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Abstract: The classical Abel-Jacobi map induces the Torelli map from a moduli space of curves of genus $ggeq 2$ into the corresponding Siegel modular variety. The goal of the talk is to explain some geometric problems related to the mapping, focusing on a conjecture of Oort on scarcity of totally geodesic subvarieties in the Torelli image.  We will also explain its relation to a rigidity problem of representation of a lattice of semi-simple Lie groups in a mapping class group.  Some new approaches will be explained.