  We define and study several new interleaving distances for persistent
cohomology which take into account the algebraic structures of the cohomology
of a space, for instance the cup product or the action of the Steenrod algebra.
In particular, we prove that there exists a persistent A-infinity-structure
associated to data sets and and we define the associated distance. We prove the
stability of these new distances for Cech or Vietoris Rips complexes with
respect to the Gromov-Hausdorff distance, and we compare these new distances
with each other and the classical one, building some examples which prove that
they are not equal in general and refine effectively the classical bottleneck
distance.
