# Program

Time | Tuesday 28th | Wednesday 29th | Thursday 30th | Friday 31st |
---|---|---|---|---|

09:00–09:45 | Laking | Pauksztello | Palu | Baur |

10:00–10:45 | Jacobsen | Vitória | Coelho Simões | Gratz |

10:45–11:15 | Coffee | |||

11:15–12:00 | Külshammer | Rock | Thomas | Todorov |

12:15–13:00 | Zvonareva | Jasso | Valdivieso Díaz | Plamondon |

13:00–15:00 | Lunch |
|||

15:00–15:45 | Fedele | Hanson | ||

16:00–16:45 | Yıldırım | Barnard | ||

16:45–17:15 | Coffee | Coffee | ||

17:15–18:00 | Pressland | Vaso | ||

18:30– | Dinner |

The conference dinner on Thursday, and all lunches will take place at CAS

## Abstracts

### Emily Barnard

#### The Combinatorics and Poset Topology of \(\tau\)-Exceptional Sequences

Let \(\Lambda\) be a \(\tau\)-tilting finite algebra. We describe how Buan and Marsh's \(\tau\)-exceptional sequences give rise to an EL-labeling of the lattice of wide subcategories for any representation-directed algebra. This result generalizes an EL-labeling of Athanasiadis for the \(W\)-noncrossing partition lattice from cluster combinatorics. We then show how \(\tau\)-exceptional sequences may be obtained from the lattice of torsion classes via a combinatorial map on bricks, which we call the kappa-map. This work is joint with Eric Hanson.

### Karin Baur

#### Friezes and Grassmannian cluster structures

In this talk, I will show how to obtain \(\mathrm{SL}_k\)-friezes using Plücker coordinates by taking certain subcategories of the Grassmannian cluster categories. These are cluster structures associated to the Grassmannians of \(k\)-dimensional subspaces in \(n\)-space. Many of these friezes arise from specialising a cluster to 1. We use Iyama–Yoshino reduction to reduce the rank of such friezes.

### Raquel Coelho Simões

#### Maximal almost rigid modules over gentle algebras

In recent work, gentle algebras were realised in terms of partial triangulations of Riemann surfaces. The surface and a partial triangulation provide geometric models for the module category and derived category of the corresponding gentle algebra. In the geometric model for the module category, indecomposable modules correspond to so-called permissible arcs. It is natural to ask whether there is any representation-theoretic meaning to triangulations of the surface by permissible arcs. It turns out these correspond to maximal almost rigid modules, which were previously studied in type A in the context of Cambrian lattices by E. Barnard, E. Gunawan, E. Meehan and R. Schiffler.

In this talk, I will discuss properties of maximal almost rigid modules over gentle algebras and corresponding endomorphism algebras. This is joint work in progress with E. Barnard, E. Gunawan and R. Schiffler.

### Francesca Fedele

#### Representation theory of "super" cluster algebras of type A

In the study of cluster algebras, computing cluster variables explicitly is an important problem. For surface cluster algebras, one can do so combinatorially using dimer covers of snake graphs. Recent work by Musiker, Ovenhouse and Zhang extend the theory to "super" cluster algebras of type A. The authors give a combinatorial formula, using double dimer covers of snake graphs to compute super lambda lengths in Penner–Zeitlin’s super Teichmuller spaces. In the classic surface cluster algebras setting, one can alternatively use a representation theoretic approach to compute cluster variables using the CC-map. Motivated by this, we introduce a representation theoretic interpretation of super lambda-lengths and a super CC-map which agrees with the combinatorial formula by Musiker, Ovenhouse and Zhang. This is a joint work in progress project with Canakci, Garcia Elsener and Serhiyenko.

### Sira Gratz

#### Pro-clusters and ind-Grassmannians

We define the notion of a pro-cluster algebra; a cluster-like structure on a subring of an appropriate inverse limit of cluster algebras. Our motivation stems from coordinate rings of ind-varieties, and we look at pro-cluster algebra structures on subrings of coordinate rings of certain ind-Grassmannians, including the Sato Grassmannian. We delve deeper into the type A case, and show that we obtain pro-cluster algebras from inverse systems of type A cluster algebras, whose pro-clusters are precisely the triangulations of a (completed) infinity-gon. This relates to the combinatorics of infinite Grassmannian cluster categories. This talk is based on work in progress with Korff and Wierzbicki, and on Wierzbicki’s PhD thesis.

