FB 6 Mathematik/Informatik/Physik

Institut für Mathematik


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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

We address the general problem of estimating the probability that a real symmetric tensor is close to rank-one tensors. Using Weyl’s tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety. More precisely, we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri-Weyl metric. These results are obtained using techniques from Random Matrix Theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices. We give a complete solution to the original problem and exhibit some novel asymptotic results in the case of rational normal curves. If time allows, we will also discuss a generalization to partially symmetric tensors due to P. Breiding and S. Eggleston. Based on a joint work with A. Cazzaniga and A. Lerario.

 

07.11.2023 um 14:15 Uhr in 93/E44

M. Sc. Victoriia Borovik (Universität Osnabrück)

Eigenvalue problem on the Grassmannian

We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian Gr(d, n) in its Plücker embedding. Our technique consists in finding a reduced Gröbner basis for the ideal of the graph of the natural birational parametrization of Gr(d, n). In the case when d=2, we obtain an explicit formula for the number of complex solutions, which involves Catalan numbers and is the volume of the Cayley sum of a Gelfand-Cetlin polytope with simplex. Moreover, we can show that the property of Plücker coordinates to form a Khovanskii basis can be lifted to the graph, which give rise to the existence of a toric degeneration of the graph. 

The motivation of the problem comes from quantum chemistry. This is a joint work with Bernd Sturmfels and Svala Sverrisdóttir.

14.11.2023 um 14:15 Uhr in 93/E44

Prof. Dr. Winfried Bruns (Universität Osnabrück)

Fusion rings from lattice points

Fusion rings are abstract versions of Grothendieck rings of certain tensor categories, i.e., categories that are endowed with a bifunctor called tensor product. The prototypical example is rhe category of finite-dimensional representations of a finite group. The corresponding fusion ring is a finite rank free algebra over the integers whose base elements correspond to isomorphism classes of irreducible representations and whose relations are defined by the decomposition of the tensor product of two irreducibles into a direct sum of irreducibles. Another source is conformal field theory, which is undoubtedly a driving force in the theory of fusion rings.The classification of fusion rings can be based on the computation of lattice points in polytopes of extremely high dimension, typically > 200. The computation is only possible since the points must satisfy quadratic equations that represent the associativity of the algebra. Normaliz now contains an efficient computation of such lattice points.Part of the results so far are reported in the paper "Classification of modular data of integral modular fusion categories up to rank 12" with Max A. Alekseyev, Sébastien Palcoux and Fedor V. Petrov (arXiv:2302.14345).

 

28.11.2023 um 14:00 Uhr in 93/E44

Anje Seyberlich (Universität Osnabrück)

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.

 

30.11.2023 um 14:15 Uhr in 93/E44

Bogdan Ichim (University of Bucharest)

On Applications of Combinatorial Learning

We present some recent results obtained by
applying combinatorial learning techniques.

05.12.2023 um 14:15 Uhr in 93/E44

M. Sc. Justus Bruckamp (Universität Osnabrück)

h*-polynomials of Cosmological Polytopes

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.

12.12.2023 um 14:15 Uhr in 93/E44

Dr. Quentin Posva (Universität Düsseldorf)

Singularities of 1-foliations in positive characteristic

While traditionally studied in differential geometry, foliations have also been considered from the point of view of algebraic geometry. It is possible to specialize the definition to positive characteristic: the resulting notion is that of 1-foliations,. In this talk, I will present recent work on the resolution of singularities of 1-foliations in dimension at most 3, present some applications, and comment on the pathologies of general 1-foliations and on some better-behaved notions.

09.01.2024 um 14:15 Uhr in 93/E44

Pierpaola Santarsiero (Universität Osnabrück)

On signature tensors and their rank

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. All signature
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.

16.01.2024 um 14:15 Uhr in 93/E44

Giulia Iezzi (RWTH Aachen)

A realisation of smooth Schubert varieties as quiver Grassmannians

Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples. We give a construction for smooth quiver Grassmannians of a specific wild quiver, which realises smooth Schubert varieties and desingularises the singular ones. This also allows us to define linear degenerations of Schubert varieties.

 

25.01.2024 um 10.00 Uhr in 35/E22

Dr. Germain Poullot (Universität Osnabrück)

Generalized permutahedra: A glimpse at the submodular cone

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).