For a k-algebra R, one can consider the ring of k-linear differential operator on R. These are thought of as higher-order analogues of derivations on R, and when R = \C[x_1,...,x_n] the ring of differential operators on R is generated by the usual partial derivatives. In contrast, differential operators on singular rings have been studied for several decades, but their explicit description is difficult, and basic properties remain mysterious. When the ring in question is the homogeneous coordinate ring of a smooth projective variety, however, we can study certain properties of the ring of differential operators via the global geometry of the variety. In particular, this allows us to demonstrate that rings with "mild" singularities need not have "nice" rings of differential operators in characteristic 0, in contrast to the situation in positive characteristic. This talk will discuss these results, and the many questions, both algebraic and geometric, that remain open.

Zoom link for the meeting: https://umich.zoom.us/j/94949291807 Speaker(s): Devlin Mallory (University of Michigan)