Thomas Willwacher: Séminaire Nicolas Bourbaki
We are delighted to announce that the Séminaire Nicolas Bourbaki has recently honoured the work of Professor Thomas Willwacher with an exposé entitled: "Derived Grothendieck-Teichmüller group and graph complexes".
In his lecture, fields medallist Maxim Kontsevich has given a broad-spectrum presentation of the work of Thomas Willwacher on graph complexes and the little disks operads, as contained in the paper "Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra" (in Inventiones Mathematicae), as well as two more recent works.
The little n-disks operads are important objects in topology, which have been introduced in the seventies and have seen numerous application in algebra, topology and mathematical physics since then. For example, they may be used to detect iterated loop spaces in topology, they govern deformation quantizations in mathematical physics, and they underly the theory of quantum groups in algebra.
Thomas Willwacher has computed the homotopy automorphisms of these little disks operads (over rationals), identifying their infinitesimal counterpart (homotopy derivations) with certain complexes of diagrams (graph complexes). In particular, this exhibits large symmetry groups acting on objects in the aforementioned fields, and also sheds light on the underlying obstruction theoretic questions. Finally, as part of the work the cohomology of the graph complex in lowest degree was computed, identifying it with the Grothendieck-Teichmüller Lie algebra.