τ-lting representation types and examples

Motivated by counting ground states in certain quantum field theories, there has been a recent interest in various notions of “tameness” with respect to stability and τ-tilting theory. We will briefly recall the interplay of stability and τ-tilting theory as developed by Brüstle-Smith-Treffinger. This will lead us to two $\tau$-tilting versions of the Brauer-Thrall conjectures I and II, and we will see how these relate to a conjectured characterization of τ-tilting finite algebras in terms of their g-vector fans due to Demonet. Then we will discuss different proposed notions of tameness in τ-tilting theory. Finally, we will see how Geiß-Leclerc-Schröer’s algebras associated to symmetrizable Cartan matrices serve as interesting examples.