Nearby cycles of automorphic étale sheaves

I will explain that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type
Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby
cycles over most of their mixed characteristics models constructed in the literature. If time permits, I will
also talk about some applications or related results. (This is joint work with Stroh.)