The commuting scheme is normal and Cohen–Macaulay (Jack Sempliner, UCLA)

Consider the moduli problem Comm_n of commuting pairs of nxn matrices (A, B) over a field k of characteristic zero. In 1955 Theodore Motzkin and Olga Taussky proved that the underlying topological space of this moduli is an algebraic subvariety of k^{2n^2}, leaving open the natural question of whether the obvious n^2 equations actually suffice to define this variety, rather than a nilpotent thickening thereof. In recent joint work with Xiangqian Yang we apply techniques from Zhu’s unipotent categorical local Langlands correspondence in the modular setting to resolve this question, and its analogue for all reductive algebraic groups and reductive Lie algebras in sufficiently large characteristics (and in characteristic zero). Along the way, we prove some results of independent interest about Zhu’s tame categorical Langlands correspondence in non-banal characteristics.