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We describe jm isomorphism of categories conjectured by Kontsevich. If M and M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya's category of Lagrangian submanifolds^on M. We prove this equivalence when M is an elliptic curve and M is its dual curve, exhibiting the dictionary in detail.

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We show that elliptic solutions of classical Yang-Baxter equation (CYBE) can be obtained from triple Massey products on an elliptic curve. We introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions. We also study some degenerations of these solutions.

1.1.1. Results on sheaves of t-structures. In section 2 we are given a nondegenerate t-structure (D(X) ≤0 , D(X) ≥0 ) with heart C = D(X) ≤0 ∩ D(X) ≥0 (see section 1.2 for definitions and a brief introduction). For a smooth projective variety S with ample line bundle L we defineAssuming that C is noetherian we prove

Abstract. In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.(called the "boundary-bulk map" in the context of topological strings), such that ch(Ē) = τĒ(id E ) and the formula (0.2) generalizes towhere α is an arbitrary endomorphism ofĒ.

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