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

**τ-cluster morphism categories of factor algebras**We take a novel lattice-theoretic approach to the τ-cluster morphism category

*T(A)*of a finite-dimensional algebra*A*and define the category via the lattice of torsion classes tors*A*. Using the lattice congruence induced by an ideal I of A we establish a functor*F*and if tors_{I}: T(A) → T(A/I)*A*is finite an inclusion*E: T(A/I) → T(A)*. We characterise when these functors are full, faithful and adjoint. As a consequence we find a new family of algebras for which*T(A)*admits a faithful group functor.**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. We prove that both conditions are satisfied for finite fans in ℝ^{2}unless one particular identification occurs. As a consequence, the classifying space of the τ-cluster morphism category of any τ-tilting finite algebra of rank 2 is a*K(π,1)*space for its picture group. As an application of the lattice structure, we show an analogous result holds for the Brauer cycle algebra of rank 3. In the final section we also offer a new algebraic proof of the relationship between an algebra and its*g*-vector fan.Link: arXiv:2311.05444

**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

**Torsion lattices and the τ-cluster morphism category - [slides]***ICRA 21*. Shanghai, China. August 2024.**A lattice of categories of an algebra - [notes]***CHARMS research school*. Versailles, France. May 2024.**τ-cluster morphism categories of factor algebras***NTNU Trondheim Algebra seminar*. Trondheim, Norway. May 2024.**Partitioned fans, hyperplane arrangements and***K(π,1) spaces*- [slides]*NCSU Algebra & Combinatorics Seminar*. online, Raleigh, NC, United States. March 2024.**The category of a partitioned fan - [slides]***Cluster Algebras and Its Applications*. Oberwolfach, Germany. January 2024.**The category of a partitioned fan***Seminar on Homological Algebra and Related Topics*. online, Buenos Aires, Argentina. December 2023.**A symmetric algebra with asymmetric wall-and-chamber structure***Lancaster PG forum*. online, Lancaster, U.K. May 2023.**An introduction to point modules***MAA MathFest 2021*. online, U.S.A. August 2021.

**The category of a partitioned fan**presented at:*Cluster Algebras and Its Applications*. Oberwolfach, Germany. January 2024.*τ-research school*. Cologne, Germany. September 2023