First PhD/master's student seminar - October 2, 2024 - Verona

Viola Conte Two-term silting complexes over gentle algebras

Abstract: Gentle algebras constitute an important and rather large subclass of finite-dimensional algebras, characterised by particularly nice bound quivers. In this talk, I will outline the main topics of my master's thesis, which focused on examining the properties of modules over gentle algebras and on providing a complete classification of their two-term silting complexes. I will show how combinatorial data can be linked to relevant information about these objects, whose study is significant since they control many of the homological properties of the algebra itself.

Anastasios Slaftsos The Q-shaped homotopy category

Abstract: A (co-)chain complex over a k-algebra A, can be viewed as an A-Mod-valued representation of the repetitive quiver of the linear oriented Dynkin quiver A_2 modulo the mesh relations. Formally speaking, this situation implies an equivalence between the category of (co-)chain complexes Ch(A) and the category of k-linear functors from the mesh category Q of A_2 to A-Mod. We denote the latter by Q,A-Mod and call it the category of “Q-shaped modules”. This abstract perspective allows one to consider different “shapes” of Q and study all of them collectively. In 2022, H. Holm and P. Jørgensen proved that this category admits two interesting abelian model structures with the same weak-equivalences and hence the same homotopy category, which they called the “Q-shaped derived category”, a notion that generalises the concept of the usual derived category. However, when someone studies categories of complexes, it is well known that there exists an intermediate step between the category of complexes and the derived category, the one known as the homotopy category of complexes. In this talk, we replace the abelian exact structure on Q,A-Mod with the object-wise split exact structure and prove that there exists an exact model structure whose homotopy category we call the “Q-shaped homotopy category” which generalises the concept of the usual homotopy category. Moreover, mimicking the category of complexes, we prove that the Q-shaped homotopy category shows up as a gluing of the Q-shaped derived category and the category of the Q-shaped acyclic objects, generalising the recollement situation introduced by H. Krause. This talk is based on a joint work in progress with Henrik Holm and Jorge Vitória.