Resolutions of differential operators of low order for an isolated hypersurface singularity

In this paper we develop a new approach for studying differential operators of an isolated singularity graded hypersurface ring $R$ defining a surface in affine three-space over a field of characteristic zero. With this method, we construct an explicit minimal generating set for the modules of differential operators of order two and three, as well as their minimal free resolutions; this expands results of Bernstein, Gel'fand, and Gel'fand and of Vigu\'e. Our construction relies, in part, on a description of these modules that we derive in the singularity category of $R$. Namely, we build explicit matrix factorizations starting from that of the residue field.