There are several notions of what I call 'sets in the middle', sets that are defined by means of some distance function and such that they lie at equal distance from one or more objects.
The most well-known of these is the cut-locus. We will define it and apply some elementary symplectic geometry to calculate it in a special case. In the second lecture we turn our attention to the cut-locus of an embedded manifold: the medial axis.
Other subjects will be the conflict set and the symmetry set. The third lecture will be about Voronoi diagrams.
Throughout we will stress how all these objects and their singularities are related to each other.