  A central problem in topological data analysis is that of computing the
homology of a given simplicial complex. Said complexes can have arbitrary large
number of simplices, as can happen, for example, if the space is the
Rips-Vietoris or Cech complex of a large data cloud. Thus, pre-processing the
simplicial complex to get a smaller complex with the same homology groups and
then applying the homology algorithm to the smaller one, has been an active
research topic in the last years. In this survey, we discuss some recent papers
that examine the complexity of this simplification via Discrete Morse Theory.
This survey was prepared as a final project for a course on Computational
Topology at The Ohio State University.
