We will discuss several counting problems in number theory.  What is the probability that a random degree d monic polynomial with integer coefficients is irreducible? How many degree d algebraic number fields have discriminant at most X?  For a given field, how many orders does it contain of discriminant at most X?  We will also briefly discuss some statistical questions about rational points in families of elliptic curves.

We will then transition to talking about similar problems over finite fields.  In particular, we will focus on questions about rational points in families of curves and surfaces over a fixed F_q.  For example, if we take two plane cubic curves what is the probability that they intersect in exactly 9 F_q-rational points?