  These series of notes serve as an introduction to some of both the classical
and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is
about studying general sets or measures which can be, in one sense or another,
approximated on all scales by well behaved spaces, typically just Euclidean
space itself. Such sets and measures turn out not to be arbitrary, and often
times come with special structure inherited from what they are being
approximated by.
  We will begin by recalling and proving the standard Reifenberg theorem, which
says that sets in Euclidean space which are well approximated by affine
subspaces on all scales must be homoemorphic to balls. These types of results
have applications to studying the regular parts of solutions of nonlinear
equations. The proof given is designed to move cleanly over to more complicated
scenarios introduced later.
  The rest of the lecture notes are designed to introduce and prove the
Rectifiable Reifenberg Theorem, including an introduction to the relevant
concepts. The Rectifiable Reifenberg Theorem roughly says that if a measure
$\mu$ is summably close on all scales to affine subspaces $L^k$, then
$\mu=\mu^++\mu^k$ may be broken into pieces such that $\mu^k$ is
$k$-rectifiable with uniform Hausdorff measure estimates, and $\mu^+$ has
uniform bounds on its mass. These types of results have applications to
studying the singular parts of solutions of nonlinear equations. The proof
given is designed to give a baby introduction to ways of thinking in more
modern PDE analysis, including an introduction to Neck regions and their
Structure Theory.
