Speaker:  Benjamin Collas (Bayreuth)
Title:  Motivic Homotopy for Moduli Stacks of Curves: Tannakian formalism I

Abstract:
The study of Geometric Galois Action of the moduli spaces of curves leads to a divisorial and a stack arithmetic of the spaces. While the theory of motives thrived on the former to produce the (unipotent) Tannakian category of mixed Tate motives - i.e. of genus 0 moduli stacks -- no motivic construction has been proposed that deals with the latter.
I will present the main obstacles to this fact, and how the homotopy definition of stacks, completed by an ad-hoc construction in Morel-Voevodsky unstable motivic homotopy, bypasses them to provide a category that is Tannakian by nature as well as not too small neither too big.