Schedule

Xiaofa Chen
Université Paris Cité
On exact dg categories

Abstract. We introduce the notion of an exact dg category and present a number of fundamental results concerning the dg nerve, the dg derived category, tensor products and functor categories with exact dg target. Inspired by recent work of Gorsky–Nakaoka–Palu and Fang–Gorsky–Palu–Plamondon–Pressland, we conclude with the formulation of 0-Auslander correspondence for exact dg categories.

Teresa Conde
Universität Bielefeld
Medley on exact Borel subalgebras

Abstract. Exact Borel subalgebras of quasi-hereditary algebras and standardly stratified algebras are an analog of Borel subalgebras of complex semi-simple Lie algebras. The aim of this talk is to give an overview on exact Borel subalgebras, with a focus on the homologically well-behaved subclass of regular exact Borel subalgebras. I will touch on topics such as their existence and uniqueness, their behaviour under recollements as well as on methods (old and new) to obtain information about these subalgebras without knowing them a priori.
This talk is partially based on joint work in progress with Julian Külshammer and also on joint work in progress with Steffen Koenig.

René Marczinzik
Universität Bonn
On Auslander-Gorenstein algebras

Abstract. We give a survey on Auslander-Gorenstein algebras and introduce the new class of dominant Auslander regular algebras as a generalisation of higher Auslander algebras. We use this to generalise cluster tilting modules and answer a question of Green about the Koszul dual of Auslander algebras.
This is joint work with Aaron Chan and Osamu Iyama.

Lang Mou
Universität zu Köln
Modulated graphs with potentials and skew-symmetrizable cluster algebras

Abstract. We propose a notion of potentials on modulated graphs, which generalizes potentials on quivers of Derksen–Weyman–Zelevinsky. From orbifold triangulations (in the sense of Felikson–Shapiro–Tumarkin), we construct families of modulated graphs with potentials and use the representations of their Jacobian algebras to categorify the skew-symmetrizable cluster algebras associated to orbifolds. The construction in particular provides in certain affine types mutations of Geiss–Leclerc–Schröer’s modulation away from acyclic clusters.
This talk is based on joint projects with Daniel Labardini-Fragoso and with Pierre-Guy Plamondon.

Yilin Wu
University of Science and Technology of China
Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

Abstract. Cluster categories were introduced in 2006 by Buan–Marsh–Reineke–Reiten–Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot to Jacobi-finite quivers with potential (2009). Later, Plamondon generalized it to arbitrary cluster algebras associated with quivers (2009 and 2011). Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, ....The work of Geiss–Leclerc–Schröer often yields Frobenius exact categories which allow to categorify such cluster algebras.
In previous work, we have constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss–Leclerc–Schröer. In this talk, we give the construction of the Higgs category and of the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka–Palu (2019). We also give the construction of a cluster character in this setting.
This is a joint work with Bernhard Keller and Chris Fraser (arXiv:2307.12279).