# Program 2021

All times are in CET (UTC+1).

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

11:00-11:05 | Introduction |

11:05-11:15 | Lidia Angeleri-Hügel |

11:20-11:30 | Jenny August |

11:35-11:45 | Andrea Solotar |

11:50-12:00 | Yu Zhou |

Coffee Break | |

12:30-12:40 | Haibo Jin |

12:45-12:55 | Alexandra Zvonareva |

13:00-13:10 | Matthew Pressland |

13:15-13:25 | Sondre Kvamme |

Lunch Break | |

14:30-14:40 | Amit Shah |

14:45-14:55 | Laertis Vaso |

15:00-15:10 | Hipolito Treffinger |

15:15-15:25 | Gustavo Jasso |

Coffee Break | |

16:00-16:10 | Emine Yıldırım |

16:15-16:25 | Raquel Coelho Simoes |

16:30-16:40 | Sira Gratz |

16:45-16:55 | Gordana Todorov |

## Abstracts

### Lidia Angeleri-Hügel

#### Cosilting mutation

We present a notion of mutation for large cosilting objects in triangulated categories. In this context, mutation is not always possible: it is controlled by properties of certain torsion pairs in the heart of the t-structure induced by the cosilting object. In the case of a two-term cosilting complex in the derived category of a finite dimensional algebra \(A\), we will explain how these constraints are reflected in the lattice \(\operatorname{tors}A\) of all torsion pairs in the category \(\bmod A\) formed by the finite dimensional \(A\)-modules. The talk is based on ongoing joint work with Rosanna Laking, Jan Šťovíček and Jorge Vitória.

### Jenny August

#### Silting for Weakly Symmetric Algebras

It is very well-known that for a symmetric finite-dimensional algebra, the concepts of silting and tilting coincide. In this short talk, I will explain to what extent this is true if you relax the symmetric condition to weakly symmetric. This is joint work with Alex Dugas.

### Raquel Coelho Simoes

#### Simple-mindedness in negative Calabi-Yau cluster categories of hereditary type

Simple-minded systems were introduced by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories. In this talk, we will consider simple-minded systems in certain orbit categories of the bounded derived category of an hereditary algebra, which can be considered to be negative Calabi-Yau cluster categories. Simple-minded systems in this setting turn out to play the role of cluster-tilting objects in the positive Calabi-Yau setup. We will give a bijection between simple-minded systems, simple-minded collections in the fundamental domain and positive (higher) noncrossing partitions. These results generalise results of Iyama-Jin for Dynkin type. This is based on joint work with David Pauksztello and David Ploog.

### Sira Gratz

#### Grassmannians, Cluster Algebras and Hypersurface Singularities

Grassmannians are objects of great combinatorial and geometric beauty, which arise in myriad contexts. Their coordinate rings serve as a classical example of cluster algebras, as introduced by Fomin and Zelevinsky at the start of the millennium.

Jensen, King and Su construct an additive categorification of these Grassmannian cluster algebras via maximal Cohen-Macaulay modules over certain plane curve singularities. Such a Grassmannian cluster category encodes key aspects of the cluster structure on the respective coordinate ring of a Grassmannian. Notably, Plücker coordinates naturally correspond to rank 1 modules. An interesting aspect of this relation is that it affords a formal connection between two famous examples of a priori unrelated ADE classifications, providing a bridge between skew-symmetric cluster algebras of finite type and simple plane curve singularities.

In this flash talk, we take the above ideas to the limit: Taking colimits of Grassmannian cluster algebras give rise to Grassmannian cluster algebras of infinite rank. We explore these structures combinatorially, and construct an infinite rank analogue of Jensen, King and Su’s Grassmannian cluster categories via maximal Cohen-Macaulay modules over certain hypersurface singularites – the Grassmannian categories of infinite rank. In particular, we investigate how Plücker coordinates are in natural correspondence with generically free modules of rank 1.

This talk is based on joint work with Grabowski, and with August, Cheung, Faber, and Schroll.

### Gustavo Jasso

#### Deriving a theorem of Ladkani

I will explain how to leverage the language of stable infinity-categories to obtain a simple proof of a theorem of Ladkani concerning derived equivalences between upper-triangular matrix rings. The methods we utilise allow us to extend Ladkani's theorem from ordinary rings to DG rings and ring spectra.

### Haibo Jin

#### Recollements and localisation theorems

For a recollement of compactly generated triangulated categories, Neeman gave a localisation theorem stating that it induces a short exact sequence, up to direct summands, of full subcategories of compact objects. In this flash talk, we consider the recollement of unbounded derived categories and give other localisation theorems on bounded derived categories and singularity categories. This is a joint work in progress with Dong Yang and Guodong Zhou.

