Geometric Aspects of Differential Equations

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Differential equations are fundamental objects within mathematics and especially within algebraic geometry. Often, fundamental aspects of a geometric problem can be described in terms of a system of differential equations. Let us mention here the principle of monodromy (analytic continuation of solutions) and the notion of a variation of periods, describing the integration of differential forms (i.e. the Hodge theory) on a family of varieties. These concepts both play an important role in the field of Mirror Symmetry of Calabi-Yau varieties with many applications to physics and enumerative mathematics. It is the aim of this proposal to develop and implement two computer algebra packages which make the above mentioned concepts accessible for explicit computation.  The first package shall deal with computational aspects of the Hodge theory of families of varieties with a special attention to the important case of Calabi-Yau varieties and related convolutions of variations of Hodge structures.  The second shall deal with the computation of the monodromy of integrable differential equations (integrable connections) on quasiprojective varieties. Thereafter, these packages shall be applied to various aspects, e.g., determination of new differential operators of Calabi-Yau type, computation of Instanton numbers or the computation of uniformizing differential equations.