A metric on a triangulated category, as developed by Neeman, provides a recipe for constructing a metric completion of the category. These completions are guaranteed to be triangulated categories as well and have recently been used to study, among other things, derived Morita theory, cluster categories, and t‑structures. The aim of this talk is to examine metric completions of bounded derived categories of hereditary rings and their connection to the concept of universal localisation. Notably, we explicitly describe the completions of bounded derived categories of hereditary finite dimensional tame algebras and hereditary commutative noetherian rings with respect to additive good metrics.

An exceptional hereditary non-commutative curve over an algebraically closed field is a weighted projective line of Geigle and Lenzing. However, over arbitrary fields, the theory of exceptional curves is significantly richer. In my talk I am going to explain the definition, examples and key properties of this class of non-commutative curves, including their invariants and their relation to squid algebras. I shall also explain a connection between real exceptional curves of tubular type and wallpaper groups, which was discovered by Lenzing many years ago. My talk is based on arXiv:2411.06222 as well as on an ongoing work in progress with Baumeister, Neaime and Schwabe.

An exceptional hereditary non-commutative curve over an algebraically closed field is a weighted projective line of Geigle and Lenzing. However, over arbitrary fields, the theory of exceptional curves is significantly richer. In my talk I am going to explain the definition, examples and key properties of this class of non-commutative curves, including their invariants and their relation to squid algebras. I shall also explain a connection between real exceptional curves of tubular type and wallpaper groups, which was discovered by Lenzing many years ago. My talk is based on arXiv:2411.06222 as well as on an ongoing work in progress with Baumeister, Neaime and Schwabe.

In this talk we will see a new definition of Coxeter-Dynkin algebras of canonical type. This generalises the existing definition in the special case where it can be described via quivers and relations. The main goal is to establish a derived equivalence to Ringel's squid algebra (and hence to the corresponding canonical algebra). Finally, we will build a bridge to geometric group theory via Saito's classification of marked extended affine root systems of codimension one.

In this talk we will see a new definition of Coxeter-Dynkin algebras of canonical type. This generalises the existing definition in the special case where it can be described via quivers and relations. The main goal is to establish a derived equivalence to Ringel's squid algebra (and hence to the corresponding canonical algebra). Finally, we will build a bridge to geometric group theory via Saito's classification of marked extended affine root systems of codimension one.

The category of finite dimensional representations of a tame hereditary algebra has a discrete part, related to the roots of an affine Kac-Moody Lie algebra, and a continuous part. The simplest case is for the Kronecker quiver, when the continuous part is precisely the projective line. Lenzing and coauthors generalised this and showed that this continuous part has the structure of a non-commutative curve, constructed using the preprojective algebra. We will revisit this important construction, strengthening their results, and providing further connections between the geometry and certain infinite dimensional representations.

The category of finite dimensional representations of a tame hereditary algebra has a discrete part, related to the roots of an affine Kac-Moody Lie algebra, and a continuous part. The simplest case is for the Kronecker quiver, when the continuous part is precisely the projective line. Lenzing and coauthors generalised this and showed that this continuous part has the structure of a non-commutative curve, constructed using the preprojective algebra. We will revisit this important construction, strengthening their results, and providing further connections between the geometry and certain infinite dimensional representations.

We introduce the notion of reflection groups of canonical type. These groups are related to the K-theoretic study of the canonical algebras of Ringel. We use the notion of a symbol introduced by Lenzing to define them. We also introduce the associated non-crossing partitions of canonical type, which are intervals of Coxeter elements equipped with a poset structure. These notions will appear in a joint work in progress with Barbara, Charly, and Igor. If time permits, I will present a research programme for the sequel in order to study these groups from a geometric group theory point of view.

We introduce the notion of reflection groups of canonical type. These groups are related to the K-theoretic study of the canonical algebras of Ringel. We use the notion of a symbol introduced by Lenzing to define them. We also introduce the associated non-crossing partitions of canonical type, which are intervals of Coxeter elements equipped with a poset structure. These notions will appear in a joint work in progress with Barbara, Charly, and Igor. If time permits, I will present a research programme for the sequel in order to study these groups from a geometric group theory point of view.

I will explain the problem appearing in the Hurwitz action on the set of reduced reflection factorizations of a Coxeter element, and present a solution to this problem. I will also explain the choice of the name "hyperbolic cover".

I will explain the problem appearing in the Hurwitz action on the set of reduced reflection factorizations of a Coxeter element, and present a solution to this problem. I will also explain the choice of the name "hyperbolic cover".

In this talk, I will report on ongoing joint work with Baumeister, Burban and Neaime. I will explain how the proofs of Hurwitz transitivity and the categorification of non-crossing partitions become very simple under the right assumptions. In other words, I will show exactly what the technical difficulties are, that were unclear when we started this project.

In this talk, I will report on ongoing joint work with Baumeister, Burban and Neaime. I will explain how the proofs of Hurwitz transitivity and the categorification of non-crossing partitions become very simple under the right assumptions. In other words, I will show exactly what the technical difficulties are, that were unclear when we started this project.