Tverberg plus constraints

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique that combines a concept of "Tverberg unavoidable subcomplexes" with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Zivaljevic and Vrecica (1992). We also get a new strengthened version of the generalized van Kampen-Flores theorem by Sarkaria (1991) and Volovikov (1996), an affine version of their "j-wise disjoint" Tverberg theorem, and a topological version of Soberon's (2013) result on Tverberg points with equal barycentric coordinates.Comment: 15 pages; revised version, accepted for publication in Bulletin London Math. Societ

Tverberg's theorem is 50 Years Old: A survey

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society

IST Austria Thesis

We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry