I am a Mathematics PhD student of Sibylle Schroll co-supervised by Hipolito Treffinger at the University of Cologne. My research area is the representation theory of finite dimensional algebras and τ-tilting theory.
You can find my CV here.
Email
mkaipel (at) uni-koeln.de
Address
Room C103
Gyrhofstraße 8a
50931 Cologne
Germany
The category of a partitioned fan
In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the τ-cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the τ-cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are K(π ,1) spaces to two sufficient conditions. By extending the definition of the picture group to a fan, we show that one condition is always satisfied for the τ-cluster morphism category and for fans underlying hyperplane arrangements. Additionally, both conditions are satisfied for fans in ℝ2 unless one particular identification occurs. As a consequence, the classifying space of the τ-cluster morphism category of any algebra with at most 3 simples is a K(π ,1) space for its picture group. In the final section we also offer a new algebraic proof of the relationship between an algebra and its g-vector fan.
Wall-and-chamber structures for finite-dimensional algebras and τ-tilting theory (lecture notes)
The wall-and-chamber structure is a geometric invariant that can be associated to any algebra. In these notes we give the definition of this object and we explain its relationship with torsion classes and τ-tilting theory.
joint with Hipolito Treffinger, arXiv:2302.12699
The category of a partitioned fan (Poster)
τ-research school, Cologne, Germany. Sep 2023. Link.
On a category associated to a partitioned fan
Cologne Algebra Group Seminar, Cologne, Germany. June 2023.
A symmetric algebra with asymmetric wall-and-chamber structure
Lancaster PG forum, online, Lancaster, U.K. May 2023.
The fan of g-vector cones of τ-rigid pairs
Cologne Algebra Group Seminar, Cologne, Germany. December 2022.
An introduction to point modules
MAA MathFest 2021, online, U.S.A. August 2021.