Thick subcategories via geometric models
I will explain some current work, in which I describe the lattice of thick subcategories of a discrete derived category. This is done using certain collections of exceptional and sphere-like objects related to non-crossing configurations of arcs in a geometric model.
Laurent Demonet, From categories of Cohen-Macaulay modules over orders to subcategories of module categories, application to cluster algebras of homogeneous coordinate rings of partial flag varieties
Algebras of generalized quaternion type and weighted surface algebras
Assume A is a finite-dimensional indecomposable algebra over some field K. We say that A is of generalised quaternion type if A is tame, symmetric, and all simple A-modules have Omega-period four. The case when the Cartan matrix of A is non-singular (so that A can have at most three simple modules) was studied in the context of blocks with quaternion defect groups; and some of these algebras also occur in work of Burban, Iyama, Keller and Reiten. In this lecture, we discuss algebras of generalised quaternion type with 2-regular quiver, these can be described by weighted surface algebras.
Cluster algebras via reflection groups
We build a geometric model of acyclic cluster algebras using reflection groups acting in linear spaces. This is a joint work with Pavel Tumarkin.
Generalized Snake Graphs for Generalized Cluster Algebras
I will talk about work in progress with Gregg Musiker, aiming to provide combinatorial interpretations of the expression of cluster variables as Laurent polynomials in certain generalised cluster algebras (as defined by Chekhov and Shapiro), in terms of snake graphs. I will provide concrete examples in various cases.
Torsion pairs in discrete cluster categories
Igusa and Todorov introduced discrete cluster categories of Dynkin type A, which generally are of infinite rank. That is, their clusters contain infinitely many pairwise non-isomorphic indecomposable objects. We study torsion pairs in these categories and provide a complete combinatorial classification. As a special case, we obtain a combinatorial classification of cluster tilting subcategories. If time allows, we will discuss a combinatorial model for mutation of torsion pairs, which yields cluster mutation of cluster tilting subcategories as a special case.
Quotients of triangulated categories and theorems of Buchweitz,
Orlov and Amiot-Guo-Keller
We give a general result which realizes a Verdier quotient of a triangulated category T by its thick subcategory as a subfactor category of T. As special cases, we obtain three theorems due to Buchweitz, Orlov and Amiot-Guo-Keller. This is based on a joint work with Dong Yang.
Karin Marie Jacobsen,
Abelian quotients of triangulated categories
We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. As an application, we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions, J is a cluster-tilting subcategory of T. In particular, this means that if T is a cluster category and X is an object in T, the functor from T to mod End(X) is only full and dense if X is a cluster-tilting object. This is joint work with Benedikte Grimeland.
Higher Nakayama Algebras
This is a report on ongoing joint work with Julian Kuelshammer. We introduce higher analogues of classical Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. These algebras admit d-cluster-tilting subcategories with a combinatorial description analogous to that of the module categories of Nakayama algebras.
Basic aspects of n-homological algebra
n-homological algebra was initiated by Iyama via his notion of n-cluster tilting subcategories. It was turned into an abstract theory by the definition of n-abelian categories (Jasso) and (n+2)-angulated categories (Geiss-Keller-Oppermann). The talk explains some elementary aspects of these notions. We also consider the special case of an n-representation finite algebra. Such an algebra gives rise to an n-abelian category which can be "derived" to an (n+2)-angulated category. This case is particularly nice because it is analogous to the classic relationship between the module category and the derived category of a hereditary algebra of finite representation type.
Diophantine equations via cluster transformations
Motivated by Fomin-Zelevinsky's theory of cluster algebras we introduce a variant of the Markov equation; we show that all natural solutions of the equation arise from an initial solution by cluster transformations.
John W. Lawson,
Mapping classes, clusters and combinatorics
Triangulations of surfaces have a cluster structure where triangle flips correspond to mutations and the surface's mapping class group has recently been shown to be isomorphic to the group of cluster automorphisms preserving this structure. We will discuss the links between these groups and the group of exchange graph automorphisms, before generalising to the skew-symmetrizable setting. This work provides a combinatorial approach to studying mapping class groups using graph automorphism groups.
Mutations of dimer models and splitting maximal modifying modules
A dimer model is a bipartite graph on the real two-torus. We obtain a quiver with potential as the dual of a dimer model. It is known that the Jacobian algebra arising from a consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a three dimensional Gorenstein toric singularity. This algebra is isomorphic to the endomorphism ring of a certain module which we call splitting maximal modifying (= MM) module. For every three dimensional Gorenstein toric singularity, such a module exists, but it is not unique. In this talk, I will show any two splitting MM modules are transformed into each other by repeating the mutation of splitting MM modules for some special cases.
Discrete triangulated categories
The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck's definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by 'negative spherical objects', which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category.
Internally Calabi-Yau algebras
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.
Weyl groups and preprojective algebras
The lecture is based on work with Iyama, Reading and Thomas. For a simply laced Dynkin diagram we consider the associated Weyl group W and the associated preprojective algebra A. We deal with the interplay between W and A. For example, there are bijections between what is called join irreducible elements in W and some special kinds of A-modules.
Raquel Coelho Guardado Simoes,
Reduction for negative Calabi-Yau triangulated categories
Iyama and Yoshino introduced a tool, called Iyama-Yoshino reduction, which is very useful in studying the generators and decompositions of positive Calabi-Yau triangulated categories. However, this technique does not preserve the required properties for negative Calabi-Yau triangulated categories. In this talk, we establish a Calabi-Yau reduction theorem for this class of categories. This is a report on joint work in progress with David Pauksztello.
Representations of Dynkin quivers over commutative Noetherian rings
Reflection groups via cluster algebras
I will describe a construction by Barot and Marsh producing presentations of finite Weyl groups from cluster algebras of finite type, and its generalization to affine Weyl groups. In particular, this leads to presentations of Weyl groups as quotients of various Coxeter groups, which has some surprising applications. I will also discuss generalizations to other mutation-finite cluster algebras. The talk is based on a joint work with Anna Felikson.
Jon M. Wilson,
The cluster structure of non-orientable surfaces.
It is well known that triangulated orientable surfaces give rise to cluster algebras. Dupont and Palesi generalised this process to non-orientable surfaces, giving birth to what they call quasi-cluster algebras. I will link these algebras to Lam and Pylyavskyy's Laurent phenomenon algebras, and will remark on the structure of finite type quasi-cluster algebras.
Cluster categories for marked surfaces: punctured case
For any triangulation of a marked surface with punctures and non-empty boundary, there is an associated generalized cluster category. In this talk, I will give a bijection between tagged curves (in the surface) and certain indecomposable objects (in the category). By interpreting dimensions of Ext^1 as intersection numbers, I will show that the cluster exchange graph of the cluster category is connected. This is based on joint work with Yu Qiu.