A Survey of Tverberg Type Problems

Abstract

Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, given a complex, all affine (or more generally, continuous) maps into some Euclidean space have some specified intersection property. Many other extensions and variations stem from this first result, such as the topological Tverberg conjecture and the colorful Tverberg theorem. Much work is still being done to generalize and extend Tverberg's theorem, which has resulted in several recent and major breakthroughs. Most notably, Frick's surprising counterexample to the topological Tverberg conjecture was only discovered in 2015, and the "constraint method" used to construct the counterexample lends itself to many other applications in proving different extensions of the topological Tverberg conjecture. In this thesis, we will look at some classical theorems that inspired Tverberg's theorem, the current state of affairs vis-a-vis the Tverberg conjecture, and other closely related problems.