Calabi-Yau manifolds and mirror symmetry

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The reflexive polytope construction of Batyrev and the subsequent work of Givental accounts for almost all cases of mirror symmetry that have been worked out so far. In algebraic geometry however, one encounters Calabi-Yau spaces that are not given as complete intersections in toric varieties and a very interesting problem is to find mirror manifolds in these cases, in order to understand how general the mirror symmetry phenomenon is. The plan is to test the range of mirror symmetry by working out more examples, in particular we want to study examples with Picard number one and try to find mirror manifolds. On the mirror side, we seek to construct Calabi-Yau manifolds with small h1,2, for example by imposing singularities, and search their moduli spaces for points of maximal unipotent monodromy.