Program

All times are in CET (UTC+1).

Time Activity
11:00-11:05 Introduction
11:05-11:15 Bin Zhu
11:20-11:30 Rudradip Biswas
11:35-11:45 Esha Gupta
11:50-12:00 Yann Palu
Coffee break
12:30-12:40 Can Wen
12:45-12:55 Fiorela Rossi Bertone
13:00-13:10 Erik Darpö
13:15-13:25 David Nkansah
Lunch break
14:30-14:40 Steffen Oppermann
14:45-14:55 Eleonore Faber
15:00-15:10 Carlo Klapproth
15:15-15:25 Esther Banaian
Coffee break
16:00-16:10 Sofia Franchini
16:15-16:25 Khrystyna Serhiyenko
16:30-16:40 Karin M. Jacobsen
16:45-16:55 Thomas Brüstle
Break
18:00-20:00 Pub Quiz

Abstracts

Esther Banaian

Varieties of Quiver Representations and Juggling Sequences

Fixing a quiver Q and a dimension vector d, the variety of representations of Q with dimension vector d can be decomposed into orbits which correspond to all vector partitions of d into dimension vectors of the indecomposable representations of Q. If we fix Q to be a Dynkin quiver, so that indecomposable representations are in bijection with positive roots, then these vector partitions are known as Kostant partitions. This latter object has been an object of recent study and in particular Benedetti, Hanusa, Harris, Morales, and Simpson showed a bijection between Kostant partitions and (multiplex) juggling sequences. Applying this perspective to the world of quiver representations allows us to label orbits with juggling sequences. We present consequences of this connection, including a novel partial order on juggling sequences and formulas for the dimension of an orbit in terms of the juggling sequence.

Esther Banaian's slides

Fiorela Rossi Bertone

Maurer-Cartan equation for gentle algebras

It is well known that the deformations of algebras are in correspondence with the Maurer-Cartan elements. In this talk, for a gentle algebra A, we will consider the \(L_\infty\) structure of the Bardzell complex and establish conditions on the quiver of A that allow us to ensure the nilpotency of the brackets and calculate the Maurer-Cartan elements of A.

This talk is based on a joint work with Monique Müller, María Julia Redondo, and Pamela Suarez.

Fiorela Rossi Bertone's slides

Rudradip Biswas

Bounded t-structures and findim


I will give a survey of my work with Chen, Manali Rahul, Parker and Zheng on bounded t-structures and a notion of finitistic dimension for triangulated categories. The focus will be on applications to artin algebras.

Thomas Brüstle

An exact structure approach to almost rigid modules

The recently introduced notion of almost rigid modules for type A is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid modules are in bijection with triangulations of a polygon. In this talk, we give a different realization of maximal almost rigid modules using a non-standard exact structure such that the maximal almost rigid modules are exactly the maximal rigid modules relative to the new exact structure.

This talk is based on joint work with Eric Hanson, Sunny Roy, and Ralf Schiffler.

Thomas Brüstle's slides

Erik Darpö

Recursive description of cluster-tilting objects in higher cluster categories of types A and D

Let \(d,d',l,l'\) be integers \(\ge2\). In case \((d'-1)/(d-1) = (l+1)/(l'+1)\in\mathbb{Z}\), the geometric description of \(d\)-cluster categories in terms of diagonals in polygons (Baur–Marsh 2008) gives rise to an embedding of objects of \(\mathscr{C}_{d'}(A_{l'})\) into \(\mathscr{C}_{d}(A_l)\).

In this talk, we shall see that this map comes from a functor, which gives rise to a recursive construction of cluster-tilting objects in larger (but lower) cluster categories from cluster-tilting objects in smaller (but higher) ones. Similar results hold for type \(D\).

Erik Darpö's slides

Eleonore Faber

Frieze patterns from the \(A_\infty\)-singularity and Penrose tilings

In this talk we investigate how to get frieze patterns from the \(A_\infty\)-curve singularity, via the category of graded maximal Cohen-Macaulay modules \(CM^{\mathbb{Z}}(\mathbb{C}[x,y]/(x^2))\). This is a Frobenius category and was studied in the context of categories for Grassmannian cluster algebras and triangulations of the infinity-gon by August, Cheung, Faber, Gratz, and Schroll. Extending the cluster character from work of Paquette and Yildirim to this setting we obtain a new type of infinite friezes that can be related to Penrose tilings. This is joint work in progress with Özgür Esentepe.

Eleonore Faber's slides

Sofia Franchini

A (-1)-Calabi-Yau version of Igusa-Todorov discrete cluster categories

Igusa-Todorov discrete cluster categories are an infinite discrete generalisation of the classical cluster category of type A. These are triangulated categories having 2-Calabi-Yau dimension. Our aim is to introduce a (-1)-Calabi-Yau version of Igusa-Todorov discrete cluster categories by stabilising the category of infinite discrete symmetric Nakayama representations.

