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).


The Catalan Mathematical Society invites participants to this first congress of a biannual series focusing on current research topics across several areas of Mathematics.

Plenary talks and thematic sessions have been selected by the Scientific Committee of the SCM. Special thanks are due to the organisers of the thematic sessions and to the local mathematical community as a whole for their support to this congress.


Societat Catalana de Matemàtiques
Institut d'Estudis Catalans
Carrer del Carme, 47
08001 Barcelona

Phone: +34 933 248 583

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