# Program

All times are in CET (UTC+1).

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

11:00-11:05 | Introduction |

11:05-11:15 | Dong Yang |

11:20-11:30 | Raphael Bennett-Tennenhaus |

11:35-11:45 | Javad Asadollahi |

11:50-12:00 | Claire Amiot |

Coffee break | |

12:30-12:40 | Peter Jørgensen |

12:45-12:55 | Monica Garcia |

13:00-13:10 | Teresa Conde |

13:15-13:25 | Jan Šťovíček |

Lunch break | |

14:30-14:40 | Henning Krause |

14:45-14:55 | Jorge Vitória |

15:00-15:10 | Ana Garcia Elsener |

15:15-15:25 | Jeanne Scott |

Coffee break | |

16:00-16:10 | Sonia Trepode |

16:15-16:25 | Emily Barnard |

16:30-16:40 | Kayla Wright |

16:45-16:55 | David Pauksztello |

Break | |

18:00-20:00 | Pub Quiz |

## Abstracts

### Claire Amiot

#### Linear Invariants for poset representations

Let \(P\) be a poset. An \(\mathbb Z\)-invariant on \(\rm mod P\) is a \(\mathbb Z\)-linear map \(K_0^{\rm sp}(\rm mod P)\to \mathbb Z^N\). In this talk, I will dicuss several examples of invariants that appear in persistence theory coming either from relative exact structures or from order embeddings. This is a joint work with Thomas Brüstle and Eric Hanson.

### Javad Asadollahi

#### On 1-Gorenstein algebras of finite Cohen-Macaulay type

In this talk, we study the 1-Gorenstein algebra \(\Lambda\) of finite Cohen-Macaulay type via the category of finitely presented functors from the category of finitely presented Gorenstein projective \(\Lambda\)-modules to the category of abelian groups. As an application, we provide a necessary and sufficient condition for \(T_3(\Lambda)\), the \(3\) by \(3\) lower triangular matrices over \(\Lambda\), to be of finite Cohen-Macaulay type.

The talk is based on a joint work with R. Hafezi and Z. Karimi.

### Emily Barnard

#### Pop-stack sorting for torsion classes

The pop-stack operator of a finite lattice L is the map that sends each element x in L to the meet of x together with the elements covered by x. We focus on the lattice of torsion classes of a tau-tilting finite algebra over a field K, where we describe the pop-stack operator in terms of certain mutations of 2-term simple-minded collections. This allows us to describe preimages of a given torsion class under the pop-stack operator. This talk will focus on examples and open questions.

### Raphael Bennett-Tennenhaus

#### String algebras over local rings: admissibility and biseriality

String algebras are classically admissible quotients of path algebras over fields. Path algebras have also been considered over any noetherian local ground ring. Raggi-Cardenas and Salmeron generalised the definition of an admissible ideal in this context. A generalisation of string algebras from my PhD thesis likewise replaced the ground field with a local ring. This definition relates to admissibility, and yields a class of rings that are biserial in a sense used by Kirichenko and Yaremenko.

In this talk I will look at an example of a string algebra over a local ring coming from metastable homotopy theory, following work of Baues and Drozd. I will also look at an example of a clannish algebra over a local ring that is related to modular representations of the Matheiu 11-group, following work of Roggenkamp. This is based on an arxiv preprint 2305.12885.

### Teresa Conde

#### Medley on exact Borel subalgebras

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. The plan is to touch on topics such as their existence and uniqueness, their behaviour under recollements as well as on methods to extract information about these subalgebras without knowing them a priori. This is partially based on joint work in progress with Julian Külshammer and also on joint work in progress with Steffen Koenig.

### Monica Garcia

#### Thick subcategories arising from semi-invariants

Determinantal semi-invariants are a specific class of regular functions defined on the varieties of modules of fixed-dimension vector over a finite-dimensional algebra. In this talk, we will recall their relation to projective presentations and their connection with semistable subcategories. We will show how these semi-invariants give rise to a certain class of thick subcategories within the extriangulated category of 2-term complexes, completing an extriangulated version of the correspondences among support \tau-tilting objects, torsion classes, and wide subcategories.

### Ana Garcia Elsener

#### Skew-Brauer graph algebras

joint work with Yadira Valdivieso (U Puebla, Mexico) and Victoria Guazzelli (UNMdP, Argentina)

We utilize surface orbifold dissections to arrive at the definition of skew-Brauer graph algebras, a class of symmetric finite dimensional algebras that generalizes the family of Brauer graph algebras. We show that the family of skew-Brauer graph algebras with multiplicity function equal to one coincides with the family of trivial extensions of skew-gentle algebras.

### Peter Jørgensen

#### The Q-shaped derived category (report on joint work with Henrik Holm)

A chain complex can be viewed as a representation of a certain quiver with relations, Q. The vertices are the integers, there is an arrow q → q − 1 for each integer q, and the relations are that consecutive arrows compose to 0. Hence the classic derived category D can be viewed as a category of representations of Q.

