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 and my Google Scholar here.


Email
mkaipel (at) uni-koeln.de

Papers and Preprints


  1. Bricks and τ-tilting theory under base field extensions with Erlend D. Børve and an appendix by Eric J. Hanson

    Let K:k be a field extension and let Λ be a finite-dimensional k-algebra. We investigate the relationship between Λ and ΛK := Λ ⊗ K with particular emphasis on various aspects of τ-tilting theory and bricks. We show that many types of objects for Λ lift injectively to the same type of object for ΛK, and many common constructions in τ-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the τ-cluster morphism category W(Λ) of Λ the τ-cluster morphism category W(ΛK) of ΛK. In particular, this establishes a faithful functor from W(Λ) to a group whenever K is of characteristic zero which has many important consequences. In the appendix, the analogous result is shown to hold whenever K is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of τ-tilting finiteness under base field extension.

    preprint - arXiv:2508.01040


  2. Mutating ordered τ-rigid modules with applications to Nakayama algebras with Aslak B. Buan and Håvard U. Terland

    A mutation operation for τ-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered τ-rigid modules, which are in bijection with τ-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.

    Submitted - arXiv:2501.13694


  3. 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 torsA. Using the lattice congruence induced by an ideal I of A we establish a functor FI: T(A) → T(A/I). If torsA is finite, FI is a regular epimorphism in the category of small categories and we characterise when FI if full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the τ-cluster morphism category up to equivalence.

    Submitted - arXiv:2408.03818


  4. 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 three sufficient conditions. We characterise when these conditions are satisfied for fans in ℝ2 and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the τ-cluster morphism category admits a faithful functor and for which the cube complexes are K(π,1) spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its g-vector fan.

    Journal of the London Mathematical Society, 111: e70071 - DOI - arXiv:2311.05444


  5. Wall-and-chamber structures for finite-dimensional algebras and τ-tilting theory (lecture notes with Hipolito Treffinger)

    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.

    Accepted for publication in Proceedings of ICRA 2022 - arXiv:2302.12699

Seminar and Conference Talks


  1. τ-tilting finiteness under base field extension

    Queer and Trans Mathematicians in Algebra and Representation Theory - Bonn, Germany - July 2025.

  2. Mutating ordered τ-rigid modules

    Algebras and Representation Theory in Germany (ARTIG) 6 - Bochum, Germany - July 2025.

  3. Mutations of ordered τ-rigid modules

    Symposium on quivers, algebras and representation theory - Trento, Italy - June 2025.

  4. Mutating τ-exceptional sequences for Nakayama algebras

    Geometric models in representation theory and beyond - Trondheim, Norway - June 2025.

  5. τ-tilting finiteness under base field extension

    Padova Algebra Seminar - Padova, Italy - May 2025.

  6. Mutating ordered τ-rigid modules

    Leeds Algebra Seminar - Leeds, U.K. - March 2025.

  7. The many approaches to the τ-cluster morphism category

    Seminar on Homological Algebra and Related Topics - Buenos Aires, Argentina - October 2024.

  8. Mutation of τ-exceptional sequences for Nakayama algebras

    Reunión Anual De La Unión Matemática Argentina 2024 - Catamarca, Argentina - September 2024.

  9. Torsion lattices and the τ-cluster morphism category

    ICRA 21 - Shanghai, China - August 2024.

  10. A lattice of categories of an algebra

    CHARMS research school - Versailles, France - May 2024.

  11. τ-cluster morphism categories of factor algebras

    NTNU Trondheim Algebra seminar - Trondheim, Norway - May 2024.

  12. Partitioned fans, hyperplane arrangements and K(π,1) spaces

    NCSU Algebra & Combinatorics Seminar - online, Raleigh, NC, United States - March 2024.

  13. The category of a partitioned fan

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

  14. The category of a partitioned fan

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

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

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

  16. An introduction to point modules

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

Posters


  1. Mutating τ-exceptional sequences presented at:

    • Advances in Representation Theory of Algebra (ARTA) X - Cologne, Germany - September 2025.

  2. The category of a partitioned fan presented at:

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