Paquette and Y {\i} ld {\i} r {\i} m recently introduced cluster-type categories for completed
infinity-gons, which are discs with an infinite closed set of marked points on their boundary …

We classify thick subcategories in a Paquette-Y\i ld\ir\im completion $\overline {\mathcal {C}}
$ of a discrete cluster category of Dynkin type $ A_ {\infty} $. To do this we introduce the …

We show that odd-dimensional projective varieties with tilting objects and only ADE-
singularities are nodal, ie they only have $ A_1 $-singularities. This is a very special case of …

Neeman shows that the completion of a triangulated category with respect to a good metric
yields a triangulated category. We compute completions of discrete cluster categories with …

We give a classification of torsion pairs, t-structures, and co-t-structures in the Paquette-
Yildirim completion of the Igusa-Todorov discrete cluster category. We prove that the aisles …

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the
Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian …

In this thesis we look at two closely related families of categories: the discrete cluster
categories of Dynkin type A∞, and their completions in the sense of Paquette and Yıldırım …