Titles and Abstracts

Claire Amiot,  Derived invariant for surface algebras
This is a joint work with Yvonne Grimeland. Let S be a surface with marked points on the boundary. Following David-Roesler and Schiffler, one can associate to each ideal triangulation $\Delta$ of S and each admissible cut $d$ of $\Delta$ a gentle algebra $\Lambda$ of global dimension 2. In this talk I will explain how to associate to the pair $(\Delta,d)$ an invariant in $H^1(S,\mathbb{Z})$, and how this invariant entirely determines the derived category $\mathcal{D}^b(\Lambda)$. I will also explain the link between this invariant and the AG (Avella-Alaminos Geiss) invariant for gentle algebras, and show why this invariant is more powerful in the case of surface algebras.

Ilke Canakci,  Resolutions of snake graphs and the bijection on the perfect matchings
I will report on a sequel of a joint work with Ralf Schiffler motivated by the expansion formula and the skein relations for surface cluster algebras. In this work, we introduce the notions of abstract snake graphs and band graphs, describe crossing snake graphs and their resolution. For labeled snake graphs associated to arcs in a surface cluster algebra, the crossing of arcs correspond to the crossing of snake graphs and the skein relations of arcs correspond to the resolutions of snake graphs.

Jan Grabowski,  Graded cluster algebras and quantum Grassmannians
I will introduce gradings on (quantum) cluster algebras in a very natural way and show how to find and classify gradings, especially in finite types where cluster categories play a significant part. On the one hand, gradings bring out some beautiful combinatorics but we will also consider a more algebraic application of gradings as part of the proof that quantum Grassmannians are quantum cluster algebras.

Osamu Iyama,  Silting reduction and Calabi-Yau reduction
This is a joint work with Dong Yang. For a rigid object P in a 2-CY triangulated category, Calabi-Yau reduction gives a smaller 2-CY triangulated category whose cluster tilting objects bijectively correspond to those in C which have P as a direct summand. Silting reduction is a similar construction for a presilting object in a triangulated category. We will show that there is a close connection between these two reductions.

Philipp Lampe,  Quantization spaces of cluster algebras
This is joint work with Florian Gellert. Berenstein and Zelevinksy have introduced quantum cluster algebras as non-commutative deformations of ordinary cluster algebras. Unfortunately, a given cluster algebra does not necessarily admit a quantization, and if a quantization exists, then it is not necessarily unique. In this talk we prove that a cluster algebra admits a quantization if and only if the initial exchange matrix has full rank. Moreover, we give an explicit and natural basis for the space of all possible quantizations.

Bernard Leclerc,  Cluster structure and categorification for Lusztig's strata in partial flag varieties (after Nicolas Chevalier's thesis)
Let X=G/P be a partial flag variety for a simple algebraic group G of type A,D,E. Lusztig has introduced a stratification of X, which yields a cell decomposition of the non-negative part of X. In his PhD thesis (University of Caen 2012), Chevalier has given conjectural cluster structures on the coordinate rings of certain of these strata, based on appropriate Frobenius subcategories of the module category of the preprojective algebra of G. I will present this conjecture, as well as some evidence supporting it. The conjecture would give new results even in the case of type A Grassmannians.

Robert Marsh,  From triangulated categories to module categories via localization
Joint work with Aslak Bakke Buan (NTNU, Trondheim). Let C be a triangulated category satisfying some mild assumptions and let T be a rigid object in C. We show how to obtain the category of finite dimensional modules over the endomorphism algebra of T by formally inverting a class of maps in C (i.e. by localising in the sense of Gabriel-Zisman). If C has Serre duality, then the quotient of C by the kernel of Hom(T,-) is a preabelian category and inverting the bimorphisms (i.e. those which are both monomorphisms and epimorphisms) gives the module category of the endomorphism algebra of T. In this case, the bimorphisms admit a calculus of left and right fractions. In the case where T is cluster-tilting, this recovers the result that the quotient of C by the image of T under the suspension functor is equivalent to the module category of its endomorphism algebra (c.f. work of Buan-M-Reiten, Iyama-Yoshino, Keller-Reiten, Koenig-Zhu).

David Pauksztello,  Torsion pairs in triangulated categories generated by spherical objects
This is a report on joint work with Raquel Coelho Simoes (Lisbon). Recently, Holm, Jorgensen and Yang studied triangulated categories T_w generated by a w-spherical object. The category T_2 can be thought of as a cluster category of type A infinity and extensions in this category can be computed by crossings of arcs in an appropriate geometric/combinatorial model. Using this model, Ng classified torsion pairs in T_2 in terms of Ptolemy diagrams. We extend extend this classification to arbitrary w. When w>0, we obtain a classification of torsion pairs in terms of Ptolemy diagrams. When w<0, the combinatorics are more complicated and we obtain a classification in terms of modified Ptolemy diagrams.

Idun Reiten,  Lattice structure for torsion classes
This talk presents joint work with Iyama, Thomas and Todorov. The theory of support tau-tilting modules introduced in a paper with Adachi and Iyama was inspired by cluster theory. Amongst other things they were shown to be in bijection with the functorially finite torsion classes denoted by ftors. For a preprojective algebra A associated with a Dynkin diagram, Mizuno showed a bijection between elements in the corresponding Coxeter group W and ftors(A), and that ftors(A), which is a poset using inclusion, is a lattice. This motivated the problem of investigating when ftors is a lattice more generally. Here we deal with path algebras and (concealed) canonical algebras.

Ralf Schiffler,  Cluster algebras and rings of snake graphs
Abstract snake graphs were introduced recently in [1] inspired by the labeled snake graphs appearing in the expansion formulas for cluster variables in cluster algebras of surface type [2]. While the labeled snake graphs are constructed from the crossing pattern of an arc in a surface with a fixed triangulation, the definition of abstract snake graphs is completely detached from triangulated surfaces and is simply given by describing the possible graphs in an elementary way. In analogy with the situation in the cluster algebras, one can define a multiplication on the free abelian group generated by all abstract snake graphs by means of taking disjoint unions modulo certain relations. In this way, one obtains a ring of abstract snake graphs which has interesting relations to cluster algebras. This is joint work with Ilke Canakci.
[1] I. Canakci, R. Schiffler : Snake graph calculus and cluster algebras from surfaces, J. Algebra, 382, (2013) 240--281
[2] G. Musiker, R. Schiffler, L. Williams : Positivity for cluster algebras from surfaces, Advances in Math. 227 (2011) 2241--2308.

Jan Schröer,  Strongly reduced components of module varieties
I will explain the concept of strongly reduced irreducible components of module varieties introduced in joint work with Christof Geiß and Bernard Leclerc, and I will discuss examples and applications.

Sibylle Schroll,  Cluster mutations and the geometry of Brauer graph algebras (joint work with Robert Marsh)
In this talk we will show that ribbon graphs have a Brauer graph structure and that any compact oriented marked surface will give rise to a unique Brauer graph algebra up to derived equivalence. The derived equivalences are induced by flips of diagonals in ideal triangulations of the surface. In the case of a disc with marked points we will give a dual construction in terms of Brauer tree algebras.

Jeanne Scott,  Combinatorics of the grassmannian's BZ-twist
I'll discuss results of joint work with R. Marsh regarding the Berenstein-Zelevinsky twist automorphism of the type A grassmannian; notably a combinatorial formula to calculate BZ-twisted Plücker coordinates using perfect matchings on a class of bipartite graphs dual to Postnikov diagrams.

Gordana Todorov,  Periodic trees, Semi-invariants, Clusters, c-vectors
In this work we consider affine quivers of type Ã_n. Periodic trees are combinatorial structures which we show to be in bijection with clusters of type Ã_n. Furthermore, the internal edges of the tree encode the c-vectors corresponding to the cluster. Consequently, these edges also define weights of the semi-invariants on the presentation spaces (virtual semi-invariants [IOTW]) of the summands of the corresponding cluster tilting object. We also assign a unipotent matrix to each tree, giving a relationship between clusters and pictures, as defined in [IO], for torsion-free nilpotent groups. This is done using Bernstein-Retakh non-commutative root space [BR].
[BR] Arkadiy Bernstein, Vladimir Retakh, Noncommutative Clusters
[IOTW] Kiyoshi Igusa, Kent Orr, Gordana Todorov, and Jerzy Weyman, Cluster complexes via semi-invariants, Compos. Math. 145 (2009), no. 4, 10011034.
[IO] Kiyoshi Igusa and Kent E. Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001), no. 6, 11251166.