Counting with p-adic integrals

Suppose we are given a multivariate polynomial over the integers. What can be said about the zeros of this polynomial modulo a natural number N? As I will explain, considering this question for all natural numbers simultaneously leads to a data set with remarkable structure and symmetry.

One route to appreciate this hidden information goes via p-adic integration. I will illustrate this approach -- hands-on and from scratch -- on a few examples.

I will also try and convince you that p-adic integrals are the tool of choice to tackle a number of other counting problems, seemingly less algebro-geometric, say of group-theoretic origin.

I will assume nothing more from my audience than what it takes to understand this abstract's first sentence.