Cours spécialisé (TA)
Contact : benoit.fresse à univ-lille.fr
Notes de cours : https://pro.univ-lille.fr/en/benoit-fresse/teaching/advanced-course-operads-graph-complexes-and-applications
The purpose of this course is to introduce students to current research questions in the domains of algebra and topology.
The first concept that will be considered in the course is the notion of an operad. To explain the idea of this notion, an operad is an object that models the structure formed by composites of operations that govern an algebra structure. The usual categories of algebras, e.g. the category of associative algebras, the category of associative and commutative algebras, the category of Lie algebras, ... can be associated to operads. There is a notion of presentation by generators and relations for operads, which reflects the classical definition of the structure of an associative algebra, of a commutative algebra, and of a Lie algebra ... in terms of a generating operation (a product, a Lie bracket, ...) that satisfies a set of relations. In this context, one of the main devices of the theory of operads is the theory of the Koszul duality, which is used to compute the syzygies (the secondary relations) associated to such presentations.
The definition of the classical categories of algebras and of the associated operads will be studied in the first part of the course. In a second step, we will focus on the study of $E_n$-operads: fundamental examples of operads, which are used to model commutativity levels that govern certain algebra structures.
Then we will explain a construction of graph complexes with the aim of computing groups of automorphisms associated to $E_n$-operads. To conclude the course, we will outline applications of graph complexes for the operadic interpretation of the Grothendieck-Teichm\"uller group (a group, defined by using ideas of the Grothendieck program in Galois theory, and which models universal symmetries of quantum groups), or the applications of graph complexes for the computation of the homotopy of knot spaces/of embedding spaces of manifolds over the rationals.