  These notes provide an introduction to the local semicircle law from random
matrix theory, as well as some of its applications. We focus on Wigner
matrices, Hermitian random matrices with independent upper-triangular entries
with zero expectation and constant variance. We state and prove the local
semicircle law, which says that the eigenvalue distribution of a Wigner matrix
is close to Wigner's semicircle distribution, down to spectral scales
containing slightly more than one eigenvalue. This local semicircle law is
formulated using the Green function, whose individual entries are controlled by
large deviation bounds.
  We then discuss three applications of the local semicircle law: first,
complete delocalization of the eigenvectors, stating that with high probability
the eigenvectors are approximately flat; second, rigidity of the eigenvalues,
giving large deviation bounds on the locations of the individual eigenvalues;
third, a comparison argument for the local eigenvalue statistics in the bulk
spectrum, showing that the local eigenvalue statistics of two Wigner matrices
coincide provided the first four moments of their entries coincide. We also
sketch further applications to eigenvalues near the spectral edge, and to the
distribution of eigenvectors.
