Quotients and parameterizations

I will discuss a theme which, at various levels of complexity, is pervasive in algebra and geometry: The attempt to parameterize algebraic or geometric objects naturally leads to the problem of constructing quotients, almost always by the action of a group. I will start by discussing elementary problems such as parameterizing finite subsets of points in the plane or conjugacy classes of matrices, showing how they lead to the so called geometric invariant theory, which also has a more differential geometric counterpart, called symplectic reduction. In the second half of the talk I will discuss the parameterization of representations of the fundamental group of a surface, and the closely related notion of moduli spaces of vector or Higgs bundles on a Riemann surface, still from the point of view of quotient constructions.