Tannakian categories, fundamental groups and Higgs bundles

After recalling the basic notions concerning profinite and proalgebraic group completions and Tannakian categories, we review how the latter can be used to define generalizations of the notion of fundamental group of a space, such as the Nori and Langer fundamental groups, and the algebraic fundamental group introduced by Simpson. Then we discuss how one can define a Tannakian category whose objects are Higgs bundles on a complex projective variety that are ``numerically flat'' in a suitable sense, and show how the Higgs fundamental group is related to a conjecture about semistable Higgs bundles