Abstracts

Ragnar-Olaf Buchweitz: Levels, dimensions and centre of a derived category

Abstract: In recent years, several authors have begun to investigate the finer structure of triangulated categories. Quite often, these questions relate to Lenzing's work on purity of modules.

Current work addresses the (strong) generating hypothesis, studied by Hovey, Lockridge et al. or Chebolu, Christensen and others, the notion of dimension as introduced by Rouquier and studied by Krause-Kussin or Chen-Ye-Chang, or the notion of levels as proposed by Bondal-van den Bergh and exploited by Avramov et al.

We intend to give a (partial) survey of such results and to show how work of Kelly, Street, Beligiannis, and Christensen builds bridges between these sometimes seemingly unrelated efforts.

Bill Crawley-Boevey: Connections for weighted projective lines

Abstract: I will discuss connections on coherent sheaves for a weighted projective line. Using a theorem of Hübner and Lenzing I will show how such connections localize to give representations of deformed preprojective algebras. I will explain how this is used in the solution of the Deligne-Simpson problem.

Jose Antonio de la Peña: Spectral methods in the representation theory of algebras

Abstract: Coxeter transformations play an important role in the study of the representation theory of certain classes of algebras. The roots of the associated Coxeter polynomial provide important information for this study. We review results on hereditary algebras, canonical algebras, extended canonical algebras and other families of algebras. We shall mention connections to Lie algebra, rings of automorphic forms, knot theory and other topics. In the study of the representation theory of these algebras and their applications, Helmut Lenzing has played a decisive role.

Lutz Hille: Weighted projective spaces, Calabi-Yau categories and reflexive polytopes

Abstract: We consider a sequence p(0),...,p(n) of positive natural numbers so that some power of the canonical divisor of the corresponding weighted projective space is trivial.

The same sequences of numbers, often called weights, also appear in toric geometry for the classification of reflexive simplices. Then one can construct a Calabi-Yau variety as a hypersurface in the corresponding toric variety.

We review some classical and several recent results related to those sequences and explain why the Calabi-Yau condition naturally appears in both constructions. In dimension 2 the correspondence is very well understood and relates equivariant sheaves on elliptic curves to weighted projective lines.

Bernhard Keller: On Grothendieck groups of cluster categories, after Y. Palu

Abstract: The Grothendieck group of the cluster category associated with a hereditary category admitting a tilting object was recently computed by Barot-Kussin-Lenzing. In this talk, we will sketch a new proof and a generalization of their result. The main tool will be the link between Calabi-Yau categories of dimensions 2 and 3 studied by Tabuada and used in joint work by the speaker with Idun Reiten. As a  by-product, we will find that any two cluster-tilting objects in an (algebraic) 2-Calabi-Yau category have the same number of pairwise non isomorphic indecomposable factors.

Mike Prest: Sheaves of rings of definable scalars

Abstract: For any ring R the finite, finite-type localisations of (mod-R,Ab) give a basis of open sets for the Gabriel-Zariski topology on pinj-R, the set of iso types of indecomposable pure-injective R-modules. The endomorphism rings of the corresponding localisations of the forgetful functor form the presheaf of rings of definable scalars over this space. All this lifts to the more general context of definable additive categories. We also describe how direct limit- and direct product-preserving functors between definable categories induce morphisms between the corresponding ringed spaces.

Idun Reiten: Tilting ideals for preprojective algebras, and 2-dimensional Calabi-Yau categories (Talk cancelled)

Abstract: In this talk we show how to use tilting ideals over (completions of) preprojective algebras to construct Calabi-Yau categories of dimension 2, and we describe cluster tilting objects in these categories. This is based on part of a project with Buan, Iyama and Scott. There is related work by Geiss, Leclerc and Schroer.

Claus Michael Ringel: Algebra ist Geometrie ist Algebra

Olivier Schiffmann: Hall algebras of higher genus curves and weighted projective lines

Andrzej Skowronski: Selfinjective algebras of quasitilted type

Abstract: A finite dimensional algebra over a field is said to be a selfinjective algebra of quasitilted type if it admits a Galois covering by the repetitive algebra of a quasitilted algebra (endomorphism algebra of a tilting object in a hereditary abelian category). The aim of the talk is to show importance of the selfinjective algebras of quasitilted type in the representation theory of selfinjective artin algebras. Invariance of the class of selfinjective algebras of quasitilted type under derived and stable equivalences will be also discussed.

Michel van den Bergh: Non-commutative resolutions of determinantal varieties

Abstract: This is joint work with Ragnar Buchweitz and Graham Leuschke. We construct a non-commutative resolution for general determinantal varieties.