  With a particular focus on explicit computations and applications of the
Koszul homology and Betti numbers of monomial ideals, the main goals of this
thesis are the following:
  Analyze the Koszul homology of monomial ideals and apply it to describe the
structure of monomial ideals. Describe algorithms to perform efficient
computations of the homological invariants of monomial ideals. Apply the theory
and computations on monomial ideals to problems inside and outside mathematics
  The thesis introduces as a main tool Mayer-Vietoris trees of monomial ideals.
