Global analysis is essentially the technique of extending results known for finite-dimensional manifolds to infinite-dimensional manifolds (typically, spaces of maps from one manifold to another). For example, on a finite-dimensional manifold the critical points of a generic smooth (Morse) function are determined by the topology of the manifold. A sample problem in global analysis is to consider the length as a function on the space of closed curves in a manifold, and classify the critical points (proving, for example, that in every homotopy class there is a length-minimizing curve). As another example we might try extending the Hopf-Rinow theorem to infinite-dimensional manifolds such as diffeomorphism groups.

To do this one needs two things: to construct a smooth manifold structure on the space of maps and functions on them, and to assume extra conditions to assume which make possible the leap from finite- to infinite-dimensional results. In this talk I will explain the most common smooth structures on these spaces (using Holder norms or Sobolev norms) starting from the basics. In addition I will sketch some of the applications to things like closed geodesics, index theorems, and continuum mechanics.