Talks

Upcoming Talks

30.07.2025: TBA (The RepNet Launch Party, York)

08.2025: Frieze patterns (Markov Numbers and the Combinatorics of Cluster Algebras, İstanbul)

Frieze patterns were introduced and developed by Coxeter and Conway in the 80s, who presented them as a kind of combinatorial game, whereby the player attempts to fill a grid with positive integers obeying certain rules. It turns out, however, that frieze patterns appear naturally in a range of mathematical problems from a number of different areas. For example, in geometry, a frieze pattern represents a positive integer valued point in a decorated Teichmüller space, or in the totally positive Grassmannian. In representation theory, the entries in a frieze pattern count submodules of quiver representations. In this series of lectures, we will look at some of these different interpretations of frieze patterns and the connections between them.

Past Talks

2025

27.06.2025: Variations on cluster categories of discs (Derived Representation Theory and Triangulated Categories, Θεσσαλονίκη)

In Dynkin type \(\mathsf{A}_n\), cluster combinatorics can be modelled by the behaviour of arcs in a disc with \(n+3\) marked points on the boundary. In particular, the corresponding triangulated cluster category has indecomposable objects in bijection with arcs between these marked points, and cluster-tilting objects in bijection with triangulations, while extension groups count crossings of arcs. As part of their work on cluster categories for general cyclically ordered sets, Igusa and Todorov extended this correspondence to the case of an infinite set of marked points having finitely many accumulation points, each two-sided and unmarked. When the accumulation points are also marked, Paquette and Yıldırım constructed a triangulated category of arcs via a Verdier quotient construction. However, its extension groups no longer count crossings, and so its (weak) cluster-tilting subcategories aren’t in bijection with triangulations of the marked surface. In this talk I will explain some joint work with İlke Çanakçı and Martin Kalck, in which we show how to repair the dictionary by passing to a relative extriangulated structure on Paquette–Yıldırım’s category. Time (and progress!) permitting, I will also comment on some connections to hyperbolic and symplectic geometry.

17.06.2025: Un cadre général pour la théorie catégorique des amas (Séminaire AG, Versailles)

La catégorification est historiquement une technique très puissante dans l’étude des algèbres amassées et des concepts liés. Typiquement, on commence par la construction d’une catégorification explicite pour une famille d’algèbres amassées, que l’on peut ensuite utiliser pour déduire des résultats pour cette famille. En revanche, dans ce travail conjoint avec Jan E. Grabowski, nous considérons plutôt une famille (assez grande !) des catégories, qui inclut les catégorifications plus spécifiques d’algèbres amassées, et essayons de reconstruire une théorie des amas dans cette généralité. Nous retrouvons les \(\mathbf{g}\)-vecteurs, \(\mathbf{c}\)-vecteurs et \(\mathcal{F}\)-polynômes, et donc les variables amassées de types \(\mathcal{A}\) et \(\mathcal{X}\). En particulier, notre formule catégorique pour les variables de type \(\mathcal{X}\) parait d’être nouvelle. De plus, nous étudions les quantifications de ce point de vue. Une conséquence de la généralité de notre cadre est que la décatégorification d’une catégorie dans notre famille n’est pas nécessairement une algèbre amassée classique dans le sens de Fomin et Zelevinsky : on trouve aussi, pour exemple, des généralisations plus récentes par Chekhov et Shapiro.

31.03–02.04.2025: Cluster categories for Grassmannians and positroid varieties (Representation Theory, Symplectic Geometry, and Cluster Algebras, Banff)

In this series of talks, I will explain the additive categorification of cluster algebra structures on the Grassmannian and more general positroid varieties. For the Grassmannian itself, the cluster structure is due to Scott, generalising a special case by Fomin and Zelevinsky, and the categorification is by Jensen, King and Su, building on an “almost-categorification” by Geiß, Leclerc and Schröer. Scott’s description of the cluster structure uses combinatorics introduced by Postnikov in his study of total positivity for the Grassmannian, and I will explain how the same combinatorics encodes several non-commutative algebras. This leads to a re-interpretation and generalisation of Jensen–King–Su’s construction, and additive categorifications of the cluster structure on the positroid variety associated to any connected positroid. The ultimate aim of the series is to explain a proof, based on the homological algebra of these categorifications, that the two most canonical cluster structures on a positroid variety quasi-coincide in the sense of Fraser, confirming an expectation of Muller and Speyer.

26.02.2025: Doubles cellules de Bruhat (Groupe de travail : Variétés de tresses, Caen)

Je vais expliquer la définition de la structure amassée sur l’anneau de coordonnées d’une double cellule de Bruhat. Je vais mettre un accent sur le cas simplement lacé, et spécialement le cas de type \(\mathsf{A}\). Ce résultat est dû à l’origine à Berenstein, Fomin et Zelevinsky, mais je donnerai une version en utilisant une combinatoire de Shen et Weng, plus adapté pour leur généralisation aux variétés doubles de Bott-Samelson (qui sera expliquée plus tard dans le groupe de travail).

21.01.2025: Cluster algebras via representation theory (Séminaire d’algèbre et de géométrie, Caen)

I will report on some joint work with Jan E. Grabowski, in which we explain how to extract cluster-theoretic data from a large family of suitable (\(2\)-Calabi–Yau, extriangulated, ...) categories. In this way, we are able to unify and generalise many existing results for particular constructions of cluster categories, give a cluster character for \(\mathcal{X}\)-cluster variables (or \(y\)-variables in Fomin–Zelevinsky’s notation), and describe a categorical analogue of the quasi-commutation data in a quantum cluster algebra. Our setting includes some examples which do not decategorify to a traditional cluster algebra in Fomin–Zelevinsky’s sense, but to the generalisations defined by Chekhov and Shapiro in the context of Teichmüller theory for orbifolds.

2024

18.12.2024: An \(\mathcal{X}\)-cluster character (The 14th AIMS Conference on Dynamical Systems and Differential Equations, أبوظبي) [slides]

The theory of cluster algebras and varieties has two sides, named \(\mathcal{A}\) and \(\mathcal{X}\) by Fock and Goncharov (and also referred to as the \(K_2\) side and the Poisson side). On the \(\mathcal{A}\) side, there is a well-developed theory of cluster characters dating back to Caldero and Chapoton, allowing \(\mathcal{A}\)-cluster variables to be computed from objects in appropriate cluster categorifications. In this talk, on joint work with Jan Grabowski, I will explain a related formula on the \(\mathcal{X}\) side, which (under the correct assumptions), computes \(\mathcal{X}\)-cluster variables from simple representations of cluster-tilting subcategories.

08–12.11.2024: Additive categorification for positroid varieties (Cluster Algebra, Representation Theory and Algebraic Geometry, 香港)

The totally non-negative Grassmannian is an important object in several stories, including Lusztig’s total positivity, and the calculation of scattering amplitudes via the amplituhedron. It has a cell decomposition, described by Postnikov, in which each cell is obtained by intersecting the totally non-negative Grassmannian with a particular subvariety of the full Grassmannian: a so-called open positroid variety. A useful tool in studying totally positive spaces is Fomin–Zelevinsky’s theory of cluster algebras, and a recent result of Galashin and Lam is that the coordinate ring of (the cone on) an open positroid variety indeed has a natural cluster algebra structure.
In this lecture series, I will describe this cluster structure and explain how to use representation-theoretic techniques to understand it, setting up a dictionary between the combinatorics and the algebra. Some of the results here are joint with Çanakçı and King. Galashin and Lam actually produce two (isomorphic, but usually not equal) cluster algebra structures on each positroid variety, which correspond in algebraic terms to the choice between left and right modules. An application of the categorification is to prove a precise relationship, called quasi-coincidence, between these two cluster algebra structures, originally conjectured by Muller and Speyer in 2017.

23.10.2024: Categorifying triangulations in completed infinity-gons (Homological Algebra Seminar, Aarhus)

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.

09.10.2024: Positroid varieties via additive categorification (Algebra Seminar, Aberdeen)

The totally non-negative Grassmannian is an important object in several stories, including Lusztig’s total positivity, and the calculation of scattering amplitudes via the amplituhedron. It has a cell decomposition, described by Postnikov, in which each cell is obtained by intersecting the totally non-negative Grassmannian with a particular subvariety of the full Grassmannian: a so-called open positroid variety. A useful tool in studying totally positive spaces is Fomin–Zelevinsky’s theory of cluster algebras, and a recent result of Galashin and Lam is that the coordinate ring of (the cone on) an open positroid variety has the structure of such an algebra, in two different but closely related ways, confirming a long-standing expectation. Muller and Speyer conjectured a precise relationship (quasi-coincidence) between these two cluster structures, which makes them equivalent from the point of view of total positivity. In this talk, I will explain how to understand the cluster structures and prove Muller and Speyer’s conjecture using the techniques of additive categorification, or in other words, of homological algebra in exact and triangulated categories.

08.08.2024: Representation theory and positroid varieties (ICRA 2024, 上海) [slides]

Positroid varieties, introduced by Knutson, Lam and Speyer based on ideas of Postnikov, are subvarieties of the Grassmannian which induce a cell decomposition of its totally non-negative part. In 2016, Muller and Speyer conjectured that the coordinate ring of each positroid variety should carry two natural cluster algebra structures, and that these should quasi-coincide (in particular, have the same cluster monomials). The first part of the conjecture was proved by Galashin and Lam in 2023, and in this talk I will explain how to prove the second part. This is based on a series of papers, including joint work with Çanakçı and King, in which the relevant cluster algebras are categorified, and various combinatorial and geometric phenomena for positroid varieties re-interpreted in representation-theoretic terms. In particular, I will explain how the quasi-coincidence of the two cluster structures is a consequence of a certain equivalence of derived categories.

04.08.2024: Categorification of positroids (Seminar, 上海)

I will give an overview of the categorification of cluster structures on positroid varieties in terms of Gorenstein projective modules over certain orders. I will begin with the construction of categorifications from dimer models on discs, and then explain some joint work with Çanakçı and King, in which these categories are related to Jensen–King–Su’s Grassmannian cluster categories, and various combinatorial aspects of positroid varieties are translated into representation theory. These results can be applied to prove a conjecture by Muller and Speyer concerning positroid cluster structures, but this part of the story will be left for my talk at ICRA.

