This is joint work with Luke Gutzwiller. The affine Grassmannian is an infinite-dimensional projective variety analogous to an ordinary Grassmannian. In particular it has an increasing filtration by Schubert varieties, indexed by the affine Weyl group mod the ordinary Weyl group and with the closure relations given by the Bruhat order. It also has a decreasing filtration by infinite-dimensional varieties of finite codimension, analogous to the dual or opposite Schubert varieties in a Grassmannian arising from the opposite Borel subgroup. These are the Birkhoff varieties, indexed by the same set as before but with the closure relations given by the opposite Bruhat order. A similar construction applies to any affine flag variety. We show that the inclusion of any Birkhoff variety in its affine flag variety is a homotopy equivalence. In particular, the inclusion induces an isomorphism on ordinary and torus-equivariant cohomology. If time permits I will discuss the Goresky-Kottwitz-MacPherson theory for Birkhoff varieties, and some questions about affine Richardson varieties (the intersection of Schubert and Birkhoff varieties).