Geometric GRT actions and Deligne’s conjecture on Hochschild cochains

Speaker: Sonja M. Farr – University of Nevada

Abstract: In 1990 V. Drinfeld introduced a pro-unipotent algebraic group $\mathrm{GRT}$, which is closely related to the absolute Galois group $\mathrm{Gal}(\overline{Q}/Q)$. It was later shown by B. Fresse that this group is isomorphic to the group of homotopy automorphisms of the rationalization of the little disks operad. Deligne’s conjecture – now a theorem with multiple proofs – asserts that the natural operations on the Hochschild cochain complex of any smooth algebraic variety lift to an action of the dg operad of little disks in a precise way. In this talk, I will present work in progress towards a new proof of this conjecture which exploits J. Lurie’s $\infty$-operadic proof of Dunn’s additivity theorem, and which aims to provide more conceptual insight on this operad action. I will also give some motivation on how Deligne’s conjecture can be utilized in studying the group $\mathrm{GRT}$.