Maximilian Kaipel
Portrait

Maximilian Kaipel

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

Papers and Preprints


  1. 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 ℝ2 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.

    arXiv:2311.05444


  2. 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

Seminar and Conference Talks


  1. The category of a partitioned fan

    Cluster Algebras and Its Applications. Oberwolfach, Germany. January 2024.

  2. The category of a partitioned fan

    Seminar on Homological Algebra and Related Topics. online, Buenos Aires, Argentina. December 2023.

  3. On a category associated to a partitioned fan

    Cologne Algebra Group Seminar. Cologne, Germany. June 2023.

  4. A symmetric algebra with asymmetric wall-and-chamber structure

    Lancaster PG forum. online, Lancaster, U.K. May 2023.

  5. The fan of g-vector cones of τ-rigid pairs

    Cologne Algebra Group Seminar. Cologne, Germany. December 2022.

  6. An introduction to point modules

    MAA MathFest 2021. online, U.S.A. August 2021.

Posters


  1. The category of a partitioned fan presented at:

    • Cluster Algebras and Its Applications. Oberwolfach, Germany. January 2024.
    • τ-research school. Cologne, Germany. September 2023
    (View here)