Abstract: This talk will present an introduction to some continuing work being conducted with Eric Friedlander, Julia Pevtsova and Andrei Suslin. The work is concerned with some basic questions about sets of commuting nilpotent operators on vector spaces. The objects that we construct generalize the class of endotrivial modules that is important in the modular representation theory of finite groups. They can also be used to construct bundles on projective spaces and Grassmannians.