Using Tools from Topology: An Example

by · Jul 23, 2016 · 1 133 views ·

The story told in this lecture starts with an innocuous little geometry problem, posed in a September 2006 blog entry by R. Nandakumar, an engineer from Calcutta, India: "Can you cut every polygon into a prescribed number n of convex pieces that have equal area and equal perimeter?'' On the path to a (partial) solution of (the d-dimensional version of) this problem, we will encounter a number of quite different areas of Mathematics that Jirka has worked on and written about. This includes Discrete and Computational Geometry, specifically Optimal Transport and weighted Voronoi diagrams, as well as LP duality. This will set up the stage for application of topological tools. Thus we solve the problem for n=2 by using the Borsuk–Ulam theorem, but for larger n, we employ Equivariant Obstruction Theory. On the way to a solution, combinatorial properties of the permutahedron turn out to be essential. These will, at the end of the story, lead us back to India, with some time travel 100 years into the past. The lecture might end by a philosophical reflection: Do we need high-power topological tools for such an elementary problem? How accessible, and how reliable, are such tools? Joint work with Pavle Blagojević.

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