09.07.2024: \(2\)-term complexes are everywhere (Simple-Mindedness, Silting, and Stability, Ambleside)

In some ways, cluster-tilting objects in cluster categories are much like projective generators for hereditary algebras, in that they generate arbitrary objects in ‘one step’. In other ways, they are spectacularly unlike such projective generators, having self extensions in infinitely many degrees (positive and negative). I will explain how to modify a cluster category in such a way that it becomes a hereditary extriangulated category, and one of its cluster-tilting objects becomes a projective generator. This is joint work with Xin Fang, Mikhail Gorsky, Yann Palu and Pierre-Guy Plamondon, and foreshadows a general result of Xiaofa Chen (summarised in slogan-form in the title).

17.06.2024: Categorical cluster ensembles (Séminaire d’Algèbre d’IMJ-PRG, Paris)

In their geometric approach to cluster theory, Fock–Goncharov and Gross–Hacking–Keel construct cluster varieties beginning with a seed datum. This consists of a lattice which contains various distinguished sublattices, has a preferred basis, and carries a partially defined bilinear form. A process of mutation allows one to construct more such seed data, and birational gluing maps between the tori dual to the lattices, leading to two cluster varieties known as \(\mathcal{A}\) and \(\mathcal{X}\). By enhancing the initial data to a cluster ensemble, in which the bilinear form is extended to the whole lattice, one also obtains a map from \(\mathcal{A}\) to \(\mathcal{X}\).
In this talk, based on joint work with Jan Grabowski, I will explain how one can obtain a seed datum, and in many cases a full cluster ensemble, from each cluster-tilting subcategory of an appropriate \(2\)-Calabi–Yau category. Furthermore, I will explain how the seed data of different cluster-tilting subcategories are related, generalising the relationship between a seed datum and its mutations.

24.04.2024: Quasi-coincidence of positroid cluster structures via categorification (Maurice Auslander International Conference, Woods Hole, MA)

I will give a brief introduction to the categorification of cluster structures on positroid varieties in the Grassmannian. A typical such variety carries a pair of cluster structures which are abstractly isomorphic, but for which different functions are cluster variables. I will explain a proof, via representation theory, that these two structures quasi-coincide, so that they do at least have the same cluster monomials. This confirms a conjecture by Muller and Speyer.

05.04.2024: Variétés positroïdes via les catégories triangulées (Séminaire de Groupes, Représentations et Géométrie, Paris)

La grassmannienne totalement non négative est un objet important dans plusieurs sujets, y compris la théorie de la positivité totale de Lusztig et dans le calcul d’amplitudes de diffusion via l’amplituèdre en théorie quantique des champs. Elle a une décomposition en cellules, décrites par Postnikov, dans laquelle chaque cellule est l’intersection de la grassmannienne totalement non négative avec une sous-variété de la grassmannienne entière appelée variété positroïde ouverte.
Un outil puissant dans l’étude des espaces positifs est la théorie des algèbres amassées de Fomin et Zelevinsky. D’après un résultat récent de Galashin et Lam, attendu depuis longtemps, l’anneau des coordonnées homogènes d’une variété positroïde ouverte a la structure d’une telle algèbre, en ce de deux façons différentes, mais reliées. En fait, Muller et Speyer ont conjecturé une relation précise (la quasi-coïncidence) entre ces deux structures amassées qui leur rend équivalentes du point de vue de la positivité totale. Dans cet exposé, je vais expliquer comment prouver leur conjecture. La preuve s’appuie sur la catégorification additive des algèbres amassées, et donc sur l’algèbre homologique dans des catégories exactes ou triangulées.

02.04.2024: Des variétés de positroïde via les catégories triangulées (Séminaire d’algèbre et de géométrie, Caen)

La grassmannienne totalement non négative est un objet important dans plusieurs sujets, y compris la théorie de la positivité totale de Lusztig, et le calcul d’amplitudes de diffusion via l’amplituèdre. Elle a une décomposition en cellules, décris par Postnikov, dans laquelle chaque cellule est l’intersection de la grassmannienne totalement non négative avec une sous-variété de la pleine grassmannienne, qui s’appelle une variété de positroïde ouverte.
Un outil puissant en étudiant des espaces positifs est la théorie des algèbres amassées par Fomin et Zelevinsky. Un résultat récent de Galashin et Lam, qui confirme une attente de longue date, est que l’anneau des coordonnées homogènes d’une variété de positroïde ouverte a la structure d’une telle algèbre, en deux façons différentes, mais relatées. Muller et Speyer ont conjecturé une relation précise (la quasi-coïncidence) entre ces deux structures amassées, qui leur rend équivalentes du point de vue de la positivité totale. Dans cet exposé, je vais expliquer comment prouver leur conjecture. Peut-être surprenant, la preuve dépend critiquement sur la catégorification additive : en d’autres termes, sur l’algèbre homologique dans des catégories exactes ou triangulées.

25.03.2024: The geometry and representation theory of frieze patterns (Workshop on Continued Fractions and \(\mathrm{SL}_2\)-tilings, Durham) [slides]

Frieze patterns were introduced and developed by Coxeter and Conway in the 80s, who presented them as a kind of combinatorial game, whereby the player attempts to fill a grid with positive integers obeying certain rules. It turns out, however, that frieze patterns appear naturally in a range of mathematical problems from a number of different areas. For example, in geometry, a frieze pattern represents a positive integer valued point in a decorated Teichmüller space, or in the totally positive Grassmannian. In representation theory, the entries in a frieze pattern count submodules of quiver representations. In this talk, I will survey some of these different interpretations of frieze patterns, and the connections between them.

19.03.2024: Des variétés de positroïde via les catégories triangulées (Séminaire AG, Online / Versailles)

12.03.2024: Positroid varieties via triangulated categories (Algebraic Topology Seminar, Warwick)

The totally non-negative Grassmannian is an important object in several stories, including Lusztig’s total positivity, and the calculation of scattering amplitudes via the amplituhedron. It has a cell decomposition, described by Postnikov, in which each cell is obtained by intersecting the totally non-negative Grassmannian with a particular subvariety of the full Grassmannian: a so-called open positroid variety. A useful tool in studying totally positive spaces is Fomin–Zelevinsky’s theory of cluster algebras. A recent result of Galashin and Lam is that the coordinate ring of (the cone on) an open positroid variety has the structure of such an algebra, in two different but closely related ways, confirming a long-standing expectation. Muller and Speyer conjectured a precise relationship (quasi-coincidence) between these two cluster structures, which makes them equivalent from the point of view of total positivity. In this talk, I will explain how to prove their conjecture. Perhaps surprisingly, the proof is highly dependent on additive categorification, or in other words, on homological algebra in exact and triangulated categories.

16.01.2024: Positroid varieties and quasi-coincidence via representation theory (Cluster Algebras and Its Applications, Oberwolfach)

Total positivity is by now a classical subject in linear algebra, having begun in earnest with the work of Gantmacher and Krein from 1937. Recent results of Postnikov and others have emphasised the importance of positivity in flag varieties, particularly the Grassmannian. A key tool in this area is Postnikov’s positroid stratification of the Grassmannian, and the cluster algebra structures on its various (open) cells, recently confirmed to exist by Galashin and Lam. In this talk, I will explain this story in the language of representation theory, with the positroid varieties and their cluster algebra structures being encoded by the representation theory of various non-commutative orders over the power series ring in one variable. Except for the top-dimensional stratum, Galashin and Lam’s construction produces two different cluster algebra structures on each open positroid, and an application of this representation-theoretic approach is a proof that these two structures quasi-coincide, as conjectured by Muller and Speyer in 2017. In particular, this means that these structures are equivalent from the point of view of total positivity.

2023

15.11.2023: Triangulations versus cluster-tilting in infinity-gons (Algebra and Geometry Seminar, Graz)

The combinatorics of triangulations of a disc with marked points on its boundary is surprisingly rich: for example, when the number of marked points is finite, the triangulations form an associahedra, and are closely related to the representation theory of type \(\mathsf{A}_n\) quivers. Understanding this combinatorics has applications in, for example, Teichmüller theory, in which a triangulation determines a coordinate system of lambda lengths on a decorated Teichmüller space, and one may be interested in the coordinate transformations relating these for different choices of triangulation.
One way of understanding these structures is to use triangulated categories. The appropriate categorical model has been defined by Caldero–Chapoton–Schiffler when the set of marked points is finite, and by Igusa–Todorov when the set is discrete with finitely many accumulation points, all two-sided. In these models, crossings of arcs are encoded by extensions in the category, and hence triangulations by maximal rigid (or weak cluster-tilting) subcategories. Recently, Paquette–Yıldırım have given a model when the accumulation points are themselves marked, violating discreteness. While their model still has indecomposable objects in bijection with arcs, crossings are no longer encoded by extensions, and so the connection to the combinatorics of triangulations appears to be lost. In joint work with İlke Çanakçı and Martin Kalck, we recover this connection by passing to an extriangulated substructure on Paquette–Yıldırım’s category, in which weak cluster-tilting subcategories are once again in bijection with triangulations. While many general results from cluster theory do not apply directly, since this extriangulated category is not Frobenius, we are able to re-prove many of their conclusions using the relationship to triangulations of the disc, or to its decorated Teichmüller space, as in work of Çanakçı–Felikson.

12.10.2023: Double Bruhat cells and generalizations (Arbeitsgemeinschaft: Cluster Algebras, Oberwolfach)

I will describe (\(\mathcal{A}\)-side, or \(\mathrm{K}_2\)) cluster algebra structures on double Bruhat cells, due to Berenstein, Fomin and Zelevinsky, and on double Bott–Samelson cells, due to to Shen and Weng, focussing on \(\mathrm{SL}_{n+1}(\mathbb{C})\). These families include interesting varieties such as base affine space, and are included in the larger set of braid varieties, which carry cluster algebra structures in general.

06.10.2023: Cluster structures on cells in flag varieties (E-RNG Wee Math Seminar, Glasgow / Glaschu)

I will give a quick introduction to the combinatorics of linear algebraic groups and flag varieties, focussing on \(\mathrm{SL}_{n+1}(\mathbb{C})\) (i.e. type \(\mathsf{A}_{n}\)), including the definition of Bruhat cells and their generalisations. I will then describe (\(\mathcal{A}\)-side, or \(\mathrm{K}_2\)) cluster algebra structures on double Bruhat cells, due to Berenstein, Fomin and Zelevinsky, and on double Bott–Samelson cells, due to to Shen and Weng, continuing to focus on \(\mathrm{SL}_{n+1}(\mathbb{C})\). These families include interesting varieties such as base affine space, and are included in the larger set of braid varieties, which carry cluster algebra structures in general.

