  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 obtain $A_{\infty}$ algebra structures and some canonically defined
deformations of such structures on the cohomology. We formulate the
$A_{\infty}$ algebraic mirror symmetry as the identification of the
$A_{\infty}$ algebras together with their canonical deformations constructed
this way on different manifolds.
