Counting points and representations

I will talk about the following questions:

1) Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?

​2)​ Given a polynomial map f:R^a–>R^b and a smooth, compactly supported measure m on R^a, does the push-forward of m by f have bounded density?

3) Given a lattice in a higher rank Lie group (say, SL(n,Z) for n>2). How many d-dimensional representations does it ​have?

I will explain how these questions are related to the singularities of certain varieties. Along the way, I'll talk about canonical singularities, random commutators, and the moduli space of local systems.

This is a joint work with Rami Aizenbud