Here are the slides of the talk “What’s an infinite dimensional manifold and how can it be useful in hospitals?” that was given by my adviser, David Mumford, at the University of Coimbra, in September 2007.

It should give you a general idea of what the field is about. The field being Computational Anatomy, or Pattern Theory (a branch of it). Some familiarity with differential geometry concepts is expected, but one can just look for some nice pictures and graphs. My favorite is the multiple image of a galaxy due to the gravitational lensing. This is probably the most powerful and the most direct observation for the curvature of our space-time.