13.30 Karin M Jacobsen

What are triangulations in the surface model?

To any gentle algebra we can associate an oriented marked surface with a dissection, giving us a geometric model of the bounded derived category, where curves taken up to homotopy correspond to indecomposable objects.

In joint work with Haugland, Schiffler and Schroll, we investigate the natural question: What do the triangulations of such a surface correspond to?

In answering the question, we arrive at a definition of maximal almost rigid subcategories in the bounded derived category. Furthermore we study interplay between algebraic and geometric mutation.

14.45 Sofia Franchini

A discrete cluster category having a negative Calabi-Yau parameter

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, i.e. their Ext spaces are symmetric.

Our aim is to introduce a (-1)-Calabi-Yau version of Igusa-Todorov discrete cluster categories. To do so, we define the category of infinite discrete symmetric Nakayama representations by using the notion of continuous Nakayama representations introduced by Rock and Zhu.

16.00 Greg Stevenson

Grothendieck categories: a survey

I'll give an introduction to Grothendieck categories and discuss some of their good properties. Some of these are well known, for instance the Gabriel–Popescu theorem, but others, such as the behaviour of the collection of localizations, are less well known.

Tea, coffee and cake will be provided.

Given a triangulated category T and a rigid subcategory R, Iyama–Yang put forward a mild technical condition that lets us compute the Verdier quotient T/thick(R). There are good reasons to extend Iyama–Yang's work to extriangulated categories, but one has to grapple with a more complicated theory of localisation. In this talk, we propose a generalisation of Iyama–Yang's work. More precisely, given an extriangulated category C and a rigid subcategory R giving rise to a generalised concentric twin cotorsion pair, we show that the Verdier quotient C/thick(R) can be expressed as an ideal quotient. If C is 0-Auslander, in the sense of Gorsky–Nakaoka–Palu, it suffices that C admits Bongartz completions. Moreover, the Verdier quotient C/thick(R) then remains 0-Auslander. The talk will be based on Section 5 in arXiv:2405.00593.

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 projectivess. Moreover, this poset is a lattice when it is finite. I will generalise these results to the case of d-term silting objects, in particular, showing that their poset is isomorphic to the poset of positive and functorially finite torsion classes in a truncated version of the derived category and to the poset of complete hereditary cotorsion classes in the homotopy category of d-term complexes of projectives. Moreover, these posets of torsion classes and cotorsion classes will indeed turn out to be lattices. I will also discuss some examples, including type A_n where these lattices are counted by the Fuss-Catalan numbers.

The Joker is a famous, very singular example of an endotrivial module over the 8-dimension subHopf algebra of the mod 2 Steenrod algebra generated by Sq^1 and Sq^2. It is known that this can be realised as the cohomology of two distinct Spanier-Whitehead dual spectra. Similarly, the double and iterated double are also realisable, but then the process stops. In the chromatic world, the double versions give rise objects whose Morava K-theory at height 2 involve endotrivial modules over the quaternion group of order 8 which lives inside the corresponding Morava stabilizer group. This gives a somewhat surprising connection between endotriviality in two different contexts. I will explain this from both an algebraic and a stable homotopy perspective, and discuss some possible generalisations and broader aspects.

An infinity-gon is a disc with infinitely many marked points in its boundary, with conditions on their accumulation points, which are unmarked. The cluster category of an infinity-gon was introduced by Igusa and Todorov, and has the property that its weak cluster-tilting subcategories are in natural bijection with the triangulations of the infinity-gon. Restricting to cluster-tilting subcategories, which must be functorially finite, requires some extra restrictions on the triangulation, given by Gratz, Holm and Jørgensen.

For complete infinity-gons, in which the accumulation points are marked, a corresponding cluster category was described by Paquette and Yıldırım (see also Cummings and Gratz), but very strong restrictions are needed on a triangulation of the completed infinity-gon for it to correspond to even a weak cluster-tilting subcategory. In this talk, based on joint work with İlke Çanakçı and Martin Kalck, I will explain how to resolve this problem, using extriangulated substructures.

Computing the structure of the Hochschild cohomology of an algebra can be hard work, but Etingof and Eu (2006) showed that it can be done surprisingly easily for preprojective algebras associated to ADE Dynkin diagrams, at least if you only want to know the graded vector space structure of each Hochschild cohomology group. Their method has since been used by Evans and Pugh (2012) on higher preprojective algebras that arise from “higher” ADE Dynkin diagrams, although the type A case was only recently completed by Morigi in his thesis (2022) up to assuming the truth of a conjectured formula for the determinant of the graded Cartan matrix of such an algebra.

In this talk, we present a generalization of the method used by Etingof and Eu (jt. with Jon W. Anundsen) obtained in part through a theory of projective resolutions of almost T-Koszul algebras (jt. with Johanne Haugland). In many cases, this generalization is easier to use, and we present applications to computations of Hochschild cohomology for other classes of higher preprojective algebras and how we can recover Morigi's results without the dependence on the conjectured formula mentioned above.

This work is in part motivated by potential applications of Hochschild cohomology to the periodicity conjecture and the conjectured acyclicity of the quivers of d-hereditary algebras.

The talk is based on joint work with Jon Wallem Anundsen and joint work with Johanne Haugland.

In joint work with Georgios Dalezios, we studied a K-linear analogy of the classical notion of Reedy category over a field K. Examples of such K-linear categories which are Hom-finite and with finitely many objects can be in a usual way encoded by finite dimensional algebras, which we call Reedy algebras. On one hand, we focused on representation theoretic properties of linear Reedy categories and especially Reedy algebras. In view of a recent result of Conde, Dalezios and Koenig, it turns out that Reedy algebras are precisely quasi-hereditary algebras with a so-called triangular decomposition. On the other hand, the raison d'être of ordinary Reedy categories is that model category structures lift to categories of diagrams of Reedy shapes, and we proved an analogous result about lifting cotorsion pairs to functor categories from linear Reedy categories.

