UC Berkeley

In this thesis we discuss the theory of vector bundles with real structure on the projective line. This extends classical work by Grothendieck classifying complex vector bundles on the projective line. In particular, we show that vector bundles with real structure can be classified in terms of the coroot lattice of GL(n), similarly to the complex case. In addition, we provide a comparison of a certain K-group of sheaves on the moduli space of vector bundles to a K- group of sheaves on the moduli space of local systems, a kind of Langlands duality statement for real bundles, and give a uniformization of the moduli space.