# Leipzig-Magdeburg seminar day

## Abstracts for the talks

**Xiangying Chen ***OvGU Magdeburg***The geometry of conditional independence structures and their Coxeter friends**

A conditional independence structure consists of triples (ij|K) which can be interpreted in many ways, for example as conditional independence between random vectors, as points with certain distances in an inner product space, as pairs not to be compared in a rank test, or as separation in graphs or topological spaces. In this talk I will discuss the geometry of these structures and their analogies in other Coxeter types.

**Volker Kaibel ***OVGU Magdeburg***Taming Polytopes by Lifting**

We cover some aspects of the theory of extended formulations, i.e., representations of polytopes that appear wild in one or the other sense as projections of higher-dimensional ones that are much tamer. Our selection of topics is biased towards parts of the work done in our own research group, where the lecture will close with a recent result related to the Hirsch-conjecture on the diameter of polytopes obtained jointly with Kirill Kukharenko.

**Julia Lindberg ***Max Planck Institute for Mathematics in the Sciences***Estimating Gaussian mixtures using sparse polynomial moment systems**

The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. Using these results, we present a homotopy method to perform parameter recovery, and therefore density estimation, for high dimensional Gaussian mixture models. The number of paths tracked in our method scales linearly in the dimension.

**Alessandro Neri ***Max Planck Institute for Mathematics in the Sciences***Linear Sperner families and strong blocking sets in coding theory**

Minimal linear codes were first introduced by Cohen and Lempel over the binary field under the name of linear intersecting codes. They later gained interest due to their application to secret sharing schemes proposed by Massey. Concretely, minimal linear codes are subspaces of vectors over a finite field such that the set of nonempty supports forms an antichain with respect to the set inclusion. In other words, they provide the vector space analogue of Sperner families. Recently, it has been shown that minimal linear codes in are in one-to-one correspondence with strong blocking sets, which are special sets of points in a projective space, such that their intersection with each hyperplane generates the hyperplane itself.

In this talk we will revise the basic theory connecting these three objects with combinatorial, algebraic and geometric flavours: linear Sperner families, minimal linear codes and strong blocking sets. We will show first properties and bounds on their cardinality and parameters. Later, we will go through some classical constructions mainly obtained as union of lines. If time allows, we will then explore new ideas for constructing strong blocking sets over any field, obtained as union of rational normal curves.

## Date and Location

**August 09, 2022**

## Scientific Organizers

**Tobias Boege**

MPI for Mathematics in the Sciences

## Administrative Contact

**Saskia Gutzschebauch**

MPI for Mathematics in the Sciences

Contact by Email