We discuss general properties of A ∞ -algebras and their applications to the theory of open strings. The properties of cyclicity for A ∞ -algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A ∞algebras and cyclic A ∞ -algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A ∞ -isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L ∞ -algebras.

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A ∞ -algebras) by closed strings (L ∞ -algebras).

We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.

We discuss general properties of classical string field theories with symmetric vertices in the context of deformation theory. For a given conformal background there are many string field theories corresponding to different decomposition of moduli space of Riemann surfaces. It is shown that any classical open string field theories on a fixed conformal background are A ∞ -quasi-isomorphic to each other. This indicates that they have isomorphic moduli space of classical solutions. The minimal model theorem in A ∞ -algebras plays a key role in these results. Its natural and geometric realization on formal supermanifolds is also given. The same results hold for classical closed string field theories, whose algebraic structure is governed by L

We discuss noncommutative solitons on a noncommutative torus and their application to tachyon condensation. In the large B limit, they can be exactly described by the Powers-Rieffel projection operators known in the mathematical literature. The resulting soliton spectrum is consistent with T duality and is surprisingly interesting. It is shown that an instability arises for any D-branes, leading to the decay into many smaller D-branes. This phenomenon is the consequence of the fact that the K-homology for a type II von Neumann factor is labeled by R.