  Matroid varieties are the closures in the Grassmannian of sets of points
defined by specifying which Pl\"ucker coordinates vanish and which don't ---
the set of nonvanishing Pl\"ucker coordinates forms a well-studied object
called a matroid. Many problems in enumerative geometry could be solved if
there were an efficient way to compute the cohomology class of a matroid
variety in an efficient way just from the combinatorial data contained in the
matroid itself. Unfortunately, in full generality, this problem is known to be
completely intractable. In this thesis, we explore two attempts to get a handle
on this problem in the special case of a better-behaved class of matroids
called positroids. First we examine a method based on degenerations on the
Grassmannian that's known to be successful for an even smaller class of
matroids called interval rank matroids, showing that it can be made to work
with some effort in rank 3 but that there is a counterexample demonstrating
that it is hopeless beyond that. Finally, we give a way of producing a number
for each matroid variety called its expected codimension that can be computed
combinatorially solely from the matroid itself and show that it agrees with the
actual codimension in the case of a positroid variety.