04.10.2023: Positroid varieties via representation theory (Algebra and Number Theory Seminar, Glasgow / Glaschu)

Total positivity is by now a classical subject in linear algebra, having begun in earnest with the work of Gantmacher and Krein from 1937. Recent results of Postnikov and others have emphasised the importance of positivity in flag varieties, particularly the Grassmannian. A key tool in this area is Postnikov’s positroid stratification of the Grassmannian, and the cluster algebra structures on its various (open) cells, recently confirmed to exist by Galashin and Lam.
In this talk, I will explain this story in the language of representation theory, with the positroid varieties and their cluster algebra structures being encoded by the representation theory of various non-commutative orders over the power series ring in one variable. Except for the top-dimensional stratum, Galashin and Lam’s construction produces two different cluster algebra structures on each open positroid, and an application of this representation-theoretic approach is a proof that these two structures quasi-coincide, as conjectured by Muller and Speyer in 2017. In particular, this means that these structures are equivalent from the point of view of total positivity.

28.09.2023: Positroid varieties via representation theory (FD Seminar, Online) [slides]

Total positivity is by now a classical subject in linear algebra, having begun in earnest with the work of Gantmacher and Krein from 1937. Recent results of Postnikov and others have emphasised the importance of positivity in flag varieties, particularly the Grassmannian. A key tool in this area is Postnikov’s positroid stratification of the Grassmannian, and the cluster algebra structures on its various (open) cells, recently confirmed to exist by Galashin and Lam.
In this talk, I will explain this story in the language of representation theory, with the positroid varieties and their cluster algebra structures being encoded by the representation theory of various non-commutative orders over the power series ring in one variable. Except for the top-dimensional stratum, Galashin and Lam’s construction produces two different cluster algebra structures on each open positroid, and an application of this representation-theoretic approach is a proof that these two structures quasi-coincide, as conjectured by Muller and Speyer in 2017. In particular, this means that these structures are equivalent from the point of view of total positivity.

14.09.2023: Quasi-coincidence of positroid cluster structures via derived categories (Silting in Representation Theory, Singularities, and Noncommutative Geometry, Oaxaca) [video]

A long-standing expectation in the study of total positivity, recently confirmed by Galashin and Lam, is that the cells in the positroid stratification of the Grassmannian have cluster algebra structures. The eventual construction actually provides two different such structures, but a conjecture by Muller and Speyer asserts that they should have a close relationship known as quasi-coincidence. In this talk, I will outline a proof of the conjecture. Despite the statement being about combinatorial geometry, the proof works by translating it into the language of representation theory, and then applying techniques from homological algebra: ultimately, the quasi-coincidence of the cluster structures follows from a statement about the derived category of a certain non-commutative Gorenstein ring.

18.08.2023: What is… an extriangulated category? (E-RNG Wee Math Seminar, Glasgow / Glaschu)

I will motivate and explain Nakaoka and Palu’s definition of extriangulated categories, a common generalisation of exact and triangulated categories. I will then briefly discuss the class of \(0\)-Auslander extriangulated categories, as defined by Gorsky, Nakaoka and Palu, and explain what these categories have to do with \(2\)-term complexes of projectives.

13.07.2023: Positroid cluster structures via derived categories (Homological Algebra and Representation Theory, Καρλόβασι)

04.07.2023: Homotopy arrow categories for finite-dimensional algebras (Nordic Congress of Mathematicians, Aalborg) [slides]

The homotopy arrow category of projectives for a finite-dimensional algebra \(A\) has as objects the morphisms between projective \(A\)-modules, and as morphisms commutative squares up to homotopy. This category has natural realisations as a full extension-closed subcategory of the triangulated category of perfect complexes of \(A\)-modules, and hence carries a canonical extriangulated structure. Indeed, it is one of the most important motivating examples of an extriangulated category.
In this talk I will describe various categorical equivalences involving the homotopy arrow category, in the context of both gentle algebras, and of cluster-tilted algebras (and their generalisations). This is joint work with Xin Fang, Mikhail Gorsky, Yann Palu and Pierre-Guy Plamondon.

13.06.2023: Homotopy arrow categories for finite-dimensional algebras (Advances in Representation Theory of Algebras IX, Kingston, ON)

10–12.05.2023: Positroid varieties: combinatorics, clusters and categorification (Mutations in Representation Theory of Algebras, Online / اصفهان)

The Grassmannian, the moduli space of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space, has certain subvarieties called open positroid varieties, with many interesting properties from the perspective of both Lie theory and total positivity. A recent theorem by Galashin and Lam (building on earlier partial results of Serhiyenko–Sherman-Bennett–Williams, Scott and Fomin–Zelevinsky) shows that the coordinate ring of each open positroid variety may be equipped with the structure of a cluster algebra in two ways, which are typically different, in the sense that different functions become cluster variables. Under an additional connectedness assumption, these cluster algebras admit Frobenius exact categorifications (by work of myself with Çanakçı and King, building on Jensen–King–Su and Baur–King–Marsh).
In this series of talks, I will explain what positroid varieties are, how their cluster algebra structures arise, and how to categorify these structures. At the end, I will show how to use the categorifications to make progress towards the proof of a conjecture by Muller and Speyer from 2017.

28.03.2023: Quasi-coincidence of cluster structures on positroid varieties (Combinatorial Aspects of Representation Theory, Oslo)

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 2017 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 and Keller.

14.03.2023: Quasi-coïncidence des structures amassées sur des variétés de positroïde (Séminaire AG, Versailles)

Les variétés de positroïde sont des sous-variétés de la grassmannienne qui apparaissent dans l’approche de Postnikov de la positivité totale. Pour cette raison et d’autres, on s’attendait depuis longtemps à ce que leurs anneaux de coordonnées portent une structure d’algèbre amassée, et cette attente a finalement été confirmée par Galashin et Lam en 2019. En fait, Galashin et Lam fournissent deux structures d’algèbre amassée (pour le prix d’une !) et bien qu’elles sont abstraitement isomorphes, elles ne sont pas égales, dans le sens où les variables amassées sont différents ensembles de fonctions sur la variété de positroïde dans chaque cas. Une conjecture de Muller et Speyer de 2017 affirme une relation précise entre ces deux structures amassées, à savoir qu’elles devraient quasi-coïncider. Cela impliquerait en particulier que les deux ensembles de monômes amassés sont en fait égaux.
Dans cet exposé, je présenterai (en anglais) une preuve de cette conjecture dans le cas générique où le positroïde est connecté. De façon peut-être surprenante, la preuve utilise la catégorification pour traduire le problème combinatoire en algèbre homologique. Plus précisément, elle utilise la catégorification des algèbres amassées par moi-même, celle des correspondances parfaites et divers autres ingrédients combinatoires de mon travail conjoint avec Çanakçı et King, et celle des morphismes quasi-amassés par Fraser et Keller.

2022

09.12.2022: Calabi–Yau algebras from consistent dimer models (GLEN Algebraic Geometry Seminar, Edinburgh / Dùn Èideann)

A dimer model is a bipartite graph drawn in a topological surface, which determines a typically non-commutative algebra. Under certain combinatorial consistency conditions, this algebra turns out to be Calabi–Yau, either on the nose or in a relative sense when the surface has boundary. I will explain this result, focussing on dimer models on the torus and on the disk. I will then describe some applications, which turn out to be quite different in the two cases: dimer models on the torus have applications in algebraic geometry, yielding resolutions of singularities of Gorenstein toric \(3\)-folds, whereas those on the disk lead to categorifications of cluster algebra structures on certain Lie-theoretic varieties.

02.11.2022: What is the Temperley–Lieb category? (Algebra Pre-Seminar, Glasgow / Glaschu)

I will describe the Temperley–Lieb category, as a monoidal category, in terms of a diagram calculus. Beginning with some preliminaries on monoidal categories, I will then define the Temperley–Lieb algebra, and finally the Temperley–Lieb category. Time-permitting, I will sketch the relationship between the Temperley–Lieb algebra and the Jones polynomial, a link invariant in knot theory.

12.10.2022: g-vectors in representation theory (Pure Maths Seminar, Lancaster)

The goal of this talk is to give a flavour of how representation theory can be applied to the recent theory of cluster algebras, introduced by Fomin and Zelevinsky. These are primarily combinatorial objects, with a great deal of complicated structure, which in the original definition must be computed recursively from some small amount of initial data. Using the cluster-theoretic notion of g-vectors as an example, I will demonstrate how homological methods from representation theory can be used to circumvent this recursive process, resulting in more direct computations.

26.08.2022: Stability, scattering and (co)silting (E-RNG Wee Math Seminar, Glasgow / Glaschu)

In this talk I will summarise, mainly through examples, part of the relationship between stability conditions (both King and Bridgeland flavoured), scattering diagrams (Kontsevich–Soibelman and Gross–Siebert), and cosilting mutation (Angeleri Hügel–Laking–Šťovíček–Vitória). The presentation is based on lecture series at ICRA 2022 by Anna Barbieri, Rosanna Laking and Hipólito Treffinger.

09.08.2022: On categorification of g-vectors (ICRA 2022, Buenos Aires) [slides]

First arising in a combinatorial form in Fomin and Zelevinsky’s theory of cluster algebras, g-vectors have two closely related representation-theoretic incarnations. The first of these is the notion of an index (or coindex) in a \(2\)-Calabi–Yau triangulated category, whereas the second involves projective presentations of modules over finite-dimensional algebras. In this talk I will explain some joint work in progress with Xin Fang, Mikhail Gorsky, Yann Palu and Pierre-Guy Plamondon, in which we show how to lift the relationship between these two computations to an equivalence of categories. The categories in question will be extriangulated, and part of the talk will serve as an introduction to such categories, via well-behaved examples. I will also explain how our results extend beyond the triangulated setting, to indices in more general stably \(2\)-Calabi–Yau categories.

21.07.2022: Categorification of positroids (Geometric and Combinatorial Methods in Homological Algebra, Aarhus)

Postnikov diagrams are combinatorial objects consisting of strands drawn in a disc, and encode initial seeds for cluster algebra structures on the coordinate ring of the Grassmannian and more general positroid varieties. In particular, such a diagram can be used to compute a cluster of Plücker coordinates, together with its exchange quiver, which then generates the relevant coordinate ring as a cluster algebra. In this talk I will explain joint work with İlke Çanakçı and Alastair King, in which we show that this combinatorial computation is in fact algebraically natural, from the point of view of Jensen, King and Su’s Grassmannian cluster category and the representation theory of a pair of algebras attached to the Postnikov diagram.

