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## WS 2023/24

### 17.10.2023 um 14:15 Uhr in 93/E44

#### Andrea Rosana (SISSA)

Random symmetric tensors close to rank-one

**FB 6 Mathematik/Informatik/Physik**

**Institut für Mathematik**

Random symmetric tensors close to rank-one

Angereicherte Ordnungspolytope von Zick-Zack Posets

Die angereicherten (engl. enriched) Ordnungspolytope sind eine erweiterte Klasse der Ordnungspolytope. Sie stützen sich auf die Theorie der angereicherten P-Partitionen. Für die Gitterpunkte dieser Polytope lässt sich eine Poset-Struktur definieren, mit welcher eine unimodulare Triangulierung beschrieben werden kann.

Insbesondere werden angereicherte Ordnungspolytope zu Zick-Zack Posets betrachtet. Hierbei lassen sich strukturelle Eigenschaften des Posets der Gitterpunkte feststellen, sowie eine rekursive Konstruktion der Gitterpunkte.

We present some recent results obtained by

applying combinatorial learning techniques.

Cosmological polytopes appear in the study of the wavefunction of the universe and therefore are of high interest in physics. There haven been several articles in the last years where cosmological polytopes have been studied. In a recent article, Juhnke, Solus and Venturello showed that cosmological polytopes admit a regular unimodular triangulation. We use these results to investigate the h*-polynomials of cosmological polytopes. For example, we can compute the h*-polynomials of cosmological polytopes of trees and cycles, where both can have multiple edges.

tensors at a particular level form the so-called universal signature variety. After introducing these objects, we will start inquiring on the rank of signature tensors and their symmetries.

The submodular cone, like a chameleon, appears in many areas of mathematics, physics, economics... under various names. In particular, it is the cone of deformations of the permutahedron: the collection of all polytopes whose normal fan coarsens the normal fan of the permutahedron (i.e. the braid fan). This cone is hard to understand, as it lives in dimension 2^n and has n * (n-1) * 2^{n−2} facets: how can one counts its rays or access its faces? After giving a general introduction on deformations of polytopes, we aim at presenting new results about some specific faces of the submodular cone, namely the ones associated with graphical zonotopes and the ones associated with nestohedra. We count the number of facets of these faces and decide when they are simplicial. Last but not least, we will draw the examples of the submodular cone for n = 3 and n = 4 (and n = 5).