Abstract: These notes are given as a reference for the minicourse on gauge theories and analysis presented by the author in UNICAMP, in August 1992. ln this course we define gauge theories and the very basic geometric ideas involved. We focus mostly on the analytical aspects of these theories and introduce concepts such as conformal invariance, subharmonicity and apriori estimates for the curvature F, gauge invariance (optimal choice of gauge), weak-compactness of Sobolev spaces of connections. We use these tools to outline proofs of removable singularities theorems for Yang-Mills fields, also on manifolds with boundary.
The minicourse is divided in four lectures, the first three corresponding to the three chapters of these notes. ln the fourth lecture, we explain Dirichlet and Neumann boundary value problems for Yang-Mills connection. We denote particular attention to the removable singularities theorem for boundary points. For the fourth lecture we refer to [10] and the last section in [11]. These notes are mostly self-contained, but not completely detailed. For more details we give further reference to the reader Ver menos