# Program 2023

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

11:00-11:05 | Introduction |

11:05-11:15 | Osamu Iyama |

11:20-11:30 | Mandy Cheung |

11:35-11:45 | Xiao-Wu Chen |

11:50-12:00 | Charley Cummings |

Coffee break | |

12:30-12:40 | Petter A. Bergh |

12:45-12:55 | Lorna Gregory |

13:00-13:10 | Julian Külshammer |

13:15-13:25 | Aran Tattar |

Lunch break | |

14:30-14:40 | Bethany Marsh |

14:45-14:55 | Eric Hanson |

15:00-15:10 | Pierre-Guy Plamondon |

15:15-15:25 | Francesca Fedele |

Coffee break | |

16:00-16:10 | Drew Duffield |

16:15-16:25 | Véronique Bazier-Matte |

16:30-16:40 | Van Nguyen |

16:45-16:55 | Maria Julia Redondo |

Break | |

19:00-21:00 | Pub Quiz |

## Abstracts

### Véronique Bazier-Matte

#### Categorification of non-orientable surfaces

Quasi-cluster algebras were defined in 2015 by Dupont and Palesi and are an analogous to cluster algebras for non-orientable surfaces. In this talk, we will first give an introduction to these quasi-cluster algebras. Then, we will associate a quiver with potential to triangulations of non-orientable surfaces and study the algebra given by this. More precisely, we use the cluster category associated to an orientable double cover of our non-orientable surface to give a correspondence between quasi-triangulations of a non-orientable surface and an analog of cluster-tilting objects.

Joint work with Aaron Chan and Kayla Wright.

### Petter A. Bergh

#### Separable equivalences, finitely generated cohomology and finite tensor categories

We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik.

### Xiao-Wu Chen

#### The injective dimension of the Jacobson radical

It is very well known that for a non-semisimple artin algebra, the projective dimension of its Jacobson radical equals its global dimension minus one. In contrast, we prove that the injective dimension of its Jacobson radical always equals its global dimension, as conjectured by Rene Marczinzik (PAMS 2020). We will also discuss two possible extensions of the mentioned result. This is joint work with Srikanth Iyengar and Rene Marczinzik.

### Mandy Cheung

#### Cluster structures for the \(A_\infty\) singularity

Jensen-King-Su give a correspondence between finite type \(A\) cluster algebras and the hypersurface singularities. The combinatorics can be described by triangulations of regular \((n+3)\)-gons. In the correspondence, the indecomposable objects are related to arcs in the polygon, and triangulations give cluster tilting objects. Further, mutations are encoded by diagonal flips. It is natural to extend this construction to the infinity-gon. We are going to see that the category of \(\mathbb Z\)-graded MCM modules over the \(A_\infty\) curve singularity carries the infinite type A cluster combinatorics. And this category has cluster tilting subcategories modeled by certain triangulations of the (completed) infinity-gon. This is a joint work with Jenny August, Eleonore Faber, Sira Gratz, and Sibylle Schroll.

### Charley Cummings

#### Left-right symmetry of finite finitistic dimension

The finitistic dimension is a numerical invariant associated to a ring. This invariant need not be finite, but the longstanding finitistic dimension conjecture asserts that it is finite for finite dimensional algebras. Thirty years ago, Happel noted that it is unknown if the finitistic dimension conjecture is left-right symmetric. That is, if an algebra having finite finitistic dimension implies that its opposite algebra also has finite finitistic dimension. We show that, heuristically, it is unknown for good reason. The left-right symmetry of finite finitistic dimension is equivalent to the finitistic dimension conjecture.

### Drew Duffield

#### Categorifying Mutations of Non-Integer-Weighted Quivers and Exchange Matrices

The mutation of (integer) exchange matrices and corresponding quiver mutation is fundamental to the theory of cluster algebras and their categorifications. However, the mutation rule can also be applied to exchange matrices over more general rings than the integers. In this talk, we will present a brief overview of how (in certain situations) we can interpret the mutations of these non-integer exchange matrices and corresponding weighted quivers in a categorical context.

### Francesca Fedele

#### Representation theory of super cluster algebras

The super cluster algebra of type \(A_n\) is a \(\mathbb{Z}_2\)-graded algebra generated by a set of even variables \(\underline{x}\) and a set of odd variables \(\underline{\theta}\). Its geometric model, introduced by Musiker, Ovenhouse and Zhang, is given by an oriented triangulation \(T\) (without internal triangles) of the disk with \(n+3\) marked points: the arcs of \(T\) are in bijection with the initial even variables and the triangles with the initial odd variables. Similarly to the classic cluster algebra case, the super cluster variables can be computed combinatorially using a snake graph formula. This work in progress project with Canakci, Garcia Elsener and Serhiyenko gives a representation theoretic interpretation of these algebras. We show the super cluster variables are in bijection with the indecomposable induced modules over the algebra \(kA_n\otimes k[\epsilon]\) by constructing a super Caldero-Chapoton map.

