Frobenius Manifolds and F-manifolds in Singularity Theory

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The aim is to study the singularities of functions on free divisors. We are particularly interested in the so-called linear free divisors - those free divisors, like the normal crossing divisor, for which Der(-log D) is generated in weight zero, and in particular is generated by a finite-dimensional Lie sub-algebra. The complement of such a divisor is a symmetric space which is the finite image of a complex Lie group.Functions on these free divisors exhibit a rich phenomenology which has striking resemblances, and also striking differences, with the classical case studied by Arnold, Saito and Brieskorn. We hope to find, in the deformation theory of these functions, further examples of Frobenius manifolds, and compare these with the Frobenius manifolds arising in quantum cohomology.