Free divisors, Gauss-Manin systems and Monodromy Calculus

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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.