Nilpotent Higgs bundles and families of flat connections

This dissertation has three largely independent parts. The first part is a gentle introduction to the moduli space of Higgs bundles, with an eye towards the nilpotent cone. In the second part we investigate [doublestruck C superscript ×]-families of flat connections whose leading term is a nilpotent Higgs field. Examples of such families include real twistor lines and families arising from the conformal limit. We show that these families have the same monodromy as families whose leading term is a regular Higgs bundle and use this to deduce that traces of holonomies are asymptotically exponential in rational powers of the parameter of the family. In the last part, in joint work with Laura Schaposnik, we use the triality of SO(8, [doublestruck C]) to study three interrelated homogeneous bases of the rings of invariant polynomials of Lie algebras, which give the bases of three Hitchin fibrations, and identify the explicit automorphisms that relate them.