13.30 Monica Garcia (Paris-Saclay University)

g-finiteness in the category of projective presentations

An algebra is said to be g-finite if it admits finitely many isomorphism classes of tau-tilting pairs. This notion was introduced and thoroughly studied by L. Demonet O. Iyama and G. Jasso, who showed that this property is equivalent to the module category admitting finitely many isomorphism classes of bricks (which is equivalent to having finitely many wide subcategories), finitely many functorially finite torsion classes, and equivalent to all torsion classes being functorially finite. Many of these concepts and their relationships have been shown to have counterparts in the extriangulated category of two-term complexes of projective modules. In this talk, we introduce new equivalent conditions to an algebra being g-finite in the context of the category of 2-term complexes. Namely, we establish that being g-finite is equivalent to the category of 2-term complexes admitting finitely many thick subcategories, finitely many complete cotorsion pairs and equivalent to all cotorsion pairs being complete.

14.45 Esther Banaian (Aarhus University)

Orbifold Markov Numbers

The Markov numbers are a family of positive integer solutions to a certain Diophantine equation, originally studied in the context of Diophantine approximation. They have been of considerable interest for the past century, due to their connections to various fields as well as Frobenius's famous (and still open) Uniqueness Conjecture. Markov numbers come in triples, and the set of all Markov triples is connected via an arithmetic rule that resembles mutation in a cluster algebra. This observation allows one to study Markov numbers in the context of both cluster theory and representation theory. After reviewing highlights of this study, we move on to discuss a related equation which is related to Chekhov and Shapiro's generalization of a cluster algebra arising from the study of Teichmüller spaces on orbifolds. This is partially based on joint work with Archan Sen (arXiv 2210.07366).

16.00 Emily Gunawan (University of Massachusetts Lowell)

Pattern-avoiding polytopes and Cambrian lattices via the Auslander–Reiten quivers

There is a bijection between type A Coxeter elements c and type A Dynkin quivers Q. For each type A Coxeter element c, we define a pattern-avoiding Birkhoff subpolytope whose vertices are the permutation matrices of the c-singletons. We show that the (normalized) volume of our polytope is equal to the number of longest chains in a corresponding type A Cambrian lattice. Our work extends a result of Davis and Sagan which states that the volume of the convex hull of the 132 and 312 avoiding permutation matrices is the number of longest chains in the Tamari lattice, a special case of a type A Cambrian lattice. Furthermore, we prove that each of our polytopes is unimodularly equivalent to the (Stanley's) order polytope of the heap H of the longest c-sorting word. The Hasse diagram of H is given by the Auslander—Reiten quiver for the quiver representations of Q. This talk is based on joint work with Esther Banaian, Sunita Chepuri, and Jianping Pan.

Tea, coffee and cake will be provided.

There is a well-known evaluation map from Hecke algebras in affine type A to those in finite type A, which allows us to pull back representations of the former to the latter. In this talk, I will explain how to categorify this to a monoidal functor from affine type A Soergel bimodules to the bounded homotopy category of finite type A Soergel bimodules, and give some applications to 2-representation theory. This is joint work with Marco Mackaay and Pedro Vaz.

In this talk we introduce a signed variant of (valued) quivers and a mutation rule, that generalizes the classical Fomin-Zelevinsky mutation of quivers. We associate to any signed valued quiver a Lie-theoretic object that is a signed analogue of the Cartan counterpart, from which we can construct root systems and a Lie algebra via a "Serre-like" presentation.

We then restrict our attention to the Dynkin case. For root systems, we show some compatibility results with the appropriate concept of mutation. Using results from Barot-Rivera and Perez-Rivera, we also show that signed quivers in the same mutation class yield isomorphic Lie algebras.

This is based on a joint project with Joe Grant (arXiv number 2403.14595).

For a finite-dimensional algebra A, being Auslander-Gorenstein is a homological condition which implies many interesting properties for the algebra and for certain subcategories of mod(A). In this talk, we will consider three well-known classes of algebras; namely, gentle, Nakayama, and monomial algebras, and aim to understand what the Auslander-Gorenstein property means in these settings. First, we will try to find a combinatorial characterisation of this homological condition, which leads us to a new class of examples of Auslander-Gorenstein algebras. Second, I will present a surprising new homological characterisation of the Auslander-Gorenstein property for these algebras. For this, a bijection between indecomposable projective and injective A-modules introduced by Auslander and Reiten plays a central role.

Frobenius algebras appear in many parts of maths and have nice properties. One can define algebra objects in any monoidal category, and there is a standard definition of when such an algebra object is Frobenius. However, this definition is quite restrictive, and it is not satisfied by an algebra object of interest, related to the preprojective algebra, in the Temperley-Lieb category at a root of unity. We will explore a more general definition of a Frobenius algebra object which covers this example, and will explore some of its properties. This is joint work with Mathew Pugh.

Higher Auslander—Reiten theory (also called higher homological algebra) was introduced by Iyama in 2007 as a generalization of classical Auslander—Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of modules categories. It turns out that many notions in algebra and representation theory have generalization to higher Auslander—Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.

In this talk I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type A_n.

Semilinear gentle algebras are path algebras over a division ring, where the underlying quiver with relations is subject to restriction similar to those for a “classical” gentle algebra. They are a type of semilinear clannish algebras studied by Bennett-Tennenhaus and Crawley-Boevey.

In this talk we will describe how the geometric models of gentle algebras extend to the semilinear case. Along the way we will also discuss how semilinear gentle algebras are nodal, and demonstrate how the Zembyk decomposition for nodal algebras can be interpreted geometrically.

Based on joint work (on the arxiv, 2402.04947) with Esther Banaian, Raphael Bennett-Tennenhaus and Kayla Wright.

