Free divisors, Gauss-Manin systems and Monodromy Calculus

镜像对称是微分几何和代数几何的一个分支，起源于弦论物理学;粗略地说，某种类型的流形（卡拉比·丘）的镜像对产生了不可区分的物理理论。这已被推广到非卡拉比亚流形。例如，n维复射影空间的镜像是（n+1）维空间中某个超曲面上的函数。每一个都与一个称为Frobenius流形的复杂结构相关联。为函数构造这个对象的过程与为流形构造这个对象的过程完全不同。这里的镜像对称在于这两个结构是同构的，尽管它们的起源如此不同。函数的过程利用了奇点理论的技术，特别是所谓的高斯-马宁连接。这是一个奇点的一个双变形的基空间上的亚纯联络，它来自于判别超曲面的补上的消失上同调群的向量丛上的自然平坦联络。为了理解这些结构并将它们推广到“非传统”类型的奇点，我们需要用一些简单的例子进行详细的具体计算。该项目的目的是进行这种计算，并从中获得尽可能多的信息。将使用的技术包括交换代数和超几何函数和微分方程的理论。

Mirror symmetry is a branch of differential and algebraic geometry which originated in the physics of string theory; roughly speaking, mirror pairs of manifolds of a certain type (Calabi Yau) give rise to indistinguishable physical theories. This has been generalised to non-Calabi-Ya manifolds. For example, the mirror of n-dimensional complex projective space is a function on a certain hypersurface in (n+1)-dimensional space. To each is associated a complicated structure called a Frobenius manifold. The procedure by which thisobject is constructed for the function is completely different from the procedure for the manifold. The mirror symmetry here resides in the fact that the two structures are isomorphic, despite having such disparate origins. The procedure for the function makes use of techniques of singularity theory, and in particular the so-called Gauss-Manin connection. This is a meromorphic connection on the base space of a versal deformation of a singularity, which comes from the natural flat connection on the vector bundle of vanishing cohomology groups over the complement of the discriminant hypersurface. In order to understand these constructions and generalise them to ''non-traditional'' kinds of singularities, it is desirable to make detailed concrete calculations with a good selection of simple examples. The aim of the project is to undertake such calculations and to derive as much information as possible from them. The techniques to be used include commutative algebra and the theory of hypergeometric functions and differential equations.