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.