Classic Auslander-Reiten theory is a neat tool used to paint a portrait of the category of modules over an Artinian ring. Nakayama functors play an important role in this painting. In suitable settings, the theory generalises to abelian categories, triangulated categories and their subcategories. In this talk, we will construct Nakayama functors on proper abelian subcategories. These categories, defined by Jørgensen in 2022, are generalisations of hearts of t-structures.

Talk is based on the following arXiv preprint: arXiv:2312.07323

Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. In joint work with Esther Banaian, Sunita Chepuri, and Sylvester W Zhang, we provide a combinatorial proof of positivity for certain cluster variables in these algebras. Our proof uses a hypergraph generalization of snake graphs, which were introduced by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. In this talk, I will explain our construction without assuming prior knowledge about cluster algebras or snake graphs.

For an associative unital ring, properties of classes of modules of projective dimension bounded by some positive integer provides large insight to both the module category over the ring, as well as intrinsic properties of the ring itself. This is exemplified by many classical results, for example, Bass' characterisation of perfect rings, or, over a commutative noetherian local ring, the Auslander-Buchsbaum formula. In this talk, we will consider a higher analogue of Bass' theorem, that is, the relationship between the big finitistic dimension and properties of the class of modules of projective dimension at most n, denoted P_n, over a commutative ring. In particular, we are interested in properties of the direct limit closure of P_n and if P_n provides minimal approximations. This is related to a recent preprint with Michal Hrbek which characterises the commutative noetherian rings for which P_n is of finite type.

String algebras are classically of path algebras over fields. Path algebras have also been considered over any noetherian local ground ring. Raggi-Cardenas and Salmeron generalised the definition of an admissible ideal in this context. A generalisation of string algebras from my PhD thesis likewise replaced the ground field with a local ring. In this talk I will explain how this definition relates to admissibility, and yields biserial rings in a sense used by Kirichenko and Yaremenko. I will also provide examples coming from metastable homotopy theory following work of Baues and Drozd. Time permitted, I will present an example of a clannish algebra over a local ring that is related to modular representations of the Matheiu 11-group, following Roggenkamp. This is based on an arxiv preprint 2305.12885.

13:30 Rosie Laking (Università degli Studi di Verona)

Cosilting sets and the Ziegler spectrum

Abstract: In this talk I will explain how torsion pairs in the category modA of finitely generated modules over an artinian ring A can be parametrised by cosilting sets, that is, maximal rigid subsets of two-term pure-injective complexes in the derived category D(A). Moreover each cosilting set determines a t-structure such that the finitely presented objects in the heart coincide with the tilt of modA at the corresponding torsion pair.

I will explain how each cosilting set is a closed subset of the Ziegler spectrum of D(A) and that the induced topology is controlled by the Serre subcategories of the associated tilt of modA. I will end by discussing the open question of whether the mutation theory of cosilting sets for a finite-dimensional algebra is determined by the isolated points?

This talk is based on joint work with Lidia Angeleri Hügel and Francesco Sentieri.

14:45 Charley Cummings (Aarhus University)

Metric completions of discrete cluster categories

Abstract: The completion of a metric space is a familiar method to generate new mathematical structures from old. Recently, Neeman emulated this idea to define the metric completion of a triangulated category. In general, these completions are difficult to compute; often one needs to use properties of ambient, already completed, triangulated categories, like the derived category. Cluster categories of Dynkin type A are triangulated categories that can be defined by a combinatorial model, and, as such, we will see that many of their completions can be computed without the use of such an ambient triangulated category. Moreover, we will see an infinite family of metric completions that can be realised combinatorially. This talk is based on joint work with Sira Gratz.

16:00 Laertis Vaso (NTNU)

τ-d-tilting theory for Nakayama algebras

Abstract: τ-tilting theory and torsion theory are established areas of interest in representation theory. Recently, there have been attempts to generalize these theories in the setting of higher homological algebra. d-torsion classes were introduced by Jørgensen, and several versions of “τ_d-tilting modules” have been introduced by different authors (Jacobsen–Jørgensen, Martínez–Mendoza, Zhou–Zu and others). The aim of this talk is to give an explicit classification of some of these higher analogues of τ-tilting modules and torsion classes for truncated linear Nakayama algebras when d>2. This classification will also be used to illustrate the different proposed notions of higher tau-tilting modules. This is joint work with Endre S. Rundsveen.

Tea, coffee and cake will be provided.

Weight structures (also known as co-t-structures) and t-structures on triangulated categories are closely related by orthogonality. I will explain how this relation can be characterized in terms of simple-minded collections and silting collections, using derived projective covers which are an analog of projective covers in triangulated categories. If time permits, I will also discuss a result about Koszul duality between simple-minded and silting collections. The talk is based on arXiv:2309.00554.

In this talk, we will discuss joint work with Moriah Elkin and Gregg Musiker about a combinatorial model for certain Grassmannian cluster algebras. The Grassmannian Gr(k,n) of k-planes in C ^n, , has a cluster structure that is not well-understood for k>2. In these algebras, Plücker coordinates ∆ I give us a subset of the cluster variables and have lovely combinatorial descriptions. However, most cluster variables are more complicated expressions in Plücker coordinates and lack such a combinatorial description. In our work, we give a graph theoretic interpretation for the Laurent expansion of cluster variables of low degree in terms of higher dimer models. This work employs SL k web combinatorics and we conjecture these webs are the key ingredient to understanding Grassmannian cluster algebras. If time permits, I would like to also pose an open problem I hope to work on (possibly with the algebraic power of Aarhus postdocs) relating our dimer combinatorics to the categorification of Grassmannian cluster algebras.

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 in an attempt to define "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.

Koszul duality has been observed in many forms across algebra, geometry and topology. This is a talk about commutative algebra, and I'll try to explain everything from the beginning.

A homomorphism f: R -> S of commutative local rings has a derived fibre F - this is a differential graded algebra over the residue field k of R, and we say that f is Koszul if F is formal and its homology H(F) = Tor^R(S,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition, how it is satisfied by many examples, and what you can do with it.

The main application is the construction of explicit free resolutions over S in the presence of a Koszul homomorphism. These are infinite free resolutions described by a finite amount of data, and so they tell you about the asymptotic homological algebra of S. This construction simultaneously generalises the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.

This is all joint with James Cameron, Janina Letz, and Josh Pollitz.

We start from the motivating toy example. For a finite dimensional vector space V, consider the graded algebras A=Sym(V) and A^!= Lambda(V*) - the symmetric and exterior algebra respectively. Their Poincare series satisfy the identity f(t) f^!(-t)=1. This is a special case of a general statement, a numeric incarnation for a pair of Koszul dual algebras.

Next, given a polytope P, we consider its incidence matrix M_P(t). Specialists in combinatorics know that it satisfies the matrix equation M_P(t) M_P(-t)= Id. Polo in mid-nineties explained this by proving that the incidence algebra for a polytope is Koszul self dual.

Recently Daria Poliakova generalized this. Given a polytope, she introduced an automorphism I_P(t) of a non-commutative power series algebra whose linear part is precisely M_P. She proved in her thesis that in some cases the automorphism is almost an involution. This result is a numeric incarnation of a surprising Koszul self-duality for a certain colored operad associated with the polytope P.

We show that automorphisms of a similar nature appear naturally in representation theory. In particular, given a finite dimensional associative algebra with a non-degenerate scalar product, we obtain a non-trivial example similar to the construction of I_P(t).

This talk will be in the commutative Noetherian local setting. If you give me a ring R of Krull dimension two, tell me that the corresponding geometric object is smooth and ask me to compute Ext^5(M,N) for two finitely generated R-modules M and N, it would take me less than two seconds to answer: it is zero. Things only get interesting if you remove the smoothness hypothesis. Then I can talk about nonzero arbitrarily higher ext modules and their annihilators. In this talk, I will tell you three related but separate theorems regarding these annihilators. The first one is joint with Akdenizli-Aytekin-Çetin and it is about a topology on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. The second one is joint with Ryo Takahashi and gives upper bounds on the dimension of singularity categories. Finally, the last one is about Frobenius structures on the category of MCM modules and is joint with Benjamin Briggs.

14:15: Thomas Brüstle (Université de Sherbrooke)

Homological approximations in persistence theory

Multiparameter persistence modules appear in topological data analysis when dealing with noisy data. They are defined over a wild algebra and therefore they do not admit a complete discrete invariant. One thus tries in persistence theory to “approximate” such a module by a more manageable class of modules. Using that approach we define a class of invariants for persistence modules based on ideas from homological algebra. This is a report on joint work with Claire Amiot, Benjamin Blanchette and Eric Hanson. No prior knowledge of topological data analysis is required.

15:15 Amit Shah (Aarhus Universitet)

Characterising Jordan-Hölder extriangulated categories via Grothendieck monoids

Abstract: The notions of composition series and length are well-behaved in the context of abelian categories. And, in addition, each abelian category satisfies the so-called Jordan-Hölder property/theorem. Unfortunately, these ideas are poorly behaved for triangulated categories. However, with the introduction of extriangulated categories, it is interesting to see what sense we can make of these concepts for extriangulated categories.

I'll present a result that characterises Jordan-Hölder, length extriangulated categories using the Grothendieck monoid of an extriangulated category. This is motivated by the exact category setting as considered by Enomoto, in which it becomes apparent that Grothendieck monoid is more appropriate to look at than the Grothendieck group. I'll present some examples coming from stratifying systems. In fact, developing stratifying systems for extriangulated categories was the original motivation of our article. This is joint work with Thomas Brüstle, Souheila Hassoun and Aran Tattar.

16:15 Henrik Holm (Københavns Universitet)

Compact and perfect objects in triangulated categories of quiver representations.

A complex of modules over a ring, A, can be viewed as an A-module valued representation of a certain quiver with relations, Q. The vertices of Q are the integers, there is an arrow q -> q-1 for each integer q, and the relations are that consecutive arrows compose to zero. Hence the classic derived category of A can be viewed as a triangulated category of representations of Q. It is well-known that the derived category is compactly generated and that the compact objects are precisely the so-called perfect complexes, i.e. complexes that are isomorphic to a bounded complex of finitely generated projective A-modules.

While complexes do play a prominent role, other types of quiver representations are important as well. It turns out that if Q is a suitably nice quiver with relations, then there exists a triangulated category whose objects are the A-module valued representations of Q. This category is called "the Q-shaped derived category" of the ring A.

We will prove that the Q-shaped derived category is always a compactly generated triangulated category. We will also extend the notion of perfect complexes (in the classic derived category) to perfect objects in the Q-shaped derived category. It turns out that, up to direct summands, perfect and compact objects in the Q-shaped derived category are the same.

The talk is based on a joint paper (arXiv:2208.13282) with Peter Jørgensen from Aarhus University.

This talk is based on the preprint https://arxiv.org/abs/2305.06031 .

Let A be a finite-dimensional algebra and T(A) the lattice of torsion classes of finitely-generated A-modules. We use the theory of stability conditions to define a partial order (the "facial semistable order") on a set of intervals in T(A). By restricting to functorially finite torsion classes, this yields a partial order on the set of all 2-term presilting complexes. This then further restricts to the oriented mutation graph on 2-term silting complexes. In this talk, we discuss the construction, cover relations, and lattice properties of the facial semistable order.

Inspired by toric geometry, I explain how to associate a cone (in the sense of convex geometry) to an abelian category; a fan to a bounded heart; a multifan to a triangulated category. One motivation for doing this is a connection to stability conditions, and a related open cone construction provides a purely convex-geometric description of Bridgeland’s stability space.

This talk will explain a combinatorial connection between exact matchings of the honeycomb (dimers), certain directed coloured graphs appearing in the representation theory of quantum groups (crystals) and the multiplication of a Schubert class by a Chern class of the tautological or quotient bundle in the small quantum cohomology ring of Grassmannians. The motivation of these combinatorial connections is the problem of finding a combinatorial/rep theoretic description of Gromov-Witten invariants.

A DGA is a chain complex with a compatible multiplication. In the case that the DGA is connective (concentrated in negative degrees) then these DGAs behave similarly to rings; they have the standard t-structure and their K-theory's are similar. Finite dimensional DGAs are a step towards studying nonconnective DGAs where many questions are still open. We say a DGA is finite dimensional if its underlying algebra is finite dimensional (not just its homology). This definition is not invariant under quasiisomorphism but the class of DGAs which are quasiisomorphic to fd ones still has some sensible properties. A result of Orlov says that if the underlying algebra of the fd DGA is semisimple then it is also semisimple as a DGA. In this sense the underlying algebra of the DGA determines some of the structure of the DGA. I'll present a result in same theme which builds on this result and mention some open problems I'm trying to apply it to.

We will explain how the cluster-tilting subcategories of a cluster category give rise to A- and X-cluster structures in the sense of Fock-Goncharov. In particular, as T varies through the cluster-tilting subcategories, the Grothendieck groups $K_0(T)$ and $K_0(\mathrm{fd}\, T)$ admit families of homomorphisms tropicalizing the birational maps used to build the corresponding A- and X-cluster varieties.

Using our framework, we construct an X-cluster character for any cluster category and relate it to the A-cluster character. We also show the existence of a canonical quantization of any Hom-finite exact cluster category, extending examples originally due to Geiß-Leclerc-Schröer.

This is joint work with Matthew Pressland (Glasgow).

The skein algebra of a marked surface admits the basis of bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis from the cluster scattering diagram by Gross-Hacking-Keel-Kontsevich. In a joint work with Travis Mandel, we show that the two bases coincide except for the once-punctured torus. Long-standing conjectures on strong positivity and atomicity follow as corollaries.

Cluster algebras are defined recursively from a set of initial data and it is a question how one writes cluster algebra elements in terms of the initial ones. There have been different ways to answer this question, i.e. compute the cluster expansion formulas for elements using such as snake graphs, T-paths or 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 algebra elements associated with arcs coming from surfaces. The method we introduce is quite efficient and also can be generalised to different settings.

In an ongoing work together with Marcelo Lanzilotta we have used the Igusa-Todorov functions to define GLIT classes and GLIT algebras, the latter being a vast family of artin algebras that satisfy the finitistic dimension conjecture. In this talk I will give the definition of a GLIT class and a GLIT algebra and explain their main properties. Furthermore, we establish when a triangular matrix algebra is GLIT and using this result we can give a partial answer to the question of whether the tensor product of GLIT algebras is again GLIT. Lastly I will give a new characterization of the finitistic dimension conjecture using GLIT classes.

We will relate homological properties of functor categories to those of finite dimensional algebras. This will allow us to generalize the no loop conjecture to nice functor categories. We present possible applications in Auslander-Reiten theory and algebraic geometry. This is joint work in progress with Martin Kalck, Nebojsa Pavic and Evgeny Shinder.

14:15-15:00: Johanne Haugland

Extension closure by indecomposables

Extension-closed subcategories of abelian categories form an important class of subcategories in representation theory. These subcategories carry a natural exact structure, and any small exact category can be seen to arise in this way. For nice classes of algebras, extensions with indecomposable end-terms can be realized geometrically as intersections of curves. In this talk, we show that when checking if a subcategory is extension-closed, it is enough to consider extensions by indecomposables. As an application, we aim to geometrically characterize extension-closed subcategories of the module category over a gentle algebra. The talk is based on joint work in progress with Sondre Kvamme and Steffen Oppermann.

15:15-16:00: Anders Sten Kortegård

Derived equivalences of self-injective 2-Calabi-Yau tilted algebras

In a $k$-linear Frobenius category $\mathscr{E}$ with a “nice” associated stable category $\mathscr{C}$, “good” maximal rigid objects have derived equivalent endomorphism algebras over $\mathscr{E}$. One may ask if the endomorphism algebras of the same objects over $\mathscr{C}$ also will be derived equivalent. We give a criteria for this being the case, and describe a two-sided tilting complex inducing this derived equivalence.

16:15-17:00: Matt Booth

Global Koszul duality

Koszul duality (a.k.a. bar-cobar duality) is a duality theory which gives an equivalence between the categories of augmented differential graded algebras, and coaugmented conilpotent differential graded coalgebras, when both are considered up to an appropriate notion of weak equivalence. More precisely, both categories admit model structures making the bar-cobar adjunction into a Quillen equivalence. Koszul duality (and variants, especially the commutative/Lie duality) pervades various different areas of mathematics, including rational homotopy theory and derived deformation theory. In this talk, I'll outline an extension of the above theory to the setting where one works with coalgebras that are no longer conilpotent (this is the `global' setting from the title). This is a report on work in progress joint with Andrey Lazarev.

In its various forms, the The New Intersection Theorem is concerned with the length of a finite free complex, that is, a complex \begin{equation*} F = 0 \leftrightarrow F_n \leftrightarrow \cdots \leftrightarrow F_1 \leftrightarrow F_0 \leftrightarrow 0 \end{equation*} of finitely generated free modules, over a local ring $(R,\mathfrak m)$. The classic version, due to Peskine and Szpiro (1973) asserts that if $HH(F)$ is non-zero and each homology module $HH_i(F)$ is of finite length, then $n \geq \dim R$ holds.

In the talk I will discuss the history of this a recent result up to a recent improvement obtained in joint work with Luigi Ferraro.

