Titles & Abstracts

Claire Amiot (Université Grenoble Alpes)

Covering techniques for Fukaya categories of surfaces and orbifolds

Abstract. This is a joint work (partially in progress) with Pierre-Guy Plamondon. Following the work of Reiten and Riedtmann, to an A∞ category with strict action of a group, we associate an A∞ category: the skew-group category. We use this construction to define the Fukaya category of an orbifold seen as a quotient of a graded surface with an action of a homeomorphism of order 2 with fixed points. This construction can also be used to describe the universal cover of the Fukaya category of a graded surface as defined by Haiden, Katzarkov and Kontsevich. And any subgroup of the fundamental group of the surface permits to study intermediate covers of the Fukaya category, and permits to define morphisms associated to intersections.

Lidia Angeleri Hügel (Università di Verona)

Generic modules arising from stability

Abstract. Let A be a finite dimensional algebra and Θ ∈ K₀(proj A) a g -vector. The category of Θ-semistable modules is a wide subcategory of mod A which can be constructed from an interval [t_Θ, ¯t_Θ] in the lattice of torsion pairs in mod A. Here t_Θ and ¯t_Θ are the semistable torsion classes introduced by Baumann, Kamnitzer and Tingley. By general results in cosilting theory we can associate to this interval a closed rigid set M_Θ in the Ziegler spectrum of the unbounded derived category D (Mod A ). The overall aim of this talk is to determine M_Θ. We will focus in particular on the case when A is tame and Θ is a g -vector that is generically indecomposable and not rigid. We will see that there exists a generic module G determining the set M_Θ. As a consequence, we will obtain some interesting properties of the interval [t_Θ, ¯t_Θ] and the category of Θ-semistable modules. The talk is based on ongoing work with Rosanna Laking and Calvin Pfeifer.

Karin Baur (Ruhr-Universität Bochum)

Comparing additive and monoidal categorifications of Grassmannians

Abstract. Both additive and monoidal categorifications for cluster algebras are studied intensively. We study how these two are linked and how this relates to the reachability conjectures. On the monoidal side, we consider the approach of Hernandez and Leclerc to work with subcategories of the category of finite dimensional modules for a quantum affine algebra of type A. On the additive side, we consider the Grassmannian cluster categories introduced by Jensen–King–Su. As an application, we are able to give a construction of non-rigid modules for these categories. This is joint work with Changjian Fu and Jian-Rong Li.

Lara Bossinger (UNAM)

Cluster algebras for scattering amplitudes

Abstract. Scattering amplitudes are functions on configuration spaces that predict the probability of particle interactions. Typically these functions are generalized (Goncharov) polylogarithms whose differential structure is encoded in their symbol alphabet. Surprisingly, Goncharov, Spradlin, Vergu and Volovichin discovered that in the toy model N = 4 super Yang Mills the letters of the symbol alphabet are given by cluster variables of Gr(4, n ), n < 8. These results have recently been extended to more general theories like QCD where the cluster structures on partial flag varieties are relevant. I present results based on collaborations with Drummond–Glew (JHEP 2023, arXiv:2212.08931), Jianrong Li (arXiv:2408.14956) and Drummond–Glew–Gürdoğan–Wright (arXiv:2507.01015). [Slides] 

Thomas Brüstle (Université de Sherbrooke and Bishop's University)

Birth and death curves for modules over k[x,y]

Abstract. The critical points of a real-valued Morse function are closely tied to the birth and death events in the persistent homology of its sublevel sets. We aim to extend this perspective from one-parameter filtrations to the setting of two-dimensional Morse functions. On the algebraic side, this amounts to studying ℤ²-graded modules over the polynomial ring k[x, y ]. We introduce formal notions of birth curves and death curves, which capture the expected behavior for direct sums of certain indecomposable modules. A central tool in our analysis is the representation theory of gentle algebras. This is joint work with Steve Oudot, Luis Scoccola, and Hugh Thomas (arXiv:2505.13412). [Slides] 

Aslak Buan (NTNU)

Mutating τ-exceptional sequences 

Abstract. Inspired by τ-tilting theory, the concept of τ-exceptional sequences was introduced in joint work with Marsh. They are sequences of certain indecomposable modules, meeting some homological criteria. For hereditary algebras, they coincide with the classical exceptional sequences. In joint work with Hanson and Marsh, a mutation operation on τ-exceptional sequences was introduced. Properties of this mutation operation will be discussed, mostly based on joint work with Hanson and Marsh. [Slides]

