Many counting problems, like
“In how many ways can a positive integer n be written as a sum of positive integers?”
“Given a polytope P, how many lattice points does the dilated polytope nP contain?”
“How many lines in a (n + 1)-dimensional space meet 2n general (n − 1)-planes?”
are solved by finding a closed form for the corresponding generating function ∑_n N_n q^n, where the N_n are the sought numbers and q is a variable. In this lecture we shall, in addition to the above questions, also address an old problem from enumerative geometry:
“How many plane curves of degree d have r singularities and pass through d(d+3)/2 - r given points in the plane?”
In this case the generating function is still unknown, but there has recently been substantial progress on the problem and its generalizations.