‘Simple-minded objects’ are generalisations of simple modules. They satisfy Schur’s lemma and a version of the Jordan-Holder theorem, depending on context. In this talk we will consider simple-minded systems (SMSs), which were introduced by Koenig-Liu as an abstraction of non-projective simple modules in stable module categories, and simple-minded collections (SMCs), a variant of SMSs first studied by Rickard in the context of derived equivalences of symmetric algebras. We will explain aspects of the theory of SMSs and SMCs, including mutation and reduction. In particular, we will see that the mutation of an SMS in a general triangulated category is again an SMS, unifying results of Dugas and Jørgensen, and that mutations of strong SMCs are again strong SMCs, giving a conceptual understanding of a result by Koenig-Yang. This talk is based on joint work with Nathan Broomhead, David Pauksztello, David Ploog and Jon Woolf.

A real matrix is called “Totally Positive” if all of its entries, minors, and determinant are positive. Lustig’s notion of total positivity generalises this to any split real semi simple Lie group and plays and important role in representation theory. I will talk about a recent generalisation of total positivity to non-split real semi simple Lie groups called Theta-Positivity due to Guichard and Wienhard, and about how noncommutative cluster algebras can encode this kind of positivity.

Oppermann and Thomas introduced the $(d + 2)$-angulated cluster category to generalise the classical cluster category to higher homological algebra. A great difficulty that arises in these categories is that cluster-tilting objects are no longer mutable at every summand, in contrast to the classical setting. In this talk we give two new ways of understanding mutability in these higher cluster categories: one from an algebraic perspective, and the other from a combinatorial perspective for the particular case of the higher Auslander algebras of type $A$.

I will talk about a new and simplified presentation of the classical pure braid group. Motivated by twist functors, the generators are given by the squares of longest elements over connected subgraphs, and the relations are either commutators or certain length 5 palindromic relations. This is based on my paper arXiv:2208.02120

Many of the classical concepts in representation theory, such as torsion classes and tilting theory, have generalisations to higher homological algebra, where the role of short exact sequences is played by exact sequences of longer length. In this talk, we explore the connection between $n$-torsion classes and $\tau_n$-tilting theory (where $n=1$ is the classical setup).

This is joint work with J. Haugland, K. Jacobsen, S. Kvamme, Y. Palu and H. Treffinger.

A tau-tilting subcategory of an abelian category is an additive contravariantly finite full subcategory satisfying certain conditions. In this talk, we study tau-tilting subcategories of an abelian category with enough projective objects from several different points of view. The talk is based on joint work with S. Sadeghi and H. Treffinger; arXiv:2207.00457 [math.RT].

The talk will also be streamed on zoom, contact the seminar organiser for details.

In this talk we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty}$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.

We will start with an introduction to singularity categories, which were first studied by Buchweitz and later rediscovered by Orlov. Then we will explain what is known about triangle equivalences between singularity categories of commutative rings, recalling results of Knörrer, D. Yang (based on our joint works on relative singularity categories. This result also follows from work of Kawamata and was generalized in a joint work with Karmazyn) and a new equivalence obtained in arXiv:2103.06584.

In the remainder of the talk, we will focus on the case of Gorenstein isolated singularities and especially hypersurfaces, where we give a complete description of quasi-equivalence classes of dg enhancements of singularity categories, answering a question of Keller & Shinder. This is based on arXiv:2108.03292.

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 over an abelian category. Consequently, we give a new characterisation of torsion classes, discuss the structure of the space of chains of torsion classes and its relation to the stability manifold. Time permitting, we will discuss some possible generalisations and future directions.

14:15-15:00 Mads Hustad Sandøy (NTNU): Quadratic monomial 2-representation finite algebras and bipartite graphs

The (extended) ADE Coxeter-Dynkin diagrams have a knack for showing up in various finite classification problems in mathematics, e.g. classifying representation finite (tame) hereditary algebras. For simple undirected graphs, they solve the problem of characterizing the simple undirected graphs with largest adjacency eigenvalue less than (or equal to) two. By work of A’Campo (1976), it is known that this is no accident.

As it turns out, both classification problems have natural higher dimensional generalizations, and I discuss some connections analogous to those shown by A’Campo. In particular, I cover recent progress on classifying quadratic monomial 2-representation finite algebras via spectral methods, connections with bipartite reflexive graphs, and an associated Diophantine equation.

15:15-16:00 Job D. Rock (Ghent): Composition Series of Arbitrary Cardinality in Abelian Categories

We generalize composition series in an abelian category to allow the multiset of composition factors to have arbitrary cardinality. Motivating examples include pointwise finite-dimensional persistence modules, Prüfer modules, and presheaves, where we generalize the notion of support used in quiver representations. A “Jordan–Hölder–Schreier” like theorem holds when objects satisfy a set of four axioms. With one additional axiom, we can prove a result that generalizes some properties of a length category. This is based on joint work (arXiv:2106.01868) with Eric J. Hanson.

16:15-17:00 Paul Smith (Washington): An overview of elliptic algebras

Elliptic algebras form a family of connected graded algebras defined over the complex numbers that depend on a pair of relatively prime positive integers $n'>k$ and a complex elliptic curve, $E$, and a point on it. They were defined in this generality by Feigin and Odesskii in 1989 and have not been studied much since then, in part because Feigin and Odesskii's papers were long on assertions and short on proofs. In joint work with Alex Chirvasitu and Ryo Kanda over the past 5 years we have proved a number of results (about 250 journal pages) about them that have put the subject on a firmer footing. For fixed $(n,k,E)$ they form a flat family of deformations of the polynomial ring on $n$ variables and share many homological properties with that polynomial ring: for example, they are Koszul and have global homological dimension $n$. Their quadratic duals are finite dimensional algebras, deformations of exterior algebras, therefore very wild, and (because they are connected) not rich in combinatorial properties. However, they are very rich in geometric connections the relevant objects being varieties built from elliptic curves. Their definition is via generators and relations involving theta functions. This means they are complicated. In particular, they have no explicit nice basis so calculations with elements are essentially impossible.