Christof Geiss (UNAM)

Cluster structures on Grothendieck rings 

Abstract. Recall, that the Grothendieck group K₀(A) of a monoidal abelian category A has a natural ring structure. Hernandez and Leclerc introduced the notion of monoidal categorifiction of cluster algebras: Roughly speaking, in this case a cluster algebra A is realized as the Grothendieck ring K₀(A) = A of a monoidal, noetherian and artinian abelian category A in such a way, that the cluster monomials are classes of (real) simple objects. This notion is motivated by the study of representations of quantum affine algebras and related structures. Conversely, this can be seen as an additional structure on the Grothendieck ring. We will review several examples of Grothendieck rings with a cluster algebra structure in this sense and point out how it leads to new formulas for q-characters of the simple representations which correspond to cluster monomials. Finally, I will report on an ongoing project with B. Leclerc and D. Hernandez, where we are developing a cluster algebra structure on the Grothendieck ring K₀(A) where A is the (integral) category A = O^{sh}_ℤ = ⊕_{μ ∈ Pˇ} O^{μ}_ℤ for the shifted quantum affine algebras (U_q)^μ (Lg) associated to a complex simple Lie algebra g. This is closely related to the QQ-system in mathematical physics. [Slides]

Sira Gratz (Aarhus University)

Metric completions of discrete cluster categories

Abstract. Methods for generating new triangulated categories from old are notoriously few and far between. Neeman’s recent innovation allows one to complete with respect to suitable metrics on a triangulated category to construct a new triangulated category. This promises the opportunity to construct a plethora of new examples of triangulated categories, and we discuss examples relevant to the study of finite dimensional algebras. Up to now, explicit computations have all taken place within an existing ambient triangulated category. In this talk, based on joint work with Charley Cummings, we present an example where the computation can be done without this crutch.

Johanne Haugland (NTNU)

Higher Koszul algebras and the (Fg)-condition

Abstract. Determining when a finite dimensional algebra satisfies the finiteness property known as the (Fg)-condition is of fundamental importance in the celebrated and influential theory of support varieties. We give an answer to this question for higher Koszul algebras, generalizing a result by Erdmann and Solberg. We discuss how this gives a connection between the (Fg)-condition and higher homological algebra, and how this significantly extends the classes of algebras for which it is known whether the (Fg)-condition is satisfied. Time permitting, we demonstrate that the condition in particular holds for an important class of algebras arising from consistent dimer models. The talk is based on joint work with Mads H. Sandøy.

Bernhard Keller (Université Paris Cité)

On triprojective algebras

Abstract. The triprojective algebra of a Dynkin diagram Δ is a triangular gluing of three copies of the corresponding preprojective algebra. It has a canonical (connective) dg enhancement: the triprojective dg algebra. We will report on joint work with Miantao Liu and with Zhenhui Ding where we show that the category of Gorenstein projective dg modules over the triprojective dg algebra is the Higgs category (in the sense of Yilin Wu), which categorifies the cluster variety of triples of flags of type Δ. We also confirm a conjecture by Merlin Christ which is crucial for his approach to the categorification of the higher Teichmüller spaces of type Δ.

Andrzej Mróz (Nicolaus Copernicus University)

Discrete derived invariants for Gorenstein algebras

Abstract. By discrete derived invariants we mean derived invariants living in the Grothendieck groups associated with the derived category of a finite-dimensional associative algebra. Among these invariants there are: homological bilinear and quadratic forms (Euler forms), certain related parameters, and the Coxeter polynomial. These invariants were successfully studied so far mainly for algebras of finite global dimension. Recall classical groundbreaking ideas of Gabriel, Bongartz, Dlab–Ringel, Kac, Brüstle–de la Peña–Skowroński (and many others) which relate the representation type of an algebra and classifications of its indecomposable modules with the definiteness and roots of its Euler (or the related Tits) form. There are also known nice results describing the properties of the module and derived category encoded in the Coxeter polynomial (results of Happel, Lenzing, de la Peña and others). In our work we study these invariants in a more general context — for Gorenstein algebras (in fact, even more generally, for certain Hom-finite triangulated categories having a Serre functor). We will present results describing the properties of discrete derived invariants, among others, for prominent examples of Gorenstein algebras: gentle algebras and Brauer graph algebras, including the complete formula for the Coxeter polynomial of a gentle algebra in terms of its Avella-Alaminos and Geiss invariant.