### Sondre Kvamme

#### Admissibly finitely presented functors for an exact category

The Auslander correspondence characterizes the category of finitely presented functors on a module category of a finite-dimensional algebra, and is an important result in representation theory. In this talk we will introduce the category of admissibly finitely presented functors and explain how it gives a version of Auslander correspondence for any exact category.

### Matthew Pressland

#### Auslander–Reiten translations for Gorenstein algebras: an observation

The category of Gorenstein projective modules for a finite-dimensional Gorenstein algebra has almost split sequences, and hence an Auslander–Reiten translation. Consequently, a Gorenstein projective module has two Auslander–Reiten translates, one in the subcategory of Gorenstein projectives, and one in the whole module category. In joint work with Sondre Kvamme, we relate these two translates, and in so doing shed new light on results of Garcia Elsener and Schiffler on Calabi–Yau tilted algebras.

### Amit Shah

#### Exangulated functors

In joint work with R. Bennett-Tennenhaus, we introduced n-exangulated functors, which are structure-preserving functors between n-exangulated categories. When n=1, these allow us to formalise statements like “short exact sequences induce triangles”. n-exangulated functors are in one-to-one correspondence with certain functors between the categories of extensions that arise from the n-exangulated categories involved. My aim of this talk is to describe some joint work with R. Bennett-Tennenhaus, J. Haugland and M. H. Sandøy, in which we are working on formalising the statement “this n-exangulated category sits inside this m-exangulated category so that their structures are compatible”.

### Andrea Solotar

#### Han's conjecture for bounded extensions

The global dimension of an associative algebra \(A\) over a field is a measure of the complexity of its representations. Han's conjecture relates the global dimension to the Hochschild homology of the algebra.

Let \(B \subseteq A\) be a bounded extension of finite dimensional algebras. I will use the Jacobi-Zariski long nearly exact sequence to show that \(B\) satisfies Han's conjecture if and only if \(A\) does, regardless if the extension splits or not. I will also give conditions ensuring that an extension by arrows and relations is bounded and comment examples of non split bounded extensions.

This talk contains joint work with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos.

### Hipolito Treffinger

#### On higher torsion classes

The study of higher homological algebra has started by Iyama in the late 2000's. This new theory quickly attracted a lot of attention, with many authors generalising classical notions to the setting of higher homological algebra. Examples of such generalisations are the introduction of n-abelian categories by Jasso and n-torsion classes by Jørgensen. Recently, it was shown by Kvamme and, independently, by Ebrahimi and Nasr-Isfahani that every small n-abelian category is the n-cluster-tilting subcategory of an abelian category.

In this talk we will talk about the relation of the n-torsion classes in an n-abelian category $\mathcal{M}$ and the torsion classes of the abelian category $\mathcal{A}$ in which $\mathcal{M}$ can be embedded. Moreover we will speak about some properties that can be deduced from this relation. Time permitting, we will also mention some results on functorially finite n-torsion classes.

Some of the results presented in this talk are part of a joint work with J. Asadollahi, P. Jørgensen and S. Schroll. The rest corresponds to an ongoing project in collaboration with J. August, J. Haugland, K. Jacobsen, S. Kvamme and Y. Palu.

### Gordana Todorov

#### Infinitesimal semi-invariant pictures and co-amalgamation

We study the local structure of the semi-invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co-amalgamation, we show that this local structure is completely described by the semi-invariant pictures of a collection of Nakayama algebras. Joint work with Eric Hanson, Moses Kim, Kiyoshi Igusa

### Laertis Vaso

#### n-cluster tilting modules for radical square zero algebras

Let \(K\) be a field. For a quiver \(Q\), we denote by \(J(Q)\) the ideal of the path algebra \(KQ\) generated by the arrows. In this talk, we give a characterisation of all bound quiver algebras of the form \(KQ/J(Q)^2\) that admit an \(n\)-cluster tilting module for some \(n>1\), based only on the shape of the quiver \(Q\).

### Emine Yıldırım

#### Discrete cluster categories of type A and beyond

Discrete cluster categories of type A are introduced by Igusa and Todorov. After briefly giving the description, we will talk about a completion of these categories. This is a joint work with Charles Paquette.

### Yu Zhou

#### A geometric model for the module category of a skew-gentle algebra

We realize skew-gentle algebras as algebras from ideal partial triangulations of punctured marked surfaces. Certain indecomposable modules (including all tau-rigid modules) correspond to tagged permissible curves, and dimensions of homomorphism spaces correspond to tagged intersections. As an application, we give a classification of support tau-tilting modules via maximal collections of non-crossing generalized tagged curves. This talk is based on joint work with Ping He and Bin Zhu.

### Alexandra Zvonareva

#### Derived equivalence classification of Brauer graph algebras

In this talk I will explain the classification of Brauer graph algebras up to derived equivalence. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. The classification relies on the connection between Brauer graph algebras and gentle algebras. We consider A-infinity trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary, this construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences. This is based on joint work with Sebastian Opper.