### Eric Hanson

#### \(\tau\)-exceptional sequences over preprojective algebras of type A

The exceptional sequences over an arbitrary finite-dimensional algebras often do not behave well (if they even exist at all). Using \(\tau\)-tilting theory, Buan and Marsh introduced a theory of \(\tau\)-exceptional sequences, which inherit many of the nice properties of the exceptional sequences over hereditary algebras. In this talk, we establish an explicit combinatorial model for the \(\tau\)-exceptional sequences over preprojective algebras of type A. We then show how this model recovers and extends known results about Reading's shard intersection order on the symmetric group. This talk is based on joint work with Xinrui You.

### Karin M. Jacobsen

#### The lattice of higher torsion classes

Higher torsion classes were introduced by Peter Jørgensen in 2016 as the generalisation of torsion classes to the setting of higher homological algebra. In joint work with August, Haugland, Kvamme, Palu and Treffinger, we showed that higher torsion classes in a d-abelian category can be characterised as subcategories closed under \(d\)-cokernels and \(d\)-extensions.

In this talk I will discuss the lattice structure of the set of higher torsion classes and some of the lattice theoretic properties it does and does not fulfil. In particular I will talk about the lattice of higher torsion classes for higher Auslander algebras, which we compute explicitly.

The talk is based on the preprint arXiv:2301.10463.

### Gustavo Jasso

#### (Non-)uniqueness of strong DG enhancements

The Derived Auslander-Iyama Correspondence, a recent theorem of Muro and myself, guarantees the existence of unique (DG) enhancements for algebraic triangulating categories that satisfy mild finiteness conditions and admit a d\(\mathbb{Z}\)-cluster tilting object. I will explain a by-product of our work that permits us to construct, to the best of our knowledge, the first examples of algebraic triangulated categories that admit a unique enhancement but not a unique strong enhancement in the sense of Lunts and Orlov. This talk is based on joint work with Fernando Muro (Sevilla).

### Julian Külshammer

#### Exact Borel subalgebras and \(A_n\)-combinatorics

Quasi-hereditary algebras were introduced by Cline–Parshall–Scott in an attempt to capture certain phenomena in Lie theory. They come equipped with a partial order on the set of isomorphism classes of simple \(A\)-modules. Hereditary algebras can be characterised by the fact that they are quasi-hereditary with respect to every (adapted) partial order. In joint work with Koenig and Ovsienko we showed that up to Morita equivalence, every quasi-hereditary algebra admits a basic regular exact Borel subalgebra, a notion inspired by Lie theory and the theory of bocses. In this talk, I will present joint work with Miemietz proving that the pair of a quasi-hereditary algebra with a basic regular exact Borel subalgebra is unique up to isomorphism. I will illustrate the results with the example of path algebras of Dynkin type A quivers obtained by Thuresson.

### Rosanna Laking

#### Mutation of cosilting pairs

In joint work with Angeleri Hügel, Stovieck and Vitória, we show that minimal inclusions of torsion classes in the category of finite-dimensional modules corresponds to irreducible mutations of associated two-term cosilting complexes of (possibly infinite-dimensional) modules. While this provides a conceptual framework for the minimal inclusions of torsion classes in terms of mutation, it is difficult to compute concrete examples.

In ongoing joint work with Angeleri Hügel and Sentieri, we are able to produce more amenable characterisations of two-term cosilting complexes and their mutations. Moreover, by passing to their zeroth cohomology, we show that cosilting complexes are in bijection with pairs \((Z, I)\) where \(Z\) is a rigid set of indecomposable pure-injective modules and I is a set of indecomposable injectives (generalising support \(\tau^{-1}\)-tilting pairs). In fact, the sets \(Z\) are closed subsets of the Ziegler spectrum and in many cases the “mutable summands” of the cosilting complex coincide with the isolated points in the induced topology. In this talk I will present this module-theoretic perspective of cosilting mutation and some resulting combinatorics coming from the family of cluster-tilted algebras of type \(\widetilde{A}\).

