Talk 8: Cohomology of Shimura varieties via the Hodge-Tate period map

Recently, a number of results have been obtained on the cohomology of Shimura varieties by studying it via a Leray spectral sequence along the Hodge-Tate period map. I will survey some of these results (some joint with Ana Caraiani, some by others) and stress in particular how they enable a study of the cohomology of the boundary. This relates to a long-standing theme of Harder's work on congruences between Eisenstein series and cusp forms, and has yielded substantial progress on the Langlands program for non-algebraic arithmetic locally symmetric spaces. If time permits, I may close with a note on another theme of this conference, motives.