### Lorna Gregory

#### Decidability and Representation Type

I will explain a conjecture of Prest which connects the representation type of a finite-dimensional algebra with the decidability of its theory of modules.

### Eric Hanson

#### Poset topology of wide subcategories

Let \(A\) be a tau-tilting finite algebra. Then each wide subcategory of \(\mathrm{mod}(A)\) can be realized as one of Jasso's \(\tau\)-perpendicular categories. In this talk, we explain how this leads to a "labeling" of minimal inclusions of wide subcategories by bricks and how this labeling relates to Buan and Marsh's \(\tau\)-exceptional sequences. We then discuss some of the combinatorial and topological properties of this labeling. This talk is based on joint work with Emily Barnard.

### Osamu Iyama

#### Title: Complete \(g\)-fans of rank \(2\)

Abstract: For each finite dimensional algebra \(A\), \(2\)-term silting complexes of \(A\) give rise to a nonsingular fan in the real Grothendieck group of \(A\), which we call the \(g\)-fan of \(A\). It is complete if and only if \(A\) has only finitely many basic \(2\)-term silting complexes. An important problem in tilting theory is to classify complete \(g\)-fans. In this talk, we give an answer for rank \(2\) case. More explicitly, we show that complete \(g\)-fans of rank \(2\) are precisely sign-coherent fans of rank \(2\). This is a joint work with T. Aoki, A. Higashitani, R. Kase and Y. Mizuno.

### Julian Külshammer

#### The monomorphism category and representations in the stable category

Given an algebra \(A\) and a quiver \(Q\), modules over \(kQ\otimes A\) can be seen as representations of \(Q\) in the category of \(A\)-modules. The associated monomorphism category is the full subcategory consisting of those representations such that for every vertex \(i\), the direct sum of the maps coming into \(i\) is a monomorphism. The origins of this exact category go back to work of Birkhoff, but recently there is been renewed interest, in particular from topological data analysis. It turns out that classification of the indecomposables in the monomorphism category can be reduced to representations of \(Q\) in the stable module category of \(A\). As a consequence, in the case of \(A\) being a radical-square zero Nakayama algebra, there is a Gabriel-type classification of finite monomorphism type. This is joint work with Nan Gao, Sondre Kvamme, and Chrysostomos Psaroudakis.

### Bethany Rose Marsh

#### Properties of tau-exceptional sequences

Joint work with Aslak Bakke Buan. Exceptional sequences over finite dimensional algebras do not always behave well: there are algebras for which complete exceptional sequences do not exist. Buan and the speaker introduced the notion of a tau-exceptional sequence and a signed version for which complete sequences always exist, motivated by the tau-tilting theory, the signed exceptional sequences of Igusa-Todorov and the link to picture groups (Igusa-Todorov-Weyman). In further recent joint work with Buan, we establish some properties of tau-exceptional sequences, including a description of a symmetric group action and a type of mutation.

### Van Nguyen

#### On deformations of quantum groups

In this talk, I will describe various ways of deforming the multiplicative structure of a graded (Hopf) algebra, how the categories of (co)modules behave under such deformations, and how these deformations can be related to each other. This is joint work with Hongdi Huang, Charlotte Ure, Kent B. Vashaw, Padmini Veerapen, and Xingting Wang.

### Pierre-Guy Plamondon

#### On the category of projective presentations

The category of projective presentations, or of 2-term complexes of projectives, behaves in many ways like a nice module category: it has enough projectives, enough injectives, it has Auslander-Reiten theory, it is hereditary, and so on. However, in many ways it is very different: most strikingly, it is not abelian but extriangulated. Nevertheless, many of the constructions from representation theory apply or can be adapted to it. In this talk, we will review many of the properties of the category of projective presentations, highlighting the similarities and differences with modules categories.

### Maria Julia Redondo

#### Infinitesimal deformations

Let \(f\) be a Hochschild 2-cocycle and \(A_f\) an infinitesimal deformation of a finite-dimensional associative \(k\)-algebra \(A\). We describe, under some conditions on \(f\), the algebra structure of the Ext-algebra of \(A_f\) in terms of the Ext-algebra of \(A\). We achieve this description by getting an explicit construction of minimal projective resolutions. This is based on joint work with L. Román and F. Rossi Bertone.

### Aran Tattar

#### Chains of torsion classes and weak stability conditions

Based on joint work-in-progress with Hipolito Treffinger. Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov's stability conditions. In this talk, we show an explicit relation between chains of torsion classes and weak stability conditions in an abelian category and discuss the structure of the space of chains of torsion classes.