Pierre-Guy Plamondon (Université Paris-Saclay)

On some affine schemes defined by a version of F-polynomials

Abstract. The F -polynomial of a representation is a generating series for Euler characteristics of its submodule Grassmannians. In this talk, I will present a variation on F -polynomials and use it to define an affine scheme with interesting properties. In simple cases, this recovers known varieties such as the moduli space of configurations of n points on a projective line. This is a report on a joint work with Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori and Hugh Thomas.

Markus Reineke (Ruhr-Universität Bochum)

Expander representations

Abstract. We discuss a definition of expander representations of quivers, generalizing dimension expanders, as a qualitative refinement of slope stability. We prove existence of uniform expander representations for any wild quiver over an algebraically closed base field, using the concept of general subrepresentations and spectral properties of Cartan matrices.

Jan Schröer (Universität Bonn)

Symmetries arising from the Auslander–Reiten formula

Abstract. For a finite-dimensional algebra A, the AR-formula tells us that the stable homomorphism spaces \overline{Hom} (M, τ(N )) and Hom (τ⁻(M ), N ) have the same dimension for all A-modules M and N. We discuss the condition that the dimensions of Hom (M, τ(M )) and Hom (τ⁻(M ), M ) coincide for all M. A prominent class of examples satisfying this condition are the Jacobian algebras associated to a quiver with potential. (This was proved by Derksen, Weyman and Zelevinsky, who also introduced Jacobian algebras.) We show that within certain classes of algebras, Jacobian algebras are the only algebras with this property. This is joint work with Grzegorz Bobiński.

Khrystyna Serhiyenko (University of Kentucky)

Reduction of syzygy categories

Abstract. A module is said to be a syzygy if it is a submodule of a projective. In the case of 2-Calabi–Yau (2-CY) tilted algebras, the non-projective syzygies form a triangulated 3-CY category. In this setting, the category of syzygies is equivalent to the category of Cohen–Macaulay modules and also the singularity category of the algebra. We investigate syzygy categories of 2-CY tilted algebras under deleting a vertex of a quiver and obtain an analog of the Iyama–Yoshino reduction. We also characterize when the syzygy category remains the same after deleting a vertex and provide applications to the study of dimer tree algebras and their skew group algebras. This is joint work with Ralf Schiffler.

Andrea Solotar (Universidad de Buenos Aires)

Hochschild cohomology of monomial algebras 

Abstract. The module and ring structure of Hochschild cohomology has been computed for several families of monomial algebras, like gentle algebras (Chaparro–Schroll–Solotar–Suárez Álvarez), radical square zero algebras (Cibils), truncated quiver algebras (Bardzell, Locateli, Marcos), among many others. For general monomial algebras, the problem is still open. Moreover, it is not known whether Happel’s question has a positive answer for monomial algebras or not. In this talk I will present some new results about this subject and also relate it with a τ-tilting analogue of this question. This is part of joint work with Claude Cibils, Marcelo Lanzilotta and Eduardo Marcos, and also with Dalia Artenstein, Janina Letz and Amrei Oswald. [Slides]

Véronique Bazier-Matte (Université Laval)

Knots and cluster algebras 

Abstract. This is joint work with Ralf Schiffler. To every knot diagram K, we associate a cluster algebra A containing a cluster x such that each cluster variable in x specializes to the Alexander polynomial of K. We identify this cluster x in A via a sequence of mutations constructed from a sequence of bigon reductions and generalized Reidemeister III moves on K. At the diagrammatic level, this sequence first reduces K to the Hopf link, then reflects the Hopf link to its mirror image, and finally reconstructs (the mirror image of) K by reversing the reduction. We prove that every diagram of a prime link admits such a sequence. [Slides]

Teresa Conde (Universität Bielefeld)

From Reedy categories to quasi-hereditary algebras

Abstract. Reedy categories play a key role in homotopy theory and in the study of diagram categories arising from model categories. Their algebraic counterpart — Reedy algebras — were recently introduced in work by Dalezios and Šťovíček as finite-dimensional algebras characterised by analogous structural properties, and they were shown to be quasi-hereditary. In this talk, we explore how Reedy algebras naturally appear as quasi-hereditary algebras admitting a triangular decomposition into a tensor product of two oppositely directed subalgebras over a common semisimple subalgebra. This decomposition also enables a characterisation of Reedy algebras in terms of idempotent ideals occurring in heredity chains. The talk is based on joint work with G. Dalezios and S. Koenig.