10.06.2022: Frieze patterns and derived categories (E-RNG Wee Math Seminar, Glasgow / Glaschu)

I will give a quick tour of some topics related to frieze patterns, with a particular focus on links to the derived category of a Dynkin quiver (and to quotients of this, such as the cluster category).

06.06.2022: Categorification for positroid varieties (Clusters and Braids Seminar, Online)

Coordinate rings of open positroid varieties in the Grassmannian carry a rich and complex combinatorial structure arising from Postnikov diagrams (also known as alternating strand diagrams, or dimer models in the disk). A recent result in this area by Galashin and Lam, confirms the conjecture that these coordinate rings are isomorphic to cluster algebras. In this talk, I will explain how these cluster algebras can be (additively) categorised, allowing for an interpretation of much of the combinatorics of Postnikov diagrams in terms of representation theory. In particular, I will explain how face labels for Postnikov diagrams (which determine clusters of Plücker coordinates) may be understood as names of modules over an appropriate algebra. Time permitting, I will also explain how the twist automorphism (or Donaldson–Thomas transformation) corresponds to the syzygy functor on the categorification. This is based on joint work with İlke Çanakçı and Alastair King.

04.06.2022: On categorification of g-vectors (Øyvind 60, Trondheim)

First arising in a combinatorial form in Fomin and Zelevinsky’s theory of cluster algebras, g-vectors have two closely related representation-theoretic incarnations. The first of these is the notion of an index (or coindex) in a \(2\)-Calabi–Yau triangulated category, whereas the second involves projective presentations of modules over finite-dimensional algebras. In this talk I will explain some joint work in progress with Xin Fang, Mikhail Gorsky, Yann Palu and Pierre-Guy Plamondon, in which we show how to lift the relationship between these two computations to an equivalence of extriangulated categories, as well as generalise to the case of indices in stably \(2\)-Calabi–Yau Frobenius exact categories. Part of the talk will serve as a brief introduction to well-behaved classes of extriangulated categories and their relationship to relative homological algebra as developed by Auslander and Solberg.

08.04.2022: Tilting: derived equivalences and mutation (E-RNG Wee Math Seminar, Glasgow / Glaschu)

I will explain the relationship between tilting modules (and complexes) and derived equivalences, focussing on the case of finite-dimensional algebras. I will then discuss two approaches to completing the class of (classical) tilting objects with respect to mutation, namely \(\tau\)-tilting and cluster-tilting.

31.03.2022: Dual dimers (Working Group, Glasgow / Glaschu)

I will describe Bocklandt’s construction of dual dimer models, and related equivalences of (\(A_\infty\)-)categories, primarily through examples.

17.03.2022: Categorification of positroids (58th ARTIN Meeting, Manchester)

The coordinate ring of the Grassmannian has a very rich combinatorial structure arising from Postnikov diagrams (also known as alternating strand diagrams, plabic graphs, or dimer models in the disk). In particular, the choice of such a diagram determines a positroid, a corresponding subvariety of the Grassmannian, and a cluster algebra structure on the coordinate ring of that variety. In this talk, I will explain how these combinatorial constructions may be interpreted in terms of representations of quivers (i.e. categorified). This is joint work with İlke Çanakçı and Alastair King.

02–04.03.2022: Dimer models: consistency, Calabi–Yau properties and categorification (Preprojective Algebras and Calabi–Yau Algebras, Online / 大阪)

A dimer model is a bipartite graph drawn in a surface. First introduced in the context of statistical mechanics, dimer models became a significant topic in string theory around fifteen years ago. In mathematics, a key development at this time was the study of consistency conditions, and the use of dimer models on the torus to construct non-commutative crepant resolutions of \(3\)-dimensional Gorenstein singularities. A key property of a consistent dimer model is that its associated non-commutative dimer algebra is \(3\)-Calabi–Yau. More recently, dimer models have reappeared, now on surfaces with boundary and sometimes called plabic graphs or Postnikov diagrams, in the context of categorifying cluster algebras with coefficients, notably the cluster structure on the Grassmannian and its positroid strata. In this lecture series, I will survey these ideas.

23.02.2022: Grassmannian twists categorified (Algebra and Number Theory Seminar, Glasgow / Glaschu)

The Grassmannian of \(k\)-dimensional subspaces of an \(n\)-dimensional space carries a birational automorphism called the twist (or sometimes the Donaldson–Thomas transformation), defined by Berenstein–Fomin–Zelevinsky and Marsh–Scott. This automorphism respects the cluster algebra structure on the coordinate ring, being a quasi-cluster automorphism in the sense of Fraser. By work of Muller–Speyer, similar results hold for positroid strata in the Grassmannian.
The cluster algebras in this picture have been categorified, by Jensen–King–Su in the case of the full Grassmannian, and by myself for more general (connected) positroid varieties. In this talk I will report on joint work with İlke Çanakçı and Alastair King, in which we describe the twist in terms of these categorifications. The key ingredient is provided by perfect matching modules, certain combinatorially defined representations for a quiver ‘with faces’, and I will also explain this construction.

15.02.2022: Grassmannian twists categorified (UNIST International Workshop on Geometry and Mathematical Physics, Online / 울산시) [video]

10.02.2022: Grassmannian twists categorified (LAGOON Seminar, Online / Edinburgh / Dùn Èideann) [video]

2021

08.12.2021: Categorification of positroids and positroid varieties (Hodge Seminar, Edinburgh / Dùn Èideann)

The Grassmannian and its totally positive part have a very rich combinatorial structure, studied by many people. In particular, Postnikov has shown how the totally positive Grassmannian is stratified by positroid varieties. Recent work of Galashin and Lam shows that the coordinate ring of each (open) positroid stratum is a cluster algebra, with this structure determined by a combinatorial object called a Postnikov diagram. In this talk I will explain how the same diagram also gives rise to representation theoretic objects which can be used to (additively) categorify this cluster algebra. This is partly joint work with İlke Çanakçı and Alastair King.

08.12.2021: Grassmannian cluster algebras (Hodge Seminar Pretalk, Edinburgh / Dùn Èideann)

For this pretalk, I will give an overview of the Grassmannian, its ring of functions and its positroid stratification. I will review the definition of a cluster algebra, and hint at its relationship to the Grassmannian: the main talk will make this connection more precise.

18.10.2021: A cluster character for \(y\)-variables (Séminaire d’Algèbre d’IMJ-PRG, Online / Paris)

Given a (Frobenius or triangulated) cluster category, I will explain how to categorify various cluster algebraic identities via lattice maps associated to pairs of cluster-tilting objects. For example, one such map is the index, well-known to categorify \(g\)-vectors. Using this formalism, I will recall the cluster character for \(x\)-variables developed by Caldero–Chapoton, Palu, Fu–Keller and others, and give a similar categorical expression for \(y\)-variables. This is joint work with Jan E. Grabowski.

23.09.2021: A cluster character for \(y\)-variables (New Developments in Representation Theory Arising From Cluster Algebras, Cambridge)

Given a cluster category, broadly interpreted, and a cluster-tilting object of this category, work of various authors (Caldero–Chapoton, Palu, Fu–Keller,...) allows us to associate a Laurent polynomial to any object of the category. Restricting to reachable rigid indecomposable objects, we recover the \(x\)-cluster variables (in Fomin–Zelevinsky’s notation) of the cluster algebra determined by the quiver of our chosen cluster-tilting object. In this talk I will report on joint work with Jan E. Grabowski, in which we give an analogous formula for the \(y\)-cluster variables by associating a rational function to each module for the endomorphism algebra of a cluster-tilting object in the category.

06.08.2021: Categorification of positroid varieties (Cluster Algebras and Related Topics, Online / 北京)

A connected Postnikov diagram \(D\), consisting of a set of crossing strands in the disc, determines both a cluster algebra structure on a corresponding positroid variety in the Grassmannian, and a (Frobenius) categorification of this cluster algebra. I will explain this construction, and its relationship to Jensen, King and Su’s Grassmannian cluster category. A key tool is provided by perfect matchings of the dimer model associated to \(D\), which correspond to certain modules for the endomorphism algebra of a cluster-tilting object in the categorification. If time permits, I will also explain how considering perfect matching modules allows us to relate Marsh and Scott’s dimer partition function for twisted Plücker coordinates to the cluster character. This is partly joint work with İ. Çanakçı and A. King.

25.02.2021: Grassmannians and cluster algebras (Plymouth Pure Mathematics Seminar, Online / Plymouth)

The Grassmannian is a projective variety parameterising \(k\)-dimensional subspaces of an \(n\)-dimensional vector space, and its coordinate ring has the rich combinatorial structure of a cluster algebra, by results of Fomin–Zelevinsky and Scott. Moreover, this cluster algebra has been categorified by Jensen–King–Su, and Baur–King–Marsh relate this categorification to dimer models, or plabic graphs, in the disc. In this talk I will explain some aspects of these constructions, and show that by using dimer models in the disc as the starting point, they can be generalised to certain open positroid subvarieties of the Grassmannian, which also have cluster algebra structures by recent work of Galashin and Lam.

17.02.2021: The cluster category of a Postnikov diagram (The TRAC Seminar, Online / Sherbrooke) [notes]

A Postnikov diagram consists of a collection of strands in the disc, with combinatorial restrictions on their crossings. Such diagrams were used by Postnikov and others to study weakly separated collections in certain matroids called positroids. In this talk I will explain how the diagram determines a cluster algebra structure on a suitable subvariety of the Grassmannian, and simultaneously provides a (Frobenius) categorification of this cluster algebra.

09.02.2021: The cluster category of a Postnikov diagram (Leeds Algebra Seminar, Online / Leeds)

08.01.2021: Auslander–Reiten translations for Gorenstein algebras: an observation (Flash Talks in Representation Theory, Online / Trondheim) [video]

The category of Gorenstein projective modules for a finite-dimensional Gorenstein algebra has almost split sequences, and hence an Auslander–Reiten translation. Consequently, a Gorenstein projective module has two Auslander–Reiten translates, one in the subcategory of Gorenstein projectives, and one in the whole module category. In joint work with Sondre Kvamme, we relate these two translates, and in so doing shed new light on results of García Elsener and Schiffler on Calabi–Yau tilted algebras.

