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 of convex pieces that have equal area and equal perimeter?" This little problem is a "sparrow", tantalizing, not as easy as one could perhaps expect, and Recreational Mathematics: of no practical use.
I will sketch, however, how this little problem connects to very serious mathematics, including Computational Geometry: For the modelling of this problem we employ insights from a key area of Applied Mathematics, the Theory of Optimal Transportation, which leads to weighted Voronoi diagrams with prescribed areas. This will set up the stage for application of a major tool from Very Pure Mathematics, known as Equivariant Obstruction Theory. This is a "cannon", and we'll have fun with shooting it at the sparrow.
On the way to a solution, I will try to illustrate the workings of Equivariant Obstruction Theory in a slide show. 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: For the last step in our (partial) solution of the sparrows problem we need a simple divisibility property for the numbers in Pascal’s triangle, which was first observed by Balak Ram, in Madras 1909.
But even if the existence problem is solved, the Computational Geometry problem is not: If the solution exists, how do you find one? This problem will be left to you. Instead, I will comment on the strained relationship between cannons and sparrows, and to this avail quote a poem by Hans Magnus Enzensberger.
Günter M. Ziegler is a professor of mathematics at Freie Universität Berlin. His current work connects combinatorics and discrete geometry (especially polytopes) with topology (in particular configuration spaces). His honours include a Leibniz Prize (2001), the MAA Chauvenet Prize (2004), and the 2008 Communicator Award, as well as an ERC Advanced Grant (2010–2015). He presented invited lectures at the ICMs in Beijing (2002), Hyderabad (2016) and Seoul (2014). From 2006 to 2008, he was President of the German Mathematical Society. He is a member of the Executive Board of the Berlin-Brandenburg Academy of Sciences and a member of the Senate of the German Science Foundation.
(Photo: Sven Paustian/Piper Verlag)