  Building on Kadeishvili's original theorem inducing $A_\infty$-algebra
structures on the homology of dg-algebras, several directions of algorithmic
research in $A_\infty$-algebras have been pursued.
  In this paper we will survey work done on calculating explicit
$A_\infty$-algebra structures from homotopy retractions; in group cohomology;
and in persistent homology.