I assume no prior knowledge of elliptic algebras.

For a skeletally small triangulated category C Chuang and Lazarev introduced the notion of a rank function on C. Such functions are closely related to functors into simple triangulated categories. In this talk, I will discuss the connection between rank functions on C and translation-invariant additive functions on its abelianization mod-C. This connection allows to relate rank functions to endofinite cohomological functors on C and, in the case when C is the subcategory of compact objects in a compactly generated triangulated category T, to endofinite objects and to the Ziegler spectrum of T. This is based on a joint work in progress with Teresa Conde, Mikhail Gorsky, and Frederik Marks.

In this talk, I will focus on the class of cluster algebras of finite type, which are classified by Dynkin diagrams. We study these commutative algebras from the point of view of singularity theory: we classify their singularities and develop constructive resolutions of these singularities over fields of arbitrary characteristics. From the same perspective, we study cluster algebras coming from a star shaped quiver, which are not of finite type, but whose singularities exhibit interesting combinatorial phenomena. This is joint work with Angélica Benito, Hussein Mourtada, and Bernd Schober.

We give a construction for a gentle algebra which can be associated with a triangulated orbifold with all orbifold points having order three. First, we discuss how features of the module category can be viewed on the orbifold. Chekhov and Shapiro demonstrate how to associate a generalized cluster algebra from a triangulated orbifold. We then compare the modules over this gentle algebra with the elements of the generalized cluster algebra from the same orbifold with triangulation.

This talk is based on joint work with Yadira Valdivieso.

Classical sequences of numbers often lead to interesting $q$-analogues. The most popular among them are certainly the $q$-integers and the $q$-binomial coefficients which both appear in various areas of mathematics and physics. It seems that $q$-analogues of rational numbers have been much less popular so far. With Valentin Ovsienko we recently suggested a notion of $q$-rationals based on combinatorial properties. The definition of $q$-rationals naturally extends the one of $q$-integers and leads to a ratio of polynomials with positive integer coefficients. I will explain the construction and give the main properties. I will mention connections with the combinatorics of posets, cluster algebras, Jones polynomials. Finally I will also present further developments of the theory, in particular I will focus on the notion of $q$-irrationals and $q$-unimodular matrices.

$n$-cluster tilting subcategories of module categories are the prototypical setting of Iyama's higher dimensional Auslander-Reiten theory. Much attention has been given to algebras of global dimension n, for which there is at most one $n$-cluster tilting subcategory. For higher global dimension the notion of $n$-cluster tilting has certain drawbacks, which motivates the stronger notion of $n\mathbb Z$-cluster tilting due to Iyama and Jasso. It has been shown by Kvamme that an $n\mathbb Z$-cluster tilting subcategory of the module category gives rise to an $n\mathbb Z$-cluster tilting subcategory of the singularity category.

In my talk I will present a classification of $n\mathbb Z$-cluster tilting subcategories for Nakayama algebras. Moreover, I will show which $n\mathbb Z$-cluster tilting subcategories they give rise to in the corresponding singularity categories.

This talk is based on joint work with Sondre Kvamme and Laertis Vaso.

In joint work with R. Bennett-Tennenhaus, we called structure-preserving functors between extriangulated categories extriangulated functors. Examples include the canonical functor from an abelian category to its derived category, and the quotient functor from a Frobenius exact category to its stable category. The first, for example, is structure-preserving in the sense that short exact sequences are sent to distinguished triangles in a functorial way. In higher homological algebra, we also see examples of structure-preserving functors, but not covered by the current terminology. E.g. $n$-cluster tilting subcategories sitting inside an ambient abelian category. In ongoing work with R. Bennett-Tennenhaus, J. Haugland and M. H. Sandøy, we have been aiming to place these kinds of more general situations in a formal framework. This has led us to take a new perspective on extrianguated functors as I will explain.

The derived category $D(A)$ of the category Mod$(A)$ of modules over a ring $A$ is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch$(A)$ of chain complexes of $A$-modules equipped with its standard model structure. One can view Ch$(A)$ as the category Fun$(Q$,Mod$(A))$ of additive functors from a special (but quite natural) small preadditive category $Q$ to Mod$(A)$. Note that the model structure on Ch$(A)$ = Fun$(Q$,Mod$(A))$ is not inherited from a model structure on Mod$(A)$ but arises instead from the "self-injectivity" of the special category $Q$. We will show that the functor category Fun$(Q$,Mod$(A))$ has two interesting model structures for many other self-injective small preadditive categories $Q$. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the $Q$-shaped derived category of $A$. We will also show that it is possible to generalize the homology functors on Ch$(A)$ to homology functors on Fun$(Q$,Mod$(A))$ for most self-injective small preadditive categories $Q$. The talk is based on a joint paper with Peter Jørgensen (arXiv:2101.06176), which has the same title as the talk.

The definition of an extriangulated category was given by Nakaoka and Palu. This notion simultaneously generalises those of an exact category and a triangulated category. Examples of extriangulated categories which need not be exact nor triangulated appear, for example, when studying pure-exact sequences in compactly generated triangulated categories.

In this talk I will discuss what are called extriangulated functors, from joint work with myself and Amit Shah. As above, the definition unifies that of exact functors between exact categories, and triangulated functors between triangulated categories. Other examples of extriangulated functors include the delta functor from an exact category to its derived category, and the restricted Yoneda functor from a compactly generated triangulated category to the appropriate functor category. Time permitted, I will explain the role of a category of extensions.

14:15-15:00. Esther Banaian (University of Minnesota): Frieze patterns from dissections

15:15-16:00. Greg Stevenson (University of Glasgow): Spaces of localizations

16:15-17:00. Sira Gratz (University of Glasgow): SL(k)-friezes

The lattice of torsion classes in the category mod A of finitely generated A-modules over a finite dimensional algebra A, usually denoted by tors A, is currently receiving a lot of attention due to its connection with silting theory and with the phenomenon of mutation. Indeed, it is shown in [1] that the functorially finite torsion classes in mod A are in bijection with two-term silting complexes, and the lattice order on these torsion classes reflects silting mutation.

