ABSTRACT FOR SHORT COURSES:
Anna Barbieri - Spaces of Bridgeland stability conditions
I will give an introduction to the notion of Bridgeland stability conditions for triangulated category and the stability manifold associated, with a focus on the case of the Ginzburg 3-Calabi-Yau category associated with a quiver arising from a triangulation of a surface.
Karin Erdmann - Tame symmetric algebras
These lectures will investigate tame symmetric algebras, which are constructed and described via surface triangulations, and which generalize naturally tame blocks of group algebras. Such blocks have groups as invariants: dihedral, or semidihedral, or quaternion.
Inspired by cluster theory, we introduce weighted surface algebras. They are almost all periodic as algebras, of period 4, and are a geometric generalization for quaternion type. We take these as a frame for several more algebras: Degenerating minimal relations gives rise to a generalization for dihedral type, and semidihedral type.
Towards a unified approach, we introduce hybrid algebras. These include all Brauer graph algebras, weighted surface algebras, and in addition many other tame symmetric algebras. Hybrid algebras are precisely the block components of idempotent algebras eAe where A is a weighted surface algebra and e an idempotent of A.
We discuss what is known about indecomposables, Auslander-Reiten components, and derived equivalence.
This is mostly joint work with Andrzej Skowrónski.
Julian Külshammer - Towards bound quivers for exact categories (online)
The proceedings of the first ICRA in 1974 contain an article by Roiter and Kleiner describing a theory of representations of semi-free differential graded categories. While the theory has been successfully applied in a number of cases, most notably in the proof of Drozd's tame-wild dichotomy, it is today not as well-known as e.g. the theory of almost split sequences described in articles by Auslander and Reiten in the same proceedings.
In this lecture series, I will explain how such semi-free differential graded categories are a convenient way to talk about certain ring extensions and provide evidence that they are a way to develop a theory of quivers and relations for exact categories. The most developed special case right now is that of the category of filtered modules over a quasi-hereditary algebra. This builds on joint work with several people over the last decade, let me in particular mention Steffen Koenig, Sergiy Ovsienko, and Vanessa Miemietz.
Rosanna Laking - Torsion pairs and mutation
The complete lattice tors-A formed by the collection of all torsion pairs in the category of finitely generated modules over a finite-dimensional algebra A encodes a wealth of combinatorial and homological information about the representation theory of A. This is due, in part, to its connections with t-structures, tau-tilting theory and stability conditions. The connection with tau-tilting theory has been a particularly powerful tool for understanding tors-A, since the functorially finite torsion pairs are parametrised by two-term silting objects in the bounded derived category of modA and the adjacent edges of the Hasse quiver are controlled by silting mutation.
In these talks we will introduce an approach to the study of tors-A that goes beyond tau-tilting theory and the functorially finite torsion classes. We will give an overview of the Demonet-Iyama-Reading-Reiten-Thomas brick labelling of the Hasse quiver in terms of simple tilts between the corresponding HRS-t-structures. By lifting these t-structures to the derived category of all modules we will see that the simple tilts induce irreducible mutations of associated (large) two-term cosilting complexes. Finally we will explain how the two-term cosilting complexes are parametrised by certain closed sets of the Ziegler spectrum and their mutations are determined by the open sets of the induced topology. The topics covered by these talks are contained in joint work with Lidia Angeleri Hügel, Ivo Herzog, Francesco Sentieri, Jan Šťovíček and Jorge Vitória.
Hipolito Treffinger - Wall-and-chamber structures of Artin algebras and τ-tilting theory
The aim of this course is to give an overview of the deep relationship between a geometric object associated to an Artin algebra, known as its wall-and-chamber structure, and the τ-tilting theoretic properties of the algebra. Being more precise, in this course we will show how several important notions of τ-tilting theory are encoded in the geometry of the wall-and-chamber structure of the algebra.
We will start this course by showing the (top-down) construction of the wall-and-chamber structure of an algebra using the stability conditions defined by King and how we can use stability conditions to calculate torsion pairs in the module category of our algebra. We then will change gears slightly to introduce τ-tilting theory and present the (bottom-up) construction of (part of) the wall-and-chamber structure of the algebra using the g-vectors of the indecomposable τ-rigid objects of the algebra.
Once these two complementaries constructions of the wall-and-chamber structure are presented, we will profit the interplay between the two to recover several important objects which are central to τ-tilting theory, including τ-tilting pairs and their mutation, (semi)bricks, torsion pairs and wide subcategories.