Women in Representation Theory: quivers, mutation and beyond – Abstracts
Cluster structures via surface combinatorics
Karin Baur (Leeds)
Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this talk, I will show how this yields combinatorial approaches to cluster structures on the coordinate ring of the Grassmannians. This approach gives rise to cluster-tilting objects corresponding to Pluecker coordinates. To obtain higher rank objects, a different strategy is needed as for example the generalisations of webs recently given by Le and Yildirm.
Simple tilts and mutations
Raquel Coelho Simões (Lancaster)
'Simple-minded objects' are generalisations of simple modules. They satisfy Schur's lemma and a version of the Jordan-Holder theorem, depending on context. In this talk we will consider simple-minded systems (SMSs), which were introduced by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories, and simple-minded collections (SMCs), introduced by Koenig-Yang as an abstraction of the simple objects in hearts of t-structures.
We will discuss when mutation of an SMS/SMC is possible and gives another SMS/SMC. This provides a conceptual understanding and unifies results by Dugas, Jorgensen and Koenig-Yang. The main tools we use are simple-minded reduction, which allows us to see that SMC mutation is compatible with SMS mutation via a singularity category construction, and simple HRS tilts of length hearts, which is of particular importance in the study of stability conditions. This talk is based on joint work with Nathan Broomhead, David Pauksztello and Jon Woolf.
Structure-preserving functors in higher homological algebra
Johanne Haugland (Trondheim)
In the research field of higher homological algebra, distinguished sequences with d middle terms play a fundamental role. The case d=1 recovers the short exact sequences and distinguished triangles of abelian and triangulated categories, and the theory corresponds to classical homological algebra.
One crucial difference between classical and higher homological algebra is that the latter framework allows simultaneous study of structures of differing "dimensions". However, describing formally what it means for higher homological structures to relate to each other in a compatible way, turns out to be quite challenging. In this talk, I will tell you about the background for studying this problem and the progress made so far. The talk is based on (ongoing) joint work with Raphael Bennett-Tennenhaus, Mads H. Sandøy and Amit Shah.
Silting in higher homological algebra
Karin M. Jacobsen (Aarhus)
Classical silting theory plays a major role in contemporary homological algebra. In particular 2-term silting complexes are closely linked with tau-tilting theory and torsion classes. Turning to higher homological algebra, we study (d+1)-term silting complexes, and show how they are linked with d-torsion classes and maximal tau-d-rigid pairs. This talk is based on joint work with August, Haugland, Kvamme, Palu, and Treffinger.
Triangulations versus cluster-tilting in completed ∞-gons
İlke Çanakçı (Amsterdam)
Building on earlier work by Igusa and Todorov, Paquette and Yıldırım introduced cluster categories for 'completed' ∞-gons, which are discs with an infinite set of marked points on their boundary. In this setting, indecomposable objects in the category are in bijection with diagonals between marked points on the disc and strong restrictions are needed on a triangulation in order for it to correspond to even a weak cluster-tilting subcategory. We propose to replace Paquette-Yıldırım's category by an extriangulated structure in which weak cluster-tilting subcategories are in bijection with (all) triangulations. This is joint work with Martin Kalck and Matthew Pressland.
Induction and restriction of 2-representations
Vanessa Miemietz (Norwich)
For a finite-dimensional algebra A with a subalgebra B, there is the well-known adjunction between induction and restriction (IndAB , ResAB) where IndAB = A ⊗B -. For 2-representations of 2-categories, an analogue of restriction is straightforward, but a lack of relative tensor product means we need to define induction differently in order to obtain an analogous 2-adjunction between induction and restriction (for finitary 2-representations of fiat 2-categories). I will explain the basics of 2-representation theory and the necessary ingredients to state this 2-adjunction.
Regular exact Borel subalgebras and Bound Quivers
Anna Rodriguez Rasmussen (Uppsala)
Let A be a quasi-hereditary algebra. In 2014, König, Külshammer and Ovsienko proved that there is always a Morita equivalent quasi-hereditary algebra R which admits a basic regular exact Borel subalgebra B. Later, Brzeziński, König and Külshammer proved that the pair (R, B) is unique up to isomorphism, and posed the question whether B is unique in R up to inner automorphism. In this talk, I will speak about recent work answering this question in the positive.
