Discrete Geometry Nexus: A Berlin Workshop

FU Berlin, November 28, 2024

Program

• 12:00 - 14:00 Lunch (together at Mensa or A3)


• 14:15 - 14:30 Introduction
• 14:30 - 14:55 Talk: Tom Baumbach
• 15:00 - 15:25 Talk: Sofía Garzón Mora


• 15:30 - 16:00 Break (with Tea / Coffee / Cake)


• 16:00 - 16:25 Talk: George Balla
• 16:30 - 16:55 Talk: Kamillo Ferry
• 17:00 - 17:10 Talk: Nikola Sadovek


• 17:30 - 23:55 Finish (self-organized drinks and dinner)

Speakers and Abstracts

• George Balla: On regular subdivisions and bounds for initial ideals
  
  Abstract: We associate a point configuration A(I) to any homogeneous ideal I. We obtain upper and lower bounds on each initial ideal of I in terms of regular subdivisions of A(I). We investigate when these bounds are exact, and it turns out that the set of all points for which this is true, is the support of a subfan of the secondary fan of A(I). This talk is based on joint work with Dan Corey, Igor Makhlin and Victoria Schleis.
  
  
• Tom Baumbach: Subspace Concentration Conditions and Cone-Volumes
  
  Abstract: We present a discrete approach to the logarithmic Minkowski problem by defining the subspace concentration polytope, which is derived from the subspace concentration condition established by Böröczky, Lutwak, Yang, and Zhang. The close relation between the subspace concentration polytope and the basis matroid polytope is explained, allowing us to apply matroid theory. Additionally, we define the cone-volume set of a given set of outer unit normal vectors that corresponds to a polytope. The equality case between the subspace concentration polytope and the cone-volume set is characterized, and we show that the closure of the latter is a semialgebraic set. Lastly, the convex hull of the cone-volume set is fully characterized in the planar case. This talk is based on joint work with Martin Henk.
  
  
• Kamillo Ferry: Waving the flag for max-linear Bayesian networks
  
  Abstract: Max-linear Bayesian networks are a type of structural equation model in statistics described by a weighted directed acyclic graph. The possible observations coming from such a MLBN all lie inside a polytrope, which is a tropical polytope that is also clasically convex. On the other hand, a tropical polytope can be seen as the image of tropical multiplication by a matrix. The weight matrix of a network can be assumed to be a lower-triangular square matrix. This allows to study MLBNs and their polytropes from the lens of flags of tropicalized linear spaces. We will look into the connection between tropical polytopes and tropical linear spaces, see the valuated matroids coming from the natural parametrization of linear spaces and how this precisely captures the relevant information of a MLBN.
  
  
• Sofía Garzón Mora: On the spectra of Fine polyhedral adjoints
  
  Abstract: Originally introduced by Fine and Reid, the Fine interior of a lattice polytope got recently into the focus of research. Based on the Fine interior, we study a modification of so-called adjoint polytopes and define the Fine adjoint polytope. We obtain a Fine Polyhedral Adjunction Theory that is better behaved than its original analogue where many existing results in Polyhedral Adjunction Theory carry over. In this talk, we will focus mainly on the finiteness of the Fine Q-codegree spectrum. Namely, we introduce the Q-codegree as an invariant which first arose in the context of toric geometry, and discuss why it can take only finitely many values for polytopes under certain conditions. We show the recent results of our classification and how to compute these values explicitly via solving a mixed LP. This is joint work with Christian Haase.
  
  
• Nikola Sadovek: A necessary and sufficient condition for k-transversals
  
  Abstract: In this talk, we will present a necessary and sufficient condition for a family of convex sets in a d-dimensional euclidean space to admit a k-dimensional affine transversal, for any non-negative k < d. This result is a common generalization of Helly's theorem (k=0) and the Goodman-Pollack-Wenger theorem (k=d-1). The proof is topological and relies on a Borsuk-Ulam-type theorem on Stiefel manifolds. This talk is based on joint work with Daniel McGinnis.
  
  

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