#### Abstract

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.