Homological Perturbation Theory and Mirror Symmetry

Abstract We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to construct A_∞ algebra structures on the cohomology, and their canonically defined deformations. Such constructions are used to formulate a version of A_∞ algebraic mirror symmetry. Partially supported by an NSF group infra-structure grant through the Geometry, Analysis and Topology (GAT) group