recent advances in the theory of smooth mod $p$ representations of a $p$-adic reductive group $g$ involve more and more derived methods.  it becomes increasingly clear that the proper framework to study smooth mod $p$ representations is the derived category $d(g)$.

i will talk about smooth mod $p$ representations and highlight their shortcomings compared to, say, smooth complex representations of $g$.  after explaining how the situation improves in the derived category, i will spend the remaining time on the left adjoint of the derived parabolic induction functor.