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Abstract:
A mediatrix is defined to be the set of all points equidistant from a given
pair of points. Of course, in Rn with its usual notion of
distance, any mediatrix is simply an (n-1)-dimensional hyperplane. On the
surface of a torus, however, there are multiple possible shapes for
mediatrices, depending on the pair of points chosen and on the metric placed
on the torus. In more elaborate
spaces, the set of possible shapes becomes significantly more complicated.
In this talk, we will show that on a closed surface, mediatrices are certain
kinds of graphs (in the sense of vertices connected by edges), and we will
discuss restrictions placed on the shape of mediatrices by the topology and
geometry of the surrounding space. We will also use related questions as an
excuse to explore some introductory Riemannian geometry. This talk is based
on joint work with J.J.P. Veerman.
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