About:
Bill CrawleyBoevey turned 60 in 2020, and a conference to celebrate his work was planned. Due to COVID, that conference had to be postponed until 2021.
Instead, an online colloquium took place: two online talks followed by an opportunity to socialise.
Contracted Preprojective Algebras, Coxeter Groups and cDV Singularities

Preprojective algebras of quivers are important objects in representation theory. In extended Dynkin case, they give noncommutative resolutions of simple singularities of dimension two. Motivated by studying noncommutative resolutions of cDV singularities, I will discuss tilting theory of a contracted preprojective algebra, which is a subalgebra eAe of a preprojective algebra A given by an idempotent e of A. It has a family of tilting modules which correspond bijectively with the chambers in the contracted Tits cone, and mutation of these tilting modules correspond to wallcrossing of chambers. Moreover, they are parametrized by certain double cosets in the Coxeter group W modulo parabolic subgroups, and mutation can be described in terms of these double cosets by using longest elements. Osamu explained some results and conjectures on the derived equivalence classes of contracted preprojective algebras. This is a joint work with Michael Wemyss.
Purity in Silting Theory
Work of Bill CrawleyBoevey from the 1990s has shown that the theory of purity and the notion of a definable subcategory studied in model theory play an important role in representation theory. Beligiannis and Krause later exported these concepts to triangulated categories. As shown by Krause, purity is an essential tool to approach the Telescope Conjecture, a problem originating in stable homotopy theory which asks when certain localizations of a triangulated category are determined by compact objects. A stronger version of this problem asks the analogous question for a tstructure. More precisely, it asks when a tstructure (U,V) with definable coaisle V is generated by a set of compact objects. Recent work by several authors shows that purity plays a crucial role also in this context and reveals important connections with silting theory. In my talk, Lidia gave a survey on these results.