November 2, 2023 
Operads and graph complexes, applications to the embedding calculus, relationships with the GrothendieckTeichmüller groups
Benoit Fresse, Lille University, France RIMS Kyoto · JP:09:00 · FR:01:00
I will survey the results of a joint research with Victor Turchin and Thomas Willwacher on applications of operads to the embedding calculus and the theory of GrothendieckTeichmüller groups. To be specific, the main results, which I will aim to explain in my talk, include:  a combinatorial description, in terms of graph complexes, of the spaces of emdeddings of Euclidean spaces in the realm of the rational homotopy theory  a description, in terms of graph complexes too, of spaces of rational homotopy automorphisms of operads, which generalize the rational GrothendieckTeichmüller group
Our work is mainly based on a study of the rational homotopy of E_noperads, a class of operads, defined by a reference model, the operads of little ndiscs (or ncubes), and which can be used to govern a hierarchy of homotopy commutative structures, from fully homotopy associative but noncommutative (n=1) up to fully homotopy associative and commutative (n=infinity). I will explain that certain combinatorial models of E_noperads, defined in terms of graphs, can be used to get a graph complex description of mapping spaces associated to E_noperads over the rationals.
The applications to the study of embeddings goes through the GoodwillieWeiss embedding calculus, which relates the spaces of embeddings of Euclidean spaces to such operadic mapping spaces. The relationship with the GrothendieckTeichmüller groups comes from the observation that the space of homotopy automorphisms of E_noperads is identified with the GrothendieckTeichmüller group 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. I could also explain a generalization of the result on the rational homotopy of emdeddings of Euclidean spaces to the case of the space of embeddings of more general manifolds into an Euclidean space.
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