Geometric Aspects of Differential Equations

微分方程是数学中的基本对象，特别是在代数几何中。通常，几何问题的基本方面可以用微分方程组来描述。让我们在这里提到的原则monodromy（解析延续的解决方案）和概念的变化的时期，描述的一体化的微分形式（即霍奇理论）的家庭品种。这两个概念在Calabi-Yau簇的镜像对称性领域中起着重要的作用，在物理学和计数数学中有许多应用。这是本建议的目的是开发和实现两个计算机代数软件包，使上述概念显式计算访问。 第一包应处理计算方面的霍奇理论的家庭的品种与特别注意的重要情况下，卡-丘品种和相关卷积的变化霍奇结构。 第二个将处理计算的monodromy可积微分方程（可积联系）的拟投射品种。此后，这些包将应用于各个方面，例如，确定新的Calabi-Yau型微分算子，计算Instanton数或计算一致化微分方程。

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.