Single commutators and beyond

Abstract: 

Operators AA and BB don't commute precisely when their commutator AB−BAAB−BA is nonzero. From quantum mechanics, we know that noncommuting operators are a fact of life and the question of which operators can arise as single commutators in various settings is of interest, {\it e.g.,} the Heisenberg commutation relations. We will describe some

classical results and some more recent results, as well as open problems and conjectures, about single commutators, sums of commutators and related topics.