The matrix equations M^2 = 0 are quadratic, so to derive the linear equation Trace(M)=0 from them requires nonalgebraic operations. Are there corresponding "surprising" equations implied by the matrix equation XY=YX? This question was posed in the '60s, and still nobody knows. Even the (normalized) volume of this space {(X,Y) : XY=YX} is very difficult to compute for large matrices, and until recently was only known to start 1,3,31,1145.
I'll talk about a bunch of related spaces of matrices, some of which are provably harder and some easier to understand than the commuting scheme {(X,Y) : XY=YX}, and the volumes of these spaces. Then I'll explain how physicists came up with the same set of numbers from a statistical mechanical model (making them much easier to compute), and why they are indeed the same.
Some of this work is joint with Paul Zinn-Justin.