Thomas Brüstle
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View article: Decomposing zero-dimensional persistent homology over rooted tree quivers
Decomposing zero-dimensional persistent homology over rooted tree quivers Open
Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is fundam…
View article: An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$
An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$ Open
Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almos…
View article: Invariants of persistence modules defined by order-embeddings
Invariants of persistence modules defined by order-embeddings Open
One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these non-to…
View article: Exact structures for persistence modules
Exact structures for persistence modules Open
We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, c…
View article: Homological approximations in persistence theory
Homological approximations in persistence theory Open
We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group g…
View article: On the lattices of exact and weakly exact structures
On the lattices of exact and weakly exact structures Open
View article: Stratifying systems and Jordan-Hölder extriangulated categories
Stratifying systems and Jordan-Hölder extriangulated categories Open
Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $Φ$ is a finite set of objects satisfying som…
View article: Stability conditions and maximal green sequences in abelian categories
Stability conditions and maximal green sequences in abelian categories Open
We study the stability functions on abelian categories introduced by Rudakov and their relation with torsion classes and maximal green sequences.Moreover, we introduce the concept of red paths, a stability condition in the sense of Rudakov…
View article: Double framed moduli spaces of quiver representations
Double framed moduli spaces of quiver representations Open
View article: Derived equivalences between skew-gentle algebras using orbifolds
Derived equivalences between skew-gentle algebras using orbifolds Open
Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain \mathbb{Z}_2 -action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection betw…
View article: Double framed moduli spaces of quiver representations
Double framed moduli spaces of quiver representations Open
Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a …
View article: Intersections, sums, and the Jordan-Hölder property for exact categories
Intersections, sums, and the Jordan-Hölder property for exact categories Open
View article: On the lattices of exact and weakly exact structures
On the lattices of exact and weakly exact structures Open
We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a…
View article: On the lattice of weakly exact structures
On the lattice of weakly exact structures Open
The study of exact structures on an additive category $\mathcal{A}$ is closely related to the study of closed additive sub-bifunctors of the maximal extension bifunctor $\mbox{Ext}^1$ on $\mathcal{A}$. We initiate in this article the study…
View article: Intersections, sums, and the Jordan-H\"older property for exact categories
Intersections, sums, and the Jordan-H\"older property for exact categories Open
We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of Hassoun and Roy. …
View article: Derived equivalences between skew-gentle algebras using orbifolds
Derived equivalences between skew-gentle algebras using orbifolds Open
Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain $\Z_2$-action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection between ske…
View article: Reduction of exact structures
Reduction of exact structures Open
View article: Wall and chamber structure for finite-dimensional algebras
Wall and chamber structure for finite-dimensional algebras Open
View article: Reduction of exact structures
Reduction of exact structures Open
Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or represent…
View article: Stability Conditions and Maximal Green Sequences in Abelian Categories
Stability Conditions and Maximal Green Sequences in Abelian Categories Open
In this paper we study the stability functions on abelian categories introduced by Rudakov in \cite{Ru} and their relation with torsion classes and maximal green sequences. Moreover we introduce a new kind of stability function which is co…
View article: Non-Leaving-Face property for marked surfaces
Non-Leaving-Face property for marked surfaces Open
We consider the polytope arising from a marked surface by flips of triangulations. Sleator, Tarjan and Thurston studied in 1988 the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations.…
View article: Stability conditions, $\\tau$-tilting Theory and Maximal Green Sequences
Stability conditions, $\\tau$-tilting Theory and Maximal Green Sequences Open
Extending the notion of maximal green sequences to an abelian category, we\ncharacterize the stability functions, as defined by Rudakov, that induce a\nmaximal green sequence in an abelian length category. Furthermore, we use\n$\\tau$-tilt…
View article: Stability conditions, $τ$-tilting Theory and Maximal Green Sequences
Stability conditions, $τ$-tilting Theory and Maximal Green Sequences Open
Extending the notion of maximal green sequences to an abelian category, we characterize the stability functions, as defined by Rudakov, that induce a maximal green sequence in an abelian length category. Furthermore, we use $τ$-tilting the…
View article: Semi-invariant pictures and two conjectures on maximal green sequences
Semi-invariant pictures and two conjectures on maximal green sequences Open
View article: Semi-invariant pictures and two conjectures on maximal green sequences
Semi-invariant pictures and two conjectures on maximal green sequences Open
We use semi-invariant pictures to prove two conjectures about maximal green sequences. First: if $Q$ is any acyclic valued quiver with an arrow $j\to i$ of infinite type then any maximal green sequence for $Q$ must mutate at $i$ before mut…