Convolutions of polynomial maps: a crossroads of algebra, geometry and analysis

Let f=(f_1,...,f_m) be a tuple of m polynomials with integer coefficients in n variables. 

Then f can naturally be considered as a complex map f:C^n->C^m, and one may approach studying f from several different perspectives:
1) Arithmetic, by counting the number of solutions to the system of equations {f_i = c_i mod r}, for different choices of integers c_i and r.
2) Geometric, by studying the geometry and singularities of the fibers (level-sets) of the complex map f:C^n->C^m.
3) Analytic, by taking a smooth measure \mu with compact support in C^n, and studying analytic 

(e.g. integrability) properties of the resulting pushforward measure f_*(\mu) on C^m.

In this talk I will explain how these different approaches relate to one another. I will then discuss a natural algebraic convolution operation 

between maps, which allows one to study the behavior of certain algebraic families of random walks using these connections. 

This is based on a series of works from recent years joint with I. Glazer.