Monica Garcia (Université du Québec à Montréal)

Infinite super friezes 

Abstract. Super friezes were introduced by S. Morier-Genoud, V. Ovsienko, S. Tabachnikov as a supersymmetric analog of classical Coxeter friezes. They show analogous properties of classical friezes: they are determined by the first non-trivial even and odd quiddity rows, they satisfy linear recurrence relations, and exhibit glide symmetry when of finite width. Moreover, as shown by G. Musiker, N. Ovenhouse and S. Zhang, all finite width super friezes arise from a decorated triangulation of a polygon, where even entries correspond to λ-lengths of arcs, and odd entries to μ -invariants of triangles in the polygon. In this talk, I will report on joint work with A. Burcroff, İ. Çanakçı, F. Fedele and V. Klász on how to construct infinite super friezes from decorated skeletal triangulations of annuli. [Slides]

Esha Gupta (Université de Versailles Saint-Quentin-en-Yvelines)

From 2-term to d-term silting complexes 

Abstract. For a finite-dimensional algebra, it is known from the seminal work of Adachi–Iyama–Reiten that two-term silting complexes are in bijection with functorially finite torsion pairs and support τ-tilting pairs in the module category. Later, more classes were added to these bijections, including complete cotorsion pairs, left-finite semibricks, and left-finite wide subcategories. In this talk, we will generalise the above bijections to arbitrary d -term silting complexes by introducing "extended module categories" or "truncated derived categories". We then provide appropriate generalisations of torsion classes, semibricks, and wide subcategories to extriangulated categories to show that d -term silting complexes are in bijection with functorially finite positive torsion pairs and complete hereditary cotorsion pairs. For algebras satisfying certain finiteness conditions, these will also be in bijection with left-finite semibricks and left-finite wide subcategories in the extended module category. This talk is partially based on a joint work in progress with Yu Zhou. [Slides]

Eric Hanson (North Carolina State University)

Para-exceptional sequences for tame hereditary algebras 

Abstract. We introduce a generalization of exceptional sequences, which we call para-exceptional sequences, for the tame hereditary algebras. These sequences, and their associated wide subcategories, serve as a representation-theoretic model for an enlargement of the non-crossing partition poset (of an affine Coxeter group W and Coxeter element c ) constructed by McCammond and Sulway. Para-exceptional sequences also provide a representation-theoretic proof that the larger poset is a combinatorial Garside structure, and in particular is a lattice. This is a report on ongoing joint work with Nathan Reading. [Slides]

Viktória Klász (Universität Bonn)

Auslander–Gorenstein algebras and the Auslander–Reiten bijection

Abstract. Auslander and Reiten discovered that every Auslander–Gorenstein algebra admits a distinguished bijection between its indecomposable projective and injective modules, now known as the Auslander–Reiten bijection. In this talk, we present a new result showing that, for certain classes of algebras, the existence of such a bijection actually characterises the Auslander–Gorenstein property. We then discuss a new, linear algebraic interpretation of the Auslander–Gorenstein property and the Auslander–Reiten bijection using Coxeter matrices and their Bruhat decompositions, based on a joint work with René Marczinzik and Hugh Thomas. This approach opens the door to extending the definition of the Auslander–Reiten bijection to algebras that are not Auslander–Gorenstein.