### Yann Palu

#### The category of walks of a gentle algebra

Generalising work by Thomas McConville, we will introduce the non-kissing complex of a gentle algebra. We will then explain how to categorify its combinatorics using some 0-Auslander exact category that we call the category of walks. The non-kissing facets are in natural bijection with the maximal rigid objects of the category of walks, and this allows us to interpret their flips as a silting mutation. This talk is based on joint works with Vincent Pilaud and Pierre-Guy Plamondon and with Mikhail Gorsky and Hiroyuki Nakaoka.

### David Pauksztello

#### Simple-minded systems and positive noncrossing partitions

Module categories have two important types of generators: projective modules and simple modules. Morita theory describes equivalences of module categories in terms of images of projective modules. Tilting theory is the generalisation of Morita theory to derived categories describing equivalences of derived categories in terms of tilting objects. Tilting, silting and cluster-tilting objects, can be thought of as "projective-minded objects".

"Simple-minded objects" are generalisations of simple modules. They satisfy Schur’s lemma and a version of the Jordan–Holder theorem, depending on context, giving rise to "simple-minded collections" and "simple-minded systems". Although the theory of simple-minded objects shows many parallels with that of projective-minded objects, it remains relatively undeveloped and is technically more challenging. In this talk I will explain the connection between simple-minded systems in negative Calabi–Yau orbit categories of hereditary algebras, simple-minded collections in their bounded derived category, and positive noncrossing partitions, generalising results of Buan–Reiten–Thomas, Coelho Simões, and Iyama–Jin. This is a report on joint work with Raquel Coelho Simões and David Ploog.

### Pierre-Guy Plamondon

#### Solving \(u\)-equations using representation theory

Varieties defined by so-called \(u\)-equations, which arise from the computation of scattering amplitudes, have an interpretation in terms of the representation theory of associative algebras, and more precisely in terms of \(g\)-vectors and \(F\)-polynomials. I will present this interpretation in this talk, and focus in particular on the case of gentle algebras associated to grids. This is based on an ongoing project with N. Arkani-Hamed, H. Frost, G. Salvatori and H. Thomas.

### Matthew Pressland

#### Quasi-coincidence of cluster structures on positroid varieties

Positroid varieties are subvarieties of the Grassmannian that appear in the context of Postnikov's approach to total positivity. For this reason and others, it was long expected that their coordinate rings should carry a cluster algebra structure, and this expectation was finally confirmed by Galashin and Lam in 2019. In fact, Galashin and Lam provide two cluster algebra structures (for the price of one!) and while they are abstractly isomorphic, they are not equal, in the sense that the cluster variables are different sets of functions on the positroid variety in each case. A conjecture by Muller and Speyer from 2016 asserts a precise relationship between these two cluster structures, namely that they should quasi-coincide. This would imply in particular that the two sets of cluster monomials are in fact equal.

In this talk, I will outline a proof of the conjecture in the generic case that the positroid is connected. Perhaps surprisingly, the proof works by using categorification to translate the problem into homological algebra. Precisely, it uses the categorification of the cluster algebras by myself, that of perfect matchings and various other combinatorial ingredients from my joint work with Çanakçı and King, and that of quasi-cluster morphisms by Fraser, Keller and Yilin Wu.

### Daisie Rock

#### Composition Series of Arbitrary Cardinality in Modular Lattices

We discuss a generalization of a composition series from the perspective of lattice theory with motivation coming from abelian categories. We show sufficient conditions to obtain a composition series in a modular lattice when the lattice has arbitrary cardinality. Then we show sufficient conditions to prove that all such composition series in a modular lattice have the same “subfactors”. Along the way we will observe the statements for abelian categories using the lattice of subobjects. We then present some applications, a “weird” example, and a non-example; plus more examples if time permits. This is joint work with Eric J. Hanson.

### Hugh Thomas

#### An application of gentle algebras to perfectly clustering words

I will begin by explaining perfectly clustering words, which are a generalization due to Simpson and Puglisi of the better-known Christoffel words. I will then explain how perfectly clustering words correspond to a natural class of brick band modules for a certain gentle algebra. This correspondence illuminates some combinatorial aspects of the brick band modules, and allows us to resolve a conjecture about perfectly clustering words. This talk is based on joint work with Benjamin Dequêne, Mélodie Lapointe, Yann Palu, Pierre-Guy Plamondon, and Christophe Reutenauer, contained in the preprint arXiv:2301.07222.