Sofia Franchini's slides

Esha Gupta

Silting objects, torsion classes, and cotorsion classes

It is known that, for a finite-dimensional algebra, the poset of two-term silting objects is isomorphic to the poset of functorially finite torsion classes in the module category and to the poset of complete cotorsion classes in the homotopy category of two-term complexes of projectives. Moreover, this poset is a lattice when it is finite. We will look at a generalisation of these results to the case of d-term silting objects for d greater than or equal to 2. We will also discuss some examples, including type \(A_n\) where the lattices of such objects are counted by the Fuss-Catalan numbers.

Esha Gupta's slides

Karin M. Jacobsen

NP-completeness and (higher) tau-tilting theory

NP-complete problems are an important class of problems in computational complexity theory; in short they are problems that are hard to solve, but their solutions are easy to verify.

In connection to work with August, Haugland, Kvamme, Palu and Treffinger, we compute the set of maximal \(\tau_{d}\)-rigid pairs algorithmically. I will explain how this can be reduced to solving the maximal clique problem, one of Karp’s 21 NP-complete problems.

Karin M. Jacobsen's slides

Carlo Klapproth

A functorial approach to n-exact categories

Jasso recently introduced n-exact categories, which generalise exact categories (in the sense of Quillen) and put an axiomatic framework to n-cluster tilting subcategories of exact categories. Generalising ideas of Enomoto, we explain how to describe n-exact structures on a given idempotent complete category through subcategories of its functor category. Considering n-exact structures on the category of (graded) projective modules over a (graded) algebra we find connections to many interesting classes of (graded) algebras arising from (non)commutative algebraic geometry, for example regular commutative rings, (generalised) Artin-Schelter regular algebras and (twisted) Calabi-Yau algebras.

David Nkansah

Rank Functions in the Framework of Higher Homological Algebra

Chuang and Lazarev introduced the concept of rank functions on triangulated categories as a generalisation of classical work by Cohn and Schofield on Sylvester rank functions. In this talk, we propose a generalisation of this notion to the broader framework of higher homological algebra.

Steffen Oppermann

Representations occurring in H_0

Inspired by problems from topological data analysis, I will ask the question which linear representations of a quiver are obtained as summands of H_0 of a representation of this quiver in topological spaces.

My aim for this talk is to motivate the question, give a few initial observations on what can happen, and finally some partial answers for particularly nice quivers.

The talk is based on joint work with Ulrich Bauer, Magnus Botnan, and Johan Steen.

Yann Palu

Flips of dissections and mutations of silting objects

The rich combinatorics of triangulations, and their link to cluster algebras, has a lot to do with the existence of flips. Karin Baur and Raquel Coelho-Simões have shown that there is a deep relationship between dissections (=partial triangulations) and gentle algebras. Our aim in this talk is to explain what the flip of a dissection is, following Garver-MacConville and Manneville-Pilaud, and to categorify it by using mutation of silting objects. The categories used are certain "0-Auslander" exact categories constructed from gentle algebras.

This is joint work with Mikhaïl Gorsky, Hiroyuki Nakaoka, Arnau Padrol, Vincent Pilaud and Pierre-Guy Plamondon.

Yann Palu's slides

Khrystyna Serhiyenko

Consistent dimer models on surfaces with boundary

A dimer model is a quiver with faces embedded in a surface, which gives rise to a certain Jacobian algebra called dimer algebra. Consistent dimer models on tori have been studied extensively in the physics literature while those on the disk are related to Grassmannian cluster algebras. We define and investigate various notions of consistency for dimer models on general surfaces with boundary. Moreover, we prove that the dimer algebra of a strongly consistent dimer model may be used to categorify the cluster algebra given by its underlying quiver. This is joint work with Jonah Berggren.

Khrystyna Serhiyenko's slides

Can Wen

The first Hochschild cohomology groups under gluings

Stable equivalences occur frequently in the representation theory of finite dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ``gluing'' two vertices. More precisely, let \(A\) be a finite dimensional algebra of the form \(kQ/I\) and \(B\) is obtained from \(A\) by gluing a source vertex and a sink vertex, then \(A\) and \(B\) are stably equivalent. In this talk, based on the Bardzell's projective resolution, we will compare the first Hochschild cohomology groups of arbitrary finite dimensional quiver algebras under gluing two arbitrary vertices. In particular, we will show that, as Lie algebras, \(HH^1(A)\) is isomorphic to a quotient of \(HH^1(B)\) under the stable equivalence induced by gluing vertices. As a generalization, we also get some very similar results under gluing arrows in monomial case. This is a joint work with Yuming Liu and Lleonard Rubio y Degrassi.

Can Wen's slides

Bin Zhu

Silting interval reduction with applications

Silting subcategories and their mutations in extriangulated categories are introduced by Adachi and Tsukamoto. We will focus on a reduction technique for silting intervals in extriangulated categories, which is called ”silting interval reduction”. We will also present some applications.

This is a report on the joint work (arXiv:2401.13513) with Jixing Pan.

2025-01-15, Johanne Haugland