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Grothendieck Groups of Discrete Cluster Categories

Cluster Categories were introduced as a way to categorify the cluster algebras that had recently been introduced by Fomin and Zelevinsky in the early part of this century. It was quickly understood that we could represent cluster categories (and cluster algebras) of Dynkin type A by the arcs between vertices of an n-gon. This idea was expanded upon by Igusa and Todorov, who introduced us to discrete cluster categories, a family of which can be modelled using triangulations of an infinitely marked circle.In this talk we compute the Grothendieck groups, and important object to study in Bridgeland stability, for this family of discrete cluster categories, as well as a closely related family of categories due to Paquette and Yildirim.