It is well known how basic Algebraic Topology and geometrization of Large Data led
to Persistence Theory, a useful tool in Data Analysis.
In this talk I will explore the other direction; how Persistence Theory suggests and motivates
refinements of some basic topological invariants, like homology and Betti numbers, and suggests
alternative descriptions of others invariants, like monodromy, of mathematical relevance and
with computational implications. The mathematics described is a part of what I refer to as an
ALTERNATIVE to MORSE-NOVIKOV theory.
The refinements proposed are in terms of configurations of vector spaces for the relevant
homologies, and in terms of polynomials for Betti numbers. The alternative description of
monodromy is computer friendly, hence without the need of infinite objects (infinite cyclic
cover). A few applications of these refinements in topology, geometric analysis and dynamics
might be indicated.