Seminar

Wednesday, 15 October 2025

• 11:00, Room B2-218
  Cyril Matousek (Aarhus): Hereditary rings and metric completions of their derived categories
  Abstract: A metric on a triangulated category, as developed by Neeman, provides a recipe for constructing a metric completion of the category. These completions are guaranteed to be triangulated categories as well and have recently been used to study, among other things, derived Morita theory, cluster categories, and t‑structures. The aim of this talk is to examine metric completions of bounded derived categories of hereditary rings and their connection to the concept of universal localisation. Notably, we explicitly describe the completions of bounded derived categories of hereditary finite dimensional tame algebras and hereditary commutative noetherian rings with respect to additive good metrics.

Friday, 17 October 2025

• 09:00, Room M4-122/126
  Igor Burban (Paderborn): Exceptional curves and real curve orbifolds
  Abstract: An exceptional hereditary non-commutative curve over an algebraically closed field is a weighted projective line of Geigle and Lenzing. However, over arbitrary fields, the theory of exceptional curves is significantly richer. In my talk I am going to explain the definition, examples and key properties of this class of non-commutative curves, including their invariants and their relation to squid algebras. I shall also explain a connection between real exceptional curves of tubular type and wallpaper groups, which was discovered by Lenzing many years ago. My talk is based on arXiv:2411.06222 as well as on an ongoing work in progress with Baumeister, Neaime and Schwabe.
• 10:30, Room M4-122/126
  Daniel Perniok (Paderborn): Coxeter-Dynkin algebras of canonical type
  Abstract: In this talk we will see a new definition of Coxeter-Dynkin algebras of canonical type. This generalises the existing definition in the special case where it can be described via quivers and relations. The main goal is to establish a derived equivalence to Ringel's squid algebra (and hence to the corresponding canonical algebra). Finally, we will build a bridge to geometric group theory via Saito's classification of marked extended affine root systems of codimension one.
  
• 11:30, Room M4-122/126
  Andrew Hubery (Bielefeld): Tame hereditary algebras, non-commutative curves, and preprojective algebras
  Abstract: The category of finite dimensional representations of a tame hereditary algebra has a discrete part, related to the roots of an affine Kac-Moody Lie algebra, and a continuous part. The simplest case is for the Kronecker quiver, when the continuous part is precisely the projective line. Lenzing and coauthors generalised this and showed that this continuous part has the structure of a non-commutative curve, constructed using the preprojective algebra. We will revisit this important construction, strengthening their results, and providing further connections between the geometry and certain infinite dimensional representations.
• 14:00, M4-122/126
  Georges Neaime (Bielefeld): Reflection groups of canonical type and their non-crossing partitions
  Abstract: We introduce the notion of reflection groups of canonical type. These groups are related to the K-theoretic study of the canonical algebras of Ringel. We use the notion of a symbol introduced by Lenzing to define them. We also introduce the associated non-crossing partitions of canonical type, which are intervals of Coxeter elements equipped with a poset structure. These notions will appear in a joint work in progress with Barbara, Charly, and Igor. If time permits, I will present a research programme for the sequel in order to study these groups from a geometric group theory point of view.
• 15:30, Room M4-122/126
  Barbara Baumeister (Bielefeld): Extended Weyl groups and their hyperbolic covers
  Abstract: I will explain the problem appearing in the Hurwitz action on the set of reduced reflection factorizations of a Coxeter element, and present a solution to this problem. I will also explain the choice of the name "hyperbolic cover".
• 16:30, Room M4-122/126
  Charly Schwabe (Paderborn): Simplicity beneath the complexity: a categorification of non-crossing partitions for exceptional curves
  Abstract: In this talk, I will report on ongoing joint work with Baumeister, Burban and Neaime. I will explain how the proofs of Hurwitz transitivity and the categorification of non-crossing partitions become very simple under the right assumptions. In other words, I will show exactly what the technical difficulties are, that were unclear when we started this project.
  

For a regular email announcement please contact birep.

