**Moduli spaces of representations**

The moduli space of representations of the fundamental group
of a compact surface in a Lie group has a tremendously rich
geometry relating to symplectic geometry, hyperkähler geometry,
integrable systems, Teichmüller theory, etc. This space can be
interpreted in purely differential-geometric terms as
a moduli space of flat connections.
Moreover, by choosing a complex structure on the surface, it can be identified
with a moduli space of holomorphic objects
known as Higgs bundles over the Riemann surface.
After explaining these correspondences,
we will illustrate the power of the holomorphic point of view to
study the topology of the moduli spaces.

*Oscar García-Prada*

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