But tors A carries important representation-theoretic information also for torsion classes which are not functorially finite. Its Hasse quiver is labeled by bricks, i.e. by modules whose endomorphism ring is a skew field. More precisely, torsion classes are adjacent if and only if they “differ” by exactly one brick, which passes from the torsion-free class of one torsion pair to the torsion class of the other [2,3]. In recent joint work with Laking, Štovíček and Vitória we have interpreted this process in terms of mutation of two-term cosilting complexes in the unbounded derived category of A.

In the present talk, I will report on ongoing joint work with Ivo Herzog, Rosanna Laking, and Francesco Sentieri which relates cosilting mutation with an exchange of indecomposable injective objects in suitable Grothendieck categories. More precisely, we will consider the hearts of the t-structures associated to two adjacent torsion pairs. The brick which labels the corresponding arrow in the Hasse quiver gives rise to a simple object in each of these hearts. We will discuss how to determine the injective envelopes of these simple objects and investigate their role in connection with mutation.

References:

[1] T. Adachi, O. Iyama, I. Reiten, τ-tilting theory, Compos. Math. 150 (2014) 415-452.

[2] E. Barnard, A. T. Carroll, S. Zhu. Minimal inclusions of torsion classes. Algebr. Comb. 2 (2019), 879-901.

[3] L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas, Lattice theory of torsion classes, arXiv:1711.01785

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the past 20 years. In this talk, we focus on Iyama and Yang’s generalised cluster categories T/T^fd, coming from n-Calabi-Yau triples. In such a construction, T is a triangulated category with triangulated subcategory T^fd and silting subcategory M. Using a different approach from Iyama and Yang, we give a deeper understanding of T/T^fd and reprove it is a generalised cluster category. In order to do so, we use more classic homological tools such as limits, colimits and a gap theorem.

Bridgeland developed stability scattering diagrams relating scattering diagrams with quiver representations. Scattering diagrams were developed as a machinery in mirror symmetry. Together with Travis Mandel, we associate tropical disks counting with quiver representations by using the stability scattering diagrams. Next, together with Yu-Wei Fan and Yu-Shen Lin, we look at the stable objects for the $A_2$ quiver. It is known that the derived Fukaya-Seidel category of the rational elliptic surface is the derived category of the $A_2$ quiver. We made use of the relation and corresponded the special Lagrangian with the stable objects in the derived category of coherent sheaves.

Let W be the Witt algebra of polynomial vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W. We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir. We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W). We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores. We then lift this classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z- \lambda)$ is a maximal Poisson ideal of Sym(Vir).

The structure of the category of modules over a commutative noetherian ring R and of its derived category is largely controlled by the prime spectrum of R. In this talk we discuss how this control extends to the structure of hearts of t-structures in the derived category. We will focus in particular on hearts arising from hereditary torsion pairs in Mod(R). These turn out to be Grothendieck categories which are derived equivalent to R and such that part of the lattice of torsion pairs can be studied using the prime spectrum of R. This talk is based on joint work with Sergio Pavon

Frieze patterns, introduced by Coxeter, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Under a certain finiteness assumption, they are in one-to-one correspondence with triangulations of polygons [Conway–Coxeter] and they come from triangulations of annuli in an infinite setting [Baur–Parsons–Tschabold]. In this talk, we will discuss a relationship between pairs of infinite friezes associated with a triangulation of the annulus and explore how one determines the other in an essentially unique way. We will also consider module categories associated with triangulated annuli where infinite friezes may be recovered using a homological formula. This is joint work with Karin Baur, Karin Jacobsen, Maitreyee Kulkarni, and Gordana Todorov

Alternating strand diagrams (as introduced by Postnikov) on the disk have been used in the study of the coordinate ring of the Grassmannian. In particular, they give rise to clusters of the Grassmannian cluster algebras (Scott) or to cluster-tilting objects of the Grassmannian cluster categories of Jensen-King-Su (Baur-King-Marsh). On the other hand, orbifolds have also been related to cluster structures as Paquette-Schiffler (or Chekhov-Shapiro for a geometric approach). Here we introduce orbifold diagrams as quotients of symmetric Postnikov diagrams and show how to associate quivers with potentials to them. This is joint work with Andrea Pasquali (Stuttgart) and Diego Velasco (Cali)

Torsion pairs are fundamental tools in the study of abelian categories, which contain important information related to derived categories and their t-structures. In this talk we will consider the lattice of torsion classes in the category of finite-dimensional modules over a finite-dimensional algebra, with a particular focus on the minimal inclusions of torsion classes.

It was shown by Adachi, Iyama and Reiten that minimal inclusions of functorially finite torsion classes correspond to irreducible mutations of associated two-term silting complexes in the category of perfect complexes. In this talk we will explain how minimal inclusions of arbitrary torsion classes correspond to irreducible mutations of associated two-term cosilting complexes in the unbounded derived category.

This talk will be based on joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitória.

Grassmannians are objects of great combinatorial and geometric beauty, which arise in myriad contexts. Their coordinate rings serve as a classic example of cluster algebras, as introduced by Fomin and Zelevinsky at the start of the millennium, and their combinatorics is intimately related to algebraic and geometric concepts such as to representations of algebras and hypersurface singularities. At the core lies the idea of generating an object from a so-called "cluster" via the concept of "mutation".

In this talk, we offer an overview of Grassmannian combinatorics in a cluster theoretic framework, and ultimately take them to the limit to explore the a priori simple question: What happens if we allow infinite clusters? In particular, we introduce the notion of a cluster algebra of infinite rank (based on joint work with Grabowski), and of a Grassmannian category of infinite rank (based on joint work with August, Cheung, Faber and Schroll)