### Bruno Vallette (Université de Nice-Sophia Antipolis), Associative algebras

#### Schedule

- Feb. 28, 2012, 17:30 - 18:30: Homotopy associative algebra
- March 1, 2012, 15:00 - 16:00: Koszul duality and Diamond Lemma

#### Abstract

*Homotopy associative algebra*

When one wants to transfer associative algebra structures though homotopy equivalences, one automatically discovers the notion of homotopy associative algebra. In order to conceptually understand such a result, and later to generalize it, I will introduce a simple operad, based on planar rooted trees. Finally, I will explain how to build a resolution of this operad, due to Jim Stasheff, which is based on polytopes, called the associaedra.

*Koszul duality and Diamond Lemma*

Starting from the symmetric algebra as the toy model for quadratic algebras, I will introduce the notion of Poincaré-Birkhoff-Witt bases. I will then describe the Koszul duality theory for associative algebras. Finally, I will show that an algebra which admits a PBW basis is Koszul. This result is based the Diamond lemma and the rewriting method thereby drawing a concrete link with the other talks of the workshop.