Higher-distance commuting varieties

The commuting variety is a well-studied object in algebraic geometry whose points are pairs of matrices that commute with one another. In this talk I present a generalization of the commuting variety by using the notion of commuting distance of matrices, which counts how many nonscalar matrices are required to form a commuting chain between two given matrices. I will prove that over any algebraically closed or real closed field, the set of pairs of matrices with bounded commuting distance forms an affine variety. I will also discuss many open problems about these varieties, and present preliminary results in these directions. This is based on joint work with Madeleine Elyze and Alexander Guterman.