On the mod $p$ and $p$-adic Jacquet-Langlands correspondence for $GL_2(Q_p)$ and $D^*$
I will discuss some consequences of the 6 functor formalism of Mann to the study of the mod $p$ and $p$-adically completed cohomology of the Drinfeld tower. This is joint work with Colmez and Niziol, and independently with Rodriguez Camargo.