Title: Representation count, rational singularities of deformation varieties, and pushforward of smooth measures

Abstract:
We will present the following  3 results:

1.  The number of n-dimensional irreducible representations of the
pro-finite group $SL(d,Z_p)$ is bounded by a polynomial on n whose
degree does not depend on d and p (our current bound for the degree is
22).

2. Let $\phi : X \to Y$ be a flat map of smooth algebraic varieties
over a local field $F$ of characteristic 0 and assume  that all the
fibers of $\phi$ are of  rational singularities. Then, the
push-forward of any  smooth compactly supported measure on $X$ has
continuous density.

 3. Let $X=\Hom(\pi_1(S),SL_d)$ where $S$ is  a surface of high enough
genus (our current bound for the genus is 12). Then $X$ is of rational
singularities.

We will also discuss the surprising relation between those results
which allowed us to prove them.