2020

06–09.10.2020: From frieze patterns to cluster categories (LMS Autumn Algebra School, Online / Edinburgh / Dùn Èideann) [notes, slides, video 1, video 2, video 3]

In this series of three talks, we will sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers, motivated by Conway and Coxeter’s combinatorial results concerning frieze patterns. The goal is to show how these more abstract theories provide a conceptual explanation for phenomena concerning friezes, principally integrality and periodicity. To this end, in the first talk we will introduce frieze patterns and their combinatorial properties, and explain Conway and Coxeter’s classification in terms of triangulated polygons. The second talk will introduce Fomin–Zelevinsky’s cluster algebras, and show how integrality of friezes follows from the Laurent phenomenon from cluster variables. In the third talk, we define cluster categories of acyclic quivers, following Buan, Marsh, Reineke, Reiten and Todorov, and explain their relationship to cluster algebras, leading to an explanation for periodicity of friezes.

16.09.2020: Cluster categories from Postnikov diagrams (Categorifications in Representation Theory, Online / Leicester) [slides]

Many rings of interest in geometry can be equipped with the additional combinatorial structure of a cluster algebra, which one would like to understand representation-theoretically by means of a cluster category. A result of Jensen, King and Su provides such a category for the cluster algebra structure on the coordinate ring of the Grassmannian, and Baur, King and Marsh show how this category is related to Postnikov diagrams, certain collections of oriented paths in a disc. In this talk I will explain how to reverse this logic, and use Postnikov diagrams to produce cluster categories. As an application, this allows us to categorify the cluster algebra structures on positroid subvarieties in the Grassmannian.

13.08.2020: From dimers in the disc to cluster categories (Dimers in Combinatorics and Cluster Algebras, Online / Ann Arbor, MI) [slides, video]

A consistent dimer model on the disc determines a cluster algebra structure on the coordinate ring of a positroid variety in the Grassmannian. I will explain how the dimer model can also be used to give a categorification of this cluster algebra, as a result of certain Calabi–Yau symmetries in the dimer algebra.

21.05.2020: Calabi–Yau properties of Postnikov diagrams (FD Seminar, Online) [slides]

A Postnikov diagram is a collection of strands in the disk, satisfying combinatorial conditions on their crossings. The diagram determines many other mathematical objects, including a cluster algebra, which Galashin and Lam have recently shown to be isomorphic to the homogeneous coordinate ring of a certain subvariety of the Grassmannian, called a positroid variety. In this talk, I will explain how to categorify this cluster algebra, using a second (non-commutative) algebra attached to the Postnikov diagram. The approach depends on studying Calabi–Yau symmetries of this algebra, and relates to work of Broomhead and others concerning dimer models on closed surfaces.

17.02.2020: Propriétés Calabi–Yau des diagrammes de Postnikov (Séminaire d’Algèbre d’IMJ-PRG, Paris)

Un diagramme de Postnikov est une collection de brins dans un disque, soumise à certaines règles concernant les croisements des brins. Le diagramme détermine beaucoup d’autres objets et notamment une algèbre amassée. Un théorème récent de Galashin et Lam affirme que cette algèbre est isomorphe à l’anneau des fonctions régulières sur une sous-variété de la grassmannienne dite variété de positroïde. Dans cet exposé, je vais expliquer la construction d’une catégorification de cette algèbre amassée basée sur une propriété de Calabi–Yau d’une autre algèbre (non commutative) associée au diagramme de Postnikov.

04.02.2020: Calabi–Yau algebras from consistent dimer models (Working Group on Dimer Models, Stuttgart)

I will explain one of the main results of Broomhead’s thesis; namely, that the dimer algebra of an algebraically consistent dimer model on the torus is a \(3\)-Calabi–Yau algebra. Moreover, it is a non-commutative crepant resolution of the toric singularity given by its centre.

2019

11.12.2019: Matrix factorisations and Knörrer periodicity (Working Group on Cohen–Macaulay Modules, Stuttgart)

I will explain how to study Cohen–Macaulay modules over a \(d\)-dimensional complete hypersurface singularity \(R\) in terms of matrix factorisations. Using this description, I will explain how Cohen–Macaulay modules over \(R\) are related to those over its \((d+1)\)-dimensional branched double cover \(R^\#\). This leads to Knörrer’s periodicity theorem, an equivalence of categories \(\underline{\operatorname{MCM}}(R)\stackrel{\sim}{\to}\underline{\operatorname{MCM}}((R^\#)^\#)\).

06.12.2019: Cluster categories via internally Calabi–Yau algebras (Darstellungstheorie Oberseminar, Bonn)

Starting from a quiver with potential \((Q,W)\), I will explain how to construct an algebra \(A\) which is \(3\)-Calabi–Yau ‘in its interior’, and a distinguished subalgebra \(B\). Whenever \(B\) is Noetherian, it is a Gorenstein algebra with \(2\)-Calabi–Yau singularity category, conjecturally equivalent to Amiot’s cluster category for \((Q,W)\). When \(Q\) is acyclic, both Noetherianity of \(B\) and this equivalence hold. The construction appears to be related to the Ginzburg dg-algebra of \((Q,W)\), and I will discuss these similarities. Moreover, the Frobenius category of Gorenstein projective \(B\)-modules is an example of an additive categorification of a cluster algebra. Time permitting, I will give more examples of internally \(3\)-Calabi–Yau algebras, which determine further categorifications of this form.

20.11.2019: Catégories amassées via algèbres de Calabi–Yau à l’intérieur (Séminaire Quantique, Strasbourg)

En commencent avec un carquois à potentiel \((Q,W)\), je vais décrire comment on construit une algèbre \(A\), qui est \(3\)-Calabi–Yau « à l’intérieur ». À son tour, \(A\) détermine une algèbre \(B\), qui est Gorenstein et qui a une catégorie de \(2\)-Calabi–Yau des singularités, quand elle est Noethérienne. Si \(Q\) n’a pas des cycles, l’algèbre \(B\) est Noethérienne, et sa catégorie des singularités est équivalente à la catégorie amassée de \(Q\). Cette construction est similaire à celle de la dg-algèbre de Ginzburg de \((Q,W)\), et je vais expliquer des connexions précises et conjecturales de cette dg-algèbre. Si le temps le permet, je donnerai autres sources des algèbres Calabi–Yau à l’intérieur.

11.10.2019: Calabi–Yau singularity categories (Algebraic Representation Theory and Related Topics / 代数表示理论及相关专题研讨会, 三亚)

Starting from the data of a quiver with potential \((Q,W)\), I will explain how to construct an algebra \(B\) which is Gorenstein with \(2\)-Calabi–Yau singularity category whenever it is Noetherian. Conjecturally, this singularity category is equivalent to Amiot’s cluster category for \((Q,W)\), and both Noetherianity of \(B\) and the equivalence are proved to hold when \(Q\) is acyclic. The construction appears to be related to the Ginzburg dg-algebra of \((Q,W)\), and I will discuss both precise and conjectural connections to this dg-algebra.

02.09.2019: Desingularising quiver Grassmannians via tilting (Tilting Theory, Singularity Categories and Non-Commutative Resolutions, Oaxaca) [video]

Given an \(A\)-module \(M\), and a dimension vector \(d\), one can define a quiver Grassmannian, a projective algebraic variety parametrising the \(d\)-dimensional \(A\)-submodules of \(M\). A famous result in geometric representation theory, obtained by several different authors, states that every projective variety \(X\) is isomorphic to such a quiver Grassmannian. In this talk I will explain how, at least in certain cases, one can use this algebraic description to construct a desingularisation of \(X\). The construction is representation-theoretic, involving a tilt of an endomorphism algebra in \(\operatorname{mod}(A)\), and the desingularising variety is again described in terms of quiver Grassmannians. This talk is based on joint work with Julia Sauter, in which we extend methodology of Crawley-Boevey and Sauter, and of Cerulli Irelli, Feigin and Reineke.

09.07.2019: Perfect matching modules for dimer algebras (SIAM Conference on Applied Algebraic Geometry, Bern) [slides]

The theory of dimer models, or bipartite graphs on surfaces, first arose in theoretical physics, and later found diverse applications in geometry and representation theory. Recently, there has been much interest in dimer models on the disk, particularly those arising from Postnikov diagrams, and their relationship to Grassmannian cluster algebras and categories. Perfect matchings of dimer models play a central role in the theory. In joint work with İlke Çanakçı and Alastair King, we provide an algebraic viewpoint on these objects, by defining and studying a module for the dimer algebra for each perfect matching. As an application, we explain the relationship between combinatorial and homological formulae for computing Grassmannian cluster variables.

11.06.2019: The Caldero–Chapoton formula as a dimer partition function (Cluster Algebras and Representation Theory, 京都)

Certain elements of a Grassmannian cluster algebra, the twisted Plücker coordinates, are expressible as ‘dimer partition functions’, i.e. as weighted sums over the set of perfect matchings of a bipartite graph, or dimer model, in the disk, via a formula due to Marsh and Scott. Using the categorification of this cluster algebra by Jensen, King and Su, the more familiar Caldero–Chapton formula provides another expression for the twisted Plücker coordinates, this time in terms of homological data in the category. I will explain how these two formulas are essentially the same, and in so doing relate combinatorial data from the dimer model to homological data in the JKS categorification. For example, a perfect matching on the dimer model encodes a module for the corresponding Jacobian algebra. This is joint work with İlke Çanakçı and Alastair King.

21.05.2019: Dominant dimension and double centraliser properties (Working Group on Double Centraliser Theorems, Stuttgart)

I will explain the homological approach to double centraliser properties, via dominant dimension and tilting, due to König, Slungård and Xi. I will then show how to apply this approach to prove classical Schur–Weyl duality.

23.03.2019: Calabi–Yau singularity categories (Interactions Between Commutative Algebra, Representation Theory and Algebraic Geometry, Münster)

I will explain a construction of Iwanaga–Gorenstein algebras having Calabi–Yau singularity categories. I will outline several connections between this construction and Buchweitz’s work, and, time-permitting, also say something about the original motivation, coming from the theory of cluster algebras.

22.01.2019: Koszulity for preprojective algebras (Working Group on Koszul Duality, Stuttgart)

Following the proof by Etingof and Eu via Hilbert series, I will show that the preprojective algebra of a non-Dynkin quiver is Koszul.

2018

02.11.2018: Calabi–Yau categories from Gorenstein algebras (Stability Conditions and Representation Theory of Finite-Dimensional Algebras, Oaxaca) [video]

Let \((Q,W)\) be a quiver with potential having finite-dimensional Jacobian algebra. I will construct from \((Q,W)\) a new algebra, which will be Gorenstein with \(2\)-Calabi–Yau singularity category whenever it is Noetherian, this singularity category being conjecturally equivalent to Amiot’s cluster category of \((Q,W)\). Both the Noetherianity and the equivalence are proved to hold when \(Q\) is acyclic. The construction appears to be related to the Ginzburg dg-algebra of \((Q,W)\), and I will discuss both precise and conjectural connections to this dg-algebra. Time permitting, I will also discuss results in other Calabi–Yau dimensions.

