# Brauer-Thrall Conjectures, Old and New!

Time: 10:30 -- Location: Salle Philippe Flajolet du LIX

\(\tau\)-tilting theory is an elegant-- but technical-- subject in representation theory of associative algebras, with motivations from cluster algebras. It was introduced by Adachi-Iyama-Reiten, in 2014. However, thanks to the recent result of Demonet-Iyama-Jasso, one can fully phrase the concept of \(\tau\)-tilting finiteness in terms of linear algebra. I adopt this elementary approach to share some new results on \(\tau\)-tilting finiteness of several families of algebras with rich combinatorics.

I begin with a review of two fundamental conjectures by Brauer and Thrall. After some historical remarks on those, I present a gentle introduction to \(\tau\)-tilting finiteness, which allows me to state a conjecture of similar nature that relies on my recent work on \(\tau\)-tilting theory. I will share my main strategy, as well as my new results on some cases.
*I will not assume any prior knowledge of representation theory of algebras!*