06.03.2025: Aaron Chan (University of Nagoya)

Title: Tilting approach for classifying quasi-hereditary structure

Abstract: The notion of quasi-hereditary algebras was introduced by Cline-Parshall-Scott, and there is an abundance of examples arising in algebraic Lie theory and non-commutative resolution of singularities. This notion is defined with respect to a poset structure on the set of simple modules, and there could be multiple possible such posets (quasi-hereditary structure).  A quasi-hereditary structure determines three sets of structural modules - the standard, costandard, and characteristic tilting modules.  The first two sets are some form of intermediate agent between the simple modules and the corresponding projective/injective respectively - the Lie theoretic origins are Verma modules and their dual.   The problem of classification of quasi-hereditary structures were studied by Goto  and also by Flores-Kimura-Rognerud.  This talk presents an on-going joint work with Adachi, Kimura, Tsukamoto, where we study a new approach on the classification problem that uses (generalised) tilting modules.  That is, we characterise the tilting modules that arise as characteristic tilting of some quasi-hereditary structure.  Moreover, we show that if all tilting modules arise as characteristic tilting, then it must be a linear Nakayama algebra with quadratic relation.  If time permits, I will show some interesting combinatorics that arise from the tilting modules of these Nakayama algebras.

 23.01.2025: Markus Kleinau (University of Bonn)

Title: Representation theory over 𝔽1

Abstract: Reinterpreting finite sets as vector spaces over 𝔽1 (the

nonexistent field with one element) yields a combinatorial degeneration

of linear algebra. Quiver representations over these new vector spaces

turn out to be closely related to graph theory. We will introduce this

topic and give a graph theoretic criterion for when all scalar

extensions of an 𝔽1-representation are indecomposable. Here the scalar extensions are representations over more familiar fields.

 16.01.2025: Norihiro Hanihara (Kyushu University)

Title: Reflexive modules and Auslander-type conditions

Abstract: Motivated by the theory of non-commutative resolutions and the results on Auslander correspondence, we study the category of reflexive modules over (commutative or non-commutative) Noetherian rings. One well-established sufficient condition for this category to behave well is that the ring should be (commutative) normal. We will explain that these nice behaviors are governed by the Auslander-type conditions which are some requirements on the minimal injective resolution of the ring. We will also discuss a Morita-type result characterizing the category of reflexive modules.

 21.11.2024: Jan Thomm (University of Cologne)

Title: Auslander–Reiten sequences in minimal A∞-structures of the module category of a representation finite algebra

Abstract: The Auslander–Reiten sequences of a representation finite algebra may be seen as the building blocks of its module category. However, any Yoneda product of a radical morphism with an Auslander–Reiten sequence will automatically be split. Considering this, the Auslander–Reiten sequences seem to be unable to ’generate’ any other short exact sequences. Enhancing the Yoneda product to an A∞-structure helps us to overcome this discrepancy. We will show that for a representation finite algebra the Ext-algebra of any basic additive generator is in fact always generated by the irreducible morphisms and Auslander–Reiten sequences as an A∞-algebra.

 14.11.2024: Tal Gottesman (University of Bochum)

Title: Fractionally Calabi-Yau Lattices: corroborating a conjecture by Chapoton

Abstract: In 2023, Chapoton published a fascinating conjecture linking combinatorial formulas, the representation theory of partially ordered sets and symplectic geometry. In 2018, Rognerud proved that Tamari lattices illustrate the first two aspects of this conjecture. In this talk I will present the lattices of order ideals of products of two chains as a first illustration all three aspects of the conjecture. In this example both the fractionally Calabi-Yau property and the link to symplectic geometry derive from the study of one family of representations defined by antichains of the lattice that have remarkable properties.

 07.11.2024: David Nkansah (University of Aarhus)

Title: Rank Functions in the Framework of Higher Homological Algebra

Abstract: Chuang and Lazarev introduced the concept of rank functions on triangulated categories as a generalisation of classical work by Cohn and Schofield on Sylvester rank functions. In this talk, we propose a generalisation of this notion to the broader framework of higher homological algebra.