Commutative algebra often abstracts geometric problems into simple questions about algebraic invariants. I will illustrate this with some open problems on the Hilbert function (a simple algebraic invariant which measures the dimensions of graded pieces of a graded ring).
Geometry enters the picture when the ring is the projective coordinate ring of a variety. When the ring has a multigrading we also get some interesting combinatorics. I will emphasize the computational and combinatorial sides of this story.