Finite element spaces for differential forms

Mixed finite elements, as developed by Raviart, Thomas, Nédélec, Brezzi and others can be given a unified presentation in the language of differential forms. As pointed out by Bossavit, the resulting spaces are related to constructs from differential topology dating back to de Rham and Whitney. This connection, developed in particular by Hiptmair, Arnold, Falk and Winther has led to the subject of Finite Element Exterior Calculus. I will try to give an account of this synthesis and present my related notion of Finite Element Systems.