The Auslander-Gorenstein condition for monomial algebras – a combinatorial and a homological characterisation
Viktória Klász (Bonn)
In this talk, we aim to get a better understanding of what it means to be Auslander-Gorenstein for quiver algebras. This is a homologically defined condition for a finite-dimensional algebra A, which implies many interesting properties for the algebra and certain subcategories of mod(A). We will focus on three well-known classes of algebras: gentle, Nakayama, and monomial algebras. First, we attempt to find a combinatorial description of this homological condition. This leads us to a new class of examples of Auslander-Gorenstein algebras. Second, we present a surprising new homological characterisation of the Auslander-Gorenstein property for these algebras. For this, the central role is played by a bijection between indecomposable projective and injective A-modules introduced by Auslander and Reiten.
Thread quivers
Emine Yıldırım (Leeds)
This is a work in progress with C. Paquette and J.D. Rock. Starting from a quiver Q, we construct a new one Q' called "thread quiver" by replacing arrows with linearly ordered posets. Then we consider a k-linear category C where objects are the union of vertices of Q and elements of linearly ordered posets, and morphisms are given by 'paths'. We study pointwise finite representations of C or quotients of C by weakly admissible ideals. Our main interest is to study the decomposition of pointwise finite representations and homological properties of such categories.
Pretorsion and torsion theories
Francesca Fedele (Leeds)
Starting from a generalisation of torsion (abelian) groups, torsion theories have been widely studied in the last 60 years for the important role they play in abelian categories. More recently, pretorsion theories have been introduced as a ``non-pointed version'' in more general categories. After introducing this framework, in this talk I will explore the connection between torsion and pretorsion theories and show how the first can be used to build examples of the second in representation theory. This is based on joint work with Federico Campanini.
Skew-group A∞-categories associated with graded orbifold surfaces
Claire Amiot (Grenoble)
Haiden Katzarkov and Kontsevich have associated a A∞-category to any graded surface with boundary and marked points giving then an alternative description of the partially wrapped Fukaya category. These categories are closely related with derived categories of gentle algebras. In a joint work with Pierre-Guy Plamondon, we use the skew-group construction of A∞-categories with strict action of a group to give an analogous construction for graded surfaces endowed with an automorphism of order two. The categories obtained are closely related with derived categories of skew-gentle algebras. The description of indecomposable objects and irreducible morphisms permits to describe some tilting objects.
When aisles meet
Sira Gratz (Aarhus)
Discrete cluster categories of type A are an exciting playing field on which to learn about infinite rank cluster combinatorics: On the one hand, they combinatorially behave, in many ways, in a familiar finite type A way. On the other hand, they exhibit new phenomena for which finite type A is "too small". One such phenomenon is the existence of non-trivial t-structures. In this talk, we describe the classification of t-structures in discrete cluster categories of type A via decorated non-crossing partitions and explain how they form a lattice under inclusion of aisles with meet given by intersection. If time permits, we will discuss the lattice of thick subcategories. This talk is based on joint work with Alexandra Zvonareva.
Metric completions of discrete cluster categories
Charley Cummings (Aarhus)
It is notoriously difficult to generate new triangulated categories from old. Despite this, Neeman recently discovered such a method that emulates the completion of a metric space. Usually, the completion of a triangulated category is difficult to compute and requires the use of another, already completed, ambient triangulated category. However, in this talk, we will see that many metric completions of discrete cluster categories can be computed directly and described by the associated combinatorial model.
This talk is based on joint work with Sira Gratz.
Combinatorics of Conway-Coxeter friezes and resolutions of plane curve singularities
Eleonore Faber (Leeds)
Conway-Coxeter friezes are arrays of positive integers satisfying a determinantal condition. Recently, these combinatorial objects have been of considerable interest in representation theory, since they encode cluster combinatorics of type A.
In this talk I will discuss a new connection between Conway-Coxeter friezes and the combinatorics of a resolution of a complex plane curve singularity: via the beautiful relation between friezes and triangulations of polygons one can relate each frieze to the so-called lotus of a curve singularity, which was introduced by Popescu-Pampu. This allows us to interprete some of the entries in the frieze in terms of invariants of the curve singularity, in particular partial resolutions. This is joint work with Bernd Schober.
Torsion pairs and the Ziegler spectrum
Rosanna Laking (Verona)
In this talk we will consider the category of modules ModA over an artin algebra A and ask how one might obtain information about its global structure. In general this is a difficult problem; even when the category modA of finitely generated A-modules is tame, the category ModA can contain a full exact abelian subcategory equivalent to modules over a wild algebra (see, for example, Ringel's article "Tame algebras are WILD"). We will give an overview of two different approaches to the problem of organising ModA: (i) studying well-behaved torsion pairs that decompose ModA into better behaved subcategories and (ii) studying indecomposable pure-injective modules with their associated topology called the Ziegler spectrum. We will then present joint work with Lidia Angeleri Hügel and Francesco Senteri in which we show that these two organising structures are intimately linked.