Ralph Morrison: The Commuting Variety: Classically and Tropically

Pairs of commuting matrices can be studied via the commuting variety, which is defined as the vanishing locus of the entries of the commutator of a pair of square matrices of variables. After reviewing this classical perspective, I will present new results for commuting matrices in the areas of min-plus linear algebra and tropical geometry. I will show that the polynomials do not tropically define the set of all matrices that commute using min-plus arithmetic, and illustrate obstructions to lifting commuting pairs of tropical matrices to points on the commuting variety over a non-Archimedean field. This is joint work with Ngoc Tran.