André Oliveira

Rank two quadratic pairs and surface group representations

Let X be a compact Riemann surface. A quadratic pair on X consists of a holomorphic vector bundle with a quadratic form which takes values in a fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the SO0(2, 3)-character variety of X.

REPRESENTATIONS OF SURFACE GROUPS IN THE PROJECTIVE GENERAL LINEAR GROUP

Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let

Topology of moduli spaces of free group representations in real reductive groups

mathcal{R}_G

The Singular Fiber of the Hitchin Map

be the moduli space of reductive representations of π1X in G. We determine the number of connected components of

Adolescents' perception of environmental features and its association with physical activity: Results from de Azorean Physical Activity and Health Study II

mathcal{R}_{{rm PGL}(n,mathbb{R})}

Longitudinal associations between motor competence and different physical activity intensities: LabMed physical activity study

, for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in

Associations between physical fitness and adherence to the Mediterranean diet with health-related quality of life in adolescents: results from the LabMed Physical Activity Study

mathcal{R}_{{rm SL}(3,mathbb{R})}

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

is homotopically equivalent to

Maximal Higgs bundles for adjoint forms via Cayley correspondence

mathcal{R}_{{rm SO}(3)}

Connectedness of Higgs bundle moduli for complex reductive Lie groups

.