  This Ph.D. thesis studies the relation between the Harder-Narasimhan
filtration and a notion of GIT maximal unstability. When constructing a moduli
space by using Geometric Invariant Theory (GIT), a notion of GIT stability
appears, which is determined by 1-parameter subgroups. This thesis shows a
correspondence between the 1-parameter subgroup giving maximal unstability from
the GIT point of view and the Harder-Narasimhan filtration for different moduli
problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank
2 tensors and quiver representations.
  The article [GSZ] contains the correspondence for torsion free coherent
sheaves, whereas [Za1] and [Za2] are devoted to finite dimensional quiver
representations and rank 2 tensors.
  In [HK], the authors explore this kind of correspondences while identifying
the Hesselink stratification on conjugacy classes of 1-parameter subgroups with
the one on Harder-Narasimhan types, showing that the Hesselink's adapted
1-parameter subgroup corresponds to the Harder-Narasimhan filtration (which is
previously prescribed on each strata). Futher work of Hoskins (c.f. [Ho1, Ho2])
continues on the same direction for other moduli problems, comprising finite
dimensional quiver representations as it is done in [Za1], and coherent sheaves
using the functorial construction of Alvarez-Consul and King (c.f. [ACK]), as
it appears in Section 3.2 of this thesis.