Future Talks

Friday, 31 October 2025

• 13:15, Room U2-232
  Gopinath Sahoo (Mumbai): tba

Friday, 07 November 2025

• 13:15, Room U2-232
  Mateusz Stroinski (Hamburg): Algebras, quivers and species in fusion and tensor categories
  Abstract: By results of Etingof and Ostrik, the theory of module categories over a finite tensor category C is equivalent to that of modules over algebras inside C. Hence it is a generalisation of the representation theory of finite-dimensional algebras, internal to C.
  The aim of this talk is to build on this observation: I will explain how to define Jacobson radicals, quivers, species and their representations, all internal to C.
  Applications I will present include a proof of a conjecture of Etingof-Ostrik on (semi)simple algebras in C, based on joint work with Kevin Coulembier (University of Sydney) and Tony Zorman (TU Dresden), as well as a variant of Gabriel's theorem on basic Morita representatives given by species in C, based on an ongoing collaboration with Edmund Heng (University of Sydney).
  
  

Friday, 21 November 2025

• 13:15, Room U2-232
  Fabian Januszewski (Paderborn): tba

Friday, 28 November 2025

• 13:15, Room U2-232
  Tilman Bauer (Stockholm): tba

Seminar Archive

Friday, 18 July 2025

• 13:15, Room B2-278
  Martin Kalck (Graz): Path into transcendence
  Abstract: Algebraic numbers are complex numbers that are roots of polynomials with rational coefficients. All other complex numbers are called transcendental. It is typically a hard question to decide whether a given complex number is transcendental.
  A more general, classical question in 'transcendental number theory' (cf. e.g. works of Lindemann and Weierstraß, Gelfond and Schneider, Baker, Wüstholz) is the following: determine the dimension of the vectorspace generated by a (finite) set of complex numbers over the algebraic numbers. For example, the vectorspace generated by 1 and π is two-dimensional since π is transcendental by Lindemann's Theorem.
  For certain complex numbers called periods, we will try to explain how this transcendence question can (sometimes) be translated into determining dimensions of certain finite dimensional algebras – in other words, into counting (equivalence classes of) paths in 'modulated' quivers (with 'multiplicities').
  The dimension formulas obtained in this way improve and clarify earlier results of Huber & Wüstholz and recover a dimension estimate of Deligne & Goncharov.
  This is based on joint work with Annette Huber (Freiburg).
  

Friday, 04 July 2025

• 13:00, Room B2-278
  Georgios Dalezios (Verona): Quasi-hereditary algebras arising from Reedy categories
  Abstract: Reedy categories form a generalization of the category of finite ordinals, with morphisms the weakly monotone functions between them. Given a Reedy category C and a Quillen model category M, there is always a model structure on the category of functors from C to M. In this talk, we study Reedy categories which are enriched over a field and we present two main results. The first is an analogue of the aforementioned result for complete cotorsion pairs and abelian model structures. In the second result, we focus on linear Reedy categories having finitely many objects, which leads us to the concept of Reedy finite-dimensional algebras. We prove that any Reedy finite-dimensional algebra is quasi-hereditary with an exact Borel subalgebra. This is joint work with Jan Stovicek.
• 14:15, Room B2-278
  Øyvind Solberg (Trondheim): Endomorphisms of modules as linear maps
  Abstract: All algebras and modules in this talk are finite dimensional over a field. A module M is non-trivially decomposable if and only there exists an endomorphism f of M such that the characteristic polynomial of f has at least two different irreducible factors. Hence, all endomorphisms of an indecomposable module has a characteristic polynomial a power of an irreducible polynomial. The talk will discuss these observations and which irreducible polynomials occur when over a finite prime field.
  This is work in progress and based on joint work with Fernando Yamauti.
  

Friday, 06 June 2025

• 13:15, Room B2-278
  Erlend Borve (Graz): tau-tilting theory and tau-tilting finiteness under scalar extension
  Abstract: Let L:k be a field extension and let A be a finite-dimensional k-algebra. The extension of scalars of A along L:k is the L-algebra A^L, obtained by tensoring A and L over k. In the early 1980s, Jensen and Lenzing showed that extension of scalars preserves many module-theoretic and homological properties, particularly when L:k is MacLane separable. In particular, representation-finiteness is preserved in this case. However, if A is tau-tilting finte, i.e. it admits only a finite number of support tau-tilting modules up to isomorphism, this need not be true for A^L. We explore some examples and counter-examples of when tau-tilting finiteness is preserved. Along the way, we explain how tau-tilting theory and related notions lift under extension of scalars.
  The talk is based on joint work in progress with Max Kaipel (Cologne).
• 14:30, Room B2-278
  Mohammad Hossein Keshavarz (Nantong): Characterizing higher Auslander(-Gorenstein) algebras
  Abstract: It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa's theorem in 1974.
  Let n be a positive integer. In this talk, by using torsion theoretic methods, we show that n-Auslander algebras can be characterized by the abelianness of the category of modules with projective dimension less than n and a certain additional property, extending the classical Auslander-Tachikawa theorem. By Auslander-Iyama correspondence a categorical characterization of the class of Artin algebras having n-cluster tilting modules is obtained.
  Since higher Auslander algebras are a special case of higher Auslander-Gorenstein algebras, the results are given in the general setting as extending previous results of Kong. Moreover, as an application of some results, we give categorical descriptions for the semisimplicity and selfinjectivity of an Artin algebra.
  Higher Auslander-Gorenstein algebras are also studied from the viewpoint of cotorsion pairs and, as application, we show that they satisfy in two nice equivalences.

