Let k be a field of characteristic p>0 and let G be a topological group admitting an open pro-p subgroup. If H=k[I\G/I] denotes the corresponding Hecke algebra, then the functor of taking I-invariants provides an adjunction between the category of smooth k-linear representations of G and the category of H-modules. When H is the pro-p Iwahori-Hecke algebra of a p-adic reductive group, this adjunction is not well-understood. This is one of the main obstacles to understanding the smooth representation theory of these groups and to the mod-p Langlands program. I will describe a recent approach to studying this adjunction using the language and techniques from the theory of model categories. I will then finish by describing the homotopy classes of simple supersingular pro-p Iwahori-Hecke modules for GL_n.This is joint work with Jan Kohlhaase