### Gordana Todorov

#### Reverse Epsilon Construction for Nakayama Algebras

Emre Sen introduced notion of “syzygy filtrations” for Nakayama algebras which is also called epsilon construction. He used it to prove statements about several homological dimensions of Nakayama algebras: \(\varphi\)-dimension, Gorenstein dimension, finitistic dimension. In this joint work we consider the process of reversing epsilon construction. When applying this, while also preserving the defect of the algebras we create Nakayama algebras which are higher Auslander algebras, Auslander–Gorenstein and finitistic Auslander algebras, by increasing homological dimensions in a controlled way.

### Yadira Valdivieso Díaz

#### Skew-Brauer graph algebras

In this talk, we define skew-Brauer graph algebras, a generalization of the well-known Brauer graph algebras.

We show that in the same way, a Brauer graph algebras is defined from a graph with extra data on each vertex and the edges attached to it, a skew-Brauer graph algebra is also defined from a graph with some extra data that captures an \(\mathbb{Z}_2\)-action on gentle algebras. We also show that the trivial extension of any skew-gentle algebra is a skew-Brauer graph algebra. Finally, we present a geometric interpretation of the notion of admissible cuts of a trivial extension of skew-gentle algebras using dissections of orbifild surfaces. This is a joint work with Ana Garcia-Elsener and Victoria Guazelli.

### Laertis Vaso

#### 2-hereditary quadratic monomial algebras via one point extensions of bipartite graphs

This is report on joint work in progress with Karin M. Jacobsen and Mads Hustad Sandøy. I will explain how we can realise the problem of classifying 2-hereditary quadratic monomial algebras as a problem in linear algebra. I will show how in a special case this reduces to a system of Diophantine equations which we can solve and give an application to spectral graph theory.

### Jorge Vitória

#### Fishing for complements

A long-standing question in silting theory is that of whether every presilting complex is a summand of a silting complex. Recently, Liu and Zhou have provided an example of a presilting complex in the bounded derived category of finite-dimensional modules over a gentle algebra that cannot be completed to a silting complex. In this talk, we focus on a strategy to build complements and on what it tells us about their existence. In particular, it allows us to see that complements exist if we allow infinite-dimensional modules to enter the picture. Moreover, the same strategy allows us to rephrase the obstacle to a completion in the finite-dimensional world in terms of co-t-structures. This talk is based on joint work in progress with David Pauksztello and Lidia Angeleri Hügel.

### Emine Yıldırım

#### Cluster Expansions via matrices

Cluster algebras are commutative rings which are defined recursively from a set of initial generators (initial cluster variables), and all the other generators (all the other cluster variables) can be written in terms of the initial ones. There are different techniques to compute these generators, i.e. compute the expansion formulas for cluster variables. For instance, we may use the combinatorics of snake graphs, T-paths or so-called CC-map in the representation theory of algebras. In a joint work with E. Kantarcı Oğuz, we compute the cluster expansion formulas using 2 by 2 matrices for the cluster variables associated with arcs coming from surfaces. The method we introduce is quite efficient and can be generalised to other settings in cluster algebras and beyond.

### Alexandra Zvonareva

#### Functorial approach to rank functions

For a skeletally small triangulated category \(\mathcal{C}\), Chuang and Lazarev recently introduced the notion of a rank function on \(\mathcal{C}\). Roughly speaking, a rank function on \(\mathcal{C}\) is an assignment to each object of \(\mathcal{C}\) of a non-negative real number such that certain natural conditions hold. Such functions are closely related to functors into simple triangulated categories. On the other hand, to each skeletally small additive category \(\mathcal{C}\) one can associate its abelianisation mod-\(\mathcal{C}\). I will discuss the connection between rank functions on \(\mathcal{C}\) and translation-invariant additive functions on mod-\(\mathcal{C}\). This connection allows to study rank functions on triangulated categories using the machinery of additive functions on abelian categories, to classify integer rank functions in terms of certain objects, and to obtain nice decompositions. This is based on a joint work in progress with Teresa Conde, Mikhail Gorsky, and Frederik Marks.