Geometric Aspects of Differential Equations

微分方程是数学尤其是代数几何中的基本对象。通常，几何问题的基本方面可以用微分方程组来描述。让我们在这里提到单态原理（解的解析延拓）和周期变化的概念，描述微分形式在一组变异体上的积分（即霍奇理论）。这两个概念在Calabi-Yau变异体的镜像对称领域中都起着重要的作用，在物理和数列数学中都有广泛的应用。本提案的目的是开发和实现两个计算机代数包，使上述概念可用于显式计算。第一个包将处理霍奇变种族理论的计算方面，特别关注Calabi-Yau变种的重要案例和霍奇结构变异的相关卷积。第二部分将处理拟射影变量上可积微分方程（可积连接）的单性计算。此后，这些软件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数或计算均匀化微分方程。

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.