General talk (this is a talk for a wide audience, colloquium type): Operads and homotopy commutative structures

Operads are algebraic structures formed by collections of operations p = p(x_1,…,x_r). Operads have been introduced in topology, in the sixties, in order to understand the associativity and commutativity defect of multiplicative structures attached to loop spaces (Boardman-Vogt, May, Stasheff). The theory of operads has been deeply renewed in the nineties, and the notion of an operad is now used as an efficient device to handle multiple structures in various domains.

In this colloquium talk, I will provide a survey of new applications of E_n-operads, which are these classes of operads introduced at the beginning of the theory for the study of loop spaces. I will notably report on: the Deligne conjecture giving the homotopy commutative structure of the deformation complex of associative algebras (McClure-Smith, Kontsevich-Soibelman, …); the applications of the Deligne conjecture to second generation proofs of the existence of deformation quantization for Poisson manifolds (Tamarkin, Kontsevich); and the relationship between E_2-operads and the pro-unipotent Grothendieck-Teichmueller group (Kontsevich, BF, Willwacher).

Extra topics include the connection with the homology of embedding spaces (Budney, Lambrechts-Turchin-Volic, Salvatore, Sinha, …), as well as the factorization (or topological chiral) homology of manifolds (Francis, Gaitsgory, Kostello, Lurie).

Seminar talk : The homotopy of E_n-operads

The E_n-operads, n = 1,2,…,infinity, used to define a hierarchy of homotopy commutative structures, from fully associative but non-commutative (n=1) up to fully associative and commutative (n=infinity), are defined by a reference model, the little discs operads. In many problems, the issue is to prove that an operad governing a given multiplicative structure is weakly-equivalent to such a reference model, and hence, forms an instance of an E_n-operad.

This lecture will be devoted to the homotopy theory of E_n-operads. I will provide a survey of recognition theorems arising from (higher/monoidal) category theory, and I will report on formality and Koszul duality issues for E_n-operads. I will also explain the relationship between the Eilenberg-MacLane iterated bar complexes and the homology theory associated to E_n-operads.

Tentative dates

Wednesday 28/08; 4:00 pm – 5:30 pm; Room:

Friday 30; 4:00 pm – 5:30 pm; Room:

Venue: Vietnam Institute for Advanced Study in Mathematics.