Period integrals and other direct images of D-modules

Abstract: This thesis consists of three papers, each touching on a different aspect of the theory of rings of differential operators and D-modules. In particular, an aim is to provide and make explicit good examples of D-module directimages, which are all but absent in the existing literature.The first paper makes explicit the fact that B-splines (a particular class of piecewise polynomial functions) are solutions to D-module theoretic direct images of a class of D-modules constructed from polytopes.These modules, and their direct images, inherit all the relevant combinatorial structure from the defining polytopes, and as such are extremely well-behaved.The second paper studies the ring of differential operator on a reduced monomial ring (aka. Stanley-Reisner ring), in arbitrary characteristic.The two-sided ideal structure of the ring of differential operators is described in terms of the associated abstract simplicial complex, and several quite different proofs are given.The third paper computes the monodromy of the period integrals of Laurent polynomials about the singular point at the origin. The monodromy is describable in terms of the Newton polytope of the Laurent polynomial, in particular the combinatorial-algebraic operation of mutation plays an important role. Special attention is given to the class of maximally mutable Laurent polynomials, as these are one side of the conjectured correspondance that classifies Fano manifolds via mirror symmetry.