Work of Bill Crawley-Boevey 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 t-structure. More precisely, it asks when a t-structure (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, I will give a survey on these results.
This talk was part of the Birthday Colloquium for Bill Crawley-Boevey on Sept 10th 2020.
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