17.09.2018: Calabi–Yau categories from Gorenstein algebras (First SWAN Workshop, Stuttgart)

I will explain how to construct Gorenstein algebras whose category of Gorenstein projective modules is stably Calabi–Yau. Moreover, I will show how (at least some of) Amiot’s \(2\)-Calabi–Yau generalised cluster categories may be realised in this way, avoiding the usual construction via dg-algebras. Time permitting, I will also discuss some examples in other dimensions.

16.08.2018: Perfect matching modules for dimer algebras (ICRA 2018, Praha) [slides]

The theory of dimer models, or bipartite graphs on surfaces, first arose in theoretical physics, and later found diverse applications in geometry and representation theory. Recently, there has been much interest in dimer models on the disk and their relationship to Grassmannian cluster algebras and categories. Perfect matchings of dimer models play a central role in the theory. In joint work with Alastair King and İlke Çanakçı, we provide an algebraic viewpoint on these objects, by associating to them modules over the dimer algebra. Studying these perfect matching modules reveals the interplay between the combinatorial, algebraic and topological aspects of dimer models. As an application, we explain the relationship between combinatorial and homological formulae for computing Grassmannian cluster variables.

29.05.2018: Stability conditions and moduli spaces of quiver representations (Working Group on Stability Conditions, Stuttgart)

I will discuss the relationship between King’s stability conditions on abelian categories, used to construct moduli spaces of representations of finite-dimensional algebras, and the stability conditions arising in Mumford’s geometric invariant theory for reductive group actions on vector spaces.

02.05.2018: Perfect matchings and modules for dimer algebras (Séminaire de Combinatoire et d’Informatique Mathématique du LaCIM, Montréal, QC)

The Grassmannian, an algebraic variety parametrising the \(k\)-dimensional subspaces of an \(n\)-dimensional space, has a very rich combinatorial and geometric structure. Much of this structure is accessible via dimer models, certain bipartite graphs drawn in a disk. For example, Marsh and Scott showed that certain special functions on the Grassmannian are expressible as ‘dimer partition functions’, i.e. weighted sums over the set of perfect matchings of a dimer model. After explaining the Marsh–Scott formula, I will report on joint work with İlke Çanakçı and Alastair King, in which we compare it to a second, more representation-theoretic, expression for the same function. This comparison allows us to deduce information about the representation theory of various algebras associated to the dimer model.

25.04.2018: A tilting viewpoint on higher Auslander algebras (Maurice Auslander International Conference, Woods Hole, MA)

To any finite-dimensional algebra, one may ‘canonically’ associate a sequence of tilting modules and a sequence of cotilting modules, the lengths of which depend on the dominant dimension of the algebra. Typically these sequences do not intersect—indeed, they do so if and only if each is the other read backwards, and this property characterises (minimal) Auslander–Gorenstein algebras, in the sense of Iyama–Solberg. Alongside this result, I will also explain some other special properties and applications of the modules appearing in these sequences. This is joint work with Julia Sauter.

17.04.2018: Almost split sequences and Cohen–Macaulay modules (Working Group on Exact Categories and Algebras of Finite Cohen–Macaulay Type, Stuttgart)

The goal of this talk is to explain two theorems of Auslander which provide motivation for subsequent talks in the working group. The first, known as the Auslander correspondence, describes the structure of module categories of representation-finite finite-dimensional algebras, in terms of homological properties of the endomorphism ring of an additive generator. The second gives a ‘geometric’ condition for certain ‘Cohen–Macaulay-like’ categories to have almost split sequences.

15.03.2018: The Caldero–Chapoton formula as a dimer partition function (Tropical Geometry Meets Representation Theory, Köln) [slides]

Certain elements of a Grassmannian cluster algebra, the twisted Plücker coordinates, are expressible as ‘dimer partition functions’, i.e. weighted sums over the set of perfect matchings of a bipartite graph, or dimer model, in the disk, via the Marsh–Scott formula. Using the categorification of this cluster algebra by Jensen, King and Su, together with the Caldero–Chapton formula, one can also give a formula for the twisted Plücker coordinates in terms of homological data in the category. I will explain how these two formulae are essentially the same, and in so doing relate combinatorial data from the dimer model to homological data in the JKS categorification. For example, a perfect matching on the dimer model encodes a module for the corresponding Jacobian algebra. This is joint work with İlke Çanakçı and Alastair King.

2017

10.11.2017: Special tilting modules and dominant dimension (Darstellungstheorietage 2017, Eichstätt)

Tilting theory is a classical subject in the representation theory of associative algebras, being useful, for example, in describing when two algebras have the same bounded derived category of modules. I will give a flavour of this theory via some particularly simple examples of tilting and cotilting modules. These modules, along with a classical invariant, the dominant dimension, can be used to characterise various families of algebras, such as Auslander algebras and their higher analogues as introduced by Iyama. This is joint work with Julia Sauter.

24.10.2017: Tilting and \(\tau\)-tilting (Working Group on \(\tau\)-Tilting Theory, Stuttgart)

Exchange phenomena, whereby each element of a maximal set of ‘compatible’ objects may be replaced by a unique different one to give a new maximal compatible set, appear throughout mathematics. I will give an overview of some results in classical tilting theory, explaining in particular that tilting objects exhibit an exchange phenomenon to some extent—given an indecomposable summand of a tilting object, there is at most one non-isomorphic indecomposable module it can be replaced with in order to obtain a new tilting object, but there may be none. As we will see in the later working group talks, support \(\tau\)-tilting objects, which generalise classical tilting objects, complete the picture, recovering a full exchange phenomenon.

27.09.2017: Dimer models and cluster categorification (Oberseminar am Institut für Algebra und Zahlentheorie, Stuttgart)

The cluster algebra structure on the homogeneous coordinate ring of a Grassmannian has been categorified by Jensen–King–Su. The interplay between the combinatorics and the representation theory can be studied via dimer models, or equivalently Postnikov diagrams, which are also fundamental to other structures on the Grassmannian, such as its positive part and matroid stratification. By comparing two formulae for computing cluster monomials, one representation theoretic (the Fu–Keller cluster character) and one combinatorial (the Marsh–Scott dimer partition function), I will explain some of these interactions. In particular, I will explain the relationship between perfect matchings on the dimer model and certain modules for its ‘cluster-tilted’ algebra. This is joint work with İlke Çanakçı and Alastair King.

18.08.2017: Derived equivalence classification of gentle algebras arising from surfaces (BIREP Summer School on Gentle Algebras, Bad Driburg)

The aim of this talk is to describe Ladkani’s derived equivalence classification of gentle algebras arising from ideal triangulations of unpunctured surfaces, via good mutations and Avella-Alaminos–Geiß invariants.

25.07.2017: Functors and morphisms determined by subcategories (Working Group on Auslander’s Philadelphia Notes, Stuttgart)

Following exposition by Krause, I will explain some aspects of the theory of morphisms determined by modules from a more functorial perspective. The main result will be a characterisation of those morphisms in the (large) module category of an arbitrary ring which are (right) determined by a subcategory of the finitely presented modules.

30.05.2017: Principal cluster categories (Oberseminar am Institut für Algebra und Zahlentheorie, Stuttgart)

I will describe constructions of \(2\)-Calabi–Yau triangulated categories and stably \(2\)-Calabi–Yau Frobenius categories. In particular, I will demonstrate how Amiot’s \(2\)-Calabi–Yau triangulated cluster category, usually defined via dg-algebras, may be realised as the stable category of the Frobenius category of Gorenstein projective modules over an ordinary algebra. One consequence of this approach is the construction of a categorification of the (polarised) principal coefficient cluster algebra associated to any acyclic quiver.

02.05.2017: Lattices of morphisms determined by modules (Working Group on Auslander’s Philadelphia Notes, Stuttgart)

In this talk I will introudce the lattice of right-equivalence classes of morphisms ending in a fixed object, and show that it is indeed a lattice. I will also introduce the ‘sublattices’ (strictly sub-meet-semilattices) of morphisms determined by a module. The second goal of the talk is to give a proof of the first main result of the seminar, namely that every morphism in the module category of an Artin algebra is right-determined by some module, so that these smaller lattices in fact cover the larger lattice.

28.03.2017: From dimer models to cluster variables: the Marsh–Scott formula revisited (Cluster Algebras in Mathematical Physics, Mainz)

A key example of a geometric cluster algebra is Scott’s cluster structure on the homogeneous coordinate ring of the Grassmannian. As was implicit in Scott’s original work, and made explicit by Baur–King–Marsh, this cluster algebra is intimately related to the combinatorics of dimer models on discs. Marsh and Scott have given a formula for computing certain cluster monomials (the twisted Plücker coordinates) in terms of these dimer models. These monomials may also be computed by Fu–Keller’s cluster character on the Grassmannian cluster category introduced by Jensen–King–Su. In joint work with İlke Çanakçı and Alastair King, we compare these formulae, yielding representation theoretic interpretations for aspects of the dimer combinatorics.

09.03.2017: Perfect matchings and the Grassmannian cluster algebra (Algebra and Number Theory Seminar, Graz)

A key example of a geometric cluster algebra is Scott’s cluster structure on the homogeneous coordinate ring of the Grassmannian. This cluster algebra admits a categorification, due to Jensen, King and Su, and it has been shown by Baur, King and Marsh that the endomorphism algebras of some special objects in this category may be described combinatorially in terms of bipartite graphs on a disc, called dimer models. Marsh and Scott give a formula computing some of the cluster monomials (the twisted Plücker coordinates) combinatorially from these dimer models. These cluster monomials may also be computed using the cluster character formula, provided in this level of generality by Fu and Keller. Understanding how these two apparently different computations arrive at the same answer reveals further connections between dimer models and homological algebra in the Grassmannian cluster category. This is joint work with İlke Çanakçı and Alastair King.

2016

09.12.2016: Classifying irreducible representations of \(\mathrm{GL}_n(\mathbb{Q}_p)\) (Affine Hecke Algebras, Bonn)

I will explain Zelevinsky’s classifiction of irreducible representations of \(G_n=\mathrm{GL}_n(\mathbb{Q}_p)\), in characteristic zero, in terms of segments of cuspidal representations for \(G_m\) with \(m\) dividing \(n\). This combinatorics will be related to the classification of finitely generated nilpotent representations of the path algebra of a linearly oriented quiver of type \(\mathsf{A}^\infty_\infty\).

