Freie Universität Berlin
The story of 3N points in a plane
Nearly 60 years ago, the Cambridge undergraduate Bryan Birch showed that any “3N points in a plane” can be split into N triples that span triangles with a non-empty intersection. He also conjectured a higher-dimensional version of this, which was proved by the young Norwegian mathematician Helge Tverberg (freezing, in a hotel room in Manchester) exactly fifty years ago.
This is the beginning of a remarkable story that I will try to survey in this lecture. Highlights include the discovery that “3N – 2 points in a plane” would have been enough (and perhaps an even better starting point); the insight that this is really a problem of combinatorial topology; the “Topological Tverberg Theorem”, proved by Bárány, Shlosman & Szücs for the case when N is a prime; a “colored version” of the problem proposed by Bárány & Larman in 1989, and finally proven in 2009 (joint with Blagojević and Matschke); and the 2014 discovery that from the Topological Tverberg Theorem one can get a lot of other results “nearly for free” (joint work with Pavle Blagojević and Florian Frick).