Galois representations attached to torsion in the cohomology of automorphic vector bundles, after Boxer

I will report on work of George Boxer in which he has constructed Hasse like invariants on the closure of Ekedahl-Ooort strata, which cut out the open stratum. This answers in the affirmative a question of Oort as to whether the open Ekedahl-Oort strata (in the minimal compactification) are affine. It also allows Boxer to attach Galois representations to torsion in the (Zariski) cohomology of automorphic vector bundles on certain Shimura varieties. This in turn leads to a new construction of Galois representations for torsion in the Betti cohomology of locally constant sheaves on certain arithmetic locally symmetric spaces.