Monday, January 23

18:00-19:00

Speaker: Benoit Fresse (Université de Lille)

Title: Rational homotopy of operads. Models of mapping spaces and applications

Abstract: I will survey research on the definition of models for the rational homotopy of operads, an application of graph complexes to the computation of mapping spaces of operads, and the applications of these results to the embedding calculus and the theory of Grothendieck-Teichmüller groups (joint work with Victor Turchin and Thomas Willwacher).

To be more specific, the main objects of this study are E_n-operads, a class of operads, defined by a reference model, the operads of little n-discs (or n-cubes), and which can be used to govern a hierarchy of homotopy commutative structures, from fully homotopy associative but non-commutative (n=1) up to fully homotopy associative and commutative (n=infinity). The definition of rational models relies on an interpretation of formality results, which I am going to explain first. Then I will explain that certain combinatorial operads, defined in terms of graphs, can be used to get a combinatorial description (a graph complex description) of mapping spaces associated to these objects.

The applications to the study of embeddings goes through the Goodwillie-Weiss embedding calculus, which relates the spaces of embeddings of Euclidean spaces to these operadic mapping spaces. The spaces of homotopy automorphisms of E_n-operads can also be identified with the Grothendieck-Teichmüller group in the case n=2, and hence, with higher dimensional generalizations of these objects in the case n>2. Depending on time and on the interest of the audience, I will address one or the other of these connections with more details.