25.11.2016: Dominant dimension and canonical tilts (BIREP Seminar, Bielefeld)

Any finite dimensional algebra with dominant dimension \(d\) admits a ‘canonical’ \(k\)-tilting module for each \(k\) from \(0\) to \(d\), each giving a derived equivalence with some algebra \(B_k\). These tilts have very special properties; for example, they never increase the global dimension. In the case of the Auslander algebra of a representation-finite algebra \(A\), Crawley-Boevey and Sauter (generalising Cerulli Irelli, Feigin and Reineke) used the tilt \(B_1\) to construct desingularisations of certain varieties of \(A\)-modules. More generally, for \(d\) at least \(2\), any algebra of dominant dimension \(d\) is the endomorphism algebra of a generating-cogenerating module \(M\) over some algebra \(A\), and many of the results for Auslander algebras have analogues in this setting. In particular, we may realise each \(B_k\) as an endomorphism algebra in the homotopy category of \(A\), an observation which we can exploit to describe rank varieties, of arbitrary finite dimensional modules over arbitrary finite dimensional algebras, as affine quotient varieties. We may also use the canonical tilting modules to give a new characterisation of \(d\)-Auslander (or, more generally, \(d\)-Auslander–Gorenstein) algebras. This is joint work with Julia Sauter.

20.10.2016: Graded Frobenius cluster categories (Workshop on Cluster Algebras and Lie Theory, Roma) [slides]

A grading of a cluster algebra is a grading of the underlying algebra such that all cluster variables are homogeneous elements. In practice, to specify a grading it is enough to give local information, by assigning degrees to the cluster variables of a single cluster in such a way that the exchange relations of that cluster are homogeneous. For cluster algebras categorified by Frobenius categories, we reinterpret gradings categorically, both globally, as functions on the Grothendieck group, and locally, as dimension vectors for endomorphism algebras of cluster-tilting objects. This is joint work with Jan Grabowski.

15.08.2016: Internally Calabi–Yau algebras (ICRA 2016, Syracuse, NY) [slides]

I will define what it means for an algebra to be internally Calabi–Yau with respect to an idempotent. This generalises the definition of a Calabi–Yau algebra by allowing the required Ext-group symmetries for modules to have a ‘restricted support’. I will explain how internally Calabi–Yau algebras are related to cluster-tilting objects in certain stably Calabi–Yau Frobenius categories, thus providing a link to the categorification programme for cluster algebras with frozen variables.

05.08.2016: Internally Calabi–Yau algebras (Oberseminar am Institut für Algebra und Zahlentheorie, Stuttgart)

I will explain what an internally Calabi–Yau algebra is, how such algebras arise from Frobenius categories with cluster-tilting objects, and vice versa. Frozen Jacobian algebras provide a source of internally \(3\)-Calabi–Yau algebras, which are related to the categorification of cluster algebras. I will show how to associate an internally \(3\)-Calabi–Yau frozen Jacobian algebra to any suitably graded quiver with potential.

15.07.2016: Internally Calabi–Yau algebras (Darstellungstheorie Oberseminar, Bonn)

I will explain what an internally Calabi–Yau algebra is, how such algebras arise from Frobenius categories modelling (higher) cluster algebras, and vice versa. Frozen Jacobian algebras provide a source of internally \(3\)-Calabi–Yau algebras, which are related to usual cluster algebras. I will show how to associate an internally \(3\)-Calabi–Yau frozen Jacobian algebra to any quiver with potential such that this potential is homogeneous with respect to some grading of the path algebra in which arrows have positive degrees.

13.06.2016: Koszul duality and equivalences of categories (Derived Deformation Theory and Koszul Duality, Bonn)

Let \(A\) be a Koszul algebra with semi-simple degree zero part \(k\) and let \(A^!=\operatorname{Ext}^\bullet_A(k,k)^{\mathrm{op}}\) be its Koszul dual. Then modules over \(A\) are related to modules over \(A^!\) via a pair of adjoint functors at the level of homotopy categories of chain complexes. Some explicit computations show that these adjoint functors do not in general descend to equivalences of the derived categories, but do induce equivalences of other, related categories (cf. the Berenstein–Gelʼfand–Gelʼfand correspondence). I will describe the induced equivalences of Fløystad, which are between categories partway between the homotopy and derived categories, and Keller (d’après Lefèvre), who gives an equivalence between the derived category of \(A\) and the ‘coderived’ category of the dual coalgebra to \(A^!\). While I will largely stick to the classical setting, I will try to indicate when results hold more generally.

03.05.2016: Internally Calabi–Yau algebras (Quivers and Bipartite Graphs, London)

To any dimer model on a closed surface, one can associate a Jacobian algebra, which, by work of Broomhead, is \(3\)-Calabi–Yau if the dimer model is consistent. A modification of the construction associates a frozen Jacobian algebra to a dimer model on a surface with boundary. In this context it is reasonable to ask when this algebra is Calabi–Yau ‘away from the boundary’, in a sense I will make precise with the definition of an internally Calabi–Yau algebra. Such algebras play an important role in the categorification programme for cluster algebras, and the main goal of the talk will be to explain this connection.

21.04.2016: Cluster categories and Calabi–Yau symmetry (MPIM Oberseminar, Bonn)

This talk consists of a brief overview of the categorification programme for cluster algebras, primarily illustrated through a running example.

11.03.2016: Internally Calabi–Yau algebras (Cluster Algebras and Geometry, Münster) [slides]

I will introduce an enlargement of the class of Calabi–Yau algebras, in which the Calabi–Yau symmetry is allowed to fail in a controlled way, determined by an idempotent. The talk will be led by examples in dimension \(3\), where there are connections to the categorification programme for cluster algebras, and to the theory of dimer models on surfaces.

22.01.2016: Internally Calabi–Yau algebras and cluster-tilting objects (BIREP Seminar, Bielefeld)

Cluster categories, which are \(2\)-Calabi–Yau triangulated categories containing cluster-tilting objects, have played a significant role in understanding the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, an analogous categorical model may be provided by a Frobenius category whose stable category is \(2\)-Calabi–Yau, although such a categorification is only known in a few cases. It is observed by Keller–Reiten that the endomorphism algebra of a cluster-tilting object in such a category has a certain relative, or internal, Calabi–Yau symmetry. In this talk, I will explain how to go in the opposite direction; given an algebra \(A\) with a suitable level of Calabi–Yau symmetry, I will explain how to construct a Frobenius category admitting a cluster-tilting object with endomorphism algebra \(A\).

2015

05.06.2015: Towards Frobenius categorification of cluster algebras (Cluster Algebras and Finite Dimensional Algebras, Leicester)

Cluster categories, which are \(2\)-Calabi–Yau triangulated categories containing cluster-tilting objects, have played a significant role in understanding the combinatorics of cluster algebras without frozen variables. However, when frozen variables do appear, analogous categorical models are only known in a few cases. In this talk, I will introduce internally \(3\)-Calabi–Yau algebras, and explain how such an algebra \(A\) can determine a Frobenius category admitting a cluster-tilting object with endomorphism algebra \(A\). As a proof of concept, I will show how this method can be used to recover Geiß–Leclerc–Schröer’s Frobenius categorifications of cluster algebra structures on unipotent cells.

08.05.2015: Cluster structures from internally \(3\)-Calabi–Yau algebras (44th ARTIN Meeting, Manchester)

I will introduce internally \(3\)-Calabi–Yau algebras, which are algebras exhibiting \(3\)-Calabi–Yau duality away from their ‘boundary’. Such algebras play a significant role in the study of Frobenius categorification of cluster algebras. In particular, I will explain how an internally \(3\)-Calabi–Yau algebra \(A\) gives rise to a Frobenius category containing a cluster-tilted object with endomorphism algebra \(A\). I will also describe aspects of Geiß–Leclerc–Schröer’s cluster structures on homogeneous coordinate rings of partial flag varieties, which provide motivation for the general theory, as well as lots of examples of internally \(3\)-Calabi–Yau frozen Jacobian algebras.

15.04.2015: Representation theory of associative algebras: An introduction (Postgraduate Geometry & Algebra Seminar, Bath)

I will introduce some aspects of the representation theory of associative algebras. The main goal of the talk will be to introduce the \(\operatorname{Ext}\) functors (by popular request!) and use them to describe how any finite-dimensional unital associative algebra is Morita equivalent to a quotient of the path algebra of a uniquely determined quiver. I also plan to discuss some Auslander–Reiten theory for hereditary algebras, and explain how to compute the Auslander–Reiten quiver in the representation-finite cases.

10.03.2015: Cluster structures from internally \(3\)-Calabi–Yau algebras (Algebra Seminar, Leeds)

I will introduce internally \(3\)-Calabi–Yau algebras, which are algebras exhibiting \(3\)-Calabi–Yau duality away from their ‘boundary’. I will explain how an internally \(3\)-Calabi–Yau algebra \(A\) gives rise to a Frobenius category containing a cluster-tilted object with endomorphism algebra \(A\). To motivate the theory, I will describe aspects of Geiß–Leclerc–Schröer’s cluster structures on homogeneous coordinate rings of partial flag varieties.

10.02.2015: Cluster structures from internally \(3\)-Calabi–Yau algebras (Geometry Seminar, Bath)

I will give a definition of an internally \(3\)-Calabi–Yau algebra. The condition generalizes the traditional \(3\)-Calabi–Yau symmetry property by allowing it to fail in a controlled way on the ‘boundary’ of the algebra. I explain how an internally \(3\)-Calabi–Yau algebra \(A\) satisfying some mild additional conditions gives rise to a Frobenius category admitting a cluster-tilting object with endomorphism algebra isomorphic to \(A\). I will also explain aspects of Geiß–Leclerc–Schröer’s construction of cluster structures on partial flag varieties, as motivation for the rest of the theory.

28.01.2015: Ice quivers with potential and internally \(3\)-Calabi–Yau algebras (Calf Seminar, London)

A dimer model, which is a bipartite graph on a closed orientable surface, gives rise to a Jacobian algebra. Under consistency conditions on the dimer model, this algebra satisfies a very strong symmetry condition; it is \(3\)-Calabi–Yau. However, the consistency condition forces the surface to be a torus. This can be avoided by allowing surfaces with boundary, on which dimer models give rise to frozen Jacobian algebras. We define a suitable modification of the \(3\)-Calabi–Yau property for these algebras, and explain some interesting cluster-theoretic results that follow from it.

