# Program 2022

The conference is planned to run from 11:00 to 17:00 (UTC+1).

Time | Activity |
---|---|

11:00-11:05 | Introduction |

11:05-11:15 | Bernhard Keller |

11:20-11:30 | Julia Sauter |

11:35-11:45 | Haruhisa Enomoto |

11:50-12:00 | Chrysostomos Psaroudakis |

Coffee Break | |

12:30-12:40 | Maitreyee Kulkarni |

12:45-12:55 | İlke Çanakçı |

13:00-13:10 | Yadira Valdivieso |

13:15-13:25 | Karin Baur |

Lunch Break | |

14:30-14:40 | Karin Erdmann |

14:45-14:55 | Zhengfang Wang |

15:00-15:10 | Job Rock |

15:15-15:25 | Rosanna Laking |

Coffee Break | |

16:00-16:10 | Martin Herschend |

16:15-16:25 | Mads H. Sandøy |

16:30-16:40 | Souheila Hassoun |

16:45-16:55 | Hugh Thomas |

# Abstracts

### Karin Baur (Leeds, UK)

#### String algebras via surface combinatorics

We will show how surfaces can be used to model string algebras and their module categories. An application of this is to characterise tau-tilting pairs. This is joint work with Raquel Coelho Simoes.

### İlke Çanakçı (VU Amsterdam, Netherlands)

#### Frieze patterns associated to pair of pants

We show that integral frieze patterns associated to triangulations of pair of pants is unitary. Representation theoretically, this means that there is a cluster-tilting object in the associated cluster category such that the frieze pattern may be recovered by a modified CC-map. This is joint with Anna Felikson, Ana Garcia Elsener and Pavel Tumarkin.

### Haruhisa Enomoto (Osaka Pref. Univ., Japan)

#### Constructing the lattice of wide subcategories

I will show that we can construct the lattice of wide subcategories of the module category of a finite-dimensional algebra from the lattice of torsion classes only using its lattice structure. This enables us to check various properties of the lattice of wide subcategories using computer, and I will give some results and conjectures on this lattice.

### Karin Erdmann (Oxford, UK)

#### Quivers for tame symmetric algebras of period 4

Several types of tame symmetric algebras of period 4 are known, including weighted surface algebras and their 'virtual mutations'. It was asked by A. Skowroński whether these might be all. In order to either prove this, or find new algebras, we study the Gabriel quivers which can occur. This is work in progress, joint with Adam Skowyrski.

### Souheila Hassoun (Northeastern, USA)

#### Double framed moduli spaces of quiver representations

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define the category of neural networks, whose objects correspond to the orbits of quiver representations, in which neural networks map input data. We also show that this category form a symmetric monoidal category.

We then prove different results, in particular that the output of a neural network depends only on the corresponding point in the moduli space. This talk is based on a joint work with Marco Armenta, Thomas Brustle and Markus Reineke: arXiv:2109.14589v2

### Martin Herschend (Uppsala, Sweden)

#### nZ-cluster tilting for Nakayama algebras

n-cluster tilting subcategories of module categories form the prototypical setting of Iyama's higher dimensional Auslander-Reiten theory. Finding algebras that admit such categories is therefore important to better understand this theory. Much attention has been given to algebras of global dimension n, for which there is at most one n-cluster tilting subcategory. For higher global dimension the notion of n-cluster tilting has certain drawbacks, which motivates the stronger notion of nZ-cluster tilting due to Iyama and Jasso. It has been shown by Kvamme that an nZ-cluster tilting subcategory of the module category gives rise to an nZ-cluster tilting subcategory of the singularity category.

In my talk I will present a classification of nZ-cluster tilting subcategories for Nakayama algebras. Moreover, I will show which nZ-cluster tilting subcategories they give rise to in the corresponding singularity categories. The classification contains some algebras of finite global dimension, some selfinjective algebras, as well as some algebras that are not Iwanaga-Gorenstein.

This talk is based on joint work with Sondre Kvamme and Laertis Vaso.

### Bernhard Keller (Paris, France)

#### Actions on cluster categories from mutations at frozens

We will illustrate how mutations at frozen vertices allow to construct group actions (e.g. braid group actions) on cluster categories and hence on cluster algebras.

### Maitreyee Kulkarni (MPI Bonn, Germany)

#### A combinatorial model for totally nonnegative partial flag varieties

Postnikov defined the totally nonnegative Grassmannian as the part of the Grassmannian where all Plücker coordinates are nonnegative. This space can be described by the combinatorics of planar bipartite graphs in a disk, by affine Bruhat order, and by a host of other combinatorial objects. In this talk, I will recall some of this story, then talk about in progress joint work, together with Chris Fraser and Jacob Matherne, which hopes to extend this combinatorial description to more general partial flag varieties.

