An algebraic approach to Harder-Narasimhan filtrations

In this article we study chains of torsion classes in an abelian category $\mathcal{A}$. We prove that chains of torsion classes induce a Harder-Narasimhan filtrations for every non-zero object $M$ in $\mathcal{A}$ and we characterise the slicings of $\mathcal{A}$ in terms of chain of torsion classes. Later, we show that all chain of torsion classes form a topological space and we study some of its properties. We finish the paper by showing that wall-crossing formulas can be deduced from chain of torsion classes.Comment: 31 pages. Added Section 8 and minor typos correcte

The size of a stratifying system can be arbitrarily large

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consists on stratifying systems of infinite size in the module category of an algebra $A$. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra $A$ can be arbitrarily large in comparison to the number of isomorphism classes of simple $A$-modules. We note that both families of examples are built using well-established results in higher homological algebra.Comment: A gap in the previous version was founded, implying that the examples constructed are stratifying system but not exceptional sequences. The note was corrected accordingl

Investigations on the Students’ Perception of an Online-Based Laboratory with a Digital Twin in the Main Course of Studies in Mechanical Engineering

During the coronavirus crisis, labs had to be offered in digital form in mechanical engineering at short notice. For this purpose, digital twins of more complex test benches in the field of fluid energy machines were used in the mechanical engineering course, with which the students were able to interact remotely to obtain measurement data. The concept of the respective lab was revised with regard to its implementation as a remote laboratory. Fortunately, real-world labs were able to be fully replaced by remote labs. Student perceptions of remote labs were mostly positive. This paper explains the concept and design of the digital twins and the lab as well as the layout, procedure, and finally the results of the accompanying evaluation. However, the implementation of the digital twins to date does not yet include features which address the tactile experience of working in real-world labs

On higher torsion classes

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes.Comment: 20 pages. Comments welcom

A geometric perspective on the τ\tau-cluster morphism category

We show how the $\tau$-cluster morphism category may be defined in terms of the wall-and-chamber structure of an algebra. This geometric perspective leads to a simplified proof that the category is well-defined.Comment: 20 pages, 5 figures. Comments welcome! v2: added a little more discussio

Sur la théorie de τ-inclinaison

Cette thèse rend compte des résultats parus dans deux articles écrit ou coécrit par l’auteur de cette thèse, à savoir [60, 24]. Ces articles sont assez indépendants entre eux. C’est pour cela qu’on peut séparer la thèse en deux parties. Dans le début de la thèse on étudie la théorie de τ-inclinaison comme une extension de la théorie d’inclinaison classique, en prouvant, par exemple, une version τ-inclinante du théorème d’inclination. Aussi, on introduit les τ-tranches. On montre que les τ-tranches généralisent d’autres tranches présentes dans la littérature. De plus, on obtient des résultats reliant les τ-tranches avec les algèbres inclinées. Dans la deuxième partie, on commence pour étudier les conditions de stabilité introduites par King et Rudakov dans [44, 52], respectivement. Notamment, on donne une description de la structure de chambres et parois d’une algèbre en utilisant les g-vecteurs des modules τ-rigides indécomposables. Pour finir, on introduit les chemins verts pour montrer que les conditions de stabilité de King et Rudakov sont compatibles