Why do matrices commute? More specifically, are there polynomial equations satisfied by the set of pairs of commuting matrices not algebraically implied by the equations AB=BA? Mel Hochster and others asked this question in the '60s, and it remains unsolved for large dimensions.(To see the problem: knowing M^2=0 tells you M is nilpotent and hence Trace(M)=0, but that linear equation doesn't lie in the ideal generated by the quadratic equations M^2=0.)
I'll talk about some related spaces of matrices that are simpler to study, which lead to some weird integer-valued invariants of permutations. Then I'll explain a statistical mechanical model that produces the same integers, but in a much more calculable manner, and use this to give a formula for the volume (or really, multidegree) of the space of commuting matrices.
This work is joint with Paul Zinn-Justin.