  The objects of interest in this thesis are positroid varieties in the
Grassmannian, which are indexed by juggling patterns. In particular, we study
affine patches on these positroid varieties. Our main result corresponds these
affine patches to Kazhdan-Lusztig varieties in the affine Grassmannian. We
develop a new term order and study how these spaces are related to subword
complexes and Stanley-Reisner ideals. We define an extension of pipe dreams to
the affine case and conclude by showing how our affine pipe dreams are
generalizations of Cauchon and Le diagrams.
