In this talk we will introduce Higgs bundles and generalized hyperpolygons, and look into different directions that have attracted attention within the area in the last decade.  In particular, we shall see how Mirror Symmetry can be seen in terms of branes within the Hitchin fibration and show that, under certain assumptions on flag types for generalized hyperpolygons, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we will also see how I have been using my geometric background to answer questions in other areas of science. Much of the talk follows recent work joint with Steven Rayan.

Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention both within mathematics and string theory. In this talk we shall consider novel correspondences between branes lying completely within the singular fibres of the Hitchin fibration, which can be understood through group isomorphisms. By considering a toy mode, the first half of the talk will introduce the audience to Higgs bundles and some of the open problems in the area -- the second half of the talk will be about work in progress with Steve Bradlow and Lucas Branco, and with Sebastian Schulz.