Lecture 11. Günter M. Ziegler: «Cannons at Sparrows»
Channel: Projekt MmF
Date: 21/08/2023

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, 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.

Speaker

Günter M. Ziegler is the President of Freie Universität Berlin, where he is also a professor of mathematics. His research focuses on discrete geometry and combinatorics. Günter has obtained many honors for his breakthrough results, including the Gottfried Wilhelm Leibniz-Preis of the German Research Foundation (DFG) in 2001 and an Advanced Grant of the European Research Council (ERC) in 2010.

Günter represented the Federal Republic of Germany at the International Mathematical Olympiad in 1981, where he won a perfect score gold medal. He is what may be called an exemplary citizen of mathematics, pushing for fair play and diversity in science and often acting as an ambassador in outreach activities and science communication events. With his colleague Martin Aigner, he has written Proofs from THE BOOK, which is recommended reading for you students in the Kharkiv-Vienna International Science School.