show how certain everywhere-regular real rational function solutions of the KP1 equation ("multi-lumps'') can be constructed via the polynomial analogs of theta-functions from singular rational curves with cusps, as studied in a recent paper of Daniele Agostini, Türkü Özlüm Çelik, and the presenter. The method we use can be understood either directly, or as a degeneration of the well-understood soliton solutions from nodal singular curves. Our main explicit example is a three-lumpsolution constructed via the polynomial analog of the theta-function from a rational curve with two cuspidal singular points, each  with semigroup <2,5>. (In the theory of curve singularities, these are known as A_4 double points.) We show
that there are similar six-lump solutions from a rational curve with two cusps, each with semigroup <2,7> (A_6 double points). We conjecture that these ideas will generalize to give similar M-lump solutions with M = N(N+1)/2 for N > 2 starting from rational curves with two singular points with semigroup <2,2N+1> (A_{2N} double points).  Finally we present a five-lump solution constructed from a rational curve with two cusps, each with semigroup <3,4>.

Online Talk: please send an email to Daniele Agostini for the link.

(exceptionally on Tuesday at 10:00 Takashi Ichikawa) (Saga): Tropical curves and KP solitons

Online Talk: please send an email to Daniele Agostini  for the link.

Tropicalization is a process that associates to an algebro-geometric object a piecewise linear polyhedral shadow that captures its essential combinatorial structure. In this talk, I will give an overview of the numerous ways on how to extract tropical information from a matrix over a non-Archimedean field. Each perspective will give rise to inherently quite different phenomena. Central instances of this rich panorama include the tropical geometry of vector bundles, logarithmic concavity results for valuated (bi-)matroids (using techniques from combinatorial Hodge theory), and the geometry of affine buildings.

This talk draws from joint work with Andreas Gross and Dmitry Zakharov; Andreas Gross, Inder Kaur, and Annette Werner; Felix Röhrle; Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis; Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond
Coles.