The project will study the mirror-symmetry conjecture in the framework of Frobenius manifolds. his conjecture has many levels, from a purely topological statement about Hodge numbers to the most advanced formulation in terms of equivalences of A-infinity categories. There is an intermediate level described by Frobenius manifolds, where the structures which are supposed to be related by this symmetry are richer than the topological picture, but also do not have the full not easily handleable categorical structure. In the physical framework there is a way to construct mirror-symmetric partners by using elementary building blocks and the two operations of tensor product and orbifolding. One of the two operations, the tensor product has been the object of previous research of R. Kaufmann and is fully understood within the theory Frobenius manifolds. The equivalent of the elementary building blocks will be a version of the miniversal unfoldings of singularities of functions with isolated critical points. The last step, however, remains to be completed; i.e. the definition of orbifolding. This amounts to studying finite group actions and defining the correct quotient in the category.
In the past few years a fruitful interaction between the mathematics and physics communities has developed driven by the subject of string theory. This has for instance united algebraic geometers and physicists in looking into questions about mirror symmetry. This is a conjectural symmetry based on physical arguments, which has striking implications for algebraic and enumerative geometry.
