On τ-representation types with examples from the representation theory of valued quivers.

In this talk, we propose a stable and a τ-reduced version of the second Brauer-Thrall conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall conjecture introduced by Mousavand and Schroll-Treffinger-Valdivieso. The latter is stated in terms of Geiß-Leclerc-Schröer’s generically τ-reduced components and provides a geometric interpretation of a question raised by Demonet. We outline implications among these conjectures and relate them to recent variations of tameness in stability and τ-tilting theory. It follows from Schroll-Treffinger-Valdivieso’s work that the conjectures are true for special biserial algebras, and we confirm them for Geiß-Leclerc-Schröer’s (GLS) algebras associated to valued quivers. If time permits, we demonstrate that the in general representation wild GLS algebras of affine type are still „tame“ from a τ-tilting perspective.