On numerically flat Higgs bundles

This thesis studies numerically flat Higgs bundles on complex projective manifolds, motivated by Bruzzo's conjecture about the vanishing of their Chern classes, and their consequent speculative relationship to harmonic bundles. The themes developed are: the spectral cover and the Grassmannian of Higgs quotients as a Quot scheme; Tannakian categories of decorated bundles and extensions of Langer's S-fundamental group scheme; families of harmonic bundles and the Gauss-Manin derivative; jump deformations of algebraic structures from the point of view of derived algebraic geometry.