Rosanna Laking (Università di Verona)

Wide intervals in cosilting theory 

Abstract. Wide subcategories of the category mod A of finite-dimensional modules over a finite-dimensional algebra A (i.e. subcategories that are closed under kernels, cokernels and extensions) are interesting abelian length subcategories of mod A that arise naturally in various contexts throughout the representation theory of A. In this talk we will be focusing on one such context: when non-trivial wide subcategories arise as intersections of a torsion class T and a torsion-free class V  in mod A. Such a pair T and V corresponds to an interval in the lattice of torsion pairs in mod A that Asai and Pfeifer call wide intervals. We will report on joint work with Lidia Angeleri Hügel and Francesco Sentieri in which we show that wide intervals are related to closed subsets of the Ziegler spectrum, as well as joint work with Lidia Angeleri Hügel, Jan Šťovíček and Jorge Vitória in which we show that they parametrise mutations of maximal rigid sets of indecomposable pure-injective modules. [Slides]

René Marczinzik (Universität Bonn)

Auslander regular algebras and the Coxeter permutation

Abstract. We survey recent developments concerning Auslander regular algebras, focusing on the role of the newly defined Coxeter permutation. We discuss its implications for the homological and combinatorial structure of incidence algebras of finite posets and Nakayama algebras, highlighting concrete examples and open problems. [Slides] 

Calvin Pfeifer (Universität zu Köln)

Serre cyclotomic algebras

Abstract. In 2013 de la Peña initiated the systematic study of algebras of cyclotomic type, that is finite-dimensional algebras of finite global dimension such that some power of their Coxeter matrix is unipotent. For example, fractionally Calabi–Yau algebras have periodic Coxeter matrices, and gained a lot of attention due to their connection with e.g. higher representation-finite algebras and Fukaya–Seidel categories. In this talk, we propose a class of algebras, which we call Serre cyclotomic, as a generalization of fractionally Calabi–Yau algebras, and as a categorification of algebras of cyclotomic type. We study dynamical properties of their Nakayama functors and the complexity of their trivial extension algebras. Based on recent work of Chang–Schroll, we characterize Serre cyclotomic gentle algebras. Finally, we provide further examples coming from generalized species. Parts of this ongoing work are joint with Sibylle Schroll.

Anna Rodriguez Rasmussen (Uppsala Universitet)

Exact Borel subalgebras of quasi-hereditary monomial algebras

Abstract. An exact Borel subalgebra of a quasi-hereditary algebra is a subalgebra whose homological properties mimick those of Borel subalgebras of Lie algebras. While the structure theory of such subalgebras uses rather abstract techniques involving A∞ algebras, in concrete cases Borel subalgebras can often be described in down-to-earth, combinatorial terms. In this talk, I will present such a description of exact Borel subalgebras for monomial quasi-hereditary algebras, building on the characterization of such algebras by Green and Schroll. In the monomial case, we obtain Reedy decompositions in the sense of Dalezios and Šťovíček, and an explicit criterion for regularity of exact Borel subalgebras. Using this criterion, we count the quasi-hereditary structures admitting regular exact Borel subalgebras for path algebras of types D and E, extending the results of Thuresson for type A.

Adam Skowyrski (Nicolaus Copernicus University)

Iterated mutations of periodic algebras

Abstract. In this talk I will present some new results concerning mutations of tame symmetric periodic algebras. Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, I applied them in a specific situation, namely, to study iterated mutations of (symmetric) periodic algebras. More precisely, for a tame symmetric algebra Λ, and an arbitrary vertex i  of its Gabriel quiver, one can define mutation μ ᵢ(Λ) of Λ at vertex i  via Okuyama–Rickard complexes. Then μ ᵢ(Λ) is again tame symmetric, and we can iterate this process. I am especially interested in studying mutations at vertices i such that the simple module S ᵢ at vertex i  is periodic of period d. The main result shows that then μ ᵢ has order d − 2, i.e. μ ᵢᵈ⁻² (Λ) = Λ under some additional assumption on the (periodic) projective–injective resolution of S ᵢ. I will try to show the main idea behind the proof and give the flavour of what is happening in the general case. Another emerging problem is to prove that the mutation class of a periodic algebra is finite. If time permits, I will explain what is known, and what are the major difficulties.

Grzegorz Zwara (Nicolaus Copernicus University)

On transversal slices of orbit closures for modules over representation finite algebras

Abstract. Let A be a representation-finite algebra and d  be a positive integer. The set mod (A, d ) of A-module structures on a d -dimensional vector space is an affine variety equipped with an action of the general linear group GL(d). Let M and N  be points in mod (A, d ) such that N  belongs to the orbit closure of M, denoted by C(M ). Based on the idea of transversal slices, we construct an affine variety C(M, N ) such that the pointed varieties (C(M ), N ) and (C(M, N ), 0) are smoothly equivalent, and dim C(M, N ) = dim C(M ) – dim C(N ).