This talk is about the (surface of the) regular octahedron equipped with its intrinsic metric, in which the distance between points is the length of the shortest path on the octahedron surface that joins them. Professor Schwartz will give a complete description of the map on the octahedron which sends a point to the (typically unique) point that is farthest away. This map has a nice geometric structure and interesting dynamics. The proof will bring up some classic geometric ideas like the developing map and the cut locus, and also some algebraic ideas like a positivity certificate for polynomials which I call the positive dominance algorithm.

No prerequisites are required, apart from possibly elementary group theory.

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