Abstract: The talk concerns recent work with Tamas Hausel in asking how SYZ mirror symmetry works for the moduli space of Higgs bundles. Focusing on C^*-invariant Lagrangian submanifolds, we use the notion of virtual multiplicity as a tool firstly to examine if the Lagrangian is closed, but  also to open up new features involving finite-dimensional algebras which are deformations of cohomology algebras. Answering some of the questions raised  requires revisiting basic constructions of stable bundles on curves.