2014

20.11.2014: Non-commutative geometry and quivers (Postgraduate Seminar Series, Bath)

A standard approach in geometry is to study a space by instead studying the ring of functions on it, replacing geometric problems by algebraic ones. These rings of functions are always commutative (meaning \(fg=gf\) for any two functions \(f\) and \(g\)) but it is sometimes helpful to consider non-commutative rings as if they were rings of functions on some ‘non-commutative space’. I will explain some applications of this point of view to resolutions of singularities. This lofty ambition will mainly serve as a pretext for introducing a few more straightforward geometric and algebraic constructions (modules, quivers and simple GIT quotients) to non-experts.

25.08.2014: Cluster automorphisms and homogeneous spaces (ICRA 2014, 三亚) [slides]

This talk will explain how to view the set of labelled seeds of a cluster algebra as a homogeneous space for the action of a group of mutations and permutations. We describe a particular class of equivalence relations on homogeneous spaces, with the property that their equivalence classes are given by the orbits of a subgroup of the automorphism group of the space. In the labelled seeds setting, one subgroup arising in this way can be identified with the group of cluster automorphisms (in the sense of Assem–Schiffler–Shramchenko) and another with the group of direct cluster automorphisms. This is joint work with Alastair King (arXiv:1309.6579).

07.08.2014: Cluster automorphisms and homogeneous spaces (Postgraduate Geometry & Algebra Seminar, Bath)

29.07.2014: Rational maps, representable functors, group schemes and partial flag varieties (Algebraic Geometry Reading Group, Bath)

This smörgåsbord talk continues the Foundations of algebraic geometry reading group. I will define and give examples of all of the objects in the title. In some cases (particularly partial flag varieties) I will also hint at links to representation theory.

29.01.2014: Quiver representations, cluster algebras and cluster categories (Imperial Junior Geometry Seminar, London)

I will discuss the representation theory of (bound) quivers, paying particular attention to why this is the same as the representation theory of finite-dimensional associative algebras with unit over an algebraically closed field. I will also describe the cluster algebra associated to a quiver, and connect its combinatorics to the representation theory via the cluster category. The discussion will mostly be restricted to the Dynkin (i.e. finite) case, where the correspondences are clearer.

21.01.2014: Quiver representations, cluster algebras and cluster categories (Postgraduate Geometry & Algebra Seminar, Bath)

2013

04.12.2013: Projective schemes (Algebraic Geometry Reading Group, Bath)

Continuing the Foundations of Algebraic Geometry reading group, this talk will define projective spaces and more general projective “varieties” by gluing a particular set of affine schemes. To make this construction more natural, it is better to glue together all affine subschemes, and this is achieved via graded rings and the Proj construction.

28.11.2013: Auslander’s theorem and the algebraic McKay correspondence (Postgraduate Geometry & Algebra Seminar, Bath)

Let \(K\) be a field, and let \(G\) be a (nice) finite subgroup of \(\mathrm{GL}(n,K)\), acting on \(S=K[x,y]\). Auslander’s Theorem states that the endomorphism ring of \(S\), as a module over the \(G\)-invariant subring \(R\), is isomorphic to the twisted group ring \(S*G\). Using this isomorphism, we can set up one-to-one correspondences between the indecomposable summands of \(S\) as an \(R\)-module, the indecomposable projective modules of the endomorphism algebra, the indecomposable projective modules of \(S*G\), and the irreducible \(K\)-representations of \(G\). I will give a sketch of these ideas.

26.11.2013: Rulers, compasses and Galois correspondences (Postgraduate Seminar Series, Bath)

This talk serves as a (very) brief introduction to Galois theory. We warm up by exploring the limitations of ruler and compass constructions, and prove that they cannot be used to trisect angles. Glossing over some technical details, we will outline the fundamental theorem of Galois theory. This theorem is the first example of a Galois correspondence. Such correspondences are somewhat ubiquitous, and we illustrate the general idea with a second (less technical) example, involving homogeneous spaces.

07.08.2013: The de Rham theorem (Differential Geometry Reading Group, Bath)

This talk will introduce singular homology and cohomology for topological spaces. On smooth manifolds, singular cohomology agrees with smooth singular cohomology, which we also discuss. We will briefly outline a proof of the de Rham theorem, which states that singular smooth cohomology agrees with de Rham cohomology.

29.07.2013: A category for Hermitian symmetric cluster algebras (Transfer Viva, Bath)

This talk will suggest how to extend Jensen, King and Su’s category for Grassmannian cluster algebras to the cluster structures on homogeneous coordinate rings of other hermitian symmetric spaces, by constructing a category for the cluster algebra in the coordinate ring of a quadric of isotropic lines in nine-dimensional projective space.

26.04.2013: Labelled seeds and mutation groups (Calf Seminar, Oxford)

This talk will introduce labelled seeds, whose definition is a modification of that of seeds of a cluster algebra. Under this new definition, the cluster algebra itself will be unchanged, but the set of labelled seeds will form a homogeneous space for a group of mutations and permutations. We will study the automorphism group of this space, and conclude that for certain mutation classes, the orbits of this automorphism group consist of seeds with “the same cluster combinatorics”, in the sense that their quivers are all related by opposing some connected components. Knowledge of cluster algebras will not be assumed, and indeed one goal is to provide an introduction to the subject, albeit in a slightly esoteric way.

09.04.2013: Labelled seeds and mutation groups (Geometry Seminar, Bath)

This talk will introduce labelled seeds, whose definition is a modification of that of seeds of a cluster algebra. Under this new definition, the cluster algebra itself will be unchanged, but the set of labelled seeds will form a homogeneous space for a free group on \(n\) involutions, thought of as a group of mutations. We will study the automorphism group of this space, and conclude that for certain mutation classes, the orbits of this automorphism group consist of seeds with “the same cluster combinatorics”, in the sense that their quivers are all related by opposing some connected components.

27.02.2013: Differential forms (Differential Geometry Reading Group, Bath)

Having seen how we can use symmetric covariant \(2\)-tensor fields to define a metric on a manifold, we will now see how to use alternating covariant tensor fields, otherwise known as differential forms. The integration of covector fields along curves is a special case of the integration of differential forms along submanifolds of higher dimension. We also discuss the exterior derivative, a natural differential operator defined on all manifolds, and say something about how it can be used to compute Lie derivatives of differential forms.

11.02.2013: Tensors and multilinear algebra (Differential Geometry Reading Group, Bath)

To continue the differential geometry reading group, this talk will explain the basics of tensor algebras, beginning with tensors on vector spaces, and proceeding to tensor fields and bundles on manifolds. The alternating and symmetric algebras will also be discussed.

05.02.2013: Rings and varieties: An introduction to algebraic geometry (Postgraduate Seminar Series, Bath)

This talk will be a gentle introduction to algebriac geometry, using commutative algebra. I will define varieties and ideals, and briefly discuss Hilbert’s Nullstellensatz, the Zariski topology, and coordinate rings. The punchline will be to use the algebra to explain why degree \(n\) polynomials over \(\mathbb{C}\) have exactly \(n\) roots, even though some of them clearly don’t; in other words, I will explain why some roots are counted more than once.

2012

16.11.2012: Cluster algebras and Grassmannians (Topics in Geometry, Bath)

This talk will describe a cluster algebra structure on the homogeneous coordinate ring of the Grassmannian, following Scott’s paper Grassmannians and cluster algebras (arXiv:0311148), in order to illustrate a general method. It is a reprise of the talk “Cluster structures on homogeneous coordinate rings of Grassmannians”.

12.11.2012: Tangent spaces and differentials (Differential Geometry Reading Group, Bath)

This talk will give the definition and some results on tangent spaces to smooth manifolds, and related them to directional derivatives of smooth functions. This construction allows us to define differentials of smooth maps between manifolds. The tangent bundle will also be defined, although the definition of a general vector bundle will be deferred. There will also be some brief comments on the categorical nature of these constructions.

26.10.2012: What is a cluster algebra? (Postgraduate Geometry & Algebra Seminar, Bath)

Cluster algebras, first described in 2000 by Fomin and Zelevinsky, occur naturally in various areas relating to geometry, topology and representation theory. This talk will give the definition of a cluster algebra and some of the first results and conjectures. The discussion will be motivated by the example of the homogeneous coordinate ring of the Grassmannian of planes in an \(n\)-dimensional complex vector space. General Grassmannians will also be discussed, with particular emphasis on the case of \(3\)-dimensional subspaces.

26.10.2012: Foundations of cluster algebras (Topics in Geometry, Bath)

This talk continues the reading seminar on cluster algbras, by giving an axiomatic framework for the phenomena observed in the examples. I will define cluster algebras, using the language of seeds and quiver/matrix mutation. The viewpoint taken in Cluster algebras I: Foundations, defining the cluster algebra via an exchange pattern on the universal cover of the exchange graph, will also be dicussed.

02.10.2012: The fundamental group and triangulations (Postgraduate Seminar Series, Bath)

The fundamental group is an important homotopy invariant of topological spaces, essentially describing the ways in which loops can be mapped into them. This talk will (loosely) explain the definition of the fundamental group and why it is a group, and then explain how to calculate it using triangulations, with some examples from \(3\)-manifold topology.

02.07.2012: Cluster structures on homogeneous coordinate rings of Grassmannians (6-Month Viva, Bath)

This talk will describe a cluster algebra structure on the homogeneous coordinate ring of the Grassmannian, following Scott’s paper Grassmannians and cluster algebras (arXiv:0311148).

06.03.2012: The structure theorem in tropical geometry (Geometry Seminar, Bath)

This talk will explain how to tropicalize a complex algebraic variety to obtain a combinatorial object known as a tropical variety. I will draw some examples of tropical varieties, and explain the structure theorem, which puts restrictions on the possible objects realizable as tropical varieties.

21.02.2012: An introduction to quiver representations (Postgraduate Seminar Series, Bath)

I will give some of the definitions and first results in the theory of quiver representations, motivated for a general mathematical audience. I will explain quivers, representations and path algebras with simple examples, and hint at the significance of simply laced Dynkin quivers.

12.01.2012: The Auslander–Reiten adjunction (Topics in Geometry, Bath)

This talk will demonstrate how the adjunction between the Auslander–Reiten translation and its “inverse” can be constructed explicitly, via the unit and counit of the adjunction.