On Deformation Quantization and Differential Operators in Positive Characteristic

This thesis consists of two papers studying noncommutative rings in positive characteristic closely related to differential operators. Their abstracts are as follows:1. Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization. 2. For k a field of positive characteristic and X a smooth variety over k, we compute the Hochschild cohomology of Grothendieck's differential operators on X. The answer involves the derived inverse limit of the Frobenius acting on the cohomology of the structure sheaf of X.