Minimal models for monomial algebras

In 1985, David Anick defined a combinatorial notion chains which can be used to compute various homological invariants of an associative algebra from a presentation of such an algebra by generators and relations that leads to a good rewriting system. In particular, for algebras with monomial relations, his construction produces those invariants directly.
In this talk, I will explain how to compute a rich algebraic structure on Anick chains leading to the explicit formula for a minimal quasi-free model for any monomial algebra. This computation relies on algebraic discrete Morse theory and on homotopy transfer formulas; those are formulas perfectly suited for homological computations where underlying chain complexes are of combinatorial nature. Prior knowledge of these techniques is not required: they will be explained along the way. Additionally, we explain how can extends the methods used for monomial algebras to algebras with a good rewriting system in the form of a conjecture and some examples.