Köln Algebra and Representation Theory Seminar

Abstract: Weight structures (also known as co-t-structures) and t-structures on triangulated categories are closely related by orthogonality. In terms of silting collections and simple-minded collections, the characterization of orthogonality is very similar to the relation of the indecomposable projectives to the simple objects in an abelian category. In my talk I will explain how this observation can be formalized using derived projective covers, which are an analog of projective covers for triangulated categories. This leads to a characterization of silting t-structures. If time permits, I will also discuss a result about Koszul duality between simple-minded and silting collections. The talk is based on arXiv:2309.00554. 

Abstract: Burban-Drozd (2017) introduced the category of triples to classify all indecomposable Cohen-Macaulay modules over certain non-isolated surface singularities. They showed that this category is representation-tame, with indecomposable objects of either band or string type. Via homological mirror symmetry, these modules correspond to loops and arcs in the Fukaya category of the mirror symplectic surface. In this talk, we explicitly describe this correspondence using the localized mirror functor developed by Cho–Hong–Lau (2017), and present its applications for algebraic operations such as the Auslander–Reiten translation. We also examine its connection with the derived category of gentle algebras associated to the same surface. This is based on joint work with Cho–Jeong–Kim–Rho (2022) and Cho–Rho (2024).

Abstract: In this talk, I will present a classification of the thick subcategories generated by string objects in the derived category of a gentle algebra. Such a thick subcategory is generated by a set of objects corresponding to a collection of arcs on the derived category’s geometric model. I will use properties of the geometric model to show that any thick subcategory generated by string objects has a generating set corresponding to a non-crossing collection of arcs. I will then discuss the poset structure of the set of non-crossing collections of arcs and how this induces a poset structure on the thick subcategories generated by string objects. This talk is based on the preprint arxiv:2502.12023.

Abstract: Brauer configuration algebras were introduced by Green and Schroll in 2017, which are generalizations of Brauer graph algebras. Each Brauer configuration algebra is defined by a combinational data called Brauer configuration. In this talk we introduce a further generalization of Brauer configuration called fractional Brauer configuration, and define the associated fractional Brauer configuration algebra (category). We also introduce various types of fractional Brauer configurations, and study the properties of associated fractional Brauer configuration algebras (categories). Then we establish a covering theory for fractional Brauer configurations, and give a connection between the covering of fractional Brauer configurations and the covering of associated fractional Brauer configuration algebras (categories). We also define Brauer $G$-sets and study their covering theory. Finally, we apply our covering theory to study the representation theory of some fractional Brauer configuration algebras. This talk is based on joint work with Nengqun Li.

Abstract: Categorical realization of collapsing subsurfaces via perverse schobers

Abstract: In Barbieri-Moller-Qiu-So's work, they studied the verdier quotient of three-Calabi-Yau categories from (decorated) marked surfaces with collapsing its subsurface. We give a quotient perverse schober of the collapsed surface to describe the Verdier quotient category. Generally, if the collapsed surface has quadratic differential with arbitrary order zeros and poles, we show the isomorphism between the principal part of two exchange graphs, where one is obtained by tilting hearts in the quotient categories and the other is by flipping mixed angulations on the collapsed surface.

Abstract: This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notion of moving vectors which was first introduced by Yanbo Li and Xiangyu Qi.  If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification.

Abstract: The derived category of any ring admits various triangulated subcategories which determine the ring up to derived Morita equivalence. This follows from seminal work of Rickard and has been revisited in recent work of Neeman, using the concept of metric completion for triangulated categories. The talk aims to explain these concepts and ideas, without going into the technicalities.

Abstract: TBA

Abstract: Exceptional sequences were first introduced in triangulated categories by the Moscow school of algebraic geometry. Later, Crawley-Boevey and Ringel considered exceptional sequences in the module categories of hereditary finite-dimensional algebras. Motivated by tau-tilting theory introduced by Adachi, Iyama, and Reiten, Jasso’s reduction for tau-tilting modules, and signed exceptional sequences introduced by Igusa and Todorov, Buan and Marsh developed the theory of (signed) tau-exceptional sequences – a natural generalization of (signed) exceptional sequences that behave well over arbitrary finite-dimensional algebras.

In this talk, we will study (signed) tau-exceptional sequences over the algebra Λ=RQ, where R is a finite-dimensional local commutative algebra over an algebraically closed field, and Q is an acyclic quiver. I will explain how (signed) tau-exceptional sequences over Λ can be fully understood in terms of (signed) exceptional sequences over kQ. 

Titles and Abstracts: Please see https://sites.google.com/view/cologne-representation-theory-/schedule-abstracts

Abstract: The partially wrapped Fukaya category of a smooth graded marked surface was introduced by Haiden, Katzarkov and Kontsevich. In a more general setting, Auroux and Jeffs defined the Fukaya category of a singular surface to be a certain A∞-quotient of the Fukaya category of a smooth surface. In this talk, we associate a quiver with relations to an admissible dissection of a graded marked surface with conical singularities. We then show that it describes the endomorphism ring of a formal generator of the A∞-quotient. By construction, the perfect derived category of this algebra is a Verdier quotient of the perfect derived category of a gentle algebra, and it provides a categorification for the contraction of a simple closed curve on a smooth surface. This is analogous to a recollement established by Chang, Jin and Schroll, which categorify the notion of cutting and pasting of surfaces. This talk is based on the preprint arXiv:2407.04374.

Abstract: We prove that, under mild assumptions, the Q-shaped derived categories introduced by Holm and Jørgensen are equivalent to derived categories of differential graded bimodules over differential graded categories. This yields new derived invariance results for Q-shaped derived categories that allow us to extend known descriptions of such categories as derived categories of differential graded bimodules over (possibly graded) algebras.

Abstract: Algebraic numbers $\overline{\mathbb{Q}} \subset \mathbb{C}$ 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 $\overline{\mathbb{Q}}$-vectorspace generated by a (finite) set of complex numbers. For example, the $\overline{\mathbb{Q}}$-vectorspace 〈1, 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 resultsof Huber & Wüstholz and recover a dimension estimate of Deligne & Goncharov.

This is based on joint work with Annette Huber (Freiburg).