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The medial axis of the region bounded by a closed curve in the plane or a closed surface in 3-space is defined, roughly, as the locus of centres of circles or spheres contained inside the region and having tangency with the boundary at two or more points. It dates back to the work of the mathematical biologist Blum in the 1970s and has received a great deal of attention from mathematicians, computer scientists, and those interested in computer vision as a "skeleton" of the region. In the plane, the skeleton is 1-dimensional and in 3-space it is 2-dimensional, yet it contains a great deal of information about the original region. The symmetry set is defined in a similar way, but we do not require the circle or sphere to lie inside the region: it is a superset of the medial axis. My own interest in this has been mainly through the application of singularity theory to the structure of the symmetry set and medial axis. I shall describe some variants of the symmetry set and medial axis which I have studied over the past few years, in collaboration with graduate students and with Guillermo Sapiro (Minnesota), Stanislaw Janeczko (Warsaw), Vladimir Zakalyukin (Moscow and Liverpool), among others. These are usually affinely invariant constructions, unlike the usual symmetry set which depends on circles or spheres and is a Euclidean construction. The different definitions have interesting and contrasting geometrical features and I shall describe some of these. I shall confine myself mostly to regions bounded by curves in the plane, and shall begin with a brief comparison between Euclidean and affine differential geometry in the plane (where, more or less, circles are replaced by conics).

This is the second of TWO TOPOLOGY SEMINARS THIS WEEK. The first is on Wednesday at 4:10.