### Rosanna Laking (Verona, Italy)

#### Simple objects in the heart and their injective envelopes

A remarkable property of the lattice of torsion classes of a finite-dimensional algebra, proved independently by Demonet-Iyama-Reading-Reiten-Thomas and Barnard-Carroll-Zhu, is that every arrow in the Hasse quiver of the lattice is labelled by a unique brick (that is, a module whose endomorphism ring is a division ring). In this talk we will show that these bricks are simple objects in the heart of an HRS-tilted t-structure. In the case where the algebra is contained in the torsion-free class, will also explain how to compute the injective envelope of these simple objects in the heart. This is joint work with Lidia Angeleri Hügel and Ivo Herzog.

### Chrysostomos Psaroudakis (AUTh, Greece)

#### Lifting recollements of abelian categories and model structures

Recollements of derived module categories have been extensively studied in representation theory (for instance, various homological invariants are computed inductively along a derived recollement). One basic source of such recollements are recollements of module categories that lift to the associated derived categories. The aim of this talk is to present a systematic method, using Quillen model structures, to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. In applications, we recover known results on lifting recollements of abelian categories to their derived counterpart, and we also show lifting of recollements in other contexts, such as lifting to homotopy categories that provide models for stable categories of Gorenstein projective modules and related categories. This is joint work with Georgios Dalezios.

### Job D. Rock (Ghent, Belgium)

#### Composition series of arbitrary cardinality in abelian categories

We generalize the notion of a composition series of a module. Instead of a finite chain of submodules we consider a totally ordered chain of submodules. We will cover the definition of this new type of composition series and some results related to representations of infinite quivers / posets (such as persistence modules). Joint work with Eric J. Hanson.

### Mads H. Sandøy (NTNU, Norway)

#### Higher Koszul duality and connections with n-hereditary algebras

Introduced by Iyama and others, n-hereditary algebras are an attempt to generalize the good properties of hereditary algebras to algebras of higher global dimension, and they come in two flavours: n-representation finite and n-representation infinite. Both exhibit connections with Koszul and almost Koszul algebras.

We describe how generalizations of the T-Koszul algebras introduced by Madsen and by Green, Reiten and Solberg yield results showing more general connections between n-hereditary algebras and certain kinds of selfinjective (or Frobenius) algebras. This is based on joint work with Johanne Haugland.

### Julia Sauter (Bielefeld, Germany)

#### Tilting subcategories in exact categories

We introduce tilting subcategories in exact categories (these are subcategories T such their perpendicular category is an exact category with enough projectives given by T and this perpendicular category is finitely coresolving). Then we discuss under which extra condition we get a derived equivalence on the bounded derived category of the exact category to the bounded derived category of the functor category on T given by those functor admitting a projective resolution by representable functors.

The aim is to unify tilting theory for exact categories (without extra assumptions like enough projectives or homological finiteness). In the end, we have a quick look at a tilting subcategory in mod(mod A) for an artin algebra A.

### Hugh Thomas (UQAM, Canada)

#### Algebro-geometric tau-tilting theory

I will (briefly!) explain, given an algebra A of finite representation type, how to obtain a system of non-linear equations whose non-negative solutions are controlled by the tau-tilting theory of A. I will speculate as to how to extend this to more general algebras, and say a word about the motivation from physics.

### Yadira Valdivieso (UDLAP, Mexico)

#### Caldero-Chapoton functions for orbifolds and snake graphs

Caldero–Chapoton functions define a relation between indecomposable objects of a module category and elements in a cluster algebra. For some families of cluster algebras, for example, the acyclic ones, Caldero-Chapoton functions form a basis of the algebras. In this talk, we describe Caldero-Chapoton functions for Jacobian algebras associated with orbifolds with orbifolds points of order three using snake graphs. This is joint work with Esther Banaian.

### Zhengfang Wang (Stuttgart, Germany)

#### A description of singularity categories via dg Leavitt path algebras

In a recent joint work with X.-W. Chen, we show that the singularity category of any finite dimensional algebra (given by a quiver with relations) is triangle equivalent to the perfect derived category of a dg Leavitt path algebra. This generalises the well-known result by Chen-Yang and Smith for singularity categories of radical-square-zero algebras. In this talk, we provide a deformation-theoretic perspective of our result.