A user's guide to topological Tverberg conjecture

http://arxiv.org/abs/1605.05141
The well-known topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for each integers $ r,d>1 $ and each continuous map
$$ f\colon\Delta\to\mathbb R^d $$
of the $ (d+1)(r-1) $-dimensional simplex $ \Delta $ there are pairwise disjoint subsimplices $ \sigma_1,\dots,\sigma_r\subset\Delta $ such that
$$ f(\sigma_1)\cap\dots\cap f(\sigma_r)\ne\varnothing. $$
A proof for a prime power $ r $ was given by I. Bárány, S. Shlosman, A. Szűcs, M. Özaydın and A. Volovikov in 1981–1996. A counterexample for other $ r $ was found in a series of papers by M. Özaydın, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, I. Mabillard and U. Wagner, most of them recent. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. We present a simplified explanation of easier parts of the arguments, accessible to non-specialists in the area, and give reference to more complicated parts.
I will also describe stronger counterexamples of http://arxiv.org/abs/1511.03501.