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.