Homological Mirror Symmetry for Singularities

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In singularity theory some up to now rather mysterious phenomena were observed: the McKay correspondence, Arnold's strange duality, Saito's duality, and a surprising relation between a priori unrelated algebraic and topological invariants, Poincar_e series on one hand and monodromy on the other hand. Most of these phenomena are not yet well understood. There is some evidence that a possible explanation for these phenomena might come from an exciting mathematical conjecture which has its origin in theoretical physics (string theory). This is the homological mirror symmetry conjecture and this is where representation theory comes into play. The main goal of the proposal is to investigate what the meaning of homological mirror symmetry for singularities is, to what extent the conjecture holds, and whether one can gain a better understanding of the above mentioned phenomena from the study of this conjecture.