RESEARCH GROUP

GEOMETRY

The preprint "Families of polytopes with rational linear precision in higher dimensions" by Isobel Davies, Eliana Duarte, Irem Portakal and Miruna-Stefana Sorea has been recently uploaded to the arXiv!

Link to the paper: arXiv:2109.08151

Abstract:
In this article we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile's theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.

In a few weeks on Wednesday 20 October, Marco will give a talk entitled 
 "Reflection length in hyperbolic Coxeter groups II" 
at the fellow's seminar of the MathCoRe here in Magdeburg.
 
See you there!

The preprint
 "Shadows in the wild -- folded galleries and their applications"
by Petra Schwer has been recently uploaded to the arXiv!

Link to the paper: arXiv:2109.02293

Abstract:
This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Folded galleries in Coxeter complexes are introduced and it is explained how they relate to retractions in Bruhat--Tits buildings and the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.