Equivalence of D-brane categories

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According to Strominger, Yau and Zaslow, mirror symmetry of Calabi-Yau manifolds should be understood in terms dual torus fibrations. On the other hand, Kontsevich proposed to formulate mirror symmetry as an equivalence of Ao-categories, one based on holomorphic D-branes and the other on special Lagrangian D-branes. These objects should be interchanged under T-duality (Fourier-Makai transform). Even for the simplest cases where the Calabi-Yau is a torus, many important questions remain nanswered. Although recently there has been progress in understanding the structure of SYZ-fibrations for certain explicit Calabi-Yau spaces, one is far from understanding the behaviour of the corresponding D-brane-categories. The purpose of the project is to find a general formulation for categorical mirror symmetry, encompassing, and to prove as much as possible for the simplest Calabi-Yau spaces like tori and quotients thereof.