This project aims to develop topics in contemporary representation theory by making novel connections between two different approaches: tilting theory and model theory. Tilting theory is a set of tools that allows us to formally compare derived categories and has applications in many areas such as Lie theory, algebraic geometry and topology. Our work will take place in the richer setting of silting theory, which encompasses traditional tilting theory but also enables the study of important abelian subcategories of the derived category that are not necessarily derived equivalent to the ring.
We will approach silting theory using techniques originating in mathematical logic, that is, we will use the model theoretic approach to purity. It has been observed for many years that there is a connection between purity and tilting theory but the underlying reasons for the link have still not been properly explained. Our new perspective aims to explain and deepen these connections. The key techniques we will use involve localisations of functor categories and their connection to a topological space called the Ziegler spectrum.