In talk I will prove the existence of an isomorphism between two families of manifolds: hyperpolygon spaces and moduli spaces of stable, rank-2, holomorphically trivial parabolic Higgs bundles over CP1 with fixed determinant and trace free Higgs field. This relationship connecting two different fields allows us to benefit from techniques and ideas from each of these areas to obtain new results. In particular, using the study of variation of moduli spaces of parabolic Higgs bundles over a curve, we describe the dependence of hyperpolygon spaces X(?) and their cores on the choice of the parameter ? and show that, when a wall is crossed, the hyperpolygon space undergoes an elementary transformation in the sense of Mukai. If time permits, I will describe how one can take advantage of the geometric description of the core components of a hyperpolygon space to obtain explicit expressions for the computation of their intersection numbers. Using our isomorphism we can obtain similar formulas for the nilpotent cone components of the moduli space of rank-2, holomorphically trivial parabolic Higgs bundles over CP^1 with fixed determinant and trace-free Higgs field.

This is joint work with Leonor Godinho, arXiv:1101.3241.