Kaveh Mousavand, Queen's University

Title: Minimal tau-tilting infinite algebras

Abstract: Originally inspired by the celebrated Brauer-Thrall conjectures, the minimal representation-infinite algebras have been studied extensively. In our recent work, we introduce "minimal tau-tilting infinite algebras" as the modern counterpart of the aforementioned classical notion. Our interest in this new family stems from an open conjecture that we proposed to give a novel algebro-geometric realization of tau-tilting (in)finiteness of algebras.

In this talk, after a brief recollection of minimal representation-infinite algebras, I introduce the minimal tau-tilting infinite algebras and highlight some of the fundamental similarities and differences between these two families. We then show that if A is a minimal tau-tilting infinite algebra, almost every tau-tilting A-module is in fact tilting. Consequently, the mutation graph of tilting A-modules is infinite but regular at almost all vertices. Furthermore, we reduce our main conjecture to a particular subfamily of minimal tau-tilting infinite algebras. Finally, from our new classification of minimal tau-tilting infinite biserial algebras, we conclude a stronger version of our conjecture. That is, we prove that a biserial algebra is tau-tilting infinite if and only if it admits a generic brick.