Schedule

Mini-courses

τ-tilting theory and τ-exceptional sequences
Bethany Marsh
University of Leeds

Abstract. τ-tilting theory was introduced by Adachi, Iyama and Reiten in a 2014 paper, building on classical tilting theory which was initiated in the 1970s. Although τ-tilting theory was motivated by the theory of cluster algebras, it nonetheless applies to arbitrary finite dimensional algebras and is a purely representation-theoretic notion. A key idea was the replacement of the rigidity of a tilting module with τ-rigidity, a vanishing property defined in terms of the Auslander-Reiten translation functor, τ, and hence the name. τ-tilting theory has good properties: in particular, every support τ-tilting module has precisely two complements. Exceptional sequences are sequences of modules over a finite-dimensional algebra with certain homological vanishing conditions which help to understand the structure of the module category. The τ-tilting version was introduced by the speaker in joint work with Buan in a 2021 paper. These talks will give an introduction to τ-tilting theory and τ-exceptional sequences.
Lecture notes. [Talks 1–4]

Varieties of modules over finite-dimensional algebras
Jan Schröer
University of Bonn

Abstract. This is an elementary introduction to some geometric aspects of representations theory. After reviewing the fundamental properties of module varieties, I turn to degenerations of modules, decomposition theorems of irreducible components and semicontinuous maps on module varieties. I will try to explain a proof of Plamondon's classification of generically tau-reduced irreducible components. For this we use a bundle construction by Derksen and Fei. If there is still time left, I give a quick introduction to convolution algebras. Convolution algebras of preprojective algebras and tau-reduced components of Jacobian algebras can be used to construct generic bases of cluster algebras and relate these to Lusztig's dual semicanonical bases.

Stability conditions in representation theory
Hipolito Treffinger
Université Paris Cité

Abstract. In recent years, stability conditions have played a major role in representation theory, in general, and in τ-tilting theory, in particular. This series of four lectures is a gentle introduction to these phenomena. The lectures will cover the following subjects.

1. What is a stability condition? In this first lecture we will start by introducing stability conditions in the sense of King and one of its generalisations, the stability conditions in the sense of Rudakov. We then will mention some of its properties and how these two notions relate. Time permitting we will speak about other two types of stability conditions: Bridgeland's stability conditions for triangulated categories and Joyce's weak stability conditions for abelian categories. 

2. Wall-and-chamber structures for finite dimensional algebras. The second lecture will be devoted to study the wall-and-chamber structure of an algebra, a geometric invariant that arises from King's stability conditions and which is deeply related with τ-tilting theory. 

3. Finiteness in τ-tilting theory. A natural aim that arises often in representation theory is to classify algebras having finitely many objects satisfying certain characteristics. In this third lecture we will explain what it means for an algebra to be τ-tilting finite and different ways to characterise these algebras. We will also speak about the relationship between this problem and the classical Brauer–Thrall conjectures. 

4. Can we tame τ-tilting infinite algebras? In this final lecture we will discuss the different attempts to define tameness in τ-tilting theory, how they relate to each other and we will show examples of algebras that illustrate the different notions of tameness. 

Research talks

Simple-mindedness in negative Calabi–Yau cluster categories of the hereditary type
Raquel Coelho Simões
Lancaster University

Abstract. "Simple-minded objects" are generalisations of simple modules. They satisfy Schur’s lemma and a version of the Jordan–Hölder theorem, depending on context, giving rise to "simple-minded collections" and "simple-minded systems". Although the theory of simple-minded objects shows many parallels with that of projective-minded objects, it remains relatively undeveloped and is technically more challenging. In this talk, I will explain the connection between simple-minded systems in negative Calabi–Yau cluster categories of hereditary algebras, simple-minded collections in their bounded derived category, and positive noncrossing partitions, which generalises previous results of Buan–Reiten–Thomas and Iyama–Jin. Buan–Reiten–Thomas's result can be viewed as a precursor to the bijection between 2-term silting objects and support τ-tilting modules. Analogously, our result could potentially be viewed as a simple-minded analogue of support τ-tilting.
This is based on joint work with David Pauksztello and David Ploog.

τ-tilting theory in higher homological algebra
Johanne Haugland
NTNU

Abstract. Torsion classes and τ-tilting theory play a crucial role in the study of finite dimensional algebras. In this talk, we explore these notions from the viewpoint of higher Auslander–Reiten theory. We first give an introduction to higher homological algebra, before discussing to what extent it is possible to lift the classical bijections between functorially finite torsion classes, maximal τ-rigid pairs and 2-term silting complexes to this higher-dimensional setup.
The talk is based on joint work in progress with Jenny August, Karin M. Jacobsen, Sondre Kvamme, Yann Palu and Hipolito Treffinger.

Lifting and restricting t-structures
Frederik Marks
University of Stuttgart

Abstract. We explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring A. More precisely, we show that every intermediate t-structure in Dᵇ(A) can be lifted to a compactly generated t-structure in D(A), by closing the respective classes under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in D(A) to restrict to an intermediate t-structure in Dᵇ(A). Finally, we discuss applications to silting theory.
This talk is based on joint work with Alexandra Zvonareva.