I shall talk about (most of) the following applications:
(Very briefly, as it is my topic in Copenhagen): Illuminated surfaces: classification of the interactions between shadows, shade boundaries, surface creases and corners, apparent contours. I shall just mention an example and say a little about the general method.
Medial axes and local symmetry in 2 and 3 dimensions: An introduction to the medial axis and associated constructions such as the symmetry set and the pre-symmetry set. Some open problems in this area.
Some other similar constructions in the plane such as those which are invariant to affine transformations, some based on 'affine arclength' and others on area. Connection with certain partial differential equations.
A little about the 3D case: medial axes in 3-dimensions.