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Tuesday, February 17, 2015

**Abstract:** Let $S$ be a closed surface of genus at least 2, and consider the moduli space of representations $\rho:\pi_1(S)\rightarrow Sp(4,\mathbb{R}).$ There is an invariant $\tau\in\mathbb{Z},$ called the Toledo invariant, which helps to distinguish connected components. The Toledo invariant satisfies a Milnor-Wood inequality $|\tau|\leq 2g-2.$ Representations with maximal Toledo invariant have many geometrically interesting properties, for instance, they are all discrete and faithful. In this talk, we will give a mapping class group invariant parameterization of all smooth connected components of the maximal $Sp(4,\mathbb{R})$ representations. Our main tool is Higgs bundles. However, to utilize Higgs bundle techniques, one has to fix a conformal structure of the surface $S,$ hence breaking the mapping class group symmetry. To restore the symmetry, we associate a unique `preferred' conformal structure to each such representation. This is done by exploiting the relationship between the associated Higgs bundles and minimal surfaces. View talk at http://youtu.be/lhJhEI1FGb8