It is an insight of Iyama and Minamoto that D is well behaved because, viewed as a small category, Q has a Serre functor. Generalising the construction of D to other quivers with relations which have a Serre functor results in the Q-shaped derived category, D_Q.

Significant parts of the theory of D can be generalised to D_Q, and we will see a few.

### Henning Krause

#### The finitistic dimension of a triangulated category

The talk is devoted to introducing and explaining the finitistic dimension of a triangulated category. For the category of perfect complexes over a ring one can show that this dimension is finite if and only if the small finitistic dimension of the ring is finite.

### David Pauksztello

#### Is convex geometry trying to teach us homological algebra?

Arising in cluster theory, the g-vector fan is a convex geometric invariant encoding the mutation behaviour of clusters. In representation theory, the g-vector fan encodes the mutation theory of support tau-tilting objects or, equivalently, two-term silting objects. In this talk, we will describe a generalisation of the g-vector fan which in some sense “completes” the g-vector fan: the heart fan of an abelian category. This convex geometric invariant encodes many important homological properties: e.g. one can detect from the convex geometry whether an abelian category is length, whether it has finitely many torsion pairs, and whether a given Happel-Reiten-Smalo tilt is length. This talk will be a report on joint work with Nathan Broomhead, David Ploog and Jon Woolf.

### Jeanne Scott

#### A diagrammatic realization of the Okada algebra

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from \(\frak{S}_n\) to \(\frak{S}_{n-1}\). In 1975 Stanley discovered a similar lattice called the Young-Fibonacci lattice which was later realized as the Bratelli diagram of a family of algebras by Okada in 1994.

In joint work with Florent Hivert (Université Paris-Sud) we realize the Okada algebra \(\frak{F}_n\) as a diagram algebra with a multiplicative/monoid basis consisting of \(n\)-strand Temperley-Lieb diagrams, each equipped with a "height" labeling of its strands. The proof involves a diagrammatic version of Fomin's Robinson-Schensted correspondence for the Young-Fibonacci lattice. This basis is cellular, which affords us with a novel, diagrammatic presentation of the irreducible representations of \(\frak{F}_n\) (i.e. cell modules).

### Jan Šťovíček

#### On integral and spectral Picard groups of quivers

Given a finite and acyclic quiver Q, the derived Pickard group (= the group of tilting complexes of kQ-kQ-bimodules) over a field k was studied by Miyachi and Yekutieli and in some cases it was shown to coincide with the automorphism group of the repetitive quiver ZQ. The aim is to explain, using work of Álvaro Sánchez, that this group of automorphisms of ZQ in fact naturally acts on representations of Q in any stable infinity category. This among others allows to compute for a class of quivers the derived Picard group over the integers (based on work of Crawley Boevey) and conjecturally also over the spectra (work in progress with Moritz Rahn).

### Sonia Trepode

#### On m-cluster tilted algebras coming from a quiver of type \(E_p\)

In this talk we study m-cluster tilted algebras coming from a quiver of type \(E_p\). We develop an algorithm to find the silting complexes in the fundamental domain in the derived category of a hereditary algebra of type \(E_p\). In particular, we get a list of all m-cluster tilted algebras of type \(E_6\) for \(m=2, 3\). We prove that, for the m-cluster tilted algebras coming from a tilting complex, the ones coming from a quiver of type \(E_6\) consist of the \(m=1,2,3\) ones. We discuss analogies and differences between m-cluster tilted algebras and 1-cluster tilted algebras.

Joint work with Natalia Bordino and Ulises Pallero from Universidad Nacional de Mar del Plata.

### Jorge Vitória

#### Fishing for complements

During 2023, the question of whether every presilting complex over a finite-dimensional algebra admits a complement (to a silting complex) has been settled: the answer is no, and counterexamples are now fairly abundant. In this Flash Talk we reframe the question of finding a complement in terms of co-t-structures and show that, if we allow infinite-dimensional silting theory to enter the picture, then the problem of finding a complement has a positive answer. This is based on joint work with Lidia Angeleri Hügel and David Pauksztello.

### Kayla Wright

#### 🍩 Representation Theory of Marked Non-Orientable Surfaces 🍩

In this short talk, we will discuss joint work with Veronique Bazier-Matte and Aaron Chan on the beginnings of an additive categorification of quasi-cluster algebras. Quasi-cluster algebras are cluster-like algebras defined from a marked non-orientable surface. We utilize the orientable double cover of a non-orientable surface to define involutions that capture the combinatorics of the covering map. Using these involutions along with symmetric representation theory developed by Derksen-Weyman and Boos-Cerulli Irelli, we define a module and cluster category to a non-orientable surface. We provide a dictionary between topological and categorical data in our main results.

### Dong Yang

#### Some examples of sms-algebras

Let Q be a finite quiver and C be the (-2)-cluster category of Q. According to a remarkable result of Jørgensen, the extension closure of a 2-simple-minded system in C is an abelian category and is equivalent to the category of finite-dimensional modules over some finite-dimensional algebra A. We call such A an sms-algebra. In this talk I will present two results on sms-algebras, in particular, which algebras are sms-algebras of Dynkin type A_n. This is a joint work with Zongzhen Xie.