Frobenius manifolds, free divisors and quiver representations

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This project is concerned with particular divisors that appear as discriminants in prehomogeneous vector spaces. The reductive algebraic group acting on those vector spaces defines in a canonical way a D-module on the dual space, a so-called tautological system. If the group happens to be an algebraic torus, then this is nothing but the well-known GKZ-system, in which case the divisor tostart with has normal crossings. Many of the interesting problems in this area are solved (at least partially) for this particular simple example.The overall aim of the project is to study Hodge theoretic as well as categorical properties of D-modules defined by prehomogenous discriminants. Moreover, the well established relation between GKZ-systems and toric mirror symmetry shall be generalized to autological systems defined by such prehomogenous actions. We seek to investigate the holonomic rank of such systems, describe certain intermediate extensions (i.e., intersection cohomology modules) of these systems and study the Hodge filtration in cases where tautological systems underly mixed Hodge modules. The categorical aspects of prehomogeneous discriminants shall be looked at by describing the category of matrix factorization of equations defining a discriminant divisor, and homological mirror symmetry statements for these categories will be discussed. The main difficulty in all these projects is that the divisors under investigation are neither toric, nor do they have isolated singularities. Hence many of the traditional methods in these areas cannot be applied directly. We hope to obtain both new insights in mirror symmetry statements beyond the toric case as well as a better understanding of the algebraic and geometric structure of singularities of prehomogeneous discriminants.