Friday, 16 May 2025

• 13:15, Room B2-278
  Simone Virili (Barcelona): A study of Sylvester rank functions via functor categories
  Abstract: Given a ring R, a Sylvester rank function on the category of finitely presented right R-modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value 1 on R. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left R-modules to Abelian groups.
  This enlargement of the setting comes with many advantages, in fact, the richer structure of the functor category makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.

Friday, 09 May 2025

• 13:15, Room B2-278
  Otto Sumray (Dresden): Quiver Laplacians and data
  Abstract: Analysing the topology or geometry of data is a common task in data analysis, using methods such as clustering, networks, and persistent homology. Often, however, we wish to embellish data with extra local information, which mathematically can take the form of a quiver representation or cellular sheaf. One example of this enriched data comes from analysis of single-cell epigenetics, and this situation can be further abstracted to a question about local versus global feature selection.
  Graph Laplacians have had great success in the analysis of networks. We introduce the quiver Laplacian as an analogue of the graph Laplacian for quiver representations, and formulate a pipeline for its application to local versus global feature selection.
  Applying it to our case study in single-cell epigenetics, we show that the eigenvalues of the quiver Laplacian can aid in feature selection to recover biologically meaningful results. To show stability of this method, we provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways.
  
• 14:30, Room B2-278
  Mikhail Gorsky (Hamburg): Counting in Calabi-Yau categories, with applications to Hall algebras
  Abstract: I will discuss a replacement of the notion of homotopy cardinality in the setting of even-dimensional Calabi--Yau categories and their relative generalizations. This includes cases where the usual definition does not apply, such as Z/2-graded dg categories. As an application of the definition in the relative case, we define a version of Hall algebras for odd-dimensional Calabi-Yau categories. I will briefly explain its relation to some previously known non-intrinsic constructions of Hall algebras. Whenever a 1CY category C is equivalent to Z/2-graded derived category of a hereditary abelian category A, our intrinsically defined Hall algebra of C realises the Drinfeld double of the twisted Hall algebra of A, thus resolving a long standing problem in this CY case. The talk is based on joint work with Fabian Haiden, arxiv:2409.10154.
  

Friday, 02 May 2025

• 13:15, Room B2-278
  Giulia Iezzi (Aachen): Quiver Grassmannians for the Bott-Samelson resolution of type A Schubert varieties
  Abstract: Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. In this talk, we construct a special quiver with relations and consider two classes of quiver Grassmannians for this quiver. For an appropriate choice of dimension vector for this quiver, we provide an isomorphism between the corresponding quiver Grassmannians and certain Bott-Samelson resolutions of type A Schubert varieties. Furthermore, for smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety.

Friday, 11 April 2025

• 13:15, Room B2-278
  Umesh V. Dubey (Prayagraj): The Balmer Spectrum of integral permutation modules
  Abstract: Balmer and Gallauer recently studied the derived category of permutation modules of finite groups over a commutative ring and computed its Balmer spectrum when the ground ring is a field.
  In our work, we compute the Balmer spectrum for the derived category of permutation modules of finite groups over a commutative Noetherian base ring. To achieve this, we employed a triangular fixed point functor recently introduced by J. Omar Gomez, which enables us to describe the underlying set of the Balmer spectrum.
  To describe the topology, we use the permutation resolution of Balmer and Gallauer to get an analog of the Koszul object that provides the required support conditions. Again following their strategies we gave a Dirac scheme structure on the Balmer spectrum of permutation modules over elementary abelian p-groups.
  In this talk, we will briefly discuss Balmer and Gallauer's construction of a Koszul object and the Dirac scheme structure on the Balmer spectrum for the elementary abelian case.
  This talk is based on an ongoing joint project with J. Omar Gomez.

For information on earlier talks please check the complete seminar archive.