Spectral graph theory has enjoyed much success in using eigenvalues of
matrices associated with a graph to understand some structural property or
bound various kinds of behavior of the graph. When two graphs share many
eigenvalues in common it can often be traced to some sort of common
structure that they share. Well known examples of this are common
coverings or equitable partitions.

We will consider another variation of this where (for the normalized
Laplacian) two graphs do not project to a common graph but share a common
``anti-covering'' (which we will define). We will also consider
anti-covers for the adjacency matrix and use it to establish the following
linear algebra result (among others): {\it Let $M$ be an $n{\times}n$
real symmetric matrix and $|M|$ be the $n{\times}n$ matrix found by taking
(entrywise) the absolute values of $M$; then there exists a nonnegative
symmetric $2n{\times}2n$ matrix ${\cal N}$ such that the spectrum of
${\cal N}$ is the union of the spectrums of $M$ and $|M|$.}

Singularities offer the opportunity to study some interesting mathematics
from a relatively simple point of view. First I'll describe what an
affine variety is and what it means for one to have a singularity. Then,
I'll discuss the work of Milnor, Brieskorn and others on the topology of a
complex hypersurface near one of its singular points, which has
connections to knot theory and exotic spheres. If I have time, I'll talk
about a special class of singularities coming from invariant theory that
were discovered by Klein, and the McKay correspondence, which gives an
unexpected connection between resolutions of singularities and the Dynkin
diagrams used to classify compact Lie groups.