In the lectures I will formulate a conjecture asserting that there is a hidden action of certain motivic cohomology groups on the cohomology of arithmetic groups. One can construct this action, tensored with $\mathbb C$, using differential forms. Also one can construct it, tensored with $\mathbb Q_p$, by using a derived version of the Hecke algebra (or a derived version of the Galois deformation rings).
I will describe these constructions and the evidence for the conjecture. The ideas and results presented here are based on papers with Prasanna and Galatius and Harris (all on the arxiv) and work in progress with Darmon, Harris, Rotger.