Title: The Space of Commuting Matrices, and Statistical Mechanics
Abstract: 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.
Title: Juggling Patterns and Gaussian Elimination
Abstract: Around 1985 three groups of jugglers independently created the same notational system for juggling patterns. This has been very useful for recording, transmitting, and categorizing old patterns, and especially for creating new ones. I’ll explain this system and demonstrate many examples.
While this theory has obvious connections to permutations, only very recently have I and my coauthors noticed it to be the natural setting for studying a certain decomposition of the space of matrices. Quite surprisingly, it is easier to understand this finite-dimensional geometry in terms of a much more familiar decomposition in infinite dimensions (the “Bruhat decomposition of the affine flag manifold”), and I’ll explain how “antimatter juggling” naturally suggested this.