We will discuss the notion of a random cover of an undirected graph and averages one can calculate for such covers. While the definition we will give for a cover is likely to be familiar to anyone who has studied a bit of algebraic topology, we will not assume everyone in the audience has the background. The notion of random we will give is quite naive, but it leads to algebraic objects we regard as very interesting, e.g., d-matchings polynomials. I will explain some known properties of these geometric and algebraic, if time permits, point out open questions which I regard as interesting.

I intend for this talk to be accessible to graduate students and encourage anyone who thinks they might be interested to attend.

Let $G$ be a finite subgroup of $GL(n,\mathbb{C})$. Then $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $\mathbb{C}$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with singular locus of codimension 1. The classical McKay correspondence relates the geometry of the resolutions of singularities of so-called Kleinian surfaces with the representation theory of finite subgroups of $SL(2,\mathbb{C})$. In particular, there is an algebraic version of this correspondence, due to M. Auslander.

In this talk we present a version of the McKay correspondence when $G$ is a finite group generated by reflections: We give a natural construction of a so-called noncommutative resolution of the coordinate ring of $D$ as a quotient of the skew group ring $A=S*G$. We will explain this construction, which allows to extend Auslander's theorem to reflection groups. This is joint work with R.-O. Buchweitz and C. Ingalls.

Speaker's web page: http://www-personal.umich.edu/~emfaber/index.html