Persistence theory as source of new invariants in topology

Persistence Theory" is the computer scientists view of Morse Theory extended to more general shapes than manifolds and more general real valued maps obtained by the  geometrization of “large data”.

Starting from this we propose for a pair (X,f), X compact ANR and f real or angle valued map a collection of computable invariants  in the form of “configurations of complex numbers"  and of "configurations of vector spaces indexed by complex numbers”  refining the homology of the space X, similar to the spectral package (eigenvalues and generalized eigenspaces) of a pair (V,T), V a complex vector space T a linear endomorphism.
They should be viewed as an additional structure on homology (cohomology) provided by a real or an angle valued map.  

We discuss their properties  and their usefulness (hopefully) in geometric analysis.